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MSE 612 (Fall 2012) 1
MSE 612 (Fall 2012)
Lecture Notes on
COMPUTATIONAL PLASTICITY AND MICROMECHANICS
YANFEI GAO, [email protected]
Department of Materials Science and Engineering University of Tennessee, Knoxville
MSE 612 (Fall 2012) 2
CONTENT CONTENT ......................................................................................................................................... 2
Objectives ................................................................................................................................... 3 Textbook ..................................................................................................................................... 3 Evaluation ................................................................................................................................... 3
1. CONSTITUTIVE BEHAVIOR ....................................................................................................... 4 1.1 Elements of Continuum Mechanics ................................................................................. 4 1.2 Continuum (Phenomenological) Plasticity ..................................................................... 16 1.3 Thermodynamics and Internal State Variables .............................................................. 26 1.4 Formulations for Large Deformation ............................................................................. 28 1.5 Applications in Materials Science and Engineering ...................................................... 30
2. FINITE ELEMENT METHOD ..................................................................................................... 31 2.1 Finite Element Analysis (FEA) Procedure ..................................................................... 31 2.2 Principle of Virtual Work ............................................................................................... 39 2.3 Finite Element Method for Linear Elastic Solids ........................................................... 43 2.4 Finite Element Method for Plastic Solids ...................................................................... 55 2.5 Implementation Procedure ............................................................................................. 59 2.6 Stress Update Algorithm ................................................................................................ 63
2.6.1 Explicit Integration Scheme .................................................................................... 63 2.6.2 Explicit (one-step) Return Mapping Algorithm (rate-independent solids) ............. 64 2.6.3 Implicit (N-R) Return Mapping Algorithm (rate-dependent solids) ....................... 65
2.7 Practicing ABAQUS UMAT Subroutine ....................................................................... 69 3. MATERIAL FAILURE ............................................................................................................... 75
3.1 Phenomenology of Fracture and Fatigue ....................................................................... 75 3.2 A Viewpoint at the Crack Tip ........................................................................................ 78 3.3 Modeling the Crack Tip Process Zone ........................................................................... 84 3.4 Stress and Strain Based Failure Criteria ......................................................................... 91
4. CRYSTAL PLASTICITY ............................................................................................................ 94 4.1 Deformation Mechanisms in Single Crystals and Polycrystals ..................................... 94 4.2 Continuum Crystal Plasticity Theory ............................................................................. 97 4.3 Finite Element Simulations .......................................................................................... 101 4.4 Applications and Further Comments............................................................................ 104
5. MATERIAL INSTABILITY ....................................................................................................... 106 5.1 Geometric vs Material Instability ................................................................................. 106
5.1.1 Necking Analysis .................................................................................................. 106 5.2 Hill-Hutchinson-Rice Theory ....................................................................................... 110 5.3 Numerical Simulations ................................................................................................. 113 5.4 Shear Band Angle Analysis in Metallic Glasses .......................................................... 117
6. MICROSTRUCTURE-BASED SIMULATIONS ............................................................................ 120 6.1 Stress Effects in Phase Transformation ........................................................................ 120
6.1.1 Energy Minimization: an Example of Precipitate Shape ...................................... 121 6.2 Non-Equilibrium Thermodynamics ............................................................................. 124 6.3 Cavitation in High Temperature Alloys ....................................................................... 130 6.4 Thin Film Mechanics ................................................................................................... 133
MSE 612 (Fall 2012) 3
Objectives Computational modeling and simulation methods will be introduced with applications in plasticity, fracture and fatigue, microstructural evolution, and material instability in engineering structural materials. Topics include the classic finite element method based on constitutive modeling, cohesive interface model, discrete dislocation dynamics, atomistic/continuum coupling techniques, and current research areas that are pertinent to the research efforts in UT and ORNL (such as shear banding behavior in metallic glass, dislocation nucleation, grain boundary modeling, etc.). The students will be provided with templates of user defined element and material subroutines for the use of ABAQUS, and will also be asked to do simple practices.
Textbook You will be primarily provided with lecture notes and directed to related papers. A main reference and basis of this lecture is “Applied Mechanics of Solids,” by A.F. Bower, Taylor and Francis Press, 2009. Also see http://solidmechanics.org. Please refer to this monograph if you are unclear with some discussions in this lecture. You can also consult the following books:
Computational Inelasticity, by J.C. Simo and T.J.R. Hughes, Springer, 1997. (This book focuses on computational mechanics and algorithm development. Our course is more on practical problems in MSE. This book is on reserve at main library.)
Introduction to Computational Plasticity, by F. Dunne and N. Petrinic, Oxford, 2005. (Although this book is good for starting students since it discusses ABAQUS UMAT, it contains a lot of incorrect descriptions on material aspects. Our library does not have it.)
Mechanical Metallurgy, by G.E. Dieter, McGraw-Hill, 1986. (This book is also required in MSE 512. We will refer to this book for material aspects, particularly on microstructure, dislocations, creep, etc.)
Evaluation Homework: A practice of ABAQUS and UMAT (30%) Course Project – Step 1 (20%):
Identify your own course project and give an in-class presentation on project background Course Project – Step 2 (50%):
Final project presentation (20%) Project report (4-page limit, 30%)
Additional Comments: Don’t be too ambitious on your course project. What I want to see is a well-defined topic
and your effort on it. Scientists must speak: be sure to present what you clearly know, and be prepared to
answer questions from non-experts. Scientists must write: a well written project report is highly desirable. Do not just pile up
numerical results without illustration and explanation.
MSE 612 (Fall 2012) 4
1. CONSTITUTIVE BEHAVIOR
1.1 Elements of Continuum Mechanics Continuum mechanics examines the mechanical behavior of a continuous body, where the discrete nature of atomic structure, microstructure, and others is not considered. For example, if we are interested in structural analysis of a bridge, we can use Mises plasticity which does not have a connection to the grain microstructure. In other words, it does not differentiate steel and cast iron; the difference only comes into the yield stress and hardening exponent in the Mises plasticity law. If we are interested in texture evolution, we need to treat individual grains as continuous bodies, and choose models that are consistent with this length scale. The microstructural features below the length scales of interest will be replaced by some kind of order parameters or state variables, such as the concentration field or dislocation density.
All the field quantities are continuous functions of time and spatial coordinates. Mathematical representation in this area often resorts to tensor calculus, which will be reviewed first. Then we will discuss the measure of strain and stress, and how to formulate a boundary/initial value problem.
Vectors and Tensors
In a Cartesian coordinate system with basis vectors xe , ye , and ze , a vector a can be
represented by its projections on the three coordinate axes, given by
x x y y z za a a a e e e , (1.1)
where ia are called components of the vector. A tensor A is given by
11 1 1 12 1 2 13 1 3
21 2 1 22 2 2 23 2 3
31 3 1 32 3 2 33 3 3
A A A
A A A
A A A
A e e e e e e
e e e e e e
e e e e e e
(1.2)
where ijA are components of the tensor, and i je e are the basis tensors.
Index notation for vector and tensor operations
Using Latin subscript , ,i x y z or 1, 2,3i , we can rewrite the above representations:
i iaa e ,
ij i jA A e e , (1.3)
where summation convention is implied on repeated indices (i.e., Einstein convention). The repeated indices are also called dummy indices. The index notation will significantly simplify the representation of vector and tensor operations. But there are some rules to follow: the same index may not appear more than twice in a product of two (or more) vectors or
tensors, such as ij jA a
free indices on each term of an equation must agree, such as ij j in t
MSE 612 (Fall 2012) 5
free and dummy indices may be changed without altering the meaning of an expression, provided that the first two rules are not violated, such as ij j in t jk k jn t
Addition
c a b , i i ic a b
C A B , ij ij ijC A B (1.4)
Dot product (scalar product)
cos , i ia b a b a b a b ,
where the first equation is definition and the second equation can be proved from Eq. (1.6).
A a b , ij j iA a b , ij jiA a A a
j jiia A a A
ik kjijA B A B
inner product : ij ijA BA B
outer product ij jiA B A B (1.5)
Kronecker delta
1,
0,i j ij
i j
i j
e e
jm j ma a (1.6)
Permutation tensor
i j ijk k e e e
1, , , 1, 2,3
1, , , 1, 2,3
0, , ,ijk
if i j k are an even permutation of
if i j k are an odd permutation of
if any of i j k arethe same
ijk imn jm kn jn mk (1.7)
Cross product (vector product)
c a b , i ijk j kc a b , (1.8)
where sin ,c a b a b and , ,a b c form a right handed triad (or called right-handed screw).
B A b , ij jmn im nB A b .
MSE 612 (Fall 2012) 6
Tensor product (dyadic product)
i jija b a b (1.9)
Convention confusion
Unfortunately, not everyone abides with the same conventions. One can usually tell the difference, but sometimes it’s very confusing. Here lists a common notation difference.
dot product A a Aa
tensor product a b ab
Apparently, one can use the combination of A a and a b , or A a and ab , or Aa and a b . But never use the combination of Aa and ab .
In this course, we will use a b for tensor product, and so that we do not need to distinguish A a and Aa , although the latter should be avoided. Thus the basis tensors are represented by i je e in Eq. (1.2).
Differential vector
iix
e
,i i ii
uu u u
x
e e
grad(v) iij
j
v
x
v
div(v) i
i
v
x
v
curl(v) kijk
j
v
x
v (1.10)
Determinant
11 12 13
21 22 23 1 2 3
31 32 33
det det ijk i j k
A A A
A A A A A A
A A A
A (1.11)
Inverse
1 A A I , 1ik kj ijA A (1.12)
Trace
11 22 33iitr A A A A A (1.13)
MSE 612 (Fall 2012) 7
Coordinate transformation
Define a transformation matrix
1 1 1
2 2 2
3 3 3
l m n
l m n
l m n
Q ,
where , ,i i il m n are direction cosines of the new coordinate basis vectors ie . Consequently, the
index notation gives
a Qa , i ij ja Q a
T A QAQ , ij ik km jmA Q A Q (1.14)
Eigenvalues and invariants
Eigenvalues and eigenvectors of a tensor are the same as that for a matrix, given by the solution of
ij j iA x x ,
or written as 0ij ij jA x . In order to obtain nontrivial solutions, the determinant should be
zero, namely,
det 0 A I , (1.15)
where I is the identity tensor. This is also related to the similarity transformation in linear algebra. Invariants of a tensor are functions of the tensor components which remain constant under a coordinate transformation. Eigenvalues and eigenvectors belong to this category. Clearly, there are an infinite number of invariants.
Kinematics and Strain Measure
Fig. 1.1: Geometric representation of a deforming solid.
e1
e2
e3
X x
u(X) dX
dx u(X+dX)
reference (original) configuration
current (deformed) configuration
MSE 612 (Fall 2012) 8
A material point in reference coordinates X in a continuum body, after deformation, is located at x . A differential length dX will be stretched and rotated, and so does a differential surface or volume element. How do we describe the shape change from the function of x X ?
Note that
d d d u X x X .
Uniaxial strain
A uniaxial measure of strain should describe the length change, thus giving rise to infinite number of definitions. Given the stretch ratio, 0l l , we can define
nominal strain: 0
0 0
1l ll
el l
,
natural strain: 0 11
l l
l
,
Lagrangian strain: 2 2
2020
1 11
2 2
l lE
l
,
Eulerian strain: 2 2
02 2
1 1 11
2 2
l l
l
(1.16)
All these definitions are the same when 1e , i.e., infinitesimal deformation.
Deformation gradient
In principle, the strain definition arises from the deformation gradient tensor,
x uF I
X X,
,i
ij ij i jj
xF u
X
, (1.17)
which contains not only the stretch ratio, but also rotation.
One approach to define the strain tensor is by the length change of a differential element. From Fig. 1.1, we have
2dS d d X X ,
2 Tds d d d d d d x x F X F X X F F X ,
so that
2 2 Tds dS d d X F F I X .
Consequently, the Lagrangian strain tensor (or called Green’s strain tensor) is defined as
MSE 612 (Fall 2012) 9
1
2T E F F I , or 1
2ij ki kj ijE F F (1.18)
There are many other kinds of strain definition, which are usually not important unless we are interested in large deformation (or called finite deformation).
Velocity gradient
The velocity at a material point during deformation is defined as
,d
tdt
x
v X , (1.19)
and the velocity gradient is thus
v
Lx
. (1.20)
We can prove that 1 L F F by using the definition of deformation gradient:
1d dd d d d d
dt dt v x F X F X F F x .
Decomposing the velocity gradient tensor into symmetric and skew parts:
L D W (1.21)
1
2T D L L , 1
2T W L L
where D is called deformation rate tensor (or stretch rate tensor), and W is called spin tensor.
Infinitesimal strain
The infinitesimal strain tensor is defined as
1
2
Tu u ε , or
1
2ji
ijj i
uu
x x
, (1.22)
where u is the displacement vector. Clearly, it is an approximate deformation measure, since
1 1
2 2j ji k k i
ij ijj i j i j i
u uu u u uE
X X X X X X
(1.23)
Similarly, the deformation rate and spin tensors are
1
2
d
dt
εD v v , 1
2 W v v (1.24)
Kinetics and Stress Measure
By definition, stress = force / area. For a continuum body under deformation, there are forces acting on the boundary surface, on the internal surfaces, and on the body, and one can
MSE 612 (Fall 2012) 10
choose a surface area in reference coordinates or that after deformation. The most useful stress measure is the Cauchy stress, as will be discussed subsequently. Usually we don’t need to distinguish the original area and deformed area unless we are concerned with finite deformation.
Cauchy (true) stress tensor
Consider a differential element in the deformed body and draw the free body diagram. The forces acting on the surfaces are used to define the stress tensor, as shown in the schematic below. The stress components are
11 12 13
21 22 23
31 32 33
σ (1.25)
Fig. 1.2: Cauchy stress components in a differential element. Only 2i are shown.
Given a stress state at a point, the force on a surface with normal n is given by the Cauchy equation:
t n σ , j i ijt n . (1.26)
Other stress measures
Kirchhoff stress: Jτ σ , (1.27)
where detJ F .
Nominal stress (or called the 1st Piola-Kirchhoff stress):
1J S F σ , (1.28)
which is a measure of force per unit area of reference solid. It is not symmetric.
Material stress (or called the 2nd Piola-Kirchhoff stress):
e1
e2
e3
σ22
σ21
σ23
MSE 612 (Fall 2012) 11
1 TJ Σ F σ F , (1.29)
which is similar to the 1st PK stress. But it is symmetric, and it is the work conjugate of the rate of change of Lagrange strain. In other words, :E Σ is the stress power per unit reference volume.
When do we consider finite deformation and the associated strain, stress, and rate measures?
For typical engineering analyses, we do not need to consider large deformation. The strain measures such as E , strain rate measures such as D and W , and stress measures such as S and Σ are only important when we are concerned with materials problems such as texture evolution, crystal slip, necking, and strain localization.
Principal stresses and stress invariants
The principal stresses (eigenvalues) can be determined from the characteristic equation
det 0 σ I ,
3 21 2 3 0I I I , (1.30)
where the three invarants are
1 iiI , 2
1
2 ii jj ij ijI , 3 det ijI . (1.31)
For the deviatoric stress, 1
3ij ij kk ijs , the three invariants are given by
1 0J , 2
1
2 ij ijJ s s , 3 det ijJ s . (1.32)
Note that 2I does not degenerate into 2J . Consequently, if we postulate that the material yield is
reached when some combination of stress components reach a critical value, this combination must be a function of the invariants; otherwise, a coordinate transformation will change your yield criterion.
Furthermore, we usually define
hydrostatic pressure: 1
3p tr σ (1.33)
von Mises equivalent stress: 3
:2e s s (1.34)
Force balance and momentum balance
This has been discussed in MSE 512. Given surface tractions *it and body force ib (force
per unit mass), the force balance in small deformation is
,ij i j jb u , (1.35)
and the boundary condition is
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*i ij jn t , at s , (1.36)
*i iu u , at u . (1.37)
The angular momentum balance leads to ij ji , so that the Cauchy stress tensor is symmetric.
Constitutive Law
In preceding sections, we have discussed the strain measure and stress measure. The difference between various materials lies at the relationship between strains (and strain rates) and stresses (and stress rates), i.e., the constitutive law.
Is the constitutive law really a law? Yes or no. If the constitutive relationship is derived from rigorous physical basis, it is a law. For example, the Young’s modulus can be calculated from atomic interactions. Most of the constitutive laws we encounter in MSE, such as Mises plasticity, have some phenomenological nature and a certain degree of microstructural connection. Although the constitutive relationship should satisfy the laws of thermodynamics, usually it involves a lot of assumptions and curve fitting to the experiments. Consequently, you will see a large number of constitutive relationships, especially for micro-plasticity in which the material microstructure is the key ingredient.
Elasticity
Elasticity means the full recovery of the shape after unloading. The most common one is the Hooke’s law (linear elasticity):
ij ijkl klc ,
ij ijkl kls , (1.38)
where ijklc is the stiffness tensor and ijkls is the compliance tensor. For single crystals, the elastic
constants ijklc clearly depend on the crystallographic orientation, so that the elastic behavior is
anisotropic. For polycrystals, if our length scale of interest is much larger than grain size (e.g., the elastic constants of a steel truss), the elastic constants actually are derived from homogenization procedure, which differ from the elastic constants of single crystals. These macroscopic elastic constants usually lead to isotropic elasticity.
For isotropic elastic material, we get
1 1 2ij ij kk ij
E
,
11ij ij kk ijE
. (1.39)
One can take an exercise of finding ij kl from the above Hooke’s law.
In addition to linear elasticity, we may have finite elasticity, meaning the elastic behavior at finite deformation. Examples include the hyper-elastic behavior of rubber. One example of rate dependent elasticity is visco-elasticity, such as Kelvin model, Maxwell model, etc.
MSE 612 (Fall 2012) 13
Yield criterion
There are two major yield criteria. The maximum shear stress yield criterion was developed by a French engineer, Tresca in 1864. And the octahedral shear stress yield criterion was developed by a German engineer, von Mises in 1913. Both theories are based on shear stress, because of the following experimental observations: (1) hydrostatic stress never causes yield, and (2) for most ductile materials, the deformation is by slip of crystal planes.
If the resolved shear stress on a glide plane is larger than a critical stress, then the material will experience permanent deformation. For polycrystalline materials, we got randomly distributed slip planes, so that if we can find any plane in solid for which 0 , plastic flow is
possible. The Tresca yield criterion may be written as
22
1,
2
1,
2
1max 0
133221
(1.40)
Note that the three principal stresses are usually ranked by 1 2 3 . Experiments suggest
that Tresca yield criterion slightly underestimates stresses required to cause yield. A possible reason is that we need many slips of crystal planes before plastic flow is noticeable.
The von Mises yield criterion says that yield occurs when the RMS value of the 3 maximum shear stresses reaches a critical value, i.e.
2 2 2
2 1 2 2 3 3 1
3 13
2 2e ij ij Ys s J , (1.41)
or
2 2 2 2 2 216
2x y y z z x xy yz zx Y .
There are two more physical interpretation of the von Mises criterion. The shear stress on the octahedral plane is
213
232
2213
1 h ,
which is also proportional to the elastic energy of distortion
2 2 2 2
1 2 3 1 2 2 3 3 1
2 2
1 1 2 1
2 3 3
1 9
2 12 h
wE
K
with 213 EK and 12E .
Plasticity models
The general procedure to devise a phenomenological plasticity model is: (1) measure the σ~ε curve using simple tests, (2) devise a criterion based on the principal stress components (or
MSE 612 (Fall 2012) 14
stress invariants), and (3) compare with more experiments involving complex stress states. The step (2) involves the generalization of failure criterion from uniaxial tension to multiaxial stress states. To this end, we have a number of phenomenological plasticity models, as categorized by
1. Hardening behavior: kinematic, isotropic, coupled
2. Rate dependence: viscoplasticity, creep
3. Geometric nonlinearity: small deformation, large deformation
For MSE students, we may be more interested in microstructure-based plasticity model. One example is the crystal plasticity, in which the discrete nature of crystalline slip is accounted for. However, the hardening behavior is still largely phenomenological.
To have a rigorous connection to the dislocation microstructural evolution is still impossible right now. There has been a persistent topic, and recent examples include discrete dislocation plasticity and strain gradient plasticity.
Failure criteria
Strictly speaking, this should NOT be considered as part of the constitutive law. The reason is obvious in fracture mechanics, where the fracture failure also depends on the specimen geometry. The constitutive law essentially gives the stress-strain relationship (together with the time history) at a single material point. If, and only if, the failure criterion involves the stress state (or strain state) at a local point, we can implement into the material constitutive law. Otherwise, we need to recourse to failure mechanics analysis.
For example, for brittle material, one can postulate that the fracture occurs when the largest principal stress reaches a critical value. This is the Rankine criterion (1858),
u 321 ,,max .
It is very simple. But the problem is that it usually does not work well.
In fact, the fracture of brittle material is based on the fracture mechanics principle, that is, the stress intensity factor of a potential crack or flaw reaches the critical value. The calculation of the stress intensity factor requires the full field stress analysis. For the fracture of ductile material, there are more sophisticated model, such as Gurson model, which describes the history of void nucleation, growth, and coalescence. The Gurson model only depends on the local stress history, and can be regarded as a constitutive law.
Formulating a Boundary/Initial Value Problem (BVP/IVP)
To pose a solvable problem in continuum mechanics, we need to have the following three ingredients, plus the appropriate boundary and initial conditions. The resulting boundary value problems (BVPs) or initial value problems (IVPs) are standard partial differential equations to solve. Analytical solutions are typically not possible for plastic solids. Numerical solution using finite element is thus very critical.
However, if you are asked to predict failure, such as fracture, fatigue, corrosion, creep fracture, etc. by using finite element simulation, you can ask back this question: what is the failure model and how can we implement that into the finite element framework? The standard finite element program only solves the PDEs, which gives nothing about failure.
MSE 612 (Fall 2012) 15
Fig. 1.3: Tripod for continuum mechanics.
If we are interested in infinitesimal deformation of a linear elastic solid under static loading, the BVP is linear. Finding analytical solution along this line is a mature field. Some level of complicacy arises if the elastic behavior is anisotropic.
From the three legs in Fig. 1.3, we could introduce nonlinearity in:
kinematic nonlinearity finite deformation
constitutive nonlinearity plasticity, etc.
kinetic nonlinearity nonlinear damping, etc.
kinematics
constitutive law
kinetics
MSE 612 (Fall 2012) 16
1.2 Continuum (Phenomenological) Plasticity In the context of continuum mechanics, the material behavior is characterized by a set of constitutive equations that prescribe the relationship of
, ,state variables, history, ...σ ε ε . (1.42)
In the past century, a large number of constitutive laws have been developed, which, although largely phenomenological, give very nice description of commonly used engineering materials. Most mechanical testing methods can be directly or indirectly used to infer the material parameters used in these laws. For example, dog-bone bar tests will give yield stress and the strain hardening exponent. Micro-hardness will give an effective flow stress, which depends on the indenter geometry. The objective of this chapter is to give an overview of common phenomenological plasticity laws (for metals and alloys). The connection to the material microstructure will be superficially discussed here, and details will be given later.
We begin by reviewing the results of a typical tensile test. For an annealed, polycrystalline metal specimen (e.g. Cu or Al), shown in Fig. 1.4 is the typical ~ curve. (We assume that the test is conducted at room temperature and at modest strains, <10%, at modest strain rates, 10-4~10-9 s-1.) In the beginning, the stress-strain relation is usually linear elastic. The stress is proportional
to the strain, and the slope is the Young’s modulus. Elastic deformation means that the deformation is fully reversible upon unloading.
If a critical stress is exceeded, the specimen is permanently changed in length on unloading. If the stress is removed from the specimen during the test, the ~ curve during unloading
has a slope equal to that of the elastic part of the ~ curve. If the specimen is re-loaded, it will initially follow the same curve, until the stress approaches its maximum value during prior loading. At this point, the ~ curve once again ceases to be linear, and the specimen is permanently deformed further.
A pre-strained solid will usually exhibit higher yield stress, thus called strain hardening or work hardening.
It is often difficult to identify the critical stress accurately, because the ~ curve starts to curve rather gradually. We often define a 0.2% proof stress as the yield stress. For materials exhibiting yield point phenomenon, we choose the lower yield point.
Materials that typically behave in a ductile manner generally have their usefulness limited by yielding, and those that typically behave in a brittle manner are usually limited by fracture.
Holding the specimen at a constant stress or constant strain leads to creep or relaxation, which is rate-dependent. Such a behavior is very sensitive to material microstructure (grain size, grain boundary strength etc.), environmental temperature, applied strain rate, etc.
A cycle of loading and unloading gives the cyclic plasticity. It’s still impossible to quantitatively predict this behavior. But phenomenological theories can be developed to fit the measurements. Usually, cyclic loading can lead to perpetual increase of plastic strain, called ratcheting or cyclic creep.
The tensile and compressive yield stresses may not be the same, called Bauschinger effect. Plastic deformation of nonporous metals is essentially incompressible, i.e., volume
preserving. …
MSE 612 (Fall 2012) 17
Fig. 1.4: Representative stress-strain curves for polycrystalline metals.
Yield Criterion and J2 Plasticity
The general procedure to devise a phenomenological yield criterion is:
(1) measure the ~ curve using simple tests, such as uniaxial tension, or thin-walled tubes under combined torsion and axial loading;
(2) devise a criterion based on the principal stress components (or stress invariants);
(3) compare with more experiments involving complex (and/or nonuniform) stress states.
Tresca and Mises Yield Criterion
Yield criterion is often represented in terms the stress invariants, such as 1 2 3, , 0f I I I .
For metals, hydrostatic stress does not cause yield (or you can argue that typical hydrostatic stress is far less than the elastic modulus). The plastic strain might be large, but volumetric strain (being elastic) is usually small (most plastic deformation is volume preserving). Therefore we don’t need 1I .
As we have discussed before, Tresca criterion gives
1 2 2 3 3 1
1 1 1max , ,
2 2 2 2Y
, (1.43)
and Mises criterion gives
2
33
2e ij ij Ys s J , (1.44)
which can also be written as
2 2 2 2 2 216
2x y y z z x xy yz zx Y . (1.45)
The prefactors in the above equations are chosen so that the uniaxial stress state leads to Y .
strain
stress
hold at constant stress
hold at constant strain
MSE 612 (Fall 2012) 18
Fig. 1.5: Under plane stress condition, Tresca criterion prescribes a polygon:
02121 ,,max , and Mises one gives an ellipse: 022
21
2212
1 .
The Mises criterion can be interpreted as the octahedral stress or the elastic energy of distortion. An octahedral plane is the material plane that makes equal angles with the principle stress directions at the considered point of the material. Therefore, the traction vector on the octahedral plane is
t n σ ,
which has the magnitude of 2 2 21 2 3
1
3t . The projection of t on the octahedral plane
normal is
1 2 3
1
3nt n σ n ,
so that the shear component on the octahedral plane is
2 2 2
1 2 2 3 3 1 2
1 2
9 3tt J . (1.46)
Fig. 1.6: Graphic representation of the yield criterion in the principal stress space.
Any arbitrary stress state can be plotted in the so-called “principal stress space”, with the three principal stresses as axes. Since hydrostatic stress is not involved in Tresca and Mises
1
2
Y
Y
Mises Tresca
elastic
yield
inaccessible
σ1
σ2
σ3
elastic
yield inaccessible
σ1 σ2
σ3
MSE 612 (Fall 2012) 19
criteria, the yield surface should have translational symmetry along 111 direction. Therefore,
we can also plot the yield surface in the so-called plane (i.e., the projection along 111 ).
Mohr-Coulomb Yield Criterion (frictional and dilatational effects)
For geomaterials such as rocks and solids and for concrete, the inelastic deformation occurs by frictional sliding over the plane of shearing, and thus the normal stress over the plane affects the yield. A simple criterion that accounts for this is the Mohr-Coulomb yield criterion:
c , (1.47)
where tan is the coefficient of internal friction, called the angle of friction, and c is the cohesive strength of the material.
In general, we represent the yield function by
2 3 1,f J J a bI ,
and the Mohr-Coulomb model can be rewritten as
1 3 1 2 2 3 1
1 2sin 2 cos sin
3 3c I , (1.48)
where 1 iiI .
Drucker-Prager Yield Criterion
Similar to the Tresca criterion, the Mohr-Coulomb criterion gives corners on the yield surface. An approximation of Mohr-Coulomb model, but without corners, is the Drucker-Prager model, giving
*2 1
10
3f J I k . (1.49)
Note that the Mohr-Coulomb yield model in Eq. (1.48) depends on all three stress invariants, while the Drucker-Prager model on the first and second stress invariants in Eq. (1.49).
Gurson-Tvergaard Yield Criterion for Porous Metals
Based on a rigid-perfectly plastic analysis of spherically symmetric deformation around a spherical cavity, Gurson (1977) suggested a yield criterion for porous metals in the form
2
2 212
2cosh 1 0
3 2 3Y
YY
Ig J f f
, (1.50)
where f is the porosity (void volume fraction) and Y is the tensile yield stress of the matrix
material. To improve the agreement with experimental data on ductile void growth, Tvergaard (1982) introduced two additional material parameters 1q and 2q , so that
MSE 612 (Fall 2012) 20
2
2 2 22 12 1 1
2cosh 1 0
3 2 3Y
YY
q Ig J f q q f
. (1.51)
The above model, also called Gurson-Tvergaard model, is often used in modeling ductile fracture of metals.
Plastic Flow
The strains can be decomposed into elastic and plastic parts:
e pij ij ij , (1.52)
where the elastic Hooke’s law gives
e pij ijkl kl ijkl kl klC C . (1.53)
While purely elastic deformation is a history independent process, in which the current stain depends only on the current stress, elastic-plastic states of strain depend on the entire history of loading and deformation. Consequently, the elastic-plastic constitutive equations are more appropriately expressed in an incremental or rate-type form by relating the rate of deformation to the rate of stress,
: p σ ε ε , (1.54)
which does not necessarily imply that the material behavior is rate dependent. Rate-dependency and history-dependency are independent from each other.
What happens after the yield? We need to determine the plastic flow direction, and the magnitude of the plastic strain. The latter can be described by an effective plastic strain (& its rate), given by,
2 2 2
1 2 2 3 3 1
2 2
3 3p p p p p p p p
ij ij . (1.55)
The direction of plastic flow is assumed to be determined from a flow potential ,
pij
ij
d d
,
p
εσ
. (1.56)
If f , it is called associated flow. In this case, the plastic strain rates are proportional to the outward normal to the yield stress in stress space, so that it is also called normality assumption.
If the yield function is given by
3
2 ij ij Yf s s ,
MSE 612 (Fall 2012) 21
0f : elastic deformation
0f : plastic deformation (1.57)
then the (associated) flow direction is
3
2ij
ij e
sf
. (1.58)
This can be proved by observing that 1
3ij ij kk ijs and 3
2ij
ij e
sf
s
, so that
3 1 3
2 3 2ijkl kl
ki lj ij klij kl ij e e
ss sf f
s
. (1.59)
Note that 32
3 3
22kl kl
kl eij ij
s sf
s s s
. The above equation leads to
2 2 3 3
3 3 2 2ij ijp p
ij ije e
s sd d d d
,
2
3p p
ij ijd d .
Now let’s examine a hindsight from the above discussion. During plastic loading, the principal components of the plastic strain rate tensor are parallel to the components of stress acting on the solid. Suppose you were to take a cylindrical shaft and pull it until it starts to deform plastically. Then, holding the axial stress fixed, apply a torque to the shaft. You may predict the occurrence of shear plastic deformation. However, experiments (done by G.I. Taylor in 1930s) show that the shaft will initially stretch, rather than rotate. This suggests that the plastic strain increment is proportional to the stress acting on the shaft, not the stress increment.
Hardening Behavior
Since gives the effective plastic strain, we can postulate that the yield function is a function of ,
, , ,... 0ij Yf .
For example, the so-called isotropic hardening gives
3
2 ij ij Yf s s . (1.60)
Physically, it means that accumulation of irreversible deformation (note that 0 ) will increase dislocation density, thus making dislocations less mobile. Some typical examples include:
perfectly plastic solid: .Y const
MSE 612 (Fall 2012) 22
linear strain hardening solid: 0Y Y h
power-law hardening: 0 1 mY Y h
An isotropic hardening law is generally not useful in situations where components are subjected to cyclic loading. It does not account for the Bauschinger effect, and so predicts that after a few cycles the solid will just harden up until it responds elastically. To fix this, an alternative hardening law allows the yield surface to translate, without changing its shape. The idea is illustrated graphically in Fig. 1.7(c). As you deform the material in tension, you drag the yield surface in the direction of increasing stress, thus modeling strain hardening. This softens the material in compression, so that this constitutive law can model cyclic plastic deformation.
For kinematic hardening, the yield surface is given by
30
2 ij ij ij ij Yf s s , (1.61)
where the evolution of the back stress ij relates to the plastic strain history. Some examples
include:
linear kinematic hardening: 3
2p
ij ijd cd ,
cyclic creep: 3
2p
ij ij ijd cd d
Consistency
Any plastic deformation beyond the yield point should satisfy 0f . Suppose that
,ij satisfies the yield condition, and the increment is given by
, , 0ij ij ij ijij
f ff d d f d d
,
so that
0ijij
f fd d
. (1.62)
MSE 612 (Fall 2012) 23
Fig. 1.7: Illustration of various hardening behavior in the plane stress space.
Tangent Modulus
Using elastic constitutive law in Eq. (1.54) and flow rule in Eq. (1.56) give
ij ijkl klkl
fd C d d
. (1.63)
1
2
2
2
(c) kinematic hardening
1
2
2
2
(b) isotropic hardening
1
2
2
2
(a) elastic-perfectly plastic
MSE 612 (Fall 2012) 24
The consistency condition in Eq. (1.62) gives
0pijkl kl kl
ij
f fC d d d
,
0ijkl klij kl
f f fc d d d
,
so that
ijkl kl
ij
ijklij kl
fc d
df f f
c
. (1.64)
Consequently, the stress increments are
: :
: :ij ijkl kl ijklkl
f dfd C d C
f f f
σ
σ σ
ε
,
: :
: :: :
ep f fd d d
f f f
σ σ
σ σ
σ ε ε
, (1.65)
where f f σ σ and f f . In terms of the index notation, we get
ijqr stkl
ij qr stepijkl ijkl
klmnop
mn op
f fC C
C Cf f f
C
(1.66)
Let’s rewrite the above equations in a form that can be directly used in computer
programming. Define a hardening modulus Yh
. The stress increments are given by
3, 0
1 1 2 2
3 3 3 3, 0
1 1 2 3 2 1 2 2 2
ij kk ij ij ij Y
ijijkl kl
ij kk ij ij ij YY Y
Ed d s s
dss dE E
d d s sE h
where , 0
0, 0
x xx
x
Viscoplasticity and Creep
MSE 612 (Fall 2012) 25
In material research community, viscoplasticity describes rate-dependent plasticity in which crystallographic slip is the dominant deformation process, while creep is typically used to describe rate-dependent plastic deformation which is mainly diffusion controlled, though dislocation activities also contribute. In mechanics, we won’t differentiate them.
In rate-dependent plasticity, there is no need to prescribe the yield surface and the loading-unloading condition (i.e., the consistency condition). Suppose that we get a uniaxial power-law creep,
1
00
mp
, (1.67)
which can be generalized into multiaxial stress states by
00
3
2
m
ijp eij
e
s
. (1.68)
Consequently, we get
,..pij ijkl kl kl ijkl kl e klC C g s . (1.69)
MSE 612 (Fall 2012) 26
1.3 Thermodynamics and Internal State Variables Discussion in the preceding sections gives us such an impression that it is quite flexible on the choice of the flow rule and hardening behavior. One question arises naturally as to whether there are restrictions on the constitutive behavior.
(1) Apparently, the constitutive law must satisfy the principle of material objectivity (or called material frame indifference). This means that the material response is independent of the observer. Typically, using stress invariants satisfies this condition.
(2) The plastic deformation must satisfy the laws of thermodynamics. This is described by the Drucker’s postulate and mechanics of dissipative process.
Drucker’s Postulate
Consider a stress ij that causes plastic flow. For stress *ij that does not reach yield or
just satisfies the yield criterion, the principle of maximum plastic resistance is given by
* 0pij ij ijd , (1.70)
which can be shown graphically in Fig. 1.8. It is a consequence of convex yield surface and normality assumption.
Fig. 1.8: Illustration of the principle of maximum plastic resistance.
The Principle of Maximum Plastic Resistance is important because it is the basis for a number of very important theorems concerning plastic deformation in solids. For example, it can be shown that the stress field in a material that obeys the Principle is always unique. In addition, the principle leads to clever techniques to estimate collapse loads for elastic-plastic solids and structures, which is an essential tool before FEA was practically available.
Constitutive models of inelastic behavior are based largely on experimental observations of plastic flow in laboratory specimens. Similar constitutive laws are used to describe very different materials, including metals, ceramics, glasses, soils and polymers. The mechanisms of deformation in these materials are very different, so it is surprising that their response is similar. One perspective on the structure of constitutive laws for inelastic solids was developed by Drucker in the 1950s. Drucker introduced the idea of a stable plastic material, stating that:
the work done in t by the increments of the boundary tractions must be positive.
elastic
yield
inaccessible
2 2
3 3
1 1
*ij
ij
ij 2 2
3 3
1 1
*ij
ij ij
MSE 612 (Fall 2012) 27
This appears very trivial; however, for plastic solids, it is equivalent to (1) convex yield surface, (2) normality condition, and (3) positive strain hardening. Consequently, most plasticity constitutive laws are very similar because of these three conditions.
Irreversible Thermodynamics and Internal State Variables
For irreversible thermodynamic process (e.g., process involving plastic or viscoelastic deformation), a set of internal variables can be introduced to describe the microstructural changes that occurred during the deformation process, such as the volume fraction of martensite phase, fraction of damage, free volume, concentration, etc.
With internal state variables, i , the rate of change of Gibbs free energy density is
ij ij i ig sT f , (1.71)
where ii
gf
can be called the generalized force, or configurational force, or thermodynamic
driving force. The evolution equations of the stable variables are
function , ,i ij jT . (1.72)
At constant stress and temperature, the direction of a spontaneous change of the thermodynamic state is in the direction of decreasing Gibbs energy,
0j jg f , (1.73)
which is a dissipative process.
Practical Concerns
Despite the thermodynamic restrictions, defining a plasticity constitutive law is largely up to your guess and curve fitting to the experiments. There does not exist strict rules on how to specify the flow (whether associate or non-associate) and hardening behavior. Your intuition is as good as, or better than, mine. Having said that, we should also admit the observations that:
(1) constitutive laws for most engineering materials have already been “guessed”;
(2) for advanced structural materials, the difficulty lies at the physical understanding of the deformation mechanism (often from molecular or microstructure-based simulations); generalization of the deformation mechanism into a set of constitutive law, if doable, is not the bottleneck.
MSE 612 (Fall 2012) 28
1.4 Formulations for Large Deformation For situations such as ductile metals, polymers, and creeping solids, we need to reformulate the above constitutive laws in terms of large deformation kinematics. The essential concepts of flow rule and hardening behavior remain the same. The difficulties include:
(1) How to choose appropriate stress and strain measures?
(2) Finite deformation involves rotation, so that we need to worry about the rotation compatibility and its evolution.
Kinematics
The deformation gradient can be decomposed into elastic and plastic parts:
e pF F F . (1.74)
where the plastic strain is assumed to shear the lattice, without stretching or rotating it, while the elastic deformation rotates and stretches the lattice. Since 1d d d x F X FF x , the velocity gradient is
1d d
d d
v xL FF
x x
, (1.75)
which can be decomposed into
1 1 1e p e e e p p e L L L F F F F F F . (1.76)
Thus the velocity gradient contains two terms, one of which involves only measures of elastic deformation, while the other contains measures of plastic deformation. We further write
L D W , e e e L D W , p p p L D W ,
where symD L , skewW L , and similarly for other terms.
Elasticity
For infinitesimal deformation, we write eij ijkl klC . For finite deformation, we write
e eij ijkl klC D , (1.77)
where the Jaumann stress rate is given by
e ed
dt
ττ W τ τW , (1.78)
and Jτ σ is the Kirchhoff stress.
Plasticity
The plastic constitutive laws for finite deformations are usually simple extensions of small strain plasticity. For example, for a finite strain, rate independent, Mises solid we set
MSE 612 (Fall 2012) 29
:1 3 3
2 2p
e e
ddt
h
ττ τ
D , (1.79)
where 3tr τ τ τ and 3
:2e τ τ .
Now the question is what about pW . For materials that do not develop a noticeable texture, we can simply set 0p W . The macroscopic stress-strain behavior is not very sensitive to the choice of pW , but any attempt to capture evolution of plastic anisotropy needs to specify this carefully.
This uncertainty can be completely avoided by using single crystal plasticity and simulating aggregates of grains, because crystal plasticity has an unambiguous definition of the plastic spin. However, the simulation cost can be astronomically high.
MSE 612 (Fall 2012) 30
1.5 Applications in Materials Science and Engineering As a material researcher, engineer, or student, with the knowledge of
constitutive law + numerical procedure to solve the BVP/IVP,
what can we do for our realistic material applications? This can be shown in Figure 1.9.
Fig. 1.9: Theme topics in this course.
In materials research, most important applications include: failure analysis and prevention, processing-structure relationship, and material design (structure-property relationship). What computational plasticity and micro-plasticity provide is just stress analysis, -- we need to make a connection of stress analysis to the mechanisms involved in material phenomena -- a theme topic we are going to illustrate in this course. To this end, we will discuss five special topics from Chapter 3 to Chapter 6.
(1) Material failure: We focus on fracture and fatigue, which have been repeatedly examined in a number of MSE courses. Here I take a different point of view, i.e., a multiscale view of the crack tip behavior, with illustrative examples on modeling the crack tip process zone.
(2) Crystal plasticity: This could be the most important tool for MSE students for microstructure-based modeling. The theory has a direct connection to the slip anisotropy (Schmid law), texture development, strengthening mechanisms, just to name a few.
(3) Material instability: We use strain localization and defect nucleation to illustrate this important line, which is viewed rather differently between mechanics and MSE communities. Buckling, necking, etc. are called geometric instability, but not material instability.
(4) Microstructure-based simulations: We discuss the effects of stress in phase transformation and microstructural evolution. For instance, the stress-assisted diffusion on the grain boundaries controls the cavitation process in high temperature alloys. We focus on a continuum framework here. At the end, noting that modern electronic and other high-technology applications often adopt thin film heterostructures, we will survey reliability issues including debonding, electromigration, interconnect design, etc.
Deformation/failure/evolution mechanisms (materials science and engineering)
Stress analysis & constitutive and failure models (applied mechanics in this course)
Standard engineering tests (materials science and engng)
Engineering applications: design rules for structural integrity
(failure prevention, reliability issues, etc)
material processing (microstructural information, stress-assisted phase transformation, etc.)
material design (strengthening mechanism, composite, etc.)
…
MSE 612 (Fall 2012) 31
2. FINITE ELEMENT METHOD The three principles of the continuum mechanics, together with appropriate boundary and initial conditions, will lead to a set of boundary/initial value problems. In the case of linear elasticity, a lot of analytical methods have been developed in the past century, most of which are based on complex analysis and are restricted to simple geometries such as half space, beam, cracks, etc. For plastic solids, in general, analytical solution is impossible. (There are cases where the slip-line theory can be applied to derive the plastic deformation field. But this is only useful for rigid-perfectly plastic behavior.) Numerical techniques, such as finite element method, finite difference method, and boundary element method, have been extensive research topics in the past several decades. Particularly, the finite element method, because of its versatile engineering applications, has become a standard engineering analysis tool. The fundamental principles and algorithms of the finite element method have been well established, so that the study of this method alone, at present, is gradually marginalized.
In Section 2.1, we will first review the finite element analysis (FEA) procedure from the standpoint of end users. FEA is based on the principle of virtual work, as discussed in Section 2.2. When the material behavior is linear elastic, the strain measure is an infinitesimal one, and the load is static, the FEA formulation leads to a set of linear algebraic equations (Section 2.3). Nonlinearity can derive from kinetics (which will not be considered in our course), kinematics (i.e., finite deformation, which will be described but not in great details in this course), and constitutive law (i.e., plasticity). A general FEA procedure to solve the plastic flow problem of a solid under force/displacement loading history is given in Section 2.4, and its implementation in ABAQUS UMAT is discussed in Section 2.5. Two examples in Section 2.6 are used to compare the explicit and implicit integration schemes. Section 2.7 gives a practice of ABAQUS UMAT.
2.1 Finite Element Analysis (FEA) Procedure As the course title suggests, we need to understand the finite element method (“computational”), the plasticity behavior (“plasticity”), and the material microstructural aspects (“micromechanics”). In the following chapters, we will cover these topics in details.
If we are merely the end users of a finite element software package, provided with (1) Geometry of the solid, (2) Boundary/initial conditions (including geometric constraints, etc), (3) History of loading and displacement boundary conditions, (4) Material behavior (i.e., constitutive law),
then we can determine (1) The deformation of the solid under a given history of applied B.C., or, (2) The stress distribution inside the solid, or (3) other physical quantities as a combination of stresses and strains, etc.
To illustrate the FEA procedure, let’s use ABAQUS as a representative example. The problem of our interest is a compact tension (CT) specimen loaded by a pair of pins. http://web.utk.edu/~ygao7/teaching/CT_teaching.cae http://web.utk.edu/~ygao7/teaching/CT_teaching.jnl
MSE 612 (Fall 2012) 32
Fig. 2.1: A representative example of boundary value problem – compact tension test.
Part (geometric information)
A CAE module is used to sketch the geometry.
(22.86,0) (63.5,0) mm
(63.5,30.48) (0,30.48)
(0,1.3361) (21.3341,1.3361)
(12.7,13.97)
(19.05,13.97)(6.35,13.97)
MSE 612 (Fall 2012) 33
Property (material behavior)
Define a set of materials, and assign them into different sections of the parts.
Assembly (as in an CAE module)
MSE 612 (Fall 2012) 34
Assemble various parts into an instance, e.g., the CT specimen plus the loading pin.
Step (history log)
Geometric information and loading conditions are one-to-many relationship.
Interaction (for contact model)
MSE 612 (Fall 2012) 35
Define (frictionless and impenetrable) contact between the loading pin and the CT specimen.
Load (B.C. and I.C.)
Symmetric B.C. along 2 0x , and displacement control of the loading pin.
Mesh (spatial discretization)
MSE 612 (Fall 2012) 36
The parts are discretized into a number of elements. Meshing is also an integral part of the CAE module. Most engineering analysis software are compatible in terms of this part. Clearly, the mesh should be design so that the mesh density is high at high stress gradients and at places of our post-processing interest. One can choose triangular and quadrilateral elements for 2D and axisymmetric problems, and tetragonal and brick-type elements for 3D problems.
Job (submitted for analysis)
It runs for 10 seconds, because we use linear elastic behavior and infinitesimal deformation.
Visualization (post-processing)
MSE 612 (Fall 2012) 37
Here we can generate a lot of color pictures to please our bosses, or to mislead their attentions. Contour plots of stress, strain, displacements, etc. are usually very straightforward to engineers. Oftentimes, we also need to extract information along a certain path. For example, the normal stress is obtained along the middle plane.
We can fit the stress distribution to 22 2K x , which is the asymptotic solution of a
crack in the framework of linear elastic fracture mechanics. Here we actually do not have a perfect crack, so we fit
022 22
2
K
x
, (2.1)
where * 0.058K E mm in this case (which is proportional to the magnitude of applied load).
0 5 10 15 20 25 30 35 40-5000
0
5000
10000
15000
20000
x
22
FEA resultskE*/(2x)+
220
k=0.058
Fig. 2.2: Stress distribution ahead the CT specimen notch.
MSE 612 (Fall 2012) 38
In another example, we study a perfect crack, and the stress field gives a perfect singular
relationship, 2K x . Here we use 250E , 0.3 , and 1K .
10-1
100
101
102
103
104
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
r
22
FEM resultstheoretical
Fig. 2.3: Stress distribution ahead of a perfect crack.
MSE 612 (Fall 2012) 39
2.2 Principle of Virtual Work The finite element method is based on the principle of virtual work. This principle can be used to derive the principle of minimum potential energy and Castigliano’s theorem for elastic solids. In essence, the principle of virtual work is the integral representation of the governing differential equations in continuum mechanics. For linear elastic solid, it is equivalent to the energy representation. It should be noted that this section is beyond the scope of most MSE students. You don’t need to read it line by line, but it will help your course project if your project is related to finite element implementation of your own constitutive law.
Let’s restrict our attention to infinitesimal deformation. Introduce a small increment of the displacement field, u x , which is continuous and twice-differentiable, and satisfies
0 u at u . (2.2)
Consequently, it is called the kinematically admissible virtual displacement field.
The principle of virtual work states that: if ij satisfies
* 0ij ij i i i i
A
dV b u dV t u dA
, (2.3)
for all virtual displacement fields, then the stress field must satisfy
, 0ji j ib , *i ij jn t on s .
And vice versa. This principle can be proved by the Gauss divergence theorem, namely,
*
*,
*,
ij ij i i i i
A
ij i j i i i i
A
ij j i ij j i i i i i
A A
dV b u dV t u dA
u dV b u dV t u dA
n u dA u dV b u dV t u dA
Physically, it means that the internal work and external work done by the equilibrium stress/force fields on the equilibrium strain/displacement fields must balance.
Principle of Minimum Potential Energy
For elastic solids, the principle of virtual work can be rewritten as the principle of minimum potential energy. That is, for arbitrary kinematically admissible displacement field, the equilibrium solution corresponds to the minimum of the potential energy:
*, ,
1
2 ijkl i j k l i i i i
A
C u u dV b u dV t u dA
. (2.4)
This is proved as follows. Suppose we perturb the displacement field by u u , then
MSE 612 (Fall 2012) 40
2 2
* 2, , , ,
1, ,
2
1
2ijkl i j k l i i i i ijkl i j k l
A
C u u dV b u dV t u dA C u u dV
u u u u
Therefore, , 0 u u because of Eq.(2.3), and 2 , 0 u u because the last term is
positive definite.
Fig. 2.4: A cantilever bending problem with unit width (in z direction).
Application of Principle of Minimum Potential Energy: Beam Bending
In Mechanics of Materials course, you may have learned the beam bending problem. Let’s revisit it from the standpoint of the principle of minimum potential energy. Assume the Kirchhoff condition, that is,
xx y , (2.5)
where w and w is the deflection (i.e., yu ). The strain energy density is therefore
2 21 1
2 2xx xx Ey w ,
and the total potential energy of the system is
2
2 2 2
0 2 0
1 1
2 2
hL L
h
Ey w dydx Pw L EIw dx Pw L
,
where 2
2
2
h
h
I y dy
is the moment of inertia.
The variation of the total potential energy is
P
x
y L
h
MSE 612 (Fall 2012) 41
0
00
0 00
L
L
LL
L
LL L
L
EIw w dx P w
EIw w EIw w dx P w
EIw w EIw w EIw wdx P w
where integration by parts (i.e., Gauss divergence theorem in 1D) has been applied. Using the principle of minimum potential energy, 0 for kinematically admissible deflection variation (note 0 0w and 0 0w ), gives the differential equations:
0EIw ,
0w L , 0EIw L P . (2.6)
The analytical solution is given by 2
36
Pxw L x
EI .
A general numerical procedure can be derived as follows. We can assume a certain form of deflection field, which is kinematically admissible. The unknown prefactors can be determined by using the minimum condition. For example, assuming 2w x cx , so that
2 22c EIL PcL .
Minimizing c gives 4
PLc
EI . Apparently, the accuracy of the solution depends on the initial
guess. Such a procedure is called Castigliano method or Ritz-Galerkin method.
Motivated by the above discussion, we can represent the entire displacement field by
some sort of combination of aw x at discrete points ax , and then the potential energy
functional becomes a function of a x . The so-called finite element method prescribes an
efficient way for this geometric representation.
Principle of Virtual Power
Many constitutive laws are presented in rate form, so that we consider:
(a) Rate of work done by Cauchy stress: ij ijdV (2.7)
(b) Rate of work done by the applied force: *i i i i
A
t u dA b u dV
(2.8)
Introduce a small increment of the velocity field, v x , which is continuous and twice-
differentiable, and satisfies
0 v at u . (2.9)
MSE 612 (Fall 2012) 42
Consequently, it is called the kinematically admissible virtual velocity field.
The principle of virtual power states that: if ij satisfies
* 0ij ij i i i i
A
dV b v dV t v dA
, (2.10)
for all virtual velocity fields, then the stress field must satisfy
, 0ji j ib ,
*i ij jn t on s .
And vice versa. This principle can be proved by the Gauss divergence theorem, namely,
*
*,
*,
ij ij i i i i
A
ij i j i i i i
A
ij j i ij j i i i i i
A A A
dV b v dV t v dA
v dV b v dV t v dA
n v dA v dA b v dV t v dA
MSE 612 (Fall 2012) 43
2.3 Finite Element Method for Linear Elastic Solids The governing field equations are given by
, ,
1
2ij i j j iu u ,
ij ijkl klC ,
, 0ij i jb ,
*i iu u on u ,
*ij i jn t on s . (2.11)
The key of finite element method is to use a set of nodal values to represent the continuous displacement field. That is, the solid is discretized into a set of nodes, which are connected through elements, as shown in Fig. 2.5. The relationship between element number and the associated nodes is called connectivity relation.
The nodal values are denoted as
aiu at a
ix , where a =1:n.
And the displacement field at an arbitrary location is determined by an interpolation scheme:
1
na a
i ia
u N u
x x .
This is a rather tedious procedure, since we don’t want to determine iu x from all the nodal
values. If the interpolation is conducted in each element, we get
1
ena a
i ia
u N u
x x , (2.12)
where x is located in the element.
Fig. 2.5: Mesh = elements + nodes.
3 2
1
7
23
10
#11
#1 #17
#5
#8 1 2,a ax x 1 2,b bx x
1 2,c cx x
1bu
2bu
1cu
2cu
1au
2au
MSE 612 (Fall 2012) 44
For the triangular element shown in Fig. 2.5, the linear interpolation is given by
1 2 1 2 1 2 1 2, , , ,a a b b c ci i i iu x x N x x u N x x u N x x u (2.13)
where
2 2 1 1 1 1 2 2
1 2
2 2 1 1 1 1 2 2
,b c b b c b
a
a b c b a b c b
x x x x x x x xN x x
x x x x x x x x
. (2.14)
The representation of 1 2,bN x x is the same as above after rotating a b and b c , and
similarly for 1 2,cN x x . For this linear interpolation function, obviously .ajN x const . The
interpolation functions satisfy
a babN x .
Finite Element Formulation from Principle of Minimum Potential Energy
The potential energy is
*, ,
1
2 ijkl i j k l i i i i
A
C u u dV b u dV t u dA
. (2.15)
Since the solid has been discretized, these integrals are evaluated in each element, given by,
e
a ba b
ijkl i ka bj l
N NC u u dV
x x
x x
*
e se
a a a ai i i i
a a
b N u dV t N u dA
x x
Therefore, the element potential energy is written as
1
2b a a a
e aibk k i i ia b a
k u u f u , (2.16)
where aibkk and iaf are called element stiffness matrix and element force matrix, given by
e
a b
aibk ijklj l
N Nk C dV
x x
x x
, (2.17)
*
e se
a a ai i if b N dV t N dA
x x . (2.18)
In terms of matrix representation, for the element #17, we can rewrite Eq. (2.16) as
MSE 612 (Fall 2012) 45
11111 1112 1131 1132 1171 11721
11211 1212 1231 1232 1271 12722
33111 3112 3131 3132 3171 31721
33211 3212 3231 3232 3271 32722
77111 7112 7131 7132 7171 71721
77212
1
2
T
e
k k k k k ku
k k k k k ku
k k k k k ku
k k k k k ku
k k k k k ku
ku
1 1 11 1 11 1 12 2 23 3 31 1 13 3 32 2 27 7 71 1 17 7 7
1 7212 7231 7232 7271 7272 2 2 2
Tu u f
u u f
u u f
u u f
u u f
k k k k k u u f
We can assemble all the unknowns into a column vector,
11
12
21
22
1
2
ˆ
..
ai
n
n
u
u
u
u u
u
u
u (2.19)
so that the total potential energy is
1
ˆ ˆ ˆ2
T Te
element
u Ku u F , (2.20)
where K and F are assembled from element stiffness and force matrices, respectively. This procedure is further illustrated in the next subsection.
Finally, the principle of minimum potential energy gives rise to a set of linear algebraic equations:
ˆ Ku F . (2.21)
Finite Element Formulation from Principle of Virtual Work
Using the principle of virtual work gives
*, , 0
s
ijkl k l i j i i i iC u u dV b u dV t u dA
, (2.22)
for all virtual displacement fields iu that satisfies 0iu on u . If iu is the exact solution of
the governing boundary/initial value problem, then the above equation should be true for all kinematically admissible virtual displacement fields.
The principle of virtual work can be written as
.. 0element
,
where
MSE 612 (Fall 2012) 46
*
..e
e se
b ab a
ijkl k ib al j
a a a ai i i i
a a
b aaibk k ia i
a b
N NC u u dV
x x
b N u dV t N u dA
k u f u
x x
x x (2.23)
and aibkk and iaf are given in Eq. (2.17) and (2.18), respectively.
Consequently, the principle of virtual work is represented by
ˆ ˆ 0T u Ku F . (2.24)
Since this is true for any virtual displacement field, we recover Eq. (2.21).
Plate-Hole Problem
In this section, we will go through a simple matlab code that implements the finite element procedure solving a plate-hole problem (Fig. 2.6). Codes can be downloaded from:
http://web.utk.edu/~ygao7/teaching/plate_hole_FEA.m
http://web.utk.edu/~ygao7/teaching/plate_hole_FEA_localK.m
-0.5 0 0.5 1 1.5 2 2.5 3-0.5
0
0.5
1
1.5
2
2.5
3
1
2
3 4 5
6
7
8 910
11
12
131415
# 1
# 2
# 3
# 4# 5
# 6# 7
# 8
# 9
#10
#11
#12#13
#14#15
#16
Fig. 2.6: A plate-hole problem, using triangular mesh and ¼ cell.
Step (i): Spatial discretization
Both nodal coordinates and node-element connectivity relationship should be prescribed. Usually this can be done by a CAE module. But here we have to construct it by ourselves.
MSE 612 (Fall 2012) 47
nodes = 1.0000 0 0.9239 0.3827 0.7071 0.7071 0.3827 0.9239 0 1.0000 1.5000 0 1.4619 0.6913 1.3536 1.3536 0.6913 1.4619 0 1.5000 2.0000 0 2.0000 1.0000 2.0000 2.0000 1.0000 2.0000 0 2.0000 elements = 1 6 2 6 7 2 2 7 3 7 8 3 3 8 4 8 9 4 4 9 5 9 10 5 6 11 7 11 12 7 7 12 8 12 13 8 8 13 9 13 14 9 9 14 10 14 15 10
Step (ii) Material properties
Here we choose 100E and 0.3 . Note that the units for length and force (or stress) are arbitrary. If you interpret L=mm and F=N, then stress should be MPa.
Step (iii) Stiffness matrix
First, let’s take a look at the local stiffness. For the element in Fig. 2.6, the three nodes are abc . The strain components are given by
1
1 1 1 211
122
2 2 2 212
1
22 1 2 1 2 1
0 0 0
ˆ0 0 0
2
a b c a
a
ba b c
eb
ca a b b c c
c
N N N u
x x x u
uN N N
x x x u
uN N N N N N
ux x x x x x
ε Bu , (2.25)
and the stress components are given, for plane-strain conditions, by
MSE 612 (Fall 2012) 48
11 11
22 22
12 12
1 0
1 01 1 2
0 0 1 2 2 2
E
σ Dε . (2.26)
The interpolation functions and their derivatives can be obtained from Eq. (2.14).
Therefore the strain energy in a given element is
1 1
ˆ ˆ2 2
e
T Telement e e e
A
W dA A σε u B DBu , (2.27)
and the element stiffness matrix is
Te ek A B DB , (2.28)
where eA is the element area. When deriving Eq. (2.27), we note that B is a constant matrix in
the linear triangular element. K_local (for element # 1)= 84.8741 55.3962 -55.9227 -26.5500 -28.9514 -28.8462 55.3962 131.4752 -36.1654 -30.1452 -19.2308 -101.3301 -55.9227 -36.1654 52.0974 7.3193 3.8252 28.8462 -26.5500 -30.1452 7.3193 16.7568 19.2308 13.3883 -28.9514 -19.2308 3.8252 19.2308 25.1262 0 -28.8462 -101.3301 28.8462 13.3883 0 87.9417 K_local (for element # 2) = 31.2195 -20.8074 5.1833 3.0485 -36.4027 17.7589 -20.8074 55.5553 -6.5669 -48.4533 27.3743 -7.1020 5.1833 -6.5669 42.3212 27.6225 -47.5045 -21.0556 3.0485 -48.4533 27.6225 65.5581 -30.6710 -17.1049 -36.4027 27.3743 -47.5045 -30.6710 83.9072 3.2967 17.7589 -7.1020 -21.0556 -17.1049 3.2967 24.2068 K_global (after assembling # 1 and # 2 elements)= 84.8741 55.3962 -28.9514 -28.8462 0 0 0 0 0 0 -55.9227 -26.5500 0 0 zeros(1,16) 55.3962 131.4752 -19.2308 -101.3301 0 0 0 0 0 0 -36.1654 -30.1452 0 0 zeros(1,16) -28.9514 -19.2308 109.0334 3.2967 0 0 0 0 0 0 -32.5775 46.6051 -47.5045 -30.6710 zeros(1,16) -28.8462 -101.3301 3.2967 112.1486 0 0 0 0 0 0 46.6051 6.2864 -21.0556 -17.1049 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) -55.9227 -36.1654 -32.5775 46.6051 0 0 0 0 0 0 83.3169 -13.4882 5.1833 3.0485 zeros(1,16) -26.5500 -30.1452 46.6051 6.2864 0 0 0 0 0 0 -13.4882 72.3121 -6.5669 -48.4533 zeros(1,16) 0 0 -47.5045 -21.0556 0 0 0 0 0 0 5.1833 -6.5669 42.3212 27.6225 zeros(1,16) 0 0 -30.6710 -17.1049 0 0 0 0 0 0 3.0485 -48.4533 27.6225 65.5581 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 zeros(1,16)
Step (iv) displacement boundary conditions
MSE 612 (Fall 2012) 49
Here let’s just consider displacement boundary conditions, so that the force vector is zero. Of course, we can apply nodal forces, boundary tractions, etc., and you can refer to any FEA textbook for details. dispBC = 1.0000 2.0000 0 6.0000 2.0000 0 11.0000 2.0000 0 5.0000 1.0000 0 10.0000 1.0000 0 15.0000 1.0000 0 13.0000 2.0000 0.1000 14.0000 2.0000 0.1000 15.0000 2.0000 0.1000
One might wonder that, since 0F , the solution of ˆ Ku F should only give trivial solution. Well, if we do not prescribe appropriate boundary conditions, the linear algebraic equations actually have many solutions, since det 0K . If we apply a displacement boundary
condition on the i -th coordinate of node a, then we need to adjust the index 2 1m a i in
the global stiffness matrix. That is,
111 12 1 1 1
121 22 2 2 2
1 2
1 2 2
.. .. 0
.. .. 0
.. .. .. .. .. .. ....
.. .. 0
.. .. .. .. .. .. ....
.. .. .. 0
m M
m M
am m mm mM i
nM M Mm
k k k k u
k k k k u
k k k k u
k k k u
is modified as
1111 12 1 1
1221 22 2 2
1 2 2
.. 0 ..
.. 0 ..
.... .. .. .. .. .. ..
0 0 .. 1 .. 0
.... .. .. .. .. .. ..
.. 0 .. ..
mM
mM
ai
nMmM M
kk k k u
kk k k u
u
kk k u
where 2M n .
For our problem, the original force vector is zero, and we prescribe a displacment 0.1 on the top surface. After the modification, we get force_global_modified = 0 0 0 0 0 0 0 0 0 0
MSE 612 (Fall 2012) 50
0 0 0 0 1.3689 12.5358 -0.1994 19.0267 0 9.8213 0 0 -1.1848 3.6712 -5.2771 0.1000 -0.3297 0.1000 0 0.1000
Step (v) Solving linear algebraic equations
There are a lot of well established methods to solve a set of linear algebraic equations with a symmetric and sparse coefficient matrix (i.e., the global stiffness matrix here). Here we simply use matlab’s own procedure to solve it.
displacements=K_global_modified\force_global_modified;
Step (vi) Post-processing
After obtaining the displacement field, we can calculate strain and stress, and any other combinations of strain and stress components.
-0.5 0 0.5 1 1.5 2 2.5 3-0.5
0
0.5
1
1.5
2
2.5
3original meshdeformed mesh
Fig. 2.7: A typical visualization of FEA results.
MSE 612 (Fall 2012) 51
The above figure plots both the original mesh and the deformed mesh. It should be noted that FEA only gives the nodal values, after solving ˆ Ku F . If we are interested in a field quantity inside an element, we need to interpolate the nodal values, using Eq. (2.12).
Practical Concerns of FEA Procedure
The general FEA procedure for linear elastic solids is clearly demonstrated in the above plate-hole example. This course deals with plastic solids, so that we shall briefly overview some practical issues in FEA.
Principle of minimum potential energy VS principle of virtual work
Using either principle of minimum potential energy or principle of virtual work is the same for elastic solids. However, as we have discussed, the principle of virtual work is more general, and will be used for plastic solids.
For rate dependent materials, the principle of virtual work is rewritten as the principle of virtual power.
Element interpolation and isoparametric mapping
The plate-hole example uses linear triangular elements, in which the interpolation functions can be derived using the nodal coordinates. The reason that it is called linear interpolation is that the interpolation function is a linear polynomial. For a 6-node triangular element, the interpolation function will be quadratic.
Deriving interpolation function in ix coordinates is extremely tedious. Usually, an
element in ix coordinates is mapped into a general shape in Fig. 2.8 by setting
1
eNa a
i j ia
x N x
, (2.29)
where j are called local coordinates. The displacement fields are then represented by
1
eNa a
i j ia
u N u
, (2.30)
which can also be regarded as a kind of mapping. To this end, we can use the same interpolation functions in Eqs. (2.29) and (2.30), which is thus called isoparametric mapping.
MSE 612 (Fall 2012) 52
Fig. 2.8: Several common elements in local coordinates, following ABAQUS.
The above figure gives several representative elements in i coordinates. Of course, one
can change the numbering sequence, locations of these points, etc., as long as all his/her codes are self-consistent. The conventions used here agree with ABAQUS manual.
For the linear triangular elements, the interpolation functions are
11
22
31 21
N
N
N
and for quadratic triangular elements,
11 1
22 2
31 2 1 2
41 2
52 1 2
61 1 2
2 1
2 1
2 1 1 1
4
4 1
4 1
N
N
N
N
N
N
and for linear quadrilateral elements,
1
2
3 1
2
1
2
3 1
2
4 5
6
1
2
3
1 2
4
1
2
3
1
2
3
4
MSE 612 (Fall 2012) 53
11 2
21 2
31 2
41 2
0.25 1 1
0.25 1 1
0.25 1 1
0.25 1 1
N
N
N
N
and for linear tetragonal elements,
1 2 3 41 2 3 1 2, , , 1N N N N
A practical concern of isoparametric mapping is the evaluation of B in Eq. (2.25), which involves a
iN x . Using the chain rule gives
a a
i
j i j
N N
x x
, (2.31)
where
1
ij ij
ξ x
x ξ,
1 1
e eaN N
a a aii i
a aj j j
NxN x x
,
and det i
j
xJ
.
Properties of the stiffness matrix
The stiffness matrix (either element or global ones) is symmetric and singular. The modified global stiffness matrix is nonsingular. Since the components of stiffness matrix is given by aibkk , so that if two nodes are not connected by any element, then the corresponding
component in the stiffness matrix is zero. Apparently, nodes are connected only through their surrounding elements, so that the stiffness matrix is sparse and banded.
Solving a set of linear algebraic equations with a gargantuan sparse, banded, and almost singular coefficient matrix is not as simple as it appears. You can refer to any textbook on numerical algebra for details on this issue. This also implies that the FEA code you wrote is usually much less robust than the commercial codes.
Gaussian quadrature
The integration in Eq. (2.27) is simple since B and other matrices are constants. In general, we have to carry out numerical quadrature. This is a rather tedious procedure, if we use too many points.
A very efficient numerical scheme for polynomials is the Gaussian quadrature, giving
1
11
MI
II
g d w g
,
MSE 612 (Fall 2012) 54
1 1
1 2 1 2 1 21 11 1
, ,N N
I JI J
I J
f d d w w f
,
where I , 1..I M are the integration points, and Iw are the integration weights. They are
tabulated below:
1M , 110, 2w ,
2M ,
11
22
0.5773502691, 1.0
0.5773502691, 1.0
w
w
,
3M ,
11
22
33
0.7745966691, 0.5555555555
0, 0.8888888888
0.7745966691, 0.5555555555
w
w
w
The quadrature method is accurate for polynomials with orders not larger than 2 1M .
>> syms x >> F=inline('x.^3-10*x.^2'); >> quad(F,-1,1) ans = -6.6667 >> x1=0.5773502691; >> ((-x1)^3-10*(-x1)^2)*1+(x1^3-10*x1^2)*1 ans = -6.6667 >> F=inline('x.^4-10*x.^2'); >> quad(F,-1,1) ans = -6.2667 >> ((-x1)^4-10*(-x1)^2)*1+(x1^4-10*x1^2)*1 ans = -6.4444
MSE 612 (Fall 2012) 55
2.4 Finite Element Method for Plastic Solids The constitutive law for elastic-plastic solid is very appealing, but how shall we apply these equations for realistic applications? Analytical solutions are impossible. Approximate analyses such as slip-line theory and limit analysis were very popular before computational plasticity becomes practically available. These techniques are essentially being phased out.
The boundary/initial value problem is given below. The governing equations are
, ,
1
2ij i j j iu u ,
, 0ij i jb ,
*i iu u , on u ,
*ij i jn t , on s , (2.32)
with the elastic-plastic constitutive law:
e pij ij ij ,
1
1eij ij kk ijE
,
3
2ijp
ije
S
, (2.33)
where 3ij ij kk ijS , 3
2e ij ijS S , and 2
3p p
ij ij . For rate-independent solid, is
obtained from the consistency condition. For rate-dependent solid, is usually specified as a function of Mises stress. The constitutive parameters involved in the flow equations will evolve according to the hardening behavior.
Note that we now have a history dependent problem. We need to specify the time variation of the applied load and boundary conditions, and our objective is to calculate the displacements, strains, and stresses as functions of time.
We shall establish the finite element formulation based on the principle of virtual power (i.e., the rate form of the principle of virtual work). If ij satisfies
*, 0
s
ij i j i i i it v dV b t v dV t t v dA
, (2.34)
for all virtual velocity fields iv (note ,ij i j ij ijv ), then the stress field must satisfy the
force balance equation and the traction boundary condition.
MSE 612 (Fall 2012) 56
(i) Spatial discretization
This is the same as that in the previous section. The unknowns are nodal displacements,
aiu t , where the nodal number a runs from 1 to NNODE. Since the problem is history
dependent, we apply the loads (or impose displacements) in a series of increments, and calculate the change in displacements and stresses during each successive increment. That is, we compute
aiu t t from the solutions available from the end of previous time step, i.e., a
iu t . The
displacement increments are interpolated as
1
x xn
a ai i
a
u N u
,
1
x xn
a ai i
a
v N v
,
, ,1
1 1
2 2
a ana a
ij i j j i i ja j i
N Nu u u u
x x
. (2.35)
(ii) Governing equations in discrete form
We need to represent every term in Eq. (2.34) by the unknowns are aiu . The second and
third terms are straightforward. In the first term, we now need to find a way to compute the stress increments caused by the displacement/strain increments during time interval t . – This is the step of stress update algorithm, or called constitutive update algorithm. – For now, we assume that this can be done, and we write
, ,aij ij kl i ij ij klt t u t t t . (2.36)
The principle of virtual work becomes
*, 0s
aa a a
ij kl i i ij
Nt dV b t t N dV t t t N dA v
x
,
and since this must hold for all kinematically admissible aiv , we must ensure that
*, 0s
ab a a
ij kl m i ij
Nu t dV b t t N dV t t t N dA
x
, (2.37)
which is a set of nonlinear equations to be solved for aiu .
(iii) Newton-Raphson iteration
Solving a set of nonlinear equations
MSE 612 (Fall 2012) 57
0i jf x (2.38)
can be done by the Newton-Raphson method. Start from an initial guess 0jx , and we can iterate
and update the solution kjx at the k -th step to 1k
jx at the 1k -th step, according to the
following equation:
1
1 1 111
2 2 2
... ... ...
k k k
k k k kj
x x f
x x x f
J , (2.39)
where
1 1
1 2
2 2,
1 2
...
...
... ... ...
i j
f f
x x
f ff
x x
J
The above procedure is derived from the Taylor expansion,
1 1 0k k k k f x f x J x x .
(iv) Linearization of Eq. (2.37)
For clarity, let’s denote a ai iw u . Since Eq. (2.37) is a set of nonlinear equations, we
need to use the Newton-Raphson iteration for root finding. Starting the solution for a generic
load step with an initial guess 0
aiw (we can use the exact solution at the end of the preceding
increment as the trial solution), we need to correct solution by a N-R iteration,
1
a a ai i ik k
w w dw .
Ideally, of course, we would want the correction to satisfy
* 0s
ab b a a
ij kl m m i ij
Nw dw dV b N dV t N dA
x
(2.40)
We linearize in bmdw to obtain a system of linear equations
* 0s
aijb a a
ij ij m kl i ikl j
Nt w d dV b N dV t N dA
x
,
which, together with Eq. (2.35) (i.e., relating kld and bkdw by b
lN x ), gives
MSE 612 (Fall 2012) 58
0b a aaibk k i iK dw R F (2.41)
with
a b
epaibk ijkl
j l
N NK C dV
x x
,
a
a bi ij ij kl m
j
NR t w dV
x
,
*
s
a a ai i iF b N dV t N dA
,
where ijepijkl
kl
C
is the material tangent (or material stiffness, or material Jacobian, or
algorithm stiffness), aibkK is the stiffness matrix, aiR is the residual vector, and a
iF is the force
vector.
(v) Stress update
We see that the key step is to integrate the plastic stress-strain equations to obtain the stress increment, σ , and the stiffness matrix ep , caused by an increment in total strain
ij applied to the specimen during a time interval t .
MSE 612 (Fall 2012) 59
2.5 Implementation Procedure Developing an elastic-plastic finite element code is a straightforward and tedious task. Based on the formulation in Section 2.4, the implementation procedure can be illustrated in the flow chart in Figure 2.9. There are two key steps:
(a) Global Newton-Raphson iteration is used to solve Eq. (2.40). In this step, given a trial
solution a
ku , the task is to linearize Eq. (2.40) into Eq. (2.41), and to update the
solution to 1
a
ku , until some convergence criterion is reached.
(b) Integration scheme and stress update subroutine. This is a module/subroutine/function that can be called in the global Newton-Raphson iteration. Since material behavior is usually nonlinear, a local Newton-Raphson iteration is usually used.
Flow Chart
We will use ABAQUS as an illustrative example. Since the load/displacement boundary conditions have been discretized into a number of time steps, we will calculate the displacement increments with given t , i.e., Step (a). Of course, if the iteration does not converge or the rate of convergence is very slow, we need to specify a smaller time step. There are a number of good criteria on the automatic time stepping. This is, however, out of our scope here, since our focus is on the usage of plasticity models, not on the finite element method itself.
In Step (a), solving Eq. (2.41) requires the information of
,a
kt σ u ,
,
ep
k t
σ
ε , (2.42)
at each Gaussian integration point of each element. These updates will then be passed to the routines that calculate element residual and element stiffness information. Using the global Newton-Raphson iteration, once a convergent solution has been found, the stress and all state variables at the element integration points must be updated, before starting the next load step.
The Step (b) is called stress/constitutive update subroutine or material integration subroutine. This step is independent of element information, global solution method, and boundary conditions. What is required is the initial condition at nt and the loading parameter ε ,
and our task is to update the stress and to calculate the material Jacobian.
In ABAQUS, the Step (b) is done by a User defined MATerial (UMAT) subroutine. Many other FEA packages provide similar user interfaces. There’s a lot more bookkeeping to do to keep track of the history dependence of the material. Specifically, it is necessary to store, and to update, the stress, accumulated plastic strain, constitutive parameters, and internal state variables.
MSE 612 (Fall 2012) 60
Fig. 2.9: The flow chart for the finite element implementation. The dashed box specifies the procedure for the ABAQUS UMAT subroutine.
Integration Scheme and Consistent Material Tangent
As discussed in Chapter 1, the constitutive law is specified in a set of partial differential equations. With a given time step t and strain increment ε , the procedure to obtain the stress increment is called integration scheme, since we need to integrate the constitutive differential equations.
N-R converge?
Given σ, ε, εp, q at time tn
Choose Δt, so that the increments of boundary conditions are known
Given nodal displacement increments a
ku
N-R converge?
Yes
Yes
No
Specify Δtnew by Abaqus automatic incrementation
Update σ, ε, εp, q at time tn+1=tn+Δt
No
Specify Δtnew in UMAT
Given σ, ε, εp, q at time tn, and Δt, Δε
(1) update stress (2) compute Jacobian
ABAQUS UMAT
compute K, R, F, and solve for 1
a
ku
integration algorithm
MSE 612 (Fall 2012) 61
Problem formulation
Given:
Stress nσ , strain nε , plastic strain pnε , constitutive and state variables nq at time nt
The total strain increment ε and time increment t
Compute:
1nσ , 1nε , 1pnε , 1nq at time 1n nt t t
The tangent modulus epijkl ij klC
Explicit integration scheme
The most straightforward integration scheme is the explicit one.
Disadvantages:
It be conditionally stable (i.e., the stability depends on the time/strain increments).
Advantages
Although such a scheme is not desirable for rate-independent case, it is usually a good starting point (and indeed commonly used) for rate-dependent materials. The reason is that no yield surface is explicitly incorporated in the rate-dependent behavior, so that uncertainties involving overshooting and drifting are not as significant and problematic as for the rate-dependent solids.
The material tangent can be easily calculated from the explicit scheme.
Material tangent, Jacobian, tangent modulus, etc.
The naming convention here can be very confusing. In Chapter 1, we have defined the tangent modulus by
ijepijkl
kl
C
,
and here we define the material tangent by
ijepijkl
kl
C
. (2.43)
They are the same since ij t and ij t are known conditions. To this end, we do not need to
differentiate various names such as material tangent, material stiffness, Jacobian, and tangent modulus.
MSE 612 (Fall 2012) 62
What we need to be careful is the so-called consistent material tangent (or called algorithm stiffness). From the integration scheme, we will get
,...ij ij kl .
If we directly take the derivatives based on this equation, the resulting material tangent is called consistent material tangent.
The biggest advantage of using consistent material tangent is the considerable enhancement of numerical stability and accuracy. However, the computational cost and implementation difficulty increase. It should also be noted that if a wrong ep
ijklC is implemented in
UMAT, the convergence rate will slow down, but the results (if obtained) are unaffected.
MSE 612 (Fall 2012) 63
2.6 Stress Update Algorithm Finite element implementation methods differ in terms of the integration scheme. The more advanced book by Simo and Hughes, Computational Inelasticity, has a complete compilation of commonly seen plasticity models. For more advanced plasticity theories, such as crystal plasticity, amorphous alloys, etc., we need to develop our own integration scheme.
2.6.1 Explicit Integration Scheme
Given , ,pn n nε ε q at nt , and the strain increments t ε ε ,
Compute 1 1 1, ,pn n n ε ε q
The constitutive law is specified as follows:
: p σ C ε ε ,
, 0f σ q ,
,p ε r σ q ,
q h ,
Kuhn-Tucker condition,
so that
: :
: :
f
f f
σ
q σ
C ε
h C r
,
:epσ ε ,
: :
: :ep f
f f
σ
q σ
r
h r
. (2.44)
The simplest method for the stress update is the forward (explicit) Euler integration scheme, given by
1n n ε ε ε ,
1p pn n n n ε ε r ,
1n n n n q q h ,
1 :epn n σ σ ε . (2.45)
This is very simple to implement, but it is conditionally stable. We will discuss (1) why we need the stress update algorithm in FEA, and (2) how to derive stable and efficient methods to do so.
MSE 612 (Fall 2012) 64
Example of uniaxial loading with isotropic hardening
Let’s use an example to illustrate why the algorithm in Eq. (2.45) is typically not desirable. We consider a uniaxial tension with linear isotropic hardening with
100E , 0 1Y , 2h .
Results with two different strain steps are shown below, giving that very fine step is needed for convergence.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-1.5
-1
-0.5
0
0.5
1
1.5
=10-4
=10-3
Fig. 2.10: Example results of an explicit integration scheme, showing dependence on step size.
2.6.2 Explicit (one-step) Return Mapping Algorithm (rate-independent solids)
(i) We define an elastic trial stress,
1 1:trial pn n n σ ε ε . (2.46)
(ii) If the yield condition is not satisfied,
1 , 0trialn nf σ q , (2.47)
then the trial stress is the new stress, and the Jacobian is the elastic stiffness matrix.
(iii) If the yield condition is satisfied, we need to return to the yield surface at *nσ . The
increment of effective plastic strain, , can be determined using the consistency condition:
* *1 1 1 1, , : : 0n n n n n n n n
n n
f ff f
σ q σ q σ σ q q
σ q, (2.48)
MSE 612 (Fall 2012) 65
where 1 1 1:trial p pn n n n σ σ ε ε , and the flow rule and hardening law give
1 ,p pn n n n ε ε r σ q , 1 ,n n n n q q h σ q . (2.49)
(iv) update stress
Fig. 2.11: Geometric interpretation of explicit one-step algorithm
2.6.3 Implicit (N-R) Return Mapping Algorithm (rate-dependent solids)
The constitutive law is given by
3
2ijp
ij effe
S
, flow rule, (2.50)
0
m
eeff
Y
, creeping law, (2.51)
1
0
0
1
n
effY Y
, hardening law, (2.52)
where 0 , 0Y , 0 , m and n are material constants. Since the plastic strain rate is proportional to
the deviatoric stress, we get p pij ije , where ije is the deviatoric (shear) components of ij .
The following UMAT procedure is essentially based on the method specified in ABAQUS theory manual. You can also refer to Prof. A.F. Bower (Brown University)’s textbook. Most constitutive laws can be implemented in similar way. In the following, the subscript n and
1n denote quantities evaluated at nt and 1nt , respectively.
Step (i): Elastic trial solution
Since hydrostatic stress does not cause plastic deformation, we need to solve for hydrostatic stress 1 1σn np tr and deviatoric stresses 1 1 3n np σ I separately, where
nσ , 0n nf σ q 1 1, 0n nf σ q
1trialnσ
*nσ
1nσ
yield surface
MSE 612 (Fall 2012) 66
1 1 2
εn n
Ep p tr
, (2.53)
1 1 1S S e S e ee p
n n n
E E
, (2.54)
and 3e ε ε Itr is the deviatoric strain increment.
Define the elastic trial stress and its Mises component by
*1 1n n
E
S S e , * * *, 1 1 1
3:
2e n n n S S , (2.55)
which can also be thought of as an elastic predictor for the deviatoric stress.
Step (ii): Integration
To calculate the plastic strain increment, we need to integrate the expression for plastic strain rate with respect to time over the interval t . Since the material is rate-dependent, the yield surface is not included. Let us use an implicit (i.e., backward Euler) method to integrate Eqs. (2.50), (2.51), and (2.52), giving
1
1
3
2p n
effen
S
e , (2.56)
, 10
, 1
m
e neff
Y n
t
, (2.57)
1
,0, 1
0
1
n
eff n effY n Y
. (2.58)
This is an implicit scheme, because the strain rate is computed based on values of stress and state variables at the end of the time interval. The implicit scheme can be shown to be unconditionally stable (you can take large time steps without encountering numerical instabilities) and also leads to symmetric material tangents, as we will see shortly.
Substitute Eq. (2.55) and Eq. (2.56) into Eq. (2.54), we get
* * 11 1 1
, 1
3
1 2 1p n
n n n effe n
E E
SS S e S , (2.59)
which shows that 1nS and *1nS are proportional to each other. Therefore, we assume
*1 1S Sn n , where the numerical factor can be determined as follows. Re-arranging Eq.
(2.59) gives
*11 1
, 1
3
2 1n
n eff ne n
E
SS S ,
MSE 612 (Fall 2012) 67
and contracting both sides of this equation gives
1
, 1
31
2 1 ee n
E
.
This representation is not desirable, since it involves unknown , 1e n .
Alternatively, we use *1 1S Sn n and Eq. (2.59) to see that
** * 1
1 1 *1
3
2 1n
n n effen
E
SS S (2.60)
Contracting both sides of this equation with *1Sn shows that
*
, 1
31
2 1 effe n
E
. (2.61)
Consequently, the elasticity constitutive law and the flow rule give
*
1 1 *, 1
31
2 1 1n n eff ne n
E E
S S S e , (2.62)
* *, 1 , 1 , 1*
, 1
31
2 1e n e n eff e ne n
E
. (2.63)
So far, we have not utilized the creeping law and the hardening behavior. Combining Eqs. (2.57) and (2.58) give
1
, 1 , 11
0 , 1 ,0
0
1
m
eff e n e nn
Y n eff n effY
t
. (2.64)
Combining Eqs. (2.63) and (2.64) lead to
1 1*, 1 ,
0 *, 1 0 0
31 1 0
2 1
n m
e n eff n eff effeff
Y e n
E
t
(2.65)
This equation can be solved for eff using Newton-Raphson iteration.
Step (iii): Update stress and other constitutive parameters
With the solution of eff from Eq. (2.65), the deviatoric stresses 1nS can be updated by
Eq. (2.62). We also need to update Y , eff , etc. in the program.
Step (iv): Material tangent
MSE 612 (Fall 2012) 68
To compute the material tangent ij kl , we fist note that
1
3ij ij
ijkl kl kl
d dS dp
d d d
(2.66)
where
1 2 kl
kl
dp E
d
(2.67)
The deviatoric term follows by differentiating the expression in Eq. (2.62), given by
** *, 11 1
1 * * *, 1 , 1 , 1
33
1 2 1 2 1e neffn n
n effe n e n e n
dEE Ed d d
S S
S e (2.68)
where *, 1e nd can be obtained by differentiating Eq. (2.55), i.e.,
** 1 1, 1 *
, 1 , 1
: :3 3
2 1 2 1n n
e ne n e n
d dE Ed
S ε S ε
, (2.69)
and effd can be computed by differentiating Eq. (2.65):
*, 1
0 0
1 1
0 0 0
3
2 1
1 11 0
e neff
Y Y
n m
eff eff effeff
effeff eff
d Ed
dt mn
. (2.70)
Finally noting that
1
3ij
ik jl ij klkl
d e
d
(2.71)
we can collect together all the relevant terms to show that
1 1
, 1 , 1 , 1
11 3 3
1 3 2 2 1 3 1 2
n neff ijep kl
ijkl ik jl ij kl ij kle n e n e n
S SE E EC
, (2.72)
where is given in Eq. (2.61) and
, 1 0
3 1 1
2 1 e n effeff eff
E
mn
. (2.73)
MSE 612 (Fall 2012) 69
2.7 Practicing ABAQUS UMAT Subroutine
The example UMAT code (taken from ABAQUS) is for 2J -type, isotropic hardening
plastic law. We test it by using a single element stretched in 3x direction, and the resulting
33 ~ 33 relationship is given in Fig. 2.12. The FORTRAN code and the input file can be
downloaded from:
http://web.utk.edu/~ygao7/teaching/umatmst3.for
http://web.utk.edu/~ygao7/teaching/uniaxial_tension.inp
To run the program, type:
abaqus job=uniaxial_tension user=umatmst3
You must have both ABAQUS and a FORTRAN complier installed on your computer.
In UMAT (Fig. 2.13), STRESS gives nσ , and we need to update STRESS to 1nσ by the
end of the subroutine. The strain increments ε are DSTRAN, as input. We need to specify the material Jacobian DDSDDE at the end of the subroutine. STATEV are used for storage and update of state variables. You can read the ABAQUS help manual for other variables. We also need to be careful about the Voigt contraction, e.g.,
11 11 12 13 14 15 15 11
22 22 23 24 25 26 22
33 33 34 35 36 33
12 44 45 46 12
13 55 56 13
23 66 23
. 2
2
2
c c c c c c
c c c c c
c c c c
symm c c c
c c
c
,
where UMAT uses STRESS(1:NDI) to denote 11 , 22 , and/or 33 , and STRESS
(NDI+1,NDI+NSHR) for the shear components.
The integration scheme is essentially the same as that in Section 2.6.3, except that the material behavior here is rate-independent. Therefore we need to test the yield condition. We are not going to talk about more details about this code. A concise algorithm is given below:
(i) formulating elastic trial stress, and elastic stiffness matrix
(ii) test the yield function, if elastic, jump to step (v)
(iii)Newton-Raphson method to determine the increment of effective plastic strain
(iv) update stress and plastic strain tensors
(v) calculate Jacobian
(vi) update STATEV
MSE 612 (Fall 2012) 70
0 0.5 1 1.5 2 2.5 3 3.5 4
x 10-3
0
50
100
150
200
250
(M
Pa)
Fig. 2.12: The uniaxial tension test and the resulting stress-strain behavior ( 33 and 33 ).
Fig. 2.13: A snapshot of the UMAT code.
MSE 612 (Fall 2012) 71
--------------------------------------------------------------------------------------------------------------------- SUBROUTINE UMAT(STRESS,STATEV,DDSDDE,SSE,SPD,SCD, 1 RPL,DDSDDT,DRPLDE,DRPLDT,STRAN,DSTRAN, 2 TIME,DTIME,TEMP,DTEMP,PREDEF,DPRED,MATERL,NDI,NSHR,NTENS, 3 NSTATV,PROPS,NPROPS,COORDS,DROT,PNEWDT,CELENT, 4 DFGRD0,DFGRD1,NOEL,NPT,KSLAY,KSPT,KSTEP,KINC) C INCLUDE 'ABA_PARAM.INC' C CHARACTER*80 MATERL DIMENSION STRESS(NTENS),STATEV(NSTATV), 1 DDSDDE(NTENS,NTENS),DDSDDT(NTENS),DRPLDE(NTENS), 2 STRAN(NTENS),DSTRAN(NTENS),TIME(2),PREDEF(1),DPRED(1), 3 PROPS(NPROPS),COORDS(3),DROT(3,3), 4 DFGRD0(3,3),DFGRD1(3,3) C DIMENSION EELAS(6),EPLAS(6),FLOW(6) PARAMETER (ONE=1.0D0,TWO=2.0D0,THREE=3.0D0,SIX=6.0D0) DATA NEWTON,TOLER/10,1.D-6/ C C ----------------------------------------------------------- C UMAT FOR ISOTROPIC ELASTICITY AND ISOTROPIC PLASTICITY C J2 FLOW THEORY C CAN NOT BE USED FOR PLANE STRESS C ----------------------------------------------------------- C PROPS(1) - E C PROPS(2) - NU C PROPS(3) - SYIELD C CALLS AHARD FOR CURVE OF SYIELD VS. PEEQ C ----------------------------------------------------------- C IF (NDI.NE.3) THEN WRITE(6,1) 1 FORMAT(//,30X,'***ERROR - THIS UMAT MAY ONLY BE USED FOR ', 1 'ELEMENTS WITH THREE DIRECT STRESS COMPONENTS') ENDIF C C ELASTIC PROPERTIES C EMOD=PROPS(1) ENU=PROPS(2) IF(ENU.GT.0.4999.AND.ENU.LT.0.5001) ENU=0.499 EBULK3=EMOD/(ONE-TWO*ENU) EG2=EMOD/(ONE+ENU) EG=EG2/TWO EG3=THREE*EG ELAM=(EBULK3-EG2)/THREE C C ELASTIC STIFFNESS C DO 20 K1=1,NTENS DO 10 K2=1,NTENS DDSDDE(K2,K1)=0.0 10 CONTINUE 20 CONTINUE C DO 40 K1=1,NDI DO 30 K2=1,NDI DDSDDE(K2,K1)=ELAM 30 CONTINUE DDSDDE(K1,K1)=EG2+ELAM 40 CONTINUE DO 50 K1=NDI+1,NTENS DDSDDE(K1,K1)=EG 50 CONTINUE C C CALCULATE STRESS FROM ELASTIC STRAINS C DO 70 K1=1,NTENS DO 60 K2=1,NTENS STRESS(K2)=STRESS(K2)+DDSDDE(K2,K1)*DSTRAN(K1) 60 CONTINUE 70 CONTINUE C C RECOVER ELASTIC AND PLASTIC STRAINS C DO 80 K1=1,NTENS EELAS(K1)=STATEV(K1)+DSTRAN(K1) EPLAS(K1)=STATEV(K1+NTENS) 80 CONTINUE EQPLAS=STATEV(1+2*NTENS) C C IF NO YIELD STRESS IS GIVEN, MATERIAL IS TAKEN TO BE ELASTIC C IF(NPROPS.GT.2.AND.PROPS(3).GT.0.0) THEN C C MISES STRESS C SMISES=(STRESS(1)-STRESS(2))*(STRESS(1)-STRESS(2)) + 1 (STRESS(2)-STRESS(3))*(STRESS(2)-STRESS(3)) + 1 (STRESS(3)-STRESS(1))*(STRESS(3)-STRESS(1))
MSE 612 (Fall 2012) 72
DO 90 K1=NDI+1,NTENS SMISES=SMISES+SIX*STRESS(K1)*STRESS(K1) 90 CONTINUE SMISES=SQRT(SMISES/TWO) C C HARDENING CURVE, GET YIELD STRESS C NVALUE=NPROPS/2-1 CALL AHARD(SYIEL0,HARD,EQPLAS,PROPS(3),NVALUE) C C DETERMINE IF ACTIVELY YIELDING C IF (SMISES.GT.(1.0+TOLER)*SYIEL0) THEN C C FLOW DIRECTION C SHYDRO=(STRESS(1)+STRESS(2)+STRESS(3))/THREE ONESY=ONE/SMISES DO 110 K1=1,NDI FLOW(K1)=ONESY*(STRESS(K1)-SHYDRO) 110 CONTINUE DO 120 K1=NDI+1,NTENS FLOW(K1)=STRESS(K1)*ONESY 120 CONTINUE C C SOLVE FOR EQUIV STRESS, NEWTON ITERATION C SYIELD=SYIEL0 DEQPL=0.0 DO 130 KEWTON=1,NEWTON RHS=SMISES-EG3*DEQPL-SYIELD DEQPL=DEQPL+RHS/(EG3+HARD) CALL AHARD(SYIELD,HARD,EQPLAS+DEQPL,PROPS(3),NVALUE) IF(ABS(RHS).LT.TOLER*SYIEL0) GOTO 140 130 CONTINUE WRITE(6,2) NEWTON 2 FORMAT(//,30X,'***WARNING - PLASTICITY ALGORITHM DID NOT ', 1 'CONVERGE AFTER ',I3,' ITERATIONS') 140 CONTINUE EFFHRD=EG3*HARD/(EG3+HARD) C C CALC STRESS AND UPDATE STRAINS C DO 150 K1=1,NDI STRESS(K1)=FLOW(K1)*SYIELD+SHYDRO EPLAS(K1)=EPLAS(K1)+THREE*FLOW(K1)*DEQPL/TWO EELAS(K1)=EELAS(K1)-THREE*FLOW(K1)*DEQPL/TWO 150 CONTINUE DO 160 K1=NDI+1,NTENS STRESS(K1)=FLOW(K1)*SYIELD EPLAS(K1)=EPLAS(K1)+THREE*FLOW(K1)*DEQPL EELAS(K1)=EELAS(K1)-THREE*FLOW(K1)*DEQPL 160 CONTINUE EQPLAS=EQPLAS+DEQPL SPD=DEQPL*(SYIEL0+SYIELD)/TWO C C JACOBIAN C EFFG=EG*SYIELD/SMISES EFFG2=TWO*EFFG EFFG3=THREE*EFFG2/TWO EFFLAM=(EBULK3-EFFG2)/THREE DO 220 K1=1,NDI DO 210 K2=1,NDI DDSDDE(K2,K1)=EFFLAM 210 CONTINUE DDSDDE(K1,K1)=EFFG2+EFFLAM 220 CONTINUE DO 230 K1=NDI+1,NTENS DDSDDE(K1,K1)=EFFG 230 CONTINUE DO 250 K1=1,NTENS DO 240 K2=1,NTENS DDSDDE(K2,K1)=DDSDDE(K2,K1)+FLOW(K2)*FLOW(K1) 1 *(EFFHRD-EFFG3) 240 CONTINUE 250 CONTINUE ENDIF ENDIF C C STORE STRAINS IN STATE VARIABLE ARRAY C DO 310 K1=1,NTENS STATEV(K1)=EELAS(K1) STATEV(K1+NTENS)=EPLAS(K1) 310 CONTINUE STATEV(1+2*NTENS)=EQPLAS C RETURN END C C
MSE 612 (Fall 2012) 73
SUBROUTINE AHARD(SYIELD,HARD,EQPLAS,TABLE,NVALUE) C INCLUDE 'ABA_PARAM.INC' DIMENSION TABLE(2,NVALUE) C C SET YIELD STRESS TO LAST VALUE OF TABLE, HARDENING TO ZERO SYIELD=TABLE(1,NVALUE) HARD=0.0 C C IF MORE THAN ONE ENTRY, SEARCH TABLE C IF(NVALUE.GT.1) THEN DO 10 K1=1,NVALUE-1 EQPL1=TABLE(2,K1+1) IF(EQPLAS.LT.EQPL1) THEN EQPL0=TABLE(2,K1) IF(EQPL1.LE.EQPL0) THEN WRITE(6,1) 1 FORMAT(//,30X,'***ERROR - PLASTIC STRAIN MUST BE ', 1 'ENTERED IN ASCENDING ORDER') CALL XIT ENDIF C C CURRENT YIELD STRESS AND HARDENING C DEQPL=EQPL1-EQPL0 SYIEL0=TABLE(1,K1) SYIEL1=TABLE(1,K1+1) DSYIEL=SYIEL1-SYIEL0 HARD=DSYIEL/DEQPL SYIELD=SYIEL0+(EQPLAS-EQPL0)*HARD GOTO 20 ENDIF 10 CONTINUE 20 CONTINUE ENDIF RETURN END
---------------------------------------------------------------------------------------------------------------------
MSE 612 (Fall 2012) 74
Homework: A practice of ABAQUS and UMAT (30%) Given:
a boundary value problem (uniaxial tension, if you like) User defined MATerial (UMAT) Subroutine on Mises plasticity (the one I gave you here)
Perform: understand the format and usage of UMAT compare results using ABAQUS alone and using ABAQUS UMAT
Hand in 1-page writing: briefly explaining Mises plasticity and UMAT showing one plot of comparison
**----------------------------------------- *USER MATERIAL,CONSTANTS=8 200.E3,.3,200.,0.,220.,.0009,220.,.0029 *DEPVAR 13, (see http://web.utk.edu/~ygao7/teaching/uniaxial_tension.inp) ** alternatively replace the above lines by the following *Elastic 200.E3, 0.3 *Plastic 200., 0. 220., 0.0009, 220., 0.0029 (see http://web.utk.edu/~ygao7/teaching/uniaxial_tension_aba.inp)
MSE 612 (Fall 2012) 75
3. MATERIAL FAILURE The most important application of mechanics of materials is to understand material failure and to design materials and structures that are capable of withstanding failures. Most engineering design rules rely on semi-empirical fracture or fatigue criteria, which have some connection to the failure mechanisms and have also been calibrated by means of standard tests. The scope here is limited to fracture and fatigue of metallic and ceramic materials.
Two excellent books are for your reference: B. Lawn, Fracture of Brittle Solids, 2nd edition, Cambridge University Press, 1993. S. Suresh, Fatigue of Materials, 2nd edition, Cambridge University Press, 1998.
3.1 Phenomenology of Fracture and Fatigue Fracture
Remember that the first thing your learn in mechanical metallurgy is that the theoretical strength of a crystalline material is about 10 , where is the shear modulus. This is almost impossible to achieve in typical materials, except for examples such as metallic whiskers, silica optical fibers, among some others. The stress-strain curves for brittle materials exhibit a tensile strength of about 100 or lower. For ductile materials, the maximum strength is even lower, and the strain at failure is quite high. Figure 3.1 show the different fracture behavior of a ductile and brittle alloy.
Fig. 3.1: Ductile versus brittle fracture of metallic materials (W.D. Callister, Materials Science and Engineering: An Introduction, 7th edition, John Wiley & Sons Inc., 2007).
The low strength of engineering materials can be interpreted by the existence of defects. Small voids, cracks, or defects inside the solid are locations with a large stress concentration, which facilitates material failure. This was observed by da Vinci (1500s) from his tensile tests on
Necking Void nucleation Void coalescence
Crack propagation Shear fracture
Cup-and-cone fracture in Al
Brittle fracture in a mild steel
Intergranular fracture
Transgranular fracture
MSE 612 (Fall 2012) 76
long and short metal wires. The longer the wire, the weaker it is, because it is more probable to find out a defect.
A famous Chinese philosopher, Han Fei Zi, wrote in his book (circa 230BC):
A dam of thousand miles long may collapse because of one ant hole,
which is a nice description of damage mechanics.
Static Fatigue
Some materials, especially brittle materials such as glasses, and oxide based ceramics, suffer from a form of time-delayed failure under steady loading, known as static fatigue, delayed fracture, or stress corrosion. It can withstand a static load for a long time, and then, without warning, breaks suddenly. Static fatigue in brittle materials is a consequence of corrosion crack growth. The highly stressed material near a crack tip is particularly susceptible to chemical attack (the stress increases the rate of chemical reaction). Material near the crack tip may be dissolved altogether, or it may form a reaction product with very low strength. In either event, the crack slowly propagates through the solid, until it becomes long enough to trigger brittle fracture. Glasses and oxide based ceramics are particularly susceptible to attack by water-vapor.
Fig. 3.2: Static fatigue crack observed in the interface of organosilicate glass and SiO2 (Tsui and Vlassak, J. Mech. Phys. Solids 2006).
Usually, the relationship between da dt versus K or G exhibits:
a threshold load, crack growth regime (sensitive to temperature and chemical concentration), rapid crack growth.
Fatigue
Mechanical engineers generally have to design components to withstand cyclic as opposed to static loading. Fatigue failure is a familiar phenomenon, but a detailed understanding of the mechanisms involved and the ability to model them quantitatively have only emerged in
MSE 612 (Fall 2012) 77
the past 50 years, driven largely by the demands of the aerospace industry. There are some forms of fatigue failure (contact fatigue is an example) where the mechanisms involved are still a mystery.
The resistance of a material to cyclic loading is characterized by plotting an “S-N” curve showing the number of cycles to failure as a function of stress. The plot normally shows different regimes of behavior, depending on stress amplitude. At high stress levels, the material deforms plastically and fails rapidly. In this regime the life of the specimen depends primarily on the plastic strain amplitude, rather than the stress amplitude. This is referred to as “low cycle fatigue” behavior. At lower stress levels, it is referred to as “high cycle” fatigue behavior. In some materials, there is a clear fatigue limit – if the stress amplitude lies below a certain limit, the specimen remains intact forever. In other materials there is no clear fatigue threshold. In this case, the stress amplitude at which the material survives a given number of cycles is taken as the endurance limit of the material.
Fig. 3.3: A typical plot of crack growth rate versus the range of applied stress intensity factor (courtesy of Dr. Gongyao Wang at UT).
Besides the S-N curve, the fatigue behavior is often described by the Paris plot, i.e., da dN versus K . We should discriminate this from the static fatigue problem.
0.5 1 10 201E-11
1E-10
1E-9
1E-8
1E-7
1E-6
Zr41.2
Ti13.8
Cu12.5
Ni10
Be22.5
(CT) Zr41.2Ti13.8Cu12.5Ni10Be22.5 (CT) Zr
41.2Ti
13.8Cu
12.5Ni
10Be
22.5 (CT)
Zr41.2
Ti13.8
Cu12.5
Ni10
Be22.5
(SE) Zr
55Cu
30Ni
5Al
10 (CT)
Zr56.2
Cu6.9
Ni5.6
Ti13.8
Nb5.0
Be12.5
(CT, C) Zr55Al10Cu30Ni5 (CCT, NC) Zr
44Ti
11Ni
10Cu
10Be
25 (CT)
2090-T81 Aluminum 300-M Steel
da/
dN
(m
/cyc
le)
K (MPam1/2)
(Gilbert et. al.)
(Schroeder et. al.)
(Flores et. al.) (Zhang et. al.)
(Nakai et. al.) (Flores et. al.)
(Fujita et. al.)
(Gilbert et. al.)
(Gilbert et. al.)
(Launey et. al.)
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3.2 A Viewpoint at the Crack Tip Fracture and fatigue involve processes at multiple time and length scales, as shown in Fig. 3.4. If we are concerned with atomic crack tip process, we need to consider defect nucleation, interatomic interactions, and discrete dislocations at length scales of a few nanometers. At microstructural length scales (grain size scale, for instance), failure initiation and growth involve complex interplay between defect evolution and material microstructure. If we are concerned with the structural failure at macroscopic scales, we need to use the fracture mechanics principle in which only the stress intensity factor is sufficient.
Fig.3.4: A multiscale point of view at the crack tip (van der Giessen and Needleman, Interface Science, 2000).
Crack Tip Elasticity
You may have learned that the crack tip elastic field can be described by the stress intensity factor. The following gives a mechanistic way of deriving this.
We assume: (1) plane strain condition, (2) a slit-like crack, (3) the crack tip is a point, and (4) the linear elasticity works all the way to the crack tip. From the strain-displacement relationship, we obtain
2 22
2 22yy xyxx
y x x y
, (3.1)
which is known as the compatibility equation. From the stress balance equation, we have
0yxxx
x y
1
xx y
, 1
yx x
,
MSE 612 (Fall 2012) 79
0yx yy
x y
2
yx y
, 2
yy x
,
and thus
1 2 0x y
1 y
, 2 x
.
Therefore, we can define the Airy’s stress function,
2
2xx y
, 2
2yy x
, 2
xy x y
. (3.2)
Using Hooke’s law, we express the compatibility equation in terms of Airy’s stress function:
4 4 4
2 24 2 2 4
2 0x x y y
, (3.3)
which is a bi-harmonic equation.
The above equations in the polar coordinates become
2
2 2
1 1rr r r r
, 2
2r
, 1
r r r
, (3.4)
and the bi-harmonic equation is
2 2 2 2
2 2 2 2 2 2
1 1 1 10
r r r r r r r r
. (3.5)
One form of Airy’s stress function that satisfies Eq. (3.5) is given by
1, cos 1 cos 1r r A B , (3.6)
where , A and B are constants to be determined by using traction boundary conditions. The stress components are
2
12
1 cos 1 cos 1r A Br
,
111 sin 1 1 sin 1r r A B
r r
.
Using the traction boundary conditions, i.e., 0r at , we get
1 cos 1 cos 1 0
1 sin 1 1 sin 1 0
A B
A B
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To obtain nontrivial solutions of A and B, the determinant must be zero, leading to the following characteristic equation:
2 1 sin 2 0 , (3.7)
and the solutions are 2
k , k=0,1,2… Taking the lowest order (i.e., 1 2 ) gives
1 3 2~ r r , 1 1 2~ r r , 1 2~u r r . (3.8)
The above analysis shows that the elastic stress fields near a crack tip must have the following asymptotic structure:
,2
Iij ij
Kr f
r
, (3.9)
where IK is a scaling parameter, called stress intensity factor (SIF). We can also derive similar
representations for Mode II and Mode III. Also note that a constant transverse stress (called T-stress) can also appear in Eq. (3.9). The SIF depends on the applied load, contact, and geometric shape. The crack tip process zone depends on K and material properties, but not on far fields.
Fig. 3.5: Schematic illustration of the K-annulus.
K-Annulus and Griffith-Irwin Fracture Mechanics
The regime around the crack tip, where the asymptotic stress fields (3.9) hold, is annular like, as shown in Fig. 3.5, thus called the K-annulus. Apparently, the upper bound of this K-field is determined by the distance from the crack tip to another geometric heterogeneity (e.g., the boundary of the solid, another crack, other defects, etc.). The lower bound of the K-field is determined by the crack tip process zone, since the elastic singular stress field cannot be
geometric boundary K-annulus
crack tip process zone
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sustained at the crack tip. The mechanism that is used to regularize the elastic singularity will give a length scale. For example, if the Mises plasticity is used, the plastic zone size is given by
2
1
2appl
pY
Kr
, (3.10)
where Y is the material yield stress. For another example, during brittle fracture (cleavage, or
negligible plastic deformation), the crack process zone is given by
2
0
1
2appl
cz
Kr
, (3.11)
where 0 is the cohesion strength of the solid.
The observation of the K-annulus allows us to pose a fracture criterion: if the applied stress intensity factor is less than a critical value, the crack will not propagate, i.e.,
appl CK K , (3.12)
where this critical value CK is called toughness. Equivalently, we can state that if the energy
release rate G is less than the fracture energy , the crack will not propagate, i.e.,
G , (3.13)
where the Irwin’s relationship gives
22
2 21
2III
I II
KG K K
E
. (3.14)
The validity of the above Griffith-Irwin fracture criterion depends on the validity of the K-annulus. With a valid K-annulus, the crack-tip process zone is solely governed by applK , and is
independent of geometric boundary conditions at far away.
For brittle solids such as glasses and ceramics, taking cohesion strength 0 10GPa and
1CK MPa m , Eq. (3.11) gives the size of the crack tip process zone:
2
9
0
1~ 10
2C
cz
Kr m
.
which is in nanometer regime. Consequently, the fraction-mechanics-based failure analysis can be applied to brittle solids. This is the Griffith fracture mechanics when ,czr a d . Thus the
design rule should focus on determining K (or G ). This is rather a routine chore. From
dimensional analysis, it must be K a , where is a geometric parameter. You can either refer to Tada’s handbook of stress analysis or use finite element to determine this. However, when sizes of materials flaws are sub-micrometer, the analysis should rely on
MSE 612 (Fall 2012) 82
statistical fracture mechanics. The material behavior under tension and compression will be radically different, as will be discussed in Section 3.4.
For high strength alloys, using 1.4Y GPa and 65CK MPa m gives
2
1~ 0.3
2C
pY
Kr mm
.
If the plastic zone is smaller than the crack size (which usually holds), we can still use the Griffith-Irwin criterion. This condition is called small scale yielding. From the crack initiation to a steady-state crack growth, the plastic zone will evolve and lead to a resistance curve, i.e., CK a . We will examine this R-curve in details in Section 3.3.3. The use of
Irwin fracture mechanics and R-curve holds under the condition of pd r a .
If pr d a , the fracture toughness scatters dramatically. This is rather due to the statistics
of plastic flow, which is different from the Weibull-type statistics in brittle solids. If we have flaws of μm~mm size in the above material, we can’t use the Griffith-Irwin
criterion. We need to explicitly model the crack process zone, which involves plastic deformation or void nucleation/growth on microstructural length scales.
For tough alloys, using 0.35Y GPa and 180CK MPa m gives
1
~ 42
Cp
Y
Kr cm
.
For typical crack sizes, the Griffith-Irwin criterion is invalid because these materials have very large plastic zone size, i.e., large scale yielding. To test the toughness, the standards of the American Society for Testing and Materials (ASTM) requires a crack length larger than 25 pr , which can lead to large specimens.
To this end, nonlinear fracture mechanics is developed. The crack-tip plastic zone again has certain asymptotic properties with a given constitutive law. For instance, with a power-law hardening solid, the plastic field near the crack tip is called Hutchinson-Rice-Rosengren (HRR) field. Such analyses avoid the use of large specimens in experiments.
For short cracks (>mm), the crack process zone usually does not involve microstructures.
Summary of Failure Analysis Methods
Depending on (1) the validity of the K-annulus and (2) the microstructural length scales involved in the problem, the failure analysis can be categorized as follows:
a valid K-annulus (e.g., crack-tip process zone size is very small)
o linear elastic fracture mechanics (i.e., Griffith-Irwin criterion, where the stress intensity factor is calculated from the known geometry of the crack and applied load)
o toughness is measured experimentally (ASTM standards)
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Under LSY, the geometric size has to be very large to ensure a valid K-annulus. Therefore, the nonlinear fracture mechanics develops a J-annulus where the asymptotic fields are based on HRR fields.
crack-tip process zone size is very large
o stress based failure criterion (e.g., Gurson-Tvergaard model for void nucleation, growth, and coalescence)
o explicit model of the crack-tip process zone (the most difficult problem, especially since microstructure-level deformation mechanisms are involved)
statistical fracture mechanics (no macroscopic cracks, but many microcracks; analysis involves Weibull statistics)
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3.3 Modeling the Crack Tip Process Zone Understanding the dissipative processes near the crack tip can lead to a quantitative description of the toughness, the development of material failure models, and design rules for desirable material properties. If we can manipulate the crack tip process zone, we can possibly tune the material toughness. For example, a fiber-reinforced composite can have a toughness that is higher than that of each constituent phase – a weak phase toughens a weak phase. Such a mechanism will be illustrated in this section.
A Simple Crack-Tip-Process Zone Model: Dugdale-Barenblatt Model
Considering the observation that plastic deformation is usually confined in a strip ahead of the crack tip, we can use the cohesive interface model to represent this nonlinear material behavior. The cohesive law is used to model the behavior of a weak interface in the solid, which separates when subjected to a sufficiently large stress. This can also be used to describe other crack tip processes. For instance, the traction-separation relationship can be used to model the crack bridging due to reinforced fibers in a composite, the atomistic interatomic interaction, the van der Waals interaction between two surfaces, etc. We can even take a phenomenological approach, that is, a cohesive zone model is introduced wherever there is a weak surface inside the solid.
Fig. 3.6: (a) Barrenblat-Dugdale model of cohesive interface. (b) A bridged central crack subjected to remote traction. (c) The strength 0max as a function of 0
*0 Ea shows two
types of bridging features: small scale bridging (SSB) and large scale bridging (LSB).
The exact shape of the cohesive interface model is usually of secondary importance, and therefore the cohesive law can be represented by the Barrenblat-Dugdale model, as shown Fig.
0
0
a2
c2
10-1
100
101
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
a0/E*
0
max
/0
LSB
SSB
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3.6(a). Along the cohesive interface, the relation between the traction and the separation is specified as
0
00
0
0
0
(3.15)
where 0 and 0 are constitutive parameters, denoted as interface strength and characteristic
length (or characteristic bridging-length). The work of adhesion is therefore 0 0 .
Since the cohesive interface model defines the relation between stress and separation, a length scale arises, namely, 00 E , (or 00
* E with 2* 1 EE ). The consequence of
this length scale is illustrated by the boundary value problem shown in Figure 3.6(b). Consider a central crack with total length c2 and un-bridged length a2 . The cohesive zone length ac 2 is determined by solving the elasticity boundary value problem (for instance, see Tada et al. 2000, §30.7). When the cohesive zone length is small ( ac ), the maximum applied stress, max
(i.e., strength), is determined by the Griffith’s criterion, i.e.,
2
2max
* *
aKG
E E
,
aE 00*
0max . (3.16)
When the cohesive zone length is large ( ac ), Griffith fracture mechanics is not applicable, and we need to match the crack opening displacement, giving
12
secln8
ln8
0
max*
0
0*
0
0
E
a
a
c
E
a. (3.17)
In conclusion, when *00 Ea <<1, the crack is of large-scale bridging (LSB) feature
(hereafter referred to as Dugdale crack). The crack interface is therefore subjected to the cohesive traction everywhere, and the theoretical strength of the interface, 0 , can be achieved.
This is very important, since the material becomes flaw insensitive. When *00 Ea >>1, the
crack is of small-scale bridging (SSB) feature (hereafter referred to as Griffith crack), and the classic fracture mechanics is valid. The above transition can also be interpreted from an energy approach. If the intended release of elastic energy *22
0 4 Ea is larger than a , the crack
initiation is unstable, i.e., crack nucleation under SSB condition.
Here is the estimate of the cohesive constitutive parameters. For atomic decohesion, 0 10GPa, 0 0.1nm, 1J/m2, and 00
* E 1nm (use
*E 100GPa). For fiber pull-out, 0 1GPa, 0 10μm, 104J/m2, and 00
* E 1mm.
For fiber cross-over, 0 10MPa, 0 100μm, 103J/m2, and 00* E 1m.
MSE 612 (Fall 2012) 86
Of course, the cohesive interface properties can be either predicted by micromechanics analysis, or measured by crack-opening displacement.
Fig. 3.7: Fiber pull-out versus fiber cross-over (Bao and Suo, Appl. Mech. Rev. 1992).
Finite Element Simulations
The nonlinear finite element procedure, developed in Chapter 2, will be applied here. The principle of virtual work is given by
int
,
s
ij i j i i i i
V
u dV T dA t u dA
, (3.18)
where the relationship between traction iT and separation i is prescribed by the cohesive
interface model.
Fig. 3.8: Schematic of cohesive interface model and cohesive element.
The normal and tangential displacement discontinuity across the cohesive interface follows as
n u u n , t
u u t , (3.19)
and the tractions are
nT n σ n , tT n σ t , (3.20)
as shown in Fig. 3.8.
T
cmax
max Tn, Δn
Tt, Δt
GP: Gaussian point in the cohesive element
GP
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Considering a 2D case and using linear interpolation, we get
1
1
2
2
3
3
4
4
t
n
t
t n
n t
n
t
n
u
u
u
u
u
u
u
u
B ,
where
1 2 1 2
1 2 1 2
0 0 0 0
0 0 0 0
N N N N
N N N N
B .
Therefore, in the global Newton-Raphson iteration to solve the field equations, the element stiffness matrix and element force vector are given by
int
* T dA
R B T , (3.21)
int
* T dA
T
K B BΔ
. (3.22)
Finally, since we work in the coordinates of ,n t , we have to rotate them back to ix
coordinates, so that
T *R Q R , T *K Q K Q ,
where Q is the rotation matrix.
Fig. 3.9: Tvergaard-Hutchinson cohesive zone model, and the mesh used in the calculation.
MSE 612 (Fall 2012) 88
Case Studies
The finite element procedure essentially requires the relationship of T , which can be
prescribed in a variety of forms. Up to now, there is no physical model that can be used to predict the parameters used in most of these cohesive models. You have freedom to choose your own model. We will illustrate several applications in the following.
Example (1): Ductile-to-brittle transition (Tvergaard & Hutchinson, J. Mech. Phys. Solids 1992)
Consider a crack subjected to a K-field with a given history of K . The ductile versus brittle behavior corresponds to the competition between crack tip blunting (by plastic flow) and crack decohesion (as modeled by the cohesive interface).
The surrounding solid is modeled by E when Y , and
1
,N
YY
YE
Therefore, a dimensional analysis gives
1 2
0 0
ˆ, , , , , ,Y
Y c c
K a aN
K R E
,
where 20 0 1K E and
2
00
1
3 Y
KR
.
The boundary conditions applied on Fig. 4.9 are given by
1 2 1 cos 2 cos 3 2
2 2 1 2 1 sin 2 sin 3 2
x
y
u K ru E
,
where 3 4 . The linear ramping displacement conditions correspond to a constant K .
Fig. 3.10: Resistance curve from the Tvergaard-Hutchinson model.
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As shown in these resistance curves (Fig. 3.10), a large interface strength will lead to a large plastic zone, thus dissipating more plastic energy. Therefore, for strong interface or soft surrounding material, the crack tip may become blunt, and the crack never propagates. If the material fails, it is either by excessive plastic flow, or other mechanisms such as void nucleation and growth.
Fig. 3.11: Coating delamination example illustrating the use of cohesive interface model (Xia, et al., Int. J. Solids Struct. 2007).
Example (2): Delamination of elastic coating
As shown in Fig. 3.11, an elastic coating is subjected to spherical indentation. During unloading, the coating can delaminate from the elastic-plastic substrate. Here we add cohesive elements in the coating-substrate interface, using the model in Fig. 3.8(a). You can download the codes from:
http://web.utk.edu/~ygao7/teaching/delamination_uel.for
http://web.utk.edu/~ygao7/teaching/delamination_fig3iv.inp
http://web.utk.edu/~ygao7/publications/xia_gao_bower_lev_cheng_ijss07.pdf
Example (3): Fatigue modeled by an irreversible, hysteretic cohesive interface model
A phenomenological way to simulate the fatigue crack growth is to incoporate damage evolution in the cohesive interface model. Here we use an irreversible, hysteretic cohesive interface model, in which, during unloading and reloading, the traction-separation relation is given by
, 0
, 0
n nn
n n
KT
K
, (3.23)
where the unloading stiffness is a constant, determined by the unloading point only, that is
unload
nunloadn
TK
(3.24)
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while the reloading stiffness is given by the following evolution equation:
, 0
, 0
nn
f
nn
a
K
K
K K
(3.25)
Clearly, during loading, K is being damaged, and the associated length scale is f
(empirically, f n in order to prevent rapid softening). During unloading, K approaches
K (empirically, a n in order to assure K K ). One representative form is given in Fig.
4.12(a).
0 0.005 0.01 0.015 0.020
10
20
30
40
50
60
70
80
90
100
n (mm)
n (M
Pa)
200 300 400 500 600 700 8000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
N (cycle)
a-a 0 (
mm
)
#4: f=0.02mm w/o overload
#4: f=0.02mm w/ overload
Fig.3.12: An irreversible, hysteretic cohesive zone model and the resulting crack growth behavior (the boundary value problem is the compact tension specimen under cyclic loading).
We have applied this model for the compact tension specimen, and calculated the fatigue crack growth rate. As shown in Fig. 3.12(b), once an overload is applied, the crack growth rate will be slowed down. It should be noted that such an approach is purely phenomenological. Its usefulness can be viewed from the crack tip field. What is implicitly modeled is the crack tip process zone (including any microstructural details of damage initiation and evolution), and what is explicitly modeled is the surrounding plasticity. When the overload effect is of our interest, we can trust our numerical simulations. If our interest is on the exponent involved in Paris law,
mda dN A K , great care is needed to differentiate the roles played by the crack tip process zone and the surrounding plastic zone.
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3.4 Stress and Strain Based Failure Criteria Apparently, the failure criterion we are going to choose should have a direct connection to the failure mechanism(s) involved in the material phenomena of our interest. If the governing failure mode is a crack which has a well defined K-annulus, then the Griffith-Irwin criterion applies, and our design rule is the analysis and calculation of stress intensity factor. In this section, we will discuss failure modes that are independent of the geometry of our engineering components (i.e., they are based on local information -- stress/strain and their history).
Brittle fracture criteria
The appropriate failure criterion for a brittle solid is the Griffith-Irwin model. For a typical brittle solid, there is a statistical distribution of micro-cracks (defects), which depends on the material selection and preparation procedures. Therefore, a brittle solid fails when
1 TS , (3.26)
while the tensile strength can be described by the Weibull model,
00
exp
m
sP V
, (3.27)
with specimen volume 0V and material constants 0 and m . The survival probability sP for a
specimen volume V is given by 0 0
exp
m
s
VP V
V
.
0 0.5 1 1.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
/0
P
m=100
m=10
m=5
Fig. 3.13: Weibull distribution with different values of exponent m.
Constitutive model for compressive failure of brittle materials
MSE 612 (Fall 2012) 92
Brittle materials are generally used under compressive stress, that is, combined hydrostatic compression and shear. Failure in compression is a consequence of distributed micro-cracking in the solid, -- large numbers of small cracks form, propagate for a short while and then arrest. Failure occurs as a result of coalescence of these cracks.
Failure in compression is less catastrophic than tension, and in some respects qualitatively resembles metal plasticity. Therefore many models along this line are based on an extension of classical plasticity theory. The total plastic strain rates are still decomposed into elastic parts and plastic (irreversible) parts. In one model, the failure surface (in contrast to the yield surface) can take the form:
31 0
2ij ij ij kk efff s s c c Y , (3.28)
where c is a material constant controlling the variation of strength with respect to the hydrostatic pressure. The plastic flow can take the associated assumption. In more complicated theories, the plastic strain rate is taken as a weighted summation of statistical distribution of micro-cracks.
It should be noted that, since compressive response of brittle solids is similar to metal plasticity, some works blindly use Mises plasticity or other theories to model experimental works such as indentation on polycrystalline ceramics. Although the observed deformation behavior appears similar, the fundamental deformation mechanisms are quite different.
Fig. 3.14: Top view and side view of indentation in heterogeneous silicon nitride with tungsten carbide sphere (radius=1.98mm, load=3000N). The indentation hardness versus the effective
indentation strain can be fitted to FEM simulations using a linear strain-hardening model. (Frischer-Cripps and Lawn, J. Am. Ceram. Soc. 1996)
Ductile fracture
Ductile fracture in tension occurs by the nucleation, growth and coalescence of voids in the material. A crude criterion is based on the accumulated plastic strain, e.g.,
MSE 612 (Fall 2012) 93
2
3p p
ij ij fd d , (3.29)
where f is the failure plastic strain measured in uniaxial tensile test.
In more complicated theories, the processes that control void nucleation, growth and coalescence are modeled in a phenomenological way. One classic model is the Gurson-Tvergaard model, in which the yield surface is given by
2
2 2 22 12 1 1
2cosh 1 0
3 2 3Y
YY
q Ig J f q q f
, (3.30)
where f is the porosity (void volume fraction), Y is the tensile yield stress of the matrix
material, and 1q and 2q are material constants to fit experimental data. Note that the yield
surface is pressure-dependent (decreasing with hydrostatic tension), and the yield stress decreases as the volume fraction of voids increases, dropping to zero when 1f .
To complete the yield surface, we add associated flow, pij
ij
gd
, and evolution of the
void volume fraction as a function of strain,
1 ,pkk kk eff effdf f d A d , (3.31)
where the first term accounts for void growth, and the second term accounts for strain controlled
void nucleation. There is no consensus on the choice of ,kk effA , so that most of these
calculations remain qualitative.
Finally we note that the material failure is a broad topic, because there exist a variety of failure mechanisms. We usually start from a qualitative mechanism understanding, and translate it into
mathematical models. Quantitative predictions from most of these deformation and failure theories are oftentimes
not trustworthy. However, we can examine the competition among various mechanisms. Plastic instability is also a failure mode, but we will discuss this later. The scope in this section is limited; many interesting topics are not addressed, including
nonlinear fracture mechanics, interface fracture, damage mechanics, etc.
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4. CRYSTAL PLASTICITY Traditional material science is dedicated to physical metallurgy, which examines the processing-structure-property relationship of metals and alloys. The connection between deformation mechanism and material microstructure has been well appreciated from the very beginning of this discipline. The classic works done in the early development of plasticity theories are by G.I. Taylor in 1930s, which have laid the foundation for crystal plasticity. Parallel to this line, we have seen many phenomenological plasticity theories for macroscopic solids, such as Mises plasticity and various hardening laws. We need to understand the applicability conditions of these two categories of theories.
4.1 Deformation Mechanisms in Single Crystals and Polycrystals One can refer to any textbook on physical metallurgy or mechanical metallurgy for details on experimental findings of deformation behavior of crystalline materials.
Slip system
For a perfect single crystal, the onset of plastic deformation occurs at the theoretical strength. Since dislocations already exist in most metals and alloys, the plastic deformation is achieved by the motion, multiplication, and annihilation of dislocations. Since dislocations are defects in the crystal structure, dislocations can not move freely in the space. Experimentally, people have found that dislocations can only move along a certain slip plane in a certain slip direction. Usually, the slip plane is the most densely packed plane since the interplanar spacing is the largest. And the slip direction is the most closely packed direction. Slip plane and slip direction form the so-called slip system.
Schmid law
The onset of plasticity occurs when the resolved shear stress RSS of a slip system reaches
a critical value, CRSS . For most single crystals, CRSS is on the order of 1MPa, because it is a
measure of resistance to the dislocation slip.
The calculation of RSS is often explained in the uniaxial tension example. If the angle
between the tensile direction and the slip direction is , and that between the tensile direction and the normal direction of the slip plane is , then the resolved shear stress is
cos cosRSS , (4.1)
MSE 612 (Fall 2012) 95
where is the uniaxial tensile stress. In arbitrary stress state, ij , the resolved shear stress on a
given slip system is
RSS ij i js m , (4.2)
where the unit vector s gives the slip direction, and the unit vector m gives the slip normal.
Strain hardening
Most metals exhibit strain hardening during cold-working. As plastic deformation continues, the increase of number and density of dislocations, as well as the evolution of dislocation microstructure, will make dislocation slip more and more difficult to proceed. In macroscopic tests of typical engineering materials, the stress-strain curves can be described by kinematic, isotropic, or other hardening laws. Typical stress-strain curves for a single crystal shows three stages of work hardening: (1) “easy glide” with low hardening rates; (2) high and constant hardening rate, where the secondary slip systems are activated; and (3) decreasing hardening rate. The hardening curves depend on the initial crystal orientation, temperature, and strain rate.
Fig. 4.1: Strain hardening behavior for single crystal copper with various initial orientations. (Haasen, Physical Metallurgy, Cambridge Press, 1996)
The most important feature of crystal plasticity is latent hardening, which means that the dislocation slip on one slip system can harden the other slip systems. There are many dislocation-based mechanisms involved in this behavior. For example, dislocations can cross slip, meaning that dislocations can move to a different slip plane. For another example, dislocations in different slip systems can form Lomer-Cottrell junctions/locks, which prevent further dislocation motion. Latent hardening behavior has been quantified for most metals.
Polycrystals
The deformation in polycrystals is more complicated. Since grains have different orientations, with the application of a uniaxial tension, the resolved shear stress varies from one grain to another. Therefore, grains with the largest resolved shear stress will yield first, and the deformation becomes nonuniform. The neighboring grains have different slip orientations, so
MSE 612 (Fall 2012) 96
that the grain boundaries must pin deformation. The grain boundary can block dislocation slip, so that polycrystals are harder than single crystals.
Fig. 4.2: Plastic incompatibility at the grain boundary (Ashby, Phil. Mag. 1970).
If each grain deforms according to their orientations and Schmid factors, without the surrounding geometric constraints, there will be overlap or void between neighboring grains. We need to introduce additional plastic deformation to accommodate such an incompatibility. The resulting additional dislocations are called geometrically necessary dislocations. Clearly, each grain should be capable of a general plastic shape change, to preserve the continuity. This means that the strain in each grain must conform to five independent components of the strain tensor (the 6th is then fixed by the condition that the volume remains constant). This necessitates five independent slip systems. A slip system is independent if the shape change it produces cannot be achieved by a combination of other slip systems. At low T, HCP metals slips only on the basal plane which contains two linearly independent Burgers vectors. These metals thus exhibit very little plasticity in the polycrystalline state, as opposed to the single crystal state. FCC and BCC crystals do have five independent slip systems.
Texture and Anisotropy
A textured polycrystalline solid, as characterized by a distribution of grain orientations, will demonstrate anisotropic deformation behavior. The most famous example along this line is the “earing” formation in deep-drawn cups. Apparently, this cannot be predicted by Mises plasticity; the simulation must have a measure of texture in it. It is not our goal here to elaborate this topic. Interested students can refer to Texture and Anisotropy, Kocks et al., Cambridge University Press, 1998.
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4.2 Continuum Crystal Plasticity Theory The framework for continuum crystal plasticity theory has been rigorously established by Hill and Rice (J. Mech. Phys. Solid, 1972). Our following discussion follows the review by Asaro (Adv. Appl. Mech. 1982). The existing crystal plasticity models are different in terms of specific forms used in the flow rule and hardening equations. As we have summarized in Section 4.1, the crystal plasticity theory must be capable of describing the Schmid law, slip anisotropy, self and latent hardening, lattice rotation, etc. In other words, we need to directly model the crystallographic dependence of the deformation behavior.
A brief review of finite-deformation kinematics is given below. Consider a displacement field u from initial configuration X to current configuration x . The deformation gradient is defined as
x uF I
X X, (4.3)
where I is identity tensor. The deformation gradient tensor can be uniquely decomposed into the product of an orthogonal tensor and a symmetric tensor, i.e. F RU VR , where T R R I . The velocity gradient is given by
1 T T1 1
2 2
v
L FF L L L Lx
. (4.4)
Refer to previous chapters for details.
Fig. 4.3: Multiplicative decomposition of deformation in crystals (Needleman, Asaro, et al., Comput. Methods Appl. Mech. Eng. 1985).
Slip kinematics
As shown in Fig. 4.3, the Kroener-Lee multiplicative decomposition is
e pij ik kjF F F , e pF F F , (4.5)
Elastic stretching and rotation, and rigid body motion
Plastic deformation due to crystalline slips
MSE 612 (Fall 2012) 98
where eF contains elastic stretching and rigid body rotation, and pF is the plastic deformation due to crystalline slip. Consequently, the velocity gradient is
1 1 1 1e e e p p e L FF F F F F F F , (4.6)
or in index form,
1 1 1 1e e e p p eij ik kj ik kj ik kl lm mjL F F F F F F F F . (4.7)
The key of the crystal plasticity is that the plastic deformation rate is a consequence of the shearing on a set of slip systems, which are characterized by unit vectors parallel to the slip
direction s , and unit vectors normal to the slip plane m . The plastic rate of deformation is given by
1
1
Np p
F F s m , (4.8)
or written as
1p pik kj i jF F s m
, (4.9)
where is the shear rate on -th slip system.
The counterpart version in infinitesimal deformation is
1
2p
ij i j j is m s m
.
Elasticity
The Cauchy stress is the force on the unit area in the deformed coordinates. However, this cannot be directly used in the constitutive law. The 2nd-type Piola-Kirchhoff stress (or called material stress) is defined as
1e eij ik kl ljT F J F , (4.10)
where det eJ F . The elastic relation is
eij ijkl klT C E , (4.11)
where the elastic Lagrange-Green strain is
1
2e e eij ki kj ijE F F . (4.12)
Flow rule
The shear rate is a function of the resolved shear stress (or called Schmid stress) and
the strength of that slip system flow . In the Peirce-Asaro-Needleman model, the plastic flow
equation is taken as a power-law form:
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0 sgn
n
flow
, (4.13)
where the Schmid stress is given by i ij jm T s .
Hardening law
The evolution of flow is given by the hardening law:
flow h
, (4.14)
where h are the slip hardening moduli. There are many kinds of proposed models for h . The
Peirce-Asaro-Needleman (PAN) model gives
2 00
0
sechs
hh h
, (no sum on ), (4.15)
with 0
tdt
. The latent hardening moduli are given by
h qh , ( ). (4.16)
Table 1: Representative values for the finite element simulations. Parameter Value Elastic constant 11c , 12c , 44c of copper 168.4, 121.4, 75.4 GPa
Characteristic strain rate 0 10-3 s-1
Stress exponent n 10 Initial yield stress 0 1 MPa
Saturated yield stress s 10 MPa
Initial hardening rate 0h 100 MPa
State I hardening rate sh 5 MPa
Latent hardening factor q 1 (PAN model), 0 (BW model)
Interaction matrix parameters f
collinear interaction Hirth lock coplanar junctions glissile junctions sessile junctions
8 8 8 15 20
Another hardening model is proposed by Bassani and Wu (1991), which describes the three stage hardening of crystalline solids. In this model, we have
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02
00
sech ;ss s
s
h hh h h h G
, (no sum on ) (4.17)
where sh is the hardening modulus during easy gliding within the stage I, and the function G is
0
; 1 tanhG f
, (4.18)
where 0 governs the transition from stage I to state II deformation, and f measures the latent
hardening behavior. To this end, q is taken to be zero because latent hardening is explicitly
accounts in f .
Infinitesimal deformation
Under infinitesimal deformation condition, the multiplicative decomposition in Eq. (4.5) becomes additive decomposition, e p
ij ij ij , where the plastic strain rates are given by
1
2p
ij i j j is m s m
, (4.19)
and the slip strain rate is again given by Eq. (4.13).
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4.3 Finite Element Simulations
The constitutive theory for crystal plasticity appears much more complicated than the 2J
plasticity, but the framework and integration scheme are essentially unchanged. That is, we shall integrate the plastic flow equation and hardening equation for the update of stress and the determination of material tangent.
Here we follow the classic work by Peirce, Asaro, and Needleman (Acta Metall. 1982). The velocity gradient is
1 1 1e p e e e p p e L L L F F F F F F , (4.20)
which can be written in the symmetric and skew parts,
L D W , e e e L D W , p p p L D W , (4.21)
where symD L , skewW L , and similarly for other terms.
Referring to Fig. 4.3 again, we get the slip direction and the slip normal direction after the deformation eF ,
* * s F s , * * * 1 *
s F F s , (4.22)
* * 1
m m F , * * * * 1
m m F F , (4.23)
so that
* 1 * 1 * * 1 * *p p p
L F F F F F s m F s m . (4.24)
The elastic relation in rate form is given by
* *
:
:
e e e e
e
d
dt
sym
ττ W τ τW D
D s m
(4.25)
where Jτ σ is the Kirchhoff stress.
For the stress update, we adopt the so-called -implicit method (explicit if =0 and implicit if =1), i.e.,
1 t t tt , (4.26)
which, together with the Taylor expansion of (4.13), gives
t flow
flow
t
. (4.27)
The hardening relation gives
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flow h
. (4.28)
Substituting the elastic constitutive law, Eq. (4.25), and the hardening law, Eq. (4.28), into the incremental flow equation, Eq. (4.27), will give us a set of nonlinear equations to
calculate from the knowledge of information at t and strain increments n
n
t t
t
dt
ε D .
Example (1): Indentation of single crystals
Fig. 4.4: Representative FEM results for indentation on copper single crystal. Slip strain on
111 011 and 111 011 respectively. The coordinates are 112 , 111 , are 110 . (Gao,
Larson, and Pharr, unpublished results.)
In this example, a single crystal copper (using PAN parameters in Table 1) is indented by a spherical indenter. Since we can normalize length by the indenter radius and the constitutive law has no length scale, feel free to interpret the length in Fig. 4.4. As shown by the finite element results, the slip strain contours on two slip systems are radical different.
Example (2): Crack tip plastic field of single crystals
The plastic zone ahead of a crack tip appears as a kidney shape if we use Mises plasticity. For a crack in a single crystal, the plastic deformation is localized into certain directions, as shown in Fig. 4.5. So the effective strain, or Mises strain, will show the sector formation. If we simulate the crack tip plastic behavior when the plastic zone size varies from less than the grain size to larger than several grains, we will see very interesting transitional behavior.
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Fig. 4.5: Slip strain contours ahead a two-dimensional crack. Analytical solution: Rice, Mech. Mater. 1984. Finite element results: Cuitino and Ortiz, Modelling Simul. Mater. Sci. Eng. 1992.
[101]
[010]
crack
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4.4 Applications and Further Comments Deformation properties of crystalline metals and alloys are far more complicated than the above dislocation-based deformation modeled in the single crystal plasticity theory. Here we briefly comment on what methods and tools we should choose in a number of commonly seen material phenomena.
Single crystal versus polycrystal
One can model a polycrystal by explicitly simulating individual grains and treating the grain boundaries as rigid interfaces, or as weak interfaces using cohesive zone model, or as evolving interfaces. Clearly, this is so tedious and not practical even with high performance computers.
One commonly used theory is to develop a polycrystal plasticity theory, as we have discussed in Chapter 1. But these theories typically do not agree on the treatment of the evolution of plastic anisotropy, i.e., equations related to plastic spin pW . Explicitly modeling all the grains using single crystal plasticity will avoid this problem.
Creep and superplasticity
The simplest (but not so useful to materials scientists) model is to fit the continuum viscoplastic law, and determine the relationship of these constitutive parameters and material microstructure. In order to understand the deformation mechanism, one needs to explicitly examine the competition between diffusional creep, dislocation creep, etc. This requires modeling on the grain level.
Scale dependence
Classic crystal plasticity has no length scale, which is based on the notion that the slip strain is achieved by statistically stored dislocations. When strain gradients are large, the resulting geometrically necessary dislocations (to accommodate the nonuniform plastic field as in Fig. 4.2) can lead to extra hardening. The gradient terms, if introduced in a constitutive model, will bring a length scale or multiple scales. The development of such a theory is one of recent research frontiers in crystal plasticity.
Connection to dislocation plasticity
When the length scale of interest is comparable to the dislocation spacing, we need to model individual dislocations explicitly. The long-range elastic interaction can be easily determined from a linear superposition scheme. The short-range interactions are typically modeled by some ad hoc rules, such as Frank-Read nucleation mechanism, dislocation junction formation, etc. Although some of these rules can be calibrated by atomistic simulations, many are still qualitative. It should also be noted that the connection between dislocation plasticity to crystal plasticity is very elusive, since the latter involves infinite number of dislocations.
Phase transformation induced plasticity
Let’s use stress-assisted martensitic transformation as an example. The corresponding plasticity theory is very similar to the slip crystal plasticity. In essence, we need to represent the plastic strain as a summation of individual crystallographic systems.
MSE 612 (Fall 2012) 105
Once a part of a material is transformed into a new phase, without surrounding constraint,
it will undergo a volumetric expansion and a shear strain on a given crystallographic
system, as characterized by the transformation shear direction m and the so-called habit-plane unit normal n . Consequently, we represent
1 1p p f f
F F m n I . (4.29)
This equation prescribes the plastic kinematics. We also need to specify f , such as
1 m
fs
,
where the transformation resistance s can be represented as a function of plastic strain.
Fig. 4.6: Decomposition of the deformation gradient for a material undergoing a stress-assisted martensitic transformation (Grujicic et al., J. Mater. Sci. 2001).
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5. MATERIAL INSTABILITY Instability means that a deformation pathway becomes unstable and cannot be attained in practice. A long cylinder under compression will buckle, so that the uniform compression (as a deformation pathway) cannot be attained in reality. A brittle solid under stress will fracture, so that an elastic deformation field is unstable. Plastic deformation can become localized in thin bands, often called shear bands. These are examples of unstable events in material phenomena.
5.1 Geometric vs Material Instability Material instability: We will examine instability that arises from material behavior, thus called material instability. For instance, plastic localization may occur due to a natural tendency of the material itself to soften at large strains. This is also called localized necking, or discontinuous bifurcation. Examples include:
localization in a Gurson solid due to the softening effect of voids at large strains; localization in a single crystal due to the softening effect of lattice rotations; dislocation nucleation due to the softening branch in the interatomic interaction; shear band in geo-materials due to the dilatation effect; strain localization in amorphous alloys.
Diffuse necking: Plastic localization may occur as a consequence of changes in specimen geometry (i.e. geometric softening). This is also called general bifurcation, or limited point bifurcation. Examples include necking and buckling.
Fig. 5.1: (left) Engineering stress-strain curve illustrating the uniaxial tensile test (W.D. Callister, Materials Science and Engineering: An Introduction, 7th edition, John Wiley & Sons Inc., 2007) (right) Micro-compression test on an Au-based metallic glass pillar at 10-3s-1 and room temperature:
SEM image of the pillar after deformation (courtesy of Prof. Nieh).
5.1.1 Necking Analysis
If we test a cylindrical specimen of a very ductile material in uniaxial tension, it will initially deform uniformly. At a critical load the specimen will start to neck, as shown in Fig. 5.1.
MSE 612 (Fall 2012) 107
Necking, once it starts, is usually unstable – there is a concentration in stress near the necked region, increasing the rate of plastic flow near the neck compared with the rest of the specimen, and so increasing the rate of neck formation. The strains in the necked region rapidly become very large, which will quickly lead to failure. Necking formation is due to geometric softening. For example, stretching a strain hardening solid (such as typically engineering alloys) will lead to necking as shown in Fig. 5.1(a).
Analysis for power-law hardening solid
Consider a cylindrical specimen with cross sectional area A. Assume that the material is perfectly plastic and has a true stress-strain curve (Cauchy stress versus logarithmic strain) that can be approximated by a power-law
0n , (5.1)
with 1n . Suppose that at some time t the specimen is subjected to a load P, and has length L, strain and cross sectional area A. We now increase the length of the specimen by an infinitesimal displacement dL. This causes an increment in logarithmic strain d dL L ,
increasing the Cauchy stress to 10
nn dL L . At the same time, the cross sectional area of
the bar decreases to A A dL L , since AL=constant to preserve volume. Note that
0AdL LdA . Consequently, the load applied to the specimen after stretching is
10 0
n ndP Ad dA An dL L A dL L , (5.2)
where the first term is the result of strain hardening, and tends to increase the load. The second term is a consequence of the lateral contraction of the bar, and tends to decrease the load. This second term is referred to as geometric softening – the effect of the specimen’s geometry is to reduce the load required to stretch the specimen. Notice that there is a critical strain crt n ,
above which 0dP , so that the load reaches a peak value at this critical strain,
max 0 0 expnP A n n .
It turns out that the point of maximum load coincides with the condition for unstable neck formation in the bar.
Considère construction
The above analysis can also be represented by the Considère condition. There is actually no need to assume Eq. (5.1). Define the stretch ratio 0L L , so that the engineering strain is
1eng . The true stress is P A . Consequently, 1 0eng engdA Ad , and
1 eng
eng
dP Ad dA A d d
, (5.3)
so that the necking condition corresponds to
1 eng
d
d
. (5.4)
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Fig. 5.2: Graphic illustration of the Considère construction in true stress versus engineering strain plot.
A general necking condition
Yet a more general analysis requires the consideration of both strain hardening and strain rate hardening. The condition for plastic stability is given by (P. Haasen, Physical Metallurgy, Cambridge University Press, 1997)
0A A . (5.5)
The symbol “ ” refers to the variational changes along the length of the specimen. The condition means that at random localized constrictions ( 0A ), the rate of necking ( A ) must be smaller than elsewhere in the specimen ( 0A ) in order that the deformation at these points does not proceed in an unstable manner.
Fig. 5.3: Perturbation of the cross-sectional area in a bar under uniaxial tension.
Using the following equations:
0P A A ,
,
L A
L A
, A
A
,
2
A A A
A A
and defining the work hardening coefficient by
P
A+δA P
σ
εeng -1
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1
, (5.6)
and the strain rate sensitivity by
1
lnm
, (5.7)
we get
11
A Am
A A m
. (5.8)
Since 0A A in tensile test, the plastic stability condition is given by
1m stable flow (5.9)
Special cases
When m , we recover the Considère construction since Eq. (5.9) becomes
d d or 1eng engd d .
When 0 as in high temperature deformation, the stable deformation can be realized only when material behavior approaches Newtonian viscous flow (i.e., 1m ).
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5.2 Hill-Hutchinson-Rice Theory A smoothly varying strain field can evolve into narrow bands inside of which the shear strain field localizes and outside of which the shear strains are almost negligible. Fig. 5.4 illustrates how this process occurs. When the stress reaches the peak stress, with the increase of applied strain, the stress increment should be negative. This can be achieved by the entire solid following the softening branch, or by a scenario in which a thin plate further deforms plastically but the surrounding material experiences elastic unloading. To see this, let’s assume that the hardening branch has material tangent , and the softening branch has material tangent * (so
that * 0 ). The shear deformation can be written as
*w w
, (5.10)
When 0 , the condition for a stable solution is
* w
w
, or *
00
w
. (5.11)
This means that the two points shown in Fig. 5.4 are the two bifurcation points. During plastic deformation, the strain inside of the shear band jumps from A to B, and that outside of it follows the elastic unloading path. If the shear band with w is infinitesimally small, the onset of shear band occurs at the peak stress.
Fig. 5.4: Shear band initiation under simple shear. (Left) Dashed lines, having slope of
w w , are tangent to the stress-train curve at the two bifurcation points. (Right) The
macroscopic load-displacement curve exhibit elastic snap-back instability.
Hill (J. Mech. Phys. Solids 1962; and subsequent works by Hutchison and Rice) has developed the general theory of bifurcation of a homogeneous elastic-plastic flow field into a band of localization deformation. If the localization occurs in a thin planar band of unit normal n , the kinematical restriction gives that the velocity gradient field, ,i jv , inside the band differs
from that outside, 0,i jv , by an expression of the form
0, ,i j i j i jv v g n , (5.12)
w
A
B
load A
displacement
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where ig is undetermined right now. The continuing equilibrium requires
0 0i ij i ijn n , (5.13)
at incipient localization, where ij is the stress rate within the band and 0ij that outside of it.
The constitutive relation in rate form is given by
,ep
ij ijkl k lC v . (5.14)
Substituting (5.14) into (5.12) and (5.13) gives
0epi ijkl l kn C n g . (5.15)
To attain a nontrivial solution, the determinant of the coefficient matrix must be zero.
In other words, the necessary criterion for the classical discontinuous bifurcation, which corresponds to the loss of ellipticity, is that the acoustic tensor has a zero eigenvalue. That is, the
2nd order acoustic tensor is singular:
det 0ep n n . (5.16)
The general condition for diffuse necking to occur is
det 0ep , (5.17)
meaning that at least one eigenvalue of the fourth order tangent modulus is zero.
Further remarks
The above analysis is valid for rate-independent solids.
The condition for Eq. (5.16) to be satisfied is very sensitivity to the constitutive laws (Rudnicki and Rice, J. Mech. Phys. Solids, 23, 371-394, 1975). The classical elastic-plastic solid with a smooth yield surface and normality of the plastic flow rule is quite resistant to localization, but deviations from the classical model can have a strong effect. For instance, localization can occur in a solid that develops a vertex on the yield surface, as arises in physical polycrystalline models based on the concept of single crystal slip. Also dilatational plastic flow and non-normality of the plastic flow rule, as induced by, for example, the porous ductile material model have a significant destabilizing effect.*
For materials with associated flow rules, the acoustic tensor is symmetric, and the loss of strong ellipticity (i.e., determinant of the symmetric part of the acoustic tensor is equal to zero) and classic discontinuous bifurcation criteria (i.e., Eq. (5.16)) identify the same first discontinuous bifurcation point.† For materials with non-associated flow, the loss of strong ellipticity criterion will predict that localization may occur prior to the point identified by the classical discontinuous bifurcation criterion.
* Thus we note strain localization is neither necessary nor sufficient for strain localization to occur. † A Hermitian (or symmetric) matrix is positive definite if and only if all its eigenvalues are positive.
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The general bifurcation (such as necking) criterion is first satisfied when the determinant of the symmetric part of the tangent modulus tensor is equal to zero. For materials with associated flow, the tangent modulus tensor is symmetric, and the above condition and Eq. (5.17) are the same. However, for materials with non-associated flow, loss of positive definiteness of the symmetric part of the tangent modulus tensor can occur prior to the condition in Eq. (5.17). More details can be found in Neilsen and Schreyer (Int. J. Solids Struct. 30, 521-544, 1993).
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5.3 Numerical Simulations According to Fig. 5.4, numerical simulations of material instability phenomena are often limited by the occurrence of an elastic snap-back instability. At the point of instability, quasi-static finite element computations are unable to converge to an equilibrium solution, which usually terminates the calculation and makes it impossible to follow the post-instability behavior. The key question is thus on how to determine a physically-meaningful equilibrium deformation pathway, particularly for history-dependent materials such as plastic solids.
Modified Riks Method
In an implicit finite element formulation (which uses Newton-Raphson iteration to solve the nonlinear equilibrium equations) one finds that the radius of convergence of the Newton-Raphson scheme reduces to zero at the point of instability. In an explicit scheme, the solution quickly diverges from the equilibrium path and leads to unphysical predictions. General-purpose schemes such as the Riks method may be used to follow the unstable branch of the solution during the snap-back. In general, these schemes require some effort to implement, and in history dependent problems (such as those involving plasticity or dislocation motion) it is not clear that following the unstable equilibrium solution necessarily leads to physically meaningful predictions. Note that this limitation is not so significant for rate-dependent solids.
Fig. 5.5: (Left) Typical unstable static response. (Right) Modified Riks algorithm. (Pictures taken from ABAQUS theory manual.)
The essence of the modified Riks method is that the solution is viewed as the discovery of a single equilibrium path in a space defined by the nodal variable and the loading parameter. As shown in Fig. 5.5, starting at A0, we move a given distance (as determined by ABAQUS’s automatic incrementation algorithm for static cases) along the tangent line to the current solution point A1, and then searching for equilibrium in the plane that passes through the point thus obtained and that is orthogonal to the same tangent line. Thus the solution is updated to A2. Graphically it can be shown that a snap-back instability can be passed over provided an
appropriate choice of 1N NA A .
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Viscosity Method
Numerical difficulties can be avoided by introducing a small viscosity in the constitutive equations. Consider a strain softening cohesive interface in Fig. 5.6. The stress is related to the separation of the interface by
max exp 1 n nn n
n n n
dT
dt
. (5.18)
The simple boundary value problem gives the displacement of the top boundary:
*
max
exp 12 n n
E U
a
, (5.19)
with *max2nE a . When decreases, i.e., a stiff interface and a compliant elastic spring,
the unstable pathway will show the snap-back instability. The introduction of a fictitious viscosity term, n , in Eq. (5.18) will give a stable solution, as shown in the right picture (Fig.
5.6). When n is made sufficiently small, a sudden load drop can be shown.
Fig. 5.6: (Left) A simple boundary value problem used to test the influence of viscosity in the cohesive interface model. (Right) For several values of fictitious viscosity, max as a function
of nU with 1 5 . (Gao and Bower, Modelling Simul. Mater. Sci. Eng. 12, 453-463, 2004).
Material Instabilities in Rate-Dependent Solids
For rate dependent solids, the plastic localization critically depends on the initial imperfection. Material instability in rate-dependent plastic solids is manifested by spectral growth of the strain field. That is, evolving from initial imperfections or small spatial fluctuations, the strain field will gradually grow into a spatial distribution that consists of high strains in certain locations and low strains elsewhere, and this strain localization process can be delayed by the material rate sensitivity. Not every imperfection can evolve into a shear band.
A general constitutive law involves the flow equation and the evolution equation of internal variables. A Mises type is given by
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3,
2ijp
ij e ke
Sf q
, ,i i e kq g q , (5.20)
where e is the Mises stress. For instance, the internal variable can be chosen as the free volume
for amorphous alloy. The stress-driven increase of free volume can soften the material.
Shear Band Width and Nonlocal Models
Post-instability simulations require a faithful measure of shear band width. Unfortunately, a local constitutive model will not give any information along this line. For rate-independent solid, the width is set by the mesh size. For rate-dependent solid, it is set by both the mesh size and initial imperfection. Fig. 5.7 gives one example showing the shear band formation following necking in a single crystal under uniaxial tension. The introduction of nonlocality (by strain gradients) in the constitutive law regularized the mesh size sensitivity.
Thermally Activated Deformation Pathway
The stable deformation pathway also shows temperature effects. As shown in Fig. 5.8(a), two atomic planes slip over each other in a uniform fashion, and the serrations are a consequence of the interface model:
max
sin 2 x xd
b dt b
, (5.21)
with Burgers vector b. Under finite temperatures, a nonuniform slip field arises at the interface, as shown by the calculations in Fig. 5.8(d). The nonuniform slip actually corresponds to a dislocation loop. The activation energy is the energy barrier between a stable solution and a saddle point solution when the applied stress is less than the critical stress (called athermal limit).
Fig. 5.7: (a) The tensile specimen considered with initial dimensions and slip systems. (b) Deformed mesh and contours of accumulated effective slip using a nonlocal theory. (Borg, Int. J. Plasticity, 23, 1400-1416, 2007).
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(a)
0 0.5 1 1.5 2 2.5 3-1.5
-1
-0.5
0
0.5
1
1.5
x/b
/ m
ax
=2, Frenkel potential
unstable solutionstable solution
(b)
(c) /max
0.4 0.5 0.6 0.7 0.8 0.9 1.0
(1
)U
act /b
3
0.0
0.5
1.0
1.5
2.0
2.5
3D
(d)
Fig. 5.8: (a) One d.o.f. model leads to a uniform slip of all the atoms on the slip plane. (b) 3D model of homogeneous dislocation nucleation under pure shear load. (c) The saddle-point
configuration of dislocation x along the x-direction at y = 0 with max . (d) The activation
energy as a function of applied shear stress.
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5.4 Shear Band Angle Analysis in Metallic Glasses Here we present an example of material instability analysis, as adapted from “On the shear band direction in metallic glasses,” by Y.F. Gao, L. Wang, H. Bei, and T.G. Nieh (Acta Materialia, 59, pp. 4159-4167, 2011).
Although detailed atomistic processes responsible for shear-band initiation in metallic glasses (or called amorphous alloys) are still an ongoing research topic, it is generally accepted that the strain localization process that transforms a homogeneous deformation field into narrow shear bands and inhomogeneous deformation is related to a stress-driven structural disordering process. The asymmetric shear-band direction under uniaxial tension/compression tests (as shown in Fig. 5.9) is typically believed to be related to the pressure sensitivity of the above process. Therefore, the deformation behavior of metallic glasses is often modeled by the Mohr-Coulomb yield criterion. On the other hand, from the continuum mechanics point of view, the onset of strain localization corresponds to a loss-of-ellipticity instability in the constitutive law, which critically depends not only on the pressure sensitivity, but also on flow normality (i.e., whether the plastic flow is associative or non-associative), Poisson’s ratio, and loading conditions. The latter factors, as will be shown here, demonstrate that the typical practice of relating the shear band angle to coefficient of internal friction in the Mohr-Coulomb model is inappropriate.
Fig. 5.9: Shear bands produced by bending a Zr52.5Al10Ti5Cu17.9Ni14.6 plate (0.6 mm × 3 mm × 15.3 mm) exhibit the asymmetric directions on the tension and compression sides.
Consider a uniaxial test with applied stress uniaxial . On an inclined plane (which makes
an angle of with the loading axis), the applied normal stress n (positive if tensile) and the
shear stress s are
2sinn uniaxial , (5.22)
sin coss uniaxial . (5.23)
The Mohr-Coulomb yield criterion is given by
s n Y , (5.24)
100 μm
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where Y is the yield stress and is a material constant and often called the coefficient of
internal friction. Substituting Eqs. (5.22) and (5.23) into Eq. (5.24) shows that the effective shear stress s n reaches a maximum at * 11
2 tan 1 under tension, or at * 112 tan 1
under compression. For a nontrivial and positive , the inclination angle of the critical plane to
first reach yield is larger (smaller) than 4 under tension (compression).
Since the tension-compression asymmetry of first-yield plane angle is consistent with the asymmetry of shear band angle, the above model has been routinely used to determine the pressure sensitivity of metallic glasses. However, such an analysis has problems for several reasons. First, a closer examination of this model suggests an inconsistency when compared to experiments. Predictions based on the Mohr-Coulomb model give the same amount of deviation of * from 4 in both tension and compression conditions. A literature compilation gives a
typical value of * 43 in compression (so that *ctg 2 0.07 ) and * 54 for tension
(so that *ctg 2 0.324 ). Such an unequal amount of deviation is routinely found in
most of metallic glasses. Second, we note that the shear band is a result of strain localization. Prior to a critical point, the deformation field is smoothly varying and stable. At the onset of strain localization, a narrow band will suddenly experience a large plastic deformation while the surrounding experiences elastic unloading. The amount of this elastic snap back depends on the material stiffness and the machine stiffness. Both the yield condition and the flow behavior are needed in a constitutive model for material instability study.
Fig. 5.10: Geometric interpretation of the coefficient of internal friction, μ, and the dilatancy factor, β (reproduced from Rudnicki and Rice, JMPS 1975). The yield surface is plotted when
the solid is subjected to a hydrostatic stress p (positive in compression) and a pure shear stress τ. The tangent defines μ, while the plastic flow direction, as indicated by the solid arrow line,
defines β.
Strain localization can be understood as instability in the macroscopic constitutive description of inelastic deformation of the material, as thoroughly discussed in Section 5.2. A smoothly varying strain field can evolve into narrow bands inside of which the shear strain field grows large and outside of which the shear strains are almost negligible. The constitutive model
β μ 1 1
p
τ
yield surface
inadmissible stress state
elastic stress state
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of a general pressure-sensitive material requires the yield surface and the flow potential. For the combination of a hydrostatic stress (positive if compressive) and a shear stress, the yield surface can be described in Fig. 5.10. The slope of the yield surface in pressure versus shear stress plane defines the coefficient of internal friction, , and the flow direction is given by the dilatancy factor, . When , the deformation is associative. A general non-associative flow, owing to possible frictional origin of the deformation resistance, gives . Rudnicki and Rice (JMPS, 1975) has used these two parameters in an incremental plasticity model (only including mean and Mises stresses in the yield function and flow potential), and then examined the material instability condition. They found that the shear band angle, as depicted in the principal stress space in Fig. 5.11, is given by
1 min0
max
tanN
N
where 11 1
3N , I
maxN
, IIN
, IIIminN
, 3mises , and
I , II , and III are principal deviatoric stresses.
Fig. 5.11: The direction of the shear band in the principal stress space. The shear-band plane is parallel to the II direction, and makes an angle of 0 with the I direction in the I III~
plane. The three principal stresses are ranked as I II III .
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6. MICROSTRUCTURE-BASED SIMULATIONS The key issue in materials science and engineering is the processing-structure-property relationship. What is called physical metallurgy typically consists of phase transformation (i.e., the processing-structure relationship) and mechanical metallurgy (i.e., the structure-property relationship). In the study of deformation and failure of structural materials, the typical strategy is to explicitly model the microstructural features while any response below these length scales of interests will be replaced by constitutive equations. Some examples are illustrated here.
6.1 Stress Effects in Phase Transformation We have learned phase diagram, solidification, and many other types of phase transformation in the introduction of materials science. The role of stresses in these phenomena can be seen from examples such as martensitic phase transformation, multiferroic materials, and stress-enhanced diffusion (e.g., hydrogen embrittlement), among many others. As an illustrative example, let’s take a look at the solidification of a single species. The driving force for such a phase transformation is the free energy reduction from liquid phase to solid phase and the penalty due to the surface energy, namely,
3 244
3 vG r G r , (6.1)
where vG is proportional to the degree of undercooling, mT T , r is the nucleus radius, and
is the interface energy. As shown in the right picture in Fig. 6.1, the rate of the phase transformation consists of the nucleation rate and growth rate. When the temperature is low, the activation energy is low, so that the nucleation rate is high. On the other hand, the growth rate is low since the mobility (or diffusivity) is low. This example clearly demonstrates the effects of energetics and kinetics on the phase transformation problem.
Fig. 6.1: (Left) Schematic diagram of a spherical solid in a liquid. (Right) The rate of phase transformation is a net result of nucleation rate and growth rate, which exhibit opposite
dependence on the temperature. (W.D. Callister, Materials Science and Engineering: An Introduction, 7th edition, John Wiley & Sons Inc., 2007).
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Figure 6.2 shows the famous microstructural development in the Fe-C system. The phase diagram gives the equilibrium phases, composition, and relative fractions of these phases, while the kinetics governs the pathway of the microstructural development. Examples in Fig. 6.2 are governed by diffusional kinetics, so that the microstructural shape and feature size are governed by cooling rate, diffusivity, heat conductivity, etc. It should be noted that when the cooling rate is high, the microstructures in Fig. 6.2 will be overtaken by the martensitic phase transformation. This is a diffusionless process and usually occurs very fast.
Fig. 6.2: Microstructural development in the Fe-C system: (a) hypoeutectoid, (b) eutectoid, and (c) hypereutectoid. (W.D. Callister, Materials Science and Engineering: An Introduction, 7th
edition, John Wiley & Sons Inc., 2007)
As we have learned in high school physics, a point mass moving on a morphological profile will be at equilibrium state if the potential energy reaches minimum. In other words, it moves on an energy landscape, with ,x y being the generalized coordinates and z being the
potential. Local valleys provide the energetically favorable (metastable) equilibrium configurations, while the trajectory on the energy landscape is governed by kinetics. The same applies to the microstructural evolution problem, in which the free energy minimum gives the energetically favorable state while the kinetics determines the intermediate states. As you can see, energetic study is usually not deterministic since we live in a non-equilibrium world.
6.1.1 Energy Minimization: an Example of Precipitate Shape
Consider an inclusion embedded inside an infinite matrix. If the elastic constants of these two phases are different, the second phase is usually called inhomogeniety. The inclusion has experienced a transformation strain, or called eigenstrain,
0 0m a a a , (6.2)
which can also be regarded as lattice mismatch strain. A stress field will be induced, as governed by the equilibrium equations:
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, , 0total mijkl kl j kl jc , (6.3)
with elastic constants ijklc . Eshelby (Proc. R. Soc. London A 1957) derived the strain energy of
this inclusion-matrix system. For a spherical inclusion,
2 38 1
3 1el mG a
, (6.4)
where a is the radius of the spherical inclusion.
The equilibrium shape of the inclusion minimizes the following free energy:
elG dA G
n , (6.5)
where n gives the surface normal of the inclusion. In general, we need to consider the following factors: (1) anisotropy in the interface energy; (2) difference in the elastic constants of the two phases; (3) elastic anisotropy; (4) inclusion size (volume); (5) interface coherency (i.e., misfit dislocation); (6) lattice diffusion; (7) interface diffusion; etc. The phase boundary should be described by a continuous surface, and the resulting kinetic equation should be solved by discretizing the surface into nodes. Such a treatment is beyond a reasonable scope of this course.
Fig. 6.3: The equilibrium shape of an elliptical cylinder (as characterized by U ) as a function of
the inclusion size (as characterized by cA A ). Beyond a critical area, the equilibrium shape
is elliptical. (Johnson and Cahn, Acta Met. 1984)
To illustrate key concepts, we follow the work by Johnson and Cahn (Acta Met. 1984) by assuming a 2D elliptical cylinder. The elastic strain energy can be solved by repeating Eshelby’s method. Minimizing the surface energy and elastic strain energy, Fig. 6.3 shows the equilibrium shape of an elliptical 2D inclusion, where
a b
Ua b
, (6.6)
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and a and b are half axes. Once the cylinder reaches a critical cross-sectional area, 1 , the equilibrium morphology changes smoothly from a circular to an elliptical cylinder, and the circular cylinder becomes unstable. Without going into too much mathematical details, we can see that from dimensional analysis, clearly, the critical feature size scales as 2
mE , because the
surface energy scales as A and elastic strain energy scales as 2mE V .
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6.2 Non-Equilibrium Thermodynamics In the non-equilibrium thermodynamic framework, we first consider a virtual change of the interface as shown in Fig. 6.4. Thus we can define the configurational force F by
nF r dA G
. (6.7)
Let nv be the actual velocity of the interface in the direction normal to the interface (i.e., the
volume of atoms added to the particle per area per time). The relationship between nv and F is
treated differently in the interface migration problem and in the diffusional phase transformation problem.
Fig. 6.4: Virtual change of an interface, nr .
Interface Migration
In this case, we are concerned with short-range material transport. Examples include grain growth, surface depositional growth, etc. The mass in a domain of interest does not necessarily conserve. In this case, the interface normal velocity nv is usually assumed to be
linearly proportional to the driving force:
nv LF . (6.8)
For example, we consider the curvature-driven, depositional growth of a solid-liquid interface (as described by a profile h ). According to Eq. (6.7), the driving force is defined with respect to the change of surface profile, so that F where is atomic volume. The
chemical potential is 0 , where is the curvature of the interface (positive if the
surface is convex). Consequently, the governing equation is
2hL h
t
, (6.9)
δrn
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since 2h when 1h .
Since this course is about computational plasticity, we can reformulate the above framework in terms of the principle of virtual work. Using Eqs. (6.7) and (6.8) gives
nn
vr dA G
L . (6.10)
Now represent the interface by a set of generalized coordinates 1q , 2q , … nq , so that the
generalized force if (also called configurational force) is defined by
1 1 2 2 ... n nG f q f q f q . (6.11)
Using the interpolation gives n i ii
r N q and n i ii
v N q . Consequently, we get a set of
linear algebraic equations:
ij j ij
H q f , (6.12)
where
1
ij i jH N N dAL
. (6.13)
Now the task becomes the solution of a set of partial differential equations.
Diffusional Process
Because of mass conservation, we have to include the continuity equation, which states that the velocity normal to the free surface is equal to the flux divergence:
0nv J . (6.14)
The diffusion flux is assumed to be linear with the driving force,
MJ F . (6.15)
For the concentration profile, the driving force is 1
F , as in the Fick’s first law.
For example, if we consider the surface-diffusion-driven morphological change of a solid-liquid interface (as described by a profile h ), the governing equation is
2 2hM h
t
. (6.16)
Cahn-Hilliard Model
To further illustrate the discussion in the above, let’s discuss the famous Cahn-Hiliard model. For a binary alloy, the total free energy consists of
2
V
G g C h C dV , (6.17)
MSE 612 (Fall 2012) 126
where the free energy of mixing is given by
ln 1 ln 1 1B
gC C C C C C
k T
. (6.18)
If >2, the free energy of mixing has two wells, leading to phase separation. In addition, if the average concentration falls into the two inflection points, we get the well known spinodal decomposition.
The Cahn-Hilliard model gives the evolution equation:
2 22
2C M g
h Ct C
, (6.19)
where M is atomic mobility and is atomic density (i.e., 1 ). A length scale can be defined from the comparison of the first two terms in the parenthesis of Eq. (6.19), given by
B
hb
k T
, (6.20)
and a time scale can be deduced as
2
B
h
M k T . (6.21)
Consequently, the normalized diffusion equation becomes
2 22C
P Ct
, (6.22)
where
ln 1 21
CP C C
C
. (6.23)
In order to solve Eq. (6.22), we consider the 2D case and use Fourier transformation,
1 2 1 2 1 1 2 2 1 2ˆ, , , , expC x x t C k k t ik x ik x dk dk
, (6.24)
so that the evolution equation becomes
2 4ˆ
ˆˆ 2C
k P k Ct
, (6.25)
where 2 21 2k k k .
Solving the ordinary differential equation in Eq. (6.25) is easy. We can use Runge-Kutta, Euler method, etc. Here we use a semi-implicit method, giving
2
1 4
ˆ ˆˆ
1 2n n
n
C k P tC
k t
. (6.26)
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50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/b
C
time: 199.9000
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/b
C
time: 299.9000
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/b
C
time: 399.9000
50 100 150 200 2500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x/b
C
time: 499.9000
Fig. 6.5: Temporal evolution of the concentration profile as in the Cahn-Hilliard model.
----------------------------------
The following matlab code solves the 2D Cahn-Hilliard model.
clear all; tic aviobj=avifile('cahn_hilliard_2d.avi','fps',5); % controlling parameters flag_refine_inc=0; flag_refine_tolerance=0.02; flag_count=0; % initial condition N=64; delta_x=2; %time_inc=0.1; time_inc=2; c0=0.5; omega=2.2; for m=1:N for n=1:N c(m,n)=c0+0.001*rand(1); end end % frequency in reciprocal space for m=0:(N/2)
MSE 612 (Fall 2012) 128
k(m+1)=2*pi/N/delta_x*m; end for m=(N/2+1):(N-1) k(m+1)=2*pi/N/delta_x*(m-N); end for m=1:N for n=1:N ksquare(m,n)=k(m)*k(m)+k(n)*k(n); end end figure; [X,Y]=meshgrid(1:1:N); % solving Cahn-Hilliard equation time_final=1000; time_output=100; time_current=0; count=0; count_output=time_output/time_inc; while time_current<time_final c_hat=fft2(c,N,N); P=log(c)-log(1-c)+omega*(1-2*c); P_hat=fft2(P,N,N); % update concentration for m=1:N for n=1:N c_hat_next(m,n)=(real(c_hat(m,n))-ksquare(m,n)*real(P_hat(m,n))*time_inc)/(1+2*ksquare(m,n)^2*time_inc); c_hat_next(m,n)=c_hat_next(m,n)+i*(imag(c_hat(m,n))-ksquare(m,n)*imag(P_hat(m,n))*time_inc)/(1+2*ksquare(m,n)^2*time_inc); end end c_next=ifft2(c_hat_next,N,N); c_next=real(c_next); if max(max(abs(c_next-c)))>flag_refine_tolerance time_inc=time_inc*0.1; disp('reduce the time increment'); flag_refine_inc=1; flag_count=count; else count=count+1; c=c_next; c_average=sum(sum(c))/N/N; if mod(count,count_output)==0 [time_current c_average] meshc(X,Y,real(c)); %contour(X,Y,real(c)); axis square xlabel('x/b','FontSize',24,'FontName','Times New Roman'); ylabel('y/b','FontSize',24,'FontName','Times New Roman'); zlabel('C','FontSize',24,'FontName','Times New Roman'); axis([1 N 1 N 0 1]); set(gca,'FontSize',16,'FontName','Times New Roman','LineWidth',2); str1=sprintf('time: %7.4f',time_current); text(0.75*N,N,1,str1,'FontSize',16,'FontName','Times New Roman'); frame=getframe(gca); aviobj=addframe(aviobj,frame); end if (flag_refine_inc==1) & (abs(flag_count-count)>4) disp('increase the time increment'); time_inc=time_inc*10; flag_refine_inc=0; end time_current=time_current+time_inc; end end aviobj=close(aviobj); toc
Stress-Assisted Microstructural Evolution
There is nothing special about the role of stress field in the microstructural evolution. What we need to do is to modify the free energy formulation by including the elastic strain energy. For example, if we consider the curvature-driven depositional growth of a solid-liquid interface, together with the effect of elastic stress field, the governing equation in Eq. (6.9) becomes
MSE 612 (Fall 2012) 129
2hL h w
t
, (6.27)
where w is the change of the strain energy density on the solid surface due to the surface roughness.
Similarly, if we consider the surface-diffusion-driven morphological change of a solid-liquid interface, together with the effect of elastic stress field, the governing equation in Eq. (6.16) becomes
2 2hM w h
t
. (6.28)
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6.3 Cavitation in High Temperature Alloys Observations of the creep fracture process have revealed that cavities nucleate mainly on grain boundaries that are normal to tensile loading. These cavities grow under the influence of high-temperature mechanisms such as grain boundary diffusion, grain boundary sliding, and creep. They then coalesce to form grain boundary microcracks, and finally intergranular fracture occurs when these microcracks link up to form a macroscopic crack. It should be noted that because cavitation occurs in an inhomogeneous, localized manner, it would induce localized stresses which, in turn, modify the resultant stress field and the stress analysis during creep fracture is a non-trivial problem. Modeling of intergranular creep fracture requires consideration at all relevant length scales.
At the atomic/molecular level, cavity nucleation can start at the triple points, the grain boundary ledges, or the second phase particles located at the grain boundary where stress concentrations exist. However, it remains a great challenge to quantify the nucleation directly from atomistic simulations and cavity nucleation is often described by a phenomenological model.
At the microscopic level, the cavity growth results from both the diffusion of atoms from the cavity surfaces into the grain boundary and the creep deformation of the surrounding grain materials. In addition, grain boundary sliding occurs in the presence of shear stress along the grain boundary, the material inside the grains is subjected to both elastic and creep deformations, and coalescence of cavities can lead to a microcrack on the grain boundary.
At the mesoscopic level, grain aggregate is considered and the microcracks can link up to form a macroscopic crack to result in fracture.
Fig. 6.6: A smear-out model of grain boundary cavitation and representative finite element simulation results (taken from Yu et al., Eng. Fract. Mech., in press, 2012).
Here we briefly introduce the Tvergaard-van der Giessen smear-out model for the grain boundary cavitation process. in which the distribution of cavities on each grain boundary (Fig. 6.6a) was replaced by a continuous varying separation as shown in Fig. 6.6b. Adopting this
MSE 612 (Fall 2012) 131
simplification, the separation of grain boundaries resulting from cavitation can be analyzed using the continuum mechanics approach. Also, the cavity on the grain boundary can generally be characterized by its radius a, half-spacing b, and spherical-caps shape parameter h that is determined by the cavity tip angle . The rate of the normal separation is
2 3
2n
V Vbu
b b
. (6.29)
The rate of void spacing change, b , is related to the strain rates in adjacent grains and the cavity nucleation rate, given by
1 1
2 2I II
b N
b N
, (6.30)
where I and II are in-plane logarithmic strain rates on the grain boundary and N is the cavity
density. Cavity nucleation occurs at the atomic scale and depends on both the grain boundary microstructure and the local stress. Different mechanisms for the cavity nucleation on the grain boundary at high temperatures have been proposed. However, there exists no unified theory to describe the complex process of cavity nucleation. A phenomenological model proposed by Tvergaard (JMPS, 1984) is illustrated here. In this model, the cavity nucleation rate is proportional to the effective creep strain-rate, c
e , and normal stress, n , on the grain boundary,
2
0
cnn eN F
for 0n , (6.31)
where nF is a material parameter and 0 is a normalization factor. Cavity growth results from
both diffusion of atoms from cavity surfaces to the grain boundary layer and the creep deformation of the surrounding grains. Hence, the volumetric cavity growth rate, V , is
1 2V V V , (6.32)
where 1V and 2V represent the contributions of atom diffusion and creep, respectively, and have
been derived previously. See van der Giessen and Tvergaard (Acta Mater, 1996) for details.
Fig. 6.7: In addition to the diffusive cavity growth, the creep plasticity in the surrounding grains contributes to the microstructural evolution (Needleman and Rice, Acta Met., 1980).
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A length scale arises from the competition between surface/grain boundary diffusion and creep plasticity in the surrounding grains. This characteristic length, L, is a function of the grain boundary diffusivity, effective stress, and effective creep strain-rate which, in turn, results in the stress- and temperature-dependences of L. This Needleman-Rice length scale (Acta Met., 1980) is given by
3L D , (6.33)
where is the remotely applied tensile stress, is the associated creep strain rate, and
b bD D kT ( b bD =grain boundary diffusion coefficient, =atomic volume, kT =energy per
atom measure of temperature). Specifically, L decreases with increasing stress and temperature. In general, the cavity growth is dominated by creep for large values of a/L, while the atom diffusion is more important for small values of a/L (e.g., a/L < 0.1).
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6.4 Thin Film Mechanics Mechanics of thin films and other heterogeneous structures plays a critical role in modern high-technology applications, such as the integrated structures in electronics. It also helps understand the processing and properties of nanostructured materials at reduced dimensions. Consequently, more and more efforts in mechanics of materials have been devoted to the above two lines of research. The students are recommended to read “Thin Film Materials” (by L.B. Freund and S. Suresh, Cambridge University Press, 2003) for extended review.
Fig. 6.8: TEM image of a interconnect structure.
Stoney equation and residual stress measurements
Residual stresses arise due to the mismatch in thermal expansion coefficients or lattice constants, or are a by-product of the materials processing, or are deliberately introduced to tune the functional properties. One can determine the residual stress from the measurement of the curvature change of the film-on-substrate system, based on the Stoney equation,
2
6
s s
f
M h , (6.34)
where the residual stress in the film is f, the curvature is , the substrate thickness is sh , and sM
is the biaxial elastic modulus of the substrate. For instance, if the residual stress arises from the lattice mismatch m , then f f mf M h .
Consequences of residual stresses
Residual stresses in the integrated, structured thin films can lead to a variety of materials phenomena and problems.
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Mechanical Failures: thin film delamination, debonding, buckling, wrinkling, etc. Clearly it is necessary to measure the interface properties, especially toughness.
Defects and Stress Relaxation: Dislocations can form to reduce the residual stress. Other means such as diffusional mass transfer can also lead to stress relaxation.
Electromechanical Failures: electromigration (electron motion induces mass transfer), voiding, stress-assisted diffusion, etc.
Heteroepitaxial Thin Film Growth: surface roughening, quantum dot growth, etc. This is also related to the stress effects in the growth of low-dimensional nanostructures.