application of finite element and constitutive models
TRANSCRIPT
1
NOTES FOR SHORT COURSE
Application of Finite Element and Constitutive Models
SOLID, STRUCTURE AND
SOIL-STRUCTURE INTERACTION:
STATIC, DYNAMIC, CREEP
THERMAL ANALYSES
By
Chandrakant S. Desai
2012
Tucson, AZ, USA
2
PREFACE
These notes present descriptions of static and dynamic finite element method, nonlinear
techniques used, various constitutive models (elastic, plastic, creep, thermal, and disturbance-
softening , procedures for determination of parameters for the constitutive models, parameters
for typical materials and interfaces, and program features for the DSC-SST2D code.
The DSC-SST2D based on the finite element method with the DSC model is considered
to be a general purpose finite element code for analysis of a wide range of problems involving
solids and interfaces or joints, subjected to thermomechanical static, cyclic (repetitive) and
dynamic loadings. The code permits a range of constitutive models for elastic, plastic, and creep
responses, microcracking leading to fracture, and fatigue and softening. As a result, the code can
be used for solutions in civil and geotechnical, mechanical and aerospace engineering,
engineering mechanics, and electronic packaging systems.
Although these notes mainly cover static problems, other codes are available for dynamic
two-dimensional analysis (DSC-DYN2D) and for dynamic three-dimensional analysis (DSC-
SST3D). Their brief descriptions are given below:
I. DSC-SST2D: Two-dimensional Computer code for Static, Dynamic, Creep and Thermal
analysis-Solid, Structures, and Soil-Structure Problems
1. Part I: Manual for Technical Background. The Notes for the Short Course herein have
been adopted from this manual.
2. Part II: User’s Guide
3. Part III: Examples Problems-Verifications and Applications
II. DSC-DYN2D: Two-Dimensional code for Dynamic and Static Analysis-Dry and
Saturated (Porous) Materials including Liquefaction
1. Part I: Manual for Technical Background
2. Part II: User’s Guide
3. Part III: Examples Problems-Verifications and Applications
III. DSC-SST3D: Three-Dimensional Computer code for Static and Coupled Consolidation
and Dynamic Analysis-Solid (Porous), Structures and Soil-Structure Problems:
1. Part I: Manual for Technical Background
2. Part II: User’s Guide
3. Part III: Examples Problems-Verifications and Applications
This manual (Part I) presents the descriptions of the DSC-SST2D code. The other two are
available in separate reports.
3
TABLE OF CONTENTS
TOPIC Page
Preface .......................................................................................................................................................... 2
Table of Contents …………………………………………………………………………..,,………… 3
Introduction ................................................................................................................................................. 6
Finite Element Method .............................................................................................................................. 7
Computational Algorithm ............................................................................................................... 8
Element Library ............................................................................................................................ 10
Constitutive Models ................................................................................................................................. 14
Nonlinear Analysis ........................................................................................................................ 16
Drift Correction ......................................................................................................................... 17
Continuous Hardening and HISS Models ................................................................................. 17
Program Features .................................................................................................................................... 19
Applied Forces ......................................................................................................................... 19
Initial or in situ Stresses ........................................................................................................... 20
Simulation of Sequences .............................................................................................................. 21
Addition of Material, or Placement or Embankment ................................................................ 21
Removal of Material or Excavation .......................................................................................... 24
Removal of Liquid (Water) or Dewatering ............................................................................... 24
Support Systems........................................................................................................................ 26
Mesh Change Option ................................................................................................................ 28
Boundary Conditions ................................................................................................................ 28
Dynamic Analysis ..................................................................................................................................... 28
Newmark Method ..................................................................................................................... 30
Wilson -Method ...................................................................................................................... 30
Mass Matrix ................................................................................................................................... 31
Absorbing Boundaries ................................................................................................................... 31
Cyclic or Repetitive Loading ......................................................................................................... 31
Creep Behavior .............................................................................................................................. 32
Material Parameters ................................................................................................................................ 32
Organization of Computer Program ...................................................................................................... 32
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Appendix I: Constitutive Models .......................................................................................................... 33
Linear and Nonlinear Elastic Models ............................................................................................ 33
Linear Elastic Model……………………………………………………………………….33
Nonlinear Elastic Models ........................................................................................................ 33
Plasticity Models ..................................................................................................................... 34
Von Mises .................................................................................................................. 35
Mohr-Coulomb ........................................................................................................... 35
Drucker Prager ........................................................................................................... 35
Modified Cam-Clay ................................................................................................... 35
Cap Model .................................................................................................................. 37
Hoek-Brown Model ................................................................................................... 39
Hierarchical Single Surface (HISS) Models .............................................................. 39
Initial Values of and ...................................................................................................... 41
Interface/Joints Element………………………………………………………………….43
Cohesive and Tensile Strengths ........................................................................................... 44
Creep Models………………………………………………………………………………44
Viscoelasticplatic (vep) or Perzyna Model ............................................................................. 46
Multicomponent DSC or Overlay Models .............................................................................. 46
Specializations of Overlay Model ........................................................................................... 50
Number of Overlays and Thicknesses ............................................................................... 51
Layered Systems with Different Material Properties .............................................................. 51
Disturbance (Disturbed State Concept – DSC) Model: Microcracking,
Degradation and Softening .................................................................................................... 53
Speciaqlizations……………………………………………………………………………55
Thermal or Initial Strains ........................................................................................................................ 55
Elastic Behavior ...................................................................................................................... 55
Plane Stress ....................................................................................................................... 56
Plain Strain ........................................................................................................................ 56
Axisymmetric .................................................................................................................... 56
Thermoplastic Behavior .......................................................................................................... 57
Thermoviscoplastic Behavior .................................................................................................. 58
DSC Model .............................................................................................................................. 61
Cyclic or Repetitive Loading .................................................................................................................... 61
Unloading ............................................................................................................................... 63
Reloading ................................................................................................................................ 66
Cyclic Hardening ..................................................................................................................... 69
Appendix II: Elasto-plastic Equations .................................................................................................. 72
Appendix III: Drift Correction and DSC Computer Algorithm ........................................................ 74
DSC Computer Algorithm ....................................................................................................... 75
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Appendix IV: Determination of Constants for Various Models ......................................................... 77
Elastic Constants ......................................................................................................................................... 77
Plasticity Constants ..................................................................................................................................... 79
Ultimate: , ........................................................................................................................... 79
Phase Change ........................................................................................................................... 81
Hardening ................................................................................................................................ 84
Nonassociative......................................................................................................................... 84
Cohesive and Tensile Strengths ........................................................................................ 86
Computer Code to Find Constants for 0- and 1-Models ..................................................................... 87
Viscoplastic and Creep Models, 0 + vp .................................................................................... 88
Mechanics of Viscoplastic Solution ........................................................................................ 88
Elastoviscoplastic: Overlay Models ........................................................................................ 92
Disturbance Model .................................................................................................................. 93
Cyclic Loading and Liquefaction............................................................................................................. 96
Cyclic or Repetitive Loadings, Unloading and Reloading ...................................................... 96
Initial Conditions ..................................................................................................................................... 98
Environmental Effects .............................................................................................................................. 98
Interface/Joint Behavior ........................................................................................................................... 98
Material Constants .................................................................................................................................... 99
Implementation and Applications ........................................................................................................... 99
Material Constants for Typical Materials: Soils, Rock, Concrete, Solders ................................ 101-107
References ........................................................................................................................................ 108-116
PART II: USER'S GUIDE ..........................................................................................................................
PART III: EXAMPLE PROBLEMS: VERIFICATIONS AND APPLICATIONS .............................
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INTRODUCTION, FINITE ELEMENT METHOD,
CONSTITUTIVE MODELS, CONSTRUCTION SEQUENCES
INTRODUCTION
Nonlinear behavior of materials involving solids and interfaces can arise due to material
or geometric nonlinearity, or both. Material nonlinearity under mechanical, thermal and other
environmental loadings, can be due to several factors such as initial state of stress, stress path
dependent response, elastic, plastic and creep strains, change in the physical state defined by
change in the density, void ratio or water content, plastic yielding or hardening, microcracking
and damage leading to softening behavior.
Problems in solid and geomechanics can involve both types of nonlinearities. However,
in the current computer procedures, only material nonlinearity is considered with two-
dimensional (2-D) (plane stress, plane strain and axisymmetric ) and three-dimensional (3-D)
idealizations. The procedures and codes can be used for stress-deformation analysis of a wide
range of problems in solid, structural, geotechnical, and mechanical engineering and electronic
packaging involving “solid” materials, interfaces and joints. The loading can be static, cyclic and
repetitive and dynamic, and the material response can include elastic, plastic and creep
deformations, microcracking and damage leading to softening or degradation, fatigue failure, and
in microstructural instabilities like liquefaction. Typical examples are also presented. Part III of
the manual covers range of applications.
Realistic solution procedures for engineering problems require appropriate provision for
initial conditions, non-homogeneities and interaction effects. Conventional methods based on
classical theories of elasticity and plasticity may not be capable to handle the above factors.
7
Hence, the approach should be to adopt improved but simplified models that are capable to allow
for factors important for a given application. Very often it becomes necessary to resort to
numerical techniques so as to allow for these factors; the finite element method (FEM) is one of
the most powerful methods to solve engineering problems, and is used herein. The FEM code
involves the unified and general approach called the disturbed state concept (DSC), which allows
for hierarchical adoption of a wide range of constitutive models: elastic, elasto-plastic,
continuous yielding, elastoviscoplastic, and disturbance (damage), depending upon the need of
the user for specific application.
FINITE ELEMENT METHOD
In this part of the report, two-dimensional static idealization is considered. Two- and
three-dimensional static and dynamic analyses are covered in other manuals.
The finite element method has been discussed in detail in books such as Desai and Abel
(1972) and Desai (1979). The method presented here is based on the displacement approach for
2-D problems, which has been adopted in the computer code. For two-dimensional typical
element (Fig. 1), the displacement components at any point are written as
q N = u (1)
where {u}T = [u v] is the vector of displacement components u and v at a point in the x- and y-
directions, respectively, [N} is the matrix of interpolation functions, {q}T = [u1 v1 u2 v2 … un vn]
is the nodal displacement vector , and n denotes the number of nodes.
The strain-displacement and stress-strain relations are given respectively by
q B = (2)
and
8
C = (3)
where {} and {} are strain and stress vectors, respectively, [B] is the strain-displacement
transformation matrix, and [C] is the constitutive matrix.
By using the principle of minimum potential energy, the element equilibrium equations
are derived and then expressed in the incremental form as
Q = q k t (4)
where [k1] is the tangent element stiffness matrix, {Q} is the element nodal load vector, {Qr} is
the vector of unbalanced or correction loads, and denotes increment. The terms in Eq. (4) can
be expressed as
V d B C B = k t
T
V
t (5)
and
Sd T N + V d X N = Q T
S
T
V 1
(6)
and
dVBQ r
T
r (7)
in which X is the body force vector, T is the surface traction vector, r is the unbalanced
or correction stress vector, V is the volume of the element, and S1 is the portion of surface on
which surface loads are prescribed. Equations (5) and (6) are usually integrated numerically by
using Gauss quadrature methods.
Computational Algorithm
A nonlinear problem is analyzed as a series of “piecewise” problems by using
incremental techniques in which the tangent constitutive matrix {C1] is updated at each load
9
(-1,-1)
(-1,-1) (1,1)
(1,-1)
t
s
Local Coordinates
4
3
2 1
t
s
Y
X
Global Coordinates
(b)4-Node Isoparametric Element
(-1,-1)
(-1,-1) (1,1)
(1,-1)
t
s
Local Coordinates
8
1
7
1
6 5
1
4
3
1 1
1
2
1
t
s
Y
X
Global Coordinates
(a)8-Node Isoparametric Element
Figure 1. Two-dimensional Isoparametric Solid Elements
10
increment, Fig. 2. A mixed procedure (Figure 2) which combines both incremental and iterative
techniques has been adopted together with improved drift correction procedure(s). In this
procedure, after applying each load increment, iterations are performed until convergence is
reached. The convergence criterion employed is based on the ratio of the norm of unbalanced
load and sum of the norm of total load and norm of equilibrating load; details are given
elsewhere (Desai, et al., 1991).
Element Library
The computer program has the provision for the following types of elements:
(i) Solid elements
(ii) Interface/joint, and
(iii) Bar elements.
(i) Solid Elements
Either 4-noded or 8-noded isoparametric finite elements as shown in Fig. 1, or infinite
elements (not operational at this time) (Damajanic and Owen, 1984) as shown in Fig. 3, can be
used. Equations (5) to (7) are used to compute element stiffness matrix and nodal load vector,
respectively. The Gauss quadrature process allows 2 or 3 point integration rules, i.e., total 4 or 9
integration points.
(ii) Joint/Interface Elements
These elements are represented by a thin layer solid element (Desai, et al., 1984; Sharma
and Desai, 1992), or zero thickness Goodman element (Goodman, et al., 1968). They can be
either 4-noded or 6-noded elements (Fig. 4) corresponding to 4-noded or 8-noded solid elements.
The shear and normal responses found from special laboratory tests are used to define the
element stiffness matrix. The constitutive laws, discussed later, are written in terms of shear
11
Figure 2. Schematic of Incremental and Iterative Technique
Load
Q1
Q2
Q3
Displacement
12
0
Y
X
6
5
4
3
2
1
Global coordinate
s
t
Local coordinate
(a) Biquadratic singly infinite element
0
Y
X
3
2
1
Global coordinate
s
t
Local coordinate
(b) Biquadratic doubly infinite element
Figure 3. Two-Dimensional Infinite Elements
13
Figure 4. Joint/Interface Elements
y
Two-Dimensional
x
t
Body 2
Body 1
(8-noded)
Thin-Layer
Element
(4- or 6-
noded)
Body 1
Body 2
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stress, , and normal stress, n. For the thin-layer solid element, the parametric study shows that
the ratio of thickness of interface element to its width of the order of about 0.01 yield satisfactory
simulation of the interface response simulated by using the thin-layer element with finite
thickness.
(iii) Bar Elements
Two types of bar elements, 2-noded linear, and 3-noded quadrilateral elements (Fig. 5),
have been used and provide compatibility with solid and joint elements. The element stiffness
matrix and computation of axial stress are given by Desai (1979) and Lightner and Desai (1979).
CONSTITUTIVE MODELS
A number of material models have been implemented in this program. They are:
(i) Linear elastic,
(ii) Nonlinear elastic (variable moduli or hyperbolic simulation),
(iii) Elasto-plastic conventional (von Mises, Drucker-Prager, Mohr-Coulomb, and Hoek-
Brown),
(iv) Elasto-plastic continuous yielding or hardening (critical state, cap),
(v) Hierarchical Single Surface (HISS) continuous yielding (0 and 1)
(vi) Viscoelastic plastic, and
(vii) Disturbed State Concept (DSC) models; details of this general and unified approach,
from which almost all of the above models can be derived as special cases, are given
later.
15
Figure 5. Bar Elements
Y
2
1
l
X
2-node bar element
2
Y
3
1
l
X
3-node bar element
16
Each of these categories may be used for solid, structural and geologic materials and
interfaces/joints, depending upon the material behavior and user’s judgment. However, the most
realistic models are considered to be those based on plasticity or viscoplasticity, in particular the
HISS models, as they include other plasticity models as special cases, and provide a number of
advantages and simplifications (Desai, et al., 1986 and Desai, 2001). The disturbed state concept
(DSC) allows for the above models as special cases, and includes microcracking, damage and
degradation or softening and stiffening or healing (Desai, 1994, 1995, 2001; Desai and Toth,
1996); stiffening is not included in this code.
Descriptions of the above models are given in Appendices I and IV.
Nonlinear Analysis
A nonlinear problem is solved by using incremental-iterative procedures with required
iterative (drift) correction and convergence schemes. The basic incremental stress-strain
equations are given by
d C = d t (8)
where {d and {d} = incremental stress and strain vectors, respectively, and [C1] is the tangent
constitutive matrix. In the case of piecewise linear approximation to nonlinear elastic behavior,
[C1] = eC1 will be composed of Et and t for solids, or knt and kst for interfaces and joints. For
elasto-plastic behavior
C C = C pt
ett (9)
where pC1 = tangent plasticity matrix (Appendix II).
The elastoplastic response forms a part of the creep or elastoviscoplastic and disturbance
(microcracking and softening) models in the DSC. Details of the models, elastoplastic, creep and
disturbance, and associated equations are given in Appendix I, together with the incorporation of
17
thermal and cyclic hardening effects. In all cases, a drift correction procedure is used with
respect to the drift of the yield surface during incremental loading. A brief description of the drift
correction procedure is given below.
Drift Correction: During each increment of loading, the stress must lie on or within the yield
surface (assuming unloading is elastic). If the increments are not very small, the stress state at the
end of an increment may not lie on the relevant yield surface leading to the problem of the drift
of the currently computed stress as shown in Figure 6. The initial stress state {A} at point A lies
on the previous yield surface, F ({A}, A) = 0, where is the hardening parameter (Appendix
I). During the next increment, yielding occurs and the state of stress moves to point B. The new
yield surface is given by F ({B}, B) = 0. Owing to the tendency to drift, the stress state
represented by point B does not necessarily lie on this new yield surface, Figure 6. This
discrepancy can be cumulative and, therefore, it is important to ensure that the stresses and the
hardening parameter, , are modified so as to lie on the yield surface.
Potts and Gens (1985) examined five different methods for drift correction. They
considered subincrements of strains for each increment, and concluded that the method which
considered hardening during drift correction gave improved results. This scheme is modified and
is described in Appendix III; it is incorporated in the program. Also incorporated is a modified
version of the scheme proposed by Ortiz and Simo (1986). Details of the modified schemes are
given by Desai and Wathugala (1987), Wathugala and Desai (1993).
Continuous Hardening and HISS Models
The classical plasticity models such as von Mises, Mohr-Coulomb and Drucker-Prager do
not allow adequately for the volumetric response, and for the existence of yielding before the
18
Figure 6. Schematic Showing Yield Surface Drift
J2D
J1
F({B},B)=0 F({A},A)=0
Drift
B
A
19
ultimate (failure) surface is reached. Hence, their use is often limited for evaluation of failure or
ultimate loads.
In the critical state and cap models, the continuous hardening or yielding parameter is
dependent only on the volumetric plastic strain, p
v . However, in the hierarchical single surface
(HISS) models, hardening is dependent on both volumetric and deviatoric plastic strain
trajectories, v and D, respectively. These models, including the viscoplastic and general
Disturbed State Concept (DSC), are described in Appendix I.
The critical state and cap models allow for yielding before failure, but do not allow for
(a) hardening due to plastic shear strains,
(b) possibility of dilation before peak stress,
(c) different strengths under different stress paths (e.g., compression and extension),
(d) nonassociative behavior for frictional materials, and
(e) involve multiple (two) yield surfaces, which can cause computational difficulties.
The HISS models that involve single continuous yield surface, removes the above
limitations, are considered to be general and more powerful. A perspective and comparison of
the HISS model with such other models as critical state, cap and Lade are given by Desai, et al.,
(1986), Desai and Hashmi (1989), Desai (1992), Desai (1994), Desai (2001).
PROGRAM (DSC-SST2D) FEATURES
The computer program has the following capabilities:
(i) Applied Forces
The program allows for three types of loads, as static, repetitive and dynamic:
a) Extenal loads – point loads and surface loads,
b) Prescribed displacements, and
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c) Prescribed temperature.
External Loads: Point loads, constant or time dependent, are prescribed at nodes,
whereas the surface loads (constant or time dependent) in the form of distributed traction or
pressure acting on the element sides, are converted to the equivalent nodal loads in the program.
Thermal Loads: Temperature increments or time-dependent temperature is applied at
nodes.
For a linear elastic analysis, total load or temperature may be applied in a single
increment, but in the case of nonlinear analysis, the total load or temperature is applied in several
increments.
Displacements: The program has an option of prescribed displacements, at nodes.
Total displacements at the nodes may be applied in a single increment for linear elastic
analysis, whereas in the case of nonlinear analysis, they are applied in several increments.
(ii) Initial or in situ Stresses
A number of options are available for computing the in situ stresses (see Part II: User’s
Guide). For example,
a) Prescribed in situ stress: The in situ stress is calculated using the expressions
(Chowdhury, 1978)
so c n i y s K =
K =
n i sK + 1 y =
oy x
yox
2oy
(10)
21
where x, y, and xy are in situ horizontal, vertical, and shear stresses, respectively, is the unit
weight of soil, Ko is the in situ ratio (x/y), y is the depth to the point of stress, and is the
slope of the side of the structure or ground surface (Figure 7).
b) Computed in situ Stresses: A finite element analysis of a soil mass is carried out for
body forces only, assuming linear elastic behavior. The computed vertical stress y is kept the
same, and the horizontal stress x and shear stress xy are computed as
n i sK + 1
so c n i s =
K =
2o
xy x
yox
(11)
For horizontal surface, xy = 0.
Simulation of Sequences
(iii) Addition of Material, or Placement Embankment
Simulation of addition of materials, which is called embankment, or placement in the
sequential construction procedure is shown in Figure 8. For each layer (lift) of embankment
placed, the equivalent nodal forces due to gravity are computed. The Young’s modulus, E, of the
material in the added lift is set to a very small value (about one percent of initial E), which
simulates a very “weak” material. The incremental displacements and stresses are computed
during each lift cycle and are added to those from the previous cycle; iterations are performed (if
necessary) to obtain the equilibrium for each lift. The displacements of the new surface of the
embankment are set to zero. The horizontal stress in the newly placed lift is calculated as the
vertical stress times the in situ stress ratio, Ko.
Note that in the program, the sign of the element material numbers in a newly placed lift
are set to negative, which assigns small value of Young’s modulus to those elements. At the end
22
xy
xy
x
y
h
y v
(a) (b)
Figure 7. Initial Stresses for Inclined Surface
23
Figure 8. Addition of Materials or Sequential Construction-
Embankment
{o}
Initial Stresses
{i}={o}+{i}
Final Lift
{1}
First Lift
Stress Free Surface
24
of computations for the lift when equilibrium is reached, the sign of the element material
numbers is changed back to positive.
(iv) Removal of Material or Excavation
Figure 9 shows schematic of the simulation of excavation process, which is similar to
cut-outs in plates, and involves removal of material(s). The elements to be excavated (removed)
for each lift are deleted from the system and iterations are performed (if necessary) until
equilibrium is obtained. This will result in a “stress free” excavated surface.
The two key features of the program are:
a) Excavated elements are deleted from the initial and changing mesh.
b) Stress-free surface is established by applying equal and opposite forces on the
excavated surface and by satisfying the equilibrium equation, Eq. (4).
The above process was proposed by Goodman and Brown (1963) and Brown and King (1966).
(v) Removal of Liquid (Water) or Dewatering, Fig. 10
Dewatering causes compression or consolidation and can be modeled by using the
coupled-consolidation theory. However, in order to provide a simpler and economical
formulation, dewatering is approximated in the program by assuming uncoupled and
instantaneous response. The main effect accounted for is the increase in effective stress due to
change in the unit weight of the soil in the dewatered elements. This increase is equal to the body
force due to the weight of water within each of the elements which is dewatered. The equivalent
nodal forces are given by:
V d N = F T
W
V
(12)
where {F} is the element nodal force vector and w is the unit weight of water.
25
Figure 9. Removal of Materials or Sequential Construction- Excavation
{o} Initial Stresses
{i}={o}+{i} Final Lift
Stress Free Surface
Nodal Point
Forces
{1} First Lift
26
Note that Eq. (12) applies only to elements which were submerged earlier and are now above the
water-table due to the dewatering. Figure 10 shows the dewatering in which only elements 1, 2
and 3 have body force loads due to dewatering, and the remaining elements are affected
indirectly by the loading from these three elements.
(vi) Support Systems
Structural Supports or Tie-Backs: Installation of support system such as tie-backs, Fig.
11, can be considered similar to the prestressing of concrete beams, and introduces compressive
stresses to counteract extension and tensile stresses. The installation of tie-backs involves four
simulation steps: drilling/boring a hole (at an angle to the horizontal), placing the tie-back,
grouting the tie-back, and then tensioning the tie-back to provide the design compressive stress.
A tie-back usually consists of either steel cables or steel reinforcing rods or other
structural supports. In the case of geotechnical systems, only the last portion of the tie-back is
grouted to form an anchor, and the rest of it is usually encased in a sheath to prevent transfer of
load to the excavated face.
In the simulation of the tie-backs installation in the FEM procedure, the first two steps are
not considered, and the procedure followed is
1. Apply a force along the direction of the tie-back equal and opposite to the tension
force in the tie-back.
2. Solve for new displacements and stresses.
3. Add the bar elements which simulate the tie-backs.
4. Set the bar elements stresses to the initial tension in the tie-back.
The order of these steps may not follow the actual construction procedure. In the
construction procedure, the bar is placed first before the tensioning force is applied. If this is
27
Figure 10. Dewatering
Initial Water Level
Final Water Level
1 2 3
4 5 6
9 7 8
28
followed in the numerical procedure, bar elements will resist the tensioning forces, which is not
correct. The wrong and correct sequences are illustrated in Fig. 11.
(vii) Mesh Change Option
During any increment of the loading, the mesh can be changed, i.e., some elements can
be added or deleted, or some nodes added or deleted and/or material number of elements is
changed. This option is used to simulate embankment construction and excavation. The material
number may be changed in the case of dewatering.
(viii) Boundary Conditions
The prescribed boundary conditions (e.g., fixity) are imposed in such a manner as to
minimize the number of equations to be solved. This is achieved by not formulating equations
corresponding to degrees-of-freedom at nodal points where displacements are zero, because of
the boundary conditions.
DYNAMIC ANALYSIS
The finite element equations for dynamic analysis are given by
t Q = q K + q C + q M (13)
Where [M], [C ] and [K] are the mass, damping and stiffnesses matrices, respectively, {q} is the
vector of nodal displacements, {Q(t)} is the vector of time dependent nodal forces and the
overdot denotes time derivative.
The mass matrix can be consistent when it is evaluated from the expression resulting
from energy considerations, while it is evaluated as lumped when the mass is lumped at nodes
and appears only on the diagonals of the matrix (Desai and Abel, 1972).
Details of the frequency and time domain solutions for the dynamic equations are given
in Desai and Abel (1972) or in other texts on the finite element method. For the time domain
29
Figure 11. Schematic of Supports or Tie Backs
2P
Wrong Sequence Correct Sequence
2P
Step 1
Step 2
2P
Physical Problem
P
P
30
analysis, Equations 13 are integrated in the time domain, particularly for nonlinear analysis, by
using various time integration schemes such as Euler, Newmark Method, and Wilson’s -
Method. In the present code, Newmark and Wilson’s -methods are used. At time tn+1 = tn + t,
where t is the time step and tn is the previous time level at which quantities are known, Eq. (13),
are derived as
Q = q K *
1 + n
* (14)
where (i) for Newmark Method
K + C t
+ M t
1 = K 2
*
(15a)
q t 1 - 2
+ q 1 - + q t
C +
q 1 - 2
1 +
t
q +
t
q M + Q = Q
nnn
n
n
2
n
1 + n
*
(15b)
in which , are integration parameters in the Newmark’s scheme. For conditional stability: 2
0.5.
(ii) for Wilson -Method
KCt
Mt
K
3
6*
2 (16a)
q 2
t + q 2 + q
t
3 C +
q 2 + q t
6 + q
t
6 M +
Q - Q + Q = Q
nnn
nnn2
n1 + nn
*
(16b)
in which is a parameter, usually taken as 1.4.
31
It is often difficult to define the damping matrix [C]. Hence, approximate procedures are
sometimes employed; in one such method, the damping matrix is expressed as (Clough and
Penzien, 1993):
M + K = C Mk (17)
where k and M are constants adopted by the user.
In the case of cyclic material behavior, the hysteretic damping is included through the
tangent stiffness matrix, [K*], and it may not be necessary to include the damping in the
analysis.
Mass Matrix
The code allows for two options: consistent mass and lumped mass. The consistent mass
matrix is fully populated and is derived from the energy formulation. In the case of lumped mass,
the matrix is diagonal and the tributary masses are lumped at the element nodes.
Absorbing Boundaries
In dynamic analysis, the waves radiating from a structure are reflected back in the mesh
(body) from the artificial or discretized end boundaries. This can cause spurious errors in the
computed response. One way to reduce this effect is to select the end boundaries far enough such
that the waves are absorbed by internal damping of the material. However, if the end boundaries
are close to the structure, it is desirable to provide for the absorption of the waves at the end
boundaries. In this code, the viscous damping model proposed by Lysmer and Kuhlemeyer
(1969) is implemented. Since this model is not very efficient in absorbing surface waves, it is
advisable to extend the (lateral) end boundaries as far as possible away from the structure.
Cyclic or Repetitive Loading
32
Details of cyclic or repetitive loading involving loading, unloading and reloading and
cyclic hardening are given in Appendix I.
Creep Behavior
The code includes the general DSC model which allows for microstructural changes
leading to fracture, failure or liquefaction and available continuum models such as elastic, plastic
and creep. For the latter, viscoelastic (ve), elasticviscoplastic (evp), and viscoelasticviscoplastic
(vevp) models can be used (Desai, 2001).
MATERIAL PARAMETERS
Appendix IV gives details for the determination of material constants for the above
models, based on appropriate laboratory tests for solids and interfaces/joints. It also gives details
of the determination of initial hardening and yield surface based on in situ stresses. Further
details for the HISS and DSC are also discussed in various references. Desai, et al. (1986), Desai
and Zhang (1987), Desai (1994, 1995, 2001), Desai, et al. (1995), Katti and Desai (1995), Desai
and Toth (1996), Desai, et al. (1997).
ORGANIZATION OF COMPUTER PROGRAM
The computer program consists of a main program and about 65 subroutines. The
program is coded in FORTRAN 90. All storage is allocated at the time of execution, and if
desired, the storage can be readily adjusted to the minimum required for the problem to be
analyzed.
33
APPENDIX I
CONSTITUTIVE MODELS
This Appendix describes various constitutive models including the unified Disturbed State
Concept (DSC).
Linear and Nonlinear Elastic Models
Linear Elastic Model
It is simplest, but probably the least applicable model for the realistic simulation of
nonlinear behavior. Its main use can be for preliminary studies, and for limited situations
involving mainly the linear behavior.
The constitutive relation for the linear elastic case is given by
C = e (I.1)
where [Cc] is the elastic constitutive matrix, which, for linear elastic and isotropic material, is a
function of two elastic constants, Young’s modulus, E, and Poisson’s ratio, [Desai and
Siriwardane (1984); Desai (2001)].
Nonlinear Elastic Models
In the computer program, hyperbolic model proposed by Kondner (1963) and formalized
by Kulhawy, et al. (1969) and Duncan and Chang (1970) is included to represent the nonlinear
elastic behavior of solid or soil materials. The tangent modulus, Et and tangent Poisson’s ratio,
t, are given by (Desai and Abel, 1972)
n i s 2 + so c c 2
- n i s- 1 R - 1
p p K = E
3
31f
2
a
3
n
at
(I.2)
and
A - 1
p / g o l F -G =
2
a3
t
34
where
n i s 2 + so c c 2
- n i s- 1 R - 1
p p K
- d = A
3
31f
a
3
n
a
31
(I.3)
1 and 3 are major and minor principal stresses, respectively, c is cohesion, is the angle of
internal friction, pa is atmospheric pressure, Rr is failure ratio, n is modulus exponent, R is
modulus number, and G, F and d are Poisson’s ratio parameters.
A total of eight parameters, K , n , Rf, c, , G, F and d are required to compute Et and
t. If the Poisson’s ratio is assumed constant, five parameters, K , n , Rf, c, and are required.
For the joint/interface elements, the normal stiffness, kn, is often assumed constant (with
a high value) for compressive normal stress and the shear stiffness, ks, is represented by the
hyperbolid model; it is expressed as (Kulhawy, et al., 1969; Desai, 1974).
n a t + c
R - 1
p K = k
ana
*f
2
a
n
n
w
*t s
*
(I.4)
where and n are shear and normal stresses, respectively, ca is adhesion, a is angle of interface
friction, w is unit weight of water and K*, n
* and *
fR are constants. Thus, for the interface, six
constants, kn, K*, n
* ca and , are required.
Plasticity Models
Various plasticity models with relevant yield criteria swhave been incorporated in the
program. The details of these criteria can be found in Desai and Siriwardane (1984), Desai
(1994), Desai, et al. (1986), Desai (1995, 2001). Here, the expressions for the yield criteria are
presented with description of parameters. Compressive stresses are assumed positive.
35
1. von Mises yield criterion
0 = - J = F yD 2 (I.5)
where J2D is the second invariant of deviatoric stress tensor, Sij, and y is the yield stress in
simple tension or compression.
2. Mohr-Coulomb yield criterion
0 = so c c - 3
n i s n i s - so c J + n i s
3
J - = F D 2
1
(I.6)
where J1 is the first invariant of the stress tensor, ij, is the angle of internal friction, c is
cohesion, and is Lode angle given by
6
6 -
J
J
2
3 3 n i s
3
1 =
5 . 1D 2
D 31 -
(I.7)
in which J3D is the third invariant of deviatoric stress tensor, Sij.
3. Drucker-Prager yield criterion
0 = k - J - J = F 1*
D 2 (I.8)
where * and k are material constants, e.g., for plane strain conditions:
n a t 12 + 9
c 3 = k ,
n a t 12 + 9
n a t =
22
* (I.9)
4. Modified Cam-clay model (Schofield and Wroth, 1968)
0 = 1 - p
p +
p p M
q = F
oo
2
2
(I.10)
where po is the semi-major size of the ellipse, Fig. I.1, M is the slope of critical state (CS) line,
and p = (1 + 23)/3 and 2D31 J 3 q . If the critical state line is considered similar to
the Mohr-Coulomb failure envelope (Eq. I.6), then
36
Figure I.1 Yield Locus for Critical State Model
dp
p
Critical State Line
Mcs
A
M
vp
J1/3
q=3J2D
2po
37
n i s n i s- so c 3
n i s3 = M
(I.11)
The size of ellipse, po, is an exponential function of the hardening parameter v = plastic
volumetric strain p
v :
p x e p = p vo co (I.12)
where pco = initial value of po,
= hardening constant =
oe1,
eo = initial void ratio,
= compression index,
= swelling index, and
v = trajectory or volumetric plastic strain.
5. Cap Model
The Cap model proposed by DiMaggio and Sandler (1971) has been adopted here. It
consists of a failure envelope (Ff) and a Cap surface (Fc), Figure I.2, the expressions for which
are
0 = J - p x e - - J = F 1
///D 2f (I.13)
and
0 = L - J + L - X - J R = F 1
22
D 22
c (I.14)
where /,
/ and
/ are material parameters, and R, X and L refer to the geometry of the cap
(Figure I.2) which are related as
L - p x e - R + L = X /// (I.15)
The yielding (hardening) defined by the cap is function of the plastic volumetric strain,
p
v , which is denoted by the hardening parameter = p
v . The hardening rule is expressed as
38
Figure I.2 Failure and Hardening Surfaces in Cap Model
Drucker-Prager Surface
Ff
Fc
Rb
J2D
J1 L Z X
von Mises Surface
39
Z+ W
- 1 n D
1 - = X
(I.16)
where D and W are material parameters, and Z is related to initial cap.
6. Hoek-Brown Model Yield Criterion (Fig. I.3)
Hoek and Brown (1980) proposed a yield (failure) criterion for rock masses as
2c3c31 s+ m - - = F (I.17)
where 1 and 3 are major and minor principal stresses, respectively, c is uniaxial compressive
strength of intact rock material, and m and s are constants which depend upon the properties of
rock and upon the extent to which it has been broken before being subjected to stresses 1 and
3. The constant m has a finite positive value which ranges from about 0.001 for highly
disturbed rock masses to about 25 for hard intact rock. The maximum value of s is unity for
intact rock, and the minimum value is zero for heavily jointed or broken rock in which tensile
strength is reduced to zero.
In terms of stress invariants, Eq. (I.17) can be written as
0 = s- 3
J m - J
3
n i s + so c m +
s o c J 4 = F c
1D 2
c
2D 2
(I.18)
where is the Lode angle (Eq. 1.7).
7. Hierarchical Single Surface (HISS) Models (Desai, et al., 1986; Desai, 1995, 2001)
Advantages of the HISS model with respect to the foregoing models are listed in Chapter
1.
The two hierarchical models, isotropic hardening with associative behavior (0 model)
and isotropic hardening with nonassociative behavior (1 model), have been incorporated in the
program.
40
Figure I.3 Hoek-Brown Model
3
1
c
Uniaxial
compresssion
Tension t
RELATIONSHIP BETWEEN
PRINCIPAL STRESS AT FAILURE
Minor principal stress or confirming pressure 3
Majo
r p
rin
cip
al
stre
ss
1 a
t fa
ilu
re
Compression Uniaxial tension
Triaxial
compresion
41
The continuous yield function (Fig. I.4) in the HISS plasticity Model:
0 = S - 1 p
J +
p
J - -
p
J = F r
m
a
1
2
a
1
n
2
a
D 2
(I.19)
where , , m and n are material parameters, pa is atmospheric pressure, Sr is the stress ratio
5.1
232
27DD JJ , and is a yield or hardening function defined as (Desai, et al. 1986; Desai and
Hashmi, 1989):
1 / a = 1 (I.20a)
or
b + b - 1 b - p x e b =
D43
D21
(I.20b)
in which a1, 1, and b1 to b4 are material constants, 2/1p
ij
p
ij dd is the trajectory of or
accumulated plastic strains, including the volumetric plastic strain (v) and deviatoric plastic
strain (D) trajectories: 2/1
D ;3/ p
ij
p
ij
p
vv EdEd ; where p
ijE = tensor of deviatoric
plastic strains.
The plastic potential function Q is expressed as
S - 1 p
J +
p
J - -
p
J = Q r
m
a
1
2
a
1
n
Q2
a
D 2
(I.21)
where
r - 1 - + = voQ (I.22)
in which /vvr , o is value of at the beginning of shear loading, and is a nonassociative
parameter. Equations I.19 and I.21 are used for the nonassociative (δ1) model.
Initial Values of and
Solution for in Eq. (I.19) leads to (Desai, et al., 1991; Desai, 1995, 2001)
42
Figure I.4 Basic, 0, and Nonassociative, 1, Models
F/
90 FQ
Ca
Ca
J1
J
2D
(a) 0 model
F/
Q/
Ca
Ca
J1
J
2D
(b) 1 model
F
Q
43
p
J
S - 1 J
J - =
a
1
n - 2
m 21
D 2
(I.23)
The initial value of = o is obtained by substituting J1, J2D, Sr based on the initial state
of stress and the knowledge of the material constants , , n and m = -0.5. Then, the initial value
of =o is found from Eq. (I-20) as the values of a1, 1, etc. are known. The value of the
volumetric plastic strain trajectory v is found from (Desai, et al., 1991):
Q
Q 3
Q
=
j ij i
2 / 1
i i
vo
(I.24)
where Q is defined based on the initial stresses. Then,
2
v
2
o D - = (I.25)
In the case of isotropic or hydrostatic initial stress:
Interface/Joint Elements
The yield function and plastic potential function for the two-dimensional case are given
by [Desai and Fishman (1991); Desai and Ma, 1992; Desai (1995)]
0
22
a
n
n
a
n
a pppF
(I.26)
0
22
a
n
n
a
n
Q
a pppQ
(I.27)
0 =
/ a = =
p
J =
D
oo
/ 1
0
a
1
n - 2
o
1
44
where and n are shear and normal stresses, respectively, n and are related to phase change
and ultimate envelope, and and Q are hardening parameters for 0 and 1, respectively. A
simple form of hardening function is given by
in which p
rdu and p
rdv are the incremental plastic shear and normal relative displacements,
respectively, a and b are hardening parameters, and Q is similar to that in Eq. (I.22).
Cohesive and Tensile Strengths
The yield function in the HISS model is extended to include cohesive or tensile strengths
by transforming the stress tensor as (Fig. I.4)
j ij i*
j i R + = (I.28a)
where R is related to cohesive or tensile strength. Details are given in Appendix IV.
Here, R can be found from empirical relations (see Appendix IV). It can also be found as
/ c= R a (I.28b)
where ac is the intercept along J2D-axis (intersection of J2D-axis and ultimate yield surface)
and is related to the cohesive strength, and is related to the slope of the ultimate yield envelope,
Fig. I.4.
Creep Models
Various models including elastoviscoplastic (evp) by Perzyna (1966) have been used to
characterize the creep behavior, Fig. I.5 (Cormeau, 1976; Owen and Hinton, 1980; Desai and
Zhang, 1987; Desai, et al., 1995; Samtani, et al., 1995). Overlay model for creep has been
proposed in (Zienkiewicz, et al., 1972; Pande, et al., 1977; Owen and Hinton, 1980). A general
v d =
v d + u d =
/ a =
prv
pr
2 pr
2 2 / 1
b
45
Figure I.5 Schematic of Strain-Time Response Under Constant Stress
t2
h
Primary creep
a
b
c
d
e
f
g
i
t1
Secondary creep
Tertiary creep
Failure
Permanent set
Time
Str
ain
0
46
approach called Multicomponent DSC (MDSC) has been proposed by Desai (2001). If the strains
in the component overlays, Fig. I.6, is assumed to be the same, the MDSC model specializes to
the overlay model.
Viscoelasticplatic (vep) or Perzyna Model
MDSC model contains various versions, such as elastic (e), viscoelastic (ve),
elastoviscoplastic (evp), and viscoelasticviscoplastic (vevp). Figure I.7(b) shows the general
rheological representation of MDSC model, from which various versions can be extracted
(Desai, 2001). For instance, the evp, Perzyna type model is shown in Fig. I.7(a), which is based
on the following expression for viscoplastic strain rate vector, vp :
} {
Q } { p v
(I.29)
N
oF
F
(I.30)
where is the fluidity parameter, is the flow function, N is the power law parameter, and Fo is
the reference value (e.g., yield stress, atmospheric constant, etc.). For associative plasticity, F
Q.
Multicomponent (MDSC) or Overlay Models
In the overlay model (Fig. I.6), the behavior of a material is assumed to be composed of
those of several overlays, each of which undergoes the same deformation (strain) and provides a
specific material characterization. The total stress field is obtained as the sum of different
contributions from each overlay. By introducing a suitable number of overlays and assigning
different material properties (parameters) to each, a variety of special models can be reproduced,
as shown below.
47
The typical strain-time (creep) relationship under constant stress is shown in Fig. I.5. The
instantaneous elastic strain. o-a, is followed by a primary creep, a-b, during which, if unloading
Figure I.6 Rheological Overly Model and Elasticviscoplastic Models
E
y
(a) Viscoplasticity
E1,1 E2,2 Ek,k
F1( y1) F2( y2) Fk (yk)
1,
N1
2,
N2 k,
Nk
(b) Overlay Model
48
occurs, an instantaneous elastic recovery, b-c, is followed by delayed elastic recovery, c-d. If the
load is continued beyond the primary creep range, secondary creep (b-e) begins which is
accompanied by irreversible deformations. Unloading at any time during b-e leaves a permanent
deformation or set (strain). On continued loading, tertiary creep begins leading to failure.
The overlay model for the two-dimensional problem is illustrated in Fig. I.6. Each
overlay can have different thicknesses and material properties. The overlays do not experience
relative motion, or they are “glued” together. Therefore, the overlay models exhibit the same
deformation under given loading.
In the MDSC (overlay) model developed here, a number of units are arranged in parallel,
Fig. I.7. This results in different stress fields, {j}, in each overlay (j) which contributes to the
total stress field {} according to the overlay thickness, tj; hence,
t = jj
k
1 =j
(I.31)
in which k is the total number of overlays in the model, and
1 = t j
k
1 =j
(I.32)
The equilibrium equations for a (finite) element become:
Q = V d t B jj
k
1 =j
T
V (I.33)
in which {Q} is the load vector.
From Eq. (I.33), the element stiffness matrix is obtained as
V d B t C B = k jj
k
1 =j
T
V
(I.34)
49
where [Cj] is the constitutive matrix. This matrix will be different for each overlay, according to
the material properties.
Figure I.7 The Overlay Model in Two-Dimensional Situation (Pande, et al.,1997)
ti 1
50
The solution procedure (see later) is then identical to that of standard viscoplasticity (Perzyna
type) involving time integration, with stress being calculated for each overlay (Owen and Hinton,
1980). It should be noted that the viscoplastic strain in an overlay will be different due to
differences in threshold yield values and flow rates, but the total strains in all overlays are the
same.
Specializations of MDSC (Overlay) Model
The material parameters for elastic, viscous and yield characterizations are shown in Fig.
I-6. By adopting different values of the parameters, the overlay model can specialize to various
versions. For instance, consider the overlay model with two viscoplastic units; such a two-
overlay model is commonly adopted; Table 1 gives examples of specializations.
Table I.1: Specializations of MDSC (Overlay) Models
Specialization
Plasticity
Model
No. of
Overlays
Thickness
Parameters
Elastic (e)1
von Mises
1
1.0 E, , , N and very
high y
Viscoelastic (ve)2
von Mises
2
0.5, 0.5
E1, 1, 1, N1, y1 = 0;
E2, 2, 2, N2, y2 =
very high
Elastoviscoplastic
(evp)3 (Perzyna type)
Any
1
1.0
E, , , N, y or F
Viscoelasticviscoplastic
(ve vp)4
von Mises
Any
1
= 2
1
0.5
0.5
E1, 1, 1, N1, y1 = 0
E2, 2, 2, N2, y2 or F
1-4The following notes show resultant models with the specific choice of parameters.
51
Notes: 1Here, as σy is high, only the elastic spring will be operational because the dashpot slider
unit will be essentially not operational.
2Here, for overlay 1 as yl = 0, only the spring and dashpot will operate, as y2 > > , only the
spring will operate in overlay 2.
3Here, with one overlay, all units are operational.
4Here, the first overlay (with y1 = 0), leads to the spring and dashpot, and, in the second
overlay, all units are operational.
Number of Overlays and Thicknesses
Usually, two overlays are sufficient and the thickness of each overlay is prescribed as 0.5.
Layered Systems with Different Material Properties
When a problem with layered material (e.g., pavement) is to be analyzed, some materials
may behave as viscoelasticviscoplastic (vevp), and others are elastic or elasto-plastic, the
following procedure can be used:
(i) For the material with vevp response, two overlays (Table I.1) can be used.
52
(ii) For the elastic response, the material is considered with one overlay and infinitely large
yield strength (Table I.1).
(iii) For the elasto-plastic response of the material, one overlay is used and the fluidity
parameter, , is taken to be very small, approximately 1/600 of fluidity parameter prescribed for
the vevp material, and N = 1.
DISTURBED STATE CONCEPT (DSC)
The DSC is considered as the culmination of various models developed previously. It is
general and unified from which most of the other models can be obtained as special cases. Its
hierarchical nature allows formulation of general constitutive matrix in computer (finite element)
procedures; hence, a chosen model can be achieved by inserting material parameters for that
model, say, elastic or continuous yield plasticity.
The DSC has been covered in a number of publications (Desai and Ma, 1992; Desai,
1995, 2001; Desai and Toth, 1996; Katti and Desai, 1995; Desai, et al., 1998a,b). Hence, brief
description is given below.
In the DSC, a deforming material element is assumed to consist of various components.
For instance, for a dry material, it is assumed to contain two components: continuum or relative
intact (RI) and discontinuum or fully adjusted (FA) phases. These components interact and
merge into each other, transforming the initial RI phase to the ultimate FA phase. The
transformation occurs due to continuous modifications in the microstructure of the material. The
disturbance or microstructural changes act as a coupling mechanism between the RI and FA
phases.
The incremental constitutive equations for the DSC can be expressed as follows:
53
ic
cc
iia
dD
dCD
dCDd
1
(I.35a)
where a,i, and c denote observed, RI and FA states, respectively, {} and {} are the stress and
strain vectors, and dD the increment (or rate) of disturbance, D.
Degradation and Softening
The disturbance can be defined on the basis observed (laboratory and/or field) behavior
in terms of stress-strain, volumetric strain, pore water pressure, ultrasonic properties as P- and S-
waves, e.g., shear wave velocity (Desai, 2001). For instance, D can be expressed (Fig. I.8) as
ci
ai
D
(I-36a)
Disturbance can be expressed in terms of an internal variable such as accumulated deviatoric
plastic strain (D) or worki:
zDA
u eDD
1 (I-36b)
where Du, A, and Z are parameters determined by using Eq. (I-35).
The continuum or RI phase can be characterized by using models based on continuum
elasticity, plasticity or viscoplasticity. For instance, the constitutive matrix [Ci] can be defined by
the HISS plasticity or conventional plasticity model. The FA part can be modeled in various
ways by assuming that FA part (i) has no strength like conventional damage model by Kachanov
(1986), (ii) has hydrostatic strength like in classical plasticity, and (iii) has strength
corresponding to the critical state (Schofield and Wroth, 1968), at which the material deforms
without change in volume or density. For instance, if we assume that the FA part has only
hydrostatic strength, defined by bulk modulus, K, Eq. (I-35a) reduces to:
54
Figure I.8 Schematic of Elastoplastic and softening (DSC) Responses
Elastoplastic(virgin)
(a) Elastoplastic Response with Unloading and Reloading
Elastoplastic(i)
(b) DSC Softening with Unloading and Reloading
Softening: Observed(a)
Fully Adjusted(c)
D
55
iii
iia
SdD
ID
dCDd
3
1 (I-35b)
where {I} is the unit vector and {S} is the vector of shear stress components. Here, it is assumed
that the mean pressure p (= Ji/3 = ii/3) and the strains are the same in the RI and FA parts. In
that case, eq. (I-35a) can be written as
dCd DSCa (I-35c)
where [CDSC
] is the general constitutive matrix and dD = {R}T {d
i}, R is derived on the basis of
the adopted yield function (Desai, 2001). The constitutive matrix is given by
icT
ciDSC
R
CDCDC
1 (I-35d)
Specializations
If D = 0, that is, the material is considered as a continuum, Eq. (I-35a) reduces to
ii dCd (I-35e)
where [Ct] can be elastic, elastoplastic, or elastoviscoplastic model.
THERMAL OR INITIAL STRAINS
Thermal and mechanical (loading) cycles are available in the finite element code. The
implementation aspects for various characterizations and cyclic (loading-unloading-reloading)
are described below.
Elastic Behavior
In the case of elastic behavior, the effect of known temperature change causing initial
strains, are given below for various two-dimensional idealizations:
56
Plane Stress
0.0
T
dTT
dTT
xy
Ty
Tx
(I.37)
where is the coefficient of thermal expansion and dT is the temperature change = T – To, To is
initial (previous) temperature and T is the current temperature.
Plane Strain
dTET
T
dTT
dTT
Tz
xy
Ty
Tx
0.0
1
1
(I.38)
where E and are the elastic parameters.
Axisymmetric
0.0
T
dTT
dTT
dTT
rz
T
Tz
Tr
(I.39)
Then the incremental elastic constitutive relation is given by
TddC
dCd
e
ee
(I.40)
where [Ce] is the elastic (tangent) constitutive matrix, and {d }, [d
e} and {d (T)} are the
vectors of total, elastic and thermal strains, respectively.
57
If the parameters E and vary with temperature, they can be expressed in terms of
temperature as (Desai, et al., 1997; Desai, 2001):
TC
t
rT
TEE
(I.41a)
C
r
rT
T
(I.41b)
where Er and r are values at reference temperature, Tr (e.g., room temperature = 300 K), and cT
and c are parameters found from laboratory tests.
Thermoplastic Behavior
The normality rule gives the increment of plastic strain vector {dp(T)} as
TQTd p , ,
(I.42)
where Q is the plastic potential function; for associative rule, Q F, where F is the yield
function. Now, the total incremental strain vector {dt} is given by
TdTdTdTd pet (I.43)
where {d(T)} is the strain vector due to temperature change. Hence,
TdQ
dTd e
(I.44a)
and
dTIQ
TC
TCd
T
e
e
0
1
e
e
d
d
(I.44b)
where 0] 1 1[0
1 I for two-dimensional case and [1 1 1 0 0 0] for three-dimensional case.
Now, the consistency condition gives
0T , , dF (I.45)
58
Therefore,
dTT
Fd
Fd
FdF
T
(I.46)
Then, use of Eqs. (I.44) and I.46) gives
2/1
o
I d
QQFQTC
F
dTTCF
dTT
FTC
T
F
T
e
T
e
T
T
e
T
(I.47a)
Therefore,
dT
QQFQTC
F
Q
T
FITC
FQ
ITC
d
QQFQTC
F
TCFQ
TCd
T
e
T
oe
T
Toe
T
e
T
e
T
e
2/1T
2/1
I
(I.47b)
The parameters in the elastoplastic model, e.g., HISS-0. can be expressed as function of
temperature as
c
r
rT
TPTP
(I.48)
where P is any parameter such as E, , Eq. (I.40); , , R, n, Eq. (I.19); a1, 1, Eq. (I.20); Pr is its
value at reference temperature Tr, and c is parameter found from laboratory tests.
Thermoviscoplastic Behavior
The total temperature dependent strain rate vector, , is assumed to be the sum of
thermoelastic strain rate, )(Te , thermoviscoplastic strain rate, )(Tvp , and the thermal strain
rate due to temperature change dT, )(T , as
TTT vpe (I.49)
Here, the thermoviscoplastic strain is contributed by rheologic or creep and temperature effects.
59
With Perzyna’s (1966) viscoplastic theory, Eq. (I.29), Eq. (I.49) can be written as
TF
F
TFTT e
o
e
(I.50)
where and are temperature dependent fluidity parameter and flow function, respectively.
Then the constitutive equations are given by
T
F
F
TFTTC
o
e
(I.51)
Viscous or creep behavior requires integration in time. The thermoviscoplastic strain rate
is evaluated from Eq. (I.29) at time step n, Fig. I.9. Then the strain rate at step (n + 1) can be
expressed by using Taylor series expansion as (Desai, et al., 1995); Owen and Hinton, 1980)
IdTGdGT
IdTT
T
dT
TT
nnnnnvp
n
nvp
n
nvpnvpnvp
21
~
~
~
1
(I.52a)
where nd~ is the stress increment, dT
n is the temperature increment, and [G1]
n, [G2]
n denote
gradient matrices at time step, n.
The increment of viscoplastic strain, nvp Td )( , can be found during the time interval
tn = tn+1- tn, Fig. I.9, as
11
nvpnvp
n
vp TTtTd (I.53)
where 0- 1. For = 0, Eq. (I.53) gives the Euler scheme, for = 0.5 the Crank-Nicolson
scheme and so on. The present code allows for = 0 and 0.5.
Now, Eq. (I.51) can be written in the incremental form as
60
Figure I.9 Time Integration for Viscoplastic Strains
nvp
tn tn+1 t
tn
nvp
vp
n+1vp
61
TdTd
dTCd
nvp
nen
~ (I.54)
Use of Eqs. (I.52) and (I.54), leads to
TdTtG
tTd TCd
n
n
n
2
n
nvp
n~
evpn
~
(I.55)
where
1
1
n
neeevp tGTCITCTC
DSC Model
In the case of the DSC model, Eq. (I.35), the RI response can be simulated as elastic, Eq.
(I.40), elastoplastic, Eq. (I.47b), or elastoviscoplastic, Eq. (I.55), which include the temperature
dependence.
With the general DSC model, Eq. (I.35), the disturbance parameters, Du, A and Z, Eq.
(I.36b) can be expressed as functions of temperature, by using Eq. (I.48). Their values
determined from tests at different temperatures, which are used to define the function in Eq.
(I.48).
CYCLIC AND REPETITIVE LOADING
Cyclic and repetitive loading, involving loading, unloading and reloading, occur in many
problems such as dynamics and earthquakes, thermomechnical response such as in electronic
packaging and semiconductor systems, and pavements. If the simulated behavior involves
continuing increase in stress along the same loading path, without unloading and reloading, Fig.
I.10, it is often referred to as monotonic or virgin loading. The unloading and reloading are often
referred to as nonvirgin loading. Loading in the opposite side, i.e., negative side of the (stress)
response, is sometimes referred to as reverse (reloading) loading. Cyclic loading without stress
62
Figure I.10 Schematic of Loading, Unloading, and Reloading
Reloading(Reverse)
Unloading
Reloading
Loading
Virgin
A
Unloading
63
reversal is often referred to one-way, while with stress reversals, it is referred to as two-way. In
the case of degradation or softening, decrease in stress beyond the peak occurs, but it is
considered different from unloading.
For the virgin loading, the constitutive equations, Eq. (I.35), apply. For nonvirgin
loading, it is required to consider additional and separate, often approximate, simulations.
In the case of elastoplastic model (e.g., HISS-0), the simulated virgin response allows
for the effect of plastic strains and plastic hardening or yielding, Fig. I.11(a). In the case of the
softening behavior, the plasticity model can simulate the RI behavior, and the use of DSC allows
for the degradation, Fig. I.11(b).
Plastic deformations can occur during unloading and reloading, and can influence the
overall response, Fig. I.11. Although models to allow for such behavior have been proposed in
the context of kinematic hardening plasticity (Mroz, et al., 1978); Somasundaram and Desai,
1988), they are often relatively complex and may involve computational difficulties. Hence,
approximate schemes that are simple but can provide satisfactory simulation have often been
used; one such method implemented in the present code, is described below.
Unloading
As indicated in Fig. I.10, the unloading response is usually nonlinear. However, as a
simplification, it is often treated as linear. Here, both linear and nonlinear elastic simulations are
included. For the nonlinear case, of which the linear simulation is a special case, the procedure
proposed by Shao and Desai (1998a,b) is used. During unloading, the following incremental
stress-strain equation is used:
dCd UL (I.56)
64
Figure I.11 Unloading Interpolation Function for CTC and SS Tests
Ge
Gb
p
e
(b) Simple Shear Test
E
e
Eu
Eb
Current
1p 1
e
1
A
(a) CTC Test
1-3
65
where [CUL
] is the elastic constitutive matrix with variable elastic unloading modulus, Eu, Fig.
I.11, and the Poisson’s ratio, , is assumed to be constant. The modulus Eu is given by
pbu EEE
111 (I.57)
where Eb is the slope of the unloading curve (response) at the point (A) of unloading, Fig.
I.11(a), and Ep is the “plastic” modulus, which is evaluated by using the following equation:
2
22
1
K
D
b
D
aa
p
JJ
pKpE
(I.58)
where K1 and K2 are constants, pa is the atmospheric pressure (used for nondimensionalization,
and b
DJ 2 and J2D and the second invariants of the deviatoric stress tensor, Sij, at the start of
unloading (point A), and at the current state during unloading, respectively.
The values of K1 and K2 are found from laboratory tests. For triaxial compression CTC:
1 > 2 = 3) and simple shear (SS) tests, their values are derived as follows:
Triaxial Compression (CTC) Test
0.1113
1
22
2
bep
D
b
D
EE
JJK
(I.59a)
1
22
12
1
2
1
3
K
a
D
b
D
p p
JJ
KK
(I.59b)
where Ee is the elastic modulus (slope) at the end of unloading and p
1 is the “plastic” strain, Fig.
I.11(a).
Simple Shear (SS) Test, Fig. I.11(b)
The relation between the elastic (Young’s) and shear moduli (G) are given by
bb GE 12 (I.60a)
66
ee GE 12 (I.60b)
Substitution of Eq. (I.60) into Eq. (I.59) and replacing DJ 2 by (shear stress) and p
1
by 1 2/3 p , where p is the “plastic” shear strain, Fig. I.11(b), leads to
0.111
2
GGK
ep
b
(I.61a)
1
2
1
2
1
1
K
a
b
p pKK
(I.61b)
where b and are the shear stresses at the point of unloading, and during unloading,
respectively.
The values of p
1 and p are evaluated by using the following equations:
be
b
p
EE
JD 11
2
32
1 (I.62a)
and
be
bp
GG
11
2
(I.62b)
Reloading
Figure I.12 shows two cases of reloading, for the one-way and two-way. In both cases,
the following constitutive equation is used:
dCRdCRd eDSCa 1 (I.63)
where R is the interpolation parameter such that 0 R 1; R = 0 for the beginning of
reloading and R = 1 at the end of reloading. Thus, at the beginning of reloading, the behavior is
elastic, given by
dCd ea (I.64a)
at the end of reloading, virgin response resumes:
67
Figure I.12 Two Reloading Cases
Eb
Ebr
=Eb
E
Unloading
Reloading
A
B
(a)Reloading case 1: A B (one-way)
Unloading
Ebr
=Ee
Reloading(Reverse) Unloading
A
B’
(b)Reloading case 2: A B’ (two-way)
68
dCd DSCa (I.64b)
The elastic modulus, ER, for the two cases, Fig. I.12, is different. For case 1, the elastic
modulus at the start of reloading, Ebr
, is given by
bbr EE (I.65a)
where Eb is the unloading slope at the beginning of unloading, Fig. I.13(a). For case 2,
ebr EE (I.65b)
where Ec is the slope at the end of unloading, Fig. I.13(b).
The interpolation parameter, R , for both cases is found as
b
D
D
J
JR
2
2 (I.66)
where b
DJ 2 and J2D are the second invariants of the shear stress tensor at the beginning of the last
unloading and current level, respectively.
In computer (finite element) analysis, the reloading stress path may be between the above
two cases. Then, a parameter, S, is defined as an indicator of the direction of reloading:
d
b
Tb dS (I.67)
where -1 S 1, {b}, {} and {d} are the stress vectors before unloading, the current stress
vector and the next stress increment respectively. S = -1 indicates case 1 reloading, while S = 1
indicates case 2 reloading. Now, Ebr
is interpolated between Eb and E as
ebbr E
S
E
S
E 2
1
2
11
(I.68a)
Then, the modulus for reloading, ER, is found as
E
R
E
R
E brR
11 (I.68b)
69
where E is the elastic modulus of the material, which is often found as (average) slope of the line
joining the unloading and end of unloading points or the initial slope, Fig. I.13(a). Then at the
beginning of reloading when R = 0, ER = E
br, which ensures smooth transition from unloading to
reloading, Fig. I.12(b). At the end of reloading (R = 1), E-R
- = E, which ensures smooth transition
from reloading to the virgin loading.
Cyclic Hardening
In the case of elastoplastic behavior, there exists a yield surface (Fo) corresponding to the
initial or past state of stress experienced by the material before the present cyclic or repetitive
load is applied, Fig. I.13. When unloading occurs, the plastic strains can change (increase or
decrease), and hence, for the reloading after the unloading, the yield surface that defines the
elastic limit usually expands from Fo to the initial surface, Fi, corresponding to each cycle N (= 1,
2, …). As a result, the magnitudes of plastic strains decrease from one cycle to the next, which is
often referred to as cyclic hardening.
For a given load or stress (increment), the final or bounding surface, Fb, can be defined
by solving the incremental constitutive equations, (I-35). In the case of repetitive loading under
constant amplitude of load (stress), Fig. I.13(b), the maximum load (Pmax) will be the amplitude
of the load (stress). In the case of cyclic (one-way) loading, Fig. I.13(c), the bounding surface,
Fb, would change for each stress increase. Note that in the repetitive load analysis, here, the time
effects are not included.
Mroz, et al. (1978) proposed a model for cyclic hardening, which was adopted by
Bonaquist and Witczak (1997) for materials in pavement structures. The approximate (modified)
method for cyclic hardening implemented in the present code is similar, and is described below.
70
Figure I.13 Cyclic Hardening Under Repeated Loading
O
B
A
C
B’
F0 F1
F2
Fb
J1
J2D
(a)Cyclic hardening
B
A C
B’
O
(c)Loading-Unloading-Reloading
O B
A
Pmax
P
Time
(b)Repeated wheel load
71
For the given load or stress increment, two bounding surfaces are defined, Fo and Fb, Fig.
I.13, and the corresponding hardening functions and parameters are o and b, Eq. (I.20), and o
and b, respectively. Here, denotes the accumulated plastic strains:
2/1
pTp dd (I.69)
where {dp} is the vector of incremental plastic strains. Then the initial yield surface parameter,
i, for a given cycle, i, is expressed as
ohoicN
b
11 (I.70a)
where hc is the cyclic hardening parameter, determined from laboratory repetitive tests. It
controls the rate of expansion of the initial yield surface, Fi, at the end of unloading for a given
cycle, N. If hc = 0, no cyclic hardening occurs.
Bonaquist and Witczak (1997) considered repeated tests involving the same stress
(amplitude) to an initially unstrained material specimen, o = 0. Then, Eq. (I.70a) becomes
bhibcN
1 (I.70b)
or ch
b N
1
where is the plastic strain trajectory up to cycle N. Plots of normalized trajectory /b vs
number of cycles are used to find hc through a least square procedure. For the granular material,
hc = 1.06 was found (Bonaquist and Witczak, 1997).
With the above formulation, the value of i, Eq. (I.70) is used to evaluate the hardening
function, i, Eq. (I.20). It is used to define the elastoplastic constitutive matrix [Cep
] = [Ci], Eq.
(I.36e), the general DSC matrix [DDSC
], Eqs. (I-35e), and (I.63), when reloading occurs
72
APPENDIX II
ELASTO-PLASTIC EQUATIONS
The incremental total strain vector {d} is the sum of incremental elastic, {de} and
plastic, {p} strain vectors, i.e.,
d + d = d pe (II.1)
The incremental elastic strain is related to the incremental stress as
d C = d ee (II.2)
where {d} is the incremental stress vector and [Ce] is the elasticity matrix.
Using the theory of plasticity, the incremental plastic strain vector is given by the flow
rule
Q = d 'p (II.3)
where / is the scalar constant of proportionality.
The consistency condition is
dF = 0 (II.4)
Equations (II.1) to (II.4) are combined to obtain the incremental stress-strain relation
d C = d p e (II.5)
where [Cep
] is the elasto-plastic constitutive matrix.
The expression for / and [C
ep] are derived as
H -
Q C
F
d C
F
=
e
T
e
T
'
(II.6)
and
73
H -
Q C
F
C
F
Q C
- C = C
e
T
e
T
e
ep e
(II.7)
where H is the term due to hardening. For non-hardening yield function, H = 0 and for hardening
yield functions, H is defined as follows:
(i) Critical State and Cap model
F
F = H
(II.8)
where
F
F =
j ij i
2 / 1
F
(II.9)
(ii) HISS model (non-assdociative)
D Q
D
Q
F +
F = H
(II.10)
where
Q
Q =
j ij i
2 / 1
Q
(II.11)
and
Q
Q =
j i Dj i D
2 / 1
D Q
(II.12)
where D denotes deviatoric part. For associative model Q F.
The elastoplastic constitutive matrix [Cep
] represents the response of the material in the
relative intact (RI) state and forms a part of the general DSC matrix, Eq. (I.35c), when
disturbance (softening or degradation) is considered.
Derivations for creep and DSC models are given by Desai (2001).
74
APPENDIX III
DRIFT CORRECTION AND DSC COMPUTER ALGORITHM
Under a given stress increment, {d}, the stresses at point B do not lie on the yield
surface, Fig. 6 (in the main text), i.e., F ({B}, B) > 0, where is the hardening function. The
stress vector {B} and B are to be corrected so that F ({B}, B) 0. The method, designated as
“correction” method by Potts and Gens (1985) and modified by Desai, et al. (1991), is described
below.
The correction is carried out by an iteration procedure. At the nth iteration, the stresses
and hardening parameters are given by
Q C e/
1nn (III.1)
dnn 1 (III.2)
vvvn dn
1
(III.3)
DDDn dn
1
(III.4)
where
HC
F
e
T
n
Q
F
, 1-n1/ (III.5)
Fd / (III.6)
FVvd / (III.7)
FDDd / (III.8)
75
0in which
F
F =
j ij i
2 / 1
F
(III.9)
3 /
F =
i i
V F
(III.10)
F
F =
j i Dj i D
2 / 1
D F
(III.11)
For non-hardening, yield function, H = 0 in Eq. III.5 and for hardening yield function, H
is given by Eq. II.8 or Eq. II.10 of Appendix II.
The derivatives
QF, and
D
FF
, are evaluated at the stress point {n-1}. The
iterations are performed until the yield function is satisfied, i.e., F ({n}, n) 0 within the
tolerance of 10-6
or less. For the first iteration, {o} is taken as {B} and o as B.
DSC Computer Algorithm
According to Eq. (I.35a), the DSC incremental finite element equations are given by
~~
o~~~
Q - QdQqdkiDSC
(III.12)
where DSC
k~
is the nonsymmetrical stiffness matrix, i
qd~
is the vector of nodal increment
displacements, ~
Q is the applied load vector,
~oQ is the balanced load vector. Incremental
iterative solution of Eq. (III.12) involves negative definite stiffness matrix in the softening zone
(Desai and Toth, 1996). However, a number of approximate but simplified strategies can be used
(Desai and Woo, 1993; Desai, et al., 1999; Desai, 2001). One such scheme is to first solve for the
76
RI response by considering only the symmetric part of DSC
k~
that defines the RI behavior. Hence,
the following RI equations are first solved:
~
1~~
i
n
i
n
i
n Qdqdk (III.13)
where i
k~
is based on elastic, elastoplastic or other suitable model for the RI behavior, i
Qd~
, is
the vector of applied loads, and n denotes incfremental step. For elasticplastic model, the drift
correction will lead to convergent solution for incremental displacements, i
in
qd1~
, which in turn
can be used for computing the RI strains, i
ind
1~ and stresses,
i
ind
1~ , Fig. III.1. Then by
considering the observed and RI strains to be at the same level, i.e., ai
inin 11 ~~
, the observed
stress, i
in 1~ , is found by using Eq. (I.35) through an iterative procedure in which the
disturbance, Eq. (I.36), is found and updated. Details of the procedure are given in Desai (2001).
77
APPENDIX IV
DETERMINATION OF CONSTANTS FOR VARIOUS MODELS
Procedures for the determination of constants for the HISS-0 and 1 models are first
described below, Desai and Wathugala (1987), Desai (1990), Desai (1994). Brief details for
determination of constants for other models, elastoviscoplastic, and disturbance (softening or
degradation) are given later.
As stated before, the constants involved in the HISS models have physical meanings and
can be determined from uniaxial, shear, hydrostatic, triaxial (cylindrical) and multiaxial (cubical)
tests.
In fact, the constants can be estimated from One Compression and One Extension
Test.
If the angles of friction in compression and extension are assumed to be equal,
i.e., c = , then three compression test can be used to find constants.
For 0 and 1 models, computer code (see below) can be used to calculate the
constants.
Schematic plots required to find the constants and brief details are given below.
Elastic Constants, Fig. IV.1
Fig. IV.1. Elastic Constants
E
1-3
1
(a) E and
v
1
78
1. Find (average) E and from unloading slopes of (1 - v) vs. 1 and v vs. 1 curves,
Fig. IV.1(a)
2. For G and K use curves in terms of oct vs. oct and 3
1J vs. v, Fig. IV.1(b).
3. Relations between elastic constants and unloading (reloading) slopes (S) for different
stress paths are given in the following Table IV.1
Figure IV.1 (continued)
G
1
oct
S
1
S
2
S
3
3
2
2 ,
3
(b) Shear Modulus, G, and Bulk Modulus,
K
(c) Slopes in Stress-Strain
Curves
v
J1/
3
K
79
Table IV.1
Figure IV.2. Ultimate Parameters: and
i
i Ultimate(Asymptotic
) J2D
J1
Compressio
n
Extension
,
22 =23
1 HC
CTE
CTC
TE RTE
PL
RT
C
SSTC
Note: The elasticity parameters can be expressed as nonlinear functions of factors such
as shear stress and mean pressure.
Plasticity Constants
Ultimate: , (Fig. IV.2, IV.3)
Test E
CTC
RTE
3S1
2
2 S1
S2 + S3
S2 + S3
4 S1
32 S1 (S2 + S3 )
4 S1 +S2 + S3 )
TC
TE
CTE
RTC
3
22 (1+)( S1 + S3 )
3S1
2 (1+)( S1 +S2 + S3 )
SS
CTC (1>2 =3), and so on.
Where SI =(average) slope of the unloading/reloading curve, oct
VS i (i=1,2,3) plot, Fig Iv.1(c)
80
Figure IV.3 Ultimate Envelopes in Different Stress Spaces
E
S
C
J2D
J1
(a) Ultimate Envelopes in J2D - J1 Space
C=Compression
S=Simple shear
E=Extension
E
S
C
(b) Ultimate Envelopes in Mohr-Coulomb (-) Space
81
1. Find ultimate (asymptotic) stresses for given stress-strain curve under initial values of J1.
Ultimate value can be found by drawing an asymptote to the curve or by taking a value of about
5 to 10% higher than stress at peak.
2. Plot DJ 2 vs. J1 for ultimate values for compression, extension and/or simple shear
paths. At least two such points are needed. If the angle of friction is compression c =
angle of friction in extension E, only one point can be sufficient.
3. Use lease square fit to find and from F = 0 with = 0.
2
E
2
c
1
tan
1
tanmm
(IV.1
m
m
p
p2
2
)(1
)(1
(IV.1b)
where
E
e
E
c
c
c
E
cp
sin3
sin
3
2tan and
sin3
sin
3
2tan ,
tan
tan.
c, s, E and c, s, E, are shown in Fig. (IV.3).
Phase Change (Fig. IV.4)
Figure IV.4. Phase Change Parameter: n
1
1
v
Contraction to Dilation
F/J1=0
J2D
J1
=0.04
0.014
(a) In J1-J2D Space
82
1. Find the state of stress at which the volume change = 0 (i.e., 1J
F
= 0).
2. Find n by substituting the stresses in the following equation:
F
1
J
J-1
2 = n
s2
1
2D
(IV.2)
(at zero volume change)
The value of n can also be found from HC test by usng the following formula (Wathugala
and Desai, 1991)
dJ 2)-(n 3 = d J 1k k1-n
1 (IV.3)
where dJ1 and dkk are increments in the J1 vs. kk curve.
Although it may depend on factors such as initial density, an average constant value of n
can be often used. For dense sands, the value of n may be around 3.0, while for loose sands and
other materials such as rock and concrete, it would be higher, often of the order of 7 to 10.
Cohesive Materials (Soils): In the case of cohesive soils, usually the (undrained) stress path may
not reach the ultimate (asymptotic) curve, and failure can occur as the phase change or the
critical state line, Fig. (IV.5), is approached (Wathugala and Desai, 1991). Then, the parameter n
is found from
n
2 =
J
J2-n
1
1m
1a
(IV.4)
where Jlm = maximum value of J1 of a yield surface and Jia = intersection of the phase change line
and the same yield surface, Fig. (IV.5). Jlm can be obtained from the effective consolidation p/ as
p 3 = J 1m (IV.5)
n can also be found from the slopes of the phase change line, SPC, and the ultimate line
(curve), SUL as
83
Figure IV.5 Phase Change Parameter for Cohesive (Soil) Materials
2
1
n
2-n =
S
S
UL
PC (IV.6)
The values of and are found by least square or an optimization procedure from:
0.5)- = (m 1 = S+ 2-n
n
1
2J
Jrm
1-
m
12D
pc
(IV.7)
where the subscript pc denotes stress quantities at the phase change; a minimum of two such
values are needed for two stress paths such as compression and extension.
Hardening Parameters: (Fig. IV.6)
Yield Surface
J
2D
J1
J1m J1a
SUL
SPC
A
Phase Change Line
Ultimate Line
84
Figure IV.6. Hardening Parameters: a1 and 1
1. For a given stress increment find p
3
p
21 d ,d , pd based on unloading modulus. Then find
.
2. Substitute the state of stress in F = 0, from which find corresponding .
3. Plot ln vs. ln for different stress-strain curves. In many cases, the results will form a
narrow band. Then draw an average straight line. The slope gives 1 and the intercept
along ln gives a1, at ℓn = 0.
If the points are scattered, it may be necessary to express a1 and/or 1 function of factors
such as initial pressure and density.
▪
▪ ▪
▪ ▪
▪
▪ ▪ ▪
▪ ▪ ▪ ▪
▪ ▪
▪ ▪ 1
ln
a1
ln
i
i
di ppp
=∫( dij dij) 1/2
; F=0 ppp
ppp
Su
Nonassociative
85
4. For nonassociative parameter , find the (constant) slope, Su, Fig. IV.6, of the final
portion of the v vs 1 curve. Use Su in the following equation to find .
-
Z
Y
)r - (1 ) - (
1 =
vo
(IV.8)
where
2/32/3
D2D3
n
1
2/1
D2
2/32/3
D2D3
n
1
2/5
D2D311
p
2/12/3
D2D3
p1n1
2/32/3
D2D3
2
1
2/1
D2
2/32/3
D2D3
11
p2
1
2/5
D2D311
p
2/12/3
D2D3
p1
)JJ271(JJ32
)JJ271(
JJJ3S
)JJ271)(3(nJY
)JJ271(
JJ32)JJ271(
SJJJ3S
)JJ271)(3(J2Z
and
3 / = / = r
pv
vv
Cohesive and Tensile Strengths (Fig. IV.7)
If a material possesses cohesive and tensile strengths, the yield function F is shifted in the
stress space shown in Fig. (IV.7). Then the transformed stress tensor, ij, is expressed as
86
ft
1
2 2 =2 3
2 (2 +R)=2 (3+R)
R
2R
2 2 =2 3 * *
Uniaxial Tensile Strength
1=1+R *
Hydrostatic Axis
Ultimate Envelope
Ultimate Envelope
Compression
Extension
Fig. IV.7 Cohesive and Tensile Strengths
ijij*
ij R + = (IV.9)
where the term R is related to c and , Fig. I.4, and ij = Kronecker delta; R = 0 for
cohesionless materials. R can be related to the uniaxial tensile strength of the material, ft. An
empirical relation is given as (Salami and Desai, 1990; Lade, 1982):
f 1.014 R f 1.003tt
(IV.10)
Once R is known, *
ij is used in F* = 0 Fig. (IV.7), and the plasticity parameters ( and ) are
found based on the modified F.
For rocks, f1 can be found from the following expression (Hoek and Brown, 1980)
s4 + m - m 2
1 = f 2
ct (IV.11a)
where c = unconfined compressive strength, s1 = 1.0 for intact rock, and parameter m is found
from compression test results.
87
The value of R can also be obtained in a simplified procedure, as
/ c= R a3 (IV.11b)
where ac = the intercept of DJ 2 - axis with respect to the ultimate yield surface and is related
to the cohesive strength, and is related to the slope of the ultimate yield surface (line).
COMPUTER CODE TO FIND CONSTANTS FOR 0- AND 1 MODELS
Based on the information above, and in Desai (2001), the parameters can be found by
using EXCEL.
A computer code has also been prepared to evaluate the parameters for the 0-- and 1-
models. Here, the user needs to input available stress-strain data, and the constants are computed
and printed out.
Viscoplastic and Creep Models, 0 + vp: Figs. (IV.8), (IV.9)
For the viscoplastic model (Samtani and Desai, 1991; Desai, et al., 1995; Perzyna, 1966):
F
F =
Q
= ij
vpd =
j i
vp
N
o
ij
oF
F
(IV.12)
where = fluidity parameter and N is power law parameter.
Thus with cohesive/tensile strength, the number of constants for
o-model will be 7 + 1* = 8 (9 with Hoek Brown ft)
and for
1-model will be 8 + 1 = 9.
*If cohesive/tensile strength is included.
88
Mechanics of Viscoplastic Solution
Figure IV.8. Mechanism of Viscoplastic Behavior
From creep tests (on rock salt), general expression for axial strain, 1, is given by
(Hermann, et al., 1980)
T ) - ( t K = pN
31q
1 (IV.13)
t=0 t=∞ (a)
A
+ B
(c-i) J1
J
2D
A
+ B
(c-ii) J1
J
2D
A
+ B
(c-iii) J1
J
2D
A
+ B
(c-iv) J1
J
2D
plastic
(d)
plastic vp
(e)
= a1/ v1
(f)
A B
F=J2D-(-J1n+J1
2)(1-Sr)
m
F
(g) F=0
σ (b)
89
where t = time, T = temperature, 1 - 3 = d = stress difference, and q, N and p are parameters.
From creep tests, Eq. (IV.13) can be established by finding the constants using least square fit.
For a rock salt, average values q = 0.4 and N = 3.0 were found. Now, a general form of rate vp
is written as (Desai and Zhang, 1987)
F
F
F t K q = N
o
1-qp v
(IV.14a)
Then the fluidity parameters can be expressed as
dt t K t
1 = 1-qt
0 (IV.14b)
t = total time during creep test, Fig. (IV.9). Then Eq. (IV.14b) can be integrated numerically
over total time, t , Fig. (IV.9a), and the average value of can be found. For the rock salt =
5.06 x 10-7
(day-1
) was found based on 22 tests on rock salt (Desai and Zhang, 1987).
90
Fig. IV.9 (continued)
Point 1
Time, Seconds(105)
Axia
l S
train
,
1
t
Point 1 Point 2 Point 1
Time, Seconds(105)
d
1 psi=6.89kPa
(a) Typical Creep Test for Rock Salt(Hermann, et al.,1980)
Point 1
Time, Seconds(105)
0
Figure IV.9 Creep Parameters: and N
t
,N
(b) Schematic of Creep Curve
91
ln
ln(F/F0)
ln
N
(c) Evaluation of Creep Parameters
F/F0
Fig. IV.9 (continued)
92
In general, the creep parameters and N can be found from laboratory creep tests.
Equation (IV.12) is expressed as (Desai, et al., 1995):
~~
F
F
T
vpTvpN
oF
F (IV.15a)
Hence,
nF
FnNn
o
(IV.15b)
The values of F/Fo and are found from test data [Fig. IV.9(b)] for various stress increments
(levels). Then ℓn vs ℓn (F/Fo) are plotted, Fig. IV.9(c). The average slope gives the value of N
and the intercept when ℓn (F/Fo) = 0 gives the value of , the fluidity parameter.
Thus for 0 + vp model, the number of
constants = 7 (8) + 2 = 9 (10).
Elastoviscoplastic: MDSC (Overlay Models
The foregoing gives details of the viscoplastic model according to Perzyna’s theory,
which is a special case of the general elastoviscoplastic (vevp) model available in the code; it is
based on the overlay model (Appendix I) and provides four options: elastic (e), viscoelastic (ve),
elastoviscoplastic (evp-Perzyna) and general vevp model.
It is useful to note that the parameters in the elastoviscoplastic models are essentially the
same as elastic, plastic and viscous, Table I.1 (Appendix I). Hence, their determination follows
the same procedures as for elastic, plastic, viscous, etc., models.
93
Some of the advantages of the MDSC (overlay model) are:
1. It allows for four hierarchical options, Table I-1.
2. The parameters are the same as those required for various characterizations such as
elastic and elastoplastic, and creep.
3. The disturbance (DSC) model including microcracking, fracture and degradation
(damage) can be used directly with the evep models to characterize the relative intact (RI)
behavior. Thus, creep effects can be integrated with disturbance (or damage).
4. The parameters have physical meanings as they are related to specific deformation
states, and hence, the need for regression (which may lose the physical meanings) is minimized.
5. The model can allow implicitly for elastic, plastic and creep strains with
microcracking, damage (or degradation) in a single framework.
6. The implementation of the models in computer (finite element) procedures is straight
forward and standard, and includes the available convergence and rebustness characteristics
(Appendices II and III).
As a result, the MDSC (overlay) model can provide an integrated and unified approach
with compactness of parameters, and can lead to significant advantages and simplification
compared to the closed-form models (e.g., Schapery, 1969, 1984).
Disturbance Model: (Fig. IV.10)
Details are given in Desai and Ma (1992), Desai (1995, 2001), Katti and Desai (1995),
Desai and Toth (1996).
In this model, the intact behavior is represented by using the 0-model (7 or 8 constants).
It can also be simulated as linear or nonlinear elastic (Desai and Toth, 1996).
94
D
Peak
Relative
Intact Behavior(i)
Observed Behavior(a)
Fully Adjusted(c)
J2D
I2D
Ultimate(Du)
Ultimate
Figure IV.10 (a) RI, Observed and FA Responses and Disturbance
Figure IV.10 (b) Schematic for Determination of A and Z
●
●
●
●
●
ln(D)
Z
1
ln(A)
ln[-
ln(
)]
D
u-D
D
95
The behavior of the material part in the fully adjusted (FA) state can be simulated in various
ways (Desai, 1995); Desai and Toth, 1995): (i) it has no strength, like in classicalo continuum
damage model (Kachanov, 1986), (ii) as a constrained liquid with no shear strength but with
hydrostatic strength, or (iii) as critical state (Roscoe, et al., 1958) when the material can carry
shear stress reached up to that state for a given hydrostatic stress and deform at constant volume.
In the present code, the constrained liquid simulation is used.
For the disturbance and softening behavior, three additional constants, Du, A and Z, are
needed in the following equations for the disturbance, D.
A - p x e - 1 - D = DZ
Du (IV.16)
where Du = ultimate disturbance and A and Z are parameters.
Disturbance D can be defined approximately as (Fig. IV.10a):
J - J
J - J = D
c2D
i2D
a2D
i2D
(IV.17)
where a, i, and c denote observed, intact and fully adjusted responses, respectively. It can also be
found from other test data such as void ratio (or volume), effective stress or pore water pressure,
and nondestructive properties such as velocities (Desai and Toth, 1996; Desai, 1995; Desai, et
al., 1998).
Now, from Eq. (IV.16)
u
u
D
DD
D
Z
A- exp (IV.18a)
u
u
D
DDn
D
ZA- (IV.18b)
and
96
D
D - D n - n = ) A ( n + )( n Z
u
u
D (IV.18c)
The values of D and D (Eq. IV.17) are found for a number of points on the stress-strain
curve and a plot of Pn (D) versus ℓ n
u
u
D
DDn is obtained. Then the slope gives Z and
the intercept gives A, Fig. (IV.10b).
Thus for the DSC model, the number of constants:
Plasticity (0) model: = 7 (8) + 3 = 10 (11)
Disturbance: = 3
CYCLIC LOADING AND LIQUEFACTION
An anisotropic hardening model (2) in the context of HISS models for sands is available
in Somasundaram and Desai (1988). A similar model (*0) for clays is available in Wathugala
and Desai (1993). These models have been implemented in dynamic coupled finite element
procedures. However, the disturbed state concept (DSC) provides a relatively simple procedure
for including the cyclic behavior. Hence, the DSC model for soils, interfaces and solders (Katti
and Desai, 1995; Desai, et al., 1995; Desai, et al., 1997; Park and Desai, 1997; Shao and Desai,
1998a,b) is implemented in separate code that allows for static and cyclic behavior of solids,
geologic materials and interfaces; it also allows identification of instability and liquefaction
(Desai, et al., 1998b). This code (DSC_DYN2D) and its documentation can be available
separately.
Cyclic or Repetitive Loadings, Unloading and Reloading
The present codes are based on use of the 0 or 1 model for virgin loading; hence, the
parameters are the same as those above for the virgin loading. The unloading and reloading are
97
simulated by using special procedures described in Appendix I. The elastic parameters involve
slopes of unloading and reloading curves (Appendix I), and cyclic hardening involves parameter
hc; they are found from appropriate laboratory tests.
INITIAL CONDITIONS (Desai, 2001)
To introduce initial (stress) conditions, the values of and need to be found to establish
the starting conditions and the corresponding yield surface.
From Eq. (I.20), can be expressed as
J
)S - (1 1
2J
J - = 1
n-2
r
m
2D
(IV.19)
where the overdot denotes nondimensional quantity using pa.
For general initial stress conditions {o}, Eq. (IV.19) is used to find = o. Then 0 is
found from
) / a( = 11/
010
(IV.20)
For hydrostatic initial stress (x = y = z; xy = yz = zx = 0), Eq. (IV.19) reduces to
= v (D = 0), and 0 is found from Eq. (IV.20).
Environmental Effects
Fluid or Water
The DSC model has been developed for saturated porous materials, and is implemented
(as stated above) for dynamic and liquefaction analysis (Park and Desai, 1997; Desai, et al.,
1998b). Here, the effective stress approach is used. Separate codes (DSC-DYN2D and DSC-
SST3D) are available for this problem.
J = 1
n-2
0
98
The DSC model has been developed for partially saturated materials by incorporating
suction (or saturation); details are given in (Desai, et al., 1996; Geiser, et al., 1997).
INTERFACE/JOINT BEHAVIOR: Fig. (IV.11)
Same framework as for “solids”
Figure IV.11. HISS Model for Interfaces/Joints
F()
P(n)
R=Roughness
200 150 175 125 100 0 75 50 25
0
5 0
4 0
30
20
1 0
(b)
Ultimate
Phase Change
Smooth Interface
=0
Phase Change
200 1500 175 125 100 0 75 50 25
0
5 0
4 0
30
20
1 0
(a)
Ultimate
Rough Interface
=0
99
The procedures for finding material constants for interfaces/joints are similar to those for
solids, and are described in Desai (1994, 2001) and other references on joints and interfaces:
Navayogarajah, et al. (1991), Desai and Fishman (1991), Desai and Ma (1992).
MATERIAL CONSTANTS
A summary of material constants in various versions of the DSC/HISS models are given
below; the first four models and the overlay model are included in the present code.
Model
Constants for
0-model
Additional
Constants
Total
0-Associative
7 (8)*
--
7 (8)
1-Nonassociative 7 (8) 1 8 (9)
0 + vp Viscoplastic 7 (8) 2 9 (10)
0 + D Disturbance 7 (8) 3 10 (11)
0 0 + vp: Disturbance 9 (10) 3 12 (13)
Temperature 7 m 7 + m (depends on
how many parameters
are functions of T)
Elastoviscoplastic: Overlay Model See Table I.1
*8 Constants if R is included.
Material constants for typical materials and interfaces/joints are given at the end of this
Appenedix.
IMPLEMENTATION and APPLICATIOINS
Various versions described before have been implemented in static and dynamic
nonlinear finite element procedures. A computer subroutine for 0- and 1-models that the users
can implement in their specific codes is given by Desai, et al. (1991).
Some of the practical problems solved and validated are stated below:
Including Verification with respect to measured responses in Field and
Laboratory
Beams
100
Footings
Piles: Static, Dynamic, Saturated Soils
Single, Group
Retaining (Reinforced) Walls
Dams and Slopes
Tunnels
Building – Foundation Systems
Nuclear Power Plant Structures
Multilayer Systems
Railway Beds
1-D,. 2-D, 3-D
Pavements
Semiconductor Chip-substrate Systems
101
MATERIAL CONSTANTS for TYPICAL MATERIALS
Material Constants for Leighton Buzzard, Munich and
McCormick Ranch Sand, (0/1-Models) (Desai, 1990; Desai and Hashmi, 1989)
Material
Constant
Leighton
Buzzard
Munich
Sand
McCormick
Ranch Sand
Elastic
Constants
E
11500 (psi)
(79328 kPa)
0.29
16500 psi
(113685 kPa)
0.36
90000 psi
(620100 kPa)
0.30
Ultimate
State
Parameter
0.1021
0.36242
0.1051
0.747
0.0519
0.36
Phase Change
Parameter
n
2.5
3.2
4.0
Hardening
Constants
b1
b2
b3
b4
0.135
450.0
0.0047
1.02
0.1258
1355.0
0.001
1.11
4.88x10-4
714.0
0.004
1.04
Nonassociative
Constant
0.29
0.35
*Usually, the hardening function, Eq. (1.20a) is used. However, when the effect of hydrostatic
(HC) and proportional loading is significant, a mixed form of can be used (Eq. I.20b)
which for HC loading reduces to
Then b1 and b2 are found from HC tests and then b3 and b4 are found from shear tests.
b + b - 1 b - p x e b =
D43
D21
b - p x e b = v21
102
MATERIAL CONSTANTS FOR SOAPSTONE FROM DIFFERENT
STRESS PATH TESTS (0-MODEL) (Salami and Desai, 1990)
Ela
stic
ity
ENGLISH UNITS SI UNITS
K 449.51 ksi 3099.37 MPa
G 614.99 ksi 4240.4 MPa
E 1327.39 ksi 9152.4 MPa
0.0792 0.0792
Cohesive
and Tensile
Strengths
R 0.155 ksi 1.067 MPa
Pla
stic
ity
Ult
imate
m -0.50 -0.50
0.0468 0.0468
0 0.74922 0.74922
1 6.8410-4
6.8410-4
Phase
Change
n 7.0 7.0
Hard
enin
g 0.747 0.747
a1 1.21510-12
1.21510-12
Note: All constants, except where indicated, are nondimensional.
103
Material Constants for Rock Salt (ov-Model)
(Desai and Zhang, 1987; Desai and Varadarajan, 1987)
K 14,989 MPa
Elasticity G 8,143 MPa
E 20,685 MPa
0.27
Cohesive/Tensile Strength R 1.79 MPa
m -0.50
Ultimate 0.0945
0.995
1 0.00049
Plasticity Phase Change n 3.0
a1 1.80910-5
Hardening 1 0.2322
Non-
associate 0.275
Viscoplastic Fluidity =5.06 10-7
per day
Parameter N=3.0
104
MATERIAL CONSTANTS FOR PLAIN CONCRETE FROM
DIFFERENT STRESS PATH TESTS (0-MODEL) (Salami and Desai, 1990)
MATERIAL CONSTANT FOR PLAIN CONCRETE
ELASTIC
CONSTANTS
ENGLISH UNITS SI UNITS
K 487.86 ksi 3363.8 MPa
G 440.36 ksi 3036.3 MPa
E 1012.82 ksi 6983.4 MPa
0.154 0.154
CONSTANTS
FOR
ULTIMATE
YIELDING
=3R 1.1833 ksi 8.1589 MPa
0.1130 0.1130
0 0.8437 0.8437
1 3.9710-4
3.9710-4
n 7.0 7.0
CONSTANTS
FOR
HARDENING
1 0.4388 0.4388
a1 6.4010-12
6.4010-12
*Here is dependent on J1, given by =0-1J1
105
Material Constants for Plain Concrete
0 + D – Disturbance Model, Desai and Woo (1993)
Constant
Value
Units
Disturbance
Du
0.875
Z
1.502
A
668.0
Plasticity
0.750
0.0678
n
5.24
a1
4.6 x 10-11
1
0.83
R
1.50
MPa
Elasticity
E (Young's Modulus)
37,000
MPa
(Poisson's Ratio)
0.25
Note: No units indicates dimensionless constant
106
Material Constants for Solder (Pb/Sn)
Material parameters for various solders (e.g., Pb/Sn) are evaluated based on available test
data; they are reported, e.g., by Desai, et al. (1997, 1998a), and Desai (2001). The elastic, plastic,
creep and disturbance parameters for 40 Pb/60 Sn solder at strain rate sec/02.0 including
temperature dependence, Eq. (I.48), are given below.
Elastic and plastic constants for Pb-Sn solders at different temperatures sec/02.0
Temperature 208K 273K 348K 373K
Ultimate Parameter
0.00083 0.00082 0.00082 0.00081
State Change
Parameter, n
2.1 2.1 2.1 2.1
Hardening Parameter
a1(10-6
)
1
1(average)
8.3
0.431
0.615
2.93
0.553
0.615
1.25
0.626
0.615
0.195
0.849
0.615
Young’s Module, E
(Gpa)
26.097 24.105 22.455 22.005
Poisson Ratio, 0.38 0.395 0.408 0.412
Thermal Expansion
Coefficient, T(1/K)
(10-6
)
2.75 2.93 3.11 3.16
Yield Stress, Y 37.241 31.724 20.690 15.172
Bonding Stress, R
(MPa)
395.456 288.168 175.196 122.105
;a
,300
)(1
1300
5.5
300
;00082.0,300
)( 300
034.0
300
;MPa67.240R,300
R)(R 300
91.1
300
;GPa45.23E,300
E)(E 300
292.0
300
4.0,300
)( 300
14.0
300
;
6300
T
24.0
300
TT 100.3,300
)(
.
107
Viscous constants for Pb-Sn solders at different temperatures
Temperature 293K 313K 333K 373K 393K
Fluidity Parameter
0.5784 2.058 3.475 4.61 6.96
Exponent, N
average
2.655
2.67
2.645
2.67
2.667
2.67
2.448
2.67
2.74
2.67
sec/8.1,300
)( 300
185.6
300
Disturbance constants for Pb-Sn Solders at different temperatures
Temperature
223K 308K 398K 423K
Plastic strain range p
Disturbance, D
Z
0.103 0.307 0.04 0.082 0.022 0.102 0.036 0.039 0.097
0.733 0.870 0.521 0.603 0.700 0.591 0.661 0.701 0.722
A 0.056 0.072 0.188 0.130 0.500 0.146 0.197 0.202 0.170
Load Drop,
A
0.026 0.062 0.068 0.054 0.007 0.069 0.046 0.039 0.058
b 0.567 0.617 0.377 0.470 0.630 0.453 0.505 0.586 0.578
102.0A,300
A)(A 300
55.1
300
Z(average) =0.676
108
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