1 probability and materials: from nano- to macro scale a workshop sponsored by the john s hopkins...
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Probability and Materials: from Nano- to Macro ScaleA Workshop Sponsored by the John s Hopkins University and the NSF CMS Division
January 5-7 2004Baltimore, Maryland
Sarah C. BaxterDepartment of Mechanical Engineering
University of South Carolina
Todd O. WilliamsTheoretical Division
Los Alamos National Laboratory
A stochastic micromechanical basis
for the characterization of
random heterogeneous materials
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• The extremes in (stochastic) microstructures typically drive important phenomena due to corresponding strong localizations
- Inelastic deformations - Viscoelasticity- Viscoplasticity
- Failure phenomena- Interfacial debonding- Cracking/damage in the phases
1. Modeling Considerations and Issues
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2. Goals and Motivation
• Develop a general, stochastic micromechanical framework for constitutive models for heterogeneous materials
–Applicable to various types of composite systems–Realistic (stochastic) microstructures–Arbitrary contrast in constituents’ elastic properties–Anisotropic local behaviors–Micro- and macro-damage–Complex rate and temperature dependent constituent behaviors
• Micromechanical theories : Predict the local and bulk behavior of heterogeneous materials based on a knowledge of the behavior of each of the component phases, the interfaces, and the microstructure
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3. Concepts from Classic MicromechanicsRepresentative Volume Fraction
RVE? Maybe
RVE? Probably not
Similar; repeating volume fraction - periodic microstructures
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When the macro conditions of homogeneous stress or homogeneous strainare imposed on an RVE the average stress and average strain are defined as:
Average Stress
Average Strain
V is the volume of the RVE.
Average Strain and Average Stress
V ijij
V ijij
dVV
dVV
1
1
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Average Strain Theorem: The average strains in the composite are the sameas the constant strains applied on the boundary.
Consider a two phase RVE where homogeneous strains are applied to the boundary. Using the definition of strain, the average strain is
Again, consider a two phase RVE, this time with homogeneous stresses applied to the boundary S. Equilibrium, in the absence of body forces, implies
The Average Stress Theorem states that the average stresses in the composite are the same as the constant stresses applied on the boundary.
Average Strain Theorem/Average Stress Theorem
0
ijij
ij ij0
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Thus the average stress is related to the average strain through effective elastic moduli, C*. A similar argument can be used to construct effective compliances.
Using formulas of linear elasticity and average strain and stress theoremsdefines the constitutive law
Effective Elastic Properties
ij(x)Cijkl* kl
dVuuxCV
Ckl
pq
kl
qpV ijpqijkl
)(
,
)(
,
*)(
2
1
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For a two phase composite with perfect bonding (c1 and c2 are volume fractions,
(and )
c1 c2 1
Building on the idea of effective moduli, and using the average strain theorem
Relationship Between Averages
and
)2(2
)1(1 ijij
ccij
ij c1ij(1) c2ij
(2)
0)2(
2
)1(
1 , ijijijijij cc
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then
and
The effective moduli can be determined if the average strain in the secondphase is known.
Relationship Between Averages
C ijkl* kl
0 C ijkl(1) kl
0 c2 C ijkl(2) C ijkl
(1) kl(2)
)1()2(
2
)1(*
0
)2(
ijklijkl
ijklijkl
kl
kl
CCc
CC
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In an RVE there is a unique relationship between the average strain in a phase and the overall strain in the composite, which can be expressed as
A1 and A2 are called strain concentration matrices, c1A1+c2A2 = I, where
I is the unit matrix. Then the effective stiffness tensor can be written as
In a two phase composite where
then
Concentration Matrices
)2(
2
)1(
1 cc )()()( iiiC
)2()2(
2
)1()1(11 CcCc
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)2()2(
2
)2(
1
)1(,, AAA
2)2(
21)1(
1* ACcACcC
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Eshelby (1957, 1959, 1961) considered the problem of an ellipsoidal inclusion in an infinite isotropic matrix. He defined two problems which he considered shouldbe equivalent
ijklC
ijklC
ijklC
ijklC
*
ij
Eshelby’s Equivalent Inclusion
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Eshelby showed that if the eigenstrains are uniform inside an ellipsoidal domainthen the total strain is uniform there too, and that the total strains are
related to the eigenstrains through Eshelby’s tensor.
ijklC
*
ij The starred strains are eigenstrainsor transformation strains, resulting from the inhomogeneity.
Transformation (Eigen) strains / stresses
ij Pijkl kl
* ,
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CS
Transformation Field Theory, Dvorak
, eigenstresses and eigenstrains respectively
Transformation field theory Dvorak & Benveniste, DvorakProc. Math. and Physical Sciences, 1992.))
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VA T (x)p (x)dV
V p
with for example
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One of the simplest models used to evaluate the effective properties of a composite, it was originally introduced to estimate the average constants of polycrystals.
For this approximation it is assumed that the strain throughout the bulkmaterial is uniform (iso-strain).
This implies that A1 = A2 = I and so
4. Concentration Tensors in Classic ModelsVoigt Approximation (1889)
2)2(
21)1(
1* ACcACcC
)2(2
)1(1
* CcCcC
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The dual assumption (to Voigt) is the Reuss Approximation which assumesthat the stress is uniform (iso-stress) throughout the phases.
This implies that B1 = B2 = I and so
Under the Voigt model the implied tractions across the boundaries of the phaseswould violate equilibrium, and under the Reuss model the resulting strains would require debonding of the phases.
Reuss Approximation (1929)
)2(2
)1(1
* ScScS
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The Dilute Approximation models a dilute suspension of spherical elastic particles in a continuous elastic phase. It assumes that the interaction between particle can be neglected. Under the assumption of spherical symmetry, ur = ur(r), uf = 0, uq = 0 the equilibrium condition reduces to
Dilute Approximation
0
22
2
,022
kkr
rrr uur
urr
ur
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One can solve for the A2 concentration tensor, or the ratio between the strains, in the second phase and the applied strain
Using this relationships the effective bulk modulus is given by
Dilute Approximation
A2 kk
(2)
kk0
3C
3D
3(1 21 )
32 2 2 41
12
111221
*
43
)43)((
K
KKKcKK
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This is the problem of an inclusion in a medium with unknown effectiveproperties. The factors (shear and bulk) are then the same as for dilute butwith effective properties replacing those of the matrix.
Effective medium
Self-Consistent Scheme
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Thus,
Which gets us to
which are the same as for dilute, but are now implicit relationships.
Self-Consistent Scheme
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***
*
0
12
)2(
12
)54(2)57(
)1(15
2
***
*
0
12
)2(
12
)54(2)57(
)1(15
)423(
)2(3*
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**
0
)2(
kk
kk
)423(
)2(3*
22
**
0
)2(
kk
kk
2
***
*
1221
*
)54(2)57(
)1(15)(
c
*
2
**
1221
*
43
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3)(
K
KKKcKK
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In this method, a single particle is embedded in a sheath of matrix which in turn is embedded in an effective medium. Solving the problem under dilation with this geometry yields
Matrix
Effective medium
Generalized Self-Consistent Scheme
114343
4343
12112
12110
)2(
KKc
KK
kk
kk
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The Mori-Tanaka method (Mori and Tanaka, 1973) was originally designed to calculate the average internal stress in the matrix of a material containing precipitates with eigenstrains. Starting with
Mori-Tanaka (Benveniste)
0
2
)2(
2
)1()2(
2
)1(*,)( AACCcCC
with,)1()2( M ,
)2(
2
)1(
1
0 cc
if
M-T assumes that where T is the concentration tensor from the dilute approximation.T can be defined by Eshebly’s tensor, P, as
TM
1)1()2(1(1) C
CCPIT
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Then the problems concentration tensor can be defined as
Mori-Tanaka (Benveniste)
1
21
)1()2(
2
)1(*)()( TcIcTCCcCC
1
212 )( McIcMA
and
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5. Background for Stochastic Formulation
Consider that a field, g, can be decomposed into mean and fluctuating parts, i.e,
the mean part is defined by
The fluctuating field is then
by assumption
Usually normalization condition
gg g
g 0
daaPagg )()(
a
adaP 1)(
g g g
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d and f are the transformation field concentration tensors and the underbar operator is defined as
When phases have the constitutive form of
Then the solution to the differential equations of continuum mechanics results ina relationship between local () and global ( fields of
, eigenstresses and eigenstrains respectively)Transformation field theory Dvorak & Benveniste, Dvorak
Proc. Math. and Physical Sciences, 1992.)
6. Background for Stochastic Formulation
C S
Ad
B f
d d(a,b) (b)P(b)db
f f (a,b)(b)P(b)db
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7. Stochastic Formulation: Part 1
Rewriting the localization equation in terms of mean and fluctuating fields
The statistics are incorporated through the overbar (mean value - mechanical concentration tensor) and underbar (transformation field concentration tensor) operations
(A A ) (d d )( )
A d
By taking the mean of both sides, it can be shown that A= I, and which implies
0d 0d
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The effective constitutive equations that result are then
SdC
ACC
C
eff
eff
effeff
C
Stochastic Formulation: Part 1
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8. Stochastic Formulation: Part 2 Hierarchal Effects
It is convenient to extend this approach by further decomposing the fluctuating fields into their phase mean and fluctuating parts. This hierarchal decomposition is based on
r
rrr gggg )ˆ(
g r r ( ˆ g r g r ) r , ˆ g r cr 1 g r r ,
g r r 0.
otherwise 0
r phasein 1r
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Hierarchal Effects
For a two phase composite, this implies that only the mechanical concentration tensoris needed. Under the additional assumption that the fluctuating parts of the local fields are zero
effeffL :
11211211
effA)LL(LLL ccc
)(A 21
T
111211
eff ccc
ˆ r ˆ A r ( (L2 L1 ) 1 (L1ˆ 1 L2
ˆ 2 ))
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9. Application
ˆ A
ˆ A 11ˆ A 12
ˆ A 13
ˆ A 21ˆ A 22
ˆ A 23
ˆ A 31ˆ A 32
ˆ A 33
ˆ A 44
ˆ A 55
ˆ A 66
Window sizes of 7x7 and 11x11 pixels. T300/2510 composite (graphite fibers in polymer matrix). The fibers transversely isotropic. The matrix isotropic.
Moving window GMC to develop a field of concentrationtensor elements. (Aboudi)
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PDFs 7 x 7 windowing used to sample
A21
A31
A22
A33
A23
A32
A44
A55
A66
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PDFs 11 x 11 windowing used to sample
A21
A31
A22
A33
A23
A32
A44
A55
A66
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10. Comparisons
7 x 7
11 x 11
A21
A31
A22
A33
A21
A31
A22
A33
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Comparisons
7 x 7
11 x 11A23
A32
A44
A55
A66
A23
A32
A44
A55
A66
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GPa Reuss 7x7 11 x 11 Voigt
C11 8.489 71.85 71.79 72.14
C22 7.946 8.619 8.247 10.32
C33 7.946 8.579 8.208 10.32
C23 4.486 4.596 4.468 5.477
C13 4.519 4.397 4.279 5.017
C12 4.519 4.406 4.288 5.017
C44 1.73 1.839 1.746 2.422
C55 1.73 1.639 1.502 2.422
C66 1.73 1.66 1.513 2.422
Stiffness Matrix - Between Bounds
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• Starting point of the analysis : Localization relations based on concentration tensors
• Statistics incorporated thru the concentration tensors–For 2 phase materials only need the mechanical concentration tensors
• Can predict the mechanical concentration tensors using only elastic properties of the phases
–Statistics independent of the history-dependent models for the phases
• Hierarchical statistical effects–Simplifies the analysis by decoupling the governing equations
11. Summary
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12. Future Work
• Generate 3D statistics –Moving windows techniques using different micromechanics models
–GMC
• Study impact of the extremes in the PDFs on predictions of the local and bulk material behavior
–Enhanced computational efficiency by simplifying PDFs appropriately
• Extend STFA to consider debonding and damage
• Start implicit implementation of STFA into ABAQUS