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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.))

1

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

2

***

*

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