untitled - ASMEHsuehetal.pdf

Chun-Jen Hsueh

Department of Mechanical and Civil Engineering,

California Institute of Technology,

Pasadena, CA 91125

e-mail: chhsueh@caltech.edu

Kaushik Bhattacharya

Department of Mechanical and Civil Engineering,

California Institute of Technology,

Pasadena, CA 91125

e-mail: bhatta@caltech.edu

Homogenization and Path

Independence of the

J

-Integral

in Heterogeneous Materials

The J-integral that determines the driving force on a crack tip is a central concept of

fracture mechanics. It is particularly useful since it is path independent in homogeneous

materials. However, most materials are heterogeneous at a microscopic scale, and the

J-integral is not necessarily path independent in heterogeneous media. In this paper, we

prove the existence of an effective J-integral in heterogeneous media, show that it can be

computed from the knowledge of the macroscopic or homogenized displacement fields

and that it is path independent in macroscopically homogeneous media as long as the

contours are large compared to the length scale of the heterogeneities. This result justi-

fies the common engineering use of the J-integral.

[DOI: 10.1115/1.4034294]

1 Introduction

It has been understood from the work of Griffith [

], Irwin [

and others that the propagation of cracks is driven by the energy

release rate, i.e., the rate of change of elastic energy with respect

to crack extension. Rice [

] showed that this energy release rate

may be described by the

-integral

(1)

where

is the stored energy density,

is the displacement,

the stress,

is the normal to the contour

enclosing the

crack tip, and

is the tangent to the crack at the crack tip. Conven-

iently, the

integral is path independent, i.e., it does not depend

on the contour

as long as the medium is homogeneous. This

makes it extremely useful, since one can choose contours along

which it is most convenient to evaluate the integrand.

However, the

integral is

not

necessarily path independent in

heterogeneous materials. And most materials are heterogeneous at

a microscopic scale. Still, the

integral has proven to be a most

useful concept. Typically, one notes that the scale of heterogene-

ities is small compared to the engineering object of interest, uses

homogenization theory to define an effective elastic medium

which is homogeneous at the engineering scale, and applies elas-

ticity theory and the

integral to this effective homogenous

medium.

Unfortunately, the relation between a “microscopic” and

“macroscopic”

integral remains open. Specifically, it is not clear

whether the

integral computed with the stress and the strain

associated with the heterogeneous medium will converge to the

integral computed with the stress and the strain associated with

the effective homogeneous medium as the contour becomes very

large. In short, it is not clear whether one can use the solutions to

the homogenized equation to compute an effective

integral.

Indeed, a casual examination of the expression for the

integral

in Eq.

(1)

suggests that the microscopic

integral will in general

be different from the macroscopic or homogenized

integral.

While we know from homogenization theory (specifically Hill’s

Lemma [

]) that the average of the microscopic stored energy

density is equal to the macroscopic stored energy density, this

does not appear to be true for the second term in the parenthesis in

the integrand of Eq.

(1)

. Specifically, the stress and the displace-

ment gradient fluctuate at the microscopic scale in a heterogene-

ous medium. Therefore, it is generally not true that the product of

their averages is equal to the average of their products; in other

words, it is unclear if

i¼

(2)

This raises an issue of using the

integral in engineering practice.

Further, the recent decades have seen an attempt to use hetero-

geneities to enhance fracture toughness. Furthermore, nature

exploits microstructure to enhance toughness of nacre and other

shells. Finally, the emergence of 3D printing and other methods of

additive material synthesis opens the possibility of exploiting

carefully controlled heterogeneity for enhancing the toughness of

materials. All of this has led to a new interest in understanding the

effective toughness of heterogeneous media (see for example,

Ref. [

] and references therein).

In this paper, we use homogenization theory in a quasi-periodic

setting to show the existence of a macroscopic

integral, and

prove that this is path independent in a macroscopically homoge-

neous material if the path is large compared to the size of the het-

erogeneities. The path independence of the

integral follows

from the fact that it is closely related to the configurational stress

tensor or the Eshelby energy momentum tensor

(3)

that satisfies

(4)

where

denotes the explicit derivative with respect to

This equation may be obtained from the equilibrium equation, and

is also referred to as the

configurational force balance

. Integrating

this equation over an annular region between two contours and

using the divergence theorem leads to the path independence of

the

integral in homogeneous materials. We show that the

homogenization of this equation retains the same form leading to

a homogenized configurational stress tensor and homogenized

integral.

We recall homogenization of the variational formulation of

elasticity in Sec.

and derive our main result in Sec.

. We con-

clude in Sec.

2 Periodic Homogenization

Consider a domain

(a bounded open set in

) with a hetero-

geneous elastic medium where the heterogeneities have a length

Contributed by the Applied Mechanics Division of ASME for publication in the

OURNAL OF

PPLIED

ECHANICS

. Manuscript received July 13, 2016; final manuscript

received July 20, 2016; published online August 22, 2016. Editor: Yonggang Huang.

Journal of Applied Mechanics

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scale

diam

in dimensionless units. The domain may

contain a smooth crack of whose length is

(1). Specifically, we

assume that the medium is

quasi-periodic

so that the stored

energy density

;

ðÞ

;

(5)

where

is the displacement gradient,

is the unit cube in

and

;

is periodic in

for each

. This is shown schematically in Fig.

: the medium appears per-

iodic if we look closely at some point

In the case of linear elasticity

;

ðÞ

ijkl

;

ðÞ

(6)

where

is the elastic modulus satisfying major

and minor symmetries, and

;

is periodic in

for each

We seek solutions of the equations of elasticity by seeking to

minimize the total energy

;

ðÞ

;

(7)

where

depends on the body force and boundary tractions among

all displacements

that satisfy the imposed displace-

ment boundary conditions. This problem is difficult because

and consequently the solution oscillates on the scale of the hetero-

geneities

. Homogenization theory [

] states that if

is small

enough and if

;

is convex in

for each

then we can

replace the problem above with the following

effective problem

minimize

E½

ðr

;

L½

(8)

among all displacements that satisfy the imposed displacement

boundary conditions where

is the effective

elastic energy density and may be obtained by solving the follow-

ing problem for each

: minimize

;

Þ¼

þr

;

(9)

overall periodic displacement fields

. Note that the

integrand of Eq.

(8)

is smooth on the scale of

and thus the solu-

tion is also expected to be smooth at that scale. It is also true that

the effective stress is given by

(10)

Further, under suitable growth and strict convexity conditions on

, the minimum is attained and unique up to an inessential trans-

lation. We call the minimum

;

.So

E½

ðr

þr

;

dydx

L½

(11)

3 Configurational Force Balance

We use the effective functional

(11)

to derive an effective con-

figurational force balance. We take

L¼

0 for convenience (i.e.,

no body force, and traction-free and displacement boundary con-

ditions), but the treatment is easily modified otherwise.

Let

minimize

defined in Eq.

(11)

for some given boundary

conditions. Recall that it is smooth on the scale of

. We now con-

sider a variation, but by rearranging the domain.

Consider a fam-

ily of rearrangments

one to one and onto for

2½

;

that satisfy

;

det

, and

Þ¼

. Set

Þ¼

ÞÞ

. Note that

is a family of perturbations of the

minimizer

with

. Therefore, the function

¼E½

(12)

has a minimum at

0 and therefore

Þ¼

(13)

We now compute

Set

;

and

Þ¼r

, and note that

¼r

ÞÞÞ ¼

ÞÞr

(14)

Þ¼

ÞÞr

þr

;

dydx

(15)

Þðr

þr

;

Jdydz

(16)

where we have changed integration variables from

inverting

to obtain

, and set

det

ðr

. Now, set

Þðr

;

þr

;

(17)

We can now calculate

ðÞ

Jdydz

(18)

where we use

to represent the explicit derivative with

respect to

and

Þ¼

to denote the total derivative of

with respect to

. Recalling the identities

;

det

¼ð

det

(19)

we obtain

ðÞ

(20)

Fig. 1 A macroscopic crack in a quasi-periodic heterogeneous

medium

This holds true for common linear elastic problems. Failure of this condition

may lead to long-range instabilities.

Such variations are known as

inner variation

in the calculus of variations.

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(21)

Further, from the unit cell problem of minimizing

(9)

, we can

infer that

(22)

We substitute Eqs.

(20)

–

(22)

into Eq.

(18)

, set

0 (so that

) and change integration variables back to

ðÞ

dydx

(23)

where

Þj

. We note that

and

are independent of

and therefore we can integrate with respect to

. We obtain by

recalling

(9)

ðÞ

(24)

(25)

where we have used the divergence theorem and the fact that

0on

(since

). Now, since

Þ¼

0 for all

and thus arbitrary

, we obtain the macroscopic

configurational

force balance

(26)

where

(27)

is the effective configurational stress tensor.

Finally, if the material is macroscopically homogeneous, i.e.,

and consequently

is independent of

, then we see from

Eq.

(26)

that

(28)

using the divergence theorem where

is any domain that is large

compared to the size of the heterogeneities (

). Now, consider a

domain with a crack in two dimensions. Given any two contours

and

that contain the crack tip, set

to be the annular region

between the contours. Now,

[

with the outward nor-

mal

and

on the two segments of the boundary. Taking the

inner product of the equation above with the tangent to the crack

tip, we obtain

(29)

or the path independence of the macroscopic

integral.

4 Conclusion

We have shown the existence of a homogenized configuration

stress tensor and configurational force balance in a quasi-periodic

medium. We have used these to show the existence of an effective

integral that may be evaluated from the macroscopic displace-

ments and stresses, and that is path independent in macroscopi-

cally homogeneous media as long as the contours are large

enough.

Acknowledgment

We gratefully acknowledge the financial support of the U.S.

National Science Foundation Award No. DMS-1535083 under the

Designing Materials to Revolutionize and Engineer our Future

(DMREF) Program.

References

[1] Griffith, A., 1921, “The Phenomena of Rupture and Flow in Solids,”

Philos.

Trans. R. Soc. A

221

(582–593), pp. 163–198.

[2] Irwin, G., 1960, “Fracture Mechanics,” Proceedings of the First Symposium on

Naval Structural Mechanics, J. Goodier and N. Hoff, eds., Pergamon Press,

New York, pp. 557–594.

[3] Rice, J., 1968, “A Path Independent Integral and the Approximate Analysis of

Strain Concentration by Notches and Cracks,”

ASME J. Appl. Mech.

(2),

pp. 379–386.

[4] Hill, R., 1963, “Elastic Properties of Reinforced Solids: Some Theoretical

Principles,”

J. Mech. Phys. Solids

(5), pp. 357–372.

[5] Hossain, M., Hsueh, C.-J., Bourdin, B., and Bhattacharya, K., 2014, “Effective

Toughness of Heterogeneous Materials,”

J. Mech. Phys. Solids

, pp. 15–32.

[6] Braides, A., 1985, “Homogenization of Some Almost Periodic Coercive

Functional,”

Rendiconti. Accademia Nazionale delle Scienze detta dei XL

, Vol. 9

(Serie V. Memorie di Matematica e Applicazioni), L’Accademia, Rome, Italy,

pp. 313–321.

[7] M

€

uller, S., 1987, “Homogenization of Nonconvex Integral Functionals and

Cellular Elastic Materials,”

Arch. Ration. Mech. Anal.

(3), pp. 189–212.

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