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 justifies the common engineering use of the J-integral

Bhattacharya, Kaushik

Hsueh, Chun-Jen

Caltech Authors - Main

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 [1], Irwin [2],
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 [3] showed that this energy release rate
may be described by the J-integral
J ¼
ð
C
ti Wdij  @uk
@xi
rkj
 
njds (1)
where W is the stored energy density, u is the displacement, r ¼
@W=@F the stress, n is the normal to the contour C enclosing the
crack tip, and t is the tangent to the crack at the crack tip. Conven-
iently, the J-integral is path independent, i.e., it does not depend
on the contour C 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 J-integral is not necessarily path independent in
heterogeneous materials. And most materials are heterogeneous at
a microscopic scale. Still, the J-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 J-integral to this effective homogenous
medium.
Unfortunately, the relation between a “microscopic” and
“macroscopic” J-integral remains open. Specifically, it is not clear
whether the J-integral computed with the stress and the strain
associated with the heterogeneous medium will converge to the
J-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 J-integral.
Indeed, a casual examination of the expression for the J-integral
in Eq. (1) suggests that the microscopic J-integral will in general
be different from the macroscopic or homogenized J-integral.
While we know from homogenization theory (specifically Hill’s
Lemma [4]) 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 ifD @uk
@xi
E
hrkji ¼
D @uk
@xi
rkj
E
? (2)
This raises an issue of using the J-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. [5] and references therein).
In this paper, we use homogenization theory in a quasi-periodic
setting to show the existence of a macroscopic J-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 J-integral follows
from the fact that it is closely related to the configurational stress
tensor or the Eshelby energy momentum tensor
Cij ¼ Wdij  @uk
@xi
rkj (3)
that satisfies
@Cij
@xj
¼ @
W
@xi
(4)
where @=@xi denotes the explicit derivative with respect to xi.
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 J-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
J-integral.
We recall homogenization of the variational formulation of
elasticity in Sec. 2 and derive our main result in Sec. 3. We con-
clude in Sec. 4.
2 Periodic Homogenization
Consider a domain X (a bounded open set inRN) with a hetero-
geneous elastic medium where the heterogeneities have a length
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received July 13, 2016; final manuscript
received July 20, 2016; published online August 22, 2016. Editor: Yonggang Huang.
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scale e 1 ¼ diamðXÞ in dimensionless units. The domain may
contain a smooth crack of whose length is O(1). Specifically, we
assume that the medium is quasi-periodic so that the stored
energy density
eW e F; xð Þ ¼ W F; x; x
e
 
(5)
where F is the displacement gradient, W : RNN  X Y ! R,
Y is the unit cube in RN and WðF; x; yÞ is periodic in y for each x,
F. This is shown schematically in Fig. 1: the medium appears per-
iodic if we look closely at some point x 2 X.
In the case of linear elasticity
W F; x; yð Þ ¼ 1
2
FijCijkl x; yð ÞFkl (6)
where C : X Y ! RN4 is the elastic modulus satisfying major
and minor symmetries, and Cðx; yÞ is periodic in y for each x.
We seek solutions of the equations of elasticity by seeking to
minimize the total energy
Ee u½  ¼
ð
X
eW e ru; xð Þdx L u½  ¼ ð
X
W ru; x; x
e
 
dx L u½  (7)
where L depends on the body force and boundary tractions among
all displacements u : X! R3 that satisfy the imposed displace-
ment boundary conditions. This problem is difficult because W
and consequently the solution oscillates on the scale of the hetero-
geneities e. Homogenization theory [6,7] states that if e is small
enough and if WðF; x; yÞ is convex in F for each x, y,1 then we can
replace the problem above with the following effective problem:
minimize
E½u ¼
ð
X
Wðru; xÞdx L½u (8)
among all displacements that satisfy the imposed displacement
boundary conditions where W : RNN  X! R is the effective
elastic energy density and may be obtained by solving the follow-
ing problem for each F, x: minimize
WðF; xÞ ¼
ð
Y
WðFþruðyÞ; x; yÞdy (9)
overall periodic displacement fields u : Y ! RN . Note that the
integrand of Eq. (8) is smooth on the scale of e 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
r ij ¼ @W
@Fij
(10)
Further, under suitable growth and strict convexity conditions on
W, the minimum is attained and unique up to an inessential trans-
lation. We call the minimum uF;x. So
E½u ¼
ð
X
ð
Y
Wðrxuþryurxu;x; x; yÞdydx L½u (11)
3 Configurational Force Balance
We use the effective functional (11) to derive an effective con-
figurational force balance. We take L ¼ 0 for convenience (i.e.,
no body force, and traction-free and displacement boundary con-
ditions), but the treatment is easily modified otherwise.
Let u minimize E defined in Eq. (11) for some given boundary
conditions. Recall that it is smooth on the scale of e. We now con-
sider a variation, but by rearranging the domain.2 Consider a fam-
ily of rearrangments zs : X! X one to one and onto for s 2 ½0; 1
that satisfy z¼ x on @X; detrxz > 08x 2 X, and z0ðxÞ ¼ x. Set
usðxÞ ¼ uðzsðxÞÞ. Note that us is a family of perturbations of the
minimizer u with u0 ¼ u. Therefore, the function
f ðsÞ :¼ E½uðzsÞ (12)
has a minimum at s¼ 0 and therefore
f 0ð0Þ ¼ 0 (13)
We now compute f 0ð0Þ.
Set ws ¼ urxus ;x and FðxÞ ¼ rxuðxÞ, and note that
rxus ¼ rxðuðzsðxÞÞÞ ¼ FðzsðxÞÞrxzsðxÞ (14)
So
f ðsÞ ¼
ð
X
ð
Y
WðFðzsðxÞÞrxzs þryws; x; yÞdydx (15)
¼
ð
X
ð
Y
WðFðzÞðrzxsÞ1 þryws; xs; yÞJdydz (16)
where we have changed integration variables from x to z by
inverting zs to obtain x ¼ xsðzÞ, and set J ¼ detðrzxsÞ. Now, set
F
s ¼ FðzÞðrzxsÞ1; Gs ¼ Fs þryws ¼ Fs þryuF
s
;xs (17)
We can now calculate
f 0 sð Þ ¼
ð
X
ð
Y
@W
@Fij
_Gsij þ
@W
@xi
_xi þW
_J
J
 !
Jdydz (18)
where we use @=@xi to represent the explicit derivative with
respect to xi and _ðqÞ ¼ dq=ds to denote the total derivative of q
with respect to s. Recalling the identities
_
A1 ¼ A1 _AA1; _detA ¼ ðdetAÞATij _Aij (19)
we obtain
_G
s
ij ¼ _F
s
ij þ
@ _w
s
i
@yj
¼ Fik zð Þ @x
s
k
@zl
 1
@ _xsl
@zm
@xsm
@zj
 1
þ @
_w
s
i
@yj
(20)
Fig. 1 A macroscopic crack in a quasi-periodic heterogeneous
medium
1This holds true for common linear elastic problems. Failure of this condition
may lead to long-range instabilities. 2Such variations are known as inner variation in the calculus of variations.
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_J
J
¼ @x
s
i
@zj
 T @ _xsi
@zj
 1
(21)
Further, from the unit cell problem of minimizing (9), we can
infer that ð
Y
@W
@Fij
@ _w
s
i
@yj
dy ¼ 0 (22)
We substitute Eqs. (20)–(22) into Eq. (18), set s¼ 0 (so that
@xsi=@zj ¼ dij) and change integration variables back to x
f 0 0ð Þ ¼
ð
X
ð
Y
Wdik  @W
@Fij
Fik
 
@ _xk
@xj
þ @
W
@xk
_xk
 
dydx (23)
where _x ¼ _xsðzÞjs¼0. We note that F and _x are independent of y
and therefore we can integrate with respect to y. We obtain by
recalling (9)
f 0 0ð Þ ¼
ð
X
Wdik  @W
@Fij
Fik
 !
@ _xk
@xj
þ @
W
@xk
_xk
 !
dx (24)
¼
ð
X
 @
@xj
Wdik @W
@Fij
Fik
 !
þ@
W
@xk
_xk
 !
_xkdx (25)
where we have used the divergence theorem and the fact that _x ¼
0 on @X (since x ¼ zsðxÞ on @X). Now, since f 0ð0Þ ¼ 0 for all zs
and thus arbitrary _x, we obtain the macroscopic configurational
force balance
@Cij
@xj
¼ @
W
@xi
(26)
where
Cij ¼ Wdij  Fki @W
@Fkj
¼ Wdij  Fkirkj (27)
is the effective configurational stress tensor.
Finally, if the material is macroscopically homogeneous, i.e., W
and consequently W is independent of x, then we see from
Eq. (26) that
0 ¼
ð
D
@Cij
@xj
dx ¼
ð
@D
CijnjdA (28)
using the divergence theorem where D is any domain that is large
compared to the size of the heterogeneities (e). Now, consider a
domain with a crack in two dimensions. Given any two contours
C1 and C2 that contain the crack tip, set D to be the annular region
between the contours. Now, @D ¼ C1 [ C2 with the outward nor-
mal n and n on the two segments of the boundary. Taking the
inner product of the equation above with the tangent to the crack
tip, we obtainð
C1
ti Wdij  @uk
@xi
rkj
 
njds ¼
ð
C2
ti Wdij  @uk
@xi
rkj
 
njds (29)
or the path independence of the macroscopic J-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
J-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
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[3] Rice, J., 1968, “A Path Independent Integral and the Approximate Analysis of
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[4] Hill, R., 1963, “Elastic Properties of Reinforced Solids: Some Theoretical
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[5] Hossain, M., Hsueh, C.-J., Bourdin, B., and Bhattacharya, K., 2014, “Effective
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[6] Braides, A., 1985, “Homogenization of Some Almost Periodic Coercive
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Homogenization and Path Independence of the J-Integral in Heterogeneous Materials

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Homogenization and Path Independence of the J-Integral in Heterogeneous Materials

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