JOURNAL OF MATERIALS SCIENCE 34 (1999)4665– 4670
Failure criteria for multiply flawed
anisotropic materials
TSUNG-LIN WU
Automative Division of Mechanical Engineering Department, Nan Tai Technology Institute,
Tainan, Taiwan 701, Republic of China
JIN H. HUANG
Department of Mechanical Engineering, Feng Chia University, Taichung, Taiwan 407,
Republic of China
E-mail: jhhuang@fcu.edu.tw
This study presents overall failure criteria for an infinite anisotropic solid containing
multiple flaws subjected to a set of uniform applied loads. Based on the inclusion method,
flaws are treated as elliptical inclusions where their elastic moduli are considered to be
zero. The explicit expression of elastic fields is obtained for a cubic crystal multiply flawed
solid through the use of the Mori-Tanaka mean field theory. The resulting expression is
further utilized to find an interaction energy function between the applied loads and flaws.
With this energy function, the energy release rates and critical stresses are acquired
separately in a closed form for Mode I, II, and III. The closed forms for energy release rates
and critical stresses reveal that they are a function of the aspect ratio and the volume
fraction of flaws, the modes of the loading, and the material properties. As an illustrated
numerical example, the energy release rates and the critical stresses that vary with both the
aspect ratio and the volume fraction of the flaws in a cubic crystal material are
discussed.
C
1999
Kluwer Academic Publishers
1. Introduction
Historically, researches in flawed materials have been
focused on the analyses of the materials containing a
single flaw. However, it has been evident that most ma-
terials contain not a single flaw but multiple flaws, and
often the flaw density is very high in some materials.
Therefore, the related investigations about the averag-
ing elastic response and overall fracture criterion of
multiply flawed materials are needed to obtain the ef-
fective fracture criteria.
Many researchers have developed the fracture crite-
ria for single crack problems. Two-dimensional crite-
rion is originallyproposed by Griffith [1]. Followed the
Griffith’s work, Sack [2] and Sneddon [3] had obtained
the criteria for penny shaped cracked body for three-
dimensional problems. Kassir and Sih [4] had investi-
gated critical stresses and surface energy of flat ellip-
soidal cracked materials. The formal derivation based
on micromechanics for the single crack problem has
been given by Willis [5], Barnett and Asaro [6], Mura
and Lin [7], Mura and Cheng [8], Huang and Liu [9]
among others, further efforts, however, must be ex-
pended to perform a failure study of a solid containing
multiple cracks or flaws subjected to applied loading in
mode I, II, and III. Accomplishing such a task would
allow us to fully exploit the advantages of materials.
Therefore, in this study, we study the closed form of
the energy release rates and the critical stresses for el-
liptical flaws involved in an infinite solid subjected to
one of three kinds of applied loading.
The objective of this work is to develop an analyt-
ical and simple approach for determining the failure
criterion for many flaws in a three-dimensional, in-
finitely extended, anisotropic medium. To this end, this
article has focused primarily on the following issues.
First, the inclusion method [10] is developed to inves-
tigate the elastic fields around an elliptical inclusion
in a three-dimensional anisotropic solid. Secondly, the
results are extended to the multiply flawed problem by
means of the equivalent inclusion method [11]. By us-
ing the Mori-Tanaka [12] mean field theory and taking
the elastic moduli of the inclusion as zero, explicit so-
lutions for equivalent eigenstrains [10] (or stress-free
transformation strains [11]) are obtained for three load-
ing modes: a uniaxial tension, an in-plane shear, and
an out-plane shear. Then, the energy release rates and
the critical stresses of the Griffith fracture criterion are
presented in closed forms for multiply flawed materi-
als subjected to these three loading modes separately.
Finally, as an illustrated numerical example, the energy
release rates and the critical stresses vary with both as-
pectratioandthevolumefractionoftheflawsin acubic
crystal medium are discussed.
2. The inclusion method
Consider an infinitely extended solid D containing an
ellipsoidal inclusion whose elastic moduli C
ijmn
are
the same as the matrix. Here the shape of inclusion is
taken as ellipsoid that is capable of treating composite
0022–2461
C
1999 Kluwer Academic Publishers
4665