The random polycrystalline microstructure of microbeams necessitates a reexamination of the crack driving force G stemming from the Griffith fracture criterion. It is found that, in the case of dead-load conditions, G computed by straightforward averaging of the spatially random elastic modulus E is lower than that obtained by correct ensemble averaging of the stored elastic energy. This result holds for both Euler-Bernoulli and Timoshenko models of micro-beams. However, under fixed-grip conditions G is to be computed by a direct ensemble averaging of E. It turns out that these two cases provide bounds on G under mixed loading. Furthermore, crack stability is shown to involve a stochastic competition between potential and surface energies, whose weak randomness leads to a relatively stronger randomness of the critical crack length.

1.
Griffith
,
A. A.
,
1921
, “
The Phenomena of Rupture and Flow in Solids
,”
Philos. Trans. R. Soc. London, Ser. A
,
221
, pp.
163
198
.
2.
Gdoutos, E. E., 1993, Fracture Mechanics: An Introduction, Kluwer, Dordrecht, The Netherlands.
3.
Chudnovsky
,
A.
, and
Kunin
,
B.
,
1987
, “
A Probabilistic Model of Brittle Crack Formation
,”
J. Appl. Phys.
,
62
(
10
), pp.
4124
4129
.
4.
Kunin
,
B.
,
1994
, “
A Stochastic Model for Slow Crack Growth in Brittle Materials
,”
Appl. Mech. Rev.
,
47
, pp.
175
183
.
5.
Altus
,
E.
,
2001
, “
Statistical Modeling of Heterogeneous Micro-Beams
,”
Int. J. Solids Struct.
,
38
(
34–35
), pp.
5915
5934
.
6.
Beran
,
M. J.
,
1998
, “
The Use of Classical Beam Theory for Micro-Beams Composed of Crystals
,”
Int. J. Solids Struct.
,
35
(
19
), pp.
2407
2412
.
7.
Ostoja-Starzewski, M., 2001, “Mechanics of Random Materials: Stochastics, Scale Effects, and Computation,” Mechanics of Random and Multiscale Microstructures, D. Jeulin and M. Ostoja-Starzewski, eds., CISM Courses and Lectures 430, Springer, Wien, pp. 93–161.
8.
Rudin, W., 1974, Real and Complex Analysis, McGraw-Hill, New York.
9.
Ostoja-Starzewski
,
M.
,
2002
, “
Microstructural Randomness Versus Representative Volume Element in Thermomechanics
,”
ASME J. Appl. Mech.
,
69
, pp.
25
35
.
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