Conserving Galerkin weak formulations for computational fracture mechanics

In this paper, a notion of invariant Galerkin-variational weak forms is proposed. Two speci c invariant variational weak forms, the J -invariant and the L-invariant, are constructed based on the corresponding conservation laws in elasticity, one of which is the conservation of Eshelby’s energy-momentum (Eshelby, Philos. Trans. Roy. Soc. 1951; 87:12; In Solid State Physics, Setitz F, Turnbull D (eds). Academic Press: New York, 1956; 331; Rice, J. Appl. Mech. 1968; 35:379). It is shown that the nite element solution obtained from the invariant Galerkin weak formulations proposed here can conserve the value of J -integral, or L-integral exactly. In other words, the J and L integrals of the Galerkin nite element solutions are path independent in the discrete sense. It is argued that by using the J -invariant Galerkin weak form to compute near crack-tip eld in an elastic solid, one may accurately calculate the crack extension energy release rate and subsequently the stress intensity factors in numerical computations, because the ux of the energy-momentum is conserved in discrete computations. This may provide an alternative means to accurately simulate crack growth and propagation. Copyright ? 2002 John Wiley & Sons, Ltd.