Finite element methodology for path integrals in fracture mechanics

Abstract

Based on the energy foundation of the path-independent integral in non-linear fracture mechanics, I* integral as the dual form of Rice's J is presented, it is also path-independent and is equivalent to J in value but it relates to the complementary energy. It is proved that, in numerical implementation, the path independence of J and I* can be ensured by using the assumed displacement finite elements and the assumed stress finite elements, respectively. Regarding the bounds of crack parameters, it is demonstrated that the lower bound of J can be estimated by the displacement compatible elements, and the upper bound of I* can be estimated by the stress equilibrium elements. In view of the difficulties in formulating stress equilibrium model, instead of it, a quasi-equilibrium model is proposed, which makes hybrid stress elements be able to estimate the bound of I*, and do not lose the characteristics of stiffness formulation. Two four-node plane elements are suggested; of them, the incompatible one can be used in incompressible/fully plastic fracture analysis, and the penalty-equilibrium one can be implemented to estimate the bound of I*. Furthermore, an incremental formulation is developed for I*, and can be extended into the calculations of ductile fracture under monotonic loading. For attestation, quite a number of numerical experiments is carried out, and some significant results are offered.1998 John Wiley & Sons, Ltd.

Accurate solution of traction free boundaries using hybrid stress membrane elements with drilling degrees of freedom

“…where (4)- (8), are, respectively, the equilibrium equations, the symmetry conditions for stress, the definition of rotation in terms of displacement gradients, the constitutive equations and the displacement boundary condition.…”

“…Reissner [16] presented a variational formulation for the boundary value problem reflected in (4)- (8). However, this formulation is inappropriate for numerical applications.…”

“…While the representation of traction free boundaries was initially attempted using ad hoc procedures [5,6], assumed stress elements proved more successful [7][8][9][10][11]. In assumed stress elements, representation of traction free boundaries can be accomplished via pre-treatment of the assumed stress field using rational constraints.…”