Cohesive Layer Models
This model employs a constitutive relationship in the form of a stress-strain relationship of the cohesive layer. The stresses and strains of the cohesive layer element are referred to axes which rotate with the element. Thus the effect of rigid body rotations is eliminated in the incremental strain – displacement relations. The model is defined in terms of the parameters describing the stress-strain relationships and the thickness of the cohesive layer. In the present context only the normal stress – strain relationship in the transverse direction is of pertinence, other relationships are assumed linear elastic. Poisson ratio is set to zero. For simplicity a nonlinear elastic relationship consisting of a linear elastic phase given by the modulus E2 followed by a phase in which stress remains constant at Omax (similar to “elastic-perfectly plastic” response) is assumed. The values of E2 and Omax must be representative of the core material through which the crack propagates. The other significant parameter is the thickness of the cohesive layer h0, the initial thickness of the cohesive layer. Ideally h0 is based on the observed dimensions of the process zone (Li, Y. et al, 2005). However the smallest possible dimension consistent with computational economy and reliability is chosen in the present work.
The cohesive material response calculations are performed in a module (UMAT) attached to the input program of Abaqus. A 4-noded plane strain/plane stress elements with reduced integration are selected for the cohesive layer for 2-D problems. Since reduced integration is used, a single integration point represents the whole element. The incremental strain (Аєп ) is based on the current cohesive layer thickness hc and so the total strain is approximately logarithmic6. As the nonlinear analysis proceeds, the value of GI is monitored as in:
GI = ^Onhc ten 0)
Failure is deemed to occur as soon as the following fracture criterion is satisfied: Gi > Gic. The main program supplies the incremental strains which are based on the current (updated) dimensions and the user supplied material subroutine (UMAT) returns to the main program material stiffness matrix and the stresses at the integration point.