#### Installation — business terrible - 1 part

September 8th, 2015

This cohesive layer model has zero initial thickness and is represented as a line in the undeformed state. Elements constituting the layer begin to acquire finite thicknesses as delamination occurs under mode I. The notion of strain is suspended and relative displacements of the nodes of the element are used to characterize the deformation and compute the nodal forces. Elements in model are 4-noded with 2 displacement degrees of freedom per node. The incremental deformation is characterized in terms of averaged normal relative displacement A81 computed from the incremental nodal displacements. The stresses in the element are related to 81 and 82 respectively via nonlinear elastic relationships consisting of two phases (a linearly varying stress phase and a constant stress phase). 810 and 820 are the proportional limits of 81 and 82 and are prescribed taking into consideration of the

sufficiently small so that the elastic strain energy contribution is small compared to the cohesive strain energy in the layer. Nodal forces and the stiffness matrix are found in terms of the stress state in the

element. These are transformed to the global axes and returned to the main program. The total strain energy stored per unit area in each mode in the element is given as a sum of incremental contributions as follows:

Gic = X °n AS1 (2)

Failure is deemed to occur as soon as the following fracture criterion is satisfied: GI > GIc.

Fig. 1 shows a typical element ABCD and its deformed configuration A’B’C’D’; the original coordinates and displacements are given by (x, y) and (u, v) respectively with appropriate subscripts. The main program supplies for each increment of loading the current nodal displacements at the end of the previous loading increment and their corresponding increments for each iteration with reference to the global coordinate system.

The current relative displacements in the normal (81) and tangential directions (S2) with respect to the current orientation of the element are then computed by transformation:

*51 *= (v2 – v1)cos0 – (u2 – u1)sin 0

*52 *= (v2 – v1)sin 0 + (u2 – u1)cos0

where:

1 2 2

Note A coincides with D and B with C in the original configuration. From the assumed stress – displacement relations, the normal and shear stresses carried by the element are determined. These in turn are employed to determine the nodal forces in terms of the element length. The tangential stiffness matrix for the element is set up by appropriate differentiation. The current nodal forces and the tangential stiffness matrix are then transformed to the global axes and returned to the main program. Note that the nodal forces are updated for every iteration.