Comparison of Variational and Generalized Plane Strain approaches for matrix cracking in general symmetric laminates

Abstract

The current research work is aimed at comparing a stress-based variational approach with generalized plane strain (GPS) model for predicting stress transfer in cracked general symmetric laminates, subject to general in-plane loading. For each model, the fundamental assumptions made on stress and displacement fields, and also procedures for defining boundary conditions and finding solutions are detailed. Moreover, a numerical comparison has been made between the results obtained from two models with regard to stresses, displacements, and thermo-elastic constants of an unbalanced symmetric cracked laminate. Finally, the accuracy and quality of both approaches are examined by checking the satisfaction of known inter-relationships among thermo-elastic constants of cracked laminates. The paper extends an existing variational model so that the effects of residual thermal stresses can be considered. It has been also shown how the displacement field can be determined for a stress-based variational approach.

A variational method for analysis of laminates with parallel arrays of intralaminar cracks

“…Several publications followed this approach, keeping Hashin's assumption of the in-plane stresses being independent of the transverse direction within each ply (eg, the works of Hajikazemi et al 10,11 ). It is worth noticing that comparison of the variational analysis applied to symmetric laminates assuming piecewise constant in-plane perturbation stress with results of the Generalized Plane Strain approach of McCartney (see, eg, the work of McCartney 12 and references therein), performed recently by Hajikazemi and McCartney, 13 demonstrated almost identical results in terms of both stress and displacement fields when sufficient ply refinement was employed.…”

“…According to the derivation in Appendix A, for (11) to hold, it suffices to set the perturbation forces N 1 and N 6 and perturbation moment M 1 constant, which is part of the more general consideration of force and moment equilibrium (12)- (13).…”

“…Substitution of (2) into the six constraints in (12) and (13) shows that the constraints are linear in the unknown functions forming f and after integration over can be written in a matrix form too…”

A variational model for free-edge interlaminar stress analysis in general symmetric and thin-ply composite laminates

“…Contrary to displacement-based approaches, the approximate stress field derived with variational approaches satisfy exactly all the necessary equilibrium equations, interlaminar continuity and boundary conditions including zero-stress conditions at the free edges as well as the top and bottom surfaces of laminates. This approach was first introduced by Hashin [14] in 1985 for the analysis of cracked two-layer cross-ply laminates under uniaxial tension and further developed by many authors for analyzing cracked laminates with general lay-ups, see for example [15][16][17][18] as the most complete variational models developed so far. Kassapoglou and Lagace [10,19] were the first who applied the variational approach for analyzing stress concentrations near free edges.…”

“…Therefore, the interlaminar stress transfer problem can be reduced to a simple boundary value problem which is dependent only on the axial or transverse coordinate of the assumed laminate. For the variational free-edge stress transfer analysis [10,19] unlike the developed variational ply cracking models [15][16][17][18], this boundary value problem leads to a set of non-homogeneous differential equations as the Euler's equation(s) from the variational calculus.…”