Variational analysis of cracked general cross-ply laminates under bending and biaxial extension

Abstract

In the current study, a variational model is developed for predicting stress transfer due to ply cracking in general cross-ply laminates subject to out-of-plane bending and biaxial in-plane loading. The model is valid for multiple ply laminates that can be nonsymmetric. Using the principle of minimum complementary energy, an optimal admissible stress field is derived that satisfies equilibrium, boundary, and traction continuity conditions. Natural boundary conditions are derived from the variational principle to overcome the limitations of the existing variational methods on the analysis of cracked general cross-ply laminates. Comparing laminates of glass/epoxy and graphite/epoxy with the available finite element results shows that stress components are in very good accordance with the analytical results. It is also shown that the existing variational models are specific cases of the current formulation. In the next step, the obtained stress field is used in conjunction with the principle of minimum complementary energy to get the effective stiffness modulus of a cracked general cross-ply laminate. It is indicated that the method provides a lower bound for the stiffness modulus of a cracked laminate.

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

“…The derived approach, when implemented together with a ply refinement technique, can be considered as a stress-based super element, where highly accurate result can be delivered and continuity of tractions and displacements at all element ply interfaces can be assured. The formulation can be enhanced to deal with un-symmetric laminates and bending loads by considering more stress perturbation functions [16]. This certainly necessitates a future detailed research work.…”

“…(14). Therefore, the result of integration over z in (16) would be written in the following form, based on the independent unknown perturbation functions: …”

“…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.…”

“…Kassapoglou and Lagace [10,19] were the first who applied the variational approach for analyzing stress concentrations near free edges. The reader is referred to the recent publications [1,2,16,17] for a detailed review of the developments in variational approach for the free-edge [1,2] and matrix cracking [16,17] stress transfer problems.…”

“…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.…”