Variational principles for generalized plane strain problems and their applications

This paper is concerned with the variational principles for the generalized plane strain problem of elasticity, which do not seem to have been documented well in the literature, hitherto. Both the total potential energy and the total complementary potential energy principles have been formulated and presented. Their counterparts in the context of generalized variational principles have also been presented. As a result, of the introduction of a discrete degree of freedom, i.e. the uniform direct strain out of the plane, which characterizes the generalized plane strain problem, a fair bit of complications arises. The minimum nature of the stationary values of the energy functionals may not be taken for granted as expected in their counterparts in conventional plane stress or strain elasticity. The expression of the total complementary potential energy obtained here has not been found before in the literature to the best of the authors' knowledge. This might be responsible for the fact that the generalized plane strain problem has been avoided in published work employing the variational principle based on the total complementary potential energy. The presently formulated total complementary potential energy functional has been applied to some classic problems in composites materials, viz. the analysis of transversely cracked laminates and the micromechanics of unidirectionally fibre-reinforced composites. Some interesting and/or new results have been obtained.