ON THE LOSS OF STABILITY OF A TWO-LAYERED PLATE MADE OF A FUNCTIONAL-GRADIENT MATERIAL WITH A NON-UNIFORM FIELD OF PRE-STRESSES

In the framework of three-dimensional nonlinear elasticity we consider linear instability of a composite plate made of functionally graded material and having initial stresses. The plae consists of two layers which were obtained as a result of flattening of an annual sector of an elastic cylinder. This deformation results in appearance of internal stresses. Thus, the plate becomes initially stressed. The initial stresses depend on the thickness coordinate, so we get inhomogeneous stress field. We have two types of inhomogeneities, the first is the inhomogeneity of the initial stresses whereas the second is the material inhomogeneity.We use the incompressible neo-Hookean material model as a constitutive relation. Despite of relatively simple form this model describes properly severe deformations of some rubber-like materials. For incompressible materials the flattening constitutes one of the so-called universal deformations, that is such deformation which is independent on the choice of constitutive relation. The material inhomogeneity is described through a dependence of the shear modulus on the thickness coordinate. Such inhomogeneity could be related to the manufacturing of the material or to further treatment. The stability was analysed using the linearization approach. We superimpose infinitesimal deformations on the finite initial one. The linearized boundary-value problem was derived and its nontrivial solutions were obtained. The solution was obtained in series of trigonometric functions. This helps to automatically satisfy a part of boundary conditions. We consider the influence of the inhomogeneity and initial stresses. We show that the initial stresses may significantly change critical deformations. For example, the loss of stability is possible due to initial stresses only.