Investigation of nonlinearity of longitudinal displacement function from mechanical and geometric characteristics of the plate

Layered composite materials are characterized by a high transverse anisotropy and low values of relations between the transverse shear modulus and the modulus of longitudinal elasticity. As a result, the behavior of longitudinal displacements and longitudinal normal stresses differs from the linear law, and the behavior of transverse tangential stresses differs from the parabolic law. The paper presents the analysis of the degree of the plate displacement nonlinearity functions depending on its elastic properties and geometric shape. The article considers a 2D square plate deformation problem. In non-dimensional terms, the authors could received a complete and demonstrative solution. A similar 3D problem is more bulky but it has no principle differences. The study of the degree of the longitudinal displacement nonlinearity functions due to elastic properties and the geometric shape of the deformable plate is based on the finite element method. The potential energy of the deformable plate is expressed through a square form of the variables with coefficients that are polynomials of dimensionless parameters such as plate size, ratio of the elasticity and shear moduli, Poisson's modulus. It is shown that the variational principle reduces the problem to the solution of the system of linear equations. As a result, the subareas of linearity and nonlinearity of the plate longitudinal displacements are constructed with an accuracy acceptable for engineering calculations of 5 %. It is necessary to consider the plate nonlinear longitudinal deformations for a length of the composite plate on order more than its thicknesses. As for steel, nonlinearity is characteristic for quite thick plates. The constructed areas of linearity and nonlinearity of the longitudinal displacements make it possible to construct the strain-stress state models with a smaller number of variables.