On the deformation of composite elastic bodies with empty pores

Abstract

The problem of refining the stress-strain state resulting from the deformation of composite elastic bodies with uniformly distributed empty pores is investigated. Based on the Lagrange variational principle and the Cowin–Nunziato micro-dilation model, the problems of static deformation of composite porous elastic bodies (rod and cylinder) are formulated taking into account surface effects in the conjugation region. In comparison with the classical formulation, equilibrium equations for non-classical stresses and additional boundary conditions and conjugation conditions for non-classical stresses and the porosity function are specified. In the course of non-dimensionalization of the problems, two small parameters are introduced: the connectivity parameter and the scale parameter responsible for the boundary effects at the boundary and in the contact region. Accurate analytical solutions of the problems for composite bodies are obtained, satisfying the corresponding boundary conditions and conjugation conditions. First, formulas are obtained for finding the distribution of the porosity function and non-classical stresses, and then formulas for finding displacements. The difference between the distribution of displacements found by solving the problems in the classical and micro-dilation formulations is shown. The influence of the connectivity parameter, the~scale parameter and the surface modulus of elasticity on the stress-strain state of composite porous elastic bodies is investigated. It is found that a local change in the porosity function occurs in the vicinity of the contact zone; with an increase in the scale parameter, a decrease in displacements and an increase in the width of the transition zone for the porosity functions are observed; with an increase in the connectivity parameter, an increase in the porosity functions and displacements and a smoother distribution of the porosity function in the contact area; when taking into account surface effects, porosity reaches a maximum value at the conjugation point, and then quickly decreases to the values calculated within the micro-dilation model in the absence of surface effects; non-classical stresses are proportional to the square of the scale parameter, at small values of which they have values much smaller than the values of classical stresses, and in the contact area they experience a jump or reach their peak in the absence of surface effects.