Determination of elastic constants based on the solution of the Lame problem

Abstract

Two-constant forms of relationships between stresses and strains in nonlinear-elastic isotropic materials are presented. Such materials can be used to dampen vibrations in building structures under dynamic loads (earthquakes, shock waves from explosions). The free energy of the considered relationships is represented as a function of algebraic invariants of the Cauchy-Green strain tensor or natural invariants of the “left” Hencky strain tensor. A method for determining the constants of the presented relationships between stresses and strains has been developed. The proposed method is based on the analysis of experimental dependencies of circumferential deformations on the outer and inner surfaces on the applied internal pressure and solutions to the Lam´e problem for a hollow cylinder in a flat deformed state. It is shown that the present constitutive relationships can be particularized by identifying the linear section of the experimental dependencies and constructing theoretical dependencies under the assumption of small deformations. Thus, the data following the linear section can be used to specify the third-order elasticity moduli of the determining relations constructed on the basis of those considered. Consequently, the methodology presented in the work can also be considered as a partial solution to the problem of particularization the relationships between stresses and strains, including third-order elasticity moduli. For the experimental data presented, it is shown that the results of particularization according to the proposed method correspond to the elasticity moduli determined by means of a classical tensile experiment. The presented method can be used both directly and for the purpose of minimizing the number of experiments in the tasks of particularization the constitutive parameters of nonlinear elasticity theory.