Testing of defining relations of nonlinear theory of elasticity in an axial strain of a hollow cylinder

An attempt to validate the accuracy of the defining relations of nonlinear theory of elasticity is made using the model of static axial strain of a hollow cylinder. A closed system of nonlinear differential equations for two unknown functions is obtained. The first function describes the cylinder points' movement in the radial direction, and the second, in the axial direction. Displacements of the inner and outer surfaces of the cylinder are specified as boundary conditions. A difference scheme for resulting system transition to a system of nonlinear equations is described. The dependences of the axial force on the outer holder displacement are obtained for three quasilinear defining relations. In the first case, the energy stress tensor is related to the Cauchy–Green deformation tensor. In the second case, the «rotated» tensor of true stresses and the Hencky tensor are used. In the third case, the incompressibility condition is imposed. It is shown that the dependence of the axial force on the axial displacement of the outer cylinder surface significantly depends on the defining relation chosen. The obtained dependencies can be used to verify the reliability of the defining relations.