Elastoplastic Axisymmetric Stress-Strain State of Laminated Shells Made of Isotropic and Transversely Isotropic Materials with Different Moduli

Abstract

A method is developed for analysis of the elastoplastic stress-strain state of laminated shells of revolution under axisymmetric loading. The shells are made of isotropic and transversally isotropic materials with different moduli. The method is based on the Kirchhoff-Love hypotheses for the whole laminate, the theory of deformation along paths of small curvature (for isotropic materials), and the theory of elasticity with different tensile and compressive moduli (transversely isotropic materials). The problem is solved by the method of successive approximations. Numerical examples are given Keywords: laminated shells of revolution, elastoplastic stress-strain state, theory of elasticity with different tensile and compressive moduli Introduction. Methods for analysis of the axisymmetric thermoelastoplastic stress-strain state (SSS) of laminated shells of revolution have been developed for isotropic and anisotropic inelastic materials deforming within [3, 4, 11, 16, etc.] and beyond [5,6,[12][13][14] the elasticity limits. The tensile and compressive moduli of these materials were assumed to be equal. It is a well known fact [2, 8, 15, etc.] that modern composite materials used in thin-walled structures often have not only anisotropy but also different tensile and compressive moduli. Therefore, to analyze the SSS of thin-walled structural members made of such materials, we need methods that would take these properties into account. There are publications, such as [1, 8, etc.], that describe methods of successive approximations and solutions to nonlinear problems for single-layer shells made of elastic heteromodulus materials. Methods for laminated shells combining isotropic and anisotropic heteromodulus materials are yet unavailable in the literature. In this connection and in support of the studies [3-6, 11-14, 16], here we set forth an approach to the elastoplastic stress-strain analysis of laminated transversely isotropic shells with different tensile and compressive moduli.1. Problem Formulation. Consider a shell of revolution with layers made of isotropic and transversely isotropic materials. The shell, which is initially unstrained and at temperature Ò = Ò 0 , is subjected to axisymmetric nonuniform heating and arbitrary loads, except for twisting. The layers are assumed to be in perfect mechanical and thermal contact. Let the shell be referred to a curvilinear orthogonal coordinate system s, , θ ς, where s is the meridional coordinate of the continuous reference surface, s s s

Linear relationship between the first invariants of the stress and strain tensors in theories of plasticity with strain hardening

“…The modern theories of plasticity with strain hardening [11][12][13][14][15], which refer to Bridgman's study [2], postulated the generalized Hooke's law in both elastic and plastic strain ranges; i.e., it is supposed that the volume of a solid does not change during elastoplastic deformation. In this case, the first invariant of the stress tensor is used as hydrostatic pressure and the first invariant of the strain tensor as volume strain; i.e., all modern theories of plasticity assume that the first invariants of the stress and strain tensors are in a linear relationship.…”

Elastoplastic deformation of elements of an isotropic solid along paths of small curvature: Constitutive equations incorporating the stress mode

“…The system of governing equations (3.9) is solved by a numerical time-stepping method and by the method of successive approximations at each step, satisfying prescribed boundary conditions [5][6][7][8]. An algorithm to solve this system of equations may be the following.…”

On two approaches to determining the axisymmetric elastoplastic stress-strain state of laminated shells made of isotropic and transversely isotropic bimodulus materials

“…The solutions of specific problems obtained by the two approaches are analyzed Keywords: laminated shell of revolution, elastoplastic stress-strain state, theory of bimodulus elasticity Introduction. A procedure to determine the axisymmetric stress-strain state (SSS) of laminated shells made of inelastic isotropic and elastic transversely isotropic materials with different tensile and compressive moduli is outlined in [8,11]. The procedure uses the theory of deformation along paths of small curvature [7] to describe the deformation of isotropic materials and the theory of bimodulus elasticity [4] to describe the deformation of transversely isotropic materials.…”