Influence of the Physical Properties of the Material on the Thermomagnetoelastic Behavior of a Flexible Conical Shell wi
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International Applied Mechanics, Vol. 56, No. 5, September, 2020
INFLUENCE OF THE PHYSICAL PROPERTIES OF THE MATERIAL ON THE THERMOMAGNETOELASTIC BEHAVIOR OF A FLEXIBLE CONICAL SHELL WITH ORTHOTROPIC CONDUCTIVITY AND JOULE HEAT L. V. Mol’chenko* and I. I. Loos**
Differential equations of thermomagnetoelasticity for flexible axisymmetric conical shells made of isotropic and orthotropic materials taking into account Joule heating are obtained. A nonlinear analysis of the thermoelastic state of a truncated conical shell is carried out for different physical properties of its material. Keywords: conical shell, thermomagnetoelasticity, orthotropy, nonlinearity, Joule heat Introduction. The action of alternating magnetic fields on metal elastic elements causes the electrodynamic volume forces and Joule heating, which, with the corresponding field parameters, cause large strains of structures [4, 6, 8, 12, 13, 15, 16, 19]. Note that taking ponderomotive forces and Joule heat into account is crucial in related problems of thermomagnetoelasticity. The effects of the interaction between mechanical strain fields and thermoelectromagnetic fields has not been studied adequately. This primarily concerns the problems of the deformation of thin-walled elements during finite displacements in a strong magnetic field, as well as current-carrying thin-walled elements. There arise very complex theoretical problems of magnetoelasticity of flexible thin shells made of real (i.e., finite-conductive) materials. Note also that the advanced technology employs structural materials that are anisotropic while undeformed or even orthotropic in specific cases. We will use here axisymmetric versions of the theory of finite-conductive isotropic and orthotropic conical shells in the microsecond range under nonstationary magnetic fields. The governing systems of equations will be represented in Lagrangian variables. Such complex problems can only be solved numerically. The basic governing equations of the electromagnetoelasticity of flexible conical shells of variable stiffness will be formulated, taking into account finite conductivity and Joule heat. 1. Problem Statement. Governing Axisymmetric Equations of Thermomagnetoelasticity of Flexible Conical Shells. Consider a truncated conical shell of variable stiffness in a magnetic field in the geometrically nonlinear case. An external r electric current of density J ex is supplied to the shell. Let the coordinate surface be the midsurface described in the undeformed state by curvilinear orthogonal coordinates ( s, q ), where s is the meridional arc length, and q is the azimuth angle. The shell will be described in an orthogonal spatial coordinate system ( s, q, g ) (Fig. 1), where g is the normal (to the midsurface) coordinate. Let us present the governing systems of thermomagnetoelasticity of a conical shell made of isotropic and orthotropic materials. 1.1. Isotropic Material. If we use the variational principle and take into account the Kirchhoff-Love hypotheses and electrodynamic hypotheses [1, 3
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