Magnetoelastic Deformation of Isotropic Variable-Stiffness Shells of Revolution: Allowing for Joule Heat and Geometrical
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International Applied Mechanics, Vol. 56, No. 2, March, 2020
MAGNETOELASTIC DEFORMATION OF ISOTROPIC VARIABLE-STIFFNESS SHELLS OF REVOLUTION: ALLOWING FOR JOULE HEAT AND GEOMETRICAL NONLINEARITY L. V. Mol’chenko*, I. I. Loos**, L. Ya. Vasil’eva, and A. Yu. Parkhomenko
The theory and derivation of the nonlinear thermomagnetoelastic equations of flexible variable-stiffness shells of revolution with allowance for Joule heat in a nonstationary magnetic field are considered. A technique for design of flexible variable-thickness shells of revolution in a magnetic field is proposed. The thermomagnetoelasticity of a truncated flexible conical axisymmetric shell with Joule heating taken into account is analyzed. Keywords: magnetic field, Joule heat, Lorentz force, variable stiffness, flexible conical shell Introduction. The theory of thermomagnetoelasticity of plates and shells is a division of elasticity theory. This is due to the fact that many members of modern engineering structures have the form of flexible plates and shells acted upon by mechanical, electromagnetic, and thermal loads [1, 3, 11]. Currently, theoretical and applied studies of nonstationary thermomechanical deformation of conductive members in magnetic fields have been actively developed [7, 8, 10, 12]. A rigorous analysis of deformation processes under thermoelectromagnetic loads should be based on the equations of mechanics, electrodynamics, and heat conduction, taking into account Joule heat. In coupled problems of magnetoelasticity, it is very important to allow for Lorentz forces and Joule heat. In what follows, we will consider the theory and technique of solving nonlinear problems of magnetoelasticity of flexible variable-stiffness shells of revolution with Joule heating taken into account. Using the technique proposed, we will analyze, as an example, the stress–strain state (SSS) of a flexible truncated conical shell of variable thickness. 1. Problem Statement. Two-Dimensional Nonlinear Magnetoelastic Equations of Flexible Variable-Stiffness Shells of Revolution. Let us solve a nonlinear magnetoelasticity problem to determine the SSS of conductive flexible variable-stiffness shells of revolution acted upon by nonstationary magnetic field and arbitrary mechanical load. Assume that the r shell is made of an isotropic material with finite conductivity s (S/m) and is in an external magnetic field H 0 (A/m). Moreover, r the shell is a conductor of uniformly distributed electric current of density J ex (A/m). The spatial magnetoelasticity equations in differential form in Lagrangian variables are the following [6, 8, 14]: r r r r r ¶B rotH = J , divB = 0, rotE = - , ¶t r r r ¶V r = r ( F + F Ù ) + div s$ , ¶t
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Mykolaiv V. O. Sukhomlynskyi National University, 24 Nikolskaya Str., Mykolaiv, Ukraine 54030; *e-mail: [email protected]; **[email protected]. Translated from Prikladnaya Mekhanika, Vol. 56, No. 2, pp. 83–94, March–April 2020. Original article submitted November 21, 2018. 198
1063-7095/20/5602-0198 ©2020 Springer Science+Business Media,
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