Solutions of coupled problem of thermomechanics for electroconductive hollow cylinder under non-stationary electromagnetic action

Abstract

A plane axisymmetrical coupled dynamic problem of thermomechanics for an electroconductive hollow cylinder under homogeneous non-stationary electromagnetic action is formulated. To construct its solution, the approximation of the determining functions the axial component of the magnetic field strength vector, the temperature, and the radial component of the displacement vector with respect to the radial variable by cubic polynomials is used. As the result, the initial-boundary problem for determining functions is reduced to a Cauchy problem with respect to the time variable for the integral characteristics of these functions. The expressions of integral characteristics are obtained in the form of a convolution of functions describing the uniform solutions and the given limit values of determining functions. As an example, the amplitude-frequency characteristics of the radial stresses in the given cylinder are analyzed numerically, with taking into account the connectivity between the temperature and the displacement fields as well as without such accounting.

Solution of the connected problem of thermomechanics for a long hollow electroconductive cylinder under the action of impulsed electromagnetic field with amplitude modulation

“…To construct the solutions of the initial boundary-value problems of the electrodynamics (5)-(7) and the connected problem of thermoelasticity (9)-(12), we use the approximation of the key functions over the radial variable by cubic polynomials [6,7]:…”

“…On the basis of the proposed methodics, described in detail in [7], we find solutions to the problem under consideration under action of the IEMF with amplitude modulation of the characteristic types described by expression (4).…”

“…Under such conditions, the calculated model for determining the thermo-stressed state of the cylinder consists of two stages [7]: at the first stage, from the equations of electrodynamics, the vector of magnetic field stress − → H in the cylinder and the predetermined by it Joule heat ejection Q and ponderomotive forces − → F are determined; and at the second stage from the equations of the connected dynamic problem of thermoelasticity, the temperature field T and the radial component U r (r, t) of the displacement vector − → U = {U r (r, t); 0; 0} and the corresponding components σ jj (j = r, ϕ, z) of the stress tensorσ are determined.…”

“…Substituting expression (4) into the found in [7] expressions of the integral characteristics Φ s (t) of key functions Φ(r, t), first we obtain the expressions: for axial component H z of the vector − → H of magnetic field stress…”

“…In the present work, using the proposed in [6,7] method which apply the cubic approximation of distributions of defining functions in a radial variable, we construct a general solution to the connected problem of thermomechanics for a long hollow electroconductive cylinder under the action of a impulse electromagnetic field with amplitude modulation.…”