Fundamental solutions of the thermoelastic equations for transversely isotropic plates

Abstract

The problem of thermoelasticity for transversely isotropic plates acted upon by concentrated heat sources is solved. The {1, 2}-order equations of thermoelasticity that incorporate the transverse shear and normal stresses are used. A bending heat source with symmetric heat transfer is considered. The dependence of thermal stress components on the thermal and thermomechanical parameters of transversely isotropic plates is studied Introduction. Fundamental solutions play a determining role in solving various boundary-value problems in the mechanics of thin-walled members, including those under concentrated and local loads, including local thermal loads.The thermoelastic state of thin-walled members subject to concentrated thermal loads has been studied since the mid 20th century [6,7]. Two models of concentrated thermal loading of thin-walled members were proposed: concentrated heating [5,12] and concentrated heat source [1, 6, 13]. These models assume an arbitrary distribution of temperature (first model) or volume heat sources (second model) along one coordinate line across the thickness. The intersection point of this coordinate line with the mid-surface is where a concentrated thermal load is applied. In the case of the first model, these are integral characteristics of temperature different from their constant values in the whole domain. In the case of the second model, they are integral heat sources responsible for the nonuniform temperature distribution in the localization of the concentrated thermal load.The arbitrary distribution f z ( )of temperature (first model) or volume heat sources (second model) across the thickness can be represented as the sum of even and odd functions. The even component is responsible for a thermoelastic membrane state [9] of the plate with symmetric heat transfer (the heat transfer coefficients on the faces of the plate are equal). Such a thermal load is called a plane one [5,16]. The odd component of f z ( ) is responsible for a thermoelastic bending state [9] of the plate with symmetric heat transfer. Such a thermal load is called a bending one [5,16].The model of concentrated heat source is more adequate since it includes the heat-conduction equations. The first model, however, is more reliable when the source strength (energy per unit time) is unknown. In this case, it is sufficient to know the temperature in the loaded area and its distribution throughout the thickness.So far, the thermoelastic state of anisotropic plates and shells under concentrated thermal loads has been analyzed within the framework of the classical theory [2,13,15,17]. The classical theory neglects transverse shear and reduction. Therefore, refined theories are relevant when it comes to modern composite materials with low shear stiffness.We will solve the thermoelastic problem for a transversely isotropic plate with a concentrated heat source using the {1, 2}-order equations of thermoelasticity [9] that allow for all the components of the stress tensor. We will examine the effect of thermal and thermo...

Analysis of fundamental solutions to the equations of statics constructed for transversal-isotropic plates

“…The fundamental solution of the specified problem is built in paper [16]. The fundamental solutions for a flat problem and bending in the case of {1,2}-approximation are received in articles [11,12].…”

“…7-8 -generalized transverse forces for the state of bending. We carried out numerical calculations, which make it possible to compare the fundamental solutions (10), (11), built on the basis of generalized theory in the variants of {1,0}-and {1,2}-approximation, respectively. These studies allowed us to estimate the refinement, introduced by the use of equations of the {1,2}-approximation for transversal-isotropic plates, exposed to the action of concentrated force, instead of applying the analogous equations of {1,0}-approximation.…”

“…Among the publications, which apply a generalized theory of the {m,n}-approximation and the procedure of mapping, built with the help of specialized G-function, those articles should be noted that deal with the construction of fundamental solutions of the equations of statics for the transversal-isotropic plates based on the {1,0}-approximation for the state of bending [8] and based on the {1,2}-approximation for the zero spin stressed state and the state of bending [9,10]. Worth noting are also the publications, in which authors obtained fundamental solutions of differential equations on thermoelasticity of the {1,0}-and {1,2}-approximation for the transversal-isotropic plates [11,12].…”

Investigation of Stress-Strain State of Transversely Isotropic Plates Under Bending Using Equation of Statics {1,2} –Approximation

“…1-3) it can be seen that with increasing shear compliance parameter E / G′, considered 10 Q , 12 Q , 1 R are increasing in magnitude. where E -Young's modulus, h -the half thickness of the plate, 9 10 -the scaling factor) where E -Young's modulus, h -the half thickness of the plate, 9 10 -the scaling factor)…”

“…Among recent publications that use the generalized theory of {m, n} -approximation, it should be noted the articles on problems of statics [4,5], as well as publications that addressed the problem of thermoelasticity [6][7][8][9].…”