Investigations On The Fractional Generalized Thermoelastic Problem With Variable Parameters

In order to overcome the shortcomings of the classical Fourier Law,scholars have developed a generalized heat transfer model that can be successfully used to describe non-stationary heat transfer.As known as the C-V models,phase lag models,and two step models and so on.However,we know that heat transfer will be accompanied by stress,which will affect the stability of the structure and even cause damage.From this point of view,the generalized thermoelastic theory considering the coupling effect of temperature-deformation is more practical.At present,the widely used ones mainly include the L-S theory based on the correction of the C-V heat wave equation,and the simultaneous introduction of two thermal relaxation times.The G-L theory established by introducing the temperature change rate in the constitutive equation and the G-N theory without considering the energy dissipation.In these theories,the energy equation is no longer parabolic.Therefore,the new hyperbolic heat equation can describe the propagation of heat in the medium at a finite velocity,and the model shows that the temperature field and the elastic field are coupled to each other.In recent years,due to the successful application of fractional calculus in stochastic dynamics and viscoelastic properties of biological tissues,some scholars have proposed using fractional calculus to modify the classical generalized thermoelastic theory to obtain a more applicable model.Among them,Sherief's and Youssef's fractional thermoelastic theories established based on fractional integrals have been widely applied and recognized.The object of generalized thermal-elasticity theory is usually an ideal elastic body model.In recent years,some scholars have found that some materials' related parameters(such as thermal conductivity)are closely related to temperature,that is,influenced by temperature changes.Can significantly affect the mechanical properties of the material.Diffusion is a phenomenon commonly found in nature.It can be defined as the random movement of a set of particles from a high concentration region to a low concentration region.In general,diffusion is common in the penetration of industrial raw materials and in the manufacture of integrated circuits,and its concentration is usually calculated using Fick's law.This is a simple matter diffusion equation in the form of the Fourier law,which does not take into account the interaction between the introduced material and the medium introduced into the medium or the influence of temperature on this interaction.Nackacki proposed the thermoelastic diffusion theory based on Fick's law,but in this theory,coupled thermoelastic theory is used,which means that the propagation velocity of thermoelastic waves is infinite.In 2004,Sherief et al.were inspired by the theory of generalized thermoelasticity and proposed a new C-V diffusion equation.A generalized thermoelastic diffusion theory was established.The predicted diffusion wave and heat wave are also propagated at a finite speed.The deficiencies of the classical Fick's law,and successfully described the mutual coupling effect of temperature field,deformation field and diffusion field.Based on Youssef's fractional generalized thermoelastic theory and Sherief's fractional generalized thermoelastic diffusion theory modified by Youssef's fractional model,the following studies were performed:(1)Considering Youssef's fractional-order generalized thermoelastic theory,the material-related parameters were studied.The influence of temperature on the dynamic response of the infinite boundary of a spherical cavity subjected to thermal shock shows that when the characteristic parameters of the material change with temperature,the dimensionless temperature,displacement,and stress are significantly affected.(2)Based on the fractional generalized thermoelastic diffusion theory,the dynamic response of an infinite structure with a spherical cavity with simultaneous thermal and chemical potential impact at the boundary is studied.The distribution of physical quantities are obtained,and the effect of fractional-order parameters are studied.