Investigation On The Dynamic Response Of The Multi-fields Coupling Problems Under The Fractional Order Generalized Thermoelasticity

As described in classical Fourier's heat conduction law,heat propagates in elastic media with an infinite speed and the heat flux vector is proportion to the temperature gradient.For the long-term heat transfer process,the heat conduction would evolve into a steady state,and it is accurate enough to characterize such heat conduction process by using the classical Fourier's heat conduction law.However,under the extreme heat transfer conditions,heat propagation will take on the so-called non-Fourier's effect,and several non-Fourier's heat conduction models were thus developed,for example,C-V heat wave model,single phase lag hyperbolic heat conduction model,hyperbolic two-step model,dual phase lag model,triple phase lag model as well as thermal mass model etc..In the development of non-Fourier's effect models,heat is assumed as wave and propagates with a finite speed.Several generalized thermoelastic theories were also proposed by combining the non-Fourier's heat conduction model with the classical thermoelastic theory.For prolems with macroscopic spatial scale while with micro temporal scale,the generalized thermoelastic theories mainly include: Lord and Shulman generalized thermoelastic theory(one thermal relaxation time introduced),Green and Lindsay generalized thermoelastic theory(two thermal relaxation times introduced)and Green and Naghdi generalized thermoelastic theory(without energy dissipation).For a number of materials,such as viscoelastic materials,porous materials,biological materials,organic materials and polymers etc.,and physical processes,such as abnormal conduction and anomalous diffusion etc.,due to the memory-dependent nature of these materials and processes,it is challengeable for the existing thermoelastic theories to properly stipulate their thermoelastic behaviors.So,fractional order calculus operator was gradually introduced into the heat conduction equation to establish the fractional order generalized themoelastic theories.In this thesis,the emphasis was placed on the interdiscipline of thermoelasticity and fractional order calculus.Based on the fractional order generalized thermoelasticity,three kinds of thermoelastic problems with macroscopic spatial scale while with micro temporal scale,including the multi-field coupling piezoelectric-thermoelastic problems,dynamic thermal shock problem of half space with spherical cavity and the two-dimensional electormagneto-thermoelastic problems of half space,were investigated.According to the dimension of the problems,the specific geometrical shape,the loading condition,the initial condition and the boundary condition etc.,the fractional order generalized thermoealstic multi-field coupling models for the above problems were formulated,and the variations of all the considered variables were obtained by effective analytical or numerical methods.The main research works are as follows:(1)For the multi-field coupling piezoelectric-thermoelastic problems with macroscopic spatial scale while with micro temporal scale and subjected to a moving heat source,three different cases were investigated by means of Laplace transform and its numerical inversion.In case one,the material properties were assumed to be constant;in case two,the material properties were assumed to be functions of coordinate,while in case three,the material properties were assumed to be functions of temperature.By numerical calculation,the non-dimensional temperature,displacement,stress and electric potential were obtained.As shown from the results,fractional order parameters and the heat source speed play very important roles on the variations of the considered variables.At the same time,the material parameters changing with coordinate of with temperature also affect the distributions of the considered physical quantities.In case one,when the fractional order parameter is constant,the absolute peak values of the non-dimensional displacement,electric potential,temperature and stress decrease with the increase of the heat source speed;When the moving heat source speed is kept constant,the absolute peak values of the non-dimensional displacement,electric potential,temperature and stress increase with the increase of the fractional order parameter.In case two,along with the increase of the fractional order parameter,the absolute values of non-dimensional displacement,stress,temperature and electric potential increase,while the heat wave velocity and stress wave velocity decrease accordingly;As the material parameters changing with coordinate increase,the absolute peak values of the non-dimensional displacement,temperature,stress and electric potential decrease.In case three,when the fractional order parameter and heat source speed are constant,the peak values of the non-dimensional temperature,displacement and stress increase with the increase of material parameters;While with the increase of the fractional order parameter,heat wave velocity and stress wave velocity decrease accordingly.(2)For the dynamic thermal shock problem of half space with spherical cavity,due to the effect of thermal shock,the half space undergoes thermal deformation,which in turn,induces displacement and stress,due to the thermoelastic coupling effect.With the extension of time,the thermoelastic coupling effect gradually spreads from the surface of the spherical cavity into the entire body.Fractional order parameter has significant influence on the distributions of various physical quantities.With the increase of fractional order parameters,the fluctuation cycles of the non-dimensional temperature,displacement,radial stress and tangential stress decrease accordingly.(3)For the dynamic response of a two-dimensional electromagnetothermoelastic half space,the normal mode method is applied to obtaining the distributions of the non-dimensional temperature,displacement and stress under thermal shock loading.As seen from the obtained results,fractional order parameter has important influence on the considered physical quantities.With the increase of fractional order parameter,thermal wave velocity decreases;To the two-dimensional problem,the variation trends of the non-dimensional displacement,temperature,stress etc.is not only dependent on the time but also on the spatial coordinates.