Wave propagation in three-dimensional fractional viscoelastic infinite solid body

Abstract

In order to analyze the wave propagation in three-dimensional isotropic and viscoelastic body, the Cauchy initial value problem on unbounded domain is considered for the wave equation written as a system of fractional partial differential equations consisting of equation of motion of three-dimensional solid body, equation of strain, as well as of the constitutive equation, obtained by generalizing the classical Hooke's law of three-dimensional isotropic and elastic body by replacing Lam\'{e} coefficients with the relaxation moduli to account for different memory kernels corresponding to the propagation of compressive and shear waves. By the use of the method of integral transforms, namely Laplace and Fourier transforms, the displacement field, as a solution of the Cauchy initial value problem, is expressed through Green's functions corresponding to both compressive and shear wave propagation. Two approaches are adopted in solving the Cauchy problem: in the first one the displacement field is expressed through the scalar and vector fields, obtained as solutions of wave equations which are consequences of decoupling the wave equation for displacement field, while in the second approach the displacement field is obtained by the action of the resolvent tensor on the initial conditions. Fractional anti-Zener and Zener model I$^{{}^{+}}$ID.ID, as well as fractional Burgers model VII, as representatives of models yielding infinite and finite wave propagation speed, are used in numerical calculations and graphical representations of Green's functions and its short and large time asymptotic behavior.