The problem of kernel determination from viscoelasticity system integro-differential equations for homogeneous anisotropic media

Abstract

We consider the problem of reconstructing the time-dependent history of the viscoelasticity medium from the viscoelasticity system of equations for an homogeneous anisotropic medium. As additional information, the Fourier image of the displacement vector for values ν = ν 0 = 0 of transformation parameter is given. It is shown that if the given functions satisfy some conditions of agreement and smoothness, the solution for the posed problem is uniquely defined in the class of a continuous functions and it continuously depends on given functions.Keywords: inverse problem, integro-differential equation, delta function, Fourier transformation, agreement condition. Received: 8 March 2016Revised: 20 April 2016 Setting up the problem and main resultWe consider the integro-differential system for x = (x 1 , x 2 , x 3 ) ∈ R 3 , t ∈ R:at the initial conditions* is the displacement vector function, * is the sign of transposition, T ij denotes the stress tensor related to the viscoelastic medium. More exactly, we have:* is the external force; ρ > 0 is the density of the medium. In equality (3), coefficients c ijkl are the elastic moduli of the medium. It is convenient and customary to describe the elastic moduli in the terms of a 6 × 6 matrix according to the following conventions relating a pair (i, j) of indices i, j = 1, 2, 3 to a single index α = 1, 2, . . . , 6 : (11) → 1, (22) → 2, (33) → 3, (23) = (32) → 4, (13) = (31) → 5, (12) = (21) → 6. This correspondence is possible due to the symmetry properties c ijkl = c jikl = c ijlk . The additional symmetry property c ijkl = c klij implies that the matrix C = (c αβ ) 6×6 of all moduli is symmetric, where α = (ij), β = (kl). We will also assume that ρ > 0, c ijkl are constants and the matrix C = (c αβ ) 6×6 is positive definite.Many important materials used in modern technologies (such as nanotechnology) are viscoelastic and anisotropic. Viscoelastic materials have the properties of viscosity and elasticity upon deformation. Some mathematical models in the field of nanotechnology are contained, for example, in articles [1, 2] (see also references in them). In mathematical modeling of processes taking place in viscoelastic materials, there is a so-called system with memory, whose behavior is not completely determined by the state at the moment, but depends on the systems entire history, and therefore, describes an integro-differential equation that contains the corresponding integral with respect to the time variable. The system of equations (1), taking into account the integral term (3) is the basic in the linear theory of viscoelastic anisotropic media.A study of inverse problems for hyperbolic integro-differential equations and systems is the subject of research by many authors. Among the problems that are closer to the present work can be identified [3][4][5][6][7][8][9]. In papers [3,4], the unique solvability and stability of the solution for the inverse problem for the identification of a memory kernel from Maxwell's system integro-differential equations for a homoge...

Analog of the Darboux problem for a loaded integro-differential equation involving the Caputo fractional derivative

“…As noted in other studies [25,26], many important materials used in modern technologies (such as nanotechnology) are viscoelastic and anisotropic. In mathematical modeling of processes taking place in viscoelastic materials, there is a so-called system with memory, whose behavior is not completely determined by the state at the moment, but depends on the systems of entire history, and therefore, describes an integro-differential equation which contains the corresponding integral with respect to the time variable.…”

“…In mathematical modeling of processes taking place in viscoelastic materials, there is a so-called system with memory, whose behavior is not completely determined by the state at the moment, but depends on the systems of entire history, and therefore, describes an integro-differential equation which contains the corresponding integral with respect to the time variable. The equation ( 1), taking into account the integro-differential operator is basic in the linear theory of viscoelastic anisotropic media [26].…”