Issue 46

A. Kostina et alii, Frattura ed Integrità Strutturale, 46 (2018) 332-342; DOI: 10.3221/IGF-ESIS.46.30 335 where С is the fourth-order stiffness tensor, el ε is the elastic strain tensor. In case of the isotropic material, the stiffness tensor has two components:     , E С С (5) where E is the Young’s modulus,  is the Poisson’s ratio. Volumetric strain due to a thermal expansion is calculated as:      0 T T T ε E (6) where  is linear thermal expansion coefficient, T is the temperature, 0 T is the initial temperature value. In case of the small strains, full strain tensor   e T ε ε ε is related to the displacement vector u by the geometric equation:       1 2 T ε u u (7) A perfect elastic body doesn’t have heat losses. This solid can be described by a linear relation (4). However, a real process of distribution of high-frequency elastic waves in solids is always accompanied by the energy dissipation related to the irreversible processes of internal friction (viscosity), thermal conductivity and diffusion. Intrinsic friction in the material is connected to the dissipative processes, which take place during the mechanical vibration of the system. Diversity of dissipative processes in the material has led to the large number of models taking into account energy dissipation during the dynamic loading. These models can be conditionally divided into two classes. The first class includes non-linear models describing hysteretic damping during the cyclic loading. A cyclic vibrational loading leads to the lag of the strain with respect to the stress (stress and strain are not in phase) and the attenuation of the elastic wave takes place. The second class includes models describing viscoelastic behavior of the material during the deformation process. In this work, both classes of the models are considered. A visco-elastic behavior of the coating is described by the Maxwell’s model. In the simplest one-dimensional case, this model can be represented as the sequential connection of an elastic spring and a viscous damper. The form of this constitutive equation in the frequency domain according to [10] is given below. In the frequency domain, the following representations of the strain and stress tensors are valid: ' ( ) i t d d real   σ σ e (8) ' ( ) i t d d real   ε ε e (9) where i d d e   σ σ and i d d e   ε ε , d σ is the deviatoric stress tensor, d ε is the deviatoric strain tensor. Then, the Hook’s law for the deviatoric stress tensor can be written as ' 2( ' '') ' el d d i   σ ε G G (10) where     G     2 2 ' 1 v v v G is the shear storage modulus,   G     2 '' 1 v v v G is the shear loss modulus,  v is the relaxation time, v G is the shear modulus. Energy dissipation averaged over the time period   2 / is expressed as   '' : ( ) v d d Q G conj ε ε (11) where  ( ) conj - complex conjugate values. Material (hysteretic) damping can be described by an isotropic loss factor  which is added to the elastic stiffness matrix: