The initial approximation for iterative solution of the problem of propagation of small elastic disturbances in 3-D stress field
Journal Title: Вісник Тернопільського національного технічного університету - Year 2015, Vol 78, Issue 2
Abstract
The present stress-strain state in the structural elements, geological systems, biological structures, etc. influences essentially their operation properties, reliability and durability. Acoustic method, the basis of interrelation between the stress-strain state and characteristics of wave field, phase velocities, in particular, is one of the ways of its detection. Therefore, development of mathematical models describing the interrelation of the parameters of sounding field in a solid body with the components of the tensor of initial deformations in it is an important scientific and practical problem. The developed earlier mathematical model of dynamics of elastic disturbances in in-homogeneously deformed continuum to solve the problem of propagation of plane harmonic waves in 3-D field inhomogeneous initial deformations is used. This model represents the Cauchy problem for a system of three differential equations of hyperbolic type with variable coefficients. To obtain solution of the problem the iteration procedure is used, in initial (zero) approximation of which the in homogeneously field of initial deformations is approximated by mean-integral values along the directions of propagation in the zero approximation the relations of acousto-elasticity are obtained. They connect the phase velocity of waves of different polarizations with distribution of the component of 3-D strain tensor on this direction. Previously the analogous expression were obtained for the cases of propagation of the sounding pulse in the body in the state of plane deformation, when the wave propagates in the flat plane of symmetry and perpendicularly to it. The written relations are used to solve numerically the problem on propagation of plane wave an infinite space with initial deformations caused by the action of the applicator of concentrated force in the coordinate origin in the axis direction. The obtained numerical estimates of relative errors of phase velocities are compared with the corresponding velocities in non-deformed medium. The constructed relations can be used to formulate the direct and inverse problems of acoustic tomography of 3-D stress-strain state in a solid body.
Authors and Affiliations
Oksana Kravchyshyn
Keywords
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