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# On the Axisymmetric Time-harmonic Lamb’s Problem for a System Comprising a Half-space and a Covering Layer with Finite Initial Strains

S.D. Akbarov1,2

Yildiz Technical University, Department of Mechanical Engineering, Yildiz Campus, 34349, Besiktas, Istanbul-TURKEY. Email:akbarov@yildiz.edu.tr
Institute of Mathematics and Mechanics of the National Academy of Sciences of Azerbaijan, 37041, Baku, Azerbaijan.

Computer Modeling in Engineering & Sciences 2013, 95(3), 173-205. https://doi.org/10.3970/cmes.2013.095.173

## Abstract

By employing the Three-dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies (TLTEWISB) the time-harmonic Lamb’s problem for a system comprising a finite pre-strained half-space and finite pre-strained covering layer made of incompressible materials is examined for the case where the material of the covering layer is stiffer than that of the half-space material. It is assumed that on the upper free face plane of the covering layer the point-located time-harmonic force acts. The elasticity relations of the materials are described through Treloar’s potential. The corresponding boundary-value problem is solved by employing the Hankel integral transformation. The corresponding inverse transformations are found (numerically) by utilizing the Sommerfeld contour. Numerical results regarding the stresses acting on the interface plane are presented and discussed. The main focus is on the frequency response of these stresses and the influence of the initial strains on them. In particular, it is established that the mechanical behavior of the forced vibration of the system under consideration is similar to that of the system comprising a mass, a parallel connected spring and a dashpot. Moreover, it is established that by increasing the stiffness of the covering layer material as well as with initial stretching of the covering layer, the “resonance” values of the stresses decrease.

## Keywords

Akbarov, S. (2013). On the Axisymmetric Time-harmonic Lamb’s Problem for a System Comprising a Half-space and a Covering Layer with Finite Initial Strains. CMES-Computer Modeling in Engineering & Sciences, 95(3), 173–205.

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