Abstract:
This paper presents a mathematical model for constructing elastic fields for transversely isotropic bodies of revolution under the conditions of the inverse problem of elasticity, where the displacements prescribed on the body surface vary over time according to a cyclic law. An axisymmetric disturbance propagates at a constant velocity along one of the elastic symmetry axes of the material. The boundary state method is used to solve the problem. Using the method of integral superposition, a relationship is established between the spatial stress-strain state of the transversely isotropic elastic body and certain auxiliary two-dimensional states. The auxiliary states are constructed based on the general solution of the plane stationary dynamic problem. A set of such plane auxiliary states is generated, and a corresponding set of spatial states is obtained by applying the transformation formulas. This set forms a finite-dimensional basis of the internal states with the desired solution expanded after orthogonalization into a Fourier series with the same coefficients. The solution of the inverse dynamic problem of elasticity is presented for a transversely isotropic circular cylinder with the kinematic boundary conditions varying according to the cosine law.
Keywords:method of boundary states, stationary isotropic problems, inverse problem of elasticity, transversely isotropic body, axisymmetric deformation.
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