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Plane waves and vibrations in the theory of elasticity for materials with a triple porosity structure
Last modified: 2017-06-09
Abstract
The concept of triple porosity media is used in many areas of applied science (e.g., geophysics, biology, biophysics, biomechanics) and engineering. The intended applications of the theories of elasticity and thermoelasticity for materials with a triple porosity structure are to geological materials such as rocks and soils, manufactured porous materials such as ceramics and pressed powders, and biomaterials such as bone.
In the present talk a new mathematical model for materials with triple (macro, meso and micro) porosity structure based on the balance of equilibrated force is presented. The governing system of field equations is expressed in terms of the displacement vector and the volume fractions fields of the three pore systems. In addition, the following results are obtained:
1. The basic properties of plane harmonic waves are established;
2. The fundamental solution of the system of equations of steady vibrations is constructed;
3. The Green's formulae and the radiation conditions are established;
4. The existence of eigenfrequencies of the internal boundary value problems (BVPs) of steady vibrations are proved;
5. The formula of the asymptotic distribution of these eigenfrequencies is obtained;
6. The uniqueness theorems for the classical solutions of the external BVPs are proved;
7. The basic properties of surface and volume potentials are established;
8. On the basis of the boundary integral equation method and the theory of singular integral equations the existence theorems for the classical solutions of the external BVPs of steady vibrations are proved.
In the present talk a new mathematical model for materials with triple (macro, meso and micro) porosity structure based on the balance of equilibrated force is presented. The governing system of field equations is expressed in terms of the displacement vector and the volume fractions fields of the three pore systems. In addition, the following results are obtained:
1. The basic properties of plane harmonic waves are established;
2. The fundamental solution of the system of equations of steady vibrations is constructed;
3. The Green's formulae and the radiation conditions are established;
4. The existence of eigenfrequencies of the internal boundary value problems (BVPs) of steady vibrations are proved;
5. The formula of the asymptotic distribution of these eigenfrequencies is obtained;
6. The uniqueness theorems for the classical solutions of the external BVPs are proved;
7. The basic properties of surface and volume potentials are established;
8. On the basis of the boundary integral equation method and the theory of singular integral equations the existence theorems for the classical solutions of the external BVPs of steady vibrations are proved.