This paper considers the problem of a semi-infinite, isotropic, linear viscoelastic half-plane containing multiple, nonoverlapping circular holes. The sizes and the locations of the holes are arbitrary. Constant or time dependent far-field stress acts parallel to the boundary of the half-plane and the boundaries of the holes are subjected to uniform pressure. Three types of loading conditions are assumed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, a force uniformly distributed over the whole boundary of the half-plane. The solution of the problem is based on the use of the correspondence principle. The direct boundary integral method is applied to obtain the governing equation in the Laplace domain. The unknown transformed displacements on the boundaries of the holes are approximated by a truncated complex Fourier series. A system of linear equations is obtained by using a Taylor series expansion. The viscoelastic stresses and displacements at any point of the half-plane are found by using the viscoelastic analogs of KolosovMuskhelishvili's potentials. The solution in the time domain is obtained by the application of the inverse Laplace transform. All the operations of space integration, the Laplace transform and its inversion are performed analytically. The method described in the paper allows one to adopt a variety of viscoelastic models. For the sake of illustration only one model in which the material responds as the standard solid in shear and elastically in bulk is considered. The accuracy and efficiency of the method are demonstrated by the comparison of selected results with the solutions obtained by using finite element software ANSYS.
This paper is concerned with the problem of an isotropic, linear viscoelastic half-plane containing multiple, isotropic, circular elastic inhomogeneities. Three types of loading conditions are allowed at the boundary of the half-plane: a point force, a force uniformly distributed over a segment, and a force uniformly distributed over the whole boundary of the half-plane. The half-plane is subjected to farfield stress that acts parallel to its boundary. The inhomogeneities are perfectly bonded to the material matrix. An inhomogeneity with zero elastic properties is treated as a hole; its boundary can be either traction free or subjected to uniform pressure. The analysis is based on the use of the elastic-viscoelastic correspondence principle. The problem in the Laplace space is reduced to the complementary problems for the bulk material of the perforated half-plane and the bulk material of each circular disc. Each problem is described by the transformed complex Somigliana's traction identity. The transformed complex boundary parameters at each circular boundary are approximated by a truncated complex Fourier series. Numerical inversion of the Laplace transform is used to obtain the time domain solutions everywhere in the half-plane and inside the inhomogeneities. The method allows one to adopt a variety of viscoelastic models. A number of numerical examples demonstrate the accuracy and efficiency of the method.
A new computational approach for calculation of the effective transverse mechanical properties of unidirectional fiberreinforced composites with linear viscoelastic matrix and elastic fibers is presented. The approach requires the knowledge of stresses outside a cluster representing the structure of composite in question. The effective properties are found from the assumption that the viscoelastic stresses at the distances far away from the cluster are the same as those from a single equivalent inhomogeneity. The approach directly takes into account the interactions between the inhomogeneities. The comparison of the results with several benchmark solutions reveals the advantages of the developed approach.
This paper presents the complex Green function for the plane-strain problem of an infinite, isotropic elastic plane containing a circular hole with surface effects and subjected to a force applied at a point outside of the hole. The analysis is based on the Gurtin and Murdoch [1975, “A Continuum Theory of Elastic Material Surfaces,” Arch. Ration. Mech. Anal., 57, pp. 291–323; 1978, “Surface Stress in Solids,” Int. J. Solids Struct., 14, pp. 431–440] model, in which the surface of the hole possesses its own mechanical properties and surface tension. Systematic parametric studies are performed to investigate the effects of both surface elasticity and surface tension on the distribution of hoop stresses on the boundary of the hole and on a line that connects the point of the applied force and the center of the hole.
The problem of thermal stress development in composite structures containing one linear isotropic viscoelastic phase is considered. The time-temperature superposition principle is assumed to be applicable to the viscoelastic media under consideration. Two methods of solution based on the reduction of the original viscoelastic problem to the corresponding elastic one are discussed. It is argued that the use of a method based on the Laplace transform is impractical for some problems, such as those involving viscoelastic asphalt binders. However, the solution can be obtained by means of the second method considered in the paper, the Volterra correspondence principle, in which the integral operator corresponding to the master relaxation modulus is presented in matrix form. The Volterra principle can be applied to the solution of viscoelastic problems with complex geometry if the analytical solution for the corresponding elastic problem is known. Numerical examples show that the proposed method is simple and accurate. The approach is suitable to the solution of problems involving viscoelastic materials, whose rheological properties strongly depend on temperature. In particular, it can be found useful in the analysis of the low-temperature thermal cracking of viscoelastic asphalt binders.
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