A transversely isotropic half- space with axis of material symmetry perpendicular to the free surface supports a flexible circular plate. The contact area of the plate and the half- space is considered to be both frictionless and unbonded (tensionless). The foundation is affected by a vertical static axisymmetric load. Detailed analysis of the interaction of these two systems is the target of this paper. With the use of ring load Green’s functions for both the plate and the continuum half- space, dual integral equations accompanied with some inequalities are obtained to model the complex boundary value problem. With the incorporation of the ring- shape finite element method, where its size gradually varies, we are capable of capturing both regular and singular solution smoothly. The validity of the combination of the analytical and numerical method is proved with comparing the results of this paper with a number of benchmark cases of both linear and nonlinear interaction of circular plate and half- space. Some new illustrations are presented to portray the aspect of the anisotropy of the half- space.

The main goal for this research is to make an in depth analysis for a transversely isotropic half-space under the effect of a circular rigid disc attached on the surface of the domain affected by a bending moment. To do so, the potential method is used to uncouple the Navier’s equilibrium equations. Utilizing both Fourier series and Hankel transforms, the potential function is determined. To find the unknown coefficients of the solution of differential equations involved in this paper, the boundary conditions are transforms into dual integral equations. The dual integral equation is analytically solved from which the displacements and stresses are determined based on displacement- and stresses-potential relations. The contact pressure is presented in a closed form format and illustrated to be discussed deeply. By numerical evaluation of the results and comparing the results for the simpler case of isotropic half-space both the validity and accuracy are shown.

The existence and extension of the cracks in structural materials is one of the issues to be considered to prevent the devastating effects of cracks. Cracks can be subjected to different types of fracture modes that considering these modes help us to predict the behavior of cracks. This paper investigates the effects of the fracture modes (opening, shearing and tearing) on the annular crack in an infinite transversely isotropic solid. In each mode, by substituting the boundary conditions into governing equations of the medium, the problem reduced to triple integral equations. With the aid of Hankel and Abel integral transforms, the triple integral equations reduced to two Fredholms integral equations which are amenable to numerical solutions. The inner and outer stress intensity factors of the annular crack are obtained for different ratios of inner-outer radius of the annular crack. Some limiting cases such as the penny-shaped crack and external crack are considered. From the results, it can be concluded that the stress intensity factors (SIFs) are independent of material properties; additionally, loads play major role in the variation of SIFs which may lead to change in the direction of crack extension.

A linear elastic transversely isotropic full-space containing three different regions, which are an upper half-space, a lower half-space and a layer in between is considered in such a way, each region contains different material. The axes of symmetry of different regions are assumed to be normal to the interface of the regions and thus parallel. An arbitrary load in frequency domain is applied on an arbitrary patch located at the interface of the upper half-space and its underneath layer. Integral formulations are presented for the determination of the displacements and stresses in all regions. By means of the complete displacement potentials introduced by Eskandari-Ghadi (2005), Fourier series in circumferential direction and Hankel transform in radial direction, the displacements and stresses are determined in Fourier-Hankel space. Then, the displacements and stresses at any point are given in the line integral form. The solution is numerically evaluated for: (i) a half-space under an arbitrary surface load, (ii) a half-space under an arbitrary buried force, (iii) a half-space fixed at the top and under an arbitrary buried force, (iv) a half-space containing a layer bonded to the top of a half-space under an arbitrary force applied at the interface of two regions, (v) a full-space under an arbitrary load in it, (vi) a bi-material full-space under an arbitrary force applied at the interface of two half-space, and (vii) a layer of finite thickness fixed at the bottom and under an arbitrary surface load. To confirm the accuracy of the numerical evaluation of the integrals involved, some of the results are compared with existing solutions.

An analytical solution is presented for displacements and stresses of a three dimensional linear hyperelastic transversely isotropic layer bonded on the top of a transversely isotropic half-space subjected to an arbitrary, time-harmonic surface tangential force. The equations of equilibrium in terms of displacements are uncoupled by using a set of two potential functions introduced by Eskandari-Ghadi (2005) for electrodynamics problems of any convex transversely isotropic domain with respect to the axis of material symmetry. The Fourier expansion and Hankel transform in a cylindrical coordinate system are employed to solve the boundary value problems for the potential functions. The development includes a set of transformed displacement-potential relations that are useful in a variety of either static or dynamic problems. To verify the accuracy of the numerical evaluation of the present solutions, comparisons with existing solutions are given. Different numerical results are also included to demonstrate the influence of the degree of the material anisotropy and the frequency of excitation on the response. Solutions presented in this paper are important in development of boundary-integral-equations to analysis both dynamic anisotropic soil-structure interaction problem and seismic waves scattering in anisotropic soils.

This paper deals with the dynamic interaction of a massless rigid circular plate and a transversely isotropic full-space in frequency domain. The plate is located in an arbitrary place in the full-space in such a way its normal vector is in the direction of the axis of material symmetry of the domain, and undergoes a vertical displacement with a frequency of. Using potential representation and Henkel integral transform, and satisfying the continuity and regularity conditions, the governing equations are transformed to dual integral equations, which can be changed to a Fredholm integral equation of second kind. The Fredholm integral equation is solved analytically in the static case, where is zero, and is numerically evaluated in the general dynamic case. The results are shown to be identical with the simple case of isotropy in both analytical form and numerical evaluation.

The main goal for this research is to analyze a transversely isotropic half-space under the effect of a circular rigid disc attached on the surface of the domain. To do so, the potential method is used to uncouple the Navier’s equilibrium partial differential equations. Utilizing Hu-Nowakii-Lekhnitskii potential functions and both Fourier series and Hankel transform, the displacements and stresses Green’s functions are determined for a surface ring load in a cylindrical coordinate system attached to the domain. Then, the effect of circular rigid disc is seen by applying some ring loads of unknown amplitudes, depends on the direction of displacement of the disc, which may be vertical, horizontal or rocking displacement. The amplitudes of the different ring loads are determined by satisfying the displacement boundary conditions. It is shown that the results are identical to the existing ones for the isotropic case. To assess the effect of anisotropy, the results are numerically evaluated for different transversely isotropic materials and compared.

In this paper an analytical solution is presented for displacements and stresses of a three dimensional linear Green elastic transversely isotropic half-space subjected to an arbitrary, time-harmonic surface tangential force. The equations of equilibrium in terms of displacements are uncoupled by using a set of two potential functions introduced by Eskandari-Ghadi (2005) for electrodynamics problems. The Fourier expansion and Hankel transform in a cylindrical coordinate system with respect to angular and radial coordinates, respectively are employed to solve the boundary value problems for the potential functions. The development includes a set of transformed stress-potential and displacement potential relations that are useful in a variety of either static or dynamic problems. To verify the accuracy of the numerical evaluation of the present solutions, comparisons with existing solutions for an isotropic material are given. Different numerical results are also included to demonstrate the influence of the degree of the material anisotropy and the frequency of excitation on the response. The solutions for transversely isotropic material are also degenerated for the isotropic material response analytically and compared with existing analytical solutions. Solutions presented in this paper are important in development of boundary-integral-equations to analysis both dynamic anisotropic soil structure interaction problem and seismic waves scattering in anisotropic soils.

Nowadays fracture behavior composites play an important role in geomechanics engineering. Also, it is common knowledge that all existing structural materials contain different inter-and intra-component defect (cracks, delaminations, etc. ). On the other hand, analytical techniques can provide a better physical interpretation of problems. In this paper, by using an analytical approach, effects of the fracture modes (opening, shearing and tearing) on a penny-shaped crack in a layer of transversely isotropic solid has been studied. The layer surfaces are fixed from displacement and the system is loaded symmetrically in each mode. In each mode, by substituting the boundary conditions into the governing equations of the medium, the problem reduced to dual integral equations. With the aid some mathematical methods, the dual integral equations are converted to a Fredholm integral equation which is amenable to numerical solution. These Fredholm integral equations are the functions of the thickness of the layer, the radius of crack and the properties of the layer. To evaluate the effect of anisotropic materials on the stress intensity factors(SIFs), several synthetic types of isotropic and transversely isotropic materials are selected. By employing a numerical method the opening, shearing and tearing SIFs for different ratios of layer thickness are obtained. The results for the opening SIF show that by increasing the the SIF decreases substasinaly. On the other hand, an increase in leads to increments in opening SIF. Also, the results demonstrate that the variation in has a negligible effect on the opening SIF. Moreover, an increasing in leads reductions in SIF. For the shearing SIF, has little effect on the results although by decreasing the the shearing SIF increases. Unlike the, the modulus of the young in the plane ( ) of the isotropy has substantial effect on the shearing SIF. An increase in leads increments in the shearing SIF. Also, by increasing the the SIF increases marginally. In the mode III, the tearing SIF is only the functions of (the shear modulus for the plane normal to the plane of isotropy) and. The results show that by reduction in the tearing SIF increases and by increasing the tearing SIF increases. An important point that can be inferred from the results is that by increasing the ratio of layer thickness to the radius of the penny-shaped crack all of the three SIFs increase, this increase for the lower thicknesses is much more in comparison to the greater thicknesses. Additionally, when the layer thickness gets higher, the stress intensity factors for all the materials tend to a constant coefficient. This means that when the layer thickness gets greater and tends to infinity, the SIFs become independent of the material of the layer.

This paper is concerned with the investigation of the interaction of a vertically-loaded disc embedded in a transversely isotropic half-space. By virtue of transform methods, the generalized mixed boundary-value problem is formulated as a set of dual integral equations, which, in turn, are reduced to a Fredholm equation of the second kind.In addition to including existing solutions for zero and infinite embedment as degenerate eases, the present analysis reveals a severe boundary-layer phenomenon which is apt to be of significance to this class of problems in general. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. To confirm the accuracy of the numerical evaluation of the integrals involved, numerical results are included for cases of different degree of the material anisotropy and compared with existing solutions. Further numerical examples are also presented to elucidate the influence of the degree of the material anisotropy on the response.