Isotropic Elastic Half-plane Research Articles

Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

A screw dislocation in a three-phase composite composed of an anisotropic elastic half-plane bonded to an isotropic elastic half-plane reinforced by a circular inhomogeneity

Using complex variable techniques, we solve the problem in anti-plane elasticity associated with a screw dislocation in a three-phase composite which is composed of an anisotropic elastic half-plane perfectly bonded to an isotropic elastic half-plane containing a circular elastic inhomogeneity. Analytical solutions are obtained for three typical cases: a screw dislocation in the isotropic half-plane in the exterior of the circular inhomogeneity, a screw dislocation inside the circular elastic inhomogeneity, a screw dislocation within the anisotropic half-plane. Explicit expressions of the image force acting on the screw dislocation are derived for each of the three cases. The numerical results indicate that at least one equilibrium position for the screw dislocation co-exists in each phase under certain conditions on the elastic constants.

УЧЕТ ТРЕНИЯ НА НАЧАЛЬНОМ ЭТАПЕ ВЕРТИКАЛЬНОГО ВНЕДРЕНИЯ ВЫПУКЛОГО УДАРНИКА В УПРУГУЮ ПОЛУПЛОСКОСТЬ

The initial stage of indentation of a convex rigid punch into isotropic elastic half-plane with friction is considered. A closed mathematical formulation in Cartesian coordinates is presented by equations of motion in the absence of a body force for homogeneous isotropic elastic medium in terms of displacement potentials, Cauchy equations for deformations, Hooke law and translational equation for the punch. Initial conditions are homogeneous. Outside of contact region surface is free from stresses. Inside contact region normal displacements of the surface of a half-space and the surface of a punch are taken to be equal and the relation between tangential and normal stresses in a form of Coulomb friction law is given. Due to the short duration of the supersonic stage and smallness of the contact region radius the tangential stress direction is considered to be point independent. Resolving functional equations are given in a form of convolutions with influence functions. The latter is the solution the original problem for half-plane with a special boundary condition in a form of the Dirac delta function. This solution is obtained in the domain of Laplace transform by time and Fourier transform by spatial coordinate. The original is constructed with the combined Fourier–Laplace inversion method. Solution is given with consideration to the initial stage characteristic property of supersonic velocity of contact region expansion (no smaller than propagation velocity of expansion-compression waves). It is shown that kinematic parameters of the punch, resulting force and discontinuities of the first kind on boundaries of the contact region are independent of friction unlike contact pressure. Calculation examples for different values of friction coefficient are presented.

Analytical Solution of Stress in a Transversely Isotropic Floor Rock Mass under Distributed Loading in an Arbitrary Direction

Rock masses with a distinct structure may present a transversely isotropic character; thus, the stress state in a transversely isotropic elastic half-plane surface is an important way to assess the behavior of the interaction between the distributed loading and the surroundings. Most previous theoretical analyses have considered a loading direction that is either vertical or horizontal, and the stress distribution that results from the effect of different loading directions remains unclear. In this paper, based on the transversely isotropic elastic half-plane surface theory, a stress solution that is applicable to distributed loading in any direction is proposed to further examine the loading effect. The consistency between the analytical solution and numerical simulations showed the effectiveness of the proposal that was introduced. Then, it was utilized to analyze the stress distribution rule by changing the Poisson’s ratio and Young’s modulus of the model. The effects of the formation dip angle on the stress state are also examined. The stress distribution, depending on the physical property parameters and relative angle, is predicted using an analytical solution, and the mechanisms associated with the transversely isotropic elastic half-plane surface subjected to the loading in any direction are clarified. Additionally, extensive analyses regarding this case study, with respect to the mechanical behavior associated with changes in the stress boundary, is available. Hence, the proposed analytical solution can more realistically account for the loading problem in many engineering practices.

Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

Dynamic response of an isotropic elastic half-plane with shear modulus varying with depth to a load moving on its surface

Abstract The dynamic response of a half-plane soil medium to a load moving on its surface is obtained analytically-numerically. The soil is assumed to be linear elastic, isotropic and continuously non-homogeneous with its shear modulus varying with depth. Both cases of the shear modulus increasing and decreasing with depth are considered. The load is assumed to be distributed and moves with constant speed. The load and the soil displacements are expanded in complex Fourier series with respect to the horizontal direction x and the time t and thus the partial differential equations governing the soil motion are reduced to ordinary differential equations with variable coefficients, which are solved analytically by the method of Frobenius. The correctness of the solution in series form is checked by using it to solve some special case problems for which there exist closed form solutions in the literature. Parametric studies are conducted to assess the effect of non-homogeneity on the response for both cases of the shear modulus increasing and decreasing with depth.

Eshelby's problem of inclusion with arbitrary shape in an isotropic elastic half-plane

Eshelby's inclusion problem in an isotropic elastic half-plane is revisited in this paper. Following Zou et al. (2012, Int. J. Solids Struct. 49(13), 1627-1636), the general solution for Eshelby's inclusion problem in a half-plane is decomposed into the fundamental solution of Eshelby's problem with the same inclusion but in a full-plane, and the solution of an auxiliary boundary value problem of the half-plane. By virtue of the Kolosov–Muskhelishvili potentials of 2D isotropic elasticity, the auxiliary solution of the half-plane problem is derived in terms of the boundary integrals in the inclusion domain. For some inclusions of particular geometries, i.e., an arbitrary N-sided polygon, an arbitrary oriented ellipse and a circle (as a special case of the ellipse), the integrals involved are carried out explicitly, and eventually the potential functions are given with elementary functions. Some specified displacement, stress and strain energy density fields are illustrated and discussed. The obtained solutions are verified in more than one ways. Compared with the known methods, the one presented in this paper is more compact and more versatile, and will no doubt be helpful in achieving more explicit solutions besides those listed in this paper. Another feature of the present work is that both the Dirichlet and Neumann boundary conditions are accounted for in a unified form.

Static analysis of shear flexible beams and frames in adhesive contact with an isotropic elastic half-plane using a coupled FE–BIE model

Shear deformable beams and frames in perfect adhesion with an isotropic elastic half-space are analysed in plane strain or plain stress states. By means of a mixed variational formulation, the beam elements are described in terms of nodal displacements and rotations using locking-free shape functions, whereas the soil substrate is represented in terms of surface tractions through a boundary integral equation that incorporates a suitable Green’s function. The formulation ensures full continuity between structure and substrate in terms of displacements and rotations.A variety of numerical examples is presented to show the effectiveness of the proposed model. For very stiff beams subjected to a point force or moment, comparisons are made with available closed-form solutions to the contact problem of a rigid indenter. For foundation beams bonded to the substrate and loaded by a vertical point force the proposed model exhibits a superior convergence rate in comparison with other standard numerical models. Moreover, the shear deformations are shown to play a crucial role on both beam displacements and soil surface tractions. Finally, the soil–structure interaction analysis of a two-bay plane frame is presented.

Diffraction of Rayleigh wave by simple surface irregularity using boundary element method

The topographic amplification effect is studied in this paper in the case of a semi-circular canyon under incident Rayleigh wave. Time-frequency domain boundary element method has been applied to investigate the wave diffraction phenomena. The model consists of a semi-circular canyon cut from an isotropic elastic half plane. The total response is decomposed into free field motion and scattered wave field. The former can be constructed analytically by superposing on incident and reflected Rayleigh waves in the half plane and the later by using boundary element method with linear elements. The analyses are performed in the frequency domain and then converted into time domain using fast Fourier transform (FFT) algorithm. It is shown that the weak and strong singularities in the governing integral equations for linear elements can be removed by analytical approaches. Different ranges of Rayleigh wavelengths from low to high are considered. The results are presented versus dimensionless frequency and distance. A limiting value for Rayleigh wavelength can be recognized beyond which the canyon has no important effects on scattering of harmonic incident waves. Furthermore, a parametric study has been performed for different values of Poisson’s ratios. The published works in the effects of this parameter are somewhat few and limited to specific values of Poisson’s ratio. Finally, the spatial variations of surface displacements for the Rayleigh wave impulse of Ricker wavelet type are obtained. The spatial distribution shows the generation of backward-scattered Rayleigh wave which its intensity depends on predominant frequency of the input impulse as well as Poisson’s ratio of the medium. Dynamic responses of the points just located at the edges of the semi-circular canyon are compared to each other which accounts for the isolation efficiency of the canyon.

Stress fields induced by a non-uniform displacement discontinuity in an elastic half plane

This paper presents the exact closed-form solutions for the stress fields induced by a two-dimensional (2D) non-uniform displacement discontinuity (DD) of finite length in an isotropic elastic half plane. The relative displacement across the DD varies quadratically. We employ the complex potential-function method to first determine the Green’s stress fields induced by a concentrated force and then apply Betti’s reciprocal theorem to obtain the Green’s displacement fields due to concentrated DD. By taking the derivative of the Green’s functions and integrating along the DD, we derive the exact closed-form solutions of the stress fields for a quadratic DD. The solutions are applied to analyze the stress fields near a horizontal DD in the half plane with three different profiles: uniform (constant), linear, and quadratic. The same methodology is applied to an inclined normal fault to investigate the effect of different DD profiles on the maximum shear stress in the half plane as well as on the normal and shear stresses along the fault. Numerical results demonstrate considerable influence of the DD profile on the stress distribution around the discontinuity.

Asymptotic analysis of stresses in an isotropic linear elastic plane or half-plane weakened by a finite number of holes

The problem of an isotropic linear elastic plane or half-plane weakened by a finite number of small holes is considered. The analysis is based on the complex potential method of Muskhelishvili as well as on the theory of compound asymptotic expansions by Maz’ya. An asymptotic expansion of the solution in terms of the relative hole radii is constructed. This expansion is asymptotically valid in the whole domain, i.e. both in the vicinity of the holes and in the far-field. The approach leads to closed-form approximations of the field variables and does not require any numerical approximation. Several examples of the interaction between holes or holes and an edge are presented.

Elastic analysis of a half-plane containing an inclusion and a void using a mixed volume and boundary integral equation method

A mixed volume and boundary integral equation method is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to a traction-free boundary. A detailed analysis of the stress field is carried out for three different geometries of the problem. It is demonstrated that the method is very accurate and effective for investigating local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.

Elastic analysis of a half-plane with multiple inclusions using volume integral equation method

A volume integral equation method (VIEM) is used to calculate the elastostatic field in an isotropic elastic half-plane containing circular inclusions subject to remote loading parallel to the traction-free boundary. The material of the inclusions may be either isotropic or anisotropic and they are assumed to be distributed in square or hexagonal array. A detailed analysis of the stress field at the interface between the matrix and one of the inclusions is carried out for different distances between the inclusion and the surface of the half-plane. The results of the calculations are compared with available results. The VIEM is shown to be very accurate and effective for investigating the local stresses in the presence of multiple inclusions. The method can be applied to multiple inclusions of arbitrary geometry and elastic properties embedded in extended isotropic elastic media.

Theoretical aspects and numerical computation of the time-harmonic Green's function for an isotropic elastic half-plane with an impedance boundary condition

This work presents an effective and accurate method for determining, from a theoretical and computational point of view, the time-harmonic Green's function of an isotropic elastic half-plane where an impedance boundary condition is considered. This method, based on the previous work done by Duran et al. (cf. [Numer. Math. 107 (2007) 295–314; IMA J. Appl. Math. 71 (2006) 853–876]) for the Helmholtz equation in a half-plane, combines appropriately analytical and numerical techniques, which has an important advantage because the obtention of explicit expressions for the surface waves. We show, in addition to the usual Rayleigh wave, another surface wave appearing in some special cases. Numerical results are given to illustrate that. This is an extended and detailed version of the previous article by Duran et al.

Radiation condition and uniqueness for the outgoing elastic wave in a half-plane with free boundary

In this Note we deduce an explicit Sommerfeld-type radiation condition which is convenient to prove the uniqueness for the time-harmonic outgoing wave problem in an isotropic elastic half-plane with free boundary condition. The expression is obtained from a rigorous asymptotic analysis of the associated Green's function. The main difficulty is that the free boundary condition allows the propagation of a Rayleigh wave which cannot be neglected in the far field expansion. We also give the existence result for this problem. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

혼합 체적-경계 적분방정식법을 이용한, 함유체와 공동을 포함한 반무한 고체에서의 탄성해석

A mixed volume and boundary integral equation method (Mixed VIEM-BIEM) is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to the traction-free boundary. A detailed analysis of stress field at the interface between the isotropic matrix and the isotropic or orthotropic inclusion is carried out for different values of the distance between the center of the inclusion and the traction-free surface boundary in an isotropic elastic half-plane containing three different geometries of an isotropic or orthotropic inclusion and a void. The method is shown to be very accurate and effective for investigating the local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.

A hybrid complex-variable solution for piezoelectric/isotropic elastic interfacial cracks

In many practical applications, piezoelectric ceramics are bonded to non-piezoelectric and insulating isotropic elastic materials such as polymer. Since the conventional form of Stroh’s formulation, on which almost all of existing works on interfacial cracks in piezoelectric media have been based, breaks down or becomes complicated for isotropic elastic materials, many solutions available in the literature cannot be directly applied to interfacial cracks between a piezoelectric material and an isotropic elastic material. The present paper is devoted to a hybrid complex-variable method which combines the Stroh’s method of piezoelectric materials with the well-known Muskhelishvili’s method of isotropic elastic materials. This method is illustrated in detail for an insulating interfacial crack between a piezoelectric half-plane and an isotropic elastic half-plane, although interface cracks between piezoelectric and isotropic elastic conductor can be analyzed in a similar way. The solution obtained generally exhibits oscillatory singularity, in agreement with a previous known result based on the Stroh’s formulation. A simple explicit condition is obtained for the bimaterial constants under which the oscillatory singularity disappears. It is expected that the hybrid complex-variable method could more conveniently handle other possible complications (such as a hole or an inclusion) inside the isotropic elastic material, because it offers explicit solutions of a single complex variable rather than several different complex-variables associated with the Stroh’s formulation.

Multiple circular nano-inhomogeneities and/or nano-pores in one of two joined isotropic elastic half-planes

The paper considers the problem of multiple interacting circular nano-inhomogeneities or/and nano-pores located in one of two joined, dissimilar isotropic elastic half-planes. The analysis is based on the solutions of the elastostatic problems for (i) the bulk material of two bonded, dissimilar elastic half-planes and (ii) the bulk material of a circular disc. These solutions are coupled with the Gurtin and Murdoch model of material surfaces [Gurtin ME, Murdoch AI. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57:291–323; Gurtin ME, Murdoch AI. Surface stress in solids. Int J Solids Struct 1978;14:431–40.]. Each elastostatic problem is solved with the use of complex Somigliana traction identity [Mogilevskaya SG, Linkov AM. Complex fundamental solutions and complex variables boundary element method in elasticity. Comput Mech 1998;22:88–92]. The complex boundary displacements and tractions at each circular boundary are approximated by a truncated complex Fourier series, and the unknown Fourier coefficients are found from a system of linear algebraic equations obtained by using a Taylor series expansion. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the nano-inhomogeneities. Numerical examples demonstrate that (i) the method is effective in solving the problems with multiple nano-inhomogeneities, and (ii) the elastic response of a composite system is profoundly influenced by the sizes of the nano-features.

체적 적분방정식법을 이용한, 다수의 함유체를 포함한 반무한 고체에서의 탄성해석

A volume integral equation method (VIEM) is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions subject to remote loading. A detailed analysis of stress field at the interface between the matrix and the central inclusion in the first column of square packing is carried out for different values of the distance between the center of the central inclusion in the first column of square packing of inclusions and the traction-free surface boundary in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions. The method is shown to be very accurate and effective for investigating the local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions.

A semi-analytical solution for multiple circular inhomogeneities in one of two joined isotropic elastic half-planes

The paper presents a semi-analytical method for solving the problem of two joined, dissimilar isotropic elastic half-planes, one of which contains a large number of arbitrary located, non-overlapping, perfectly bonded circular elastic inhomogeneities. In general, the inhomogeneities may have different elastic properties and sizes. The analysis is based on a solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. Apart from round-off, the only errors introduced into the solution are due to truncation of the Fourier series. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the inhomogeneities. Numerical examples are included to demonstrate the effectiveness of the approach.