Antiplane Strain Problem Research Articles

Uniformity of anti-plane stresses inside a nonlinear elastic elliptical or parabolic inhomogeneity

We study the anti-plane strain problem associated with a p-Laplacian nonlinear elastic elliptical inhomogeneity embedded in an infinite linear elastic matrix subjected to uniform remote anti-plane stresses. A full-field exact solution is derived using complex variable techniques. It is proved that the stress field inside the elliptical inhomogeneity is nevertheless uniform. The uniformity of stresses is also observed inside a p-Laplacian nonlinear elastic parabolic inhomogeneity.

Stresses in a monoclinic elastic plate placed upon an irregular monoclinic elastic half space

Objective: To study the stresses due to strip load in a monoclinic elastic plate placed over an irregular monoclinic elastic half space. Method: Antiplane strain problem with different interfacing boundary conditions namely “perfectly welded interfacing, smooth-rigid interfacing, rough-rigid interfacing” has been considered and Fourier Transformation is used on the equilibrium equations to obtain the solutions. Findings: The displacements and stressesare obtained for each boundary condition and the variation of shearing stresses with horizontal distance are studied due to different size of rectangular shaped irregularity at any point of the medium consisting of monoclinic elastic plate placed upon an irregular monoclinic elastic half space. The comparison between the stresses for different types of interfacing has been made graphically. Novelty: The static deformation of two or more connected elastic mediums with irregularities due to surface loads has been studied earlier but just a few have showed graphic representation of stresses with irregularity in different sizes. Keeping in mind the shortcomings of earlier work done, the present paper visualizes shearing stresses in both theoretical and graphical manners completely. Keywords: Monoclinic; strip loading; anti plane strain; rectangular irregularity

Inversion Symmetry of the Solutions of Basic Boundary-Value Problems of Two-Dimensional Elasticity Theory for a Wedge

By using the Mellin integral transformation, we obtain the solutions of basic two-dimensional problems for an elastic wedge. It is shown that, in the case of boundary conditions symmetric under the inversion transformation, some components of the solutions of plane strain problems and the solutions of antiplane strain problems for a wedge also have the inversion symmetry.

Singular integral equations with two fixed singularities and applications to fractured composites

A symmetric characteristic singular integral equation with two fixed singularities at the endpoints in the class of functions bounded at the ends is analyzed. It reduces to a vector Hilbert problem for a half-disc and then to a vector Riemann-Hilbert problem on a real axis with a piecewise constant matrix coefficient that has two points of discontinuity. A condition of solvability and a closed-form solution to the integral equation are derived. For the Chebyshev polynomials of the first kind in the right hand-side, the solution of the integral equation is expressed in terms of two nonorthogonal polynomials with associated weights. Based on this new generalized spectral relation for the singular operator with two fixed singularities an approximate solution to the complete singular integral equation is derived by recasting it as an infinite system of linear algebraic equations of the second kind. The method is illustrated by solving two problems of fracture mechanics, the antiplane and plane strain problems for a finite crack in a composite plane. The plane is formed by a strip and two half-planes; the elastic constants of the strip are different from those of the half-planes. The crack is orthogonal to the interfaces, and it is located in the strip with the ends lying in the interfaces. Numerical results are reported and discussed.

Wave propagation in micropolar monoclinic thermoelastic half space

Propagation of waves in a micropolar monoclinic medium possessing hermoelastic properties based on the Lord- Shulman (L-S),Green and Lindsay (G-L) and Coupled thermoelasticty (C-T) theories is discussed. The investigation is divided into two sections, viz., plane strain and anti-plane strain problem. After developing the solution, the phase velocities and attenuation quality factor have been derived and computed numerically. The numerical results have been plotted graphically.

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing an anti-plane strain inclusion of arbitrary cross-sectional shape prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for anti-plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite anti-plane strain elastic body. The solution reduces to that of the infinite-domain anti-plane strain inclusion problem when the boundary effect is not considered. The problem of a circular cylindrical inclusion embedded concentrically in a finite cylindrical elastic matrix undergoing anti-plane strain deformations is analytically solved by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical linear elasticity-based Eshelby tensor for the circular cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is small and that the boundary effect can dominate when the inclusion volume fraction is high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.

Dual integrals in small strain elasticity with body forces

Dual integrals of small strain elasticity are derived and related to the energy release rates associated with a defect motion in the presence of body forces. A modified energy momentum tensor is used, which includes a work term due to body forces, and which yields simple expressions for the configurational forces in terms of the J k , L k , and M integrals. Since the complementary potential energy does not include body forces explicitly, the complementary energy momentum tensor has the same structure as in the elasticity without body forces. The expressions for the nonconserved J k , L k , and M integrals, and their duals, are related to the volume integrals whose integrands depend on body forces and their gradients. If body forces are spatially uniform, the conservation laws J k = J ˆ k = 0 hold for both 2D and 3D problems, and L 3 = L ˆ 3 = 0 for the antiplane strain problems. The conservation law M = M ˆ = 0 holds if body forces are absent, or if they are homogeneous functions of particular degree in spatial coordinates. To cite this article: V.A. Lubarda, X. Markenscoff, C. R. Mecanique 336 (2008).

Anti-plane Green's functions and cracks for piezoelectric material with couple stress and electric field gradient effects

This research is concerned with the anti-plane strain problems of polarized ceramics with both the couple stress and electric field gradient effects. This theory possesses two characteristic lengths l1 and l2 which are determined explicitly. In addition the two characteristic lengths can be either positive real or complex conjugate with positive real part. We first investigate the electroelastic field induced by a static line force and a line charge. It is found that the displacement and the electric potential are regular at the point where the line force and line charge are located. We then consider the near-tip asymptotic electroelastic field for a mode III crack. The analysis demonstrates that the near-tip asymptotic electroelastic field is governed by two parameters B and D. The total stresses and in-plane electric displacements exhibit the stronger r−3/2 singularity near the crack tip; while the couple stresses, the out-of-plane electric displacement, and those associated with electric quadrupole densities exhibit the weaker r−1/2 singularity near the crack tip.

The asymptotic expansion of elastic fields in the vicinity of the contour of a plane crack at the interface of two materials

The stress-strain state in the neighbourhood of the front of a plane crack at the interface of two dissimilar half-spaces of ideally elastic isotropic materials is investigated. The form of the asymptotic expansions of the projections of the displacement vector onto the axis, directed along the tangent, the principal normal and the binormal to the crack contour is obtained. It is shown that asymptotic expansions of the projections of the displacement vector onto directions corresponding to the tangent and principal normal, beginning with the second-order term of the expansion, include both terms with half-integer and complex powers of the distances to the crack contour. This indicates that these projections of the solutions of the three-dimensional problem have singularities, defined by the solutions of both the antiplane and plane strain problems of cracks at the interface of materials. The singularities of the projection of the displacement vector on to the binormal correspond to the singularities of the solution of the plane strain problem.

Some generalized anti-plane strain problems for an inhomogeneous elastic halfspace

This paper develops a generalized formulation of the theory of anti-plane deformations of a linear elastic solid, the Lame constants of which depend on a single spatial variable. The generalized theory is applied to the study of a generalized screw dislocation, a semi-infinite crack and a finite crack located in a halfspace region where the elastic inhomogeneity is depth dependent. The problems examined in this paper are relevant to the modelling of cracks located in inhomogeneous geological materials and the study of surface defects associated with functionally graded materials.

On interface crack propagation with variable velocity for longitudinal shear problems

The antiplane strain problem of straight interface crack propagation between two elastic half-spaces under arbitrary variable loading is considered. The crack edge is specified as an arbitrary smooth function of time. It is assumed that the crack speed is less than the smaller of the shear wave velocities of two media. An integral transform method and factorization technique are used to solve the problem. The solutions are worked out for semi-infinite crack and finite crack problems. The dynamic stress intensity factors at the crack tip of the moving interface crack are given and it is found that the stress intensity factor of the interface crack is slightly higher than that in the homogeneous medium with slower shear wave velocity.

Antiplane strain problem of diamond inclusion

In this paper, the singular stress fields created by an antiplane deformation at an inclusion corner are studied. It is shown that these singular stress fields can be separated into two independent types: a symmetric type with the stress singularity of 1/Υ 1−λ1 and a skew-symmetric type with the stress singularity of 1/Υ 1−λ2. These two types of the singular stress field can not occur simultaneously at the corner. If G2 G1, there exists only the singularity of the symmetric type. A general expression of stress fields in the vicinity of the corner is presented. In the expression the singular stress fields for the symmetric type and the skew-symmetric type are defined in terms of the constants KIII, λ1 and KIII, λ2, respectively. KIII, λ2 and KIII, λ2 have to be determined from the complete boundary conditions of the given problem. For the problem of an infinite plate containing a diamond inclusion and subjected to a uniform longitudinal shear stress at infinity, the values of KIII, λ1 and KIII, λ2 are obtained by body force method. In the body force method, the investigated stresses are simulated by the superposition of the fundamental stress fields due to point forces. In order to obtain accurate solutions, the basic density functions of the distributed point forces are used, so that the stress singularities at the corner tip can be simulated by the point forces.

Geotechnical applications of a two-dimensional elastodynamic displacement discontinuity method

A general two-dimensional elastodynamic displacement discontinuity method is used to model a variety of application problems. The plane strain problems are: the elastodynamic motions induced on a cavity by shear slip on a nearby crack; the dynamic response of a backfilled versus a non-backfilled tabular excavation or stope; and the interaction between two inclined stopes and a slipping fault. An antiplane strain problem is also presented, where slip occurs along a crack near the surface of a half plane. These problems indicate the wide potential of this method in investigating elastodynamic effects, especially for crack-like geometries.

A Doubly Periodic Rectangular Array of Fiber-Matrix Interfacial Cracks Under Longitudinal Shearing

The antiplane strain problem of a unidirectional fiber composite consisting of a doubly periodic rectangular array of fibers containing interfacial cracks in an infinite matrix is considered. The interfacial cracks are assumed to exhibit the same periodicity as the fibers. The periodicity of the geometry allows the use of a unit cell in the formulation of the problem. The governing weakly singular integral equation of the mixed boundary value problem permits an explicit solution which contains a set of unknown constants. The unknown constants are then determined by satisfying the boundary conditions on the external surfaces of the unit cell through the method of least squares. The stress intensity factor is calculated for various crack lengths, fiber volume fractions, and fiber spacings. Unlike the plane strain or plane stress deformation, the oscillations in stress and displacement around the interface crack tip are absent in the current antiplane strain problem.

Static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads

The transfer matrix approach is used to solve the problem of static deformation of an orthotropic multilayered elastic half-space by two-dimensional surface loads. The general problem is decoupled into two independent problems. The antiplane strain problem and the plane strain problem are considered in detail. Integral expressions for displacements and stresses at any point of the medium due to a normal line load and a shear line load, acting parallel to a symmetry axis, are obtained. In the case of a uniform half-space, closed form analytic expressions for displacements and stresses are derived. The procedure developed is quite easy and convenient for numerical computations.

Antiplane problems in composite anisotropic materials with an inclined crack terminating at a bimaterial interface

The antiplane strain problem for bonded dissimilar half planes of general anisotropic material containing an inclined crack terminating at the bimaterial interface is considered. The surfaces of the crack can be subjected to traction-traction, traction-displacement or displacement-displacement boundary conditions. The dependence of the order of the stress singularity on the inclined angle and material constants is studied. If the effective crack angle and the effective material constant are introduced for the anisotropic case, then the characteristic equation which determines the order of the stress singularity has the same functional form as the isotropic case. Explicit solutions for the order of stress singularity are obtained for some special cases. It is found that the order of the stress singularity is always real for all the cases studied in this paper. This is a quite different character from the in-plane case in which the complex type of stress singularity might exist. The angular distribution of stresses near the crack tip and the exact full field stress solutions are also investigated.

Analysis of dissimilar anisotropic wedges subjected to antiplane shear deformation

The antiplane strain problem of two dissimilar anisotropic composite wedges of arbitrary angles that are bonded together along a common edge is considered. The surfaces of the wedges can be subjected to traction-traction, traction-displacement or displacement-displacement boundary conditions. The dependence of the order of the stress singularity on the wedge angles and material constants is studied. The angular distribution of stresses at the apex and the exact full field stress solutions are also investigated. Explicit solutions for the order of stress singularity are obtained for some special cases. It is found that the order of the stress singularity is always real for the antiplane dissimilar anisotropic wedge problem. This is quite different from the in-plane case, in which the complex type of stress singularity might exist.

The stress field in an edge cracked plate in anti-plane deformation

The anti-plane strain problems of an edge cracked elastic plate under 1) uniform surface displacement load and 2) surface line loads are solved using conformal mapping techniques. The solutions yield stress, strain and displacement distributions and stress intensity factors in the plate.

Asymptotic Fields of a Perfectly-Plastic, Plane-Stress Mode II Growing Crack

The asymptotic near-tip stress and velocity fields are presented for a plane-stress Mode II crack propagating quasi-statically in an elastic-perfectly plastic Mises solid. The solution is found to have fully continuous stress and velocity fields, and a configuration similar to that of the anti-plane strain problem: a singular centered fan plastic sector ahead of the crack, followed by an elastic unloading sector and a constant stress plastic sector extending to the crack flank. The impossibility of a plane-stress Mode I crack solution having these properties is also discussed.

Harmonic waves in a periodically laminated medium

A stiffness method using the continuity of displacement and traction at the interfaces of a periodically laminated composite medium and the Floquet's Theory has been presented here for studying harmonic wave propagation in a layered composite. Both plane strain and antiplane strain problems have been studied. The equations have been developed for both isotropic and anisotropic layers. Numerical results are presented for the dispersion spectrum for propagation in a periodic two-layered medium and these are shown to compare well with available exact results. Finally, numerical results are presented for a boron-aluminum composite medium, which is modelled as a composition of anisotropic and isotropic layers.