Homogeneous Elastic Half-plane Research Articles

Elastic Solutions to 2D Plane Strain Problems: Nonlinear Contact and Settlement Analysis for Shallow Foundations

The classical Neumann boundary value problem of an isotropic, homogeneous elastic half-plane under plane strain conditions is readdressed as the limiting case of the fully three-dimensional problem. Analytical solutions of the stress and strain tensors are obtained by taking the limit from known three-dimensional solutions. It is shown that the displacement fields for the plane strain problem are not well defined. A small number of simple expressions are developed, which provide a general solution for linearly-varying traction over arbitrary regions on the boundary. A simple, efficient, and rapidly convergent algorithm is developed which uses these solutions as analytic elements and provides a solution approach to the general boundary value problem. The method is verified against known solutions for Hertzian contact between parallel cylinders. Two numerical examples are presented for the analysis of shallow foundation systems. In the first, the boundary conditions are informed by analytical elastoplastic calculations and a strain influence analysis is performed and compared with the Schmertmann method. Subsequently, empirical laboratory contact traction distributions measured by Bauer et al., in both the normal and tangential directions are employed as boundary conditions for an analysis of the underlying stress field.

An Analytical Solution for an Arbitrary Cavity in an Elastic Half-Plane

The complex variable method is used for an analytical solution for an arbitrary cavity in a homogeneous elastic half-plane. Proposing a method for solving the coefficients of the function by conformally mapping the region that contains an arbitrary cavity onto a circular ring utilizing a mapping function, which can be used for arbitrary shape cavity. Considering the stress-free condition of the upper surface, the relationship between two analytic functions is derived; consequently, the problem of solving two analytic functions is attributed to solving one analytic function, and the solve process for the analytic function in general case is given in detail, finally, achieving the solution of an arbitrary shape cavity in an elastic half-plane. The effects of cavity shape and depth were analyzed to the required terms of analytic function under a given accuracy. Examples for a vertical-wall semicircle cavity and a horseshoe cavity are given through the present method, and verification example is given by ANSYS software.

A contact problem in couple stress thermoelasticity: The indentation by a hot flat punch

It is well known, that thermo-elastic effects may have significant results upon the macroscopic response in the mechanics of contact. On the other hand, as the scales in the contact system reduce progressively (micro to nano-scales), the internal material lengths become important and their effect upon the macroscopic response cannot be ignored. The present work extends the classical contact solution for a hot flat punch indenting a homogeneous elastic half-plane, where heat conduction is permitted (Comninou et al., 1981), to the analogous case of an indented microstructured solid. The behavior of the indented material is modeled through the couple-stress elasticity theory, which introduces characteristic material lengths and is appropriately modified in order to incorporate the thermal effects. The problem formulation is based on singular integral equations, resulted from a treatment of the mixed boundary value problems via integral transforms and generalized functions. The results show significant departure from the predictions of classical thermoelasticity showing that the microstructural characteristics of the material should not be ignored.

Interfacial shear stress analysis of bar and thin film bonded to 2D elastic substrate using a coupled FE–BIE method

In the present work, a Finite Element–Boundary Integral Equation (FE–BIE) coupling method is proposed to investigate the problem of axially loaded thin structures bonded to a homogeneous elastic half-plane. Making use of a mixed variational formulation including the Green function of the substrate, the axial displacement of the bar is interpolated using Lagrange polynomials of first or second order, whereas the interfacial shear stress is approximated by piecewise constant functions. Bars subject to different load conditions are investigated, including the case of a bar partially detached from the substrate. The strength of interfacial stress singularities is investigated in detail.

Interaction between an Embedded Crack and an Interface Crack in Nonhomogeneous Coating System

A general methodology is constructed for the fundamental solution of a crack in the homogeneous half-plane interacting with a crack at the interface between the homogeneous elastic half-plane and the nonhomogeneous elastic coating in which the shear modulus varies exponentially with one coordinate. The problem is solved under plane strain or generalized plane stress condition using the Fourier integral transform method. The stress field in the homogeneous half plane is evaluated by the superposition of two states of stresses, one is associated with a local coordinate system in the infinite fractured plate, while the other in the infinite half plane defined in a structural coordinate system.

A Crack in the Homogeneous Half Plane Interacting with a Crack at the Interface Between the Nonhomogeneous Coating and the Homogeneous Half-Plane

A fundamental solution is established for a crack in a homogeneous half-plane interacting with a crack at the interface between the homogeneous elastic half-plane and the nonhomogeneous elastic coating in which the shear modulus varies exponentially with one coordinate. The problem is solved under plane strain or generalized plane stress conditions using the Fourier integral transform method. The stress field in the homogeneous half plane is evaluated by the superposition of two states of stresses, one of which is associated with a local coordinate system in the infinite fractured plate, while the other one in the infinite half plane defined in a structural coordinate system.

Surface subsidence associated with long wall mining: Two and three dimensional boundary element model

Abstract Systematic measurements of surface displacements associated with longwall mining made by British Coal (formerly National Coal Board - NCB) are compared with NCB results calculated in the present study. Three aspects of subsidence are considered: maximum subsidence, surface configuration and applicability of two-dimensional models to three-dimensional cases. The model presented in this paper is based on elementary solution presented by Golecki (1978, 1979) for stresses and displacements in an infinite homogeneous elastic halfplane or half space, subjected at its boundary by a discontinuous uniform displacement. Two and three dimensional models were developed using those solutions. The model was used for an analysis of ground subsidence as a function of the span and height of the mined openings, assuming various Poisson’s ratios. Results of two-dimensional models, representing openings of infinite length were compared with measurements compiled by the NCB which correspond to longwall mines. The comparison shows that for deep mines, use of small Poisson’s ratios renders close agreement between calculated and measured subsidence, whereas for shallow mines use of large Poisson’s ratios gives the best agreement. These results lead to the conclusion that subsidence in shallow mines is associated with a state of failure extending to the ground surface. This entails an increase in volume (dilatancy) and is therefore best represented by a model with a large Poisson’s ratio. On the other hand failure in deep mines is confined to a relatively limited zone above the subsiding roof. Computed subsidence over the longwall face (rib subsidence) appears to be consistently larger than that measured in the field. This may be attributed to one fundamental difference between the model and the prototype; while the model assumes closure over the entire longwall span (using a step function) actual closure occurs only at some distance behind the longwall face. Results of the three dimensional model indicate that subsidence associated with a square groundplan is smaller than that associated with a two dimensional opening. Subsidence over square openings amounts to less than 70% of that calculated in two dimensional models of equal relative span. For openings with a rectangular groundplan, subsidence increases rapidly with increasing length. For shallow openings with a length to width ratio exceeding 2.5, maximum subsidence is more than 95% of that in an opening of equal width but of infinite length.

A non-stationary dynamical periodic contact problem for a homogeneous elastic half-plane

The problem of determining the contact stresses under a periodic system of stamps located on the boundary of a homogeneous elastic half-plane and moving under the effect of a load, identical for all stamps, that is arbitrary in time, is investigated. The problem reduces to solving a Fredholm integral equation of the first kind for the Laplace transform of the contact stresses. The stresses are sought in the form of a double expansion in Chebyshev polynomials of the linear coordinate and Laguerre polynomials of time. The coefficients of the expansions are determined recursively from an infinite quasiregular system of linear algebraic equations. Despite the fact that the static periodic contact problems of the theory of elasticity, on the one hand (/1–6/, say), and dynamic problems for a finite number if stamps on the other (see the survey in /17/), have been studied repeatedly by different investigators, so far as we know, the plane non-stationary dynamical periodic contact problem has still not been examined at all.