Plane Elasticity Problems Research Articles

Slit Maps in the Study of Equal-Strength Cavities in $n$-Connected Elastic Planar Domains

The inverse problem of plane elasticity on $n$ equal-strength cavities in a plane subjected to constant loading at infinity and in the cavities' boundaries is analyzed. By reducing the governing boundary value problem to the Riemann--Hilbert problem on a symmetric Riemann surface of genus $n-1$, a family of conformal mappings from a parametric slit domain onto the $n$-connected elastic domain is constructed. The conformal mappings are presented in terms of hyperelliptic integrals, and the zeros of the first derivative of the mappings are analyzed. It is shown that for any $n\ge 1$ there always exists a set of the loading parameters for which these zeros generate inadmissible poles of the solution.

Load paths visualization in plane elasticity using load function method

The paper proposes a load function method for load path visualization and load flow calculation in plane elasticity problems. The load functions are directly derived from 2D equilibrium equations. The stresses are written in terms of the load functions and the load flow is calculated using the load function gradient. The load functions are defined using generalized Beltrami representation. A constructive proof is given for the existence of load functions. A set of Poisson's equations in terms of load functions and stress components are developed and an efficient numerical procedure to solve them is discussed. Numerical results show the proposed load function method is able to define the load paths and obtain the load flow using an Airy stress function for problems with available closed form solution, and also can be easily integrated into the numerical approaches to calculate load paths for complex problems.

Discontinuous Solutions of a Doubly Periodic Problem for a Piecewise Homogeneous Plane with Interphase Defects

Discontinuous solutions of equations of a plane elasticity problem for a piecewise homogeneous plane obtained by joining two dissimilar strips are found using the Airy stress function. In the general case, doubly periodic interphase defects are on their connection lines, on which displacements and stresses are discontinuous. Based on the solutions obtained, a system of governing singular integral equations is derived for a particular problem where the defect is a finite crack. This system is solved by the method of mechanical quadratures. Numerical calculations are also carried out.

The uniform convergence of the eigenfunctions expansions of the biharmonic operator in closed domain

The mathematical models of the various vibrating systems are partial differential equations and finding the solutions of such equations are obtained by developing the theory of eigenfunction expansions of differential operators. The biharmonic equation which is fourth order differential equation is encountered in plane problems of elasticity. It is also used to describe slow flows of viscous incompressible fluids. Many physical process taking place in real space can be described using the spectral theory of differentiable operators, particularly biharmonic operator. In this paper, the problems on the uniform convergence of eigenfunction expansions of the functions from Nikolskii classes corresponding to the biharmonic operator are investigated.

Periodic Contact Problems in Plane Elasticity: The Fracture Mechanics Approach

In this study, the concept of the fracture mechanics is used to solve the: (i) frictionless purely normal contact and (ii) the similar material contact under the mutual actions of the normal and tangential load. Considering the contact region is simply connected, the out-of-contact regions can be treated as periodic collinear cracks. Through evaluating the stress intensity factor (SIF), we are able to obtain the size and location of the contact/out-of-contact region. Then, the normal traction, shear traction and interfacial gap can be directly determined by the Green's function of the periodic collinear crack. In the case of frictionless purely normal contact, the new approach is applied to two classic problems, namely, the Westergaard problem (sinusoidal waviness punch) and the periodic flat-end punch problem. Then, the sinusoidal waviness contact pair in the full stick and the partial slip conditions under the mutual actions of the normal and tangential loads are solved by the newly developed approach.

A stress/displacement Virtual Element method for plane elasticity problems

The numerical approximation of 2D elasticity problems is considered, in the framework of the small strain theory and in connection with the mixed Hellinger–Reissner variational formulation. A low-order Virtual Element Method (VEM) with a priori symmetric stresses is proposed. Several numerical tests are provided, along with a rigorous stability and convergence analysis.

Stochastic spline fictitious boundary element method for random vibration analysis of plane elastic problems with structural uncertainties

A stochastic spline fictitious boundary element method (SFBEM) is proposed for random vibration analysis of plane elastic problems with both structural and loading uncertainties modeled as random fields and random processes, respectively. Two sets of governing differential equations with respect to the means and deviations of dynamic responses are derived by including the first order terms of deviations. As these equations are in similar forms to those of deterministic plane elastostatic problems, they can be solved by the traditional procedure of SFBEM with deterministic fundamental solutions, leading to the mean and covariance solutions to dynamic responses. To deal with the domain integrals of equivalent loads involved in the covariance solution, the random fields and processes considered are effectively represented by Karhunen–Loeve (KL) expansion in conjunction with the Galerkin projection. Numerical examples indicate that the results of the proposed method agree well with those of Monte-Carlo simulation (MCS) with small variations of structural parameters, and the present approach has higher efficiency than the perturbation stochastic finite element method (FEM) with KL expansion techniques.

The M-Integral based failure description on elasto-plastic materials with defects under biaxial loading

This paper presents a M-Integral based failure description of the elasto-plastic materials containing multiple defects, such as voids, subjected to biaxial loading conditions. A non-dimensional Π-parameter is defined, and a novel failure criterion based on the Π-parameter is proposed to assess the multi-defected media. Analytical, numerical and experimental investigations are conducted to evaluate the Π-parameter for different types of problems in order to elucidate the Π-parameter based failure criterion. The explicit definition of the Π-parameter is first derived from the case of elastic inclusion problems in plane elasticity being subject to biaxial loading conditions. And then the Π-parameter is regarded as a damage driving force, and its critical value indicates the occurrence of failure of elasto-plastic materials with defects under biaxial loading. This Π-parameter based failure criterion is further validated through a series of numerically elasto-plastic analyses of cruciform biaxial specimens containing multiple and randomly distributed local defects. In addition, an experimental procedure on evaluating the Π-parameter for elasto-plastic materials is proposed by using the digital image correlation.

Convergence estimates for a plane elasticity problem solved by the Galerkin boundary integral formulation with NURBS

A classical mixed boundary-value problem of linear elasticity in two dimensions in the Galerkin boundary integral formulation is considered. We prove a-priori error estimates of the solution to this problem by the isogeometric adaptive method using NURBS. The estimates include approximation of the boundary of the domain.

The angular deformation of the ring with reference to the centrifugal forces

In the paper the boundary plane elasticity problem for the ring is solved usingMuskhelishvili complex potentialsmethod. The ring rotates uniformly under the influence of tangential forces, and the centrifugal forces of inertia are taken into account. Themutual rotation angle of the boundary points is obtained by analyzing the displacement field.

Stress analysis near the tips of a transverse crack in an elastic semi-strip

The plane elastic problem for a semi-strip with a transverse crack is investigated. The initial problem is reduced to a one-dimensional continuous problem by use of an integral transformation method with a generalized scheme. The one-dimensional problem is first formulated as a vector boundary problem, and then reduced to a system of three singular integral equations (SIEs). The system is solved by use of an orthogonal polynomial method and a special generalized method. The contribution of this work is the consideration of kernel fixed singularities in solving the system. The crack length and its location relative to the semi-strip’s lateral sides are investigated to simplify the problem’s statement. This simplification reduces the initial problem to a system of two SIEs.

A new analytical method for the evaluation of transverse displacements and stresses in plane problems of elasticity

Numerous solutions of the plane theory of elasticity, derived over the past hundred years, provide a very good approximation to the in-plane stress components. However, an accurate evaluation of the out-of-plane (or transverse) stresses is often impossible as the three-dimensional stress field may not follow either plane stress or plane strain assumptions, specifically, near the edges formed by notches, cracks and other stress concentrators. For relatively simple geometries and loading conditions, the 3D stress and displacement fields can be analytically evaluated using the first-order plate theory. This theory can be considered as an elementary extension of the plane theory of elasticity and it retains the simplicity of the two-dimensional formulation. However, the derivation of exact solutions within this theory is far more difficult. In this paper, we present the governing equations of the first-order plate theory in arbitrary curvilinear coordinates, and based on these equations we develop a simplified method for the evaluation of the out-of-plane displacements and stress components. Several examples are considered to demonstrate the effectiveness and the accuracy of the new method. Due to simplicity it can be applied to investigate many 3D elastic problems involving sharp wedges and plate geometries weakened by cracks, notches and other stress concentrators. All these problems currently represent a great challenge for analytical approaches.

Adaptive Poly-FEM for the analysis of plane elasticity problems

ABSTRACTIn this work, we present an adaptive polygonal finite element method (Poly-FEM) for the analysis of two-dimensional plane elasticity problems. The generation of meshes consisting of n − sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set, is generated whereby the method also includes tessellation of nonconvex domains. In this work, we propose a region by region adaptive polygonal element mesh generation. A patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a − posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the Poly-FEM. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains, we resort to classical Gaussian quadrature applied to triangular subdomains of each polygonal element. Numerical examples of two-dimensional plane elasticity problems are presented to demonstrate the efficiency of the proposed adaptive Poly-FEM.

The geometry of a dyke swarm as a result of dyke interaction with each other and with external stresses

In the framework of plane elastic problem we use a numerical approach to study the forms of opening and paths of growth of three parallel magma fractures as a model of dyke swarm formation and development. As expected, the internal dynamic mechanism of dyke interaction distorts their shapes in comparison with a single dyke shape, and curves their paths combining them into a divergent or a convergent system. The external dynamic mechanism of regional stress tends to align the growing dyke paths in parallel to the axis of the maximum compressive stress. The external and internal mechanisms compete with each other. The impact of the internal mechanism is stronger when the ratio of the distance between dykes to their length is less, the initial parallel dyke shift in relation to each other is larger, and the differential regional stress is less. Under the opposite conditions, the external mechanism prevails.

Determine of Shale Tensile Strength by Using Brazilian Disc Test

The disc specimen subjected to compression is proposed for testing tensile strength of brittle materials such as rocks. The Brazilian disc test is an indirect measurement method of mechanical properties of rock, which theoretical foundation is the elasticity solution of stress of pressed disc. However, the distribution of contact load acting on the disc is not exactly known. Based on the complex function theory for elastic plane problem, the complex solution of stress is given under the condition that the distribution of contact load is general distribution. The theoretical formula of tensile strength is also presented. Discuss the influence of seven kinds of contact load distributions and different contact angles on stress field. Through the Brazilian disc test for shale from the depth of eight hundred, we obtain the tensile strength.

Estimate of rock mass stability in surface–borehole mining of high-grade iron ore

Under consideration is the estimate of rock mass stability around underground openings generated as a result of hydraulic borehole mining of iron ore. The authors use analytical solutions of two plane elasticity problems on stress state of infinite media with the zone of weakening in the form of one or two circular holes, given initial stresses are set in the study domains.

Three-dimensional one-mode penetration of elastic waves through a doubly periodic array of cracks

The paper is devoted to the calculation of the reflection and transmission coefficients, when a plane wave is incident on a two-dimensional grating with an infinite doubly periodic array of rectangular cracks in the elastic material. In the one-mode frequency range the problem is reduced to an integral equation holding over the single chosen crack. The semi-analytical method previously introduced for scalar and elastic plane problems gives explicit representations for the wave field and the scattering parameters.

OUP accepted manuscript

The interaction of a thin rigid inclusion with a finite crack is studied. Two plane problems of elasticity are considered. The first one concerns the case when the upper side of the inclusion is completely debonded from the matrix, and the crack penetrates into the medium. In the second model, the upper side of the inclusion is partly separated from the matrix, that is the crack length $2a$ is less than $2b$, the inclusion length. It is shown that both problems are governed by a singular integral equation of the same structure. Derivation of the closed-form solution of this integral equation is the main result of the paper. The solution is found by solving the associated vector Riemann-Hilbert problem with the Chebotarev-Khrapkov matrix coefficient. A feature of the method proposed is that the vector Riemann-Hilbert problem is set on a finite segment, while the original Khrapkov method of matrix factorization is developed for a closed contour. In the case, when the crack and inclusion lengths are the same, the solution is derived by passing to the limit $b/a\to 1$. It is demonstrated that the limiting case $a=b$ is unstable, and when $a<b$, and the crack tips approach the inclusion ends, the crack tends to accelerate in order to penetrate into the matrix.

A stabilizedP1-nonconforming immersed finite element method for the interface elasticity problems

We develop a new finite element method for solving planar elasticity problems involving heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the ‘broken’ Crouzeix–Raviart P 1 -nonconforming finite element method for elliptic interface problems [D.Y. Kwak, K.T. Wee and K.S. Chang, SIAM J. Numer. Anal. 48 (2010) 2117–2134]. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method [D.N. Arnold, SIAM J. Numer. Anal. 19 (1982) 742–760; D.N. Arnold and F. Brezzi, in Discontinuous Galerkin Methods. Theory, Computation and Applications , edited by B. Cockburn, G.E. Karniadakis, and C.-W. Shu. Vol. 11 of Lecture Notes in Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101; M.F. Wheeler, SIAM J. Numer. Anal. 15 (1978) 152–161.]. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace–Young condition along the interface of each element. We prove optimal H 1 and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lame parameters μ and λ and locking free as λ → ∞.

A Kirchhoff type equation in a nonlinear model of shell vibration

The paper deals with the study of the initial boundary value problem for the Donnell‐Mushtari‐Vlasov nonlinear system of differential equations, which describes large deflections of a rectangular shallow shell. The sought functions and are respectively longitudinal and transverse displacements of points of the shell midsurface. It is assumed that Dirichlet and Neumann type homogeneous conditions are fulfilled for functions u1 and u2. After expressing u1 and u2 through w, i.e. after constructing the relations with a help of Fourier series method, we come to a nonlinear integro‐differential hyperbolic equation for the transverse displacement function w, which by its structure is analogous to Kirchhoff string, Woinowsky‐Krieger beam and Berger plate equations. These equations are united under the common name – Kirchhoff type equations. The advantage of the described technique consists in splitting the initial boundary value problem at the end into two subproblems. From the integro‐differential equation we first find w and using the explicit relations we construct u1 and u2. The issue of the use of this approach in the general case not restricted by the rectangular domain and Dirichlet and Neumann conditions on the boundary is discussed. It is shown that in order to obtain formulas we have to solve a linear plane problem of elasticity with respect to the functions u1 and u2.