Interaction between two neighboring circular holes in a prestretched simply supported orthotropic strip under bending

In this work, the influence of initial stretching of a simply supported plate-strip containing two circular holes on the stress concentration around the holes caused by bending of the strip is examined using the finite-element method. The mathematical formulation of the corresponding boundary-value problem is presented within the frame work of the three-dimensional linearized theory of elasticity (TDLTE) under a plane strain state. The material of the plate-strip is linearly elastic, homogeneous, and orthotropic. The numerical results obtained in investigating the influence of the initial stretching and the location of holes on the stress concentration are presented. In particular, it is established that the initial stretching significantly decreases the stress concentration at some characteristic points on the contour of the holes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 827–838, November–December, 2008.     

The stress state of an anisotropic plate with two arbitrarily located elliptic holes or cracks

Using Lekhnitskii's method of complex potentials we study the stress state of an anisotropic plate with two arbitrary elliptic holes. We consider the cases when one or both of the holes become narrow slit-cracks, when the cracks extend to the edge of the hole, and when two cracks intersect or form a broken two-link crack. We give the results of numerical studies. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 24, 1993, pp. 33–43.     

负压激励下含椭圆孔高弹体的屈曲分析

基于数值模拟与理论分析,研究了含周期性椭圆孔二维结构的屈曲行为。针对不同的屈曲模态,建立理论模型进行模态分析。结果表明,改变孔的几何参数,椭圆孔结构的屈曲模态会随之发生转换,理论分析与数值结果吻合良好。此外,在数值模拟中,与位移加载不同,负压激励下的单胞需要考虑力边界条件的修正,以确保其满足完备性条件。已有工作在单胞选择中常存在问题,导致错误结果。针对上述问题研究了不同单胞所对应的边界条件,并结合有限结构进行了分析与讨论。     

Effect of Holes on the Eigenmodes of Reinforced Shells of Rotation

The vibrational eigenmodes of reinforced shells of rotation with holes are examined. A solution is constructed on the basis of the linear theory of thin elastic shells using the Ritz method. Results are presented from numerical studies of the effect of holes and reinforcements on the eigenmodes and forms of the vibrations. The results are compared with experimental data.     

Reconstruction of Mobilities for Electrons and Holes in Semiconductors

From a simplified approximate semiconductor model, we develop a 1-D identification problem to recover the mobilities for electrons and holes in semiconductors based on the LBIC technique, and cast it as an optimization problem. Its solution is defined by the minimal point of some objective functional. On this argumentation, we derive the gradient operator of objective functional and the necessary condition for the solution of inverse problem. Our result provides a numerical approach to reconstruct the mobilities for electrons and holes in semiconductors.     

Shear strength of webs with holes

In lightweight design the webs of beams which are loaded by bending are often provided with holes. Aim of this article is to present a method to calculate the shear strength of webs with certain holes. The semi-analytical method is derived with respect to the results of geometrically and physically non-linear numerical simulations in which the shear strength of the web is calculated. The method consists of a plastic failure approach for rather thick webs and a framework analogy to model the failure of a solid web after onset of buckling. To take the influence of the holes into account the shear strength of the solid web is reduced as a function of geometry parameters. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of the fundamental problems of the theory of elasticity for a half-plane with holes and cracks

We obtain the general solution of the fundamental problems of the theory of elasticity for an isotropic half-plane with a finite number of arbitrarily situated elliptic holes whose boundaries may intersect or form rectilinear cuts or boundaries of curvilinear holes. On the rectilinear boundary the first problem and the second or mixed problem of the theory of elasticity are defined. We use general expressions obtained previously by the author for the complex potentials generated by solving the problem of linear coupling for cuts in a multiconnected region, conformal mappings, and the method of least squares. The problem is reduced to solving a system of linear algebraic equations. The results of numerical experiments are given for a half-plane with a crack in the case of the first fundamental problem and the action of various loads. Two figures, two tables. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 157–171.     

On a numerical scheme for curved crack propagation based on configurational forces and maximum dissipation

A numerical scheme is presented to predict crack trajectories in two dimensional components. First a relation between the curvature in mixed–mode crack propagation and the corresponding configurational forces is derived, based on the principle of maximum dissipation. With the help of this, a numerical scheme is presented which is based on a predictor–corrector method using the configurational forces acting on the crack together with their derivatives along real and test paths. With the help of this scheme it is possible to take bigger than usual propagation steps, represented by splines. Essential for this approach is the correct numerical determination of the configurational forces acting on the crack tip. The methods used by other authors are shortly reviewed and an approach valid for arbitrary non–homogenous and non–linear materials with mixed–mode cracks is presented. Numerical examples show, that the method is a able to predict the crack paths in components with holes, stiffeners etc. with good accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of Steady Heat Conduction in Regions with Circular Holes by a Special Boundary-integral Method

Special boundary-integral equations developed in an earlierpaper are used here to analyse problems of steady heat conductionin regions with circular holes. In this approach, the solutionon the boundary of each hole is expanded in circular harmonicsand the unknown coefficients are determined by a special systemof boundary-integral equations. Approximate equations, whichretain only the first harmonic of the solution expansion oneach hole, are solved analytically for three problems involvingsymmetric arrangements of holes inside a circle. These resultscompare extremely well with an available exact solution in onecase and with numerical solutions of an exact integral equationin several others.     

A posteriori error analysis for elliptic pdes on domains with complicated structures

Summary. The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates. Mathematics Subject Classification (2000):65N30, 65R20, 73C50Received: 28, June 2001     

On the contact interaction of dissimilar anisotropic half-planes with thermally insulated gaps at the interface

We obtain a system of singular integral equations for the problem of thermoelastic contact of two dissimilar anisotropic half-planes with a system of thermally isolated gaps on the line of contact. Analysis of the contact interaction parameters was carried out for half-planes made of identical materials using the condition that on the surface of one of them there is a symmetric hole of prescribed shape. We give a scheme for numerical solution of the problem for holes of different profiles. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 117–124.     

Free-Edge Stresses around Holes in Laminates by the Boundary Finite-Element Method

In the present paper the boundary finite-element method is presented as a highly efficient technique for the numerical investigation of the free-edge stresses around a circular hole in laminates. In this method, as in the boundary element method, only the boundary needs to be discretized, whereas the element formulation in essence is finite-element based. The surface discretization provides a high numerical efficiency and requires less computation time compared to finite-element analyses. Numerical results for the concentration of interlaminar stresses at holes in composite laminates show a very good agreement with comparative finite-element calculations.     

Unsteady thermal stresses near a curvilinear hole in a plate with heat transfer under its warming by a heat flow

We describe an algorithm of determining quasistatic thermal stresses in multiply connected plates with heat transfer, induced by the disturbance of heat flow near holes. Our approach is based on the Laplace transformation and a modified relation of its numerical conversion. The boundary-value problems for the Helmholtz equation, from which the Laplace transform is determined, are solved using the method of boundary integral equations. We solve the integral equations by the method of mechanical quadratures. The results of calculation of nonstationary temperature fields and stresses induced by them in a strip with small holes of different shape are also presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 105–111, January–March, 2008.     

Inverse problems of the plane theory of elasticity : PMM vol. 38, n 6, 1974, pp. 963–979

In connection with the fact that failure of a structure ordinarily starts at sites of the most acute stress concentrations near cavitiies, it is of interest to determine the shape of the equally strong outlines of holes on which the technologically inevitable stress concentration would be least as compared with all other outlines.An effective exact solution of some inverse plane problems of the theory of elasticity concerning the determination of equally strong outlines of holes is proposed. A formulation of the problem is given first and the fundamental relationships are presented. Then the general problem for any number of holes in an infinite plane is reduced to a standard Dirichlet problem for the exterior of the same number of parallel slits on a parametric plane. An effective exact solution is found by this method for the case of one and two holes as well as for the case of periodic and doubly-periodic series of holes. The question of application of the solutions obtained to the theory of a minimum weight structure is considered.     

Forced vibrations of a multiply connected isotropic plate

On the basis of A. I. Aleksandrovich's investigations the present authors worked out an approximate method of solving problems of forced flexural vibrations of a multiply connected isotropic plate. For an elliptic plate with circular holes numerical results are presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 105–111, 1987.     

Asymptotics of the solution of Laplace’s equation with third-type boundary conditions on the boundaries of two small holes

A boundary value problem for Laplace’s equation in a bounded domain with two small holes is considered. Third-type boundary conditions are set on the boundaries of the holes. A Neumann condition is specified on the outer boundary of the domain. A uniform asymptotic approximation of the solution is constructed and justified up to an arbitrary power of a small parameter.     

Stress concentration in a spherical shell with two neighboring circular holes

We have solved analytically the problem of the stressed state in an isotropic shallow spherical shell with two circular holes. The cases of the close location of holes of both equal and unequal radii in the shell under the action of internal pressure have been investigated numerically. We have detected a considerable increase in stresses on the bulkhead between the holes at their approach.     

Dynamical complexity induced by Allee effect in a predator–prey model

In this paper, we investigate the complex dynamics induced by Allee effect in a predator–prey model. For the non-spatial model, Allee effect remains the boundedness of positive solutions, and it also induces the model to exhibit one or two positive equilibria. Especially, in the case with strong Allee effect, the model is bistable. For the spatial model, without Allee effect, there is the nonexistence of diffusion-driven instability. And in the case with Allee effect, the positive equilibrium can be unstable under certain conditions. This instability is induced by Allee effect and diffusion together. Furthermore, via numerical simulations, the model dynamics exhibits both Allee effect and diffusion controlled pattern formation growth to holes, stripe–hole mixtures, stripes, stripe–spot mixtures, and spots replication. That is to say, the dynamics of the model with Allee effect is not simple, but rich and complex.     

Risk averse elastic shape optimization with parametrized fine scale geometry

Shape optimization of the fine scale geometry of elastic objects is investigated under stochastic loading. Thus, the object geometry is described via parametrized geometric details placed on a regular lattice. Here, in a two dimensional set up we focus on ellipsoidal holes as the fine scale geometric details described by the semiaxes and their orientation. Optimization of a deterministic cost functional as well as stochastic loading with risk neutral and risk averse stochastic cost functionals are discussed. Under the assumption of linear elasticity and quadratic objective functions the computational cost scales linearly in the number of basis loads spanning the possibly large set of all realizations of the stochastic loading. The resulting shape optimization algorithm consists of a finite dimensional, constraint optimization scheme where the cost functional and its gradient are evaluated applying a boundary element method on the fine scale geometry. Various numerical results show the spatial variation of the geometric domain structures and the appearance of strongly anisotropic patterns.     

Numerical solution of the transport equations for charges in semiconducting devices

We propose a method of solving the problem of determining the concentration fields of electrons and holes and the electric potential field for semiconducting devices. We conduct an analysis of the numerical solutions of problems obtained on various finite-element grids.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 89–94.