Dynamical (time-harmonic) axisymmetric interface stress field in the finite pre-strained half-space covered with the finite pre-stretched layer

Within the framework of the piecewise homogeneous body model with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies (TLTEWISB), the dynamical (time-harmonic) stress field in the initially finite strained half-space covered with an initially and finitely stretched layer is investigated. It is assumed that on the upper free face of the covering layer the point located force which acts is harmonic with respect to time. The corresponding boundary contact problem is solved by employing the Hankel integral transformation. Moreover, it is assumed that the material of the layer and half-space are incompressible and elastic relations for those are given through the Treloar’s potential. In the case where the initial strains are absent in the layer and half-space the considered problem formulation and solution to that coincide with the corresponding ones of the classical linear theory of elasticity for an incompressible body. The algorithm for obtaining numerical results is proposed. The numerical results regarding the stresses acting on the interface plane are presented. These results are obtained for the case where the stiffness (distortion wave velocity) of the covering layer material is less (greater) than that for the half-space material. In this case, the main attention is focused on the dependencies between the values of the stresses and frequency of the external force and also the influence of the initial strains on these dependencies. In particular, it is established that the “resonance” values of the frequency of the external force increase, but the absolute maximum values of the stresses decrease significantly with the amount of the initial tension of the covering layer.     

Hybrid asymptotic-direct approach to the problem of finite vibrations of a curved layered strip

A multi-stage approach for the mathematical modeling in the field of nonlinear problems of mechanics of thin-walled structures is the subject of the present paper. A combination of the asymptotic, direct, and numerical methods for consistent and efficient analysis of problems of structural mechanics is presented on the example of plane problem of finite vibrations of a thin curved strip with material inhomogeneity. The method of asymptotic splitting allows for a consistent dimensional reduction of the original two-dimensional continuous problem as the thickness is small: the leading-order solution of the full system of equations of the theory of elasticity results in a one-dimensional formulation of the reduced theory and a problem in the cross-section. The direct approach to a material line extends the results to the geometrically nonlinear range. The appropriate finite element formulation allows for practical applications of the theory; with the numerical solution of the reduced problem, we restore the distributions of stresses, strains, and displacements over the thickness. Numerically and analytically investigated convergence of the solutions of various problems in the original (two-dimensional) and reduced (one-dimensional) models as the thickness tends to zero justifies the analytical conclusion that the curvature and variation of the material properties over the thickness do not require special treatment for classical Kirchhoff’s rods. Further terms of the asymptotic expansion lead to a model with shear and extension, in which curvature appears in a nontrivial way.     

A two-dimensional homogenized model for a pile of thin elastic sheets with frictional contact

The present paper deals with the computational simulation of a pile of thin sheets. The sheets are not laminated or glued, but they interact by frictional contact. In general, it is not possible to perform a full-scale finite element contact computation for piles containing thousands of sheets; the problem size becomes too large, and numerical solution methods suffer from severe convergence problems due to the large number of strongly coupled contact conditions. In this paper, a macroscopic material model is presented for the two-dimensional case. The pile of sheets is homogenized by introducing an effective anisotropic constitutive law, which is motivated by formulations of the theory of elasto-plasticity. This macroscopic material law models the behavior of a pile of sheets, allowing for no tensile stresses in the direction normal to the sheets and obeying Coulomb??s law of friction in the tangential contact plane. Applying this macroscopic material model, an equivalent homogeneous body can be treated using much coarser discretizations. Computational results for the problems are provided, and a comparison with simplified contact computations is done.     

Indirect Trefftz method for boundary value problem of Poisson equation

Trefftz method is the boundary-type solution procedure using regular T-complete functions satisfying the governing equation. Until now, it has been mainly applied to numerical analyses of the problems governed with the homogeneous differential equations such as the two- and three-dimensional Laplace problems and the two-dimensional elastic problem without body forces. On the other hand, this paper describes the application of the indirect Trefftz method to the solution of the boundary value problems of the two-dimensional Poisson equation. Since the Poisson equation has an inhomogeneous term, it is generally difficult to determine the T-complete function satisfying the governing equation. In this paper, the inhomogeneous term containing an unknown function is approximated by a polynomial in the Cartesian coordinates to determine the particular solutions related to the inhomogeneous term. Then, the boundary value problem of the Poisson equation is transformed to that of the Laplace equation by using the particular solution. Once the boundary value problem of the Poisson equation is solved according to the ordinary Trefftz formulation, the solution of the boundary value problem of the Poisson equation is estimated from the solution of the Laplace equation and the particular solution. The unknown parameters included in the particular solution are determined by the iterative process. The present scheme is applied to some examples in order to examine the numerical properties.     

A new boundary integral equation method of three-dimensional crack analysis

Introducing the mode II and mode III dislocation densities W ₂(y) and W ₃(y) of two variables, a new boundary integral equation method is proposed for the problem of a plane crack of arbitrary shape in a three-dimensional infinite elastic body under arbitrary unsymmetric loads. The fundamental stress solutions for three-dimensional crack analysis and the limiting formulas of stress intensity factors are derived. The problem is reduced to solving three two-dimensional singular boundary integral equations. The analytic solution of the axisymmetric problem of a circular crack under the unsymmetric loads is obtained. Some numerical examples of an elliptical crack or a semielliptical crack are given. The present formulations are of basic significance for further analytic or numerical analysis of three-dimensional crack problems.     

Stress intensity factors of an elliptical crack or a semi-elliptical crack subject to tension

This paper is concerned with the analysis of stress intensity factors of a semi-infinite body with an elliptical or a semi-elliptical crack subject to tension. Analysis is based on the body force method [1] which has been applied to the various plane stress problems. In this paper the method is extended to three-dimensional problems. The numerical calculations are performed for various shapes and configurations of ellipses and the results are in agreement with the two-dimensional cases by M. Isida asb/a→0. The stress intensity factor of a semi-elliptical crack in a plate of finite width is also discussed.     

Two-dimensional version of Sternberg and Al-Khozaie fundamental solution for viscoelastic analysis using the boundary element method

In this work a general and concise two-dimensional fundamental solution is obtained for quasi-static linear viscoelastic problems using the boundary element method. For this purpose, the three-dimensional fundamental displacement, derived by Sternberg and Al-Khozaie from the generalization of Navier equation, is integrated with respect to z-coordinate. A time formulation is constructed from the viscoelastic Reciprocity Principle, defined in terms of the Stieltjes integral and the material functions are acquired by means of Boltzmann's rheological model. The collocation method and a semi-analytical procedure for the singular boundary integral are employed to the numerical analysis of the boundary integral. The Gaussian quadrature, the analytical method and an incremental approach are used to deal with the convolution integral. As the latter has presented the best performance, it is employed in most analyses of the examples. Finally, numerical results of problems, found in the literature, are presented in order to validate the formulation and the two-dimensional fundamental solution.     

Finite deformation post-buckling analysis involving inelasticity and contact constraints

This paper is concerned with the numerical solution of large deflection structural problems involving finite strains, subject to contact constraints and unilateral boundary conditions, and exhibiting inelastic constitutive response. First, a three-dimensional finite strain beam model is summarized, and its numerical implementation in the two-dimensional case is discussed. Next, a penalty formulation for the solution of contact problems is presented and the correct expression for consistent tangent matrix is developed. Finally, basic strategies for tracing limit points are reviewed and a modification of the arc-length method is proposed. The good performance of the procedures discussed is illustrated by means of numerical examples.     

Advanced boundary element analysis of two- and three-dimensional problems of elasto-plasticity

An advanced formulation of the boundary element method has been developed for inelastic analysis based on an initial stress approach. The iterative solution algorithm makes use of an accelerated initial stress approach in which the past history of initial stresses are used to obtain an initial estimate for the current increment. In the present analysis the geometry and functions are represented by higher order (quadratic) shape functions to model complex geometries and rapid functional variations accurately. The methods of numerical integration of the kernels, particularly the singular type, are substantially improved by devising suitable automatic sub-segmentation routines that incorporate the recent developments in mapping procedures. The formulations have been implemented for two-dimensional plane stress, plane strain and three-dimensional elasto-plasticity problems.     

Shape optimization in frictionless contact problems

A finite element approach for shape optimization in two-dimensional (2-D) frictionless contact problems is presented in this work. The goal is to find the shape that gives a constant distribution of stresses along the contact boundary. The whole formulation, including mathematical model for the unilateral problem, sensitivity analysis and geometry definition is treated in a continuous form, independently of the discretization in finite elements. Shape optimization is performed by direct modification of geometry through B-spline curves and an automatic mesh generator is used at each new configuration to provide the finite element input data for numerical analysis and sensitivity computations. Using augmented-Lagrangian techniques (to solve the contact problem) and an interior-point mathematical-programming algorithm (for shape optimization), we obtain several results reported at the end of the article.     

A computational method for frictional contact problem using finite element method

Using the finite element method a numerical procedure is developed for the solution of the two-dimensional frictional contact problems with Coulomb's law of friction. The formulation for this procedure is reduced to a complementarity problem. The contact region is separated into stick and slip regions and the contact stress can be solved systematically by applying the solution technique of the complementarity problem. Several examples are given to demonstrate the validity of the present formulation.     

穿孔管阻性消声器横向模态和声学特性计算与分析

应用二维有限元法计算穿孔管阻性消声器的横向模态,利用数值模态匹配法计算其传递损失,推导了相应的公式并编写了计算程序。对于圆形同轴穿孔管阻性消声器的传递损失,数值模态匹配法计算结果与三维有限元法计算结果以及实验值吻合良好,表明了二维有限元法计算穿孔管阻性消声器横向模态和数值模态匹配法预测消声性能的准确性。进而分析孔径、穿孔率、吸声材料的密度和穿孔管偏移对圆形直通穿孔管阻性消声器横向模态和消声特性的影响。结果表明,孔径减小、穿孔率增大,或者穿孔管偏移量增大均能使消声器有效的平面波区域变宽,高频消声效果变好,但中频消声效果变差;增加吸声材料的填充密度则能提高消声器中高频的消声量。     

The interphase finite element

Mesomodelling of structures made of heterogeneous materials requires the introduction of mechanical models which are able to simulate the interactions between the adherents. Among these devices is quite popular the zero thickness interface (ZTI) model where the contact tractions and the displacement discontinuities are the primary static and kinematic variables. In some cases the joint response depends also on the internal stresses and strains within the thin layer adjacent to the joint interfaces. The interphase model, taking into account these additional variables, represents a sort of enhanced ZTI. In this paper a general theoretical formulation of the interphase model is reported and an original finite element, suitable for two-dimensional applications, is presented. A simple numerical experiment in plane stress state condition shows the relevant capabilities of the interphase element and allows to investigate its numerical performance. Some defects related to the shear locking of the element are resolved making use of well known numerical strategies. Finally, further numerical application to masonry structures are developed.     

Application of mixed variational formulations based on the finite element method to the solution of problems on natural vibrations of elastic bodies

We consider mixed variational formulations and the application of the mixed approximations of the finite element method to the solution of problems on natural vibrations of elastic bodies. To solve the generalized spectral problem, three forms of the mixed variational formulations are proposed. The correctness and stability of mixed variational formulations for displacements, strains and stresses are investigated. Matrix equations of the mixed method are given whose solution is performed using the modified algorithm of the steepest descent method. The results of calculations for natural frequencies of free vibrations of a straight and a circular beam are presented that are obtained in the solution of the problem in a two-dimensional formulation based on the classical and mixed finite-element method approaches. __________ Translated from Problemy Prochnosti, No. 2, pp. 121–140, March–April, 2008.     

On some mixed finite element methods for plane membrane problems

An analysis of some quadrilateral dual mixed finite element methods for plane membrane problems is presented. The methods are based on a variational formulation which a priori does not involve symmetric stresses. After having presented the governing equations of the problem under discussion in the linear framework, a detailed analysis of a method already proposed in Cazzani and Atluri (1993) is performed. In particular, a result setting the equivalence of the method and another one involving symmetric stresses is established. Two other methods, this time not equivalent to any symmetric stress method, are presented and for them an analysis is outlined. Finally, some numerical tests showing the method performances are provided.     

On the thermal problem for composite beams using a finite element semi-discretisation

The paper deals with steady state thermo-elastic problems in beam-like structures and it is composed of three theoretical sections. The first part presents a two-dimensional finite element procedure to compute the temperature distribution within a beam cross section subjected to prescribed boundary conditions. It allows the beam cross section to be modelled taking into account any kind of thermal anisotropy or inhomogeneity.

The second part is devoted to the structural thermo-elastic problem in a beam having arbitrary non-homogeneous, anisotropic material properties over the cross section but constant along the axis; the extension of a well-known semi-discretisation procedure to take into account anisotropic thermal expansion coefficients is presented. In this way it is possible to compute strains and stresses related to temperature distributions on the cross section computed, using the method outlined in the first part of the paper.

The third part describes the procedure to evaluate thermal equivalent loads suitable for a three dimensional frame analysis.

Some examples are presented and the results are compared either with their theoretical counterparts or with numerical results obtained from a full three-dimensional finite element analysis.     

Generalized RayleighWave Dispersion Analysis in a Pre-stressed Elastic Stratified Half-space with Imperfectly Bonded Interfaces

Within the framework of the piecewise homogeneous body model the influence of the shear-spring type imperfect contact conditions on the dispersion relation of the generalized Rayleigh waves in the system consisting of the initially stressed covering layer and initially stressed half plane is investigated. The second version of the small initial deformation theory of the three-dimensional linearized theory of elastic waves in initially stressed bodies is applied and the elasticity relations of the materials of the constituents are described by the Murnaghan potential. The magnitude of the imperfectness of the contact conditions is estimated through the shear-spring type parameter. Consequently, the influence of the imperfectness of the contact conditions on the generalized Rayleigh wave propagation velocity is studied through the influence of the values of this parameter. Numerical results on the action of the imperfectness of the contact conditions and the influence of the initial stresses in the constituents on the wave dispersion curves are presented and discussed. In particular, it is established that the magnitude of action of the imperfectness of the contact conditions under the influence of the initial stresses on the wave propagation velocity cannot be limited with corresponding ones obtained in the case where the contact between the constituents is complete and in the case where this contact is full slipping one. The possible application of the obtained results on the geophysical and geotechnical engineering is also discussed.     

On the theory of thin bodies

It is shown that, in accordance with the law of plane sections representing the basis of the theory of thin bodies, two analogies are true: a nonstationary analogy and a stationary one. Within the framework of the stationary analogy, a new-type expansion reducing a three-dimensional problem to a stationary two-dimensional one is introduced. The asymptotic conditions of validity of the stationary and nonstationary analogies were determined and the boundary-layer conceptions for both cases were compared. Solutions of the internal and external problems on a two-dimensional flow in a narrow zone were obtained in the closed form. An asymptotically correct mathematical model of a flow in a film is proposed. The stationary and nonstationary analogies for a swirling flow around a body and a flow around a rotating body were determined. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 5, pp. 45–54, September–October, 2007.     

An experimental–numerical hybrid technique for three dimensional stress analysis

The paper describes a numerical method for determining the stress distribution in the interior of a three-dimensional body using experimentally determined surface stresses, and the interior displacements from surface displacements. The normal and shear stresses inside the body are obtained by solving Laplace's equation in terms of sum of normal stresses together with the three-dimensional compatibility equations in terms of stresses, using the finite difference technique, when the stresses on the surface of the body are known. On the other hand if surface displacements are known (from which strain components could be determined) then displacement components in the interior of a body can be determined by solving Laplace's equation in terms of sum of normal strains together with the three-dimensional equilibrium equations in terms of displacements. It is shown that axi-symmetric problems can also be solved in an identical way by transforming the equations into cylindrical co-ordinates. The application of the method has been illustrated through several examples.     

A new error indicator based on stresses for three-dimensional elasticity

As of now, most of the error indicators available for elastostatic problems are computed in terms of quantities that do not necessarily have the most relevant physical meaning from the point of view of a mechanical engineer. Furthermore, only few of them have been extended to three-dimensional problems where all the advantages of the boundary element method (BEM) over other numerical techniques are more evident. In this work, a new efficient and reliable error indicator for three-dimensional elastostatic problems is presented based on ideas previously developed by the authors for the two-dimensional case. This error indicator directly estimates the error in the numerical solution for the boundary stresses and exploits in its formulation the high accuracy in the nodal values for those quantities provided by Hermite-like higher order boundary elements. The basic idea behind the computation of the new error indicator is to compare, on an element-by-element basis, two different numerical solutions. The first solution is obtained from an analysis using Hermite-like elements. The second one is obtained by using some of the degrees of freedom of the Hermite-like elements to approximate the field variables inside the elements using conventional Lagrangian shape functions. In this sense, it is assumed that the bigger the difference among these two solutions, the bigger the error in the stresses computed with the Hermite-like elements. Since both solutions are obtained from just one analysis, the computational cost of the proposed error indicator is very low. Three numerical examples are presented to show that the results obtained using this new error indicator with an-h-adaptive strategy are satisfactory and very promising.