基于解析试函数的各向异性材料厚薄通用板单元

该文采用满足Kirchhoff假设的薄板理论,推导了各向异性材料系列解析试函数,并利用该系列解析试函数构造了一个四边形应力杂交板单元。首先,该文从薄板理论的基本方程出发,推导了各向异性材料薄板中面挠度w应满足的特征微分方程。然后,从该方程出发求得w的系列特征通解,由w特征通解可进一步求得广义位移、广义应变和广义应力的解析试函数。同时,根据广义应力利用平衡条件构造了相应的横向剪力解析试函数。最后,根据已有的广义应力和横向剪力解析试函数构造了一个四边形应力杂交板单元ATF-PH4。数值算例表明:上述方法构造出的单元模型有很好的精度、收敛性,且对网格畸变不敏感,同时能较好地解决板单元的厚薄通用性问题。     

A quadrilateral hybrid-Trefftz plate bending element for the inclusion of warping based on a three-dimensional plate formulation

A plate formulation, for the inclusion of warping and transverse shear deformations, is considered. From a complete thick and thin plate formulation, which was derived without ad hoc assumptions from the three-dimensional equations of elasticity for isotropic materials, the bending solution, involving powers of the thickness co-ordinate z, is used for constructing a quadrilateral finite plate bending element. The constructed element trial functions, for the displacements and stresses, satisfy, a priori, the three-dimensional Navier equations and equilibrium equations, respectively. For the coupling of the elements, independently assumed functions on the boundary are used. High accuracy for both displacements and stresses (including transverse shear stresses) can be achieved with rather coarse meshes for thick and thin plates.     

弹性介质模糊有限元控制方程的快速解法

线弹性模糊有限元方法是分析弹性介质体模糊特性对结构响应产生不确定性影响的有效方法。即使对弹性介质体而言,模糊有限元控制方程的求解时间问题也是困扰其推广应用的主要障碍。为获得可靠可行的模糊有限元控制方程的快速求解方法,在深入研究弹性介质体的模糊源特点基础上,提出当引起结构模糊特性的力学参数为单源模糊数时,可以利用单源模糊数的运算特点来求解模糊有限元的控制方程,进而利用合成运算求解结构的模糊位移和模糊应力的分布。推导了基于单源模糊数运算的弹性介质模糊应力和模糊位移的计算表达式。应用模糊有限元求解的区间解法和快速解法对算例进行比较分析,结果表明了快速解法的正确性。     

Exact solution for axisymmetric deformation of laminated transversely isotropic annular plates

Summary Based on the three-dimensional theory of elasticity, this paper presents the state space equation for axisymmetric deformation of a laminated transversely isotropic annular plate. The finite Hankel transform is then introduced and applied to the state space equation. Four exact solutions corresponding to four specified boundary conditions are obtained and expressions for displacements and stresses are presented. Numerical results are finally compared with those obtained by the classical plate theory, the Reissner plate theory and the finite element method.     

A simple cubic displacement element for plate bending

Early attempts to construct a triangular finite element for plate bending problems from a compatible cubic displacement field are not entirely satisfactory. The present paper shows how an accurate plate element can be achieved using independent cubic polynomial assumptions for the internal and boundary displacements in conjunction with a modified potential energy principle. This approach yields a simple algebraic formulation with favourable connection quantities at the element vertices which will appeal to practical users of the conventional finite element displacement method. Moreover, in Appendix I it is shown that the cubic element is identical to a previous hybrid stress element with linear internal bending and twisting moments and cubic boundary displacements. The stresses obtained from the former hybrid finite element solution therefore satisfy the strain compatibility conditions exactly. This remarkable result has an important significance in the theory of hybrid finite elements.     

厚板问题的位移法分析

本文从弹性体的三维 L—N方程出发,导出了平板问题的普遍形式解.作为实例,给出了半无限板、半无限长板条两个边值问题的严格解析解.用本文的方法分析平板,所得结果能反映边界位移和应力沿厚度方向的变化情况.     

Combining X-FEM and a multilevel mesh superposition method for the analysis of thick composite structures

The use of simultaneous multiple plate models offers an attractive and alternative solution to full scale three-dimensional finite element method for the global–local analysis of laminated composite structures. In this paper, an approach is proposed where the less accurate plate model, used to carry out the analysis at the global level, is enhanced by more accurate and complex plate models in each laminate subregion where more accurate transverse stress or strain estimation is required (the local level).The total displacement is represented as the superposition of the displacements of a number of plate models. By appropriately defining boundaries to the enhancing model/region, it is demonstrated that the superposition of displacements can be used to locally enrich the solution where accurate through-the-thickness stresses are required. In this manner, a computationally efficient global model can be used to determine gross displacements, and potentially the enriched models can be used to determine stresses at lamina interfaces for the accurate prediction of localized phenomena such as damage initiation and growth. The model is implemented combining an extended FEM (X-FEM) and multilevel mesh superposition approach (MMSA). Extra degrees-of-freedom are added to the model to represent the additional displacement fields, and the meshing process remains independent for each field.The displacements and stresses computed by this approach are compared to literature data and analytical solutions for various plate geometries and loads showing an excellent correlation. Morevoer, the results showed, as expected, that the accuracy of the approximation is improved by the proposed approach compared to using the global plate model alone.     

Finite element analysis of rubber-like materials by a mixed model

A Reissner type variational principle is utilized to formulate a mixed finite element model for a finite-strain analysis of Mooney-Rivlin rubber-like materials. An incremental and stationary Lagrangian formulation is adopted. The functional consists of incremental displacements and incremental hydrostatic and distortional stresses as variables. In the finite element formulation the displacements are interpolated in terms of nodal displacements while the two different strss components are approximated independently. The stress parameters for the distortional stresses are eliminated at the element level and the resulting matrix equations for each incremental solution involve the incremental nodal displacements and the average hydrostatic pressure in each element as unknowns. Four-node quadrilateral plane stain elements were used to analyze the inflation of an infinitely long thick-walled cylinder subjected to internal pressure. Both resulting displacements and stresses are found to converge to exact values as the magnitude of the loading increments is decreasing.     

A mixed least squares method for solving problems in linear elasticity: formulation and initial results

In this paper a mixed least squares finite element method for solving problems in linear elasticity is proposed. The developed numerical technique allows the use of separate unknowns for displacements and stresses, discontinuous interpolation functions for displacements, and the resulting linear system has a symmetric and positive definite coefficient matrix. The approximate solution of the linear elasticity problem is obtained by minimization of a least squares functional based on the constitutive equations and equations of equilibrium. The proposed method is implemented in an original computer code written in C programming language. Its performance is tested on classical examples from theory of elasticity with well-known exact analytical solutions. Results from the implementation of a constant displacement-bilinear stress element and bilinear displacement-bilinear stress element are discussed.     

A method for linear elasto-static analysis of multi-layered axisymmetrical bodies using Hankel's transform

 A method to determine the distribution of stresses and displacements in an infinite, linear, elastic, multi-layered medium subjected to static axisymmetric loading is presented in this work. By using axisymmetric governing equations, Hankel's transform and matrix analysis, the methodology gives a clearly arranged way to calculate the stresses and displacements in the medium. A numerical method for Hankel's transform is employed to perform the calculation. Two representative examples are studied. The results can be utilized as a fundamental solution for boundary element methods for the linear, elasto-static, axisymmetric multi-layered problem with a little modification.     

Frictional contact analysis of composite materials

In this paper, the highly non-linear frictional contact problems of composite materials are analysed. A proportional loading, the potential contact zone method and finite element analysis are used to solve the problems. A tree-like searching method is used to obtain the solution of the parametric linear complementary problem, which may overcome the anisotropic properties of contact equations caused by composite materials. In the frictional contact analysis of composite materials, the distributions of normal contact pressures, tangential contact stresses and relative tangential displacements are presented for different contact material systems and different coefficients of friction. The results show that the solutions in the paper have good agreement with Hertzian solutions. The influence of different contact material systems and different coefficients of friction on the contact stresses and displacements is large. As a numerical example, ball-indentation tests of composite materials are modelled by the three-dimensional finite element method.     

Application of the moment scheme of the finite element method to the solution of problems with initial stresses in the incremental elasticity theory

We discuss application of the finite element method to the solution of problems with initial stresses within the elasticity theory. Based on the incremental theory of deformable solids, the relationships of the finite element method are derived to calculate the stiffness matrix coefficients for a prestressed spatial element of the serendip family with quadratic approximation of displacements. The calculation of the stressed state of an eccentrically compressed beam and a round plate under conditions of longitudinal-transverse bending is carried out. Comparison of the numerical results with analytical solutions is presented. The variation in the compression and shear strains of a cylindrical damper is studied depending on the degree of deformation and the sequence of load application. __________ Translated from Problemy Prochnosti, No. 3, pp. 131–143, May–June, 2006.     

On free vibration of a functionally graded piezoelectric rectangular plate

Summary On the basis of three-dimensional theory equations of transversely isotropic piezoelasticity, two independent state equations with variable coefficients are derived. To this end, separation formulae for displacements and shear stresses are employed. A laminated approximation is used to transform the state equations to the ones with constant coefficients in each layer. The free vibration problem of a piezoelectric rectangular plate with a functionally graded property is then investigated. Discussion on the boundary conditions is presented.     

Stability of laminated orthotropic plates upon slippage of layers

The article deals with a laminated orthotropic plate. Between the layers there is rigid contact as well as contact with frictionless slippage. On the end faces at the level of the neutral surface of the slipping layers the plate is loaded by tangential loads. The model of loss of stability is based on the hypotheses on displacements of the plate that take transverse shear strain into account. Normal displacements of the plate do not depend on the transverse coordinate. The hypotheses derived by iteration make it possible to satisfy the conditions of interlaminar contact, both rigid and with frictionless slippage, with respect to displacements and stresses, and also the conditions on the outer surfaces of the plate. With the aid of Lagrange's variational principle we obtained differential equations under conditions of the precritical and the postcritical state, and also the corresponding boundary conditions. The equations of the postcritical state are written in terms of displacements. On the basis of the presented equations the problem of the stability of a homogeneous orthotropic rectangular plate with pin support along the periphery is solved. It is shown that the results obtained with the suggested model coincide with the known results. The article also examines a square three-layered orthotropic plate under conditions of frictionless slippage contact between the layers and pin support along the periphery. The central layer is loaded by tangential loads on the end faces at the level of its neutral surface. It is shown that when slippage is taken into account, the critical forces are substantially reduced.Translated from Problemy Prochnosti, No. 8, pp. 35–41, August, 1994.     

Crack opening displacement in plate with bonded repair patch

Mathematical techniques are extended to compute crack opening displacements in a cracked plate with an adhesively bonded composite patch. The plate and the patch are considered as orthotropic materials. The problem is reduced to the solution of integral equations. A software program is written to compute shear stresses in adhesive, stress intensity factors in the plate and the crack openings at the centreline of the crack. The effects of adhesive thickness, adhesive modulus, patch thickness and plate thickness on crack openings are investigated. A test program is carried out to obtain crack opening displacements in plate with bonded patch. A good agreement with analytical predictions is obtained. The effects of patches bonded on one or both sides of a plate on stress intensity factors are evaluated.     

Nonlinear analysis via global–local mixed finite element approach

A computational algorithm, based on the combined use of mixed finite elements and classical Rayleigh–Ritz approximation, is presented for predicting the nonlinear static response of structures; The fundamental unknowns consist of nodal displacements and forces (or stresses) and the governing nonlinear finite element equations consist of both the constitutive relations and equilibrium equations of the discretized structure. The vector of nodal displacements and forces (or stresses) is expressed as a linear combination of a small number of global approximation functions (or basis vectors), and a Rayleigh–Ritz technique is used to approximate the finite element equations by a reduced system of nonlinear equations. The global approximation functions (or basis vectors) are chosen to be those commonly used in static perturbation technique; namely a nonlinear solution and a number of its path derivatives. These global functions are generated by using the finite element equations of the discretized structure. The potential of the global–local mixed approach and its advantages over global–local displacement finite element methods are discussed. Also, the high accuracy and effectiveness of the proposed approach are demonstrated by means of numerical examples.     

Numerical modeling of micro- and macro-behavior of viscoelastic porous materials

This paper presents a numerical method for solving the two-dimensional problem of a polygonal linear viscoelastic domain containing an arbitrary number of non-overlapping circular holes of arbitrary sizes. The solution of the problem is based on the use of the correspondence principle. The governing equation for the problem in the Laplace domain is a complex hypersingular boundary integral equation written in terms of the unknown transformed displacements on the boundaries of the holes and the exterior boundaries of the finite body. No specific physical model is involved in the governing equation, which means that the method is capable of handling a variety of viscoelastic models. A truncated complex Fourier series with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the boundaries of the holes. A truncated complex series of Chebyshev polynomials with coefficients dependent on the transform parameter is used to approximate the unknown transformed displacements on the straight boundaries of the finite body. A system of linear algebraic equations is formed using the overspecification method. The viscoelastic stresses and displacements are calculated through the viscoelastic analogs of the Kolosov–Muskhelishvili potentials, and an analytical inverse Laplace transform is used to provide the time domain solution. Using the concept of representative volume, the effective viscoelastic properties of an equivalent homogeneous material are then found directly from the corresponding constitutive equations for the average field values. Several examples are given to demonstrate the accuracy of the method. The results for the stresses and displacements are compared with the numerical solutions obtained by commercial finite element software (ANSYS). The results for the effective properties are compared with those obtained with the self-consistent and Mori–Tanaka schemes.     

A quadrilateral hybrid stress element for Mindlin plates based on incompatible displacements

An hybrid stress element formulation based on internal, incompatible displacements is used to develop efficient Mindlin plate elements. The 4-node quadrilateral Mindlin plate element is derived from a modified energy functional. Both displacements and stresses are defined in the natural co-ordinate interpolation system. The assumed stress field is obtained by tensor transformation and so chosen as to ensure that the element is co-ordinate invariant and stable. Shear locking is avoided through an appropriate identification of the internal, incompatible displacement field. The role played by incompatible displacements in the formulation of hybrid stress elements for thin and moderately thick plates is discussed. Numerical applications are presented to illustrate the accuracy and reliability of the suggested Mindlin plate element.     

Through-the-thickness stress predictions for laminated plates of advanced composite materials

A finite element formulation is developed with emphasis primarily focused on providing stress predictions for thin to moderately thick plate (shell) type structures. Plate element behaviour is specified by prescribing independently the neutral surface displacements and rotations, thus relaxing the Kirchhoff hypothesis. Numerical efficiency is achieved due to the simplicity of the element formulation, i.e. the approach yields a displacement dependent multi-layer model. In-plane layer stresses are determined via the constitutive equations, while the transverse shear and short-transverse normal stresses are determined via the equilibrium equations. Accurate transverse stress variations are obtained by appropriately selecting the displacement field for the element. A selective reduced integration technique is utilized in computing element stiffness matrices. Static and spectral (eigenvalue) tests are performed to demonstrate the element modelling capability.     

Solutions of a twelfth order thick plate theory

Summary A system of equilibrium equations governing a twelfth-order theory for the bending of thick plates is shown to be equivalent to a biharmonic equation together with four Helmholtz equations. These equations are closely related to equations derived by Cheng for an elasticity based thick plate theory. Detailed comparisons between the solutions for the displacements and stresses predicted by the approximate plate theory and an exact theory give some basis for deciding the applicability of the plate theory. As an example of the application of the solution procedure presented here, some earlier results for the decay parameters for the end problem for finite width plates are extended to the present case of twelfth-order plate theory.With 5 Figures