An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain

 In this paper, an integral equation method to the inclusion-crack interaction problem in three-dimensional elastic medium is presented. The method is implemented following the idea that displacement integral equation is used at the source points situated in the inclusions, whereas stress integral equation is applied to source points along crack surfaces. The displacement and stress integral equations only contain unknowns in displacement (in inclusions) and displacement discontinuity (along cracks). The hypersingular integrals appearing in stress integral equation are analytically transferred to line integrals (for plane cracks) which are at most weakly singular. Finite elements are adopted to discretize the inclusions into isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements and the crack surfaces are decomposed into discontinuous quadratic quadrilateral elements. Special crack tip elements are used to simulate the variation of displacements near the crack front. The stress intensity factors along the crack front are calculated. Numerical results are compared with other available methods. Received: 28 January 2002 / Accepted: 4 June 2002 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7101/99 E).     

Invariant 8-node hybrid-stress elements for thin and moderately thick plates

Eight-node hybrid-stress elements are developed for the analysis of plates ranging from arbitrarily thin to moderately thick. The displacement behaviour is characterized by a transverse displacement and independent cross-section rotations, which are interpolated using the 8-node Serendipity shape functions. All components of stress are included; alternative elements are developed which differe in the form of the inplane distribution of the stresses. Elements are sought for whic the stiffiness is invariant and of correct rank, and whic show on signs of deterioration in the thin-plate limit. A discussion of the prospects for developing a 4-node element with these characteristics is also presented. Example problems are used to compare the performance of the 8-node elements including convergence behaviour, intraelement stress distributions and optimal sampling locations, and range of applicability in terms of plate thickness ratio.     

Elastoplastic analysis using the 14-node brick element family

Typical brick-based finite elements in current use are the 8-and 20-node members of the serendipity group. For displacement analyses of solids the 8-node element can be quite stiff in certain deformation modes, while the 20-node element can be quite expensive to use, involving as it does 60 degrees of freedom and a fairly high order of quadrature to avoid spurious eigenmodes of the element stiffness. In this paper a family of intermediate 14-node elements is investigated. Derivation of their properties can be considerably assisted by computer algebra. Performance is evaluated for elastic and elastoplastic problems.     

Axisymmetric quadrilateral elements for large deformation hyperelastic analysis

In this paper, axisymmetric 8-node and 9-node quadrilateral elements for large deformation hyperelastic analysis are devised. To alleviate the volumetric locking which may be encountered in nearly incompressible materials, a volumetric enhanced assumed strain (EAS) mode is incorporated in the eight-node and nine-node uniformly reduced-integrated (URI) elements. To control the compatible spurious zero energy mode in the 9-node element, a stabilization matrix is attained by using a hybrid-strain formulation and, after some simplification, the matrix can be programmed in the element subroutine without resorting to numerical integration. Numerical examples show the relative efficacy of the proposed elements and other popular eight-node elements. In view of the constraint index count, the two elements are analogous to the Q8/3P and Q9/3P elements based on the u–p hybrid/mixed formulation. However, the former elements are more straight forward than the latter elements in both formulation and programming implementation.     

The use of ANSYS to calculate the behaviour of sandwich structures

In this article, we use different models to compute displacements and stresses of a simply supported sandwich beam subjected to a uniform pressure. 8-node quadrilateral elements (Plane 82), multi-layered 8-node quadrilateral shell elements (Shell 91) and multi-layered 20-node cubic elements (Solid 46) are used. The influence of mesh refinement and of the ratio of Young's moduli of the layers are studied. Finally, a local Reissner method is presented and assessed, which permits an improvement in the accuracy of interface stresses for a high ratio of Young's moduli of the layers with Plane 82 elements.     

复合材料双曲度大网格高精度单元研究

基于四节点Timoshenko层合梁理论和八节点薄壳单元相关理论, 通过自编四节点板单元和八节点壳单元, 实现了加筋前后双曲度复合材料薄壳单元的大网格高精度计算。结果表明: 八节点单元精度稍高, 但四节点单元分析的时间更短, 因而效率更高, 更具有工程实用性; 筋条网格尺寸大小对计算精度影响小。研究成果可用于大规模复合材料双曲度结构设计和分析。     

非协调元性能分析的两个定理

在构造非协调元的过程中,必须遵守一定的构造规律。本文从基本力学观点出发,提出并证明了两个定理。定理一、如果某种类型的有限单元共有n个独立参与整体刚度运算的自由度,则该单元最多只能精确模拟n种弹性力学基本解。该定理说明了单元的精度从根本上受自身自由度限制的,并指出了现有的四边形四结点单元发展空间不大,而四边形八结点Q8单元以及三维八结点H8单元仍然具有较大的发展余地。定理二则认为四边形四结点内参型非协调元如果能够通过小片试验,则不可能在任意畸变状态下精确表示纯弯场。该定理表明了畸变问题的尝试是有限制的。以上的结论虽然是针对非协调元的构造来提出的,但从论证过程看,应对其它类型的有限单元也适用。定理一和定理二对于今后新型有限元的发展可以起到一定的指导作用。     

Deformation mechanisms in negative Poisson's ratio materials: structural aspects

Poisson's ratio in materials is governed by the following aspects of the microstructure: the presence of rotational degrees of freedom, non-affine deformation kinematics, or anisotropic structure. Several structural models are examined. The non-affine kinematics are seen to be essential for the production of negative Poisson's ratios for isotropic materials containing central force linkages of positive stiffness. Non-central forces combined with pre-load can also give rise to a negative Poisson's ratio in isotropic materials. A chiral microstructure with noncentral force interaction or non-affine deformation can also exhibit a negative Poisson's ratio. Toughness and damage resistance in these materials may be affected by the Poisson's ratio itself, as well as by generalized continuum aspects associated with the microstructure.     

Holographic study of conventional and negative Poisson's ratio metallic foams: elasticity,yield and micro-deformation

An experimental study by holographic interferometry is reported of the following material properties of conventional and negative Poisson's ratio copper foams: Young's moduli, Poisson's ratios, yield strengths and characteristic lengths associated with inhomogeneous deformation. The Young's modulus and yield strength of the conventional copper foam were comparable to those predicted by microstructural modelling on the basis of cellular rib bending. The re-entrant copper foam exhibited a negative Poisson's ratio, as indicated by the elliptical contour fringes on the specimen surface in the bending tests. Inhomogeneous, non-affine deformation was observed holographically in both foam materials.     

Plane strain finite element simulation of shear band formation during metal forming

A plane strain finite element formulation and solution procedure for shear band failure during the plane strain metal forming process are developed and presented. The large strain elastic-plastic formulation includes a 5-node 10-degree-of-freedom (d.o.f.) ‘crossed-triangle’ element, a 4-node 8-d.o.f. element with selective reduced integration, an 8-node 16-d.o.f. element and a 4-node 8-d.o.f. element with an embedded shear band. The formulation includes an elastic-plastic material model with a modified Gurson yield function and combined isotropic-kinematic hardening. The solution procedure is based on a Newton–Raphson incremental-iterative method with an orthogonal projection of zero or negative eigen-modes when required. Two different examples of plane strain tension test are studied with results compared with available numerical solutions to evaluate the present formulation and solution procedure of the four different elements. The results demonstrate that both types of the 4-node quadrilaterals are comparable to the 5-node crossed-triangle element as well as the 8-jiode element. To further validate and to demonstrate the predictive capability and practical applicability of the present development, two plane strain metal forming examples are investigated. The first application is a numerical simulation of a sheet-stretching test with results compared with experimental data for a commercially pure aluminium–magnesium 5182-O sheet. The load vs. extension history and the through-thickness strain are compared. The good agreement suggests that it is possible to numerically determine the parameters needed for the modified Gurson yield function. The second application is a numerical simulation of the formation of dead metal zones in the extrusion process. A plane strain extrusion of a short aluminium billet through straight-sided dies is presented and characteristic features of the formation of dead metal zone are observed.     

Plane isoparametric hybrid-stress elements: Invariance and optimal sampling

Formulation and applications of the hybrid-stress finite element model to plane elasticity problems are examined. Conditions for invariance of the element stiffness are established for two-dimensional problems, the results of which are easily extended to three-dimensional cases. Next, the hybrid-stress functional for a 3-D continuum is manipulated into a more convenient form in which the location of optimal stress/strain sampling points can be identified. To illustrate these concepts, 4- and 8-node plane isoparametric hybrid-stress elements which are invariant and of correct rank are developed and compared with existing hybrid-stress elements. For a 4-node element, lack of invariance is shown to lead to spurious zero energy modes under appropriate element rotation. Alternative 8-node elements are considered, and the best invariant element is shown to be one in which the stress compatibility equations are invoked. Results are also presented which demonstrate the validity of the optimal sampling points, the effects of reduced orders of numerical integration, and the behaviour of the elements for nearly incompressible materials.     

Explicit expressions of eigenvalues and eigenvectors for transversely isotropic piezoelectric materials

Summary.  Explicit expressions of eigenvalues and associated eigenvectors in Stroh's theory [1] for transversely isotropic piezoelectric materials under generalized plane deformation are presented. It is shown that the practical calculation for some piezoelectric ceramics yields all eigenvalues which are distinct from each other for generalized x-z plane deformation, where z is the poling axis of a transversely isotropic piezoelectric material. Some numerical results based on the explicit expressions are given, and the orthogonality and closed relation between matrices A and B in Stroh's theory [2], [3] for piezoelectric material are verified numerically. Received May 8, 2002; revised December 14, 2002 Published online: May 20, 2003 The paper is supported by the doctorate foundation of Xi'an Jiaotong University. The science foundation of the Shaanxi Province of China is appreciated, too.     

Assessment of 4-node EAS–ANS shell elements for large deformation analysis

In this work we derive and mutually compare several 4-node shell hyperelastic finite elements for large deformation analysis. The elements are based on Reissner–Mindlin shell theory. They use the enhanced assumed strain (EAS) concept for enhancement of membrane and bending displacement-compatible strains and the assumed natural strain (ANS) concept for treatment of transverse shear strains. The elements differ from each other by the number of membrane and bending EAS parameters. An optimal number of these parameters is suggested on the basis of numerical results.     

GFWRP管增强混凝土柱的轴压泊松比

为完善和发展GFWRP管增强混凝土柱的设计与计算理论,设计并制备了不同管径、壁厚及缠绕角的GWFRP管混凝土柱试件。进行了轴心压缩试验并对试验结果回归分析,得到GFWRP管混凝土柱轴心压缩过程中转折点及峰值点的泊松比变化方程和相应的应变预测公式,得到组合结构轴压全过程的泊松比变化预测方程。可由给定的组分材料性能预测组合构件的极限泊松比和极限应变,达到预测实际工程中GFWRP管混凝土柱抗力与变形性能的目的。通过试验及计算结果对GFWRP管混凝土与传统钢管混凝土的泊松比变化趋势进行了对比分析,发现两者变化规律完全不同。进一步探讨了GFWRP约束管的纤维缠绕角对组合结构轴压泊松比的影响。结果表明,纤维缠绕角与轴压泊松比近似成反比关系。为制定合理的GFWRP复合材料增强混凝土柱规范提供了理论依据。      

Modified and Trefftz unsymmetric finite element models

The unsymmetric finite element method employs compatible test functions but incompatible trial functions. The pertinent 8-node quadrilateral and 20-node hexahedron unsymmetric elements possess exceptional immunity to mesh distortion. It was noted later that they are not invariant and the proposed remedy is to formulate the element stiffness matrix in a local frame and then transform the matrix back to the global frame. In this paper, a more efficient approach will be proposed to secure the invariance. To our best knowledge, unsymmetric 4-node quadrilateral and 8-node hexahedron do not exist. They will be devised by using the Trefftz functions as the trial function. Numerical examples show that the two elements also possess exceptional immunity to mesh distortion with respect to other advanced elements of the same nodal configurations.     

Composite delamination modelling using a multi-layered solid element

This paper focuses on the latest development of a solid hexahedron element for composite delamination analysis. The 8-node solid is derived from a 20-node hexahedron. It is transformed into two physical independent 4-node shell elements according to the propagation of delamination process within the element.     

Eigenvalues and stable time steps for the bilinear Mindlin plate element

Eigenvalues are obtained for the 4-node Mindlin plate element with one-point quadrature. Both isotropic and some anisotropic matenals are considered, but for the latter only bounds are obtained on the eigenvalues. These eigenvalues provide stable time steps for explicit time integration algorithms.     

带转角自由度的四边形六面体单元

本文提出了三个带转角自由度单元，其中一个平面四边形单元，两个空间六面体单元。对平面单元每个结点有两个线位移自由度、一个转角自由度；对空间单元，每个结点有三个线位移自由度、三个转角自由度。这些单元列式简单，其中两个无多余零能模式，数值计算表明，它们的计算精度高。     

A Serendipity cubic-displacement hybrid-stress element for thin and moderately thick plates

A hybrid-stress element is developed for the analysis of thin and moderately thick plates. The independent transverse displacement and rotations are interpolated by the 12-node cubic Serendipity shape functions. All components of stress are included and 36β stress assumption is used. The element stiffness possesses correct rank and numerical results indicate that the element does not lock in the thin-plate limit. Results obtained using the present element are compared with those obtained using a 12-node assumed-displacement based Mindlin plate element with reduced integration; the present hybrid-stress element is shown to yield superior accuracy for all cases considered. In addition, the accuracy of the present element is compared against that of analogous 4-node and 8-node hybrid-stress Mindlin plate elements.     

Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements

In this paper we discuss and compare three types of 4-node and 9-node finite elements for a recently formulated finite deformation shell theory with seven degrees of freedom. The shell theory takes thickness change into account and circumvents the use of a rotation tensor. It allows for the applicability of three-dimensional constitutive laws and equipes the configuration space with the structure of a vector space. The finite elements themselves are based either on a hybrid stress functional, on a hybrid strain functional, or on a nonlinear version of the enhanced strain concept. As independent variables either the normal and shear resultants, the strain tensor related to the deformation of the midsurface, or the incompatible enhanced strain field are taken as independent variables. The fields of equivalence of these different formulations, their limitations as well as possible improvements are discussed using different numerical examples. Received 10 December 1998