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。     

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.     

Modeling of relaxation of viscoelastic stresses in multi-layered thin films/substrate systems due to thermal mismatch

Assume the ratio of the total axial rigidity of thin films to that of the substrate is smaller than 0.02, an approximate closed-form solution for viscoelastic stresses in multi-layered thin films/substrate systems due to thermal mismatch is derived. This is achieved by utilizing the analogy between the governing field equation of elasticity and the Laplace transform with respect to time of the viscoelastic field equation. Based on two solutions, simplified solutions for relaxation of residual stresses distributed in multiple layers of thin films deposited on a thick substrate are obtained. The effect of the thickness, thermal expansion coefficient, Young's modulus, and viscosity coefficient of the substrate and thin films on the relaxation of residual stresses is considered. This simplified solution can be applied to some special cases such as one layered or periodic multi-layered thin films on a thick substrate.     

Stability analysis of an explicit finite element scheme for plane wave motions in elastic solids

 Expressions for critical timesteps are provided for an explicit finite element method for plane elastodynamic problems in isotropic, linear elastic solids. Both 4-node and 8-node quadrilateral elements are considered. The method involves solving for the eigenvalues directly from the eigenvalue problem at the element level. The characteristic polynomial is of order 8 for 4-node elements and 16 for 8-node elements. Due to the complexity of these equations, direct solution of these polynomials had not been attempted previously. The commonly used critical time-step estimates in the literature were obtained by reducing the characteristic equation for 4-node elements to a second-order equation involving only the normal strain modes of deformation. Furthermore, the results appear to be valid only for lumped-mass 4-node elements. In this paper, the characteristic equations are solved directly for the eigenvalues using <ty>Mathematica<ty> and critical time-step estimates are provided for both lumped and consistent mass matrix formulations. For lumped-mass method, both full and reduced integration are considered. In each case, the natural modes of deformation are obtained and it is shown that when Poisson's ratio is below a certain transition value, either shear-mode or hourglass mode of deformation dominates depending on the formulation. And when Poisson's ratio is above the transition value, in all the cases, the uniform normal strain mode dominates. Consequently, depending on Poisson's ratio the critical time-step also assumes two different expressions. The approach used in this work also has a definite pedagogical merit as the same approach is used in obtaining time-step estimates for simpler problems such as rod and beam elements. Received: 8 January 2002 / Accepted: 12 July 2002 The support of NSF under grant number DMI-9820880 is gratefully acknowledged.     

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。     

Evaluation of Residual Stresses in Prismatic Rods with Unknown Young's Moduli

We propose a procedure for the evaluation of residual axial stresses in the surface layers of prismatic rods with unknown Young's moduli.     

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.     

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.     

Superconvergent patch recovery with equilibrium and conjoint interpolant enhancements

The superconvergent patch derivative recovery method of Zienkiewicz and Zhu is enhanced by adding the squares of the residuals of the equilibrium equation and natural boundary conditions. In addition, a new conjoint polynomial for interpolating the local patch stresses over the element which significantly improves the local projection scheme is presented. Results show that in the 4-node quadrilateral, the equilibrium and boundary condition residuals usually improve accuracy but not the rate of convergence, whereas in the 9-node quadrilateral, results are mixed. The conjoint polynomial always improves the accuracy of the derivative field within the element as compared to the standard nodal interpolation, particularly in 4-node quadrilaterals.     

Alternative reduced integration avoiding spurious modes for 8-node quadrilateral and 20-node hexahedron finite elements

Integrating the isoparametric 8-node quadrilateral and the 20-node hexahedron elements with Gauss integration based on the 3 point rule produces stiff elements. The excessive stiffness is mainly due to locking phenomenon. One remedy to partly remove locking consists in using reduced integration. Mostly, 2 × 2 or 2 × 2 × 2 integration, respectively, is employed. The lower order integration introduces spurious element modes, however. These modes may deteriorate solutions for finite element models. To overcome this drawback alternative reduced integration procedures are presented. A 5-point rule for the quadrilateral is described. 9-point and 21-point procedures are introduced for the hexahedron. The performance of these procedures is studied by some test problems.     

An adaptive h-refinement using transition element for plate bending problems

An adaptive h-refinement procedure utilizing compatible 5- to 7-node quadrilateral transition elements together with 4-node elements is presented for Reissner–Mindlin plate bending problems. By introducing the transition elements which have physical midside nodes, some difficulties associated with imposing displacement constraints on irregular nodes to enforce interelement compatibility in common adaptive h-refinement on quadrilateral mesh are easily overcome. Numerical examples are presented to show the effectiveness of using the transition elements for the adaptive h-refinement.     

A new hybrid-mixed variational approach for Reissner–Mindlin plates. The MiSP model

A valuable variational approach for plate problems based on the Reissner–Mindlin theory is presented. The new MiSP (Mixed Shear Projected) approach is based on the Hellinger–Reissner variational principle, with a particular representation of transversal shear forces and transversal shear strains. The approximations of the shear forces are derived from those of the bending moments using the corresponding equilibrium relations. The shear strains are defined in terms of the edge tangential strains that are projected on the element degrees of freedom. Two finite elements are developed on the MiSP approach basis: 3-node triangular element MiSP3 and 4-node quadrilateral element MiSP4. Both elements can be considered as the most simple among the existent mixed elements. A modified MiSP model with a derived 4-node element is also presented. Numerical experiments are presented which show that the MiSP elements do not exhibit shear locking and give excellent results for thick and thin plates. They also pass the patch test for a general triangle and quadrilateral. © 1998 John Wiley & Sons, Ltd.     

Unified and mixed formulation of the 4-node quadrilateral elements by assumed strain method: Application to thermomechanical problems

In this paper, a class of ‘assumed strain’ mixed finite element methods based on the Hu–Washizu variational principle is presented. Special care is taken to avoid hourglass modes and shear locking as well as volumetric locking. An unified framework for the 4-node quadrilateral solid and thermal as well as thermomechanical coupling elements with uniform reduced integration (URI) and selective numerical integration (SI) schemes is developed. The approach is simply implemented by a small change of the standard technique and is applicable to arbitrary non-linear constitutive laws including isotropic and anisotropic material behaviours. The implementation of the proposed SI elements is straightforward, while the development of the proposed URI elements requires ‘anti-hourglass stresses’ which are evaluated by classical constitutive equations. Several numerical examples are given to demonstrate the performance of the suggested formulation, including static/dynamic mechanical problems, heat conduction and thermomechanical problems.     

A study of the size-dependent elastic properties of ZnO nanowires and nanotubes

We present our calculations of the Young's modulus of ZnO nanowires and nanotubes by using the empirical Buckingham-type potential. Our results indicate that the Young's moduli of ZnO nanowires increase as the diameters decrease, and the Young's moduli of ZnO nanotubes increase as the thicknesses decrease. Furthermore, we find that such size-dependent elastic properties mainly arise from the lateral facets of the nanowires and nanotubes. In particular, for a ZnO nanotube with a thin wall, the Coulomb interaction between the ions of the outer and inner atomic layers plays an important role in the Young's moduli of the surface atomic layers.     

The reference coordinates and distortion measures for quadrilateral hybrid stress membrane element

The skew Cartesian coordinate system determined by the Jacobian of the isoparametric transformation evaluated at the origin can be shown to be a geodesic coordinate system at the origin. By using a theory in differential geometry, inverse relations of the isoparametric coordinate transformation can be derived and expressed in terms of these geodesic coordinates. In the formulation of hybrid stress finite elements, it is suggested as a new strategy for assumed stresses that such coordinates be used as the reference coordinates. The theory described is exemplified by its applications to the 4-node hybrid stress membrane elements. A set of new distortion-measuring parameters for the quadrilateral element are also proposed based on such theory.     

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.     

广义协调六结点平面曲边单元研究

主要运用广义协调原理,针对计算平面曲边单元的有限元算法进行了研究,并且利用点、周混合协调条件构造了三种高性能六结点曲边单元。第一、二种单元在平面直角坐标内分别采用解析试函数和完全三次多项式构造,第三种单元在六结点等参单元Q6的基础上附加广义协调泡状位移而成。这三种单元均能通过强式分片试验,并且显示了良好的计算精度和抗畸变能力。     

A family of simple and robust finite elements for linear and geometrically nonlinear analysis of laminated composite plates

A family of simple, displacement-based and shear-flexible triangular and quadrilateral flat plate/shell elements for linear and geometrically nonlinear analysis of thin to moderately thick laminate composite plates are introduced and summarized in this paper.

The developed elements are based on the first-order shear deformation theory (FSDT) and von-Karman’s large deflection theory, and total Lagrangian approach is employed to formulate the element for geometrically nonlinear analysis. The deflection and rotation functions of the element boundary are obtained from Timoshenko’s laminated composite beam functions, thus convergence can be ensured theoretically for very thin laminates and shear-locking problem is avoided naturally.

The flat triangular plate/shell element is of 3-node, 18-degree-of-freedom, and the plane displacement interpolation functions of the Allman’s triangular membrane element with drilling degrees of freedom are taken as the in-plane displacements of the element. The flat quadrilateral plate/shell element is of 4-node, 24-degree-of-freedom, and the linear displacement interpolation functions of a quadrilateral plane element with drilling degrees of freedom are taken as the in-plane displacements.

The developed elements are simple in formulation, free from shear-locking, and include conventional engineering degrees of freedom. Numerical examples demonstrate that the elements are convergent, not sensitive to mesh distortion, accurate and efficient for linear and geometric nonlinear analysis of thin to moderately thick laminates.     

Thermoelastic analysis of cracked woven GFRP laminates at cryogenic temperatures

We deal with the thermal-mechanical response of multi-layered G-11 woven glass/epoxy laminates with cracks and temperature-dependent properties under tension at liquid helium temperature (4 K). The composite material is assumed to be in generalized plane strain. Cracks are located in the fill fiber bundles and are assumed to span the width of the fill fiber bundles. Finite element model is used to study the influence of residual thermal stresses on the mechanical behavior of multi-layered woven laminates with cracks. Numerical calculations are carried out, and the Young's modulus and stress distributions near the crack tip are shown graphically.