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.     

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.     

Three-dimensional hybrid-stress isoparametric quadratic displacement elements

Two 20-node quadratic displacement three-dimensional isoparametric elements are developed based on the hybrid-stress model. The elements differ in the stress interpolation used. In one case, the stress polynomials are selected to correspond approximately to the strain polynomials obtained from the displacement field, and a 57β stress field results. In the other element, complete cubic polynomials are used which are forced to satisfy the equilibrium equations and stress compatibility equations, and a 69β stress field results. Both elements possess correct rank, but only the 69β element is invariant. Results obtained using these two elements, and the corresponding 20-node assumed-displacement element, are compared and the 69β element is shown to be the better element. The 2 × 2 × 2 Gauss stations are also verified to be the optimal sampling points for these elements.     

Hybrid stress reduced-Mindlin elements for thin multilayer plates

A hybrid-stress formulation of isoparametric elements for the analysis of thin multilayer laminated composite plates is presented. The element displacement behaviour is characterized by laminate reference surface inplane and transverse displacements and laminate non-normal cross-section rotations; as a result, simple C^o interpolation of displacement and rotation can be used, and the number of degrees-of-freedom is independent of the number of layers. All components of stress are included and are related to a set of laminate stress parameters, the number of which is independent of the number of layers. Attention is restricted here to thin laminates, for which it is shown that the contributions of transverse shear stress and transverse normal stress to the internal complementary energy can be neglected. As a result of this reduction, a modified stiffness-formation algorithm can be used which provides a significant improvement in computational efficiency. The formulation presented is used to develop an 8-node isoparametric thin multilayer plate element. The resulting element is naturally invariant, of correct rank, and non-locking in the thin plate limit.     

Improved hybrid-stress axisymmetric elements includng behaviour for nearly incompressible materials

A series of 4-node axisymmetric solid-of-revolution elements with quadrilateral cross-section are developed based on the hybrid-stress finite element model. The displacement interpolation is identical, and the elements differ in the stress field chosen; alternative schemes for selection of stress distribution are investigated. The performance of these elements is assessed and compared for the example problems of a thick cylinder and thick sphere under internal pressure. Of particular importance are convergence, intra-element stress distributions, and element performance as the incompressible state is approached.     

Three-dimensional 8-node and 20-node refined hybrid isoparametric elements

In this paper a new hybrid variational principle with independent variables of strain, stress and displacement and with a weaker constraint condition of interelement continuity is proposed. Based on this functional, a general formulation of a refined hybrid isoparametric element method has been established by the orthogonal approach. The present formulation is a rational approach to be adopted for deriving high-performance three-dimensional hybrid isoparametric elements even up to the higher-order 20-node element. Several numerical examples are presented to show that the present elements RGH₈(8-node) and RGH₂₀(20-node) have high accuracy, excellent computational efficiency and less sensitivity to mesh distortion.     

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).     

FINITE ELEMENT ANALYSIS OF CLOSURE BEHAVIOUR OF FATIGUE CRACKS IN RESIDUAL STRESS FIELDS

An elastic-plastic finite element analysis with high order elements is performed to examine closure behaviour of fatigue. cracks in residua1 stress fieids and the numerical results are then compared with experimental results. The finite element analysis, performed under plane stress using 8-node isoparametric elements, can predict fatigue crack closure behaviour through residual stress fields very well. The crack opening and closing behaviour through a compressive residual stress field is found to be complicated and influenced by the applied load magnitude and the location of the crack tip. Three different types of crack opening behaviour, namely, normal, unsymmetric partial and symmetric partial crack opening behaviour are observed through a compressive residual stress field. The partial crack opening stress intensity factor including the partial crack opening effect is recommended for the prediction of fatigue crack growth through a compressive residual stress field.     

Elastic-plastic analysis of plates by the hybrid-stress model and initial-stress approach

Two alternative hybrid-stress-based functionals are examined for the incremental elastic-plastic static analysis of single layer plates. Material nonlinear effects are incorporated via the initial-stress approach so that an equivalent nodal force vector is defined and the stiffness remains constant throughout the incremental loading. The alternative functionals differ in the incremental stress which is assumed to satisfy equilibrium; in the first, it is the actual stress increment, and in the second it is the elastic stress increment. Results are presented for two example problems, and comparisons of the alternative functionals and plausible iteration schemes are given. The effects of variation of pertinent solution parameters are also shown. A 4-node hybrid-stress plate element based on a Mindlin-type displacement field is used for most cases; however, limited results are also presented using an 8-node plate element, thus permitting comparisons of the relative efficiencies of the two elements.     

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.     

Control of spurious mechanisms for 20-node and transition sub-integrated hexahedral elements

In this paper, control of spurious mechanisms for sub-integrated 20-node and transition hexahedral elements is devised by an assumed stress approach. The higher-order stress modes for stabilizing the sub-integrated elements are identified by examining the spurious mechanisms. With an admissible simplification of the flexibility matrix, row vectors of the leverage matrix can be used as stabilization vectors. Numerical examples for the stabilized 20-node element are presented. Accuracy of the derived element is far better than the fully integrated displacement elements. Meanwhile, the former also consumes marginally less CPU time than the latter.     

Isoparametric hybrid hexahedral elements for three dimensional stress analysis

Based on a new functional in which displacements, strains and stresses are taken as independent variables, a set of three 8-node hexahedral hybrid elements Q_S11-1, Q_S11-2 and Q_S11-3 is developed. The adoption of separated stress and displacement variables proves to be effective in improving the accuracy of the elements. The new elements are all capable of yielding converging results, and they all possess the properties of having no zero energy deformation modes and of being co-ordinate invariant. From the numerical example of a beam under bending it is concluded that exact solutions can be obtained for right prism elements, while good results are still attainable for severely distorted elements. The relationship between the new hybrid elements and the conventional displacement elements is also explored in this paper.     

Relations between incompatible displacement model and hybrid stress model

An examination of the variational formulations confirms the similarity between the incompatible displacement model and the assumed stress hybrid model that was pointed out by Irons in 1972. But the basic differences between the two are also identified. For 8-node solid elements the assumed stress terms obtained through a rational procedure also agree with those deduced by Irons purely from his physical insight.     

General behaviour of collapsed 8-node 2-D and 20-node 3-D isoparametric elements

This paper presents a simple and thorough technique to examine the general behaviour of two-dimensional (2-D) 8-node and 3-D 20-node isoparametric elements, collapsed at one side. With this method the conditions for the well-known quarter-point and half-point elements, frequently used in fracture mechanics applications to model the crack tip behaviour, are arrived at in a natural way. It also allows for an in-depth analysis of the Jacobian determinant, and can easily be applied to other types of elements.     

Boundary elements for three-dimensional elastic crack analysis

In this paper 8-node traction singular boundary elements are employed to represent displacement and traction variations in the vicinity of the crack front in three-dimensional geometries. The numerical procedure suggested for evaluating the singular integrals extending over these special elements is described. The efficiency and accuracy of the special elements and integration procedure are demonstrated by the results obtained in a simple test problem whose analytical solution is known. The interaction of two circular coplanar cracks embedded in an infinite medium under uniform tension loading is also analysed. Finally, the stress intensity factor variation computed for a semi-circular inner surface crack in a pressurized cylinder is presented.     

粗网格划分下的箱梁三维实体有限元分析方法

针对现有箱梁分析方法普遍存在的计算精度与计算效率之间矛盾的问题,提出了粗网格划分下的箱梁三维实体有限元分析方法。在充分考虑箱梁受力变形特点的基础上,以修正的Hellinger-Reissner变分原理为基础,通过合理引入非协调位移插值项,构造出直角坐标系下的六面体八结点杂交应力单元8N21β和柱坐标系下的六面体八结点杂交应力单元8N21βc,分别用于粗网格划分下的直箱梁和曲线箱梁的三维实体有限元分析。数值算例表明:8N21β单元和8N21βc单元在粗网格划分下具有较高的计算精度,能有效提高箱梁三维实体有限元分析的计算效率。     

Finite-rotation shell elements via mixed formulation

Starting from a tensorial five-parametric finite-rotation shell theory a family of mixed finite elements is developed on the basis of a Reissner-Mindlin type functional. The family developed contains 4-node and 9-node quadrilateral shell elements. In each of them the displacement approximation is combined with various force variable interpolations in order to improve flexibility for numerical applications. The so-called difference vector occurring in the shell theory is expressed in terms of new rotational degrees of freedom which permit a unique determination of this variable in every deformed position. The corresponding constraints are then satisfied at the element level numerically. Due to the underlying theory the numerical models developed are able to predict the physical 2D force variables accurately. Their capability to deal with strongly nonlinear situations is demonstrated by several examples where numerical results due to Kirchhoff-Love type elements are also included for a systematical comparison.     

内参型非协调元合理位移场的研究

本文研究了内参型非协调元附加内部自由度的有效数目，结合平面四结点元讨论了有效附加内部非协调位移的合理形式。     

A Simple Locking-Alleviated 3D 8-Node Mixed-Collocation C0 Finite Element with Over-Integration,for Functionally-Graded and Laminated Thick-Section Plates and Shells,with & without Z-Pins

Following previous work of [Dong, El-Gizawy, Juhany, Atluri (2014)], a simple locking-alleviated 3D 8-node mixed-collocation C0 finite element (denoted as CEH8) is developed in this study, for the modeling of functionally-graded or laminated thick-section composite plates and shells, without using higher-order or layer-wise zig-zag plate and shell theories which are widely popularized in the current literature. The present C0 element independently assumes an 18-parameter linearly-varying Cartesian strain field. The independently assumed Cartesian strains are related to the Cartesian strains derived from mesh-based Cartesian displacement interpolations, by exactly enforcing 18 pre-defined constraints at 18 pre-selected collocation points. The constraints are rationally defined to capture the basic kinematics of the 3D 8-node C0 element, and to accurately model each basic deformation mode of tension, bending, shear, and torsion. A 2x2x2 Gauss quadrature is sufficient for evaluating the stiffness matrix of CEH8 C0 3D elements for homogeneous materials, but over-integration (with a higher-order Gauss Quadrature, a layer-wise Gauss Quadrature, or a simple Trapezoidal Rule in the thickness direction) is used for functionally-graded materials or thick-section laminated composite structures with an arbitrary number of laminae. Through several numerical examples, it is clearly shown that the present CEH8 3D C0 element can accurately capture the stress distribution of FG and thick laminated structures with an arbitrary number of laminae even when only one element is used in the thickness direction. In stark contrast to the higher-order or layer-wise zig-zag plate and shell theories, with assumptions for displacement or stress fields in the thickness direction, which may require complicated C1 finite element, the present C0 element can accurately compute the jumps in bending stresses at the interfaces of layers, while the out-of plane normal and shear stresses can be accurately recovered by exploring the equilibrium equations of 3D linear elasticity. By adding the contributing stiffness of z-pins into the stiffness matrix of CEH8, it is also demonstrated that the presently developed method can be used to study the effect of using z-pin reinforcements to reduce the inter-laminar stresses of composite structures, in a very simple and computationally-efficient manner.     

Stability of collapsed 8-node 2-D and 20-node 3-D isoparametric finite elements

The collapse of ordinary finite elements to generate the desired strain singularity at the crack tip for fracture mechanics applications can lead to unwanted additional singularities in that area. Although neither the quarter-point nor the half-point 8-node two-dimensional (2-D) and 20-node three-dimensional (3-D) elements exhibit this behaviour, the present article proves that arbitrarily small deviations from the quarter-point element can be constructed which do have additional singularities. Since the general behaviour of the half-point element is not affected by small modifications, this element is better suited to match complex body geometries.