Implementation of assumed strain degenerated shell elements

The implementation of the new degenerated shell elements with assumed transverse shear and membrane strains is described in this paper. An explanation of locking behaviour and also the location of sampling points for the assumed strain fields is given in the present work. In the formulation of the new elements, assumed transverse shear strains in the natural coordinate system are used to overcome the shear locking problem and assumed membrane strains in the orthogonal curvilinear coordinate system are applied to avoid membrane locking behaviour. Both linear and nonlinear problems are considered.     

The discrete strain gap method and membrane locking

The discrete shear gap (DSG) method, initially proposed for the elimination of transverse shear locking in plate and shell finite elements, is extended to a more general concept, rephrasing ‘DSG’ as ‘discrete strain gap’. We focus on the application of the method to the problem of membrane locking in beam and shell finite elements in the present paper. It turns out that a straightforward extension of the original DSG method is suitable to avoid membrane locking for both triangles and quadrilaterals. Moreover, there are strong indications that the presented idea includes the potential for a general formulation of locking-free structural finite elements, equally well suited for beams, plates, shells and solids.     

Generation of an eight node brick element using Papcovitch-Neuber functions

Some conventional finite elements suffer from drawbacks, such as shear locking, membrane locking, etc. To overcome them researchers have developed various techniques, termed as tricks by some and variational crimes by others. Many attempts have been made, but satisfactory explanations for why some of these techniques work have not been obtained, especially in the case of solid elements. This paper attempts a simple non-conforming solid element using assumed displacement fields which satisfy the Navier equation exactly. Its behaviour under simple loadings like bending, torsion and tension is examined and comparisons are made with existing elements.     

基于假定应变法的协同转动四边形复合材料壳单元

为对复合材料层合板壳结构进行精确的大变形数值模拟,提出一种采用假定应变法的能分析层合结构大转动问题的协同转动四边形壳单元.该方法在建立有限元公式时引入假定应变法以克服膜闭锁和剪切闭锁的不利影响.与其他能分析大转动问题的复合材料壳单元相比,在新的协同转动框架中采用矢量型转动变量,可大大降低在非线性增量求解过程中更新转动变量的难度,且能得到对称的单元切线刚度矩阵,提高单元的计算效率.分析两个典型算例,并与其他学者的结果进行对比,结果表明在计算层合结构大转角问题时拥有较好的精度和收敛性.     

Postbuckling analysis of functionally graded plates subject to compressive and thermal loads

Postbuckling analysis of functionally graded ceramic–metal plates under edge compression and temperature field conditions is presented using the element-free kp-Ritz method. The first-order shear deformation plate theory is employed to account for the transverse shear strains, and the von Kármán-type nonlinear strain–displacement relationship is adopted. The effective material properties of the functionally graded plates are assumed to vary through their thickness direction according to the power-law distribution of the volume fractions of the constituents. The displacement fields are approximated in terms of a set of mesh-free kernel particle functions. Bending stiffness is estimated using a stabilised conforming nodal integration approach, and, to eliminate the membrane and shear locking effects for thin plates, the shear and membrane terms are evaluated using a direct nodal integration technique. The solutions are obtained using the arc–length iterative algorithm in combination with the modified Newton–Raphson method. The effects of the volume fraction exponent, boundary conditions and temperature distribution on postbuckling behaviour are examined.     

A simple isoparametric three-node shell finite element

A three-node isoparametric shell finite element including membrane and bending effects is proposed. The element is based on the degenerated solid approach and uses an assumed strain method to avoid shear locking. An intermediate convected covariant frame is used in order to construct the modified shear strain interpolation matrix. Validation tests show that shear locking is avoided and that a reduced integration procedure can be used without any loss of accuracy which is useful for the numerical efficiency.     

Improvements in the membrane behaviour of the three node rotation-free BST shell triangle using an assumed strain approach

In this paper an assumed strain approach is presented in order to improve the membrane behaviour of a thin shell triangular element. The so called Basic Shell Triangle (BST) has three nodes with only translational degrees of freedom and is based on a Total Lagrangian Formulation. As in the original BST element the curvatures are computed resorting to the surrounding elements (patch of four elements). Membrane strains are now also computed from the same patch of elements which leads to a non-conforming membrane behaviour. Despite this non-conformity the element passes the patch test. Large strain plasticity is considered using a logarithmic strain–stress pair. A plane stress behaviour with an additive decomposition of elastic and plastic strains is assumed. A hyperplastic law is considered for the elastic part while for the plastic part an anisotropic quadratic (Hill) yield function with non-linear isotropic hardening is adopted. The element, termed EBST, has been implemented in an explicit (hydro-)code adequate to simulate sheet-stamping processes and in an implicit static/dynamic code. Several examples are given showing the good performance of the enhanced rotation-free shell triangle.     

Geometrically exact assumed stress–strain multilayered solid-shell elements based on the 3D analytical integration

This paper presents a family of geometrically exact assumed stress–strain four-node curved solid-shell elements with six displacement degrees of freedom per node by using the first-order equivalent single-layer theory. The proposed finite element formulation is based on the new strain–displacement relationships written in general reference surface coordinates, which are objective, i.e., invariant under rigid-body motions. This is possible because displacement vectors of the bottom and top surfaces of the shell are introduced and resolved in the reference surface frame. To overcome shear and membrane locking and have no spurious zero energy modes, the assumed strain and stress resultant fields are invoked. In order to circumvent thickness locking, three types of the modified material stiffness matrix extracted from the literature are employed and compared. All three elemental stiffness matrices have six zero eigenvalues and require only direct substitutions. Besides, they are evaluated by applying the 3D analytical integration that is very economical and allows using extremely coarse meshes.     

A study of moderately thick quadrilateral plate elements based on the absolute nodal coordinate formulation

Finite element analysis using plate elements based on the absolute nodal coordinate formulation (ANCF) can predict the behaviors of moderately thick plates subject to large deformation. However, the formulation is subject to numerical locking, which compromises results. This study was designed to investigate and develop techniques to prevent or mitigate numerical locking phenomena. Three different ANCF plate element types were examined. The first is the original fully parameterized quadrilateral ANCF plate element. The second is an update to this element that linearly interpolates transverse shear strains to overcome slow convergence due to transverse shear locking. Finally, the third is based on a new higher order ANCF plate element that is being introduced here. The higher order plate element makes it possible to describe a higher than first-order transverse displacement field to prevent Poisson thickness locking. The term “higher order” is used, because some nodal coordinates of the new plate element are defined by higher order derivatives. The performance of each plate element type was tested by (1) solving a comprehensive set of small deformation static problems, (2) carrying out eigenfrequency analyses, and (3) analyzing a typical dynamic scenario. The numerical calculations were made using MATLAB. The results of the static and eigenfrequency analyses were benchmarked to reference solutions provided by the commercially available finite element software ANSYS. The results show that shear locking is strongly dependent on material thickness. Poisson thickness locking is independent of thickness, but strongly depends on the Poisson effect. Poisson thickness locking becomes a problem for both of the fully parameterized element types implemented with full 3-D elasticity. Their converged results differ by about 18 % from the ANSYS results. Corresponding results for the new higher order ANCF plate element agree with the benchmark. ANCF plate elements can describe the trapezoidal mode; therefore, they do not suffer from Poisson locking, a reported problem for fully parameterized ANCF beam elements. For cases with shear deformation loading, shear locking slows solution convergence for models based on either the original fully parameterized plate element or the newly introduced higher order plate element.     

Superelastic shell structures modelling: Part I: Element formulation

A C⁰ three-node shell finite element belonging to the assumed shear strain elements family is extended to account for large strains when a rotating frame formulation is adopted to describe the material behaviour. Within an incremental method associated with the Newton iterative scheme, a strain measure is defined and interpolated in an intermediate configuration assuming a linear interpolation of the incremental geometric transformation. This strain measure allows the definition, in a rotated configuration, of a constitutive incremental strain obtained from a material cumulated tensorial strain. The proposed approach is validated herein considering several elastic finite strains examples.     

Sheet metal forming and springback simulation by means of a new reduced integration solid-shell finite element technology

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the field of sheet metal forming. Working with one integration point in the shell plane and an arbitrary number of integration points in thickness direction, highly non-linear stress states over the sheet thickness can be incorporated in an efficient way. In order to avoid volumetric locking and Poisson thickness locking at the level of integration points the enhanced assumed strain (EAS) concept with only one EAS degree-of-freedom is implemented. A key point of the formulation is the construction of the hourglass stabilization by means of different Taylor expansions. This leads to the advantage that the sensitivity with respect to mesh distortion is noticeably reduced. The hourglass stabilization includes the assumed natural strain (ANS) concept and a kind of B-Bar method. So transverse shear locking and volumetric locking are eliminated.The finite element formulation incorporates a finite strain material model for plastic anisotropy as well as non-linear (Armstrong–Frederick type) kinematic and isotropic hardening. In this context the plastic anisotropy can be modeled by representing the yield surface and the plastic flow rule as functions of so-called structural tensors. The integration of the evolution equations is performed by means of an exponential map exploiting the spectral decomposition. The element formulation and material model have been implemented into the commercial code ABAQUS/Standard by means of the UEL interface for user-defined elements. Using an implicit time integration scheme numerical results for classical deep drawing simulations as well as springback predictions are presented in comparison to experimental measurements.     

Monomial test: Testing the flexural behavior of the degenerated shell element

Shear locking of the degenerated shell element is a result of the development of spurious shear strain by the element, when subjected to high order Kirchhoff displacement fields. The shear locking phenomenon is analyzed in this paper using an analytical test. The effects of the integration scheme, order of the element, order of the modeled Kirchhoff field and the element distortion are analyzed explicitly and quantitatively for the four-node, eight-node and nine-node degenerated shell elements.     

A woven reinforcement forming simulation method. Influence of the shear stiffness

A simulation method is proposed for forming processes of fabrics used as reinforcements of composite materials. It uses specific finite elements made of woven material. The nodal interior loads are deduced from yarn tensile strain energy and woven cell shear energy. A picture frame shear test is presented. Optical measures permit to analyse the strains in yarns at microlevel for the different stages of the shear test. The influence of shearing in the formulation is studied on two forming simulations. It is shown that taking shear into account permits the appearance and the development of wrinkles when the locking angle is overcome.     

A robust non-linear solid shell element based on a mixed variational formulation

The paper is concerned with a geometrically non-linear solid shell finite element formulation, which is based on the Hu-Washizu variational principle. For the approximation of the independent displacement, stress and strain fields, the strain field is additively decomposed into two parts. Due to the fact that one part of the strain field is interpolated in the same manner as proposed by the enhanced assumed strain (EAS) method, it is denoted as EAS field. The other strain field is approximated with the same interpolation functions as the stress field. In contrast to the EAS concept the approximation spaces of the stresses and the enhanced assumed strains are not orthogonal. Consequently the stress field is not eliminated from the finite element equations. For the displacements tri-linear shape functions are considered. Shear locking and curvature thickness locking are treated using assumed natural strain interpolations. A static condensation leads to a simple low order hexahedral solid shell element. Numerical tests show that the present model is very robust and allows larger load steps than an EAS solid shell element.     

MITC elements for a classical shell model

The formulation of nine-node mixed-interpolated shell elements based on a classical mathematical shell theory is presented, taking into account some fundamental considerations for the finite element analysis of shells. The elements are based on the mixed interpolation of tensorial components approach (MITC), but the assumed covariant strain fields are applied only for the membrane and shear components. Two different types of elements are considered, depending on whether or not geometric approximations are included in the formulation. The performance of the proposed elements is illustrated with a well-established test problem––the Scordelis-Lo roof.     

Shear locking reduction in eight-noded tri-linear solid finite elements

A new quadrature rule for eight-noded solid finite elements is suggested. The formulation reduces those locking problems associated with the classical full and selectively-reduced Gauss integration schemes and it is based on an element local coordinate system where the shear strain components are averaged in groups of four integration points.This work is mainly interesting in contexts of plasticity and explicit time integration, where low order elements often are preferable.     

An edge-based smoothed finite element method (ES-FEM) with stabilized discrete shear gap technique for analysis of Reissner–Mindlin plates

An edge-based smoothed finite element method (ES-FEM) for static, free vibration and buckling analyses of Reissner–Mindlin plates using 3-node triangular elements is studied in this paper. The calculation of the system stiffness matrix is performed by using the strain smoothing technique over the smoothing domains associated with edges of elements. In order to avoid the transverse shear locking and to improve the accuracy of the present formulation, the ES-FEM is incorporated with the discrete shear gap (DSG) method together with a stabilization technique to give a so-called edge-based smoothed stabilized discrete shear gap method (ES-DSG). The numerical examples demonstrated that the present ES-DSG method is free of shear locking and achieves the high accuracy compared to the exact solutions and others existing elements in the literature.     

A generalised technique for fracture analysis of cracked plates under combined tensile,bending and shear loads

The objective of this paper is to propose a generalized technique called numerically integrated modified virtual crack closure integral (NI-MVCCI) technique for fracture analysis of cracked plates under combined tensile, bending and shear loads. NI-MVCCI technique is used for post-processing the results of finite element analysis (FEA) for computation of strain energy release rate (SERR) components and the corresponding stress intensity factor (SIF) for cracked plates. NI-MVCCI technique has been demonstrated for 4-noded, 8-noded (regular and quarter-point) and 9-noded (regular and quarter-point) isoparametric plate finite elements. These elements are based on Mindlin’s plate theory that considers shear deformation. For all the elements, reduced integration/selective reduced integration techniques have been employed in the studies. In addition, for 9-noded element assumed shear interpolation functions have been used to overcome the shear locking problem. Numerical studies on fracture analysis of plates subjected to tension–moment and tension–shear loads have been conducted employing these elements. It is observed that among these elements, the 9-noded Lagrangian plate element with assumed shear interpolation functions exhibits better performance for fracture analysis of cracked plates.     

Degenerated plate-shell elements with refined transverse shear strains

Although traditional three-dimensional plate-shell elements relax the constraint so that normal cross-sections remain normal to the neutral plane during transverse shear deformation, the section is still constrained to remain plane. The work reported here relaxes this constraint by introducing shape functions across the thickness to approximate the transverse shear strain field in the thickness direction. These shape functions are treated in the manner of generalized angles undergoing small deformations. They are added as new degrees of freedom to the ordinary displacement field of the degenerated shell elements. The field is still able to simulate large deformation behavior of the element. Each shape function yields two independent variables, one in each direction. In this work, two types of shape functions are proposed allowing a parabolic transverse shear strain, as well as an unsymmetric transverse shear strain distribution in the thickness direction. These two modes of deformation are particularly important in the case of diffused material failure in shell structures. Displacement field representation and finite element formulation based on a total Lagrangian approach are given. Examples are presented demonstrating the applicability of this element in a variety of problems.     

Simple and effective elements based upon Timoshenko–Mindlin shell theory

The simple and effective mixed models are developed for the analysis of multilayered anisotropic Timoshenko–Mindlin-type shells. The effects of the transverse shear and transverse normal strains, and laminated anisotropic material response are included. The precise representation of rigid body motions in the displacement patterns of curved shell elements is considered. This consideration requires the development of the strain–displacement equations of the Timoshenko–Mindlin-type theory with regard to their consistency with the rigid body motions. The fundamental unknowns consist of six displacements and eleven strains of the face surfaces of the shell, and 11 stress resultants. The element characteristic arrays are obtained by using the Hu–Washizu mixed variational principle. Numerical results are presented to demonstrate the high accuracy and effectiveness of the developed mixed models and to compare their performance with other finite-element models reported in the literature.