An assumed covariant strain based 9-node shell element

A new shell element, which is free from serious locking problems and which does not possess hourglass modes, is proposed. Solutions obtained with this element exhibit good convergence and are satisfactorily insensitive to mesh distortion. The element also exhibits good convergence for plates with variable thickness. This element is based on the use of assumed covariant strains which are obtained from the covariant strain field defined with respect to the element natural co-ordinate system. Only the linear version for thin shell cases is considered. The element performance is tested by application to several standard plate and shell problems. A problem involving variable thickness is also presented.     

A simple unsymmetric 4-node 12-DOF membrane element for the modified couple stress theory

In this work, the recently proposed unsymmetric 4-node 12-DOF (degree-of-freedom) membrane element (Shang and Ouyang, Int J Numer Methods Eng 113(10): 1589-1606, 2018), which has demonstrated excellent performance for the classical elastic problems, is further extended for the modified couple stress theory, to account for the size effect of materials. This is achieved via two formulation developments. Firstly, by using the penalty function method, the kinematic relations between the element's nodal drilling DOFs and the true physical rotations are enforced. Consequently, the continuity requirement for the modified couple stress theory is satisfied in weak sense, and the symmetric curvature test function can be easily derived from the gradients of the drilling DOFs. Secondly, the couple stress field that satisfies a priori the related equilibrium equations is adopted as the energy conjugate trial function to formulate the element for the modified couple stress theory. As demonstrated by a series of benchmark tests, the new element can efficiently capture the size-dependent responses of materials and is robust to mesh distortions. Moreover, as the new element uses only three conventional DOFs per node, it can be readily incorporated into the standard finite element program framework and commonly available finite element programs.     

An assumed stress hybrid curvilinear triangular finite element for plate bending

An assumed stress hybrid curvilinear triangular finite element is described which is based upon the Kirchhoff theory of plate bending. The derivation extends the assumed stress hybrid technique to curvilinear boundaries where the twelve connectors are related to those of an equilibrium rectilinear element and to Semiloof. The solution process demands only first derivatives of the shape functions. The element is subjected to various patch tests for constant bending, e.g. where the central element is in close approximation to a circle. All tests are passed for stress couples and vertex displacements, but values of the remaining connectors do not resemble exact results. Patch tests for rigid-body movements are passed exactly in every respect.     

An assumed-stress hybrid 4-node shell element with drilling degrees of freedom

An assumed-stress hybrid/mixed 4-node quadrilateral shell element is introduced that alleviates most of the deficiencies associated with such elements. The formulation of the element is based on the assumed-stress hybrid/mixed method using the Hellinger-Reissner variational principle. The membrane part of the element has 12 degrees of freedom including rotational or ‘drilling’ degrees of freedom at the nodes. The bending part of the element also has 12 degrees of freedom. The bending part of the element uses the Reissner-Mindlin plate theory which takes into account the transverse shear contributions. The element formulation is derived from an 8-node isoparametric element by expressing the midside displacement degrees of freedom in terms of displacement and rotational degrees of freedom at corner nodes. The element passes the patch test, is nearly insensitive to mesh distortion, does not ‘lock’, possesses the desirable invariance properties, has no hidden spurious modes, and for the majority of test cases used in this paper produces more accurate results than the other elements employed herein for comparison.     

A formulation for the 4-node quadrilateral element

A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.     

A co-rotational 8-node assumed strain shell element for postbuckling analysis of laminated composite plates and shells

 The formulation of a nonlinear composite shell element is presented for the solution of stability problems of composite plates and shells. The formulation of the geometrical stiffness presented here is exactly defined on the midsurface and is efficient for analyzing stability problems of thin and thick laminated plates and shells by incorporating bending moment and transverse shear resultant forces. The composite element is free of both membrane and shear locking behaviour by using the assumed natural strain method such that the element performs very well as thin shells. The transverse shear stiffness is defined by an equilibrium approach instead of using the shear correction factor. The proposed formulation is computationally efficient and the test results showed good agreement. In addition the effect of the viscoelastic material is investigated on the postbuckling behaviour of laminated composite shells. Received: 6 February 2002 / Accecpted: 6 January 2003 ID=" Present address: School of Civil Engineering, Asian Institute of Technology     

An assumed displacement hybrid finite element model for linear fracture mechanics

This paper deals with a procedure to calculate the elastic stress intensity factors for arbitrary-shaped cracks in plane stress and plane strain problems. An assumed displacement hybrid finite element model is employed wherein the unknowns in the final algebraic system of equations are the nodal displacements and the elastic stress intensity factors. Special elements, which contain proper singular displacement and stress fields, are used in a fixed region near the crack tip; and the interelement displacement compatibility is satisfied through the use of a Lagrangean multiplier technique. Numerical examples presented include: central as well as edge cracks in tension plates and a quarter-circular crack in a tension plate. Excellent correlations were obtained with available solutions in all the cases. A discussion on the convergence of the present solution is also included.     

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     

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.     

Two hybrid stress membrane finite element families with drilling rotations

Two new assumed stress membrane finite element families with two translational and one rotational degree of freedom per node are presented. The families, denoted by 8β(M) and 8β(D), are rank sufficient and invariant, and are derived using a unified Hu–Washizu like variational formulation. A recent stress mode classification method is used to illustrate the construction of the stress interpolation matrices. In each family, a number of new formulations are derived. Numerical results are presented. Copyright © 2001 John Wiley & Sons, Ltd.     

A different view of the assumed stress hybrid method

A somewhat unconventional, yet simple, derivation of the assumed stress hybrid formulation is presented in which its relationship with displacement and stress-based finite element models is demonstrated. This gives insight into the hybrid method and shows that it represents a least energy fit of an equilibrium stress field to a displacement model strain field. By this means the attributes of both fields may be exploited in extending the analysis scope of hybrid models.     

Classification of stress modes in assumed stress fields of hybrid finite elements

A classification method is presented to classify stress modes in assumed stress fields of hybrid finite element based on the eigenvalue examination and the concept of natural deformation modes. It is assumed that there only exist m (=n−r) natural deformation modes in a hybrid finite element which has n degrees of freedom and r rigid-body modes. For a hybrid element, stress modes in various assumed stress fields proposed by different researchers can be classified into m stress mode groups corresponding to m natural deformation modes and a zero-energy stress mode group corresponding to rigid-body modes by the m natural deformation modes. It is proved that if the flexibility matrix [H] is a diagonal matrix, the classification of stress modes is unique. Each stress mode group, except the zero-energy stress mode group, contains many stress modes that are interchangeable in an assumed stress field and do not cause any kinematic deformation modes in the element. A necessary and sufficient condition for avoiding kinematic deformation modes in a hybrid element is also presented. By means of the m classified stress mode groups and the necessary and sufficient condition, assumed stress fields with the minimum number of stress modes can be constructed and the resulting elements are free from kinematic deformation modes. Moreover, an assumed stress field can be constructed according to the problem to be solved. As examples, 2-D, 4-node plane element and 3-D, 8-node solid element are discussed. © 1997 John Wiley & Sons, Ltd.     

Free and forced vibration analysis of laminated composite plates and shells using a 9-node assumed strain shell element

The natural frequencies of isotropic and composite laminates are presented. The forced vibration analysis of laminated composite plates and shells subjected to arbitrary loading is investigated. In order to overcome membrane and shear locking phenomena, the assumed natural strain method is used. To develop a laminated shell element for free and forced vibration analysis, the equivalent constitutive equation that makes the computation of composite structures efficient was applied. The Mindlin-Reissner theory which allows the shear deformation and rotary inertia effect to be considered is adopted for development of nine-node assumed strain shell element. The present shell element offers significant advantages since it consistently uses the natural co-ordinate system. Results of the present theory show good agreement with the 3-D elasticity and analytical solutions. In addition the effect of damping is investigated on the forced vibration analysis of laminated composite plates and shells.     

On the incompressible constraint of the 4-node quadrilateral element

The standard plane 4-node element is written as the summation of a constant gradient matrix, usually obtained from underintegration, and a stabilization matrix. The split is based on a Taylor series expansion of element basis functions. In the incompressible limit, the ‘locking’-effect of the quadrilateral is traced back to the stabilization matrix which reflects the incomplete higher-order term in the Taylor series. The incompressibility condition is formulated in a weak sense so that the element displacement field is divergence-free when integrated over the element volume. The resulting algebraic constraint is shown to coincide with a particular eigenvector of the constant gradient matrix which is obtained from the first-order terms of the Taylor series. The corresponding eigenvalue enforces incompressibility implicitly by means of a penalty-constraint. Analytical expressions for that constant-dilatation eigenpair are derived for arbitrary element geometries. It is shown how the incompressible constraint carries over to the element stiffness matrix if the element stabilization is performed in a particular manner. For several classical and recent elements, the eigensystems are analysed numerically. It is shown that most of the formulations reflect the incompressible constraint identically. In the incompressible limit, the numerical accuracies of the elements are compared.     

Stabilization by non-conforming modes: 9-node membrane element with drilling freedom

A simple technique of stabilizing spurious zero energy modes by non-conforming modes is presented. The non-conforming modes are derived by removing one or a number of nodes from the original element, and hence the associated stiffnesses are used to stabilize a uniform reduced integrated 9-node element. Three models were pursued to control the communicable spurious modes. The proposed model has been on the basis of including drilling freedoms and retaining much of the excellent performance of the original 9-node element with uniform reduced integration. The drilling freedoms are incorporated by constraining the true continuum mechanics definition of rotation at the reduced integration points by the penalty function method. A patch test was applied and a considerable number of problems were tested to investigate the performance of the new element.     

A polygonal finite element for plate bending problems using the assumed stress approach

The assumed stress distribution approach is used to derive the stiffness matrix of a plate-bending element of general polygonal shape having any number of nodes. The effect of assuming various numbers of unknown coefficients in the stress distributions is examined and the convergence properties of the resulting elements compared with others derived form assumed displacements.     

Development of a 4-node finite element for the computation of nano-structured materials

The molecular structure of a material determines its mechanical, thermal and chemical properties. Thus, to better understand characteristic mechanical properties like damping behavior or softening, in principle, one just has to model the interactions of a sufficiently large number of atoms. Various force field approaches have been proposed for that purpose, which are based on molecular-dynamic simulations, or rather quantum-mechanical ab initio calculations. They provide the potential energy of a structure in dependence of sort and number of chemical and physical bonds. In general, the different energy forms can be represented by nonlinear normal, bending and torsional springs which suggests the use of a finite element code. However, standard finite elements like truss, beam or shell elements are not very applicable because of the interaction of many atoms and, considered from a mechanical perspective, the absence of rotational degrees of freedom. For example, a bending of beam elements would lead to unrealistic constraints of neighboring molecular groups. In order to overcome this disadvantage, a new 4-node finite element is introduced, which uses only translational degrees of freedom and therefore is capable of representing the different energy forms exactly.     

8-node hexahedral unsymmetric element with rotation degrees of freedom for modified couple stress elasticity

A new C⁰ 8-node 48-DOF hexahedral element is developed for analysis of size-dependent problems in the context of the modified couple stress theory by extending the methodology proposed in our recent work (Shang et al., Int J Numer Methods Eng 119(9): 807-825, 2019) to the three-dimensional (3D) cases. There are two major innovations in the present formulation. First, the independent nodal rotation degrees of freedom (DOFs) are employed to enhance the standard 3D isoparametric interpolation for obtaining the displacement and strain test functions, as well as to approximatively design the physical rotation field for deriving the curvature test function. Second, the equilibrium stress functions instead of the analytical functions are used to formulate the stress trial function whilst the couple stress trial function is directly obtained from the curvature test function by using the constitutive relationship. Besides, the penalty function is introduced into the virtual work principle for enforcing the C¹ continuity condition in weak sense. Several benchmark examples are examined and the numerical results demonstrate that the element can simulate the size-dependent mechanical behaviors well, exhibiting satisfactory accuracy and low susceptibility to mesh distortion.     

24-DOF quadrilateral hybrid stress element for couple stress theory

A 24-DOF quadrilateral hybrid stress element for couple stress theory is proposed in this study. In order to satisfy the equilibrium equation in the domain of the element, the $21\beta $ Airy stress functions are chosen a assumed stress interpolation functions, and beam functions are adopted as the displacement interpolation functions on the boundary. This element can satisfy weak C $^{0}$ continuity with second-order accuracy and weak $\hbox {C}^1$ continuity simultaneously. So the element can pass the enhanced patch test of a convergence condition. Moreover, the reduced integration and a stresses smooth technique are introduced to improve the element accuracy. Numerical examples presented show that the proposed model can pass the $\hbox {C}^{0-1}$ enhanced patch test and indeed possesses higher accuracy. Besides, it does not exhibit extra zero energy modes and can capture the scale effects of microstructure.     

A class of equivalent enhanced assumed strain and hybrid stress finite elements

The interrelation of a certain class of finite elements based either on the Enhanced Assumed Strain (EAS) concept or the Hybrid Stress (HS) method is addressed. It is shown that both concepts lead to identical elements, if the material law is strongly satisfied in the Hu-Washizu principle. Conditions for the spaces of admissible functions are derived and the equivalence of the resulting weak formulations is proved. A `recipe' for the selection of trial functions of corresponding elements is given, and a class of equivalent EAS- and HS-elements is presented.