Stress resultant geometrically non-linear shell theory with drilling rotations. Part III: Linearized kinematics

A consistent formulation of the geometrically linear shell theory with drilling rotations is obtained by the consistent linearization of the geometrically non-linear shell theory considered in Parts I and II of this work. It was also shown that the same formulation can be recovered by linearizing the governing variational principle for the three-dimensional geometrically non-linear continuum with independent rotation field. In the finite element implementation of the presented shell theory, relying on the modified method of incompatible modes, we were able to construct a four-node shell element which delivers a very high-level performance. In order to simplify finite element implementation, a shallow reference configuration is assumed over each shell finite element. This approach does not impair the element performance for the present four-node element. The results obtained herein match those obtained with the state-of-the-art implementations based on the classical shell theory, over the complete set of standard benchmark problems.     

Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity

We develop a variational principle based on recent advances in couple-stress theory and the introduction of an engineering mean curvature vector as energy conjugate to the couple stresses. This new variational formulation provides a base for developing a couple-stress finite element approach. By considering the total potential energy functional to be not only a function of displacement, but of an independent rotation as well, we avoid the necessity to maintain C¹ continuity in the finite element method that we develop here. The result is a mixed formulation, which uses Lagrange multipliers to constrain the rotation field to be compatible with the displacement field. Interestingly, this formulation has the noteworthy advantage that the Lagrange multipliers can be shown to be equal to the skew-symmetric part of the force-stress, which otherwise would be cumbersome to calculate. Creating a new consistent couple-stress finite element formulation from this variational principle is then a matter of discretizing the variational statement and using appropriate mixed isoparametric elements to represent the domain of interest. Finally, problems of a hole in a plate with finite dimensions, the planar deformation of a ring, and the transverse deflection of a cantilever are explored using this finite element formulation to show some of the interesting effects of couple stress. Where possible, results are compared to existing solutions to validate the formulation developed here.     

A highly efficient membrane finite element with drilling degrees of freedom

In this paper, the development of a new quadrilateral membrane finite element with drilling degrees of freedom is discussed. A variational principle employing an independent rotation field around the normal of a plane continuum element is derived. This potential is based on the Cosserat continuum theory where skew symmetric stress and strain tensors are introduced in connection with the rotation of a point. From this higher continuum theory a formulation that incorporates rotational degrees of freedom is extracted, while the stress tensor is symmetric in a weak form. The resulting potential is found to be similar to that obtained by the procedure of Hughes and Brezzi. However, Hughes and Brezzi derived their potential in terms of pure mathematical investigations of Reissner’s potential, while the present procedure is based on physical considerations. This framework can be enhanced in terms of assumed stress and strain interpolations, if the numerical model is based on a modified Hu-Washizu functional with symmetric and asymmetric terms. The resulting variational statement enables the development of a new finite element that is very efficient since all parts of the stiffness matrix can be obtained analytically even in terms of arbitrary element distortions. Without the addition of any internal degrees of freedom the element shows excellent performance in bending dominated problems for rectangular element configurations.     

Dual-mixed hp finite element methods using first-order stress functions and rotations

 A two-field dual-mixed variational formulation of three-dimensional elasticity in terms of the non-symmetric stress tensor and the skew-symmetric rotation tensor is considered in this paper. The translational equilibrium equations are satisfied a priori by introducing the tensor of first-order stress functions. It is pointed out that the use of six properly chosen first-order stress function components leads to a (three-dimensional) weak formulation which is analogous to the displacement-pressure formulation of elasticity and the velocity-pressure formulation of Stokes flow. Selection of stable mixed hp finite element spaces is based on this analogy. Basic issues of constructing curvilinear dual-mixed p finite elements with higher-order stress approximation and continuous surface tractions are discussed in the two-dimensional case where the number of independent variables reduces to three, namely two components of a first-order stress function vector and a scalar rotation. Numerical performance of three quadrilateral dual-mixed hp finite elements is presented and compared to displacement-based hp finite elements when the Poisson's ratio converges to the incompressible limit of 0.5. It is shown that the dual-mixed elements developed in this paper are free from locking in the energy norm as well as in the stress computations, both for h- and p-extensions. Received 22 October 1999     

Genuinely resultant shell finite elements accounting for geometric and material non-linearity

A general theoretical framework is presented for the fully non-linear analysis of shells by the finite element method. The governing equations are derived exclusively in terms of resulting quantities through a logical and straightforward descent from three-dimensional continuum mechanics without appealing to simplifying assumptions (hence the name genuinely resultant). As a result, the underlying theory is statically and geometrically exact, and it naturally includes small strain and finite strain problems of thin as well as thick shells. The underlying mathematical structure and the variational formulation of the theory are examined. This appears to be crucial for the development of computational procedures employing the Newton-Kantorovich linearization process and the Galerkin type discretization method. The treatment of finite rotations through an arbitrary parametrization of the rotation group and the interpolation procedure of SO(3)-valued functions underlying the construction of finite element basis are other issues studied in this paper. A numerical analysis is presented in order to assess the effectiveness of the proposed formulation. Small strain problems as well as finite strain deformation of rubber-like shells undergoing finite rotations are considered. Special attention is devoted to the assessment of the relevance of the drilling degree-of-freedom and highly non-uniform through-the-thickness deformation in the case of shells made of incompressible material.     

A consistent theory of finite stretches and finite rotations, in space-curved beams of arbitrary cross-section

 Attention is focused in this paper on the development of a consistent finite deformation beam theory, and its mixed variational formulation. The shearing deformation, as well as cross-sectional warping displacement, are taken into account in this formulation. Beginning with the equilibrium equations of 3-D continuum body, we obtain the linear momentum balance (LMB), angular momentum balance (AMB) and director momentum balance (DMB) conditions of the beam. The conjugate relationships between the strain and stress measures are obtained through the stress power, in which the AMB condition plays an important role. The use of the strain measures proposed herein, leads to the strain energy function which is invariant under a rigid-body motion. The present formulation is shown to be objective by using a numerical example. On the basis of Atluri's variational principle, we develop a mixed type variational functional for a space-curved beam, undergoing arbitrarily large rotations and arbitrarily large stretches. A choice of a proper finite rotation vector, and unsymmetric curvature strains, makes it possible for constructing a consistent variational principle. The use of the present functional always leads to a symmetric tangent stiffness. The mixed variational functional developed herein leads to a powerful tool for obtaining accurate numerical results of 3-D space-curved beams, undergoing arbitrarily large stretches and rotations. Received 22 November 2000     

On non-linear dynamics of shells: implementation of energy-momentum conserving algorithm for a finite rotation shell model

Continuum and numerical formulations for non-linear dynamics of thin shells are presented in this work. An elastodynamic shell model is developed from the three-dimensional continuum by employing standard assumptions of the first-order shear-deformation theories. Motion of the shell-director is described by a singularity-free formulation based on the rotation vector. Temporal discretization is performed by an implicit, one-step, second-order accurate, time-integration scheme. In this work, an energy and momentum conserving algorithm, which exactly preserves the fundamental constants of the shell motion and guaranties unconditional algorithmic stability, is used. It may be regarded as a modification of the standard mid-point rule. Spatial discretization is based on the four-noded isoparametric element. Particular attention is devoted to the consistent linearization of the weak form of the initial boundary value problem discretized in time and space, in order to achieve a quadratic rate of asymptotic convergence typical for the Newton–Raphson based solution procedures. An unconditionally stable time finite element formulation suitable for the long-term dynamic computations of flexible shell-like structures, which may be undergoing large displacements, large rotations and large motions is therefore obtained. A set of numerical examples is presented to illustrate the present approach and the performance of the isoparametric four-noded shell finite element in conjunction with the implicit energy and momentum conserving time-integration algorithm. © 1998 John Wiley & Sons, Ltd.     

A formulation for frictionless contact problems using a weak form introduced by Nitsche

In this paper a finite element formulation for frictionless contact problems with non-matching meshes in the contact interface is presented. It is based on a non-standard variational formulation due to Nitsche and leads to a matrix formulation in the primary variables. The method modifies the unconstrained functional by adding extra terms and a stabilization which is related to the classical penalty method. These new terms are characterized by the presence of contact forces that are computed from the stresses in the continuum elements. They can be seen as a sort of Lagrangian-type contributions. Due to the computation of the contact forces from the continuum elements, some additional degrees-of-freedom are involved in the stiffness matrix parts related to contact. These degrees-of-freedom are associated with nodes not belonging to the contact surfaces.     

A FINITE ELEMENT FORMULATION FOR 3-D CONTINUA USING THE CO-ROTATIONAL TECHNIQUE

The co-rotational technique is described for the three-dimensional analysis of continua. The technique exploits the proven technology of the best continua elements for linear analysis which are embedded into a formulation that applies an element-attached local co-ordinate frame that continuously moves and rotates with the element. The geometric non-linearity is then incorporated via the rotation of this local system. The method uses similar procedures to those recently described for 2-D continua elements but introduces concepts from a more conventional ‘continuum mechanics’ approach. The general framework for the co-rotational procedure is kept. However, a much neater formulation is derived, which readily allows the extension from two to three dimensions.     

Dynamic finite element formulation for Cosserat elastic plates

A displacement and rotation‐based dynamic finite element formulation for Cosserat plates is discussed in detail in this paper. Special attention is given to the validation of the element through adequate benchmarks and patch tests. The choice of the interpolation functions and the order of integration of the stiffness and the mass matrices are extensively argued. The possibility of local system deficiencies is explored, and a shear locking investigation specifically elaborated for Cosserat plates is carried out. It is shown how the present formulation has interesting computational properties as compared to a classical continuum‐based formulation and how it can provide suitable results despite the use of reduced integration. An example of application of the finite element is given, in which the natural frequencies of a masonry panel modelled by means of discrete elements are computed and compared with the finite elements solution. Copyright © 2014 John Wiley & Sons, Ltd.     

Finite deformation formulation of a shell element for problems of sheet metal forming

An elastic-plastic thin shell finite element suitable for problems of finite deformation in sheet metal forming is formulated. Hill's yield criterion for sheet materials of normal anisotropy is applied. A nonlinear shell theory in a form of an incremental variational principle and a quasi-conforming element technique are employed in the Lagrangian formulation. The shell element fulfills the inter-element C ¹ continuity condition in a variational sense and has a sufficient rank to allow finite stretching, rotation and bending of the shell element. The accuracy and efficiency of the finite element formulation are illustrated by numerical examples.     

Smart element method I. The Zienkiewicz–Zhu feedback

A new error control finite element formulation is developed and implemented based on the variational multiscale method, the inclusion theory in homogenization, and the Zienkiewicz–Zhu error estimator. By synthesizing variational multiscale method in computational mechanics, the equivalent eigenstrain principle in micromechanics, and the Zienkiewicz–Zhu error estimator in the finite element method (FEM), the new finite element formulation can automatically detect and subsequently homogenize its own discretization errors in a self‐adaptive and a self‐adjusting manner. It is the first finite element formulation that combines an optimal feedback mechanism and a precisely defined homogenization procedure to reduce its own discretization errors and hence to control numerical pollutions. The paper focuses on the following two issues: (1) how to combine a multiscale method with the existing finite element error estimate criterion through a feedback mechanism, and (2) convergence study. It has been shown that by combining the proposed variational multiscale homogenization method with the Zienkiewicz–Zhu error estimator a clear improvement can be made on the coarse scale computation. It is also shown that when the finite element mesh is refined, the solution obtained by the variational eigenstrain multiscale method will converge to the exact solution. Copyright © 2004 John Wiley & Sons, Ltd.     

Variational methods in irreversible thermoelasticity: theoretical developments and minimum principles for the discrete form

A variational approach for fully coupled dynamic irreversible thermoelasticity is developed for continua, which considers both the conservative and dissipative character in terms of mixed variables. By introducing a consistent variational scheme for the spatial and temporal discretization of the governing equations, a mixed continuum element is established under the Hamiltonian-Lagrangian formalism. The proposed method leads to the development of minimum principles in discrete form with the proper selection of state variables and temporal action sum operators. Consequently, this novel mixed variational formulation can provide the basis for a class of optimization-based methods for irreversible thermomechanics. Several applications are considered to demonstrate the robustness of the proposed variational approach, including transient dynamic response of thermoelastic media due to surface heating caused by ramp- and step-type heat fluxes, and a sequence of laser pulses.     

An integral method for extremely low frequency magnetic shielding

A novel approach for analyzing conducting shields of extremely low frequency magnetic fields in linear media is presented. It consists of an integral formulation based on the cell method, expressed in terms of network-like loop currents and magnetic vector potential line integrals on the shield surface. This formulation leads to a considerable reduction of field problem variables, thus limiting the amount of allocated memory and speeding-up the numerical procedure compared to other differential and integral techniques. Eddy currents are computed first, then the magnetic vector potential and the magnetic flux density distributions are evaluated by applying the superimposition principle. A detailed comparison between this method and a three-dimensional finite element method code demonstrates the accuracy of the results and the advantages of the method.     

Large displacement analysis of three-dimensional beam structures

An updated Lagrangian and a total Lagrangian formulation of a three-dimensional beam element are presented for large displacement and large rotation analysis. It is shown that the two formulations yield identical element stiffness matrices and nodal point force vectors, and that the updated Lagragian formulation is computationally more effective. This formulation has been implemented and the resulted of some sample analyses are given.     

A consistent shell theory for finite deformations

Summary For shells of finite deformations, a non-linear theory will be derived using the Kirchhoff-Love assumption. Its derivation is accomplished by a variational procedure ensuring a consistent formulation. Special attention is confined to the correct derivation of the dynamic boundary conditions which succeeds by introduction of a rotation vector connected with the rotational movement of the normal vector of the middle surface. The paper closes with the operator formulation of the theory which demonstrates the characteristic properties of the non-linear theories in a very general manner.With 1 Figure     

A HYBRID STRESS QUADRILATERAL SHELL ELEMENT WITH FULL ROTATIONAL D.O.F.S

A 4-node hybrid stress quadrilateral shell element with 3 rotational d.o.f.s per node is presented. The mid-surface displacement of the element is founded on Allman's rotation. The equal-rotation mode intrinsic to the rotation is suppressed by a stabilization vector. The assumed stress field and the stabilization scalar is chosen such that membrane locking can be avoided. Computational efficiency of the element is improved by employing orthogonal stress modes and admissible matrix formulation. Popular benchmark tests are attempted and the results are found to be satisfactory. © 1997 by John Wiley & Sons, Ltd.     

Finite element formulation of the Ogden material model with application to rubber-like shells

The present contribution proposes a variational procedure for the numerical implementation of the Ogden material model. For this purpose the strain energy density originally formulated in terms of the principal stretches is transformed as variational quantities into the invariants of the right Cauchy–Green tensor. This formulation holds for arbitrary three-dimensional deformations and requires neither solving eigenvalue problems nor co-ordinate system transformations. Particular attention is given to the consideration of special cases with coinciding eigenvalues. For the analysis of rubber-like shells this material model is then coupled with a six parametric shells kinematics able to deal with large strains and finite rotations. The incompressibility condition is considered in the strain energy, but it is additionally used as 2-D constraint for the elimination of the stretching parameter at the element level. A four node isoparametric finite element is developed by interpolating the transverse shear strains according to assumed strain concept. Finally, examples are given permitting to discuss the capability of the finite element model developed concerning various aspects. © 1998 John Wiley & Sons, Ltd.