Friedrich Gruttmann

A 4-node finite shell element for the implementation of general hyperelastic 3D-elasticity at finite strains

Abstract In this paper a finite shell element for large deformations is presented based on extensible director kinematics. The essential feature is an interface to arbitrary three-dimensional material laws. The non-linear Lagrangian formulation is based on the three-field variational principle, parametrized with the displacement vector, enhanced Green-Lagrangian strain tensor and second Piola Kirchhoff stress tensor. The developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches. Furthermore, plane stress response is approximately recovered in the asymptotic case of vanishing thickness. A number of example problems investigating large deformation as well as finite strain applications are presented. Compressible and incompressible hyperelastic materials of the St. Venant-Kirchhoff, Neo-Hookean and Mooney-Rivlin type are particularly used.

A continuum based three-dimensional shell element for laminated structures

Abstract In this paper a continuum based three-dimensional shell element for the nonlinear analysis of laminated shell structures is derived. The basis of the present finite element formulation is the standard eight-node brick element with tri-linear shape functions. Especially for thin structures under certain loading cases, the displacement based element is too stiff and tends to lock. Therefore we use assumed natural strain and enhanced assumed strain methods to improve the relatively poor element behaviour. The anisotropic material behaviour of layered shells is modeled using a linear elastic orthotropic material law in each layer. Linear and nonlinear examples show the applicability and effectivity of the element formulation.

THEORY AND NUMERICS OF THREE-DIMENSIONAL BEAMS WITH ELASTOPLASTIC MATERIAL BEHAVIOUR ∗

A theory of space curved beams with arbitrary cross-sections and an associated finite element formulation is presented. Within the present beam theory the reference point, the centroid, the centre of shear and the loading point are arbitrary points of the cross-section. The beam strains are based on a kinematic assumption where torsion-warping deformation is included. Each node of the derived finite element possesses seven degrees of freedom. The update of the rotational parameters at the finite element nodes is achieved in an additive way. Applying the isoparametric concept the kinematic quantities are approximated using Lagrangian interpolation functions. Since the reference curve lies arbitrarily with respect to the centroid the developed element can be used to discretize eccentric stiffener of shells. Due to the implemented constitutive equations for elastoplastic material behaviour the element can be used to evaluate the load-carrying capacity of beam structures. Copyright

Theory and numerics of thin elastic shells with finite rotations

SummaryA bending theory for thin shells undergoing finite rotations is presented, and its associated finite element model is described. The kinematic assumption is based on a Reissner-Mindlin theory. The starting point for the derivation of the strain measures is the polar decomposition of the material deformation gradient. The work-conjugate stress resultants and stress couples are integrals of the Biot stress tensor. This tensor is invariant with respect to rigid body motions and therefore appropriate for the formulation of constitutive equations. The rotations are described by using Eulerian angles. The finite element discretization of arbitrary shells is performed using isoparametric elements. The advantage of the proposed shell formulation and its numerical model is shown by application to different non-linear plate and shell problems. Finite rotations can be calculated within one load increment. Thus the step size of the load increment is only imited by the local convergence behaviour of Newtons method or the appearance of stability phenomena.ÜbersichtEs wird eine Biegetheorie elastischer Schalen mit endlichen Drehungen dargestellt und die numerische Behandlung mit der Methode der finiten Elemente beschrieben. Dazu müssen die Verzerrungen und die schwache Form des Gleichgewichts angegeben werden. Es wird eine Reissner-Mindlin-Kinematik zugrunde gelegt. Die Verwendung des Greenschen Verzerrungstensors führt im Schalenraum bei endlichen Drehungen auf sehr komplizierte Ausdrücke. Daher wird von der polaren Zerlegung des materiallen Deformationsgradienten ausgegangen. In diesem Fall sind die Kräfte und Momente Spannungsresultierende des Biot-Spannungstensors, der invariant gegenüber Starrkörperbewegungen und somit für ein Stoffgesetz geeignet ist. Als Drehkinematen werden Eulerwinkel verwandt. Um allgemeine Schalengeometrien beschreiben zu können, wird eine isoparametrische Elementformulierung gewählt. Endliche Drehungen können in den berechneten Platten- und Schalenproblemen in einem Lastschritt aufgebracht werden. Die Lastschrittweite wird nur durch das Lösungsverfahren und das Auftreten von Stabilitätspunkten beschränkt.

Shear stresses in prismatic beams with arbitrary cross-sections

In this paper the approximate computation of shear stresses in prismatic beams due to Saint–Venant torsion and bending using the finite element method is investigated. The shape of the considered cross-sections may be arbitrary. Furthermore, the basic co-ordinate system lies arbitrarily to the centroid, and not necessarily in principal directions. For numerical reasons Dirichlet boundary conditions of the flexure problem are transformed into Neumann boundary conditions introducing a conjugate stress function. Based on the weak formulation of the boundary value problem isoparametric finite elements are formulated. The developed procedure yields the relevant warping and torsion constants. Copyright

A Geometrical Nonlinear Eccentric 3D-Beam Element with Arbitrary Cross-Sections

Abstract In this paper a finite element formulation of eccentric space curved beams with arbitrary cross-sections is derived. Based on a Timoshenko beam kinematic, the strain measures are derived by exploitation of the Green-Lagrangean strain tensor. Thus, the formulation is conformed with existing nonlinear shell theories. Finite rotations are described by orthogonal transformations of the basis systems from the initial to the current configuration. Since for arbitrary cross-sections the centroid and shear center do not coincide, torsion bending coupling occurs in the linear as well as in the finite deformation case. The linearization of the boundary value formulation leads to a symmetric bilinear form for conservative loads. The resulting finite element model is characterized by 6 degrees of freedom at the nodes and therefore is fully compatible with existing shell elements. Since the reference curve lies arbitrarily to the line of centroids, the element can be used to model eccentric stiffener of shells with arbitrary cross-sections.

Delamination growth analysis in laminated structures with continuum based 3d-shell elements and a viscoplastic softening model

In this paper we consider the simulation of delaminations in composite structures. For this purpose we discuss two aspects of a numerical treatment. The first one is the formulation of an accurate 3D-shell element to describe the global as well as the local behaviour in laminates in a proper way. Here a refined eight-node brick element is presented. The modifications of the element are based on assumed natural strain – and enhanced assumed strain methods. This element is used in the second part of the paper to describe delamination within an interface element with a small but non-vanishing thickness. Thus, strains, stresses and the delamination criterion are calculated in a standard manner from the displacement field and the material law. We introduce – based on an extension of the delamination criterion of Hashin (J. Appl. Mech. 47 (1980) 329–334) – an inelastic material model with softening. Here, the critical energy-release rate Gc is the crucial parameter to describe the damage behaviour. Furthermore, a viscoplastic regularization with strain rates according to an approach of Duvaut and Lions is used to prevent negative stiffness parameters in the consistent tangent operators. Numerical calculations show the successful application of the 3D-shell element and the delamination concept.

A simple finite rotation formulation for composite shell elements

In this paper we derive a simple finite element formulation for geometrical nonlinear shell structures. The formulation bases on a direct introduction of the isoparametric finite element formulation into the shell equations. The element allows the occurrence of finite rotations which are described by two independent angles. A layerwise linear elastic material model for composites has been chosen. A consistent linearization of all equations has been derived for the application of a pure Newton method in the nonlinear solution process. Thus a quadratic convergence behaviour can be achieved in the vicinity of the solution point. Examples show the applicability and effectivity of the developed element.

Finite element analysis of Saint–Venant torsion problem with exact integration of the elastic–plastic constitutive equations

Abstract In this paper torsion of prismatic bars considering elastic–plastic material behaviour is studied. Based on the presented variational formulation associated isoparametric finite elements are developed. The unknown warping function is approximated using an isoparametric concept. The elastic–plastic stresses are obtained by an exact integration of the rate equations. Thus the ultimate torque can be calculated in one single load step. This quantity describes the plastic reserve of a bar subjected to torsion. Furthermore, for linear isotropic hardening no local iterations are necessary to compute the stresses at the integration points. The numerical results are in very good agreement with available analytical solutions for simple geometric shapes. The arbitrary shaped domains may be simply or multiple connected.

Elastic and plastic analysis of thin-walled structures using improved hexahedral elements

In this paper a continuum based 3D–shell element for the nonlinear analysis of thin-walled structures is developed. Assumed natural strain method and enhanced assumed strain method are used to improve the relative poor element behaviour of a standard hexahedral displacement element. Different elastic and inelastic constitutive laws are considered. The anisotropic material behaviour of layered shells is modeled using a hyperelastic orthotropic material law in each layer. Furthermore, finite multiplicative J2-plasticity is discussed. The models are characterized by an interface to the material setting of the boundary value problem. Several examples show the applicability and efficiency of the developed element formulation.