Peter Betsch

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.

On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells

This paper concerns the computational treatment of large rotations with application to the finite element discretisation of smooth shells. Formulations based on various rotational parametrizations are reviewed and classified with respect to their update structure: category (II) is based on total rotational degrees of freedom leading to an additive update structure, whereas category (I) relies on linearized rotational degrees of freedom leading to a multiplicative update structure. Based on this classification, new formulations are developed. Among them are the Rodrigues formula applied within category (II) with additive update structure for the two components of the rotation vector and two successive elementary rotations (in the sense of Euler angles) applied within category (I) yielding a singularity-free formulation. The numerical examples confirm that every rotational parametrization when applied within category (II) leads to singularities, whereas the application within category (I) enables the calculation of overall rotations unrestricted in size without any singularity.

Constrained integration of rigid body dynamics

In the present paper rigid body dynamics is formulated as mechanical system with holonomic constraints. This approach offers the appealing possibility to deal with finite rotations without employing any specific rotational parameterization. The numerical discretization of the underlying system of differential algebraic equations is treated in detail. The proposed algorithm obeys major conservation laws of the underlying continuous system such as conservation of energy and angular momentum. In addition to that, the constraints on the configuration and momentum level are fulfilled exactly. Two numerical examples are dealt with to assess the performance of the constrained algorithm.

Conservation properties of a time FE method. Part I: time-stepping schemes forN-body problems

In the present paper one-step implicit integration algorithms for the N-body problem are developed. The time-stepping schemes are based on a Petrov–Galerkin finite element method applied to the Hamiltonian formulation of the N-body problem. The approach furnishes algorithmic energy conservation in a natural way. The proposed time finite element method facilitates a systematic implementation of a family of time-stepping schemes. A particular algorithm is specified by the associated quadrature rule employed for the evaluation of time integrals. The influence of various standard as well as non-standard quadrature formulas on algorithmic energy conservation and conservation of angular momentum is examined in detail for linear and quadratic time elements. Copyright

Numerical implementation of multiplicative elasto-plasticity into assumed strain elements with application to shells at large strains

Alternative formulations of isotropic large strain elasto-plasticity are presented which are especially well suited for the implementation into assumed strain elements. Based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts three distinct eigenvalue problems related to the reference, intermediate and current configuration are investigated. These eigenvalue problems are connected by similarity transformations which preserve the eigenvalues. They play an important role in the subsequent development of alternative constitutive formulations and the corresponding finite element implementation. The developed constitutive procedures rely on the right Cauchy–Green tensor, or equivalently on the Green–Lagrangian strain tensor, rather than the deformation gradient. Consequently, they can be applied directly to assumed strain elements. Specifically, we are concerned with efficient low order shell elements for which the assumed strain method has proven to be extremely powerful to overcome spurious locking effects.

A DAE Approach to Flexible Multibody Dynamics

The present work deals with the dynamics of multibody systems consisting ofrigid bodies and beams. Nonlinear finite element methods are used to devise a frame-indifferent spacediscretization of the underlying geometrically exact beam theory. Both rigid bodies and semi-discrete beams are viewed as finite-dimensional dynamical systems with holonomic constraints. The equations of motion pertaining to the constrained mechanical systems under considerationtake the form of Differential Algebraic Equations (DAEs).The DAEs are discretized directly by applying a Galerkin-based method.It is shown that the proposed DAE approach provides a unified framework for the integration of flexible multibody dynamics.

A nonlinear extensible 4-node shell element based on continuum theory and assumed strain interpolations

SummaryA quadrilateral continuum-basedC0 shell element is presented, which relies on extensible director kinematics and incorporates unmodified three-dimensional constitutive models. The shell element is developed from the nonlinear enhanced assumed strain (EAS) method advocated by Sino & Armero [1] and formulated in curvilinear coordinates. Here, the EAS-expansion of the material displacement gradient leads to the local interpretation of enhanced covariant base vectors that are superposed on the compatible covariant base vectors. Two expansions of the enhanced covariant base vectors are given: first an extension of the underlying single extensible shell kinematic and second an improvement of the membrane part of the bilinear element. Furthermore, two assumed strain modifications of the compatible covariant strains are introduced such that the element performs well even in the case of very thin shells.

Energy-Momentum Conserving Schemes for Frictionless Dynamic Contact Problems

Dynamic contact problems in elasticity are treated within a finite element framework by employing the well-established node-to-segment method. A new formulation of the algorithmic forces of contact is proposed which makes possible the design of energymomentum conserving integrators. The numerical example presented herein indicates that the present approach provides enhanced numerical stability.

Original article: Galerkin-based energy-momentum consistent time-stepping algorithms for classical nonlinear thermo-elastodynamics

This paper presents energy-momentum consistent time-stepping schemes for classical nonlinear thermo-elastodynamics, which include well-known energy-momentum conserving time integrators for elastodynamics. By using the time finite element approach, this time-stepping schemes are not restricted to second-order accuracy. In order to retain the first and second law of thermodynamics in a discrete setting, the equations of motion are temporally discretised by a Petrov-Galerkin method, and the entropy evolution equation by a new Bubnov-Galerkin method. The new aspect in this Bubnov-Galerkin method is the used jump term, which is necessary to avoid numerical dissipation beside the local physical dissipation according to Fouriers law. The stress tensor in the obtained enhanced hybrid Galerkin (ehG) method is approximated by a higher-order accurate discrete gradient. As additional new features of a monolithic solution strategy, this paper presents a convergence criterion and an initializer routine, which avoids scaling problems in the primary unknowns and leads to a more rapid convergence for large time steps, respectively. Representative numerical examples verify the excellent performance of the ehG time-stepping schemes in comparison to the trapezoidal rule, especially concerning rotor dynamics.

A localization capturing FE-interface based on regularized strong discontinuities at large inelastic strains

The objective of this work is the development of a finite element interface formulation tailored to capture localization within geometrically nonlinear solid mechanics. In this context, strong discontinuities are considered as the final failure mechanism within localization problems. The failure kinematics are governed by a jump in the nonlinear deformation map across a discontinuity surface. In the finite element discretization the interface element is endowed with these kinematics. As a consequence, the interface stiffness is dominated by the weighted spatial localization tensor. Based on these developments a localization capturing procedure is advocated and is computationally highlighted.