Ekkehard Ramm

Strategies for Tracing the Nonlinear Response Near Limit Points

For the prebuckling range an extensive literature of effective solution techniques exists for the numerical solution of structural problems but only a few algorithms have been proposed to trace nonlinear response from the pre-limit into the post-limit range. Among these are the simple method of suppressing equilibrium iterations, the introduction of artificial springs, the displacement control method and the “constant-arc-length method” of Riks/Wempner. It is the purpose of this paper to review these methods and to discuss the modifications to a program that are necessary for their implementation. Selected numerical examples show that a modified Riks/Wempner method can be especially recommended.

Shear deformable shell elements for large strains and rotations

Well-known finite element concepts like the Assumed Natural Strain (ANS) and the Enhanced Assumed Strain (EAS) techniques are combined to derive efficient and reliable finite elements for continuum based shell formulations. In the present study two aspects are covered: The first aspect focuses on the classical 5-parameter shell formulation with Reissner–Mindlin kinematics. The above-mentioned combinations, already discussed by Andelfinger and Ramm for the linear case of a four-node shell element, are extended to geometrical non-linearities. In addition a nine-node quadrilateral variant is presented. A geometrically non-linear version of the EAS-approach is applied which is based on the enhancement of the Green–Lagrange strains instead of the displacement gradient as originally proposed by Simo and Armero. In the second part elements are derived in a similar way for a higher order, so-called 7-parameter non-linear shell formulation which includes the thickness stretch of the shell (Buchter and Ramm). In order to avoid artificial stiffening caused by the three dimensional displacement field and termed ‘thickness locking’, special provisions for the thickness stretch have to be introduced.

A unified approach for shear-locking-free triangular and rectangular shell finite elements

A new concept for the construction of locking-free finite elements for bending of shear deformable plates and shells, called DSG (Discrete Shear Gap) method, is presented. The method is based on a pure displacement formulation and utilizes only the usual displacement and rotational degrees of freedom (dof) at the nodes, without additional internal parameters, bubble modes, edge rotations or whatever. One unique rule is derived which can be applied to both triangular and rectangular elements of arbitrary polynomial order. Due to the nature of the method, the order of numerical integration can be reduced, thus the elements are actually cheaper than displacement elements with respect to computation time. The resulting triangular elements prove to perform particularly well in comparison with existing elements. The rectangular elements have a certain relation to the Assumed Natural Strain (ANS) or MITC-elements, in the case of a bilinear interpolation, they are even identical.

Adaptive topology optimization of elastoplastic structures

Material topology optimization is applied to determine the basic layout of a structure. The nonlinear structural response, e.g. buckling or plasticity, must be considered in order to generate a reliable design by structural optimization. In the present paper adaptive material topology optimization is extended to elastoplasticity. The objective of the design problem is to maximize the structural ductility which is defined by the integral of the strain energy over a given range of a prescribed displacement. The mass in the design space is prescribed. The design variables are the densities of the finite elements. The optimization problem is solved by a gradient based OC algorithm. An elastoplastic von Mises material with linear, isotropic work-hardening/softening for small strains is used. A geometrically adaptive optimization procedure is applied in order to avoid artificial stress singularities and to increase the numerical efficiency of the optimization process. The geometric parametrization of the design model is adapted during the optimization process. Elastoplastic structural analysis is outlined. An efficient algorithm is introduced to determine the gradient of the ductility with respect to the densities of the finite elements. The overall optimization procedure is presented and verified with design problems for plane stress conditions.

Displacement dependent pressure loads in nonlinear finite element analyses

Abstract Often pressure loading is falsely identified as a nonconservative load leading automatically to nonsymmetric load stiffness matrices. The present paper discusses in detail the conditions when a pressure load is conservative and when it is not. The essential part is a clear classification of the load definition. Here either body attached or space attached loads are considered. The load stiffness matrices are derived for a pressure loaded curved surface in space. Several numerical examples are given; among these are linear and nonlinear buckling analyses of beams, rings and shells.

On the physical significance of higher order kinematic and static variables in a three-dimensional shell formulation

Abstract In recent years, considerable attention has been given to the development of higher order plate and shell models. These models are able to approximately represent three-dimensional effects, while pertaining the efficiency of a two-dimensional formulation due to pre-integration of the structural stiffness matrix across the thickness. Especially, the possibility to use unmodified, complete three-dimensional material laws within shell analysis has been a major motivation for the development of such models. While the theoretical and numerical formulation of so-called 7-parameter shell models, including a thickness stretch of the shell, has been discussed in numerous papers, no thorough investigation of the physical significance of the additional kinematic and static variables, coming along with the extension into three dimensions, is known to the authors. However, realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures, e.g. concerning loading and kinematic boundary conditions, and a correct interpretation of the results. In the present paper, the significance of kinematic and static variables, appearing in a 7-parameter model proposed by Buchter and Ramm (1992a) are discussed. It is shown, how these quantities ‘refine’ the model behavior and how they can be related to the ‘classical’ variables, such as ‘curvatures’ and ‘stress resultants’. Furthermore, the special role of the material law within such a formulation is addressed. It is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness, to ensure correct results. In this context it is found, that the considered 7-parameter model can be regarded as ‘optimal’ with respect to the number of degrees of freedom involved.

Adaptive topology optimization

Topology optimization of continuum structures is often reduced to a material distribution problem. Up to now this optimization problem has been solved following a rigid scheme. A design space is parametrized by design patches, which are fixed during the optimization process and are identical to the finite element discretization. The structural layout is determined, whether or not there is material in the design patches. Since many design patches are necessary to describe approximately the structural layout, this procedure leads to a large number of optimization variables. Furthermore, due to a lack of clearness and smoothness, the results obtained can often only be used as a conceptual design idea.To overcome these shortcomings adaptive techniques, which decrease the number of optimization variables and generate smooth results, are introduced. First, the use of pure mesh refinement in topology optimization is discussed. Since this technique still leads to unsatisfactory results, a new method is proposed that adapts the effective design space of each design cycle to the present material distribution. Since the effective design space is approximated by cubic or Bézier splines, this procedure does not only decrease the number of design variables and lead to smooth results, but can be directly joined to conventional shape optimization. With examples for maximum stiffness problems of elastic structures the quality of the proposed techniques is demonstrated.

Constraint Energy Momentum Algorithm and its application to non-linear dynamics of shells

Abstract A modified time-integration method is developed for non-linear structural dynamics combining controllable algorithmic dissipation of higher modes and, at the same time, conservation of energy as well as linear and angular momentum. Although these features seem to be actually inconsistent, numerical stability with large time steps for dynamic buckling and snap-through problems as well as for dynamical systems with smooth solutions is guaranteed. Arbitrary implicit, one-step time-integration schemes with Newmark approximations are suitable for use as basic algorithms for the proposed method. The desired conservation attributes are obtained if the basic algorithm is augmented by energy and momentum constraints. The potential of the suggested algorithm is demonstrated by selected applications to non-linear dynamics of shell structures.

Continuous and discontinuous modelling of cohesive-frictional materials

Computational models for failure in cohesive-frictional materials with stochastically distributed imperfections.- Modeling of localized damage and fracture in quasibrittle materials.- Microplane modelling and particle modelling of cohesive-frictional materials.- Short-term creep of shotcrete - thermochemoplastic material modelling and nonlinear analysis of a laboratory test and of a NATM excavation by the Finite Element Method.- Thermo-poro-mechanics of rapid fault shearing.- A view on the variational setting of micropolar continua.- Macromodelling of softening in non-cohesive soils.- An experimental investigation of the relationships between grain size distribution and shear banding in sand.- Micromechanics of the elastic behaviour of granular materials.- On sticky-sphere assemblies.- Cohesive granular texture.- Micro-mechanisms of deformation in granular materials: experiments and numerical results.- Scaling properties of granular materials.- Discrete and continuum modelling of granular materials.- Difficulties and limitation of statistical homogenization in granular materials.- From discontinuous models towards a continuum description.- From solids to granulates - Discrete element simulations of fracture and fragmentation processes in geomaterials.- Microscopic modelling of granular materials taking into account particle rotations.- Microstructured materials: local constitutive equation with internal lenght, theoretical and numerical studies.- Damage in a composite material under combined mechanical and hygral load.

From particle ensembles to Cosserat continua: homogenization of contact forces towards stresses and couple stresses

In the present contribution, a transition from the dynamics of single particles to a Cosserat continuum is discussed. Based on the definition of volume averages, expressions for the macroscopic stress tensors and for the couple stress tensors are derived. It is found that an ensemble of particles allows for a non-symmetric macroscopic stress tensor and, thus, for the existence of couple stresses, even if the single particles are considered as standard continua. Discrete element method simulations of a biaxial box are used for the validation of the proposed homogenization technique.