On a virtual element formulation for trusses and beams

The virtual element method (VEM) was developed not too long ago, starting with the paper [2] related to elasticity in solid mechanics. The virtual element method allows to revisit the construction of different elements; however, it has so far not applied to one-dimensional structures like trusses and beams. Here we study several VEM elements suitable for trusses and beams and show that the virtual element methodology produces elements that are equivalent to well-known finite elements but also elements that are different, especially for higher-order ansatz functions. It will be shown that these elements can be easily incorporated in classical finite element codes since they have the same number of unknowns as finite beam elements. Furthermore, the formulation allows to compute nonlinear structural problems undergoing large deflections and rotations. © 2022, The Author(s)

Fe2-homogenization of micromorphic elasto-plastic materials

In this work, a homogenization strategy for a micromorphic–type inelastic material is presented. In the spirit of FE2, a representative volume element is attached to each macroscopic quadrature point. Due to the inherent length scale of the micromorphic continuum, size effects for inelastic behavior are obtained on RVE–level. A focus is placed on the computation of the homogenized algorithmic tangent. It is determined via sensitivity analyses with respect to the boundary conditions imposed on the RVE. Following this procedure, costly single–scale computations with dense meshes can be replaced by a robust homogenization approach with optimal convergence rates

Systematic Fitting and Comparison of Hyperelastic Continuum Models for Elastomers

Hyperelasticity is a common modeling approach to reproduce the nonlinear mechanical behavior of rubber materials at finite deformations. It is not only employed for stand-alone, purely elastic models but also within more sophisticated frameworks like viscoelasticity or Mullins-type softening. The choice of an appropriate strain energy function and identification of its parameters is of particular importance for reliable simulations of rubber products. The present manuscript provides an overview of suitable hyperelastic models to reproduce the isochoric as well as volumetric behavior of nine widely used rubber compounds. This necessitates firstly a discussion on the careful preparation of the experimental data. More specific, procedures are proposed to properly treat the preload in tensile and compression tests as well as to proof the consistency of experimental data from multiple experiments. Moreover, feasible formulations of the cost function for the parameter identification in terms of the stress measure, error type as well as order of the residual norm are studied and their effect on the fitting results is illustrated. After these preliminaries, invariant-based strain energy functions with decoupled dependencies on all three principal invariants are employed to identify promising models for each compound. Especially, appropriate parameter constraints are discussed and the role of the second invariant is analyzed. Thus, this contribution may serve as a guideline for the process of experimental characterization, data processing, model selection and parameter identification for existing as well as new materials

Virtual element formulation for gradient elasticity

The virtual element method has been developed over the last decade and applied to problems in solid mechanics. Different formulations have been used regarding the order of ansatz, stabilization of the method and applied to a wide range of problems including elastic and inelastic materials and fracturing processes. This paper is concerned with formulations of virtual elements for higher gradient elastic theories of solids using the possibility, inherent in virtual element methods, of formulating C1-continuous ansatz functions in a simple and efficient way

A Virtual Element Method for Contact Modeling and Dynamics

Decreasing resources and limited energy results in a greater demand for virtual development processes and efficient product development. This trend points out the importance of digitalization and the subsequent need for efficient and accurate numerical prediction methods for product development. Due to their flexibility, numerical methods are gradually and steadily replacing physical tests in industrial product developments. The finite element method is perhaps the most well-known and widely used numerical method in industry and science. Increasing computer capabilities and further developments of these methods in recent years have increased the amount of application fields, including civil, automotive, naval, space and geo-technical engineering. However, along with complex geometries the spatial discretization of the domain emerges as a very time consuming step. Due to the fact that the classical finite element method is restricted to basic regular shaped element topologies, a more general choice of element shapes would give more flexibility. Within mesh-based methods, polygonal methods are a helpful alternative and showed great performance in engineering and science. However, most of these methods seem to need more computational effort and beside the aforementioned advantage of flexible element shapes, disadvantages appear as well. A relatively new method, the virtual element method, promises great numerical properties and can be seen as a generalization of the classical finite element method. All new methods need to be investigated for different applications in engineering and science before they can be applied commercially. This work deals with the application of the virtual element method to dynamic and elastoplastic material behavior. To deal with elastic and plastic incompressibility, a mixed virtual element formulation is presented as well. As a further development, the virtual element method is used to model three dimensional contact with different contact discretizations. A new projection algorithm is developed to manipulate the mesh at the contact interface, such that a very simple and efficient node-to-node contact formulation can be used. Various numerical examples for all aforementioned applications are performed, including benchmark problems such as the classical patch test. For comparison purposes, different finite element formulations are also adopted. As a final example, all models, including plasticity, dynamics and contact, are coupled to model mechanical impact.Eine Verringerung von Ressourcen und die damit einhergehende Energieknappheit f ¨uhren zu einem erh¨ohten Bedarf an virtuellen Entwicklungsprozessen und effizienter Produktentwicklung. Dieser Trend verdeutlicht die Bedeutung der Digitalisierung und den daraus resultierenden Bedarf an effizienten und hoch genauen numerischen Vorhersagemethoden f ¨ur die Produktentwicklung. Aufgrund ihrer Flexibilit¨at und mit steigenden Rechnerkapazit¨aten ersetzen numerische Methoden allm¨ahlich und stetig physikalische Tests in der industriellen Produktentwicklung. Die Finite Elemente Methode ist vielleicht die bekannteste und am weitesten verbreitete numerische Methode in Industrie und Wissenschaft. Durch die zunehmenden Rechnerkapazit ¨aten und die Weiterentwicklung dieser Methoden in den letzten Jahren hat sich die Zahl der Anwendungsbereiche vergr¨oßert. Numerische Methoden werden unter anderem im Bauwesen, im Automobilbau, in der Schifffahrt, in der Luft- und Raumfahrt und in der Geotechnik eingesetzt. Bei komplexen Geometrien erweist sich jedoch die r¨aumliche Diskretisierung des Gebiets als ein sehr zeitaufw¨andiger Prozess. Da die klassische Finite Elemente Methode auf einfache, regelm¨aßig geformte Elementgeometrien beschr¨ankt ist, w¨urde eine allgemeinere Auswahl von Elementgeometrien mehr Flexibilit¨at bieten. Innerhalb der netzbasierten Methoden sind polygonale Methoden eine hilfreiche Alternative und haben sich bereits in Industrie und Wissenschaft bew¨ahrt. Allerdings scheinen die meisten dieser Methoden einen h¨oheren Rechenaufwand zu erfordern, und neben dem bereits erw¨ahnten Vorteil der flexiblen Elementgeometrien treten auch gewisse Nachteile auf. Eine relativ neue Methode, die Virtuelle Elemente Methode, verspricht gute numerische Eigenschaften und kann als eine Verallgemeinerung der klassischen Finite Elemente Methode angesehen werden. Wie bei allen neuen Methoden m¨ussen auch hier verschiedene Anwendungen in der Industrie und Wissenschaft untersucht werden, bevor die Methode kommerziell eingesetzt werden kann. Diese Arbeit befasst sich mit der Anwendung der Methode der virtuellen Elemente auf dynamisches und elasto-plastisches Materialverhalten. Um elastische und plastische Inkompressibilit¨at zu behandeln, wird auch eine gemischte virtuelle Elementformulierung vorgestellt. In einem weiteren Schritt wird die Virtuelle Elemente Methode zur Modellierung dreidimensionaler Kontaktprobleme mit verschiedenen Kontaktdiskretisierungen verwendet. Es wird ein neuer Projektionsalgorithmus vorgestellt, welcher das Netz an der Kontaktschnittstelle so manipuliert, dass eine sehr einfache und effiziente Knoten-zu-Knoten Kontaktformulierung verwendet werden kann. Es werden verschiedene numerische Beispiele f ¨ur alle oben genannten Anwendungen behandelt, darunter auch Benchmark-Probleme wie der klassische Patch-Test. Um einen geeigneten Vergleich durchzuf¨uhren, werden die entwickelten Formulierungen mit verschiedene Finite Elemente Formulierungen verglichen. Als letztes Beispiel werden alle Modelle, einschließlich Plastizit¨at, Dynamik und Kontakt, gekoppelt, um einen mechanischen Stoß zu modellieren

Discrete element model for general polyhedra

We present a version of the Discrete Element Method considering the particles as rigid polyhedra. The Principle of Virtual Work is employed as basis for a multibody dynamics model. Each particle surface is split into sub-regions, which are tracked for contact with other sub-regions of neighboring particles. Contact interactions are modeled pointwise, considering vertex-face, edge-edge, vertex-edge and vertex-vertex interactions. General polyhedra with triangular faces are considered as particles, permitting multiple pointwise interactions which are automatically detected along the model evolution. We propose a combined interface law composed of a penalty and a barrier approach, to fulfill the contact constraints. Numerical examples demonstrate that the model can handle normal and frictional contact effects in a robust manner. These include simulations of convex and non-convex particles, showing the potential of applicability to materials with complex shaped particles such as sand and railway ballast. © 2021, The Author(s)

Powder compaction with polygonal particles built from radially extending one-dimensional frictional devices

Powder compaction is a major ingredient in a wide range of production techniques for objects of every day importance. Diverse applications – from pharmaceutical tablets to metallic and ceramic parts, where compaction is usually a part in the sintering process – are covered by current technology. The aim of cold compaction is to increase the relative density of the part. Due to the granular nature of the powder material compaction is random process which requires careful mastering and comprehensive understanding. The experimental access to compaction processes, even as simple ones as uniaxial compaction, are limited. Therefore simulation of compaction processes offers the opportunity to improve understanding of powder compaction. The Direct Element Method (DEM) simulates the powder as individual particles. These particles are distinct from each other and the forces applied to the entire powder are equilibrated by the contacting force between the particles. Usually, an explicit time integration, with or without considering dynamic effects, allows the particles to move and the powder to be compacted. Especially for metallic powders the plastic deformation of individual particles plays an important role and has a perceptible influence up to the macroscopic scale of the whole part. This leads to efforts to formulate a DE method in which individual particles are discretized as distinct FE models. Yet, such approach is deemed too costly for the simulation entire parts. Therefore a new approach for plastic particle deformation has been devised. The particles are simulated as ’hedgehogs’ of one-dimensional frictional devices. The frictional devices form spikes that extend in a radial way from the center of the particle. The tips of the spikes are connected and the connections form the edges of the polygonal particles. The contacts between the particles are found by geometric means as intersections of the spikes of one particle with the edges of the particle’s contact partner. This indents the spike and its frictional device generates forces that act on the two particles. Multiple contacts between two particles are allowed and concave particles can be treated intrinsically. Preliminary results indicate that this new approach to model plastic particles might bridge the gap between sufficiently realistic behaviour of the plastic particles and low computational effort. However, for now, the results are based on two-dimensional proofof-concept simulations and, for the sake of simplicity, the springs in the spikes are linear and the friction is rate-independent and perfectly plastic. These model assumption already offer considerable liberties to tune the plastic behaviour of the particles to experimental results

Non-local ductile damage formulations for sheet bulk metal forming

A ductile damage model for sheet bulk metal forming processes and its efficient and accurate treatment in the context of the Finite Element Method is presented. The damage is introduced as a non-local field to overcome pathological mesh dependency. Since standard elements tend to show volumetric locking in the bulk forming process a mixed formulation is implemented in the commercial software simufact.forming to obtain better results.DFG/SFB/TR 7