Peter Wriggers

Leibniz Universität Hannover · Institute of Continuum Mechanics

Systems composed of numerous particles, as granular materials, can be simulated by the discrete element method (DEM). There are numerous versions of DEM considering particle shapes as spheres, superellipsoids, polyhedra and others. Classically, particles are considered rigid and the only flexibility present in the model is local, embedded in the co...

Locking effects can be a major concern during the numerical modelling of elastic materials, especially for large strains. Those effects arise from volumetric constraints such as incompressibility or anisotropic effects of the underlying material class. One particular solution strategy is to employ mixed formulations, which provide solutions tailore...

The virtual element method (VEM) provides new ways of deriving discretization for problems in structural and solid mechanics, starting with the contribution by Beirão da Veiga et al. (2013) for elastic solids. Interestingly, the virtual element method allows also to revisit the construction of different elements which have the same shape as finite...

Atherosclerosis is a disease in blood vessels that often results in plaque formation and lumen narrowing. It is an inflammatory response of the tissue caused by disruptions in the vessel wall nourishment. Blood vessels are nourished by nutrients originating from the blood of the lumen. In medium-sized and larger vessels, nutrients are additionally...

In this paper, a novel higher stabilization-free virtual element method is proposed for com-pressible hyper-elastic materials in 2D. Different from the most traditional virtual element formulation, the method does not need any stabilization. The main idea is to modify the virtual element space to allow the computation of a higher-order polynomial 2...

The Node-to-Segment (NTS) method enhanced computational contact mechanics by enabling contact analysis with large deformations with contact location as part of the unknowns. Despite having shortcomings, it is still to this day largely employed by the industry for contact analysis between continuous surfaces due to its low computational cost and sca...

In this work, we present a first-order stabilization-free virtual element method (SFVEM) for three-dimensional hyperelastic problems. Different from the conventional virtual element method, which necessitates additional stabilization terms in the bilinear formulation, the method developed in this work operates without the need for any stabilization...

In this paper, a novel first- and second-order stabilization-free virtual element method is proposed for two-dimensional elastoplastic problems. In contrast to traditional virtual element methods, the improved method does not require any stabilization, making the solution of nonlinear problems more reliable. The main idea is to modify the virtual e...

We develop low-order triangular virtual elements for linear Kirchhoff–Love shells from an engineering point of view. Flat element geometry is considered, which enables a direct shell discretization with no need for a curvilinear coordinate system or predefined initial mapping. Along with the assumed linearity of the problem, the superposition of th...

Electro-active materials are classified as electrostrictive and piezoelectric materials. They deform under the action of an external electric field. This research work presents a variational-based computational modeling approach for the failure prediction of ferromagnetic materials. To solve this problem, a coupling between magnetostriction and mec...

Ultra high performance fibre reinforced concrete (UHPFRC) demonstrates intricate failure mechanisms like fibre bending and yielding, mortar cracking, crushing and spalling, as well as fibre-mortar interfacial debonding. These are often beyond the resolutions of conventional experiments as well as homogeneous models and motivate the need for more re...

Phase field models can effectively capture complicated crack evolution characteristics such as propagation, bifurcating, intersecting and merging. However, the simulation of three-dimensional (3D) quasi-brittle fracture remains a challenge due to large nonlinear equation systems and significant computational costs, which are often intractable by it...

Contact detection is one of the most important steps to be handled during numerical simulations of particle systems. It is extremely dependent on particle geometry, which governs the mechanical behavior of systems and the computational cost of simulations. NURBS are an option to model the boundary of particles with nonstandard geometries, here call...

In the context of a multibody numerical environment, when aiming at representing the contact interaction between bodies one needs the inclusion of special mathematical formulations. They can be geometrically described as surface‐to‐surface or line‐to‐line contacts, for instance. The last one is usually the natural choice when addressing beam‐to‐bea...

This chapter provides an introduction to differential geometry of embedded surfaces and curves in the three-dimensional Euclidean space. The geometry of manifolds of dimension two or curved surfaces is extensively used in many branches of physics and engineering. Examples of which include general relativity and structural mechanics. The geometry of...

Vector and tensor analysis or calculus is prerequisite for many applications in science and engineering. Examples of which include differential geometry, electromagnetic field theory and continuum mechanics. It is an important branch of mathematics that studies differentiation and integration of vector and tensor fields which will extensively be us...

Recall from Chap. 1 that scalars were zeroth-order tensors with only one component and vectors or first-order tensors were characterized by three components in three-dimensional spaces. The order of a tensor was consistently increased in Chap. 2 wherein a tensor of order two, characterized by nine components, was declared as a linear mapping that t...

Tensors are characterized by their orders. Scalars are zeroth-order tensors with only magnitude and no other characteristics. They are extensively used in physical relations and usually designated by small and capital lightface Latin or Greek letters; examples of which include mass, time and temperature. In contrast, vectors are physical quantities...

Vector algebra was briefly discussed in the previous chapter not only to introduce the concept of vector and represent its important relationships but also as a primary mathematics for an introduction to tensor algebra. Consistent with vectors, second-order tensors or simply tensors are geometric objects that aim at describing linear relation betwe...

Eigenvalues (or characteristic values or principal values ) and eigenvectors (or principal axes or principal directions ) are extensively used in many branches of physics and engineering; examples of which include quantum mechanics, control theory and stability analysis. They represent mathematical objects that are associated with a linear transfor...

In a continuation of vector and tensor calculus, this chapter mainly studies the actions of gradient , divergence and curl operators on vectors and tensors. Needless to say that these differential operators are the workhorses of vector and tensor analysis. Recall that Chap. 6 dealt with the gradients of tensor functions which were non-constant tens...

So far tensorial variables have been expressed with respect to a Cartesian coordinate frame defined by an origin and the standard basis vectors. Sometimes, the symmetry of a problem demands another set of coordinates. For instance, cylindrical coordinates are usually used when there is symmetry about the cylindrical axis or spherical coordinates ar...

This chapter contains two sections. The first section deals with well-known theorems involving integrals of tensorial field variables to complete vector and tensor calculus started from Chap. 6. Specifically, the integral theorems of Gauss and Stokes are studied. They are of central importance in mathematics of physics because they eventually appea...

So far we have formulated ways to compute the projection of an ansatz function \(\boldsymbol{u}_h\) onto a polynomial space \(\boldsymbol{u}_\pi \) for the virtual element method using a linear or a quadratic order interpolation. These projections provide the basis for the discretization of linear and nonlinear partial differential equations relate...

Heterogeneous materials (such as composites, bones, wood, concrete and metallic-polycrystalline materials) consist of complicated constituents across scales with complex material response. These materials, even with similar properties at macroscopic level, can behave differently at micro-scale. The response of such materials is often related to non...

Almost all engineering applications include structural parts that act on other elements through an area of contact. In many cases the behaviour at the contact interface has an influence on the performance of machines, looking for example at friction and wear, or on production processes like metal forming. Therefore it is necessary to capture these...

This chapter summarizes the basic relation needed to formulate the deformation of solids in the linear and nonlinear range. It is subdivided into the sections kinematics, balance laws, variational formulations and constitutive equations. This part of the book is not meant for studying continuum mechanics, it only summarizes results that provide ess...

Virtual elements have so far been formulated for different problems in applied engineering and physics. One of the first papers was on the general mathematical formulation of the method

Beams and plates serve as individual structures or structural members of many technical constructions such as buildings, airplanes, cars, or ships. Thus, many analytical and numerical simulation schemes were developed over the last century that can predict kinematical quantities, like deflections and rotations, and stress resultants, like normal fo...

Many engineering problems require an analysis that takes into account more than one field. These are known as coupled problems which can be split in two distinct categories. The first category relates to problems in which the physical domains overlap and coupling occurs via the differential equations of the different physical phenomena. In the seco...

Many discretization schemes—like finite differences, finite element and boundary element methods—can be applied to solve problems related to linear and nonlinear dynamics.

Fracture processes occur in different environments and define the lifespan of many engineering structures. To illustrate the effects of fracture in more detail one can think of its negative implications like failure of large structures, e.g. bridges, ships and vessels. Further instances are shattered glass, a broken leg, a ruptured aorta, a torn sa...

The application of virtual elements covers a wide range in solid mechanics. Besides pure elastic behaviour, material models can be introduced that include path dependency and thus history variables. The treatment of such constitutive models needs a special approach with respect to algorithms and the formulation of the virtual element method. The de...

One of the first applications of virtual elements in engineering was related to elasticity. It had the aim to construct a general methodology that could be applied to problems involving two-dimensional linear elastic solids under general loading conditions, see Beirão da Veiga et al. (2013). After these first developments and formulations of the ne...

The formulation of virtual elements for heat conduction problems, see (2.42), or more general for the Poisson equation (\(-\Delta \,\theta = f\)) was one of the starting points of this method, see Beirão da Veiga et al. (2013) and Beirão da Veiga et al. (2014). Since Laplace, Poisson and the elasticity equations are of elliptical nature, the ansatz...

To investigate dynamic fracture mechanisms of quasi-brittle materials, this work proposes a rate-dependent phase field model that integrates both macroscopic viscoelasticity and micro-viscosity to reflect the rate effects by free water and unhydrated inclusions. Based on the unified phase field theory, the model introduces a linear viscoelastic con...

A novel mixed enriched finite element model is developed for coupled non‐linear thermo‐hydro‐mechanical simulation of fractured porous media with three‐phase flow and thermal coupling. Simulation of induced acoustic emission (AE) and microseismic emission (ME) due to tensile fracturing and shear slip instability of pre‐existing fracture interfaces...

This paper presents a mathematical model for arterial dissection based on a novel hypothesis proposed by a surgeon, Axel Haverich, see Haverich (Circulation 135(3):205–207, 2017. https://doi.org/10.1161/circulationaha.116.025407). In an attempt and based on clinical observations, he explained how three different arterial diseases, namely atheroscle...

Purpose: For analyzing structures’ nonlinear dynamic behaviors, it is broadly accepted to use time integration methods and model the nonlinearities using iterative methods. The iterations may however fail. With attention to the sources of computational errors, in 2015 and 2022, the authors proposed continuation of the analysis, even when the iterat...

Stochastic phase-field models can reproduce the nonlinearity and randomness of the mechanical behavior of quasi-brittle materials within a unified macroscopic continuum framework. To study the probabilistic characteristics of microscopic and macroscopic strength within the stochastic phase-field model and their relationship, a series of numerical r...

This paper deals with the mathematical modeling of bacterial co-aggregation and its numerical implementation in a FEM framework. Since the concept of co-aggregation refers to the physical binding between cells of different microbial species, a system composed of two species is considered in the modeling framework. The extension of the model to an a...

This work is placed in the scope of vehicle protection from blast load applied to a vehicle following the detonation of an IED, through the use of a fluid filled cladding. The importance of the confinement of the fluid in this cladding and the free surface conditions is highlighted. In particular, impulse spreading and its link to the location of t...

The virtual element method (VEM) provides new ways of deriving discretization for problems in structural and solid mechanics, starting with the contribution by Beir˜ao da Veiga et al. (2013) for elastic solids. Interestingly, the virtual element method allows also to revisit the construction of different elements which have the same shape as finite...

We present a general framework for the analysis and modelling of frictional contact involving composite materials. The study has focused on composite materials formed by a matrix of rubber and synthetic or metallic fibres, which is the case of standard tires. We detail the numerical treatment of incompressibility at large deformations that rubber c...

Material modeling using modern numerical methods accelerates the design process and reduces the costs of developing new products. However, for multiscale modeling of heterogeneous materials, the well-established homogenization techniques remain computationally expensive for high accuracy levels. In this contribution, a machine learning approach, co...

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 formu...

The thermal-induced failure mechanism of the bearing outer-ring guiding-surface is investigated within this work when subjected to cyclic impact and sliding actions. The paper combines numerical simulations and experi- mental analysis. A high-speed bearing oil interruption experiment is carried out for testing the severe damage of the bearing steel...

The contact between bodies is a complex phenomenon that involves mechanical interaction, frictional sliding and heat transfer, among others. A common (and convenient) approach for the mechanical interaction in a finite element framework is to directly use the geometry of the elements to formulate the contact. The main drawback lies in the sharp cor...

In many aspects, elastomers and soft biological tissues exhibit similar mechanical properties such as a pronounced nonlinear stress–strain relation and a viscoelastic response to external loads. Consequently, many models use the same rheological framework and material functions to capture their behavior. The viscosity function is thereby often assu...

Damage in soft biological tissues causes an inflammatory reaction that initiates a chain of events to repair the tissue. This work presents a continuum model and its in silico implementation that describe the cascade of mechanisms leading to tissue healing, coupling mechanical as well as chemo-biological processes. The mechanics is described by mea...

Bisher haben wir bei der Untersuchung des Verhaltens von festen Körpern ein linear elastisches Verhalten angenommen. Viele Materialien, wie Metalle, zeigen jedoch bei höheren Spannungen ein plastisches Verhalten, was nach einer Entlastung durch das Auftreten bleibender (plastischer) Verformungen gekennzeichnet ist. Anderen Werkstoffe, wie Polymere,...

Bei zahlreichen Bauteilen sind die Abmessungen der Querschnitte klein gegen die Länge. Sie können dann näherungsweise als linienhafte Körper betrachtet werden. Zwei solcher Tragwerke haben wir schon kennengelernt, nämlich Stäbe (gerade Achse, Last in Richtung der Stabachse, vgl. Band 1, Kap. 6) und Balken (gerade Achse, Last senkrecht zur Balkenach...

In Band 3 haben wir freie und erzwungene Schwingungen von mechanischen Systemen mit einem bzw. mit zwei Freiheitsgraden behandelt. Solche Systeme mit endlicher Zahl von Freiheitsgraden nennt man auch diskrete Systeme. Die Beschreibung ihrer Schwingungsbewegung führt auf gewöhnliche Differentialgleichungen. In diesem Kapitel wollen wir nun Schwingun...

In Band 2 haben wir uns schon mit Problemen der Elastostatik befasst, wobei wir uns dort auf die Untersuchung von Stäben und Balken beschränkt haben. Um weitergehende Fragen behandeln zu können, werden hier die Grundlagen der linearen Elastizitätstheorie zusammengestellt. Das Beiwort „linear“ deutet dabei an, dass sich diese Theorie auf das linear-...

Die Formulierung mechanischer Probleme führt auf Gleichungen, die für konkrete Aufgabenstellungen gelöst werden müssen. Diese Gleichungen können je nach Fragestellung algebraische Beziehungen, Differentialgleichungen oder Variationsgleichungen sein. Beispiele dafür finden sich in den Bänden 1 – 3 und in den vorangegangenen Kapiteln dieses Buches. A...

Dieses Kapitel führt in die Grundlagen der Hydrostatik und Hydrodynamik ein. Behandelt wird nach den Eigenschaften von Flüssigkeiten zunächst die Druckverteilung in schweren, ruhenden Flüssigkeiten konstanter Dichte. Es wird gezeigt, wie man daraus die resultierenden Kräfte auf ebne und gekrümmte Flächen sowie auf schwimmende Körper berechnet, wobe...

Wir beschäftigen uns in diesem Kapitel mit der statischen Stabilität elastischer Tragwerke. Hierunter wollen wir die Untersuchung von Gleichgewichtslagen auf deren Stabilität verstehen (vgl. Band 1, Abschnitt 8.5). Die Betrachtungen werden zunächst an einfachen Stab-Feder-Modellen durchgeführt, an denen man viele wesentliche Phänomene erkennen kann...

This work presents a robust non-deterministic free vibration analysis for engineering structures with random field parameters in the frame of stochastic finite element method. For this, considering the randomness and spatial correlation of structural physical parameters, a parameter setting model based on random field theory is proposed to represen...

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...

Die Materialmodellierung von Beton mittels moderner numerischer Methoden beschleunigt den Entwurfsprozess von Bauwerken erheblich. Bei der Multiskalenmodellierung eines solch heterogenen Materials sind jedoch die etablierten Homogenisierungsverfahren weiterhin sehr rechenintensiv, insbesondere bei hohen Genauigkeitsanforderungen. In diesem Beitrag...

The magnesium alloy LAE442 showed promising results as a bone substitute in numerous studies in non-weight bearing bone defects. This study aimed to investigate the in vivo behavior of wedge-shaped open-pored LAE442 scaffolds modified with two different coatings (magnesium fluoride (MgF 2 , group 1)) or magnesium fluoride/calcium phosphate (MgF 2 /...

In this contribution, the Virtual Element Method (VEM) with a linear ansatz is applied to a computational crystal plasticity framework in a micro-structural environment. Furthermore, a simple anisotropic energetic contribution, based on invariant-formulations of tensorial deformation measures and structural tensors, is presented for the cubic elast...

The shear instability of the bearing outer-ring guiding-surface is investigated within this work when subjected to cyclic impact and sliding actions. The paper combines numerical simulations and experiments. A high-speed bearing oil interruption experiment is carried out for testing the severe damage of the bearing steel at high-speed impact-slidin...

For nonlinear time history analysis, employing a time integration method and some nonlinearity iterative method (for implicit analyses) is a broadly accepted practice. In 2015, the authors proposed a change in the analysis, according to which, when the nonlinearity iterations do not converge, the analysis proceeds to the next integration step. In t...

Stoß: Als Stoß bezeichnet man ein plötzliches Aufeinanderprallen zweier Körper.

Die folgenden Formeln und Aufgaben beschränken sich auf Schwingungen von linearen Systemen mit einem Freiheitsgrad.

Die Lage eines Punkte P im Raum wird durch den Ortsvektor.

Die Bewegung eines starren Körpers lässt sich aus einer Translation und einer Rotation zusammensetzen.

Vielfach ist es zweckmäßig, die Bewegung eines Punktes P nicht in Bezug auf ein festes Koordinatensystem (x, y, z) sondern in Bezug auf ein bewegtes System (ξ, η, ζ) zu beschreiben.

Das Geschwindigkeitsfeld v(x(t), t) beschreibt die Bewegung einer Flüssigkeit. Der Vektor x weist jedem Ort in der Flüssigkeit eine Geschwindigkeit v zur Zeit t zu.