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This paper aims to describe the application of the virtual element method (VEM) to contact problems between elastic bodies.

Polygonal elements with arbitrary shape allow a stable node-to-node contact enforcement. By adaptively adjusting the polygonal mesh, this methodology is extended to problems undergoing large frictional sliding.

The virtual element is well suited for large deformation contact problems. The issue of element stability for this specific application is discussed, and the capability of the method is demonstrated by means of numerical examples.

This work is completely new as this is the first time, as per the authors’ knowledge, the VEM is applied to large deformation contact.

In certain cases, traction–separation laws do not reflect the behaviour sufficiently so that thin volumetric elements, Internal Thickness Extrapolation formulations, bulk material projections or various other approaches are applied. All of them have disadvantages in the formulation or practical application.

Damage within thin layers is often modelled using at cohesive zone elements (CZE). The constitutive behaviour of cohesive zone elements is usually described by traction–seperation laws (TSLs) that consider the (traction separation) relation in normal opening and tangential shearing direction. Here, the deformation (separation) as well as the reaction (traction) are vectorial quantities.

In this contribution, a CZE is presented that includes damage from membrane modes.

Membrane mode-related damaging effects that can be seen in physical tests that could not be simulated with standard CZEs are well captured by membrane mode–enhanced cohesive zone elements.

Several mechanical models can be employed for the analysis of thin walled structures. A bending indicator is developed for a nonlinear adaptive process and applied to thin walled axisymmetric shell problems incorporating membrane and bending elements. If the structural response of the model including bending is available the error made by the reduced membrane model is easy to evaluate. Thus an indication for the bending has to be found from the structural response of the reduced membrane model. This is done by an approximation of the rotations of the membrane part of the structure and by an evaluation of the bending energy leading to the bending indicator with these approximated rotations. Finally a criterion for the change of the models is proposed based on the bending indicator and some examples are presented.

The paper aims to introduce an efficient contact detection algorithm for smooth convex particles.

The contact points of adjacent particles are defined according to the common‐normal concept. The problem of contact detection is formulated as 2D unconstrained optimization problem that is solved by a combination of Newton's method and a Levenberg‐Marquardt method.

The contact detection algorithm is efficient in terms of the number of iterations required to reach a high accuracy. In the case of non‐penetrating particles, a penetration can be ruled out in the course of the iterative solution before convergence is reached.

The algorithm is only applicable to smooth convex particles, where a bijective relation between the surface points and the surface normals exists.

By a new kind of formulation, the problem of contact detection between 3D particles can be reduced to a 2D unconstrained optimization problem. This formulation enables fast contact exclusions in the case of non‐penetrating particles.

Considers the problem of stability of the enhanced strain elements in the presence of large deformations. The standard orthogonality condition between the enhanced strains and constant stresses ensures satisfaction of the patch test and convergence of the method in case of linear elasticity. However, this does not hold in the case of large deformations. By analytic derivation of the element eigenvalues in large strain states additional orthogonality conditions can be derived, leading to a stable formulation, regardless of the magnitude of deformations. Proposes a new element based on a consistent formulation of the enhanced gradient with respect to new orthogonality conditions which it retains with four enhanced modes volumetric and shear locking free behaviour of the original formulation and does not exhibit hour‐glassing for large deformations.