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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.

This article presents a geometrically non-linear finite element formulation for the analysis of planar two-layer beam-columns taking into account the inter-layer slip and uplift.

The co-rotational method is adopted, in which the motion of the element is decomposed into a rigid body motion and a small deformational one. The geometrically linear formulation can be used in the local frame and automatically be transformed into a geometrically nonlinear one. In co-rotational frame, both layers are assumed to be discretely connected at the element ends. Slips and uplifts are assumed to be small. Consequently, the condition of non interpenetration between the layers can be treated using a node-to-node contact algorithm. The resolution methods such as penalty (PM) and augmented Lagrangian method (ALM) with Uzawa updating scheme can be used.

The non-penetration condition between the layers of composite beams can be formulated by using contact law. It is found that despite a low convergence rate of augmented Lagrangian method compared to penalty method, the former prevents the unrealistic penetration. Besides, it is shown that the buckling load of the composite beam-column is largely affected by the uplift stiffness of the connectors.

The proposed finite element model is capable of simulating accurately the geometrically non-linear behavior of planar two-layer beam-columns taking into account the inter-layer slip and uplift. Regarding uplift, the non-penetration condition is strictly enforced by considering rigorous contact conditions at the interface. The constraint problem is solved using the penalty method or the augmented Lagrangian method with the Uzawa updating scheme.

The finite element analysis of deformation of viscoplastic material involves contact between the tool and the workpiece. Here unilateral contact condition with the possibility of nodes originally in contact, losing contact subsequently, is analysed in non‐steady state forming processes. Friction has been taken into consideration through a potential function. Node to node contact is analysed and contact forces at the node are used to decide if the node is to be released. Two different algorithms are presented for treating the nodal contact condition. The one step explicit method with projections on the surface of contact was already implemented in the FORGE2^® software. An implicit scheme is proposed and compared with the existing scheme. The advantages of this scheme are numerically shown by solving some examples. It is observed that the volume losses are reduced. This makes it possible to use larger time steps or increase the computational accuracy.

Contact problems are among the most difficult ones in mechanics. Due to its practical importance, the problem has been receiving extensive research work over the years. The finite element method has been widely used to solve contact problems with various grades of complexity. Great progress has been made on both theoretical studies and engineering applications. This paper reviews some of the main developments in contact theories and finite element solution techniques for static contact problems. Classical and variational formulations of the problem are first given and then finite element solution techniques are reviewed. Available constraint methods, friction laws and contact searching algorithms are also briefly described. At the end of the paper, a bibliography is included, listing about seven hundred papers which are related to static contact problems and have been published in various journals and conference proceedings from 1976.

In the first part of this series of papers on the combined finite/discrete element simulation of shot peening processes, different contact interaction laws for 2D cases are extensively studied with special attention given to the proper selection of the parameter values involved, which is one of the key issues for successful direct simulation. In addition, computational issues including contact forces, partial contact, energy dissipation, and rheological representation are addressed. Numerical examples for a single shot impact system simulated by the coupled finite/discrete element method using different interaction laws are provided to verify the proposed approaches. The results are also compared with those obtained by using only finite element methods. Findings obtained by performing 2D simulations will, in the subsequent article, be used in realistic computational simulations of 3D shot peening processes.

New contact boundary modelling is achieved with a basic set of 2 and 3 dimension contact primitives. Contact constraints are originally introduced in the variational equations and associated Newton—Raphson scheme via an external penalty formulation using primitive equations. Consequently, penalty part of external load vector and tangent stiffness matrices are developed for all contact primitives. In this way, contact prescribed boundary displacements are also taken into account. Contact treatment is then completed with Newton—Raphson elements for elastic and plastic regularized friction constitutive models. In this paper, the process is extended to elastoplastic models. Finally, we propose a self acting procedure with contact algorithms (interiority, sliding and contact loss) and related subroutines for implementation in finite element framework. We illustrate these developments by means of two‐dimensional open die forging and three‐dimensional plate coining typical benchmarks with reference to bulk elastoplastic and viscoplastic constitutive models.

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A numerical procedure is described for the elastic—plastic finite element analysis of crack propagation with branching. Constraint equations are used to model crack closing and sliding. Constraint conditions are imposed by using a penalty method for the self‐similar crack propagation and an elimination method for the off‐axis propagation. The contact condition is examined during plasticity iterations. The use of multiple constraints at the crack branching point to determine the mode of contact is discussed in detail. The method is then applied to (i) the self‐similar crack growth in a single‐edge notch specimen, (ii) the self‐similar propagation followed by interfacial splitting in a center‐cracked 0° composite plate, and (iii) the bifurcation of a crack in a compact tension specimen.

The purpose of this paper is to focus on the creation of an isothermal elastic ring-liner model to highlight, through stresses, the occurrence of the plastic deformation in certain crank angles under extreme dry lubrication conditions.

The stresses that are exported from this analysis are pointing out not only the necessity for an elastoplastic model to be created, but also the importance of predicting the correct friction coefficient, as pointed out by both the contact surface stress and that in depth of the two bodies in contact.

The comparison between the finite element model and the adhesion mathematical model of Johnson, Kendall and Roberts seals the importance to calculate the interaction forces, acting on the common solid surface, in the pursuit of defining a propriate contact patch. Additionally, a three-dimensional ring model is built, highlighting the importance of the modeling surface’s micro asperities for a solid stress analysis. Also, numerical experiments are conducted, in contact with the cylinder and a piston ring made of an iron alloy and of two different plating materials, such us Chromium (Cr) and Chromium‒Nickel Alloy (CrN). The ability to calculate the stress concentration factor is also described.

A three-dimensional ring model is built, highlighting the importance of the modeling surface’s micro asperities for a solid stress analysis. Also, numerical experiments are conducted, in contact with the cylinder and a piston ring made of an iron alloy and of two different plating materials, such us Chromium (Cr) and Chromium‒Nickel Alloy (CrN). The ability to calculate the stress concentration factor is also described.

To present a numerical model of squeeze casting process.

The modelling consists of two parts, namely, the mould filling and the subsequent thermal stress analysis during and after solidification. Mould filling is described by the Navier‐Stokes equations discretized using the Galerkin finite element method. The free surface is followed using a front tracking procedure. A thermal stress analysis is carried out, assuming that a coupling exists between the thermal problem and the mechanical one. The mechanical problem is described as an elasto‐visco‐plastic formulation in an updated Lagrangian frame. A microstructural solidification model is also incorporated for the mould filling and thermal stress analysis. The thermal problem is solved using enthalpy method.

During the mould‐filling process a quasi‐static arbitrary Lagrangian‐Eulerian (ALE) approach and a microstructural solidification model were found to be applicable. For the case of the thermal stress analysis the influence of gap closure, effect of initial stresses (geometric nonlinearity), large voids and good performance of a microstructural model have been demonstrated.

The model can also be applied to the simulation of indirect castings. The final goal of the model is the ability to simulate the forming of the material after mould filling and during the solidification of the material. This is possible to achieve by applying arbitrary contact surfaces due to the sliding movement of the cast versus the punch and die.

The presented model can be used in engineering practice, as it incorporates selected second‐order effects which may influence the performance of the cast.

During the mould‐filling procedure a quasi‐static ALE approach has been applied to SQC processes and found to be generally applicable. A microstructural solidification model was applied which has been used for the thermal stress analysis only. During the thermal stress analysis the influence of gap closure and initial stresses (geometric nonlinearity) has been demonstrated.