A geometric interpretation of frictional contact mechanics

The work-conjugacy of pressures and tangential tractions with so-called “gap”and “stick” constraints is deduced in order to delineate a rigid-plastic model of a frictional interface. This is accomplished by pursuing a differential-geometric view of the two surfaces that comprise the frictional interface. Given that contact is described in the current configuration, Lie derivatives are shown to be the natural means of establishing the work-conjugacy between tractions and constraints.     

Level set based multi-scale methods for large deformation contact problems

We consider the numerical simulation of contact problems in elasticity with large deformations. The non-penetration condition is described by means of a signed distance function to the obstacle's boundary. Techniques from level set methods allow for an appropriate numerical approximation of the signed distance function preserving its non-smooth character. The emerging non-convex optimization problem subject to non-smooth inequality constraints is solved by a non-smooth multiscale SQP method in combination with a non-smooth multigrid method as interior solver. Several examples in three space dimensions including applications in biomechanics illustrate the capability of our methods.     

Computational implementation of stress integration in FE analysis of elasto-plastic large deformation problems

In this paper, we compare different numerical implementation algorithms for the rate type constitutive equation and present an integration scheme based on the physical meaning of the stress. Numerical implementation of various schemes is investigated in conjunction with the return mapping algorithm and the conditions to maintain plastic consistency. Jaumann and Truesdell rates are taken as the objective stress rates in the constitutive equation. An alternative numerical treatment for rate of deformation tensor D_ij is presented and is shown to maintain incremental objectivity. Numerical examples included a single element under rigid body rotation, a necking bifurcation of a bar in tension and a punch indentation process. It is shown that the use of Truesdell stress rate with specific numerical integration procedure gives more accurate results than other procedures presented.     

Qualitative analysis of 3D elastostatic contact problems with orthotropic Coulomb friction and solution-dependent coefficients of friction

This paper analyzes a discrete form of 3D contact problems with local orthotropic Coulomb friction and coefficients of friction which may depend on the solution itself. The analysis is based on the fixed-point reformulation of the original problem. Conditions guaranteeing the existence and uniqueness of discrete solutions are established. Finally, numerical results of a model example are presented.     

Finite element simulation for dynamic large elastoplastic deformation in metal powder forming

In this paper, a transient dynamic analysis of the powder compaction process is simulated by a large displacement finite element method based on a total and updated Lagrangian formulation. A combination of the Mohr–Coulomb and elliptical yield cap model, which reflects the stress state and degree of densification, is applied to describe the constitutive model of powder materials. A Coulomb friction law and a plasticity theory of friction in the context of an interface element formulation are employed in the constitutive modelling of the frictional behaviour between the die and powder. Finally, the powder behaviour during the compaction of a plain bush, a rotational flanged and a shaped tip component are analysed numerically. It is shown that the updated Lagrangian formulation, using a combination of the Mohr–Coulomb and elliptical cap model, can be effective in simulating metal powder compaction.     

Homogenization of a viscoelastic matrix in linear frictional contact

The paper is devoted to study of acoustic wave propagation in a partially consolidated composite material containing loose particles. Friction of particles against the consolidated part of the material causes mechanical energy dissipation. This situation is modelled by assuming that the medium has a periodic microstructure changing rapidly on the small scale ε. Each of the periodic microscopic cells is composed of a viscoelastic matrix containing a rigid particle in frictional contact with the matrix. We use the methods of two‐scale convergence to obtain effective acoustic equations for the homogenized material. The effective equations are history‐dependent and contain the body force term, reminiscent of the well‐known Stokes drag force. Copyright © 2004 John Wiley & Sons, Ltd.     

A finite strain contact model for mixed lubricated surfaces in forming processes

For an accurate simulation of forming processes, it is of paramount importance to model the different lubrication regimes that can develop at the contact interface. These might vary from zone to zone of the forming piece, and from one regime to another, resulting in forces of different nature and magnitude. In these cases, the use of the classical Coulomb friction law will be clearly not sufficient to capture, in a suitable manner, the variety of forces applied on the forming piece.     

A new element for analyzing large deformation of thin Naghdi shell model. Part II: Plastic

In this paper a new element is developed that is based on Cosserat theory. In the finite element implementation of Cosserat theory shear locking can occur, especially for very thin shells. In the present investigation the director vector is constrained to remain perpendicular to the mid surface during deformation. It will be shown that this constraint yields accurate results in very large deformation of thin shells also the rate of convergency is very good. For plastic formulation, the model introduced by Simo is used and it has been reduced for constrained director vector and the consistent elasto-plastic tangent moduli is extracted for finite element solution. This model includes both kinematic and isotropic hardening. For numerical investigations an isoparametric nine node element is employed then by linearization of the principle of virtual work, material and geometric stiffness matrices are extracted. The validity and the accuracy of the proposed element is illustrated by the numerical examples and the results are compared with those available in the literature.     

A homotopy analysis solution to large deformation of beams under static arbitrary distributed load

In this article, homotopy analysis method (HAM) is employed to investigate non-linear large deformation of Euler–Bernoulli beams subjected to an arbitrary distributed load. Constitutive equations of the problem are obtained. It is assumed that the length of the beam remains constant after applying external loads. Different auxiliary parameters and functions of the HAM and the extra auxiliary parameter, which is applied to initial guess of the solution, are employed to procure better convergence rate of the solution. The results of the solution are obtained for two different examples including constant cross sectional beam subjected to constant distributed load and periodic distributed load. Special base functions, orthogonal polynomials e.g. Chebyshev expansion, are employed as a tool to improve the convergence of the solution. The general solution, presented in this paper, can be used to attain the solution of the beam under arbitrary distributed load and flexural stiffness. Ultimately, it is shown that small deformation theory overestimates different quantities such as bending moment, shear force, etc. for large deflection of the beams in comparison with large deformation theory. Finally, it is concluded that solution of small deformation theory is far from reality for large deflection of straight Euler–Bernoulli beams.     

A new element for analyzing large deformation of thin Naghdi shell model. Part 1: Elastic

One of the best approaches for modeling large deformation of shells is the Cosserat surface. However, the finite-element implementation of this model suffers from membrane and shear locking, especially for very thin shells. The basic assumption of this theory is that the mid-surface of the shell is regarded as a Cosserat surface with one inextensible director. In this paper, it is shown that by constraining the director vector normal to the mid-surface, besides very good and accurate results, shear locking is also eliminated. This constraint is in fact a limiting analysis of the Cosserat theory in which Kirichhoff’s hypothesis is enforced. Numerical solution is performed using nine-node isoparametric element. The principal of virtual work is used to obtain the weak form of the governing differential equations and the material and geometric stiffness matrices are derived through a linearization process. The validity and the accuracy of the method are illustrated by numerical examples.     

On a solvability of hydro-mechanical problem based on contact problem with visco-plastic friction in Bingham rheology

This paper deals with the solvability and numerical solution of contact problem with a local visco-plastic friction in the visco-plastic Bingham rheology. The model problem discussed represents a simple hydro-mechanical model of the global project on a security of regions endangered by great hurricanes and deluges. The main goal of the idea of this project is to connect the climatic observations and the corresponding climatic models with the thermo-hydro-dynamic and the thermo-hydro-mechanic models, with the possibility to estimate future destructions of such endangered regions with landslides of unstable slopes. The investigated mathematical model is based on the visco-plastic Bingham rheology. The numerical approach is based on the semi-implicit scheme in time and the FE approximation in space. The algorithm is shortly discussed.     

Computable upper bounds for the constants in Poincaré‐type inequalities for fields of bounded deformation

We collect various Poincaré‐type inequalities valid for fields of bounded deformation and give explicit upper bounds for the constants being involved. Copyright © 2011 John Wiley & Sons, Ltd.