Frictionless Unilateral Contact Research Articles

Quadratic Discontinuous Galerkin Finite Element Methods for the Unilateral Contact Problem

Abstract In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori error estimator is addressed. The suitable construction of the discrete Lagrange multiplier 𝝀 𝒉 {\boldsymbol{\lambda_{h}}} and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution 𝒖 {\boldsymbol{u}} . Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.

Self-consistent approximations for the tangential-displacement correction to the incremental indentation stiffness

An axisymmetric problem of unilateral frictionless contact between a paraboloidal or conical indenter and a transversely isotropic elastic half-space is considered in the refined formulation by accounting for the tangential (radial) displacements of the surface points of the elastic body. Using the idea of self-consistent approximation, a system of two coupled relations for the main contact variables (contact force, indenter displacement, and contact radius) is derived. The concept of the true contact radius is discussed, and it has been argued that the incremental stiffness relation should be expressed in its terms. Correction factors for the force–displacement relation and the incremental indentation stiffness (as a function of the true contact radius) are evaluated in explicit form.

Adaptive Quadratic Finite Element Method for the Unilateral Contact Problem

In this paper, we present and analyze a posteriori error estimates in the energy norm of a quadratic finite element method for the frictionless unilateral contact problem. The reliability and the efficiency of a posteriori error estimator is discussed. The suitable decomposition of the discrete space $${\mathbf{V^h_0}}$$ and a discrete space $${\mathbf{Q^h}}$$ , where the discrete counterpart of the contact force density is defined, play crucial role in deriving a posteriori error estimates. Numerical results are presented exhibiting the reliability and the efficiency of the proposed error estimator.

Homogenization and numerical algorithms for two-scale modeling of porous media with self-contact in micropores

The paper presents two-scale numerical algorithms for stress–strain analysis of porous media featured by self-contact at pore level. The porosity is constituted as a periodic lattice generated by a representative cell consisting of elastic skeleton and a void pore. Unilateral frictionless contact is considered between opposing surfaces of the pore. For the homogenized model derived in our previous work, we justify incremental formulations and propose several variants of two-scale algorithms which commute iteratively solving of the micro- and the macro-level contact subproblems. A dual formulation which takes advantage of the assumed microstructure periodicity and a small deformation framework, is derived for the contact problems at the micro-level. This enables to apply the semi-smooth Newton method. For the global, macrolevel step two alternatives are tested; one relying on a frozen contact identified at the microlevel, the other based on a reduced contact associated with boundaries of contact sets. Numerical examples of 2D deforming structures are presented as a proof of the concept.

Pointwise A Posteriori Error Control of Discontinuous Galerkin Methods for Unilateral Contact Problems

Abstract In this article, we derive a reliable and efficient a posteriori error estimator in the supremum norm for a class of discontinuous Galerkin (DG) methods for the frictionless unilateral contact problem between two elastic bodies. The proposed error estimator generalizes the basic residual type estimators for the linear problems in linear elasticity taking into account the nonlinearity on a part of the boundary. The analysis hinges on the super- and sub-solutions constructed by modifying the discrete solution appropriately, and it is carried out in a unified manner which holds for several DG methods. The terms arising from the contact stresses in the error estimator vanish on the discrete full contact set. We illustrate the performance of the proposed error estimator via several numerical experiments in two dimensions.

Existence and Convergence Results for an Elastic Frictional Contact Problem with Nonmonotone Subdifferential Boundary Conditions

The goal of this paper is to study a mathematical model of a nonlinear static frictional contact problem in elasticity with the mixed boundary conditions described by a combination of the Signorini unilateral frictionless contact condition, and nonmonotone multivalued contact, and friction laws of subdifferential form. First, under suitable assumptions, we deliver the weak formulation of the contact model, which is an elliptic system with Lagrange multipliers, and which consists of a hemivariational inequality and a variational inequality. Then, we prove the solvability of the contact problem. Finally, employing the notion of H-convergence of nonlinear elasticity tensors, we provide a result on the convergence of solutions under perturbations which appear in the elasticity operator, body forces, and surface tractions.

Контактная задача об изгибе двух балок с внутренним шарниром

The joint bending of two Bernoulli–Euler’s beams is considered. Each beam has one end fixed and the other free. The beams have the different lengths and thicknesses. The long beam is loaded by the concentrated force. This beam is composite as it includes the internal joint. There is the frictionless unilateral contact between the beams. The elastic lines of the beams are to be found. This problem is reduced to finding of the density of forces of interaction between the beams and the constant that describes the unknown term in the displacement of the unrestrained part of the composite beam. The mathematical formulation of this contact problem is propounded. The density is assumed to be the sum of piecewise continuous function and delta-functions describing the concentrated forces. The uniqueness of the solution of the problem is proved and the analytical solution is constructed. Two possible contact patterns are found out. The former is contact at one point at the end of the short beam. The latter is contact at the same point and at one more point located at the unrestrained part of the composite beam. The coordinate of this point is the root of the cubic equation. The obtained analytical solution is used for the optimization of the structure. The optimization problem is to find the beams thicknesses that minimize the maximum stress for the given loading, beams lengths and the overall deflection. This problem is solved numerically for some values of the given parameters. The hypothesis of the equal-stressed optimum structure is set up on the basis of the numerical results. This hypothesis enables to construct the analytical solution of the optimization problem.

Depth-sensing indentation of spherical particles on corrugated substrates — An asymptotic model

An AFM-based nanoindentation of a spherical particle absorbed on a corrugated surface of rigid substrate is considered in the framework of continuum mechanics. The corresponding problem of unilateral frictionless contact with multiple contact zones is formulated using isotropic infinitesimal elasticity. Both the AFM probe surface and the surfaces of rigid supports, which are exposed to contact with the elastic sphere, are assumed to be spherical. The method of matched asymptotic expansions is utilized for constructing the leading order asymptotic model of multiple contact. Explicit formulas are derived for the force-displacement relationship and the incremental indentation stiffness. It is shown that the difference between the one-point contact and the multi-point contact (e.g., with two and three local supports) is found to be significant. The case of tight hexagonal packing of the support spheres is considered in detail and illustrated by a comparison with the Hertzian theory of local contact.

Homogenization and numerical modelling of poroelastic materials with self-contact in the microstructure

We present a two-scale homogenization-based computational model of porous elastic materials subject to external loads inducing the self-contact interaction at the pore level. Microstructures under consideration are constituted as periodic lattices generated by a representative cell consisting of a solid skeleton and a pore. On its surface, the unilateral frictionless contact appears when the porous material is deformed. We focus on microstructures with rigid inclusions whereby the contact process involves opposing surfaces on the rigid and the compliant skeleton parts. A macroscopic model is derived using the periodic unfolding homogenization and the method of oscillating test functions. An efficient algorithm for the two-scale computational analysis is proposed for the numerical model obtained using the finite element discretization of the homogenized model. For this, a sequential linearization of the two-scale elasticity problem leads to the consistent effective elasticity tensor yielding consistent stiffness matrices of the macroscopic incremental formulation. The micro-level contact problem attains the form of a nonsmooth equation solved using the semi-smooth Newton method without any regularization, or problem relaxation. Numerical examples of two-dimensional deforming structures are presented as a proof of the concept. The proposed modelling approach can be extended to treat self-contact in structures subject to finite deformation.

Robust optimization of structures subjected to frictionless unilateral contact with uncertain initial gaps

Robust optimization of structures subjected to frictionless unilateral contact with uncertain initial gaps

Tykhonov well-posedness of a frictionless unilateral contact problem

We consider a frictionless contact problem, Problem , for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem . Then we consider a perturbation of Problem , which could be frictional, governed by a small parameter . This perturbation leads in a natural way to a family of sets . We prove that Problem is well-posed in the sense of Tykhonov with respect to the family . The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem . Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.

Exploiting Lagrange duality for topology optimizationwith frictionless unilateral contact

This paper presents tractable reformulations of topology optimization problems of structures subject to frictionless unilateral contact conditions. Specifically, weconsider stiffness maximization problems of trusses and continua. Based on the Lagrange duality theory, we derive formulations that do not involve complementarity constraints. It is often that a structural optimization problem with contact conditions is formulated as a mathematical programming problem with complementarityconstraints (MPCC problem). However, MPCC usually requires special treatment for numerical solution, because it does not satisfy standard constraint qualifications. In contrast, to the formulation presented in this paper, we can apply standard optimization approaches. Numerical experiments on trusses and continua are performed to examine efficiency of the proposed approach.

Well-posedness of a class of generalized mixed hemivariational–variational inequalities

The purpose of this paper is to investigate a class of generalized mixed hemivariational–variational inequalities of elliptic type in a Banach space. A well-posedness result for the inequality is obtained, including existence, uniqueness, and stability of solution. The approach is based on a set-valued fixed point theorem combined with the theory of nonsmooth analysis. An elastic contact problem with the constitutive law involving a convex subdifferential inclusion is considered as an illustrative application, in which the contact boundary conditions are described by a unilateral frictionless contact condition, and a bilateral multivalued frictional contact law.

Boundary Integral Formulation of Frictionless Contact Problems Based on an Energetic Approach and Its Numerical Implementation by the Collocation BEM

A unified methodology to solve problems of frictionless unilateral contact as well as adhesive contact between linear elastic solids is presented. This methodology is based on energetic principles and is casted to a minimization problem of the total potential energy. Appropriate boundary integral forms of the energy are defined and the quadratic problem form of the contact problem is proposed. The problem is solved by the collocation boundary element method (BEM). To solve the quadratic problem two algorithms are developed, both being variants of the well-known conjugate gradient algorithm. The difference between them is given by an explicit construction or not of the quadratic-problem matrix. This matrix has the same physical meaning as the stiffness matrix commonly used in the context of the finite element method (FEM). Both symmetric and non-symmetric formulations of this matrix are presented and discussed, showing that the non-symmetric one provides more accurate results. The present procedure, in addition to its interest by itself, can also be extended to problems where dissipative phenomena take place such as friction, damage and plasticity. Elements of the numerical implementation are briefly presented and the numerical solution of some standard problems of frictionless contact are given and compared to those obtained by other well-known BEM and FEM procedures for contact problems.

The Local Average Contact (LAC) method

The Local Average Contact (LAC) method allows the handling of nonmatching meshes in an easy way by averaging locally the interpenetration between the contacting bodies. In this paper we consider the frictionless unilateral contact problem and several numerical experiments involving two and three-dimensional bodies discretized with various linear and quadratic finite elements. We also present a convergence analysis of the method in the geometrical nonconforming case in which the boundary points of the candidate contact areas do not coincide.

A mixed variational formulation for a class of contact problems in viscoelasticity

We consider a deformable body in frictionless unilateral contact with a moving rigid obstacle. The material is described by a viscoelastic law with short memory, and the contact is modeled by a Signorini condition with a time-dependent gap. The existence and uniqueness results for a weak formulation based on a Lagrange multipliers approach are provided. Furthermore, we discuss an efficient algorithm approximating the weak solution for the more general case of a two-body contact problem including friction. In order to illustrate the theory we present two numerical examples in 3D.

Subdifferential inclusions for stress formulations of unilateral contact problems

We consider two classes of inclusions involving subdifferential operators, both in the sense of Clarke and in the sense of convex analysis. An inclusion that belongs to the first class is stationary while an inclusion that belongs to the second class is history-dependent. For each class, we prove existence and uniqueness of the solution. The proofs are based on arguments of pseudomonotonicity and fixed points in reflexive Banach spaces. Then we consider two mathematical models that describe the frictionless unilateral contact of a deformable body with a foundation. The constitutive law of the material is expressed in terms of a subdifferential of a nonconvex potential function and, in the second model, involves a memory term. For each model, we list assumptions on the data and derive a variational formulation, expressed in terms of a multivalued variational inequality for the stress tensor. Then we use our abstract existence and uniqueness results on the subdifferential inclusions and prove the unique weak solvability of each contact model. We end this paper with some examples of one-dimensional constitutive laws for which our results can be applied.

Rate-independent elastoplasticity at finite strains and its numerical approximation

Gradient plasticity at large strains with kinematic hardening is analyzed as quasistatic rate-independent evolution. The energy functional with a frame-indifferent polyconvex energy density and the dissipation is approximated numerically by finite elements and implicit time discretization, such that a computationally implementable scheme is obtained. The nonself-penetration as well as a possible frictionless unilateral contact is considered and approximated numerically by a suitable penalization method which keeps polyconvexity and simultaneously bypasses the Lavrentiev phenomenon. The main result concerns the convergence of the numerical scheme toward energetic solutions. In the case of incompressible plasticity and of nonsimple materials, where the energy depends on the second derivative of the deformation, we derive an explicit stability criterion for convergence relating the spatial discretization and the penalizations.

An effective asymptotic method in the axisymmetric frictionless contact problem for an elastic layer of finite thickness

A brief review of asymptotic methods to deal with frictionless unilateral contact problems for an elastic layer of finite thickness is presented. Under the assumption that the contact radius is small with respect to the layer thickness, an effective asymptotic method is suggested for solving the unilateral contact problem with a priori unknown contact radius. A specific feature of the method is that the construction of an asymptotic approximation is reduced to a linear algebraic system with respect to integral characteristics (polymoments) of the contact pressure. As an example, the sixth‐order asymptotic model has been written out. Copyright © 2015 John Wiley & Sons, Ltd.

Enriched beam model for slender prismatic solids in contact with a rigid foundation

The present paper deals with an enhanced beam model for semi-infinite prismatic solid in contact with a frictionless rigid support. To circumvent the solution of a plane elasticity problem, we propose an extension of the refined beam theory of Baluch et al. (1985) [6] which accounts for the contact interactions with a frictionless rigid foundation. We combine in a certain way two-dimensional elasticity (plane stress) and the classical engineering beam theory with the aim to derive a refined beam model which takes into account the influence of transverse normal strain as well as contact conditions. The results obtained with the proposed analytical model are compared against semi-analytical solutions for receding contact problems and against refined F.E. analyzes. It is shown that the proposed model provides accurate results and is an efficient tool to tackle problems involving semi-infinite beams in frictionless unilateral contact with a rigid foundation.