Anders Klarbring

Formulation and Comparison of Algorithms for Frictional Contact Problems

This paper presents two algorithms for solving the discrete, quasi-static, small-displacement, linear elastic, contact problem with Coulomb friction. The algorithms are adoptions of a Newton method for solving B-differentiable equations and an interior point method for solving smooth, constrained equations. For the application of the former method, the contact problem is formulated as a system of B-differentiable equations involving the projection operator onto sets with simple structure; for the application of the latter method, the contact problem is formulated as a system of smooth equations involving complementarity conditions and with the non-negativity of variables treated as constraints. The two algorithms are numerically tested for two-dimensional problems containing up to 100 contact nodes and up to 100 time increments. Results show that at the present stage of development, the Newton method is superior both in robustness and speed. Additional comparison is made with a commercial finite element code.

Derivation of a model of adhesively bonded joints by the asymptotic expansion method

Abstract The bonding of two solids by a thin adhesive layer is considered. The asymptotic expansion method is used to derive a model where the displacements vary linearly and the stresses are constant through the thickness of the adhesive. Thus, the adhesive layer is represented as a material surface. Several variational formulations of the model are given and the existence of a unique solution is shown. The occurrence and nature of a boundary layer is also discussed.

An introduction to structural optimization

Mechanical and structural engineers have always strived to make as efficient use of material as possible, e.g. by making structures as light as possible yet able to carry the loads subjected to the ...

A mathematical programming approach to three-dimensional contact problems with friction

Abstract The present study is concerned with the numerical treatment of linear elastic contact problems with friction. Making the restriction that the contact surface is constant during the loading history, we obtain a variational inequality formulation, which justifies a finite element approximation. The finitedimensional problem so obtained is shown to be equivalent to a problem of mathematical programming—namely, a parametric linear complementarity problem (LCP) involving derivatives. A solution procedure for such problems has previously been suggested. In this paper, the conditions under which this procedure defines a unique solution map, which describes the evolution of contact stresses and displacements for a prescribed load history, are given. As an application of the theory presented, the indentation of a halfspace by a flat-ended square punch is considered. Both the loading cases of a vertical displacement of the punch that increases monotonically and the gradual removal of the same displacement are considered.

Derivation and analysis of a generalized standard model for contact, friction and wear

A model for mechanical contact including friction, wear and heat generation is proposed. By defining an internal state variable for the wear process, a generalized standard model for contact, friction and wear is derived from the principle of virtual power and the fundamental laws of thermodynamics. Within the frame of the generalized standard model some specific constitutive models are presented. For instance, a free energy corresponding to an extension of Signorinis unilateral contact conditions to include the wear process at the interface and having a linear tangential compliance between the relative tangential displacement and the tangential contact traction is suggested. Furthermore, a dual pseudo-potential with a friction and wear limit criterion in agreement with Coulombs law of friction and Archards law of wear is given. In order to study existence and uniqueness questions, this pair of free energy and dual pseudo-potential is analysed in a one point elastic quasi-static contact problem with two degrees of freedom and thermal effects neglected. The so-called rate problem is solved.

A mathematical programming approach to contact problems with friction and varying contact surface

Abstract The numerical treatment of contact problems with friction is considered. Starting from the algebraic stiffness equation of two discretized linear elastic bodies and taking the contact and friction laws into account we derive a new type of Linear Complementarity Problem (LCP). A principal pivot algorithm for this problem is suggested. It is then shown how the evolution of displacements and contact stresses for a prescribed load history is obtained. As an application of the method we solve a three-dimensional punch indentation problem.

Optimization of Structures in Unilateral Contact

This article gives a review of optimization of structures in mechanical contact. Emphasis is put on linear elastic structures in frictionless contact. In particular, for optimization problems where an energy objective is used, a unified framework is given in parallel with the review. Papers related to optimal control of variational inequalities or dealing with pure sensitivity analysis are treated in less detail. Problems involving friction are also reviewed at a less detailed level. It is explained why structural optimization problems involving contact cannot be treated within classical smooth optimization theory and how they relate to modern fields such as nonsmooth optimization and mathematical programs with equilibrium constraints (MPECs). Throughout the article, discrete and continuous problems are treated in parallel. This review article includes 106 references.

Asymptotic Modelling of Adhesive Joints

An asymptotic approach is presented for modelling of adhesive joints in a layered structure. We consider the case when two elastic layers are bonded together by a thin and soft layer of glue. Differential equations for the displacements (including the displacement jump across the thin interface layer) in a compound thin beam are derived.

Thermoelastic frictional contact problems: Modelling, finite element approximation and numerical realization

Abstract In this paper frictional contact including frictional heat generation and heat transfer across the contact interface is studied. A model for the behaviour of the contact interface is proposed and combined with the conventional equations of linear thermoelasticity for the bulk material of the contacting bodies. A finite element discretization in space and a finite difference discretization in time are introduced, and an iterative algorithm is proposed to deal with the coupled thermal-mechanical nature of the problem. Finally, several numerical examples are given.

A mechanochemical 3D continuum model for smooth muscle contraction under finite strains

This paper presents a modelling framework in which the mechanochemical properties of smooth muscle cells may be studied. The activation of smooth muscles is considered in a three-dimensional continuum model which is key to realistically capture the function of hollow organs such as blood vessels. On the basis of a general thermodynamical framework the mechanical and chemical phases are specialized in order to quantify the coupled mechanochemical process. A free-energy function is proposed as the sum of a mechanical energy stored in the passive tissue, a coupling between the mechanical and chemical kinetics and an energy related purely to the chemical kinetics and the calcium ion concentration. For the chemical phase it is shown that the cross-bridge model of Hai and Murphy [1988. Am. J. Physiol. Cell Physiol. 254, C99-C106] is included in the developed evolution law as a special case. In order to show the specific features and the potential of the proposed continuum model a uniaxial extension test of a tissue strip is analysed in detail and the related kinematics and stress-stretch relations are derived. Parameter studies point to coupling phenomena; in particular the tissue response is analysed in terms of the calcium ion level. The model for smooth muscle contraction may significantly contribute to current modelling efforts of smooth muscle tissue responses.