P.D. Panagiotopoulos

Comparison of two methods for the solution of a class of nonconvex energy problems using convex minimization algorithms

Nonmonotone, possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying (super)potentials the problems generally have nonunique solutions (stable or unstable). In this paper we propose two methods for the solution of the hemivariational inequality problem. The first method is based on the decomposition of the nonconvex superpotential into convex constituents. The second one uses an iterative scheme in order to approximate the hemivariational inequality problem with a sequence of variational inequality problems. Both methods are based on the solution of convex subproblems and constitute an effective, reliable and versatile family of numerical algorithms for large scale hemivariational inequalities. Finally, the two methods are applied to solve the same problem and the obtained results are compared.

Numerical treatment of problems involving nonmonotone boundary or stress-strain laws

Abstract In order to describe the softening behavior of the materials, nonmonotone possible multivalued laws have been recently introduced. These laws are derived by nonconvex, generally nonsmooth energy functions called superpotentials that give rise to hemivariational inequalities. Due to the lack of convexity and the nonsmoothness of the underlying superpotentials these problems have generally nonunique solutions. On the other hand, problems involving monotone laws lead to variational inequalities that can be easily treated using modern convex minimization algorithms. The present paper proposes a new method for the solution of the nonmonotone problem by approximating it using monotone ones. The proposed method finds its justification in the approximation of a hemivariational inequality by a sequence of variational inequalities. This approach leads to effective reliable and versatile numerical alogrithms for large-scale hemivariational inequalities. The numerical method proposed is illustrated using appropriate examples.

Unilateral contact, friction and related interactions in cracks. The direct boundary integral method

Abstract The paper presents a theory for the study of cracks having a given geometry by taking into account all types of actions of monotone type like unilateral contact and friction phenomena between the two crack sides. The arising problems are of a non-classical nature, due to the interface conditions expressed in terms of non-differentiable convex superpotentials. The direct B.I.E.M. is extended appropriately in order to treat this type of problem. The developed method is illustrated by a numerical example concerning the calculation of stress intensity factors under the unilateral contact and friction interface conditions.

Fractal Geometry in Contact Mechanics and Numerical Applications

The contribution to the present volume deals with the study of the influence of fractal geometry on contact problems. After a short presentation of the new mathematical tools and methods used for the correct consideration of the fractal geometry we study unilateral contact and friction problems, adhesive contact problems in interfaces of fractal geometry and finally crack problems of fractal geometry. Numerical applications illustrate the uheory. This contribution contains also an advanced mathematical section concerning the nature of the forces on a fractal boundary.

On the numerical treatment of the delamination problem in laminated composites under cleavage loading

Abstract A method for the effective numerical treatment of the delamination problem in laminated composites under cleavage loading is herein proposed. The interlaminar interface mechanical behaviour is described by means of the so-called complete laws which are non-monotone and possibly multivalued force/ displacement laws including jumps (or in general, decreasing branches) corresponding to the discontinuous strength reduction. These complete laws that take into account the development of delamination phenomena in a quasistatic way are derived by non-convex energy functions, called delamination superpotentials which in turn, lead to the formulation of the principle of virtual work for the laminated composite in a hemivariational inequality form and to the generalisation of the principle of minimum potential energy as a substationarity principle. Applying an appropriate finite element discretisation scheme to the laminated composite, the respective discrete problem is formulated which describes the response of the structure taking into account the development of the delamination phenomenon. The numerical treatment of the latter problem is successfully performed by applying a new algorithm that approximates the nonmonotone law by a sequence of monotone ones. The performed numerical applications presented in the last part of the paper and several analogous numerical experiments exhibit very good convergence properties.

Friction laws of fractal type and the corresponding contact problems

Abstract The aim of the present paper is the study of contact problems allowing for debonding, i.e. unilateral contact problems, with fractal type friction laws. The iterative method proposed here is based on the description of a fractal as the attractor of a contractive operator T : C 0 → C 0 . Using this method, the fractal friction law is approximated by a sequence of nonmonotone classical C 0 -curves. The numerical treatment of each arizing nonmonotone problem is accomplished by an advanced solution method which approximates the nonmonotone stress—strain problem by a sequence of monotone problems.

THE FEM AND BEM FOR FRACTAL BOUNDARIES AND INTERFACES. APPLICATIONS TO UNILATERAL PROBLEMS

Abstract The scope of the present paper is the study of structures involving boundaries and interfaces of fractal geometry. The geometry is analysed here by a sequence of classical geometry problems formulated by means of C 0 and C 1 -interpolating functions. These approximations of the fractal geometry are combined with the FEM and the BEM in order to calculate the stress and displacement fields in fractal structures.

On the fractal fracture in brittle structures : Numerical approach

Abstract The aim of the present paper is to study the influence of the irregular fracture of structures, consisting of brittle materials, on the arising stress and strain fields. This type of fracture, depends strongly on the microstructure of the body and on the loading applied on the structure. In this paper brittle fracture phenomena are modelled by means of fractal geometry, which describes with great accuracy the arising fracture patterns. Not only the fractality of the form of the arising cracks but also the fractality of the induced friction law at the interface is considered here. This fact takes into account the radomness of the interface asperities causing the friction forces. According to the fractal model introduced in this paper, the fractal interface of a crack or the fractal stress-strain or reaction-displacement law is considered to be the unique ‘fixed point’ of a given Iterative Function System (I.F.S.) or the graph of a fractal interpolation function (F.I.F.). On the fractal interface nonmonotone contact and friction conditions are assumed to hold. The methods developed here extend the classical FEM to the case of fractal interfaces and to the case of fractal nonmonotone stress-strain laws. Numerical applications from the static analysis of brittle structures with prescribed crack geometry and crack interface laws are included in order to illustrate the theory.

A hemivariational inequality approach to the rock interface problem

Abstract In the present paper a method is presented for the analysis of rock joints and interfaces. The mechanical response of rock joints and interfaces involves phenomena of nonclassic nature as debonding and slip along the interfaces, which can be completely described by nonmonotone possibly multivalued (e.g. sawtooth) phenomenological constitutive laws, the so-called complete laws, which result from the respective experimental stress-stain diagrams. Due to the lack of monotonicity of these laws, the rock interface problem cannot be studied by means of the classical variational methods and therefore, the need for a generalization of the classical variational theories to cover also such constitutive laws arises. This can be achieved by means of the notion of generalized gradient which was already applied to the solution of a large number of structural analysis problems involving nonconvex and nondifferentiable energy functions. First the interface behaviour is modelled through nonconvex superpotentials describing the debonding and/or stick/slip phenomena. Then the hemivariational inequalities are formulated and some methods for the numerical treatment are proposed.

Saddle-supported pipelines: Computation of the pressure distribution on the pipe-saddle interface

Abstract In the present paper a new method is proposed which concerns the accurate computation of the unilateral contact forces and displacements between a circular cylindrical shell and its supporting saddles. The herein proposed method is based on the formulation of a linear complementarity problem (LCP) (or equivalently of a quadratic programming problem (QPP)) which numerically describes the behaviour of an elastic body with unilateral contact boundary conditions and leads to a variational inequality formulation of the problem at hand. The previously mentioned LCP is a direct extension of the Force Method of Structural Analysis and has the great advantage of leading to a straightforward and accurate computation of the distribution of stresses on the pipe-saddle interface.