P. D. Panagiotopoulos

Quasidifferentiability and Nonsmooth Modelling in Mechanics, Engineering and Economics

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On the rigid body displacements and rotations in unilateral contact problems and applications

Abstract This paper deals with the treatment of rigid body displacements and rotations in unilateral contact problems. In the presence of rigid body modes the equivalent formulations of the problem, i.e. the variational inequality, the quadratic programming and the linear complementarity formulation involve positive semidefinite matrices. The basic differences from the classical bilateral problems is that in unilateral problems the rigid body modes must be compatible with the inequality constraints. This is a fairly difficult problem to solve and in its generality was open until now. In this paper a systematic analysis including convergence results is presented with respect to some methods already in use, namely the method of coupling with additional elastic bodies or ‘soft’ springs and the method of consideration of certain nodes as fixed. Moreover a new linear complementarity formulation of the problem which explicitly includes the rigid body ‘displacements’ is proposed and is numerically treated by a complementary pivoting technique. Necessary and sufficient conditions for the solution of the problem are derived and the theory is illustrated by examples from structural analysis and from robotics.

Existence Theorems of Hartman–Stampacchia Type for Hemivariational Inequalities and Applications

We give some versions of theorems of Hartman-Stampacchias type for the case of Hemivariational Inequalities on compact or on closed and convex subsets in infinite and finite dimensional Banach spaces. Several problems from Nonsmooth Mechanics are solved with these abstract results.

Neural networks for computing in fracture mechanics. Methods and prospects of applications

Abstract A neural network is proposed and studied for the treatment of fracture mechanics problems. Both the cases of classical cracks and of cracks involving Coulombs friction or detachment (unilateral contact) interface conditions are considered. For the first case, the Hopfield model is appropriately modified, whereas for the second case, a neural model is proposed covering the case of inequalities. For this model, new results generalizing the results of Hopfield and Tank are obtained. Numerical applications illustrate the theory. Finally, the parameter identification problem for fractured bodies is formulated as a supervised learning problem.

A variational inequality approach to the inelastic stress-unilateral analysis of cable-structures

Abstract The present paper deals with the inelastic, stress-unilateral analysis of cable-structures undergoing large displacements. The response of the cable-structures to load- and initial strain-increments is described by a set of equations and inequalities, which are formulated compactly as variational inequalities. Two dual extremum principles—generalizations of the minimum of the potential and complementary energy—with inequalities as subsidiary conditions are derived. The numerical calculation is preformed iteratively by using the decomposition techniques of the multilevel optimization and applying in the resulting substructures the algorithms of non linear optimization. As an example the elastic, elastic-perfectly plastic and workhardening behaviour of a cable-structure is examined and some useful results from the comparison of the solutions are obtained.

The delamination effect in laminated von Kármán plates under unilateral boundary conditions. A variational ― hemivariational inequality approach

This paper deals with the delamination effect for laminated plates undergoing large displacements (v. Kármán plates). The interaction between the laminae due to the binding material as well as the delamination effect are described by means of a nonmonotone, possibly multivalued law, while on the boundary of each lamina general unilateral boundary conditions obeying monotone laws are assumed to hold. The interface and the boundary laws are written in terms of nonconvex and convex superpotentials, respectively. The problem is written in the form of a variational-hemivariational inequality. Certain results on the existence and the approximation of the solution of this problem are obtained by means of compactness, monotonicity and average value arguments.

Steel T-Stub connections under static loading: an effective 2-D numerical model

Abstract The proposed numerical model concerns the simulation of the structural behaviour of steel bolted T-stub connections in the case where the development of zones of plastification, as well as unilateral contact effects on the interfaces between connection members and bolts, are taken into account. Within such a framework, an effective two-dimensional finite element model capable of describing plasticity, large displacement and unilateral contact effects is proposed. The model constitutes an easy-to-use and accurate numerical model for the analysis of steel connections subjected to tensile loading, and is a simplification of a respective three-dimensional one and aims to reduce in a reliable way the huge computational effort required for the analysis of fine meshes of discretized steel bolted connections. The validity of the assumptions that led to the proposed 2-D model is demonstrated by comparing the numerical results with those obtained by laboratory tests.

Fractal geometry and fractal material behaviour in solids and structures

SummaryThe present paper discusses certain methods which permit us to consider the influence of the fractal geometry and the fractal material behaviour in solid and structural mechanics. The method of fractal interpolation function is introduced and the fractal quantities (boundary geometry, interface geometry and stress-strain laws) are considered as the fixed points of a given set-valued transformation. Our first aim here is to define the mechanical quantities on fractal sets using some elementary results of the theory of Besov spaces. Then we try to extend the classical finite element method for the case of fractal bodies and fractal boundaries and corresponding error estimates are derived. The fractal analysis permits the formulation and the treatment of complicated or yet unsolved problems in the theory of deformable bodies.ÜbersichtDiskutiert werden Methoden, die es erlauben, den Einfluß von fraktaler Geometrie und fraktalem Materialverhalten in der Festkörper- und Strukturmechanik zu betrachten. Die Methode der fraktalen Interpolationsfunktion wird eingeführt; die fraktalen Größen (Randgeometrie, Grenzflächengeometrie und Spannungs-Dehnungsgesetze) werden als Fixpunkte einer gegebenen mengenwertigen Transformation betrachtet. Das erste Ziel ist die Definition der mechanischen Größen auf fraktalen Mengen, wofür einige grundlegende Ergebnisse der Theorie der Besov-Räume herangezogen werden. Weiterhin wird die klassische Finite-Element-Methode auf fraktale Körper und fraktale Ränder erweitert und zugehörige Fehlerabschätzungen werden abgeleitet. Die fraktale Betrachtung gestattet die Formulierung und Behandlung komplizierter oder noch ungelöster Probleme der Theorie deformierbarer Körper.

A linear complementarity approach to the frictionless gripper problem

This article deals with static unilateral phenomena arising in grasping mechanisms and caused by the possible loss of contact between elastic grippers and rigid objects. The problem is formulated as a linear complementarity problem (LCP) that is derived from the equilibrium equations of forces and the compatibility equations for the displacements, as well as from the complementarity inequality restriction describing the contact. A variant of the infeasibility-reducing, complementary pivot al gorithm of Lemke is used for the numerical solution of the arising LCP. Form and force closure conditions are fulfilled in an auto matic way from the solution of the aforementioned LCP. The used algorithm guarantees rapid convergence to the solution of the problem. The theory is illustrated by numerical examples.

Transient elastodynamics around cracks including contact and friction

In this paper, some numerical investigations of fast transient elastodynamic problems are presented, which consider especially the interaction of elastic waves with cracks, including contact and friction. For this purpose, a boundary element formulation in the time domain is coupled with contact mechanics models previously proposed and tested by the authors in the framework of elastostatic contact problems. In the numerical examples, qualitative and quantitative effects of the interaction between elastic waves and non-classical, unilateral cracks are briefly discussed.