Unilateral Boundary Conditions Research Articles

Unilateral problems in nonlinear, three-dimensional elasticity

In 1977 John Ball obtained a series of very important existence results in hyperelasticity. The objective of the present paper is to extend his results in two directions, first by considering unilateral boundary conditions corresponding to contact without friction, secondly by allowing “locking” constraints in the set of admissible deformations.

On the existence of optimal shapes in contact problems

The optimal shape design of a two-dimensional elastic body on a rigid frictionless foundation is analyzed. The problem is to find the boundary part of the body where the unilateral boundary conditions are assumed, in such a way that the total energy of the system in the equilibrium state will be minimized. The solvability of the problem is proved.

Stefan problems with the unilateral boundary condition on the fixed boundary. II

On considere des problemes de Stefan a 1 phase a 1 dimension avec la condition limite unilaterale sur la frontiere fixee

Stefan problems with the unilateral boundary condition on the fixed boundary. II

On considere des problemes de Stefan a 1 phase a 1 dimension avec la condition limite unilaterale sur la frontiere fixee

An old elasticity problem in a unilateral setting

A limiting case of the Michell problem involving an elastic wedge is the unbounded solid with a semi-infinite cut, the tip of which is subjected to a concentrated force. For the limiting case, the classical solution leads to overlapping of material whenever the component of the force along the axis of symmetry is directed away from the cut, and the problem must be solved anew using unilateral boundary conditions. The required mathematics is simple, and the subject is suitable for classroom discussion. Two examples are solved explicitly, and additional exercise problems are suggested.

Regularity of solutions of quasilinear elliptic equations with unilateral boundary conditions

Regularity of solutions of quasilinear elliptic equations with unilateral boundary conditions

On general boundary value problems and duality in linear elasticity. I

The equilibrium state of a deformable body under the action of body forces is described by the well known conditions of equilibrium, the straindisplacement relations, the constitutive law of the linear theory and the boundary conditions. The authors discuss in detail the boundary conditions. The starting point is the general relation between the vectors of stress and displacement on the boundary which can be expressed in terms of a subgradient relation. It is shown that this relation includes as special cases all known classical, bilateral and unilateral boundary conditions. Further, the principle of virtual displacements and the principle of minimum of the potential energy are established and it is shown that these principles are equivalent to the original boundary condition problem.

A Variational Inequality Approach to the Friction Problem of Structures with Convex Strain Energy Density and Application to the Frictional Unilateral Contact Problem

ABSTRACT The analysis of structures with friction boundary conditions is considered. The stress-strain relations are generally monotone nonlinear allowing for jump, locking, and slacking effects, and are derived from a convex strain energy density by “subdifferentiation.” It is proved that for the considered boundary value problem the principles of virtual and of complementary virtual work hold in an inequality form which constitutes a variational inequality. The theorems of minimum potential and complementary energy are proved to be valid for this type of boundary conditions. These theorems are used to formulate the analysis as a nonlinear programming problem. A numerical example concerning the calculation of a structure having friction boundary conditions combined with unilateral contact boundary conditions illustrates the theory.

Cauchy and periodic unilateral problems for aging linear viscoelastic materials

Our aim in this paper is to study the Gauchy-Signorini problem and the periodic Signorini problem for a class of linear viscoelastic materials with aging and fading memory defined in Section 1. In classical linear elasticity with infinitesimal displacements the Signorini problem corresponds to contact without friction and gives rise to unilateral boundary conditions studied with variational inequalities (see [12] and its bibliography). The results obtained in Section 3 for the Cauchy-Signorini problem complete the work of Duvaut [I I] for hereditary linear viscoelastic materials with fading memory but the study of the periodic Signorini problem seems new even in this particular case. A particular feature of the periodic problem for Maxwell-type materials, whose constitutive laws can be written

Dual finite element analysis for elliptic problems with obstacles on the boundary. I

For an elliptic model problem with non-homogeneous unilateral boundary conditions, two dual variational formulations are presented and justified on the basis of a saddle point theorem. Using piecewise linear finite element models on the triangulation of the given domain, dual numerical procedures are proposed. By means of one-sided approximations, some a priori error estimates are proved, assuming that the solution is sufficiently smooth. A posteriori error estimates and two-sided bounds for the energy are also deduced.

Convex analysis and unilateral static problems

In this paper by the use of “convex analysis” the general nonlinear unilateral static problem of structural mechanics is formulated. By means of the “subdifferentiability” concept the unilateral nonlinear boundary conditions and the unilateral nonlinear constitutive laws are formulated as “subdifferential relations”, permitting the derivation of the variational inequalities of virtual and complementary virtual work. Thus the minimum potential and complementary energy theorems are proved. These minimum theorems are used to formulate the analysis as a nonlinear programming problem. Numerical examples on structures with nonlinear unilateral constitutive laws and nonlinear unilateral boundary conditions illustrate the theory.