Multivalued Law Research Articles

Delamination of composites as a substationarity problem: Numerical approximation and algorithms

A method for the effective numerical treatment of the delamination problem in laminated composites is proposed. The interaction between the laminae due to the binding material is described by means of a nonmonotone, possibly multivalued law. Such laws express a variety of limit phenomena related to the discontinuous loss of resistance at the interface. By using the methods of Nonsmooth Mechanics, the problem is written as a hemivariational inequality and is transformed into a substationarity problem for nonconvex nonsmooth energy functions. Using the appropriate finite element discretization, the latter problem is formulated in finite dimensions. A ‘domain decomposition’ type numerical approximation is proposed and a new algorithm is developed for the numerical treatment of the delamination problem. Numerical examples verify the theoretically predicted stability, accuracy and speed of the method and illustrate its properties and applicability.

Convex multilevel decomposition algorithms for non‐monotone problems

AbstractA convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non‐monotone, possibly multivalued laws. The theory is developed here for a model structure with non‐monotone interface or boundary conditions. First the non‐monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non‐linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non‐convex min‐min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,1 is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non‐monotone stick‐slip frictional effects illustrate the theory.

On debonding and delamination effects in adhesively bonded cracks —A boundary integral approach

The present paper studies the influence of adhesives on the behaviour of cracks in two-dimensional linear elastic bodies. Especially the delamination and debonding effects are studied. The adhesive material is assumed to introduce non-monotone, possibly multivalued laws which can be described via non-convex superpotentials. The direct boundary integral equation method is extended for this problem. It gives rise to two equivalent multivalued integral equations holding on each crack. Numerical examples concerning the resulting stress intensity factors illustrate the theory.

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. Karman 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.

Semicoercive hemivariational inequalities. On the delamination of composite plates

In this paper semicoercive hemivariational inequalities are studied in the framework of a concrete mechanical problem: the delamination effect of laminated plates. The interlaminar bonding forces are described by a nonmonotone multivalued law which may be written as the generalized gradient of a nonconvex superpotential in the sense of F. H. Clarke. Then necessary conditions are proved for the existence of the solution, as well as sufficient conditions using compactness and average value arguments.

A variational-hemivariational inequality approach to the laminated plate theory under subdifferential boundary conditions

In this paper the delamination problem for laminated plates is studied. A nonmonotone multivalued law is introduced in order to describe the interlaminar bonding forces. This law is written as the generalized gradient in the sense of F. H. Clarke of an appropriately defined nonconvex superpotential. Moreover, monotone boundary conditions of the subdifferential type are assumed to hold. The problem is formulated as a variational-hemivariational inequality expressing the principle of virtual work in inequality form. By using compactness and monotonicity arguments, the existence and the approximation of the solution of this inequality are investigated.

Laminated Orthotropic Plates under Subdifferential Boundary Conditions. A Variational‐Hemivariational Inequality Approach

AbstractThis paper deals with an orthotropic laminated plate concerning the delamination problem. A nonmonotone multivalued law simulates the mechanical behaviour of the binding material. This law is written in terms of a generalized gradient (in the sense of F. H. Clarke) of an appropriately defined nonconvex superpotential. Moreover, monotone, e.g. frictional slipallowing boundary conditions of subdifferential type are assumed to hold, which are obtained from a convex superpotential through subdifferentiation. The problem is formulated as a variational‐hemivariational inequality which physically expresses the principle of virtual work in inequality form. The existence and the approximation of the solution of this inequality is discussed by an appropriate combination of compactness, monotonicity and average value arguments.