Plate theory and complementary displacement method

Abstract

With the complementary displacement method introduced in this article regarding plates, it is possible to deduce a constitutive law of a mechanical model from that of the usual three-dimensional (3-D) model. The model studied here is that of a plate undergoing infinitesimal transformations. The method is based on an appropriate kinematic formulation: it decomposes the field of displacements into the sum of a principal displacement, which allows the usual plate theory concepts to be introduced, and a complementary displacement which has been greatly neglected in the classical approach.

The classical hypotheses, whether kinematic like those of Kirchhoff or Reissner, or static like that of plane constraints, are replaced here by the sole hypothesis that the laws of external forces are plate laws: the forces are independent of the complementary displacement and develop no virtual work in a virtual complementary displacement. This being assumed, we show the existence of a global constitutive law linking the fields of generalised deformations and generalised constraints, and also the identical nature of plate equilibrium and three-dimensional solid problems.

A local constitutive law for plates can only be approached provided the plate is thin enough for its vast majority to be far enough from the edge. The most natural way is to assume that the field of generalised deformations is constant; hence the complementary displacement is the solution of an ordinary differential equation which, when solved, gives the constitutive law sought. The cases to which the equation is applied are an elastic isotropic material, producing classical results, and an elastic sandwich plate, leading to new results. Finally, when the field of generalised deformations is polynomial, the fields of complementary displacements and constraints can be found by polynomial identification, thus providing Saint-Venant polynomial solutions, as well as the stiffness matrix of a plate finite element.

Introduction

In classical works on Strength of Materials the usual models of beams, plates, shells etc., are presented as approximations of the common model of reference which is the 3-D continuous medium. The universal character often attributed to the latter can be questioned, for example when evaluating the behaviour of a cable which has several levels of strands. It remains, nevertheless, indispensable to have at our disposal clear relations between the mechanical properties of each and, more precisely, to be able to deduce the constitutive law of the model chosen from that of the 3-D model.

To achieve this goal, textbooks on Strength of Materials use various hypotheses named after Bernoulli, Kirchhoff, Mindlin or Reissner. All consist of restricting the kinematics by, for example, assuming that the normals to the mean surface remain rectilinear and normal to the mean surface (Kirchhoff) , and must be accompanied by static hypotheses, i.e. relative to efforts, e.g. that of plane stresses.

Moreover, most authors do not specify whether the quantities which they neglect, e.g. transverse shearing, should be regarded as small or null: for the sake of coherence it might be advisable, in the first case to verify after the fact, the supposed smallness, and in the second to provide the corresponding constraint equation and introduce the corresponding unknown constraint effort.

The purpose of this article is to present, in the case of plates submitted to infinitesimal transformations, a general method for establishing links between Strength of Materials models and the 3-D solid. It consists of changing the kinematic representation of the latter by decomposing the total displacement field into the sum of a principal displacement and a complementary displacement.

The former, which verifies the classical kinematic hypotheses such as Kirchhoffs, is the usual displacement of the new model; it bears the concepts of the new model, e.g. that of the curve of the mean surface and consequently that of the bending moment. The latter, being the difference between the total displacement of the 3-D solid and the principal displacement, makes it possible to establish the precise link between the two problems by solving the boundary value problem which naturally arises from the chosen approach.

In fact, the traditional textbook approaches almost ignore the complementary displacement, and it is necessary to turn to journals to find articles which seriously address the 3-D problem, such as those by Koiter (1989) , Koiter and Simmonds (1972) , Ladeveze, 1976, Ladeveze, 1980 , Levinson (1980) , Nair and Reissner (1977) , Reissner (Reissner, 1975, Reissner, 1976, Reissner, 1985) , Touratier (1991) , Rychter (1987) or Verchery (1974) .

Besides articles of highest quality, as those of the authors quoted above and probably some others, the bibliography of plates is huge and could be counted, if necessary, in cubic meters! Nevertheless, good theoretical papers are not numerous and are concerned with elastic behaviour. On the contrary the present study presents a general approach to plate theory, independent of the tridimensional constitutive equation even if, of course, the application examples given in the last part suppose an elastic behaviour and thus make possible comparisons with classical results.

Two hypotheses are introduced here in order that the 3-D solid problem be replaced by a 2-D problem from which the complementary displacement has disappeared. The first is both kinematic and static and concerns the laws of external efforts: they are independent of the complementary displacement and develop no virtual work in any virtual complementary displacement. This property will subsequently be referred to as P: such an effort law is in fact a plate effort law. In practice, most laws of volume force, such as gravity, include property P; however, this is not the case for surface forces, and particularly for supporting forces.

The two aspects of P are, as we shall see, closely linked to Saint Venants principle, according to which it is possible to replace a force law by a related law possessing the property, and it can even be said that the principle consists essentially of expliciting this possibility. According to the hypothesis that all external load laws possess this property, we establish the equivalence between the 3-D problem and a plate problem, to arrive at a global constitutive law which links a field of generalised stresses (e.g. bending moment) with a field of generalised strains (e.g. curvature) . This is where the boundary value problem for the complementary displacement arises.

The hypothesis that the external load laws possess property P replaces the usual hypotheses, whether they be static such as that of plane stresses, or kinematic like those of Kirchhoff, Mindlin–Reissner or those which have enriched the kinematics of the latter.

The second hypothesis leads to the definition of a constitutive law applicable to the interior of the plane domain occupied by the plate, and to the calculation of an appropriate constitutive law. It consists once again of referring to Saint Venants principle in order to replace the boundary value problem by an infinite field problem: as soon as one is sufficiently far from the edge, the internal behaviour is independent of the boundary conditions. Only at this stage does the hypothesis concerning the thinness in relation to the transversal dimensions of the plate intervene implicitly; it is thereby possible to assert that the vast majority of the plate is, in effect, sufficiently far from the edge. Since the famous article by Toupin (1965) on beams, several authors have contributed remarkable works on this question, such as those listed in the bibliography at the end of this article under the names of Knowles (1966) , Knowles and Horgan (1969) , Horgan (1982) , Horgan and Knowles (1983) , and Ladeveze (1983) .

Nevertheless, no accurate local constitutive law exists for the new model, and we find ourselves confronted with two possibilities: either we seek to create a mechanical model of the plate with its own local constitutive equation or we seek a constitutive equation for a plate finite element.

In the first alternative the obvious approach is to establish the law for a constant field of generalised strains: namely to solve an ordinary differential equation which will give, depending on the generalised strains, the secondary field, followed by that of the stresses, and finally that of the generalised stresses.

In the finite element approach the situation is very different: for a start, the problem of the edge is irrelevant, except if it is to be located at the boundary. Moreover, the field of generalised strains is given in polynomial form depending on the generalised displacements at the nodes; so, in the case of linear elasticity, it is possible to identify exactly the fields of secondary displacements and 3-D stresses in polynomial form, as we shall show, to a degree equal to or less than two, which should normally be feasible to any degree.

These solutions are given, obviously, for an elastic, homogeneous and isotropic material. They are also given for one of the most interesting applications of the theory: the treatment of plates composed of several orthotropic layers. The vital contribution of formal calculation tools (here Mathematica) to the implementation of the method must be underlined here; if publication of the theory, which was conceived in the 70s, was postponed, it was because the tools necessary to implement it were not available at the time. Manual polynomial identification would indeed be a Herculean task.

This article includes only the purely mechanical aspects of the theory, in order to be more easily read by those who have no particular taste for the mathematical notions which are detailed in another article to be published later on where the functional analysis aspects and the questions of duality in coupling the 3-D model and the plate model are dealt with. It will be demonstrated, in particular, that the choice of functional spaces for the latter may be deduced from the choice made for the former, and that it is possible to deduce also two important closedness theorems for the plate from the fact that they have already been established for the 3-D solid.