Truss modular beams with deformation energy depending on higher displacement gradients.

Abstract : Until now, no third gradient theory has been proposed to describe thehomogenized energy associated to a microscopic structure. In this paper, weprove that this is possible using pantographic type structures. Theirdeformation energies involve combinations of nodal displacements having theform of second or third order finite differences. We establish the Gamma-convergence of these energies to second and third gradient functionals.Some mechanical examples are provided so as to illustrate the specialfeatures of these homogenized models.
 

Keywords : Second gradient theory, Third gradient theory, Homogenization, Gamma-convergence, Finite differences, Modular trussbeam, Pantograph.

Introduction :  A formalized theory for constitutive equations in continuum mechanics has been first developed by Noll. In the framework of the aforementioned axiomatization it was proven that the second principle of thermodynamics does not allow any dependence of stress tensor on second gradient of placement sothat the new concept of interstitial working had to be introduced .

However another possible formalization of continuum mechanics, based on theprinciple of virtual power and stemming from the d'Alembert conception of mechanics, is possible.

Following the classification formalized by Germain the mechanical material behavior of bodies can be characterized by the expression of internal (deformation) energy in terms of the displacement gradients. Cauchy 3-D materials coincide with first gradient materials: their deformation is described by, and their deformation energy depends on, the first gradient of displacement only.

The deformation energy of second gradient 3-D materials , instead, depends also on the second gradient of displacement. Let us call  e(u) the symmetric part of the gradient of the displacement u and w(u) its skew part. We will say that a second gradient material 3-D is incomplete if its internal energy depends only on the first gradient of u and on the gradient of w(u). These materials are also called ``3-D Materials with couple stresses'' :  microrotations in considered bodies are modeled introducing in the constitutive equations the aforementioned dependence on the gradient of w(u). Such a modeling assumption has been subsequently improved introducing microstructural kinematical descriptors.

Incomplete second gradient materials have been studied for a long time. Precursor of incomplete second gradient models is the elastica introduced by Euler, Bernouilli and Navier at the beginning of XVIII century: it is a 1-D model in which i) the attitude of the beam sections is kinematically described by the gradient of the vertical displacement field, ii) contact couple depends on the second derivative of the vertical displacement, iii) the deformation energy depends on the gradient the attitude and therefore on the second gradient of displacement.

Another example of 1-D model in which higher order derivatives of displacement must be introduced is given by the theory of Vlasov describing the twist of thin-walled beams. In Vlasov's homogenized model the phenomenon at the micro-level to be accounted for is the warping of beam sections and the corresponding deformation energy is shown to be depending on first and second gradients of twist angle.

The first (incomplete) second gradient 3D-model is due to E. and F. Cosserat (beginning of XIX century): in it the deformation energy explicitly depends on the gradient of w(u). More recently incomplete second gradient materials has been introduced to model granular solids. Complete 3-D models have been introduced for describing capillary phenomena and begin to be extensively used in the theory of damage and plasticity as they provide a more accurate description of transition zones and, from a mathematical point of view, as they lead to regularized well-posed problems. The regularizing properties of second gradient models are also exploited in the description of the mechanical behavior of elastic crystals .

Complete and incomplete second gradient materials have fundamentally different behaviors. While, in incomplete models, boundary conditions fix the displacement and the rotation w(u) (or their dual quantities force and moment), in complete models one has to fix also the normal part of e(u) or its dual quantity called double force  to which not all mechanicians are accustomed: indeed the only widely used double contact action is Vlasov's bimoment needed for describing the external action at the extremities of thin walled twisted tubes.

It is remarkable that mathematically established relationship between 1-D or 2-D second gradient models and Cauchy materials have been investigated only when the dependence of deformation energy on second gradient of displacement can be related, at the micro-level, to variations of attitude. Indeed, the limit analysis 3D-1D or 2D-1D of plates or beams leads only to such type of second gradient models. Is there any fundamental physical reason for that? In our opinion, this is probably due to the desire of remaining in a standard framework. For more details about these rigorous results we limit ourselves to refer to Bensoussan, Cioranescu, Cioraplate, Trabucho, Telega,...  In technical theories (which supply the mechanical ground for the aforementioned mathematical results) of beams (for an extensive historical discussion we refer to Benvenuto, who traces back to Maxwell and de Saint-Venant  the first of these analyses the macro-models are related to micro-models using several identification procedures. That one which seems to be more encompassing is based on identification in expended power: one postulates a macroscopic and a microscopic model, a kinematic correspondence between the deformations and assumes that the power expended in corresponding motions coincides. In this way one obtains, in terms of micro properties of the beam, the coefficients of the macro constitutive equations, the form of which has been postulated at the beginning. The very nature of this procedure\shows how the properties of the macro-model, in general, are not obtained as a result of the homogenization process, but are, instead, assumed a priori.

Here we present a microscopic model which leads to the simplest macroscopic second gradient one: the one-dimensional planar beam already studied by Casal . The structure we consider, i.e. the pantographic one, is simple and the reader may have already experimented it when handling a corkscrew. We assume that the considered pantograph is made by a very large number n of small modules and we study its limit behavior when n tends to infinity. In other words we study the homogenized model for the pantograph. Computation of equilibrium state is straightforward and we prove rigorously, using the technique of Gamma-convergence that the homogenized model is really a second gradient model. Considering different equilibrium situations, we recall the principal features of this model and we get an evident and self-explanatory interpretation for its special features, in particular for the notion of double force.

To our knowledge, no structure has already been described as a third gradient body. To find such a body is a problem closely related to the previous one. Indeed, it is relatively easy, once one has obtained a complete second gradient body, to construct a third gradient body. We do it and describe a structure, based on the pantograph, which we call Warren-type pantographic structure. Its homogenized energy is rigorously proved to correspond to a third gradient bending beam. This beam has an unusual behavior which we shortly describe.

As theorems and proofs are quite similar in both cases we decided to group them in a single theorem. This theorem has is own interest: it states that, for any k>0, a quadratic functional of finite differences of order k converges to a quadratic functional of k-th derivative. This convergence is proved in the sense of Gamma-convergence with respect to the weak* topology of measures. In that way, our result does not depend of the choice of any interpolation of displacements which have physical meaning at nodes only. We identify the external forces which are admissible for the considered structure: it is a class of distributions of order k which contains, in particular, any distribution of order k-1. For instance, for a second gradient beam (k=2), the derivative of a Dirac distribution is admissible. This corresponds to the notion of punctual double force .