Compatibility Conditions for Discrete Elastic Structures

The theory of plane, elastic trusses is reconsidered from the viewpoint of the continuum theory of elastic media. A main difference between continuum and discrete theories is the following: In the continuous case all quantities are declared throughout the whole body, whereas in the discrete case each quantity has its own “carrier”. In a truss, for instance, displacements and applied forces are declared in the nodes while strain and stress live in the members of the truss. The paper deals with the proper formulation of compatibility conditions for a truss. They are derived along the same lines as St.-Venant’s compatibility conditions of plane elasticity, i. e. by stipulating that Cesàro’s integrals are path independent. Compatibility conditions can be formulated at each inner node of a truss, and they relate the strains of all members which constitute the rosette surrounding the inner node. 1. Continuous and discrete elastic systems Continuum theories are usually developed from physical models that are discrete in nature. A continuous distribution of dislocations, for instance, would hardly be conceivable, if we had not a clear idea of an isolated dislocation. Even the notion of stress as a distributed force follows the example of a single force. Within the framework of a continuum theory, however, discrete quantities appear as singularities and are formally less convenient to handle than their continuous counterparts. By the process of homogenization the underlying discrete ideas are transformed into a continuum theory. The resulting partial differential equations do not admit closed-form solutions, in general. To solve them numerically a discretization process is invoked, which approximates the continuum by a discrete system. In this sense a continuum theory is squeezed between the underlying discrete physical model and the discrete numerical approximation. The general structure of a physical theory should be perceptible independently of the discrete or continuum formulation. A balance equation, for instance, has a genuine physical meaning whether the model is continuous or discrete. The theory of a discrete elastic structure, be it a crystal lattice, a finite-element system or an elastic truss, should exhibit the same fundamental laws as continuum elasticity theory. The general form of the fundamental equations can be represented most suggestively by a so-called TONTI diagram [6, 7]. Figure 1 shows the TONTI diagram of plane, linear elasticity theory. If we consider a plane, discrete elastic system, we should encounter the same physical laws, although in a rather different formal garment. This paper deals with the governing equations of plane, elastic trusses with special emphasize of the compatibility conditions, which are derived along the same lines as ST.-VENANT’s compatibility conditions of plane elasticity, i. e. by stipulating that CESÀRO’s integrals are path 37 38 M. Braun independent. The theory of plane, elastic trusses is reconsidered from the viewpoint of the continuum theory of elastic media. Mathematically a truss is considered as an oriented 2-complex, on which displacement, strain, etc. are defined. In contrast to the continuous body the mechanical quantities in a truss are not available everywhere in the body, each quantity resides on its own “carrier”: Displacements and applied forces are declared in the nodes while strain and stress live in the members of the truss. It will be shown that the compatibility conditions are attached to “rosettes”, ı. e. inner nodes that are completely surrounded by triangles of truss members. To consider trusses from the point of view of elasticity theory is not at all new. KLEIN and WIEGHARDT [4] have presented such an exposition even in 1905, and they rely on earlier works of MAXWELL and CREMONA. Meanwhile, however, trusses have become more a subject of structural mechanics and the more theoretical aspects have been banned from textbooks. As an exception a manuscript by RIEDER [5] should be mentioned, in which the cross-relations between electrical and mechanical frameworks are studied in great detail. 2. Trusses Mechanically a truss is a system of elastic members joint to each other in hinges or nodes without friction. The truss is loaded by forces acting on the nodes only. The appropriate mathematical model of a truss is a 1-complex consisting of 0-simplexes (nodes) and 1-simplexes (members), which are “properly joined” [3]. The subsequent analysis gives rise to two extensions of this model, namely (i) each member is given an orientation, which u displacement vector