Structural rigidity

In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges.

Contents 1 Definitions 2 Mathematics of rigidity 3 History 4 Notes 5 References

Definitions

Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges.

There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the structure will not flex by even an amount that is too small to be detected even in theory. (Technically, that means certain differential equations have no nonzero solutions.) The importance of finite rigidity is obvious, but infinitesimal rigidity is also crucial because infinitesimal flexibility in theory corresponds to real-world minuscule flexing, and consequent deterioration of the structure.

A rigid graph is an embedding of a graph in a Euclidean space which is structurally rigid.^[1] That is, a graph is rigid if the structure formed by replacing the edges by rigid rods and the vertices by flexible hinges is rigid. It is also possible to consider rigidity problems for graphs in which some edges represent compression elements (able to stretch to a longer length, but not to shrink to a shorter length) while other edges represent tension elements (able to shrink but not stretch). A rigid graph with edges of these types forms a mathematical model of a tensegrity structure.

Mathematics of rigidity

The fundamental problem is how to predict the rigidity of a structure by theoretical analysis, without having to build it. Key results in this area include the following:

• In any dimension, the rigidity of rod-and-hinge linkages is described by a matroid. The bases of the two-dimensional rigidity matroid (the minimally rigid graphs in the plane) are the Laman graphs.
• Cauchy's theorem states that a three-dimensional convex polyhedron constructed with rigid plates for its faces, connected by hinges along its edges, forms a rigid structure.
• Flexible polyhedra, non-convex polyhedra that are not rigid, were constructed by Raoul Bricard, Robert Connelly, and others. The bellows conjecture, now proven, states that any continuous motion of a flexible polyhedron must preserve its volume.

However, in many other simple situations it is not yet always known how to analyze the rigidity of a structure mathematically despite the existence of considerable mathematical theory.

History

One of the founders of the mathematical theory of structural rigidity was the great physicist James Clerk Maxwell. The late twentieth century saw an efflorescence of the mathematical theory of rigidity, which continues in the twenty-first century.

Notes

References

• Alfakih, Abdo Y. (2007), On dimensional rigidity of bar-and-joint frameworks. Discrete Applied Mathematics, Vol. 155, No. 10, pp. 1244–1253.

• Connelly, Robert (1980), The rigidity of certain cabled frameworks and the second-order rigidity of arbitrary triangulated convex surfaces. Advances in Mathematics, Vol. 37, pp. 272–299.

• Crapo, Henry (1979), Structural rigidity. Topologie Structurale (Structural Topology), Vol. 1, pp. 26–45.

• Maxwell, J. C. (1864), On reciprocal figures and diagrams of forces.

Philosophical Magazine (4th Series), Vol. 27, pp. 250–261.

• Rybnikov, Konstantin, and Zaslavsky, Thomas (2005), Criteria for balance in abelian gain graphs, with applications to piecewise-linear geometry. Discrete and Computational Geometry, Vol. 34, No. 2, pp. 251–268.

• Whiteley, W. (1988), "The union of matroids and the rigidity of frameworks", SIAM Journal on Discrete Mathematics 1 (2): 237–255, doi:10.1137/0401025 .