Pseudo-rigid continua: basic theory and a geometrical derivation of Lagrange's equations

Abstract

Pseudo–rigid bodies are regarded here as globally constrained three–dimensional homogeneous continua. The constraint reaction stresses play a fundamental role in maintaining the homogeneity of the deformation field in pseudo–rigid bodies, and the theory is formulated in a manner that makes this role explicit. Our derivation of Lagrange's equations is patterned after geometrical derivations recently given for particle systems and rigid bodies. The pseudo–rigid body is represented by an abstract particle P moving in a higher–dimensional Euclidean space, called Hertzian space, the metric of which is determined by the radius of gyration of the body. The dynamical equations for the pseudo–rigid body are transformed into a single balance equation in Hertzian space. In the presence of holonomic constraints, the particle P is confined to move in a manifold, the configuration manifold, imbedded in Hertzian space. The geometry of the configuration manifold is Riemannian. Lagrange's equations emerge as the covariant components of the balance equation taken along the coordinate directions in the configuration manifold. Non–holonomic constraints are also considered.