Differential equations to describe elasticity are derived without the use of stress or strain. The points within the body are the independent parameters instead of strain and surface forces replace stress tensors. These differential equations are a continuous analytical model that can then be solved using any of the standard techniques of differential equations. Although the equations do not require the definition stress or strain, these quantities can be calculated as dependent parameters. This approach to elasticity is simple, which avoids the need for multiple definitions of stress and strain, and provides a simple experimental procedure to find scalar representations of material properties in terms of the energy of deformation. The derived differential equations describe both infinitesimal and finite deformations.

I. Todhunter, “A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the Present Time,” Cambridge University Press, Cambridge, 1886.

H. H. Hardy and H. Shmidheiser, “A Discrete Region Model of Isotropic Elasticity,” Mathematics and Mechanics of Solids, Vol. 16, No. 3, 2011, pp. 317-333. http://dx.doi.org/10.1177/1081286510391666

R. S. Rivlin and D. W. Saunders, “Large Elastic Deformations of Isotropic Materials. VII. Experiments on the Deformation of Rubber, Philosophical Transactions of the Royal Society of London. Series A,” Mathematical and Physical Sciences, Vol. 243, No. 865, 1951, pp. 251-288. http://dx.doi.org/10.1098/rsta.1951.0004

L. A. Wood and G. M. Martin, “Compressibility of Natural Rubber at Pressures Below 500 kg/cm2,” Journal of Research of the National Bureau of Standards, Vol. 68A, No. 3, 1964, p. 259.