Linear Algebra Provides a Basis for Elasticity without Stress or Strain
Author(s) H. H. Hardy*
ABSTRACT
Linear algebra provides insights into the description of elasticity without stress or strain. Classical descriptions of elasticity usually begin with defining stress and strain and the constitutive equations of the material that relate these to each other. Elasticity without stress or strain begins with the positions of the points and the energy of deformation. The energy of deformation as a function of the positions of the points within the material provides the material properties for the model. A discrete or continuous model of the deformation can be constructed by minimizing the total energy of deformation. As presented, this approach is limited to hyper-elastic materials, but is appropriate for infinitesimal and finite deformations, isotropic and anisotropic materials, as well as quasi-static and dynamic responses.

Cite this paper
Hardy, H. (2015) Linear Algebra Provides a Basis for Elasticity without Stress or Strain. Soft, 4, 25-34. doi: 10.4236/soft.2015.43003.
References
[1]   [1] Todhunter, I. (1886) A History of the Theory of Elasticity and of the Strength of Materials from Galilei to the Present Time. Cambridge University Press, Cambridge.

[2]   [2] Spencer, A.J. (1980) Continuum Mechanics. Dover, New York.

[3]   [3] Rivlin, R.S. and Saunders, D.W. (1951) 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, 243, 251-288. http://dx.doi.org/10.1098/rsta.1951.0004

[4]   [4] Hardy, H.H. and Shmidheiser, H. (2011) A Discrete Region Model of Isotropic Elasticity. Mathematics and Mechanics of Solids, 16, 317-333. http://dx.doi.org/10.1177/1081286510391666

[5]   [5] Hardy, H.H. (2013) Euler-Lagrange Elasticity: Differential Equation for Elasticity without Stress or Strain. Journal of Applied Mathematics and Physics, 1, 26-30. http://dx.doi.org/10.4236/jamp.2013.17004

[6]   [6] Gilbert, J.D. (1970) Elements of Linear Algebra. International Textbook Company, Scranton.

[7]   [7] Gelfand, I.M. and Fomin, S.V. (1991) Calculus of Variations. Dover, New York.

[8]   [8] Landau, L.D. and Lifshitz, E.M. (2005) Theory of Elasticity, Course of Theoretical Physics. Volume 7, Elsevier, London.

[9]   [9] Hardy, H.H. (2014) Euler-Lagrange Elasticity with Dynamics. Journal of Applied Mathematics and Physics, 2, 1183-1189. http://dx.doi.org/10.4236/jamp.2014.213138

[10]   [10] Truesdale, C. and Noll, W. (2004) The Non-Linear Field Theories of Mechanics. Springer-Verlag, New York. http://dx.doi.org/10.1007/978-3-662-10388-3

[11]   [11] Wu, H.-C. (2004) Continuum Mechanics and Plasticity. CRC Press, New York. http://dx.doi.org/10.1201/9780203491997

[12]   [12] Kelly, P. (2015) Mechanics Lecture Notes Part III: Foundations of Continuum Mechanics. http://homepages.engineering.auckland.ac.nz/~pkel015/SolidMechanicsBooks/Part_III/

Top