Euler-Lagrange Elasticity with Dynamics
Author(s)
H. H. Hardy*
ABSTRACT
The equations of Euler-Lagrange elasticity describe elastic deformations without reference to stress or strain. These equations as previously published are applicable only to quasi-static deformations. This paper extends these equations to include time dependent deformations. To accomplish this, an appropriate Lagrangian is defined and an extrema of the integral of this Lagrangian over the original material volume and time is found. The result is a set of Euler equations for the dynamics of elastic materials without stress or strain, which are appropriate for both finite and infinitesimal deformations of both isotropic and anisotropic materials. Finally, the resulting equations are shown to be no more than Newton's Laws applied to each infinitesimal volume of the material.
The equations of Euler-Lagrange elasticity describe elastic deformations without reference to stress or strain. These equations as previously published are applicable only to quasi-static deformations. This paper extends these equations to include time dependent deformations. To accomplish this, an appropriate Lagrangian is defined and an extrema of the integral of this Lagrangian over the original material volume and time is found. The result is a set of Euler equations for the dynamics of elastic materials without stress or strain, which are appropriate for both finite and infinitesimal deformations of both isotropic and anisotropic materials. Finally, the resulting equations are shown to be no more than Newton's Laws applied to each infinitesimal volume of the material.
KEYWORDS
Elasticity, Stress, Strain, Infinitesimal Deformations, Finite Deformations, Discrete Region Model
Elasticity, Stress, Strain, Infinitesimal Deformations, Finite Deformations, Discrete Region Model
Cite this paper
Hardy, H. (2014) Euler-Lagrange Elasticity with Dynamics. Journal of Applied Mathematics and Physics, 2, 1183-1189. doi: 10.4236/jamp.2014.213138.
Hardy, H. (2014) Euler-Lagrange Elasticity with Dynamics. Journal of Applied Mathematics and Physics, 2, 1183-1189. doi: 10.4236/jamp.2014.213138.
References
[1] Maugin, G.A. (2013) Continuum Mechanics through the Twentieth Century. Springer, London.
http://dx.doi.org/10.1007/978-94-007-6353-1 [2] Truesdell, C. and Noll, W. (2009) The Non-Linear Field Theories of Mechanics. Springer, London. [3] Srinivasa, A.R. and Srinivasan, S.M. (2004) Inelasticity of Materials. World Springer, New York. [4] Pedregal, P. (2000) Variational Methods in Nonlinear Elasticity. Siam, Philadelphia. [5] Hardy, H.H. (2013) Euler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain. Journal of Applied Mathematics and Physics, 1, 26-30. [6] Todhunter, I. (1886) A History of the Theory of Elasticity and of the Strength of Materials from Galileo to the Present Time. Vol. 1, Cambridge University Press, New York. [7] Shabana, A.A. (2008) Computational Continuum Mechanics. Cambridge University Press, New York, 131.
http://dx.doi.org/10.1017/CBO9780511611469.005 [8] Spencer, A.J. (1980) Continuum Mechanics. Dover, Mineola, New York. [9] Ogden, R.W. (1984) Non-Linear Elastic Deformations. Dover, Mineola, New York. [10] 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
[1] Maugin, G.A. (2013) Continuum Mechanics through the Twentieth Century. Springer, London.
http://dx.doi.org/10.1007/978-94-007-6353-1 [2] Truesdell, C. and Noll, W. (2009) The Non-Linear Field Theories of Mechanics. Springer, London. [3] Srinivasa, A.R. and Srinivasan, S.M. (2004) Inelasticity of Materials. World Springer, New York. [4] Pedregal, P. (2000) Variational Methods in Nonlinear Elasticity. Siam, Philadelphia. [5] Hardy, H.H. (2013) Euler-Lagrange Elasticity: Differential Equations for Elasticity without Stress or Strain. Journal of Applied Mathematics and Physics, 1, 26-30. [6] Todhunter, I. (1886) A History of the Theory of Elasticity and of the Strength of Materials from Galileo to the Present Time. Vol. 1, Cambridge University Press, New York. [7] Shabana, A.A. (2008) Computational Continuum Mechanics. Cambridge University Press, New York, 131.
http://dx.doi.org/10.1017/CBO9780511611469.005 [8] Spencer, A.J. (1980) Continuum Mechanics. Dover, Mineola, New York. [9] Ogden, R.W. (1984) Non-Linear Elastic Deformations. Dover, Mineola, New York. [10] 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