Abstract
In this work, we study the transformation properties of the local form of the material (referential) balance of energy equation under the superposition of arbitrary material diffeomorphisms. For this purpose, the tensor analysis on manifolds is utilized. We show that the material balance of energy equation, in general, cannot be invariant; in fact an extra term appears in the transformed balance of energy equation, which is directly related to the work performed by the configurational stresses. By making the fundamental assumption that the body and the ambient space manifolds are always related in the course of deformation and by utilizing the metric concept, we determine this extra term. Building on this, we derive several constitutive equations for the material stress tensor. The compatibility of these constitutive equations with the second law of thermodynamics is evaluated. Finally, we postulate that the material balance of energy equation is covariant, and we study this case in detail, as well.
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References
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (original edition by Prentice Hall, 1983) (1994)
Doyle T.C., Ericksen J.L.: Nonlinear Elasticity Advances in Applied Mechanics. Academic Press, New York (1956)
Yavari A., Marsden J.E., Ortiz M.: On spatial and material covariant balance laws in elasticity. J. Math. Phys. 47, 1–53 (2006)
Gurtin M.E.: The nature of configurational forces. Arch. Ration. Mech. Anal. 131, 67–100 (1995)
Valanis K.C.: The concept of physical metric in thermodynamics. Acta Mech. 113, 169–184 (1995)
Valanis K.C., Panoskaltsis V.P.: Material metric, connectivity and dislocations in continua. Acta Mech. 175, 77–103 (2005)
Panoskaltsis V.P., Soldatos D., Triantafyllou S.P.: The concept of physical metric in rate-independent generalized plasticity. Acta Mech. 221, 49–64 (2011)
Simo J.C., Marsden J.E., Krishnaprasad P.S.: The Hamiltonian structure of elasticity. The convected representation of solids, rods and plates. Arch. Ration. Mech. Anal. 104, 125–183 (1988)
Stumpf H, Hoppe U.: The application of tensor algebra on manifolds to nonlinear continuum mechanics—invited survey article. Z. Angew. Math. Mech. 77, 327–339 (1997)
Ganghoffer J.F.: Differential geometry, least action principles and irreversible processes. Rend. Sem. Mat. Univ. Pol. Torino 65, 171–203 (2007)
Bishop R.L., Goldberg I.: Tensor Analysis on Manifolds. Dover, New York (1980)
Panoskaltsis, V.P., Soldatos, D.: A phenomenological constitutive model of non-conventional elastic response. Int. J. Appl. Mech. 05(04) (2013). doi:10.1142/S1758825113500361
Gurtin M.E.: On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients. Int. J. Plast. 19, 47–90 (2003)
Gurtin, M.E., Anand, L.: The decomposition F = F e F p, material symmetry, and plastic irrotationality for solids that are isotropic–viscoplastic or amorphous. Int. J. Plast. 21, 1686–1719 (2005)
Panoskaltsis V.P., Soldatos D.: On spatial covariance, second law of thermodynamics and configurational forces in continua. Entropy 16, 3234–3256 (2014). doi:10.3390/e1606323
Gotay, M.J., Isemberg, J., Marsden, J.E., Montgomery, R. with the collaboration of Sniatycki J. and Yasskin, P. B.: Momentum Maps and Classical Fields. Part I: Covariant Field Theory (2003). arXiv:Physics/9801019v2
Rahouadj R., Ganghoffer J.F., Cunat C.: A thermodynamic approach with internal variables using Lagrange formalism. Part I: General framework. Mech. Res. Commun. 30, 109–117 (2003)
Rahouadj R., Ganghoffer J.F., Cunat C.: A thermodynamic approach with internal variables using Lagrange formalism. Part 2: Continuous symmetries in the case of the time–temperature equivalence. Mech. Res. Commun. 30, 119–123 (2003)
Romero I.: A characterization of conserved quantities in non-equilibrium thermodynamics. Entropy 15, 5580–5596 (2013)
Ganghoffer, J.F.: Symmetries in mechanics: from field theories to master responses in the constitutive modeling of materials. In: Ganghoffer, J.F., Mladenov, I. (eds.) Similarity and Symmetry Methods, Applications in Elasticity and Mechanics of Materials, pp. 271–351. Springer, Berlin (2014)
Perzyna P.: On the thermomechanical foundations of viscoplasticity. In: Lindolm, U.S. (ed.) Mechanical Behavior of Materials under Dynamic Loads, pp. 61–76. Springer, Wien (1968)
Lubliner J.: On the structure of the rate equations of material with internal variables. Acta Mech. 17, 109–119 (1973)
Lubliner J.: An axiomatic model of rate-independent plasticity. Int. J. Solids Struct. 16, 709–713 (1980)
Szekeres P.: A Course in Modern Mathematical Physics: Groups, Hilbert Space and Differential Geometry. Cambridge University Press, New York (2004)
Fock V.: Three lectures on relativity theory. Rev. Mod. Phys. 29, 325–333 (1957)
Norton J. D.: Did Einstein stumble? The debate over general covariance. Erkenntnis 42, 223–245 (1995)
Bernstein B.: Hypo-elasticity and elasticity. Arch. Ration. Mech. Anal. 6, 90–104 (1960)
Simo J.C., Pister K.S.: Remarks on rate constitutive equations for finite deformation problems: computational implications. Comput. Methods Appl. Mech. Eng. 46, 201–215 (1984)
Xiao H, Bruhns O.T., Meyers A.: On the existence and uniqueness of the integrable exactly hypoelastic equation \({\dot {\tau}^{{\ast}}=\lambda ({\rm tr}{\mathbf {D}}){\mathbf {I}}+{2} \mu {\mathbf {D}}}\) and its significance to finite inelasticity. Acta Mech. 138, 31–50 (1999)
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Panoskaltsis, V.P., Soldatos, D. Material covariant constitutive laws for continua with internal structure. Acta Mech 227, 881–898 (2016). https://doi.org/10.1007/s00707-015-1436-x
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DOI: https://doi.org/10.1007/s00707-015-1436-x