Abstract
This paper presents constitutive theories for finite deformation of homogeneous, isotropic thermoelastic solids in Lagrangian description using Gibbs potential. Since conservation of mass, balance of momenta and the energy equation are independent of the constitution of the matter, the second law of thermodynamics, that is, entropy inequality, must form the basis for all constitutive theories of the deforming matter to ensure thermodynamic equilibrium during the evolution (Surana and Reddy in Continuum mechanics, 2012; Eringen in Nonlinear theory of continuous media. McGraw-Hill, New York, 1962). The entropy inequality expressed in terms of Helmholtz free energy is recast in terms of Gibbs potential. The conditions resulting from the entropy inequality expressed in terms of Gibbs potential permit the derivation of the constitutive theory for the strain tensor in terms of the conjugate stress tensor and the constitutive theory for the heat vector. In the work presented here, it is shown that using the conditions resulting from the entropy inequality, the constitutive theory for the strain tensor can be derived using three different approaches: (i) assuming the Gibbs potential to be a function of the invariants of the conjugate stress tensor and then using the conditions resulting from the entropy inequality, (ii) using theory of generators and invariants and (iii) expanding Gibbs potential in the conjugate stress tensor using Taylor series about a known configuration and then using the conditions resulting from the entropy inequality. The constitutive theories resulting from these three approaches are compared for equivalence between them as well as their merits and shortcomings. The constitutive theory for the heat vector can also be derived either directly using the conditions resulting from the entropy inequality or using the theory of generators and invariants. The derivation of the constitutive theory for the heat vector using the theory of generators and invariants with the complete set of argument tensors yields a more comprehensive constitutive theory for the heat vector. In the work, we consider both approaches. Summaries of the constitutive theories using parallel approaches (as described above) resulting from the entropy inequality expressed in terms of Helmholtz free-energy density are also presented and compared for equivalence with the constitutive theories derived using Gibbs potential.
Similar content being viewed by others
References
Surana, K.S., Reddy, J. N.: Continuum mechanics (2012) (Manuscript of the textbook in preparation)
Eringen A.C.: Nonlinear Theory of Continuous Media. McGraw-Hill, New York (1962)
Eringen A.C.: Mechanics of Continua. Wiley, New York (1967)
Gurtin M.: The continuum mechanics of coherent two-phase elastic solids with mass transport. Proc. R. Soc. A Math. Phys. Eng. Sci. 440(1909), 323–343 (1993)
Schapery R.A.: Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. J. Appl. Phys. 35, 1451–1465 (1964)
Shapiro N.Z., Shapley L.S.: Mass action laws and the Gibbs free energy function. J. Soc. Ind. Appl. Math. 13, 353–375 (1965)
Landel R.F., Peng S.T.J.: Equations of state and constitutive equations. J. Rheol. 30, 741–765 (1986)
Hassanizadeh S.M.: Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries. Adv. Water Resour. 13, 169–186 (1990)
Stevens R.N., Guiu F.: Balance concepts in the physics of fracture. Proc. R. Soc. A: Math. Phys. Eng. Sci. 435, 169–184 (1991)
Lustig, S.R., Shay, Jr., R.M., Caruthers, J.M.: Thermodynamic constitutive equations for materials with memory on a material time scale. J. Rheol. 40 (1996)
Maire J.F., Chaboche J.L.: A new formulation of continuum damage mechanics (CDM) for composite materials. Aerosp. Sci. Technol. 1, 247–257 (1997)
Gray W.G., Hassanizadeh S.M.: Macroscale continuum mechanics for multiphase porous-media flow including phases, interfaces, common lines and common points. Adv. Water Resour. 21, 261–281 (1998)
O’Rielly O.M.: On constitutive relations for elastic rods. Int. J. Solids Struct. 35, 1009–1024 (1998)
Fischer F.D., Oberaigner E.R., Tanaka K., Nishimura F.: Transformation induced plasticity revised an updated formulation. Int. J. Solids Struct. 35, 2209–2227 (1998)
Houlsby G.T., Puzrin A.M.: A thermomechanical framework for constitutive models for rate-independent dissipative materials. Int. J. Plast. 16, 1017–1047 (2000)
Houlsby, G.T., Puzrin, A.M.: Rate-dependent plasticity models derived from potential functions. J. Rheol. 46 (2002)
Lubarda V.A.: On thermodynamic potentials in linear thermoelasticity. Int. J. Solids Struct. 41, 7377–7398 (2004)
Zhao, J., Sheng, D., Collins, I.F.: Thermomechanical formulation of strain gradient plasticity for geomaterials. J. Mech. Mater. Struct. 1, (2006)
Rajagopal K.R., Srinivasa A.R.: A Gibbs-potential-based formulation for obtaining the response functions for a class of viscoelastic materials. Proc. R. Soc. A: Math. Phys. Eng. Sci. 467, 39–58 (2010)
Bridges, C.: Implicit rate-type models for elastic bodies: development, integration, linearization & application. PhD thesis, Texas A&M University, (2011)
Prager W.: Strain hardening under combined stresses. J. Appl. Phys. 16, 837–840 (1945)
Reiner M.: A mathematical theory of dilatancy. Am. J. Math. 67, 350–362 (1945)
Rivlin R.S., Ericksen J.L.: Stress-deformation relations for isotropic materials. J. Rat. Mech. Anal. 4, 323–425 (1955)
Rivlin R.S.: Further remarks on the stress-deformation relations for isotropic materials. J. Rat. Mech. Anal. 4, 681–702 (1955)
Todd J.A.: Ternary quadratic types. Philos. Trans. R. Soc. Lond. Ser. A: Math. Phys. Sci. 241, 399–456 (1948)
Wang C.C.: On representations for isotropic functions, part I. Arch. Rat. Mech. Anal. 33, 249 (1969)
Wang C.C.: On representations for isotropic functions, part II. Arch. Rat. Mech. Anal. 33, 268 (1969)
Wang C.C.: A new representation theorem for isotropic functions, part I and part II. Arch. Rat. Mech. Anal. 36, 166–223 (1970)
Wang C.C.: Corrigendum to ‘Representations for Isotropic Functions’. Arch. Rat. Mech. Anal. 43, 392–395 (1971)
Smith G.F.: On a fundamental error in two papers of C.C. Wang, ‘On Representations for Isotropic Functions, Part I and Part II’. Arch. Rat. Mech. Anal. 36, 161–165 (1970)
Smith G.F.: On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors. Int. J. Eng. Sci. 9, 899–916 (1971)
Spencer A.J.M., Rivlin R.S.: The theory of matrix polynomials and its application to the mechanics of isotropic continua. Arch. Rat. Mech. Anal. 2, 309–336 (1959)
Spencer A.J.M., Rivlin R.S.: Further results in the theory of matrix polynomials. Arch. Rat. Mech. Anal. 4, 214–230 (1960)
Spencer, A.J.M.: Theory of Invariants. Chapter 3 ‘Treatise on Continuum Physics, I’. In: Eringen, A.C. (ed.). Academic Press, London (1971)
Boehler J.P.: On irreducible representations for isotropic scalar functions. J. Appl. Math. Mech. Zeitschrift für Angewandte Mathematik und Mechanik 57, 323–327 (1977)
Zheng Q.S.: On the representations for isotropic vector-valued, symmetric tensor-valued and skew-symmetric tensor-valued functions. Int. J. Eng. Sci. 31, 1013–1024 (1993)
Zheng Q.S.:) On transversely isotropic, orthotropic and relatively isotropic functions of symmetric tensors, skew-symmetric tensors, and vectors. Int. J. Eng. Sci. 31, 1399–1453 (1993)
Surana, K.S., Nunez, D., Reddy, J. N., Romkes, A.: Rate constitutive theory for ordered thermofluids. Continuum Mech. Thermodyn. (2012). doi:10.1007/s00161-012-0257-6
Surana, K.S., Nunez D., Reddy, J.N., Romkes, A.: Rate constitutive theory for ordered thermoelastic solids. Ann. Solid Struct. Mech. (2012) (in print)
Surana, K.S., Moody, T., Reddy, J.N.: Rate constitutive theories in Lagrangian description for thermoelastic solids. Mech. Adv. Mater. Struct. (2013) (in print)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Surana, K.S., Mendoza, Y. & Reddy, J.N. Constitutive theories for thermoelastic solids in Lagrangian description using Gibbs potential. Acta Mech 224, 1019–1044 (2013). https://doi.org/10.1007/s00707-012-0805-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-012-0805-y