Article
MSC:
74A15,
74A20,
74D10,
76A02,
76A05,
76A10 | MR 2491852 | Zbl 1200.76005 | DOI: 10.1007/s10492-009-0010-z
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
constitutive relations; Lagrange multiplier; Helmholtz potential; rate of dissipation; viscoelasticity; Burgers' fluid; maximum entropy production
constitutive relations; Lagrange multiplier; Helmholtz potential; rate of dissipation; viscoelasticity; Burgers' fluid; maximum entropy production
Summary:
Most three dimensional constitutive relations that have been developed to describe the behavior of bodies are correlated against one dimensional and two dimensional experiments. What is usually lost sight of is the fact that infinity of such three dimensional models may be able to explain these experiments that are lower dimensional. Recently, the notion of maximization of the rate of entropy production has been used to obtain constitutive relations based on the choice of the stored energy and rate of entropy production, etc. In this paper we show different choices for the manner in which the body stores energy and dissipates energy and satisfies the requirement of maximization of the rate of entropy production that can all describe the same experimental data. All of these three dimensional models, in one dimension, reduce to the model proposed by Burgers to describe the viscoelastic behavior of bodies.
References:
[1] Burgers, J. M.: Mechanical considerations-model systems-phenomenological theories of relaxation and of viscosity. First Report on Viscosity and Plasticity Nordemann Publishing Company New York (1935).
[2] Green, A. E., Naghdi, P. M.: On thermodynamics and the nature of the second law. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 357 (1977), 253-270. DOI 10.1098/rspa.1977.0166 | MR 0462241
[3] Itskov, M.: On the theory of fourth-order tensors and their applications in computational mechanics. Comput. Methods Appl. Mech. Eng. 189 (2000), 419-438. DOI 10.1016/S0045-7825(99)00472-7 | MR 1781866 | Zbl 0980.74006
[4] Málek, J., Rajagopal, K. R.: A thermodynamic framework for a mixture of two liquids. Nonlinear Anal.--Real World Appl. 9 (2008), 1649-1660. MR 2422570 | Zbl 1154.76311
[5] Maxwell, J. C.: On the dynamical theory of gases. Philos. Trans. Roy. Soc. London 157 (1867), 49-88. DOI 10.1098/rstl.1867.0004
[6] J. Murali Krishnan, Rajagopal, K. R.: Thermodynamic framework for the constitutive modeling of asphalt concrete: Theory and applications. J. Mater. Civ. Eng. 16 (2004), 155-166. DOI 10.1061/(ASCE)0899-1561(2004)16:2(155)
[7] Oldroyd, J. G.: On the formulation of rheological equation of state. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 200 (1950), 523-591. DOI 10.1098/rspa.1950.0035 | MR 0035192
[8] Rajagopal, K. R.: Multiple configurations in continuum mechanics. Report Vol. 6 Institute for Computational and Applied Mechanics, University of Pittsburgh Pittsburgh (1995).
[9] Rajagopal, K. R.: On implicit constitutive theories. Appl. Math. 48 (2003), 279-319. DOI 10.1023/A:1026062615145 | MR 1994378 | Zbl 1099.74009
[10] Rajagopal, K. R., Srinivasa, A. R.: Mechanics of the inelastic behavior of materials. Part II: Inelastic response. Int. J. Plast. 14 (1998), 969-995. DOI 10.1016/S0749-6419(98)00041-2
[11] Rajagopal, K. R., Srinivasa, A. R.: A thermodynamic framework for rate type fluid models. J. Non-Newtonian Fluid Mech. 88 (2000), 207-227. DOI 10.1016/S0377-0257(99)00023-3 | Zbl 0960.76005
[12] Rajagopal, K. R., Srinivasa, A. R.: On the thermomechanics of materials that have multiple natural configurations. Part I: Viscoelasticity and classical plasticity. Z. Angew. Math. Phys. 55 (2004), 861-893. DOI 10.1007/s00033-004-4019-6 | MR 2087769 | Zbl 1180.74006
[13] Rajagopal, K. R., Srinivasa, A. R.: On the thermomechanics of materials that have multiple natural configurations. Part II: Twinning and solid to solid phase transformation. Z. Angew. Math. Phys. 55 (2004), 1074-1093. DOI 10.1007/s00033-004-4020-0 | MR 2100532
[14] Rajagopal, K. R., Srinivasa, A. R.: On thermomechanical restrictions of continua. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 460 (2004), 631-651. DOI 10.1098/rspa.2002.1111 | MR 2034660 | Zbl 1041.74002
[15] Rajagopal, K. R., Srinivasa, A. R.: On the thermodynamics of fluids defined by implicit constitutive relations. Z. Angew. Math. Phys. 59 (2008), 715-729. DOI 10.1007/s00033-007-7039-1 | MR 2417387 | Zbl 1149.76007
[16] Rao, I. J., Rajagopal, K. R.: On a new interpretation of the classical Maxwell model. Mech. Res. Comm. 34 (2007), 509-514. DOI 10.1016/j.mechrescom.2007.07.001 | MR 2372417 | Zbl 1192.74058
[17] Ziegler, H.: Some extremum principles in irreversible thermodynamics. In: Progress in Solid Mechanics, Vol. 4 I. N. Sneddon, R. Hill North Holland New York (1963). MR 0163470
Search
Partner of
