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Modeling a class of geological materials

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Abstract

In this paper we consider the development of rate type viscoelastic models from two different perspectives, using the Helmholtz potential based on the deformation gradient and the rate of entropy production, and the Gibbs potential based on the stress and the rate of entropy production. These two methods are not equivalent and one cannot always be obtained from the other by using a Legendre transformation. Also, it is not even possible for certain classes of materials to define a Helmholtz potential for bodies belonging to such classes, and for certain other classes to define a Gibbs’ potential for bodies belonging to them. The Helmholtz potential formulation has been shown to be capable of modeling generalizations of the Maxwell, Oldroyd-B, Burgers’ models as well as models that can be cast within the context of the conformation tensor. These models can also describe the evolution of the anisotropy of such materials during the deformation process. The Gibbs’ potential formulation is particularly well suited to the development of models wherein the material moduli that characterize the body are dependent on the invariants of the stress. Thus, such models can be used to describe bodies whose material moduli depend on the mean normal stress and could be very useful in characterizing geological materials. Such a framework would also be particularly relevant to describing the problem of compaction of asphalt concrete as the material property of the compacted body would depend on the mean normal stress which effects the compaction.

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Notes

  1. It is not necessary to restrict ourselves to the requirement that the body possess instantaneous elastic response. In fact, there is considerable experimental evidence to show that certain asphalts do not exhibit instantaneous elastic response. The thermodynamic framework that is outlined in this paper has been used to describe such response [16].

  2. It is not clear whether by pressure in these studies one means the mean normal stress in the fluid or the thermodynamic pressure. In fact, in Andrade’s paper, and the work of Bridgman [21], the pressure of the confining fluid is treated as the pressure that appears in the formula for the viscosity and the not the pressure within the actual fluid of interest. The pressure of the confining fluid is treated as though it were the pressure in the fluid of interest whose dependence of viscosity on pressure is being determined. Moreover, the pressure is considered to be the same everywhere in the flow domain, and while this is clearly incorrect in that the pressure field will vary over the flow domain, it might however be a very good approximation to being that within the confining fluid. It is also worth observing that the viscosity is inferred by assuming that the drag that an object suffers is the Stokes drag.

  3. A viscoelastic solid model whose material moduli depend on the Lagrange multiplier that enforces the constraint of incompressibility (not the mean normal stress) has been investigated by Rajagopal and Saccomandi [25]. However, the model has not been derived within a thermodynamic framework.

  4. The notion of “natural configuration” was discussed by Eckart [52] within the context of the inelastic response of solids. He however did not discuss issues concerning the material symmetry of the natural configuration and how it evolves. The notion of natural configuration plays a critical role in the response of entropy producing bodies and a detailed discussion of the notion can be found in the report by Rajagopal [53].

  5. If the deformations are not homogeneous, one cannot always unload to κ p(t)(B) such that κ p(t)(B) is stress-free and geometrically contiguous, and \({\bf F}_{\kappa_{p({\rm t})}}\) is in general not the gradient of a function. One then needs to work within the context of non-Euclidean geometry to define these concepts [see 38, 39, for a more detailed discussion].

  6. In general G is not the gradient of a mapping but is a mapping from appropriate tangent space at \({\bf X} \in \kappa _{R}(B)\) to the tangent space at \({\bf x} \in \kappa_{t}(B). \)

References

  1. Burgers, J.: Mechanical considerations model systems phenomenological theories of relaxation and of viscosity. In: First Report on Viscosity and Plasticity, 2nd edn. Nordemann Publishing Company, Inc., New York (1939)

  2. Saal, R., Labout, J.: Rheological properties of asphaltic bitumens. J. Phys. Chem. 44(2), 149–165 (1940)

    Article  Google Scholar 

  3. Lethersich, W.: The mechanical behavior of bitumen. J. Soc. Chem. Indus. 61, 101–108 (1942)

    Article  Google Scholar 

  4. Frohlick, H., Sack, R.: Theory of the rheological properties of dispersion. Proc. R. Soc. Lond. A: Math. Phys. Sci. 186, 415–430 (1946)

    Article  Google Scholar 

  5. Krishnan, J.M., Rajagopal, K.R.: Review of the uses and modeling of bitumen from ancient to modern times. Appl. Mech. Rev. 56, 149–214 (2003)

    Article  Google Scholar 

  6. Detlef, W., Falk, A.: Viscoelastic perturbations of the earth: significance of the incremental gravitational force in models of glacial isostasy. Geophys. J. Int. 117(3), 864–879 (1994)

    Article  Google Scholar 

  7. Weertman, J., White, S., Cook, A.H.: Creep laws for the mantle of the earth [and discussion]. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 288(1350), 9–26 (1978)

    Google Scholar 

  8. Peltier, W., Wu, P., Yuen, D.: The viscosities of the earth mantle. In: Stacey F, Paterson M, Nicholas A (eds) Anelasticity in the Earth. American Geophysical Union, Colorado (1981)

  9. Yueu, D.A., Peltier, W.: Normal modes of the viscoelastic earthy geophys. J. R. Astron. Earth 69, 495–526 (1982)

    Google Scholar 

  10. Miller, G.: Generalized Maxwell bodies and estimate of mantle viscosity. Geophys. I: R. Astron. Earth 87, 1113–1141 (1986)

    Google Scholar 

  11. Rumpker, G., Wolf, D.: Viscoelastic relaxation of a Burgers half-space: implications for the interpretation of the Fennoscandia uplift. Geophys. J. Int. 124, 541–555 (1996)

    Article  Google Scholar 

  12. Krishnan, J.M., Rajagopal, K.R.: Thermodynamic framework for the constitutive modeling of asphalt concrete: Theory and applications. J. Mater. Civil Eng. 16(2), 155–166 (2004)

    Article  Google Scholar 

  13. Krishnan, J.M., Rajagopal, K.R.: Triaxial testing and stress relaxation of asphalt concrete. Mech. Mater. 36(9), 849–864 (2004)

    Article  Google Scholar 

  14. Saradhi, K., Eyad, M., Rajagopal, K.R.: A thermomechanical framework for modeling the compaction of asphalt mixes. Mech. Mater. 40(10), 846–864 (2008)

    Article  Google Scholar 

  15. Ravindran, P., Krishnan, J.M., Masad, E., Rajagopal, K.R.: Modeling sand-asphalt mixtures within a thermodynamic framework; theory and applications to torsion experiments. Int. J. Pavement Eng. 10, 115–131 (2009)

    Article  Google Scholar 

  16. Karra, S., Rajagopal, K.R.: A thermodynamic framework to develop rate-type models for fluids without instantaneous elasticity. Acta Mech. 205, 105–119 (2009)

    Article  MATH  Google Scholar 

  17. Atul, N., Deshpande, A.P., Krishnan, J.M., Rajagopal, K.R.: Experiments and modeling of torque and normal force relaxation of bituminous materials. To be submitted to Mech. Mater. (2011)

  18. Chockalingam, K., Saravanan, U., Krishnan, J.M.: Characterization of petroleum pitch using steady shear experiments. Int. J. Eng. Sci. 48(11), 1092–1109 (2010)

    Article  Google Scholar 

  19. Gauss, C.F.: Uber ein neues allgemeines grundgesatz der mechanik. Journal four die Reine und Angewandte Mathematik 4, 232–235 (1829)

    Article  MATH  Google Scholar 

  20. Rajagopal, K.R., Srinivasa, A.R.: On the nature of constraints for continua undergoing dissipative processes. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 461(2061), 2785–2795 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Bridgman, P.W.: The Physics of High Pressures. Macmillan Co, NY (1931)

    Google Scholar 

  22. Barus, C.: The fusion constants of igneous rock iii the thermal capacity of igneous rock, considered in its bearing on the relation of melting-point to pressure. Phil. Mag. 35, 296–307 (1893)

    Google Scholar 

  23. Dowson, D., Higginson, G.R.: Elastohydrodynamic Lubrication: The Fundamentals of Roller and Gear Lubrication. Pergamon, Oxford (1966)

    Google Scholar 

  24. Krishnan, J.M., Rajagopal, K.R., Masad, E., Little, D.N.: Thermomechanical framework for the constitutive modeling of asphalt concrete. Int. J. Geomech. 6(1), 36–45 (2006)

    Article  Google Scholar 

  25. Rajagopal, K.R., Saccomandi, G.: The mechanics and mathematics of the effect of pressure on the shear modulus of elastomers. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 465, 3859–3874 (2009)

    Article  MATH  Google Scholar 

  26. Singh, H., Nolle, A.W.: Pressure dependence of the viscoelastic behavior of polyisobutylene. J. Appl. Phys. 30(3), 337–341 (1959)

    Article  Google Scholar 

  27. McKinney, J.E., Belcher, H.V.: Dynamic compressibility of polyvinylacetate and its relation to free volume. J. Res. Nat. Bur. Stand. A: Phys. Chem. A 67(1), 43–53 (1963)

    Google Scholar 

  28. Ivins, B.R., Sammis, C., Yoder, C.: Deep mantle viscous structure with prior estimate and satellite constraint. J. Geophys. Res. 98, 4579–4609 (1993)

    Article  Google Scholar 

  29. Sahaphol, T., Miura, S.: Shear moduli of volcanic soils. Soil Dyn. Earthq. Eng. 25(2), 157–165 (2005)

    Article  Google Scholar 

  30. Karra S., Prusa V., Rajagopal K.R.: On Maxwell fluid with relaxation time and viscosity depending on the pressure. Int. J. Non-linear Mech. (2010). doi:10.1016/j.ijnonlinmec.2011.02.013

  31. Onsager, L.: Reciprocal relations in irreversible processes. i. Phys. Rev. 37(4), 405–426 (1931)

    Article  MATH  Google Scholar 

  32. Onsager, L.: Reciprocal relations in irreversible processes ii. Phys. Rev. 38(12), 2265–2279 (1931)

    Article  MATH  Google Scholar 

  33. Prigogine, I.: Introduction to Thermodynamics of Irreversible Processes, 3rd edn. Interscience, New York (1967)

  34. Glansdorf, P., Prigogine, I.: Thermodynamic Theory of Stability, Structure, Fluctuation. Wiley, New York (1971)

    Google Scholar 

  35. Rajagopal, K.R., Srinivasa, A.R.: On thermomechanical restrictions of continua. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 460, 631–651 (2004a)

    Article  MathSciNet  MATH  Google Scholar 

  36. Rajagopal, K.R., Srinivasa, A.R.: A thermodynamic framework for rate type fluid models. J. Non-Newton. Fluid Mech. 88, 207–227 (2000)

    Article  MATH  Google Scholar 

  37. Kannan, K., Rajagopal, K.R.: A thermomechanical framework for the transition of a viscoelastic liquid to a viscoelastic solid. Math. Mech. Solids 9, 37–59 (2004)

    MathSciNet  MATH  Google Scholar 

  38. Rajagopal, K.R., Srinivasa, A.R.: Mechanics of the inelastic behavior of materials—part 1, theoretical underpinnings. Int. J. Plast. 14(10-11), 945–967 (1998)

    Article  MATH  Google Scholar 

  39. Rajagopal, K.R., Srinivasa, A.R.: Mechanics of the inelastic behavior of materials. part ii: Inelastic response. Int. J. Plast. 14(10-11), 969–995 (1998)

    Article  MATH  Google Scholar 

  40. Prasad, S.C., Rao, I.J., Rajagopal, K.R.: A continuum model for the creep of single crystal nickel-base superalloys. Acta Mater. 53(3), 669–679 (2005)

    Article  Google Scholar 

  41. Prasad, S.C., Rajagopal, K.R., Rao, I.J.: A continuum model for the anisotropic creep of single crystal nickel-based superalloys. Acta Mater. 54(6), 1487–1500 (2006)

    Article  Google Scholar 

  42. Srinivasa, A.R., Rajagopal, K.R., Armstrong, R.W.: A phenomenological model of twinning based on dual reference structures. Acta Mater. 46, 1235–1248 (1998)

    Article  Google Scholar 

  43. Lapczyk, I., Rajagopal, K.R., Srinivasa, A.R.: Deformation twinning during impact—numerical calculations using a constitutive theory based on multiple natural configurations. Comput. Mech. 21(1), 20–27 (1998)

    Article  MATH  Google Scholar 

  44. Rajagopal, K.R., Srinivasa, A.R.: On the thermomechanics of shape memory wires. ZAMP 50(3), 459–496 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  45. Kannan, K., Rao, I.J., Rajagopal, K.R.: A thermomechanical framework for the glass transition phenomenon in certain polymers and its application to fiber spinning. J. Rheol. 46(4), 977–999 (2002)

    Article  Google Scholar 

  46. Rao, I.J., Rajagopal, K.R.: A thermodynamic framework for the study of crystallization in polymers. ZAMP 53(3), 365–406 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  47. Barot, G., Rao, I.: Constitutive modeling of the mechanics associated with crystallizable shape memory polymers. ZAMP 57(4), 652–681 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  48. Barot, G., Rao, I.J., Rajagopal, K.R.: A thermodynamic framework for the modeling of crystallizable shape memory polymers. Int. J. Eng. Sci. 46, 325–351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Kannan K., Rajagopal K.R. (2010) A thermodynamic framework for chemically reacting systems. Zeitschrift fur Angewandte Mathematick und Physik 62, 331–362 (2011)

    Google Scholar 

  50. Rajagopal, K.R., Srinivasa, A.R.: On the thermomechanics of materials that have multiple natural configurations—part I: viscoelasticity and classical plasticity. Zeitschrift Fur Angewandte Mathematik und Physik 55(5), 861–893 (2004b)

    Article  MathSciNet  MATH  Google Scholar 

  51. 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. Zeitschrift Fur Angewandte Mathematik und Physik 55(6), 1074–1093 (2004c)

    Article  MathSciNet  MATH  Google Scholar 

  52. Eckart, C.: The thermodynamics of irreversible processes iv, the theory of elasticity and anelasticity. Phys. Rev. 73(4), 373–382 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  53. Rajagopal, K.R.: Multiple configurations in continuum mechanics. Technical Reports. Reports of the Institute for Computational and Applied Mechanics, University of Pittsburgh, Pittsburgh (1995)

  54. Krishnan, J.M., Rajagopal, K.R.: On the mechanical behavior of asphalt. Mech. Mater. 37(11), 1085–1100 (2005)

    Article  Google Scholar 

  55. Oseen, C.: Theory of liquid crystals. Trans. Faraday Soc. 29, 883–899 (1953)

    Article  Google Scholar 

  56. Frank, F.: On the theory of liquid crystals. Faraday Soc. 25, 19–28 (1958)

    Article  Google Scholar 

  57. Ericksen, J.: Conservation laws of liquid crystals. Trans Soc. Rheol. 5, 23–34 (1961)

    Article  MathSciNet  Google Scholar 

  58. Leslie, F.: Some constitutive equations for liquid crystals. Arch. Rat. Mech. Anal. 28, 265–283 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  59. Leslie, F.M.: Some thermal effects in cholesteric liquid crystals. Proc. R. Soc. Lond. A 307, 359–372 (1968)

    Article  Google Scholar 

  60. DeSilva, C., Kaloni, P.: Theory of oriented fluids. Phys. Fluids 13, 1708–1716 (1970)

    Article  MATH  Google Scholar 

  61. Doi, M.: Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. Polym. Phys. 19, 229–243 (1981)

    Article  Google Scholar 

  62. Rajagopal, K.R., Srinivasa, A.R.: Modeling anisotropic fluids within the framework of bodies with multiple natural configurations. J. Non-Newton. Fluid Mech. 99(2-3), 109–124 (2001)

    Article  MATH  Google Scholar 

  63. Simo, J., Hughes, T.: Computational Inelasticity. Springer, New York (1998)

    MATH  Google Scholar 

  64. Truesdell, C.A.: Hypo-elasticity. J. Ration. Mech. Anal. 4, 323–425 (1955)

    Google Scholar 

  65. Truesdell, C., Noll, W.: The Non-Linear Field Theories of Mechanics, 2nd edn. Springer, Berlin (1992)

    Google Scholar 

  66. Rajagopal, K.R., Srinivasa, A.R.: On a class of non-dissipative materials that are not hyperelastic. Proc. R. Soc. Lond. A: Math. Phys. Eng. Sci. 465, 495–500 (2009)

    MathSciNet  Google Scholar 

  67. Rajagopal K.R., Srinivasa A.R. (2010) A Gibbs potential based formulation for obtaining the response function for a class of viscoelastic materials. Proc. R. Soc. Lond. A doi:101098/rspa20100136

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  1. Department of Mechanical Engineering, Texas A&M University, College Station, TX, USA

    K. R. Rajagopal

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Rajagopal, K.R. Modeling a class of geological materials. Int J Adv Eng Sci Appl Math 3, 2–13 (2011). https://doi.org/10.1007/s12572-011-0029-8

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