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Implications and Constraints of Time-Independent Poisson Ratios in Linear Isotropic and Anisotropic Viscoelasticity

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Abstract

It is proven that time-independent viscoelastic Poisson ratios (PR) can only exist under separation of variable solutions which severely limits the class of applicable problems to quasi-static ones with incompressible homogeneous materials and non-moving boundaries under separable stress or displacement boundary conditions without any thermal expansions. Therefore, composites which are inherently anisotropic and sandwich structures which are nonhomogeneous and anisotropic are generally precluded from having time-independent PRs. Equal time variations for material properties in all directions are shown to be another simultaneous requirement instead of the incompressibility condition for achieving time-independent PRs. However, such restricted models lead to physically unrealistic bulk moduli responses when compared to experimentally determined relaxation moduli and are not generally achievable in current real materials. Consequently, viscoelastic materials are best characterized in terms of relaxation or creep functions, moduli or compliances rather than combinations of the latter with Poisson's ratios. Additionally, the assumption of constant PRs in problems involving thermal and chemical expansions, such as curing and manufacture of viscoelastic composites, is shown to be unjustified and insupportable. The distinct viscoelastic PR definitions, as found in the literature, are examined and classified into five categories. It is further shown that each is inherently unrelated to the others and all are always time-dependent, unless the above extremely limiting conditions are imposed. An extensive literature review indicates that experimental results overwhelmingly confirm the time dependent nature of viscoelastic PRs as no constant experimentally observed PRs were reported.

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REFERENCES

  1. W.T. Kelvin, On the elasticity and viscosity of metals. Proc. Royal Soc. 14(1865) 289–297.

    Google Scholar 

  2. J.C. Maxwell, On the dynamical theory of gases. Phil. Trans. Royal Soc. London A 157(1867) 49–88.

    Google Scholar 

  3. W. Voigt, Theoretischen Studien über das Elasticitatsverhaltnisse der Kristalle. Abhandlung Geselschaft der Wissenschaft zu Göttingen 34(1887) 52–79.

    Google Scholar 

  4. V. Volterra, Sulle equazioni integrodifferenzialli dela teoria dell'elastica. Atti Reale Accademia Nazionale dei Lincei 18(1909) 295–301.

    Google Scholar 

  5. T. Alfrey, Jr., Non-homogeneous stress in viscoelastic media. Quart. Appl. Math. 2(1944) 113–119.

    Google Scholar 

  6. T. Alfrey, Jr., Mechanical Behavior of High Polymers. Interscience Publishers, New York (1948).

    Google Scholar 

  7. D.R. Bland, The Theory of Linear Viscoelasticity. Pergamon Press, New York (1960).

    Google Scholar 

  8. R.M. Christensen, Theory of Viscoelasticity – An Introduction, 2nd edn. Academic Press, New York (1981).

    Google Scholar 

  9. B.D. Coleman and W. Noll, The foundations of linear viscoelasticity. Rev. Modern Phys. 33(1961) 239–249.

    Google Scholar 

  10. B.D. Coleman, Thermodynamics of materials with memory. Arch. Rational Mech. Anal. 17(1964) 1–46.

    Google Scholar 

  11. A. Drozdov, Viscoelastic Structures – Mechanics of Growth and Aging. Academic Press, New York (1998).

    Google Scholar 

  12. J.D. Ferry, Viscoelastic Properties of Polymers.Wiley, New York (1961).

    Google Scholar 

  13. B. Gross, Mathematical Structure of the Theories of Viscoelasticity. Hermann, Paris (1953).

    Google Scholar 

  14. M.E. Gurtin, Variational principles in the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 13(1963) 179–185.

    Google Scholar 

  15. M.E. Gurtin and E. Sternberg, On the linear theory of viscoelasticity. Arch. Rational Mech. Anal. 11(1962) 291–356.

    Google Scholar 

  16. M.E. Gurtin and E. Sternberg, A reciprocal theory of anisotropic viscoelastic solids. SIAM J. 11(1963) 607–613.

    Google Scholar 

  17. A.C. Pipkin, Lectures on Viscoelasticity Theory. Springer-Verlag, Berlin (1972).

  18. M. Renardy, W.J. Hrusa and J.A. Nohel, Mathematical Problems in Viscoelasticity. Longman, New York (1987).

    Google Scholar 

  19. N.W. Tschoegl, The Phenomenological Theory of Linear Viscoelasticity. Springer-Verlag, Berlin (1989).

    Google Scholar 

  20. H.H. Hilton and S. Yi, The significance of (an)isotropic viscoelastic Poisson ratio stress and time dependencies. Inter. J. Sol. Struct. 35(1998) 3081–3095.

    Google Scholar 

  21. S.D. Poisson, Mémoire sur l'équilibre et le mouvement des corps élastiques. Mémoires de l'Académie Royal des Sciences de l'Institut de France 8(1829) 357–570.

    Google Scholar 

  22. S.D. Poisson, Addition au mémoire sur l'équilibre et le mouvement des corps élastiques. Mémoires de l'Académie Royal des Sciences de l'Institut de France 8(1829) 623–627.

    Google Scholar 

  23. K. Ravi-Chandar, Simultaneous measurement of nonlinear bulk and shear relaxation behavior. In: Proceedings of the 2nd International Conference on Mechanics of Time-Dependent Materials, SEM, Ljubljana, Slovenia (1998), pp. 30–31.

    Google Scholar 

  24. K. Ravi-Chandar, Inelastic deformation in polymers under multiaxial compression. Mechanics of Time-Dependent Materials 4(2000) 333–357.

    Google Scholar 

  25. H.H. Hilton, On the inadmissibility of separation of variable solutions in linear anisotropic viscoelasticity. Internat. J. Mech. Comp. Mat. Struct. 3(1996) 97–100.

    Google Scholar 

  26. H. Bertilsson, M. Delin, J. Kubát, W.R. Rychwalski and M.J. Kubát, Strain rates and volume changes during short-term creep of PC and PMMA. Rheologica Acta 32(1993) 361–369.

    Google Scholar 

  27. A.D. Drozdov, Mechanics of Viscoelastic Solids. Wiley, New York (1998).

    Google Scholar 

  28. W.N. Findley, J.S. Lai and K. Onaran, Creep and Relaxation of Nonlinear Viscoelastic Materials with an Introduction to Linear Viscoelasticity. North-Holland, New York (1976).

    Google Scholar 

  29. W. Flügge, Viscoelasticity. Springer-Verlag, New York (1975).

    Google Scholar 

  30. H.H. Hilton, An introduction to viscoelastic analysis. In: E. Baer (ed.), Engineering Design for Plastics, Reinhold Publishing Corp., New York (1964), pp. 199–276.

    Google Scholar 

  31. Yu.N. Rabotnov, Elements of Hereditary Solid Mechanics. Mir, Moscow (1980).

    Google Scholar 

  32. N. Kh. Arutyunyan, Some Problems in the Theory of Creep.Gostekizdat, Moscow (1952) (in Russian). Also Pergamon Press (1966).

    Google Scholar 

  33. P.B. Benham and D. McCammond, Studies of creep and contraction ratio in thermoplastics. Plastics and Polymers 39(1971) 130–136.

    Google Scholar 

  34. A.M. Freudenthal and L.A. Henry, On “Poisson's ratio” in linear visco-elastic propellants. In: M. Summerfield (ed.), American Rocket Society Solid Propellant Rocket Conference, Princeton University (1960), pp. 33–66.

  35. A.M. Freudenthal, One-dimensional response and coefficient of thermal expansion in timesensitive materials. Acta Technica 41(1962) 415–449.

    Google Scholar 

  36. D. Göritz, Messung der Volumenänderung bei uniaxialen Dehnen. Colloid and Polymer Sci. 260(1982) 193–197.

    Google Scholar 

  37. W.G. Gottenberg and R.M. Christensen, Some interesting aspects of general linear viscoelastic deformation. Trans. Soc. Rheology 7(1963) 171–180.

    Google Scholar 

  38. I.H. Hwang, Thermo-viscoelastic behavior of composite materials. Ph.D. Thesis, University of Washington, Seattle (1990).

    Google Scholar 

  39. Y.K. Kim and S.R. White, Stress relaxation during cure of 3501-6 epoxy resin. In: Proc. Symp. Design and Manufacture of Composites, 1995 ASME Annual Winter Meeting, MID-69-1 (1995), pp. 43–56.

  40. R.S. Lakes, The time dependent Poisson's ratio of viscoelastic cellular materials can increase or decrease. Cellular Polymers 10(1991) 466–469.

    Google Scholar 

  41. L.E. Nielsen, Stress dependence of Poisson's ratio and the softening temperature of plastics. Trans. Soc. Rheology 9(1965) 243–254.

    Google Scholar 

  42. L.E. Nielsen and R.F. Landel, Mechanical Properties of Polymers and Composites, 2nd edn. Marcel Dekker, New York (1994).

    Google Scholar 

  43. D.J. O'Brien, P.T. Mather and S.R. White, Viscoelastic properties of an epoxy resin during cure. J. Comp. Mat. 35(2001) 883–904.

    Google Scholar 

  44. K.G. Popov and K.B. Khadzhov, Relationships between relaxation characteristics and creep for isotropic materials. Mech. Comp. Mat. 16(1980) 6–10.

    Google Scholar 

  45. J.M. Powers and R.M. Caddell, The macroscopic volume changes of selected polymers subjected to uniform tensile deformation. Polymer Eng. Sci. 12(1972) 432–436.

    Google Scholar 

  46. Z. Rigbi, The value of Poisson's ratio of viscoelastic materials. Appl. Polymer Symp. 5(1967) 1–7.

    Google Scholar 

  47. F. Schwarzl, Linear viscoelastic behaviour of isotropic materials – transient measurements. Kolloid Zeitschrift 148(1956) 47–57.

    Google Scholar 

  48. A.J. Staverman and F. Schwarzl, Linear deformation behaviour of high polymers. In: H.A. Stuart (ed.), Die Physik der Hochpolymeren4, Springer-Verlag, New York (1956), pp. 1–95.

    Google Scholar 

  49. V.K. Stokes and H.F. Nied, Lateral strain effect during the large extension of thermoplastics. Polymer Eng. Sci. 28(1988) 1209–1218.

    Google Scholar 

  50. P.S. Theocaris, Creep and relaxation contraction ratio of linear viscoelastic materials, J. Mech. Phys. Solids 12(1964) 125–138.

    Google Scholar 

  51. P.S. Theocaris, Influence of plasticizer on Poisson's ratio of epoxy polymers. Polymer 20(1979) 1149–1152.

    Google Scholar 

  52. G.V. Vinogradov and A.Ya. Malkin, Rheology of Polymers – Viscoelasticity and Flow of Polymers. Mir, Moscow (1980).

    Google Scholar 

  53. S.R. White and A.B. Hartman, Effect of cure state on stress relaxation in 3501-6 epoxy resin. ASME J. Eng. Mat. Technology 119(1997) 262–265.

    Google Scholar 

  54. J.G. Williams, Stress Analysis of Polymers, 2nd edn. Ellis Horwood, Chichester (1980).

    Google Scholar 

  55. R. de Prony, Essai experimental et analytique. J. l'École Polytechnique de Paris 1(1795) 24–76.

    Google Scholar 

  56. H.H. Hilton and S.B. Dong, An analogy for anisotropic, nonhomogeneous, linear viscoelasticity including thermal stresses. In: Development in Mechanics, Pergamon Press, New York (1964), pp. 58–73.

    Google Scholar 

  57. M.A. Biot, Linear thermodynamics and the mechanics of solids. In: Proc. Third US Nat. Congr. Appl. Mech.(1958), pp. 1–18.

  58. H.S. Tsien, A generalization of Alfrey's theorem in viscoelastic media. Quart. Appl. Math. 8(1950) 104–106.

    Google Scholar 

  59. H.H. Hilton, Thermal distributions without thermal stresses in nonhomogeneous media. ASME J. Appl. Mech. E 26(1959) 137–138.

    Google Scholar 

  60. H.H. Hilton and H.G. Russell, An extension of Alfrey's analogy to thermal stress problems in temperature dependent linear viscoelastic media. J. Mech. Phys. Sol. 9(1961) 152–164.

    Google Scholar 

  61. H.H. Hilton and J.R. Clements, Formulation and evaluation of approximate analogies for temperature-dependent linear viscoelastic media. In: Thermal Loading and Creep, Institution of Mechanical Engineers, London (1964), pp. 6-17–6-24.

    Google Scholar 

  62. A.E. Green and W. Zerna, Theoretical Elasticity. Oxford Univ. Press, New York (1968).

    Google Scholar 

  63. S. Yi, H.H. Hilton and M.F. Ahmad, Nonlinear thermoviscoelastic analysis of interlaminar stresses in laminated composites. ASME J. Appl. Mech. 63(1966) 218–224.

    Google Scholar 

  64. A.D. Drozdov, Finite Elasticity and Viscoelasticity – A Course in the Nonlinear Mechanics of Solids. World Scientific, Singapore.

  65. Z. Li, Effective creep Poisson's ratio for damaged concrete. Int. J. Fract. 66(1994) 189–196.

    Google Scholar 

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  1. Aeronautical and Astronautical Engineering Department, National Center for Supercomputing Applications, University of Illinois at Urbana-Champaign, 104 South Wright Street, MC–236, Urbana, IL, 61801-2935, U.S.A.

    Harry H. Hilton

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Hilton, H.H. Implications and Constraints of Time-Independent Poisson Ratios in Linear Isotropic and Anisotropic Viscoelasticity. Journal of Elasticity 63, 221–251 (2001). https://doi.org/10.1023/A:1014457613863

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