The elusive and fickle viscoelastic poisson's ratio and its relation to the elastic-viscoelastic correspondence principle

Journal of Mechanics of Materials and Structures

Volumn 4, Issue 7-8, 2009, Pages 1341-1364

  Hilton, Harry H ^a  

^a University of Illinois at Urbana Champaign   (United States)

Author keywords

Bernoulli Euler beams; Correspondence principle; Material characterization; Poisson's ratio; Timoshenko shear coefficients; Viscoelasticity

Indexed keywords

EID: 73849094239     PISSN: 15593959     EISSN: None     Source Type: Journal    
DOI: 10.2140/jomms.2009.4.1341     Document Type: Article

References (65)

1. T. Alfrey, Jr., "Non-homogeneous stress in viscoelastic media", Quart. Appl. Math. 2 (1944), 113-119.
2. T. Alfrey, Jr., Mechanical behavior of high polymers, High Polymers 6, Interscience, New York, 1948.
3. I. V. Andrianov, J. Awrejcewicz, and L. I. Manevich, Asymptotical mechanics of thin-walled structures: a handbook, Springer, Berlin, 2004. Pages 244-245.
4. H. di Bernedetto, B. Delaporte, and C. Sauzéat, "Three-dimensional linear behavior of bituminous materials: experiments and modeling", Int. J. Geomech. (ASCE) 7:2 (2007), 149-157.
5. C. W. Bert, "Simplified analysis of static shear factors for beams of nonhomogeneous cross section", J. Compos. Mater. 7:4 (1973), 525-529.
6. 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", Rheol. Acta 32:4 (1993), 361-369.
7. M. Bieniek, L. A. Henry, and A. M. Freudenthal, "One-dimensional response of linear visco-elastic media", pp. 155-174 in Selected papers by Alfred M. Freudenthal: civil engineering classics, ASCE, New York, 1981.
8. T.-M. Chen, "The hybrid Laplace transform/finite element method applied to the quasi-static and dynamic analysis of viscoelastic Timoshenko beams", Int. J. Numer. Methods Eng. 38:3 (1995), 509-522.
9. R. M. Christensen, Theory of viscoelasticity: an introduction, 2nd ed., Academic Press, New York, 1982.
10. G. R. Cowper, "The shear coefficient in Timoshenko's beam theory", J. Appl. Mech. (ASME) 33 (1966), 335-340.
11. A. M. Freudenthal and L. A. Henry, "On Poisson's ratio in linear viscoelastic propellants", pp. 33-66 in Solid propellant rocket research: a selection of technical papers based mainly on a symposium of the American Rocket Society (Princeton University, 1960), Progress in Astronautics and Rocketry 1, Academic Press, New York, 1960.
12. M. Giovagnoni, "On the direct measurement of the dynamic Poisson's ratio", Mech. Mater. 17:1 (1994), 33-46.
13. W. G. Gottenberg and R. M. Christensen, "Some interesting aspects of general linear viscoelastic deformation", J. Rheol. 7:1 (1963), 171-180.
14. H. T. Hahn, "Simplified formulas for elastic moduli of unidirectional continuous fiber composites", Compos. Technol. Rev. 2 (1980), 5-7.
15. H. H. Hilton, "An introduction to viscoelastic analysis", pp. 199-276 in Engineering design for plastics, edited by E. Baer, Reinhold, New York, 1964.
16. H. H. Hilton, "On the inadmissibility of separation of variables solutions in linear anisotropic viscoelasticity", Mech. Compos. Mater. Struct. 3:2 (1996), 97-100.
17. H. H. Hilton, "Implications and constraints of time-independent Poisson ratios in linear isotropic and anisotropic viscoelasticity", J. Elasticity 63:3 (2001), 221-251.
18. H. H. Hilton, "Comments regarding 'Viscoelastic properties of an epoxy resin during cure' by D. J. O'Brien, P. T. Mather and S. R. White", J. Compos. Mater. 37:1 (2003), 89-94.
19. H. H. Hilton, "Viscoelastic Timoshenko beam theory", Mech. Time-Depend. Mat. 13:1 (2009), 1-10.
20. H. H. Hilton and J. R. Clements, "Formulation and evaluation of approximate analogies for temperature dependent linear viscoelastic media", pp. 17-24 (Section 6) in Thermal loading and creep in structures and components (London, 1964), Proc. Institution of Mechanical Engineers 178/3L, Inst. Mech. Eng., London, 1964.
21. H. H. Hilton and S. B. Dong, "An analogy for anisotropic, nonhomogeneous, linear viscoelasticity including thermal stresses", pp. 58-73 in Developments in mechanics: proceedings of the 8th Midwestern Mechanics Conference (Case Institute of Technology, 1963), vol. 2, edited by S. Ostrach and R. H. Scanlan, Pergamon, Oxford, 1965.
22. H. H. Hilton and A. R. A. El Fouly, "Designer auxetic viscoelastic sandwich column materials tailored to minimize creep buckling failure probabilities and prolong survival times", in 48th AIAA/ASME/ASCE/AHS Structures, Structural Dynamics and Materials Conference (Honolulu, HI, 2007), AIAA, Reston, VA, 2007. Paper #2007-2400.
23. H. H. Hilton and J. J. Piechocki, "Shear center motion in beams with temperature-dependent linear elastic or viscoelastic properties", pp. 1279-1289 in Proceedings of the 4th U.S. National Congress of Applied Mechanics (Berkeley, CA, 1962), vol. 2, edited by R. M. Rosenberg, ASME, New York, 1962.
24. 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. Solids 9:3 (1961), 152-164.
25. H. H. Hilton and C. F. Vail, "Bending-torsion flutter of linear viscoelastic wings including structural damping", pp. 1461-1481 in 34th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference (La Jolla, CA, 1993), vol. 3, AIAA, Reston, VA, 1993. Paper #1993-1475.
26. H. H. Hilton and S. Yi, "The significance of anisotropic viscoelastic Poisson ratio stress and time dependencies", Int. J. Solids Struct. 35:23 (1998), 3081-3095.
27. 0.3P", Appl. Phys. Lett. 79:25 (2001), 4160-4162.
28. Z.-H. Jin, "Some notes on the linear viscoelasticity of functionally graded materials materials", Math. Mech. Solids 11:2 (2006), 216-224.
29. [Z.-H. Jin and G. H. Paulino, "A viscoelastic functionally graded strip containing a crack subjected to in-plane loading", Eng. Fract. Mech. 69:14-16 (2002), 1769-1790.
30. M. Klasztorny, "Constitutive compliance/stiffness equations of viscoelasticity for resins", pp. 245-254 in Computational methods in materials characterisation (Santa Fe, NM, 2003), edited by A. A. Mammoli and C. A. Brebbia, High Performance Structures and Materials 6, WIT, Southampton, 2004.
31. S.-C. Ko, S. Lee, and C.-H. Hsueh, "Viscoelastic stress relaxation in film/substrate systems: Kelvin model", J. Appl. Phys. 93:5 (2003), 2453-2457.
32. R. S. Lakes, "The time dependent Poisson's ratio of viscoelastic cellular materials can increase or decrease", Cell. Polymers 10 (1991), 466-469.
33. R. S. Lakes and A. Wineman, "On Poisson's ratio in linearly viscoelastic solids", J. Elasticity 85:1 (2006), 45-63.
34. R. S. Lakes, J. L. Katz, and S. S. Sternstein, "Viscoelastic properties of wet cortex bone, I: Torsional and biaxial studies", J. Biomech. 12:9 (1979), 657-675.
35. E. H. Lee, "Stress analysis in viscoelastic materials", Quart. Appl. Math. 13 (1955), 665-672.
36. L. Librescu and K. N. Chandiramani, "Recent results concerning the stability of viscoelastic shear deformable plates under compressive edge loads", Solid Mech. Arch. 14 (1989), 215-250.
37. L. Librescu and N. K. Chandiramani, "Dynamic stability of transversely isotropic viscoelastic plates", J. Sound Vib. 130:3 (1989), 467-486.
38. C. van Loan, Computational frameworks for the fast Fourier transform, Frontiers in Applied Mathematics 10, SIAM, Philadelphia, 1992.
39. D. J. Mead and R. J. Joannides, "Measurement of the dynamic moduli and Poisson's ratios of a transversely isotropic fibre-reinforced plastics", Composites 22:1 (1991), 15-29.
40. T. Nakao, T. Okano, and I. Asano, "Theoretical and experimental analysis of flexural vibration of the viscoelastic Timoshenko beam", J. Appl. Mech. (ASME) 52 (1985), 728-736.
41. J. Noh and J. Whitcomb, "Effect of transverse matrix cracks on the relaxation moduli of linear viscoelastic laminates", J. Compos. Mater. 37:6 (2003), 543-558.
42. D. J. O'Brien, P. T. Mather, and S. R. White, "Viscoelastic properties of an epoxy resin during cure", J. Compos. Mater. 35:10 (2001), 883-904.
43. Z. Olesiak, "Thermal stresses in thin-walled cylindrical viscoelastic shells with temperature dependent properties", pp. 407-415 in Theory of plates and shells: selected papers, presented to the Conference on the Theory of Twoand Three-dimensional Structures (Smolenice, 1963), edited by J. Brilla and J. Balaš, Slovak Academy of Sciences, Bratislava, 1966.
44. G. H. Paulino and Z.-H. Jin, "Correspondence principle in viscoelastic functionally graded materials", J. Appl. Mech. (ASME) 68:1 (2001), 129-132.
45. G. H. Paulino and Z.-H. Jin, "Viscoelastic functionally graded materials subjected to antiplane shear fracture", J. Appl. Mech. (ASME) 68:2 (2001), 284-293.
46. A. C. Pipkin, Lectures on viscoelasticity theory, Applied Mathematical Sciences 7, Springer, New York, 1972.
47. S. D. Poisson, "Mémoire sur l'équilibre et le mouvement des corps élastiques", pp. 357-570 and 623-627 in Mémoires de l'académie royal des sciences de l'institut de France, vol. 8, Didot, Paris, 1829.
48. D. Qvale and K. Ravi-Chandar, "Viscoelastic characterization of polymers under multiaxial compression", Mech. Time-Depend. Mat. 8:3 (2004), 193-214.
49. K. Ravi-Chandar, "Simultaneous measurement of nonlinear bulk and shear relaxation behavior", pp. 30-31 in 2nd International Conference on Mechanics of Time Dependent Materials: Proceedings (Pasadena, CA, 1998), edited by I. Emri and W. G. Knauss, SEM Center for Time Dependent Materials, Ljubljana, 1998.
50. K. Ravi-Chandar and Z. Ma, "Inelastic deformation in polymers under multiaxial compression", Mech. Time-Depend. Mat. 4:4 (2000), 333-357.
51. W. T. Read, Jr., "Stress analysis for compressible viscoelastic materials", J. Appl. Phys. 21:7 (1950), 671-674.
52. R. A. Schapery, "Approximate methods of transform inversion for viscoelastic stress analysis", pp. 1075-1085 in Proceedings of the 4th U.S. National Congress of Applied Mechanics (Berkeley, CA), vol. 2, edited by R. M. Rosenberg, ASME, New York, 1962.
53. P. Shrotriya, Dimensional stability of multilayer circuit boards, Ph.D. thesis, University of Illinois at UrbanaChampaign, Department of Theoretical and Applied Mechanics, 2000.
54. P. Shrotriya and N. R. Sottos, "Creep and relaxation behavior of woven glass/epoxy substrates for multilayer circuit board applications", Polymer Compos. 19:5 (1998), 567-578.
55. A. Shtark, H. Grozbeyn, G. Sameach, and H. H. Hilton, "An alternate protocol for determining viscoelastic material properties based on tensile tests without use of Poisson's ratio", in Proceedings of the 2007 International Mechanical Engineering Congress and Exposition (Seattle, WA, 2007), ASME, New York, 2007. Paper #IMECE2007-41068.
56. S. Sim and K.-J. Kim, "A method to determine the complex modulus and Poisson's ratio of viscoelastic materials for FEM applications", J. Sound Vib. 141:1 (1990), 71-82.
57. M. P. Singh and A. S. Abdelnaser, "Random vibrations of externally damped viscoelastic Timoshenko beams with general boundary conditions", J. Appl. Mech. (ASME) 60:1 (1993), 149-156.
58. D. Therriault, Directed assembly of three-dimensional microvascular networks, Ph.D. thesis, University of Illinois at Urbana-Champaign, Department of Aerospace Engineering, 2003.
59. N. W. Tschoegl, "Time dependence in material properties: an overview", Mech. Time-Depend. Mat. 1:1 (1997), 3-31.
60. N. W. Tschoegl, W. G. Knauss, and I. Emri, "Poisson's ratio in linear viscoelasticity: a critical review", Mech. Time-Depend. Mat. 6:1 (2002), 3-51.
61. H. S. Tsien, "A generalization of Alfrey's theorem in viscoelastic media", Quart. Appl. Math. 8 (1950), 104-106.
62. G. V. Vinogradov and A. Y. Malkin, Rheology of polymers: viscoelasticity and flow of polymers, Mir, Moscow, 1980.
63. J. M. Whitney and R. L. McCullough, Delaware composites design encyclopedia, 2: Micromechanical materials modeling, edited by L. A. Carlsson and J. W. Gillespie, Jr., Technomic, Lancaster, PA, 1990.
64. Q. Zhu, Dimensional accuracy of thermoset polymers composites: process simulation and optimization, Ph.D. thesis, University of Illinois at Urbana-Champaign, Department of Aerospace Engineering, 2000.
65. Q. Zhu, P. Shrotriya, N. R. Sottos, and P. H. Geubelle, "Three-dimensional viscoelastic simulation of woven composite substrates for multilayer circuit boards", Compos. Sci. Technol. 63:13 (2003), 1971-1983.