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On Poisson’s ratio in linearly viscoelastic solids. (English) Zbl 1098.74013
Summary: Poisson’s ratio in viscoelastic solids is in general a time-dependent (in the time domain) or a complex frequency-dependent quantity (in the frequency domain). We show that the viscoelastic Poisson’s ratio has a different time dependence depending on the test modality chosen; interrelations are developed between Poisson’s ratios in creep and relaxation. The difference, for a moderate degree of viscoelasticity, is minor. Correspondence principles are derived for the Poisson’s ratio in transient and dynamic contexts. The viscoelastic Poisson’s ratio need not increase with time, and it need not be monotonic with time. Examples are given of material microstructures which give rise to designed time-dependent Poisson’s ratios.

74D05 Linear constitutive equations for materials with memory
Full Text: DOI
[1] Pipkin, A.C.: Lectures on Viscoelasticity Theory. Springer, Berlin Heidelberg New York (1972) · Zbl 0237.73022
[2] Adams, R.D., Peppiatt, N.A.: Effect of Poisson’s ratio strains in adherends on stresses of an idealized lap joint. J. Strain Anal. 8, 134–139 (1973) · doi:10.1243/03093247V082134
[3] Ferry, J.D.: Viscoelastic Properties of Polymers, 2nd edn. Wiley, New York (1970)
[4] Lakes, R.S.: Viscoelastic Solids. CRC, Boca Raton, Florida (1998)
[5] Hilton, H.H.: Implications and constraints of time-independent Poisson’s ratios in linear isotropic and anisotropic viscoelasticity. J. Elast. 63, 221–251 (2001) · Zbl 1009.74013 · doi:10.1023/A:1014457613863
[6] Tschoegl, N.W., Knauss, W., Emri, I.: Poisson’s ratio in linear viscoelasticity – A critical review. Mech. Time-Depend. Mater. 6, 3–51 (2002) · doi:10.1023/A:1014411503170
[7] Lu, H., Zhang, X., Knauss, W.G.: Uniaxial, shear, and Poisson relaxation and their conversion to bulk relaxation. Polym. Compos. 18, 211–222 (1997) · doi:10.1002/pc.10275
[8] Kugler, H., Stacer, R., Steimle, C.: Direct measurement of Poisson’s ratio in elastomers. Rubber Chem. Technol. 63, 473–487 (1990) · doi:10.5254/1.3538267
[9] Sokolnikoff, I.S.: Mathematical Theory of Elasticity. Krieger, Malabar, Florida (1983) · Zbl 0499.73006
[10] Read Jr., W.T.: Stress analysis for compressible viscoelastic materials. J. Appl. Phys. 21, 671–674 (1950) · Zbl 0041.53803 · doi:10.1063/1.1699729
[11] Gross, B.: Mathematical Structure of the Theories of Viscoelasticity. Hermann, Paris (1968) · Zbl 0052.20901
[12] Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, London (1982) · Zbl 0045.26402
[13] Chen, C.P., Lakes, R.S.: Holographic study of conventional and negative Poisson’s ratio metallic foams: Elasticity, yield, and micro-deformation. J. Mater. Sci. 26, 5397–5402 (1991) · doi:10.1007/BF02403936
[14] Gurtin, M.E., Sternberg, E.: On the linear theory of viscoelasticity. Arch. Ration. Mech. Anal. 11, 291–356 (1962) · Zbl 0107.41007 · doi:10.1007/BF00253942
[15] Anderson, D.L.: Theory of the Earth. Blackwell, Boston, Massachusetts (1989)
[16] Gibson, L.J., Ashby, M.F.: Cellular Solids, 2nd edn. Cambridge University Press, Cambridge, UK (1997) · Zbl 0723.73004
[17] Weiner, J.H.: Statistical Mechanics of Elasticity. Wiley, New York (1983) · Zbl 0616.73034
[18] Kolpakov, A.G.: On the determination of the averaged moduli of elastic gridworks. Prikl. Mat. Meh. 59, 969–977 (1985)
[19] Lakes, R.S.: Advances in negative Poisson’s ratio materials. Adv. Mater. (Weinheim, Germany) 5, 293–296 (1993) · doi:10.1002/adma.19930050416
[20] Lakes, R.S.: Foam structures with a negative Poisson’s ratio. Science 235, 1038–1040 (1987) · doi:10.1126/science.235.4792.1038
[21] Lakes, R.S.: The time dependent Poisson’s ratio of viscoelastic cellular materials can increase or decrease. Cell. Polym. 11, 466–469 (1992)
[22] Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941) · JFM 67.0837.01 · doi:10.1063/1.1712886
[23] Zener, C.: Internal friction in solids. I. Theory of internal friction in reeds. Phys. Rev. 52, 230–235 (1937) · JFM 63.1341.03 · doi:10.1103/PhysRev.52.230
[24] Nye, J.F.: Physical Properties of Crystals. Oxford University Press, London, UK (1976) · Zbl 0079.22601
[25] Zener, C., Otis, W., Nuckolls, R.: Internal friction in solids. III. Experimental demonstration of thermoelastic internal friction. Phys. Rev. 53, 100–101 (1938) · JFM 64.1421.01 · doi:10.1103/PhysRev.53.100
[26] Christensen, R.M.: Restrictions upon viscoelastic relaxation functions and complex moduli. Trans. Soc. Rheol. 16, 603–614 (1972) · Zbl 0257.73029 · doi:10.1122/1.549265
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