Journal of Elasticity 50: 87–89, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.
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Poisson’s Ratio for Orthorhombic Materials PH. BOULANGER1 and M. HAYES2 1 Département de Mathématique, Université Libre de Bruxelles, Campus Plaine C.P. 218/1,
1050 Brussels, Belgium 2 Department of Mathematical Physics, University College Dublin, Dublin 4, Ireland e-mail:
[email protected] Received 18 December 1997 Abstract. An example of an orthorhombic material is constructed such that even though the strainenergy density is positive definite, Poisson’s ratio may take an arbitrarily large positive value for one pair of orthogonal directions and take an arbitrarily small negative value for another pair of orthogonal directions. Mathematics Subject Classifications (1991): 73C02, 73B10, 73B40. Key words: elasticity, moduli, anisotropic material, orthorhombic symmetry, Poisson’s ratio.
1. Introduction Homogeneous elastic materials are considered. It is assumed throughout that the stored energy density for each material considered is positive definite. For an isotropic material Poisson’s ratio is a constant which must lie within the range (−1, 12 ). For anisotropic homogeneous elastic materials Poisson’s ratio, σ (m; n), is not constant but depends upon the direction n of the applied tension and upon m, orthogonal to n, the direction in which the ‘lateral contraction’ is measured. Here, it is shown, for a particular choice of orthorhombic material that Poisson’s ratio may be arbitrarily large and positive for one choice of m and n and may be arbitrarily small and negative for the same n and a different m. For an orthorhombic material the stress-strain relations are txx = c11 exx + c12 eyy + c13 ezz , tyy = c12 exx + c22 eyy + c23 ezz , tzz = c13 exx + c23 eyy + c33 ezz , tyz = c44 eyz , tzx = c55 ezx , txy = c66 exy ,
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where tij are the stress components, eij are the infinitesimal strain components and the c’s are constant. If the material is subject to a tension T along n, so that tij = T ni nj (T constant; n · n = 1), then Poisson’s ratio σ (m; n) where m · m = 1, m · n = 0, is the ratio of the laterial contraction −eij mi mj along m to the longitudinal extension eij ni nj along n: σ (m; n) = −
eij mi mj . ers nr ns
Because the stored energy density 6, given by 26 = tij eij is positive definite it follows that c44 , c55 , c66 > 0 , and the matrix (C) =
c11 c12 c13
c12 c22 c23
c13 c23 c33
!
is positive definite. 2. Special Example Let c44 , c55 , c66 be arbitrary positive quantities and let 2L2 0 L E 3E 3 E2 , (C) = 0 L L2 L
E2 L
E
where L and E are positive quantities both with the dimensions of stress. This matrix C is positive definite. So the stored energy density is positive. Now suppose that the only nonzero stress is txx = T . Then n = (1, 0, 0) and 3E 3 E2 e + ezz = 0, yy L2 L Lexx +
exy = exz = eyz = 0,
E2 eyy + Eezz = 0, L
so that if m = (0, cos θ, sin θ) then eij mi mj = eyy cos2 θ + ezz sin2 θ L L = −3 sin2 θ + cos2 θ exx , 2E E
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and so σ (0, cos θ, sin θ; 1, 0, 0) =
L 2E
L 3 sin2 θ − cos2 θ . E
Thus, σ (0, 1, 0; 1, 0, 0) = −L2 /(2E 2 ), σ (0, 0, 1; 1, 0, 0) = 3L/(2E). For (L/E) very large, both are very large in absolute value but one is positive and the other is negative. Of course σ takes all values between −L2 /(2E 2 ) and 3L/(2E) as θ changes. This simple artificial example illustrates that, in general, for homogeneous anisotropic elastic materials for orthorhombic symmetry with positive definite stored energy density, Poisson’s ratio need not be bounded either above or below – it may be arbitrarily large and positive, or arbitrarily large in absolute value but negative, whereas the situation for isotropic elastic materials is that Poisson’s ratio is a constant and lies within the range (−1, 12 ). It has been shown by Hayes and Shuvalov [1] for cubic materials that Poisson’s ratio is either bounded above or bounded below. Reference 1. Hayes, M. and A. Shuvalov, On the extreme values of Young’s modulus, the shear modulus, and Poisson’s ratio for cubic materials. Journal of Applied Mechanics (to appear).
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