MT-1620 al.2002 If this is extended to the three-dimensional case and applied over infinitesimal areas and lengths we get the relation between stress and strain known as Generalized hookes law mn nnpp where e is the "elasticity tensor mpg How many components does this appear to have? m,n,p,q=1,2,3 3x 3X3x3=81 components But there are several symmetries 1. sin (energy considerations (symmetry in switching first two indices 2. Since (geometrical considerations) → Paul A Lagace @2001 Unit 4 -p 10MIT - 16.20 Fall, 2002 If this is extended to the three-dimensional case and applied over infinitesimal areas and lengths, we get the relation between stress and strain known as: Generalized Hooke’s law: σmn = Emnpq εpq where Emnpq is the “elasticity tensor” How many components does this appear to have? m, n, p, q = 1, 2, 3 ⇒ 3 x 3 x 3 x 3 = 81 components But there are several symmetries: 1. Since σmn = σnm (energy considerations) ⇒ Emnpq = Enmpq (symmetry in switching first two indices) 2. Since εpq = εqp (geometrical considerations) ⇒ Emnpq = Emnqp Paul A. Lagace © 2001 Unit 4 - p. 10