Question

(3) Elastic constants: (a) Consider a cube of isotropic material and show that the bulk modulus Eq. (3.5) is related to Young's modulus Eq. (3.1) and Poisson's ratio Eq. (3.6) by K 3(1-2v) (b) What happens when v reaches its upper limit of 0.5? (4) Elastic constants: We have stated that the numerical value of Poisson's ratio is always between +0.5 and -1, but we have proven only the upper limit. Use Eq. (3.10) to argue why -1 is the lower boundary for the Poisson ratio.

Answer 1

strain. 1. Relation between YK and bop up Let us consider the following t o stress-Strais table such that t h e long. Stroain as well as volume p lot Extension by P Stress along storain along zu ox OY O Ž Ox Oy oʻ2 p o o -6 -6 on po 6t 2 - 68 0 0 P-62 -6 i total extension (e) along ox=b (1-26) OY (1-26) n " " OZ = 1 (1-26) "If & be the uniform volume strain, we have 8=30=31 (1-26) Now, K = f f = 3 (1-26) on Y = 3k(1-26)

6 what happens who What happens when yor 6 reaches it's upper limit o.s? - - when & reaches to it € When & reaches to it's upper limit of 5 her if you stretch the object by appling stress then the object can not change. e when = 0.5 the object becomes incompressible. 4 we know that, Y = 36:(1-2) again we know that, y = 2n (1+2) where, y = young modulus, sa poisson ratio K = Bulk modulus, n = axial modulus as tre so, we can say Y,K, n are the (1-27) 70 or 2 x ½ and (1+r) > 0 ts) Thus & is lower bounded to rl.