Question

(30p) Consider a two-dimensional strain state defined by 612 = € and all normal strains zero. Write the strain and the stress tensors in this coordinate system (use Hooke's law). The shear modulus is G. Find the principal strains and the principal directions of strain. Write the stress and strain tensors in these principal directions. Are these also the principal directions of stress? You don't need to solve the eigenvalue problem for stress to answer this question. Find the effective stiffness with respect to the tensile load in the principal directions, in terms of Young's modulus E and Poisson ratio v. Write this expression in terms of G. This represents the link between G and E and v.

Answer 1

# defined Consider a by E12 = & two and dimensional strain rated all normal strains zero, Evony: Ex-Le-to Erateral - Ey Ad or or at 1 do Strain tensor [Eboo Kex . Vxy/2 | By Hooke's haw] Strain tensor for plain strain problem Principal strain & principal directions of strain [Ende Ext 84] + 4 [8x ] 60128 + Yay sh20 E., 2 = 1 / [(Extey) & (Ex-Ey)² + 4 ( Vxy J² No, there is no principal direction of stress. Expression for Ex under tensile loading • E = du - Entley - 5) Ex= S (-MOT] =) En = b = 1 [op - Hon]

Mis Poisson's ratio.al Expression in terms of G E is young modules TE=2G Cd + E, K, G o locust VI M so (E = 3K (1-2) E = 9KG 3 KfG .........................