Question

ME378K Homework 3 1. Compute the modulus of elasticity for copper and aluminum single crystals in the 100 1 10s, and c111> directions. Compare these values with the polycrystalline moduli. Material Copper Sux 10.11 Pa.l 1.50 1.57 0.63 1.33 130 GPa Aluminum 0.57 3.51 70.3 GPa 2. A 10 mm cube of single crystal copper (see Problem 1 for compliance data) is oriented as shown and loaded along the sample (unprimed) coordinate system. Crystal axes are given by a". Assuming elastic deformation only Boundary Conditions: The origin has zero displacement, points on the x1-x2 plane do not displace vertically, and points on the x2 axis are constrained to slide along the x2 axis. , compute the displacements (ui, u2. u3) for x3. a 0 0 -2 ksi 10-24 45

Answer 1

We know that copper and �aluminium are cubic materials. So, �1/E = s_11 - 2[(s_11 - s_12) - 1/2 s_44] (L^2_1 L^2_2 + L^2_2 L^2_3 + L^2_3 L^2_1) �where l_1, l_2, l_3 are direction cosines. �(l^2_1 l^2_2 + l^2_2 l^2_3 + l^2_3 l^2_1) for < 100 > = 0 �(l^2_1 l^2_2 + l^2_2 l^2_3 + l^2_3 l^2_1) for < 110 > = 1/4 �(l^2_1 l^2_2 + l^2_2 l^2_3 + l^2_3 l^2_1) for < 111 > = 1/3 �copper: �s_11 = 1.50 times 10^-11 pa^-1 s_12 = -0.63 times 10^-11 pa^-1 �s_44 = 1.33 times 10^-11 pa^-1 �(s_11 - s_12) - 1/2 s_44 = [(1.50 + 0.63) - 1/2 times 1.33] times 10^-11 pa^-1 \r\n (s_11 - s_12) -1/2 s_44 = 1.465 timies 10^-11 pa^-1 �1/E_100 = 1.50 times 10^-11 pa^-1 - (2 times 1.465 times 10^-11 pa^-1 times �1/E_100 = 1.50 times 10^-11 pa^-1 �E_100 = 66.7 G pa �1/E_110 = [1.50 - (2 times 1.465 times 1/4)] times 106-11 pa^-1 �1/E_110 = 0.7675 times 10^-11 pa^-1 �E_110 = 130.3 G Pa �1/E_111 = [1.50 - (2 times 1.465 times 1/3)] times 106-11 pa^-1 �