A member whose material properties remain unchanged (invariant) under rotations of 90° abo...

A member whose material properties remain unchanged (invariant) under rotations of 90° about axes (x, y, z) is called a cubic material relative to axes (x, y, z) and has three independent elastic coefficients (C1, C2, C3). Its stress–strain relations relative to axes (x, y,z)are (a special case of Eq. 3.50)

σxx = C1 ϵxx + C2 ϵyy + C2 ϵzz

σyy = C2 ϵxx + C1 ϵyy + C2 ϵzz

σzz = C1 ϵxx + C2 ϵyy + C1 ϵzz

σxy = C3 γxy

σxz = C3 γxz

σyz = C3 γyz

Although in practice aluminum is often assumed to be an isotropic material (E = 72 GPa and v = 0.33), it is actually a cubic material with C1 = 103 GPa, C2 =55 GPa, and C3 = 27.6 GPa. At a point in an airplane wing, the strain components are ϵxx = 0.0003, ϵyy = 0.0002, ϵzz = 0.0001, ϵxy = 0.00005, and ϵxz = ϵyz = 0.

a. Determine the orientation of the principal axes of strain.

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b. Determine the stress components.

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c. Determine the orientation of the principal axes of stress.

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d. Calculate the stress components and determine the orientation of the principal axes of strain and stress under the assumption that the aluminum is isotropic.

 (3.50)