Description
1. [50 points] (Stretch) A state of deformation known as simple shear occurs when F is given
by the component matrix:
F =
1 g 0
0 1 0
0 0 1
é
ë
ê
ê
ê
ù
û
ú
ú
ú
Given = 0.5,
1.1. Find left Cauchy-Green deformation tensor B and right Cauchy-Green deformation tensor C.
1.2. Find eigenvalues (e1, e2, and e3) of B and C. Are they identical?
1.3. Find principal stretches (1, 2, 3) and principal stretch directions (b1, b2, and b3).
1. 4. Verify that B = 1
2
b1 b1+2
2
b2 b2+3
2
b3 b3
1.5. Calculate three invariants and their alternative set (i.e., normalized form).
2.1 [20 points] (Hyperelastic material) Derive
expressions for the Cauchy stress and the
Nominal stress for an incompressible, NeoHookean material subjected to
2.1.1 Uniaxial tension (e1-directional stretch
is )
2.1.2 Equibiaxial tension (e1- and e2-directions
stretches are )
2.2 [10 points] Repeat problem 2.1 for a Mooney-Rivlin material.
2.3. [10 points] Repeat problem 2.1 for an Arruda-Boyce material.
AA 530: SOLID MECHANICS Out: Nov. 2, 2021
HW #4 Due: Nov. 9, 2021
2
2.3 [10 points] Repeat problem 2.1 for a Ogden material.