1. [50 points] (Stretch) A state of deformation known as simple shear occurs when F is given by the component matrix:
é 1 g 0 ù ê ú
F=ê 0 1 0 ú ê 0 0 1 ú ë û
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 = 12b1 b1+22b2 b2+32b3 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, Neo-
Hookean 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.
2.3 [10 points] Repeat problem 2.1 for a Ogden material.
