[SOLVED] AA530 Homework 4

1.  (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,

• Find left Cauchy-Green deformation tensor B and right Cauchy-Green deformation tensor C.
• Find eigenvalues (e_1, e_2, and e₃) of B and C. Are they identical?
• Find principal stretches (₁, ₂, ₃) and principal stretch directions (b_1, b_2, and b₃).

1. Verify that B = ₁²b₁ b₁+₂²b₂ b₂+₃²b₃ b₃

1.5. Calculate three invariants and their alternative set (i.e., normalized form).

2.1  (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  Repeat problem 2.1 for a Mooney-Rivlin material.

2.3. Repeat problem 2.1 for an Arruda-Boyce material.

2.3Repeat problem 2.1 for a Ogden material.

2