[SOLVED] AA530 - SOLID MECHANICS  - HW 6

Description

Viscoelastic material – Maxwell model) Plot the stress of this viscoelastic material as a function of time t (0 to 600 s), given ε₀ = 0.1, k = 0.1 GPa, and η = 20 GPa-s.

2. [10 points] (Viscoelastic material – Voigt-Kelvin model) Given a viscoelastic material, we impose constant stress σ₀ at t = 0. Derive the constitutive equation using the Voigt-Kelvin model (i.e., express strain ε as a function of stiffness k, damping factor η, time t, and given stress σ₀).

3. [10 points] (Viscoelastic material – Voigt-Kelvin model) Plot the strain of this viscoelastic material as a function of time t (0 to 600 s), given σ₀ = 1 MPa, k = 0.1 GPa, and η = 20 GPa-
4. Which behavior does this model represent, retarded elastic behavior or steady-state creep behavior? (The response might not be exactly same as what we have covered in the class. But you can qualitatively judge which behavior this model represents).

4. [40 points] (Constitutive relationship for viscoelastic materials) A floor is covered with a pad with thickness h of viscoelastic material, as shown in the figure. The pad is perfectly bonded

to the floor, so that . The pad can be 1 idealized as a viscoelastic solid with time independent bulk modulus K, and has a shear modulus that can be approximated by

G t( )=G_∞+G e₁ ^{−t t/ 1} .  The surface of the pad is subjected to a history of displacement

. Assume out-of-plane strain  is uniform throughout the thickness of the pad.

• Calculate the history of stresses (all six components) induced in the pad by

• Calculate the history of stresses (all six components) induced in the pad by u t( ) = 0 t < 0 u t( ) =u₀sinωt t > 0

(Note )

• Plot the stress history in 1.2. using appropriate parameters of your choice, such

that the curves show both transient and steady-state responses.

4.4  Assume that the pad is subjected to a displacement u t( ) =u₀sinωt for long enough for the cycles of stress and strain to settle to steady state. Calculate the total energy dissipated per unit area of the pad during a cycle of loading.

5. [15 points] (Tresca yield criterion) Derive a mathematical expression for a 3D Tresca yield criterion, given the uniaxial yield stress σ₀ = 200 MPa. Plot the envelop in 3D space using Matlab.

6. [15 points] (Von Mises yield criterion) Plot the envelop of the Von-Mises yield criterion in 3D space using Matlab, given the uniaxial yield stress σ₀ = 200 MPa.