1. [10 points] (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-
s. 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.
4.1 Calculate the history of stresses (all six components) induced in the pad by
4.2 Calculate the history of stresses (all six components) induced in the pad by u t( ) = 0 t < 0 u t( ) =u0sinωt t > 0
(Note )
4.3 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.
