Let's solve **Question 8(a): Derive Voigt model for stress relaxation as well as creep condition.** --- # Voigt (Kelvin-Voigt) Model for Viscoelasticity The **Voigt model** (also known as the Kelvin-Voigt model) consists of a **spring (elastic element)** and a **dashpot (viscous element)** connected **in parallel**. ## 1. **Constitutive Equation** Let: - \( \sigma(t) \) = stress at time \( t \) - \( \epsilon(t) \) = strain at time \( t \) - \( E \) = Young's modulus of the spring - \( \eta \) = viscosity of the dashpot **Voigt Model equation:** \[ \sigma(t) = E \epsilon(t) + \eta \frac{d\epsilon(t)}{dt} \] --- ## 2. **Creep Condition (Constant Stress)** **Creep** is the increase of strain over time when a constant stress is suddenly applied. - Apply constant stress: \( \sigma(t) = \sigma_ \) for \( t \geq \) - Initial condition: \( \epsilon() = \) (assume initially unstrained) **Differential Equation:** \[ \sigma_ = E\epsilon(t) + \eta \frac{d\epsilon(t)}{dt} \] **Rearrange:** \[ \frac{d\epsilon(t)}{dt} + \frac{E}{\eta}\epsilon(t) = \frac{\sigma_}{\eta} \] **Solution (Using Integrating Factor):** Let \( \tau = \frac{\eta}{E} \) (relaxation time). \[ \epsilon(t) = \frac{\sigma_}{E} \left( 1 - e^{-t/\tau} \right) \] ### **Interpretation** - At \( t = \): \( \epsilon() = \) (no instantaneous deformation). - As \( t \rightarrow \infty \): \( \epsilon(\infty) = \frac{\sigma_}{E} \) (reaches a steady value). **Creep curve**: Exponential approach to the steady strain. --- ## 3. **Stress Relaxation Condition (Constant Strain)** **Stress relaxation** is the decrease in stress under a constant applied strain. - Apply constant strain: \( \epsilon(t) = \epsilon_ \) for \( t \geq \) - Initial stress: \( \sigma() = E\epsilon_ \) **From the model:** \[ \sigma(t) = E\epsilon_ + \eta \frac{d\epsilon_}{dt} \] But, since \( \epsilon_ \) is constant, \( \frac{d\epsilon_}{dt} = \): \[ \sigma(t) = E\epsilon_ \] ### **Interpretation** - **No stress relaxation** occurs in the Voigt model for a constant strain; the stress remains constant. --- ## 4. **Summary Table** | Condition | Differential Equation | Solution | Behavior | |---------------------|--------------------------------------------------------|----------------------------------|--------------------------------------| | **Creep** | \( \sigma_ = E\epsilon + \eta \frac{d\epsilon}{dt} \) | \( \epsilon(t) = \frac{\sigma_}{E}(1-e^{-t/\tau}) \) | Strain increases to steady value | | **Stress Relaxation** | \( \sigma(t) = E\epsilon_ \) (as \( d\epsilon/dt = \)) | \( \sigma(t) = E\epsilon_ \) | Stress stays constant | --- # **Final Answers** ### **Voigt Model for Creep:** \[ \boxed{ \epsilon(t) = \frac{\sigma_}{E}\left[1 - \exp\left(-\frac{E}{\eta} t\right)\right] } \] ### **Voigt Model for Stress Relaxation:** \[ \boxed{ \sigma(t) = E \epsilon_ } \] (No relaxation: stress remains constant for a constant strain.)