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Question #42212

Difficulty: 8/10 | 🟢 Free

The Question

A viscoelastic material is subjected to a cyclic stress $\sigma(t) = \sigma_0 \sin(\omega t)$ . Due to the elastic after-effect, the material does not respond instantaneously. The strain $\epsilon(t)$ and strain rate $\dot{\epsilon}(t)$ are related to the applied stress by the Kelvin-Voigt model equation: $\eta \dot{\epsilon} + k \epsilon = \sigma(t)$ , where $\eta$ is the coefficient of internal viscosity and $k$ is the elastic modulus. After the system reaches a steady state, what is the total energy dissipated per unit volume of the material during one complete cycle of loading?

Options

A.
$\frac{\pi \sigma_0^2 \eta \omega}{k^2 + \eta^2 \omega^2}$
B.
$\frac{\pi \sigma_0^2 k}{k^2 + \eta^2 \omega^2}$
C.
$\frac{2\pi \sigma_0^2 \eta \omega}{k^2 + \eta^2 \omega^2}$
D.
$\frac{\pi \sigma_0^2 \eta \omega}{2(k^2 + \eta^2 \omega^2)}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Elastic After-Effect|Hysteresis Loop Area|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy dissipation in viscoelastic materials experiencing the elastic after-effect.

A common trap is treating the hysteresis loop as a static geometric area or incorrectly applying quasi-static elastic energy formulas (like $1/2 \times \text{stress} \times \text{strain}$ ).

Students must recognize that the loop area is dynamically generated by the phase lag $\delta$ between stress and strain.

By failing to integrate $\oint \sigma d\epsilon$ over the time-delayed path or ignoring the steady-state phase shift $\tan\delta = \eta\omega/k$ , students miss the frequency-dependent nature of the dissipation and often select dimensionally similar but physically incorrect distractors.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Solids - Class 11 Part 2
Concept	Elastic After-Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free