Question

Shear stress (\( \sigma \)) and shear viscosity (\( \eta \)) are plotted as functions of the shear rate (\( \dot{\gamma} \)) for idealized "solid-like with yielding (1)" and "liquid-like (2)" materials. 
 

Associate the shear stress and viscosity plots with the appropriate material responses.

• A. P-2, Q-1, R-2, S-1
• B. P-1, Q-2, R-2, S-2
• C. P-1, Q-2, R-2, S-1
• D. P-2, Q-1, R-1, S-2

Hint:

For shear stress and viscosity plots, a "solid-like with yielding" material shows a linear increase in stress with shear rate, while a "liquid-like" material shows a decrease in viscosity with increasing shear rate.

Answer 1

The Correct Option is B

We are asked to associate shear stress (\( \sigma \)) and shear viscosity (\( \eta \)) plots with the appropriate material responses for two types of idealized materials: "solid-like with yielding" and "liquid-like." Let's break down the behavior of each plot.

Step 1: Understanding the shear stress plot.
- The plot \( P \) represents the shear stress (\( \sigma \)) as a function of shear rate (\( \dot{\gamma} \)). - In a material that is "solid-like with yielding" (Option 1), the shear stress increases linearly with the shear rate until a yield point is reached, at which the material starts to flow. This is a typical behavior for materials that have a certain threshold before they begin to flow. - The plot \( P \) is therefore associated with material response \( 1 \), which corresponds to a "solid-like with yielding" material. Hence, we associate \( P \) with Option 1.

Step 2: Understanding the viscosity plot.
- The plot \( Q \) represents the shear viscosity (\( \eta \)) as a function of shear rate (\( \dot{\gamma} \)).
- A "liquid-like" material shows a typical decrease in viscosity as the shear rate increases, indicating that it flows more easily at higher rates. This is characteristic of materials that do not have a yield point and are more fluid-like.
- The plot \( Q \) aligns with the behavior of a liquid-like material, hence it is associated with Option 2.

Step 3: Analyzing the behavior of \( R \) and \( S \).
- The plot \( R \) for shear viscosity in a liquid-like material exhibits a continuous decrease in viscosity with increasing shear rate.
- The plot \( S \), which shows shear viscosity as a function of shear rate, typically exhibits a constant or near-constant behavior in materials that do not have a yield stress.
- Therefore, both \( R \) and \( S \) correspond to behavior seen in liquid-like materials, and they are associated with Option 2.

Step 4: Conclusion.
By analyzing the plots and understanding the behaviors of the materials, we conclude that the correct association is: - \( P \) corresponds to "solid-like with yielding," hence \( P-1 \).
- \( Q \) corresponds to "liquid-like," hence \( Q-2 \).
- \( R \) corresponds to "liquid-like," hence \( R-2 \).
- \( S \) corresponds to "liquid-like," hence \( S-2 \).

Final Answer: \text{(B) P-1, Q-2, R-2, S-2}