Question

In a linearly hardening plastic material, the true stress beyond the initial yielding

• A. Decreases linearly with the true strain
• B. Increases linearly with the true strain
• C. Remains constant with true strain
• D. Increases exponentially with true strain

Hint:

Break down the technical term: 1. Hardening: Stress must *increase* to cause further deformation (eliminates options A and C). 2. Linearly: The increase follows a straight-line trend with a constant slope (eliminates option D). Therefore, it must increase linearly!

Answer 1

The Correct Option is B

Concept: When analyzing the plastic behavior of metals beyond their initial elastic limit, constitutive material models are used to simplify real stress-strain behavior. A linearly hardening plastic material (often modeled as an elastic-linearly plastic or rigid-linearly plastic material) exhibits strain hardening (work hardening) during plastic deformation. In this model, once the true stress ($\sigma$) exceeds the initial yield strength ($\sigma_y$), further plastic deformation requires an increase in stress due to dislocation interactions within the crystal lattice. For a linear hardening model, this relationship is expressed mathematically by the linear flow equation: \[ \sigma = \sigma_y + H \cdot \varepsilon_p \] Where:

• $\sigma$ represents the current true stress value.

• $\sigma_y$ represents the initial tensile yield strength.

• $H$ represents the constant plastic hardening modulus (slope of the stress-strain curve in the plastic zone).

• $\varepsilon_p$ represents the plastic true strain component.
This equation shows that beyond initial yielding, the true stress increases as a linear function of the strain.

Step 1: Interpret the phrase "linearly hardening".
The term "linearly hardening" specifies that the rate of strain hardening remains constant. This means the slope ($\frac{d\sigma}{d\varepsilon}$) of the stress-strain curve in the post-yielding plastic regime is a constant positive value ($H$).

Step 2: Evaluate the behavior described by each option.

• Option (A): Incorrect. A decrease in stress with strain would indicate strain softening or strain-localization (necking), not hardening.

• Option (B): Correct. The stress increases along a straight line with a constant positive slope relative to the strain.

• Option (C): Incorrect. A constant stress with increasing strain describes a *perfectly plastic* material ($\sigma = \sigma_y$), which has zero hardening.

• Option (D): Incorrect. An exponential increase occurs in power-law hardening models ($\sigma = K\varepsilon^n$), not linear models.
Thus, the true stress increases linearly with the true strain beyond initial yielding, matching Option (B).