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

The correct answer is: increases linearly with the true strain

Detailed Explanation:

To understand the behavior of a linearly hardening plastic material, we look at its behavior in both the elastic and plastic regions:

1. Elastic Behavior:
Initially, the material behaves elastically. The relationship between stress ($\sigma$) and strain ($\epsilon$) is linear and is governed by Hooke's Law up to the yield point (${\sigma }_y$).

2. Plastic Behavior and Yielding:
Once the stress exceeds the initial yield strength (${\sigma }_y$), the material undergoes plastic (permanent) deformation. In this regime, most metals undergo work hardening (strain hardening), meaning they become stronger as they deform, requiring higher stress for further deformation.

3. Linear Hardening Model:
For a linearly hardening plastic material, the relationship between true stress and true strain beyond the initial yield point is assumed to be linear. The equation governing this behavior in the plastic range is:
$\sigma ={\sigma }_y+E_p(\epsilon -{\epsilon }_y)$
where:
- $\sigma$ is the true stress,
- ${\sigma }_y$ is the initial yield stress,
- $E_p$ is the constant plastic tangent modulus (or hardening modulus),
- $\epsilon$ is the true strain,
- ${\epsilon }_y$ is the true strain at the yield point.

Since the plastic tangent modulus $E_p$ is a constant positive slope, the true stress increases linearly with the true strain beyond the initial yield point.