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

Correct Answer:
The correct option is increases linearly with the true strain.

Analysis and Explanation:

The attached image shows a stress-strain diagram with two distinct linear regions. The vertical axis is labeled with the symbol $\sigma$ (representing true stress), and the horizontal axis is labeled with the symbol $\epsilon$ (representing true strain).

For a material modeled as a linearly hardening plastic material (or bilinear material), the behavior is divided into two stages:

1. Elastic Stage:
Before reaching the yielding point, the material deforms elastically. In this region, the true stress is directly proportional to the true strain, governed by Hooke's Law:

$$\sigma =E\epsilon$$

where $E$ is the Young's modulus (the steeper initial slope visible in the graph).

2. Plastic Stage (Beyond Yielding):
Once the true stress exceeds the initial yield stress (${\sigma }_y$), the material yields and enters the plastic deformation regime. For a linearly hardening material, the post-yield behavior is represented by a second straight line with a constant positive slope, known as the tangent modulus ($E_t$). The relationship between the true stress and true strain in this region is expressed as:

$$\sigma ={\sigma }_y+E_t(\epsilon -{\epsilon }_y)$$

where ${\epsilon }_y$ is the yield strain and $E_t$ is the post-yield tangent slope. Because $E_t$ is constant, the true stress beyond initial yielding increases linearly with the true strain, as depicted by the second upward-sloping linear segment in the diagram.