Understanding True Stress vs Engineering Stress

When analyzing the mechanical behavior of materials, especially under tensile load, it's important to distinguish between engineering stress and true stress. Both relate the applied load to a measure of the material's cross-section, but they differ in which cross-sectional area is used in the calculation.

Defining Engineering Stress

Engineering stress is defined as the applied load divided by the original cross-sectional area of the material specimen. It's represented by the symbol $\sigma_e$. The formula is:

$$ \sigma_e = \frac{P}{A_0} $$

Here, $P$ represents the instantaneous load applied to the material, and $A_0$ is the initial, undeformed cross-sectional area.

Defining True Stress

True stress, conversely, is defined as the applied load divided by the instantaneous cross-sectional area of the material specimen as it deforms. It's represented by the symbol $\sigma_t$. The formula is:

$$ \sigma_t = \frac{P}{A} $$

In this formula, $P$ is the same instantaneous load, but $A$ is the actual, current cross-sectional area of the material at the point of measurement, which changes during deformation.

Comparing the Two Stresses

Consider what happens to a material, like a metal, during a tensile test. As the load increases and the material stretches, its cross-sectional area typically decreases. This reduction in area is particularly noticeable in ductile materials, often leading to a phenomenon called 'necking' where the cross-section narrows significantly before fracture.

• Engineering stress calculation uses a constant denominator ($A_0$).
• True stress calculation uses a denominator ($A$) that decreases as the material elongates.
• Since $A$ becomes smaller than $A_0$ during deformation ($A < A_0$), the value of $\sigma_t$ will be greater than $\sigma_e$ for the same load $P$.

Conclusion on Stress Values

Given that the instantaneous cross-sectional area ($A$) shrinks as the load ($P$) is applied and the material deforms, the true stress ($\sigma_t = P/A$) will always be greater than the engineering stress ($\sigma_e = P/A_0$) after the initial loading phase where deformation begins ($A < A_0$). Therefore, the true stress experienced by a material is higher than the engineering stress at a given load, especially once deformation starts.