The relationship between true shear stress, σ and engineering stress σ0 is given by (where, ϵ is the conventional strain)
Understanding True Stress and Engineering Stress
In the study of material behavior under load, particularly during tensile testing, we often differentiate between engineering stress and true stress. These measures differ based on whether the original or instantaneous cross-sectional area is used in the calculation.
Defining Key Terms
- Engineering Stress (σ0): This is the stress calculated using the original cross-sectional area ($A_0$) of the specimen and the applied force ($F$). The formula is: \({\sigma _0} = \frac{F}{{{A_0}}}\).
- True Stress (σ): This is the stress calculated using the instantaneous cross-sectional area ($A$) at the moment the force is applied. The formula is: \(\sigma = \frac{F}{A}\).
- Conventional Strain (ϵ): Also known as engineering strain, it is defined as the change in length ($\Delta L$) divided by the original length ($L_0$). The formula is: \(\in = \frac{{\Delta L}}{{{L_0}}} = \frac{{L - {L_0}}}{{{L_0}}}\), where $L$ is the instantaneous length. From this, we can write $L = {L_0}(1 + \in)$.
Deriving the Relationship
To find the relationship between true stress and engineering stress, we typically assume that the volume of the material remains constant during plastic deformation. This is a reasonable assumption for many metals.
Assuming constant volume, the original volume ($V_0 = A_0 L_0$) is equal to the instantaneous volume ($V = A L$).
\[{A_0}{L_0} = AL\]We can rearrange this equation to find the instantaneous area \(A\):
\[A = {A_0}\frac{{{L_0}}}{L}\]We know from the definition of conventional strain that \(L = {L_0}(1 + \in)\). So, we can substitute this into the equation for \(A\):
\[A = {A_0}\frac{{{L_0}}}{{{L_0}(1 + \in)}}\] \[A = \frac{{{A_0}}}{{1 + \in}}\]Now, let's substitute this expression for \(A\) into the definition of true stress (\(\sigma = \frac{F}{A}\)):
\[\sigma = \frac{F}{{\left( {\frac{{{A_0}}}{{1 + \in}}} \right)}}\] \[\sigma = \frac{F}{{{A_0}}}(1 + \in)\]We know that \({\sigma _0} = \frac{F}{{{A_0}}}\) is the engineering stress. Substituting this into the equation gives us the relationship between true stress and engineering stress:
\[\sigma = {\sigma _0}(1 + \in)\]To express this as a ratio of true stress to engineering stress, we divide both sides by \({\sigma _0}\):
\[\frac{\sigma }{{{\sigma _0}}} = 1 + \in\]Conclusion
The relationship between true stress ($\sigma$) and engineering stress ($\sigma_0$), where $\varepsilon$ is the conventional strain, assuming constant volume, is given by:
\[\frac{\sigma }{{{\sigma _0}}} = 1 + \in\]This means the ratio of true stress to engineering stress is equal to one plus the conventional strain.
Comparing this derived relationship with the given options, we find that it matches the first option.
Important Questions from Stress and Strain
Dimensional formula for stress is
Unit of stress in SI unit is
A hollow steel column has to carry an axial load of 2,00,000 kg and the ultimate stress for the steel column is 4800 kg/cm 2and allows a load factor of 4. What is the sectional area of the column?
When a body is subjected to two equal and opposite pulls, as a result of which the body tends to extend its length, the stress and strain induced are
Stress at any point in a material is defined as -
