Understanding True Stress and True Strain Graphs: A Comprehensive Guide

Understanding True Stress and True Strain Graphs: A Comprehensive Guide

Introduction to Stress and Strain

In materials science and engineering, stress and strain are fundamental concepts used to understand the response of materials to external forces. Engineers typically work with engineering stress and engineering strain, which are calculated based on the initial dimensions of the material. However, in certain applications, it is more accurate to use true stress and true strain, which take into account the actual deformation of the material during testing.

Engineering Stress and Strain

Engineering stress is defined as the force applied (in Newtons, N) per unit cross-sectional area (in square meters, m2) at a given point along the length of a bar. It is calculated using the initial cross-sectional area and original length of the material:

[ sigma_{text{engineering}} frac{F}{A_0} ]

where:

(F) is the applied force. (A_0) is the initial cross-sectional area of the material.

Engineering strain, on the other hand, measures the fractional change in length of the material and is calculated using the original length of the material:

[ varepsilon_{text{engineering}} frac{Delta L}{L_0} ]

where:

(Delta L) is the change in length. (L_0) is the original length of the material.

True Stress and True Strain

True stress and true strain are more accurate representations of the material's behavior during deformation. True stress is calculated using the actual instantaneous cross-sectional area and length of the material, which changes as deformation occurs. The formulas for true stress and true strain are as follows:

True Stress

To calculate true stress:

[ sigma_{text{true}} sigma_{text{engineering}} times (1 varepsilon_{text{engineering}}) ]

This means:

(sigma_{text{true}}) is the true stress. (sigma_{text{engineering}}) is the engineering stress. (varepsilon_{text{engineering}}) is the engineering strain.

True Strain

True strain is calculated using the natural logarithm of the ratio of the current length to the original length:

[ varepsilon_{text{true}} ln(1 varepsilon_{text{engineering}}) ]

This means:

(varepsilon_{text{true}}) is the true strain. (varepsilon_{text{engineering}}) is the engineering strain.

Plotting True Stress-True Strain Curve

Once you have calculated the true stress and true strain values using the above formulas, you can plot a graph of true stress versus true strain. This curve, known as the true stress-true strain curve, provides a more accurate representation of the material's behavior under deformation, especially at higher levels of strain where the differences between engineering and true values become more significant.

The process to plot the true stress-true strain curve involves the following steps:

Load Measurement: Apply a load to the material in incremental steps.

Length and Diameter Measurement: Measure the change in length and diameter of the test bar at each step.

True Stress Calculation: Calculate the true stress at each step using the formula:

[ sigma_{text{true}} frac{F}{A_i} ]

where:

(F) is the applied force. (A_i) is the instantaneous cross-sectional area of the material.

True Strain Calculation: Calculate the true strain at each step using the formula:

[ varepsilon_{text{true}} lnleft(frac{L_i}{L_0}right) ]

where:

(L_i) is the current length. (L_0) is the original length.

Graph Plotting: Plot the true stress on the y-axis and true strain on the x-axis.

The true stress-true strain curve can be plotted using software like Excel or specialized engineering analysis tools like MATLAB, ANSYS, or ABAQUS.

Conclusion

By following these steps and converting your engineering stress and strain data to true stress and true strain, you can generate a true stress-true strain graph that provides a more precise understanding of the material's mechanical properties during deformation. This more accurate representation is particularly important in applications where precise material behavior is critical.

Keywords: true stress, true strain, engineering stress, engineering strain