True Stress and True Strain

Stress n. An applied force or system of forces that tends to strain or deform a body.

Strain n. A deformation produced by stress.

Stress has units of a force measure divided by the square of a length measure, and the average stress on a cross-section in the tensile test is the applied force divided by the cross-sectional area. Similarly, we may approximate the strain component along the long axis of the specimen as the change in length divided by the original reference length.

It sounds simple enough, but you should realize that there are still some choices to make. Specifically, what area should be used for the cross-sectional area? Should you use the original area or the current area as the load is applied? By the same token, should changes in length always be compared to the original length of the specimen?

The answer is that we will define different types of stress strain measures according to the way we perform the calculations. Engineering stress and strain measures are distinguished by the use of fixed reference quantities, typically the original cross-sectional area or original length. More precisely,

$$\begin{array}{c} \boldsymbol{\sigma}_E = \frac{P}{A_0}, \boldsymbol{\epsilon}_E = \frac{\Delta l}{l_0}. \end{array}$$ (1)

Engineering vs. True

Engineering stress and strain measures incorporate fixed reference quantities. In this case, undeformed cross-sectional area is used. True stress and strain measures account for changes in cross-sectional area by using the instantaneous values for area, giving more accurate measurements for events such as the tensile test.

In most engineering applications, these definitions are accurate enough, because the cross-sectional area and length of the specimen do not change substantially while loads are applied. In other situations (such as the tensile test), the cross-sectional area and the length of the specimen can change substantially. In such cases, the engineering stress calculated using the above definition (as the ratio of the applied load to the undeformed cross-sectional area) ceases to be an accurate measure. To overcome this issue alternative stress and strain measures are available. Below we discuss true stress and true strain.

Figure 3: Engineering stress measures vs. true stress measures. The latter accounts for the change in cross-sectional area as the loads are applied.

True Stress: The true stress is defined as the ratio of the applied load (P) to the instantaneous cross-sectional area (A):

$${\boldsymbol\sigma}_T = \frac{P}{A}.$$ (2)

True stress can be related to the engineering stress if we assume that there is no volume change in the specimen. Under this assumption,

$$A\cdot l = A_0 \cdot l_0,$$ which leads to:

$${\boldsymbol\sigma}_T = \frac{P}{A} = \frac{P}{A_0} \cdot \frac{l}{l_0} = {\boldsymbol\sigma}_E (1+{\boldsymbol\epsilon}_E ).$$ (3)

True Strain: The true strain is defined as the sum of all the instantaneous engineering strains. Letting

$$d\epsilon = \frac{dl}{l},$$ (4)

the true strain is then

$${\boldsymbol\epsilon}_T = \int d\epsilon = \int\limits_{l_0}^{l_f} \frac{dl}{l} = \frac{l_f}{l_0}.$$ (5)

where $l_f$ is the final length when the loading process is terminated. True strain can also be related back to the engineering strain, through the manipulation

$${\boldsymbol\epsilon}_T = \frac{l_f}{l_0} = \frac{l_0 + \Delta l}{l_0} = (1 + {\boldsymbol\epsilon}_E)$$ (6)

In closing, you should note that the true stress and strain are practically indistinguishable from the engineering stress and strain at small deformations, as shown below in Figure 4. You should also note that as the strain becomes large and the cross-sectional area of the specimen decreases, the true stress can be much larger than the engineering stress.

Key note:

As the strain becomes large and the cross-sectional area of the specimen decreases, the true stress can be much larger than the engineering stress.

Figure 4: Engineering stress-strain curve vs. a true stress-strain curve.

Exercise: The first first movie file of the tensile test is shown below. Pause the movie at the beginning, and use the scroll to view the specimen at different times. At what point in the test do you begin to notice a real change in the cross-sectional area? Roughly what percentage of the total time of the test has elapsed?

Quick Quiz: What is the true strain of a steel rod that has been stretched from 10 inches to 10.18 inches?

.018
1.018
-4.017
0.01784