structural:reformatting
Discussion

Since we disregard the change of area during deformation above, the true stress and strain curve should be re-derived. For deriving the stress strain curve, we can assume that the volume change is 0 even if we deformed the materials. We can assume that:

Then, the true stress can be expressed as below:

Additionally, the true strain ε can be expressed as below:

Then, we can express the value as

Thus, we can induce the plot in terms of σ T {\displaystyle \sigma _{T}} and ε E {\displaystyle \varepsilon _{E}} as right figure.

Additionally, based on the true stress-strain curve, we can estimate the region where necking starts to happen. Since necking starts to appear after ultimate tensile stress where the maximum force applied, we can express this situation as below:

so this form can be expressed as below:

It indicates that the necking starts to appear where reduction of area becomes much significant compared to the stress change. Then the stress will be localized to specific area where the necking appears.

Additionally, we can induce various relation based on true stress-strain curve.

1) True strain and stress curve can be expressed by the approximate linear relationship by taking a log on true stress and strain. The relation can be expressed as below:

Where K {\displaystyle K} is stress coefficient and n {\displaystyle n} is strain-hardening coefficient. Usually, the value of n {\displaystyle n} has range around 0.02 to 0.5 at room temperature. If n {\displaystyle n} is 1, we can express this material as perfect elastic material.

2) In reality, stress is also highly dependent on the rate of strain variation. Thus, we can induce the empirical equation based on the strain rate variation.

Where K ′ {\displaystyle K'} is constant related to the material flow stress. ε T ˙ {\displaystyle {\dot {\varepsilon _{T}}}} indicates the derivative of strain by the time, which is also known as strain rate. m {\displaystyle m} is the strain-rate sensitivity. Moreover, value of m {\displaystyle m} is related to the resistance toward the necking. Usually, the value of m {\displaystyle m} is at the range of 0-0.1 at room temperature and as high as 0.8 when the temperature is increased.

By combining the 1) and 2), we can create the ultimate relation as below:

Where K ″ {\displaystyle K''} is the global constant for relating strain, strain rate and stress.

3) Based on the true stress-strain curve and its derivative form, we can estimate the strain necessary to start necking. This can be calculated based on the intersection between true stress-strain curve as shown in right.

This figure also shows the dependency of the necking strain at different temperature. In case of FCC metals, both of the stress-strain curve at its derivative are highly dependent on temperature. Therefore, at higher temperature, necking starts to appear even under lower strain value.

All of these properties indicate the importance of calculating the true stress-strain curve for further analyzing the behavior of materials in sudden environment.

4) A graphical method, so-called "Considere construction", can help determine the behavior of stress-strain curve whether necking or drawing happens on the sample. By setting λ = L / L 0 {\displaystyle \lambda =L/L_{0}} as determinant, the true stress and strain can be expressed with engineering stress and strain as below:

Therefore, the value of engineering stress can be expressed by the secant line from made by true stress and λ {\displaystyle \lambda } value where λ = 0 {\displaystyle \lambda =0} to λ = 1 {\displaystyle \lambda =1}. By analyzing the shape of σ T − λ {\displaystyle \sigma _{T}-\lambda } diagram and secant line, we can determine whether the materials show drawing or necking.

On the figure (a), there is only concave upward Considere plot. It indicates that there is no yield drop so the material will be suffered from fracture before it yields. On the figure (b), there is specific point where the tangent matches with secant line at point where λ = λ Y {\displaystyle \lambda =\lambda _{Y}}. After this value, the slope becomes smaller than the secant line where necking starts to appear. On the figure (c), there is point where yielding starts to appear but when λ = λ d {\displaystyle \lambda =\lambda _{d}}, the drawing happens. After drawing, all the material will stretch and eventually show fracture. Between λ Y {\displaystyle \lambda _{Y}} and λ d {\displaystyle \lambda _{d}}, the material itself does not stretch but rather, only the neck starts to stretch out.