True Stress - True Strain Curve: Part Four


This article describes strain-hardening exponent and strength coefficient, materials constants which are used in calculations for stress-strain behaviour in work hardening, and their application in some of the most commonly used formulas, such as Ludwig equation.

The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power curve relation


where n is the strain-hardening exponent and K is the strength coefficient. A log-log plot of true stress and true strain up to maximum load will result in a straight-line if Eq. (10) is satisfied by the data (Fig. 1).

The linear slope of this line is n and K is the true stress at e = 1.0 (corresponds to q = 0.63). The strain-hardening exponent may have values from n = 0 (perfectly plastic solid) to n = 1 (elastic solid), see Fig. 2. For most metals n has values between 0.10 and 0.50, see Table 1.

It is important to note that the rate of strain hardening ds /de, is not identical with the strain-hardening exponent. From the definition of n



Figure 2. Log/log plot of true stress-strain curve

Figure 3. Various forms of power curve s=K* e n

Table 1. Values for n and K for metals at room temperature

Metal Condition n K, psi
0,05% C steel Annealed 0,26 77000
SAE 4340 steel Annealed 0,15 93000
0,60% C steel Quenched and tempered 1000oF 0,10 228000
0,60% C steel Quenched and tempered 1300oF 0,19 178000
Copper Annealed 0,54 46400
70/30 brass Annealed 0,49 130000

There is nothing basic about Eq. (10) and deviations from this relationship frequently are observed, often at low strains (10-3) or high strains (1,0).

One common type of deviation is for a log-log plot of Eq. (10) to result in two straight lines with different slopes. Sometimes data which do not plot according to Eq. (10) will yield a straight line according to the relationship


Datsko has shown how e0, can be considered to be the amount of strain hardening that the material received prior to the tension test.

Another common variation on Eq. (10) is the Ludwig equation


where s0 is the yield stress and K and n are the same constants as in Eq. (10). This equation may be more satisfying than Eq. (10) since the latter implies that at zero true strain the stress is zero.

Morrison has shown that s0 can be obtained from the intercept of the strain-hardening point of the stress-strain curve and the elastic modulus line by

The true-stress-true-strain curve of metals such as austenitic stainless steel, which deviate markedly from Eq. (10) at low strains, can be expressed by

where eK is approximately equal to the proportional limit and n1 is the slope of the deviation of stress from Eq. (10) plotted against e. Still other expressions for the flow curve have been discussed in the literature.

The true strain term in Eqs.(10) to (13) properly should be the plastic strain

ep= etotal- eE= etotal- s/E







The Total Materia Extended Range includes a unique collection of stress-strain curves and diagrams for calculations in the plastic range for thousands of metal alloys, heat treatments and working temperatures. Both true and engineering stress curves are given for various strain rates where applicable.

Finding a stress-strain graph in the database is simple and takes only seconds.

Enter the material of interest into the quick search field. You can optionally narrow your search by specifying the country/standard of choice in the designated field and click Search.

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After selecting the material of interest to you, click on the Stress-Strain diagrams link to view data for the selected material. The number of available stress-strain diagrams is displayed in brackets next to the link.

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Because Total Materia stress-strain curves are neutral across standard specifications, you can review stress-strain diagrams by clicking the appropriate link for any of the subgroups.

Besides the stress-strain curves at different temperatures, stress and strain data are given in a tabular format which is convenient for copying to, for example, a CAE software.

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It is also possible to view stress-strain curves and data for other working temperatures.

To do this, simply insert a new temperature into the ‘Enter temperature’ field within the defined range.

After clicking the Calculate button, a new curve is plotted and values in the table now correspond to the temperature you have defined. See example below for 250°C.

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