True Stress - True Strain Curve: Part One


During stress testing of a material sample, the stress–strain curve is a graphical representation of the relationship between stress, obtained from measuring the load applied on the sample, and strain, derived from measuring the deformation of the sample. The nature of the curve varies from material to material.

A typical stress-strain curve is shown in Figure 1. If we begin from the origin and follow the graph a number of points are indicated.

Figure 1: A typical stress-strain curve

Point A: At origin, there is no initial stress or strain in the test piece. Up to point A Hooke's Law is obeyed according to which stress is directly proportional to strain. That's why the point A is also known as proportional limit. This straight line region is known as elastic region and the material can regain its original shape after removal of load.

Point B: The portion of the curve between AB is not a straight line and strain increases faster than stress at all points on the curve beyond point A. Point B is the point after which any continuous stress results in permanent, or inelastic deformation. Thus, point B is known as the elastic limit or yield point.

Point C & D: Beyond the point B, the material goes to the plastic stage till the point C is reached. At this point the cross- sectional area of the material starts decreasing and the stress decreases to point D. At point D the workpiece changes its length with a little or without any increase in stress up to point E.

Point E: Point E indicates the location of the value of the ultimate stress. The portion DE is called the yielding of the material at constant stress. From point E onwards, the strength of the material increases and requires more stress for deformation, until point F is reached.

Point F: A material is considered to have completely failed once it reaches the ultimate stress. The point of fracture, or the actual tearing of the material, does not occur until point F. The point F is also called ultimate point or fracture point.

If the instantaneous minimal cross-sectional area can be measured during a test along with P and L and if the constant-volume deformation assumption is valid while plastic deformation is occurring, then a true stress-strain diagram can be constructed.

The construction should consist of using the equation ε = log (L/L0) for true strain in the elastic and initial plastic regions and the equation ε= log (A0/A) once significant plastic deformation has begun. In practice, however, the equation ε= log (A0/A) is used for the entire strain range since the amount of error introduced is usually negligible. The equation σ = P/A for true stress is valid throughout the entire test.

Accurate construction of a true stress-strain diagram becomes exceedingly difficult if the plastic deformation cannot be assumed to occur at a constant volume.

Finally, it should be mentioned that the neither the true stress-strain diagram nor the engineering stress-strain diagram accounts for the fact that the state of stress within the necked-down region is multiaxial. There is no simple way to account for the effect of this multiaxial state of stress on the material’s response, so it is customarily ignored. A schematic representation of an engineering stress-strain diagram and the corresponding true stress-strain diagram can be found in Figure 2.

Figure 2: Material properties determined from stress-strain diagrams: (a) Engineering stress-strain diagram; (b) determination σ by the offset method; (c) true stress-strain diagram

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This article belongs to a series of articles. You can click the links below to read more on this topic.

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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