Eren BillurPreparing the Hardening Curve
November 1, 2022Comments
A hardening curve, also known as the true plastic strain vs. true stress curve and essentially only a tensile-test result tweaked with some formulas, is required for all metal forming simulations. Most commercially available software would ask the user to enter this data until reaching 1.0 true plastic strain (ɛp) or 178-percent engineering plastic strain. No automotive sheet material can elongate that much at room temperature, so what should we do?
Understanding the Hardening Curve
A tensile test involves recording force and extension values (Fig. 1a), with this data then converted to an engineering stress-strain curve by dividing the force by the initial cross-section area and the extension by the original gauge length. Most of the material specifications (elastic modulus, yield stress, universal tensile strength, uniform and total elongation) are defined on the curve as shown in Fig. 1b (see recent Metal Matters columns in MetalForming for detailed explanations of these specifications).
Engineering stress-strain curves gain wide use for design purposes. However, analysis of plastic deformation, such as with metal forming processes, requires true stress-strain curves. True stress-true strain curves can be drawn only until reaching the end of uniform elongation (shown as a square in Fig. 1c). The data gathered after necking must be deleted (dashed line), as the formula used to develop the curve does not remain valid in this region. Lastly, we take the elastic strain out to reveal the true plastic strain-true stress curve, or hardening curve (Fig. 1d).
For a typical automotive sheet metal, the hardening curve (Fig. 1d) can be determined only until 0.15 true plastic strain. For deep-draw-quality steels, the maximum true plastic strain can slightly exceed 0.25. Most software, however, requires this data until 1.0 true plastic strain. The most common method to achieve this: Extrapolate the data by fitting an equation to the data you know, and trying to predict how the true stress may evolve at high (true plastic) strains.
Two main classes of material models exist: unbounded and saturation models. Generally, we use unbounded models for most steels and saturation models for aluminum. Unbounded models, unlike saturation models, feature a strain-hardening exponent “n” (Table 1). You may have heard that n is not constant for aluminum, or that aluminum cannot have a single n-value for aluminum. Also, advanced high-strength steel (AHSS) grades do not have a constant n-value. 
How to decide on a model? Use unbounded for low-strength steels, saturation for aluminum and mixed models for AHSS, or check all models to see which fits best. In any case, different models may yield huge variations at true plastic strain of 1. Fig. 2 shows almost a 20-percent difference between two material models. With no further information, a user may not be sure of springback results nor press-force estimations. 
A hydraulic bulge test involves clamping a square sheet of 200 by 200-mm square sheet of material with a lock bead, and hydraulic pressure introduced to freely bulge the material. During the test, digital images provide for strain measurements/calculations with pressure recorded for stress calculations. The minimum necking strain in a bulge test measures 0.36, and with increased strain hardening (n-value), the necking strain increases. This can reduce the need for extrapolation, as reported by many researchers. Use caution when using the hydraulic bulge test on tough materials such as new-generation AHSS. These material sheets, with high elongation together with high strength, may burst at high-enough internal pressure and release great amounts of energy. 
Even using a 3D DIC system may not deliver a hardening curve to 1.0 true plastic strain. We have recently developed a computer code that fits all of the hardening-curve models listed in Tables 1 and 2. The extrapolation ends at 1.0 true plastic strain and checks for the median value. Then, the minimum and maximum values are recorded to determine the error margin. Furthermore, the code provides filtered raw data until the end of experimental data—in our case, the beginning of the local neck—and then continues extrapolating with (almost) no interruption.
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