Introduction

Over the years, empirical relationships have been developed to relate hardness to strength for different metals (Ref 1-3). The authors of the current study have developed empirical correlations to determine reasonable values of yield strength and ultimate tensile strength of steels, based on their hardness (Ref 1). These correlations were based on a large number of experimental studies, and encompassed over 150 data points from 21 different Advanced Steel Processing and Products Research Center theses.

Both of the strength correlations showed a linear dependence of strength on hardness. Yield strength, s ys, is expressed as

$$s_{\text{ys}} = - 90. 7 + 2. 8 7 6H,$$
(1)

where H is the diamond pyramid hardness, or Vickers hardness, of the steel. Similarly, the ultimate tensile strength, s uts, is

$$s_{\text{uts}} = - 9 9. 8 + 3. 7 3 4H.$$
(2)

In Eq 1 and  2, strength has units of MPa and hardness has units of kgf/mm2.

In a second study, Pavlina and Van Tyne (Ref 4) developed an empirical correlation between the hardness and the uniform elongation of steel. The uniform plastic strain, εpl-u, is given by

$$\upvarepsilon_{\text{pl - u}} = 0. 4 6 3 - 0. 4 3 9\left( { 1 - e^{ - 0.00 7 4 2H} } \right).$$
(3)

These three correlations were combined to show how the parameters for the stress-strain flow curve of steel can be obtained from a single hardness measurement. The stress-strain behaviors originally examined include the Holloman, Ludwig, and Voce hardening models. Although the method to obtain these parameters has been previously presented, tabulated values for the material model parameters as a function of hardness were not given. The method developed in Ref 4 has also been applied to the Swift hardening model in the present paper.

The objective of the present paper is to provide easy look-up tables that engineers can use to determine the parameters for different types of constitutive flow behavior, based on a single hardness value. These tables may be of particular interest to the simulation community, as access to material behavior parameters may be limited or extensive mechanical testing may not be possible. In these cases, using the tabular values presented in the current study may allow for a reasonably accurate prediction of the stress-strain behavior.

Hardening Models

In order to determine the parameters of a simple hardening model for a plastic stress-strain flow curve of a material, at least two (and often three) of the material’s mechanical property values must be known. Three properties that can be used in determining these parameters are the yield strength, the ultimate tensile strength, and the uniform plastic strain. Pavlina and Van Tyne (Ref 1, 4) have provided empirical correlations of these three mechanical properties to hardness for a wide range of steels. These equations are given as Eq 1 to 3. A method to determine the parameters for three different hardening models has also been developed (Ref 4). The methodology can be found in the original paper. The goal of the present work is to use that methodology to provide a set of tables that can be used to determine the parameters for four different stress-strain hardening models from a single hardness test. The four hardening models are presented below.

Holloman Hardening

The Holloman power-law hardening model is given by

$$\upsigma = K_{\text{H}}\upvarepsilon_{\text{pl}}^{{n_{\text{H}} }} ,$$
(4)

where σ is the true stress, εpl is the plastic strain component, KH is the strength coefficient, and n H is the strain hardening exponent. The two parameters in Eq 4 are K H and n H. Table 1 uses the method of Pavlina and Van Tyne (Ref 3) to provide specific values of K H and n H as a function of hardness. As a result, from a single hardness measurement, the two parameters can be found in the table and can then be used to quantify the stress-strain behavior by the Holloman hardening model.

Table 1 Holloman hardening parameters
Full size table

Ludwig Hardening

The Ludwig hardening model provides a modification of the Holloman equation and incorporates the yield strength into the simple power-law hardening model. The Ludwig power-law hardening model is expressed as

$$\upsigma =\upsigma_{0} + K_{\text{L}} \varepsilon_{\text{pl}}^{{n_{\text{L}} }} ,$$
(5)

where σ0 is the pre-stress (or yield strength) of the material and K L and n L are the strength coefficient and strain hardening exponent, respectively. The three parameters in Eq 5 are σ0, K L, and n L. These values are different from the values found in the Holloman equation. Table 2 provides specific values for σ0, K L, and n L as a function of hardness.

Table 2 Ludwig hardening parameters
Full size table

Swift Hardening

The Swift hardening model provides a second modification of the Holloman equation and introduces a strain offset, ɛ0, to account for the initial flow strength of the material. The Swift model is given by

$$\upsigma = K_{\text{S}} \left( {\varepsilon_{0} + \varepsilon_{\text{pl}} } \right)^{{n_{\text{S}} }} ,$$
(6)

where K s and n S are the strength coefficient and strain hardening exponent, respectively. Because the pre-strain term is positive, the flow curve defined by the Swift model is simply the Holloman flow curve shifted in strain space. The three parameters in Eq 6 are ɛ0, K s, and n S. These values are different from the values for the other two power-law hardening models (i.e., the Holloman and Ludwig models). Table 3 provides specific values for ɛ0, K s, and n S as a function of hardness.

Table 3 Swift hardening parameters
Full size table

Voce Hardening

The Voce hardening law represents exponential hardening behavior, and is given by

$$\upsigma =\upsigma_{\infty } - \left( {\upsigma_{\infty } -\upsigma_{\text{o}} } \right)e^{{ - B\upvarepsilon_{\text{pl}} }} ,$$
(7)

where σ∞ is the saturation stress, σ0 is the yield strength, and B is related to the decay constant of the system. The three parameters are σ∞, σ0, and B. Table 4 gives values for these three parameters as a function of hardness.

Table 4 Voce hardening parameters
Full size table

Discussion

In Tables 1 to 4, the first column is the hardness in HV and the other columns provide parameter values for the specific hardening model. The tables should be used as follows: (1) a hardness test is performed on the steel of interest; (2) a hardening model is chosen to characterize the flow behavior of the steel; and (3) the appropriate table is consulted, so that the values of the parameters in the model are obtained.

Because these tables are based on empirical correlations, there is no guarantee that they will provide a perfect fit for the steel of interest. Their use becomes important when a flow model is needed, especially in simulation work and in situations where only a limited amount of physical material is available. By performing the hardness test and using these tables, the engineer can make a reasonable approximation for the actual behavior, which is based on the averaged behavior of many steels, as determined by the empirical correlations.

Remarks on Uncertainty

There is a range of uncertainty associated with the three hardness correlations. The nature of this type of empirical engineering analysis will always have a relatively high degree of uncertainty. The data used to develop the correlations have come from steels with large variations in chemical composition, grain size, phase or constituent fractions, and processing. Each of these variations affects the strain hardening behavior of the steel—and likewise, these variations will influence the measured hardness. Thus, the correlations based on hardness will have an uncertainty that is larger than that of carefully controlled experiments. Nevertheless, when no or little mechanical data exist for a steel, the tables developed in the current study provide a reasonable means to quantify the expected approximate stress-strain value of a steel.

The tables developed in the current study provide a prediction of the average properties or the average mechanical response of steel from a single hardness measurement. Although these tables should not replace actual experimental stress-strain curves, in situations where experiments or material is limited, the tables provide a good engineering method to quantify the hardening response of steels based on a single hardness value. These tables should not be used if sufficient material is available for appropriate testing; but they do allow for the use of flow models for steels, especially in simulations where only hardness values are available. In sheet metal forming simulations, there is often only limited material available to the person performing the computer modeling. Although they are not perfect, the tabular values developed in the current study can give an analyst confidence that the simulations represent the reasonable response of the steel, based upon averaged values that have been empirically correlated over a wide range of steel grades.

Summary

Correlations for yield strength, ultimate tensile strength, and uniform elongation with respect to hardness were developed for steels over a wide range of compositions. These correlations were combined to predict the stress-strain flow curve from a single hardness test for four different hardening models: Holloman, Ludwig, Swift, and Voce. Tables for the parameters for these four hardening models were calculated and are presented. These tables allow an engineer to obtain a reasonable set of parameters for one of the hardening models, based on a single value of hardness.