# Constitutive models characterisation for XC48 steel in compression by a parametric identification technique

Bertrand Langrand, Eric Markiewicz, Pascal Drazetic, Pascal Geoffroy, J.L.

Petitniot, Jacky Fabis

► To cite this version:

Bertrand Langrand, Eric Markiewicz, Pascal Drazetic, Pascal Geoffroy, J.L. Petitniot, et al. Constitutive models characterisation for XC48 steel in compression by a parametric identification technique. International Journal of Materials and Product Technology, 1997, 12 (4-6), 10.1504/IJMPT.1997.036379. hal-04911406

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# Constitutive models characterisation for XC48 steel in compression by a parametric identification technique

B. Langrand*, E. Markiewicz,* P. Drazetic*, P. Geoffroy**, J.L. Petitniot** and J. Fabis**

*Industrial and Human Control and Mechanical Engineering Laboratory, Mechanical Engineering Research Group, URA CNRS D1775, University of Valenciennes, B.P. 311-59304 Valenciennes Cedex, France

**Fluid Mechanical Institute of Lille, Entity of ONERA, 59800 Lille, France

**Abstract:** This paper deals with a parametric identification technique, based on conjugate gradient methods. As part of this research, compression tests are performed on a XC48 steel (AFNOR norm 1048AISI), which is strain rate sensitive. Two kinds of specimen, tubular and cylindrical 'dumbbells', are tested at different strain rates, from $10^{-3}$ to $130s^{-1}$, and for impact velocities ranging from 5 mm/mn to 6 m/s. The parameters of the Johnson-Cook constitutive model are evaluated directly from the XC48 steel compression stress-strain diagrams expressed in terms of true stress and strain measurements. These parameters are next used as input data for a FE crash simulation of the tubular specimen. Comparison with experimental results prove the validity of the identified parameters.

**Keywords:** compression testing, parametric identification technique, XC48 steel.

**Reference** to this article should be made as follows: Langrand, B., Markiewicz, E., Drazetic, P., Geoffroy, P., Petitniot, J.L. and Fabis, J. (1997) 'Constitutive models characterisation for XC48 steel in compression by a parametric identification technique', *Int. J. of Materials and Product Technology*, Vol. 12, Nos. 4-6, pp. 428-446.

## 1 Introduction

Classical numerical simulations relative to crash analysis, ballistic tests and metal sheet stamping, require the knowledge of dynamic material behaviour. Mathematical models enable one to reproduce the experimental physical behaviour as accurately as possible. Program packages, available on the open market, offer several constitutive models which allow one to take the strain rate effects in a particular plastic strain domain into account. One of the problems encountered by users is the implementation of the suitable model and in particular the choice of the model parameters. Jones [1] shows how important it is to use the constitutive models suitable for the strain domain and for
the strain rate. Moreover, the use of these models is still inadequate because the dynamic behaviour of many materials is not completely defined.

The material strain rate sensitivity has been widely studied. Of particular interest is Klepaczko's work [2] in the case of shear at strain rates from $10^{-3}$ to $10^5$ s$^{-1}$, Jones's work [1] on mild steel at strain rates from the quasi-static to $10^3$ s$^{-1}$, Johnson's work [3, 4] for different strain rates, temperature and materials, and Bois's and Grave's work [5] for armour steel 30 NCD16 and Taylor cylinder impact tests. In the strain rate range between the quasi-static to $500$ s$^{-1}$, there is a lack of literature on the constitutive models characterisation. Paradoxically, this strain rate range is that mostly met in the automotive and railway crash simulations.

This paper suggests an identification technique of constitutive models parameters. The method is based on conjugate gradients method [6, 7]. Compression tests are performed on two kinds of specimen, tubular and cylindrical 'dumbbells'. Experimental data is obtained for a XC48 steel (AFNOR norm 1048AISI). Parameters identification is done directly from the compression experimental true stress-strain diagrams, using the Johnson-Cook [3] model. A FE crash simulation is next performed on the tubular specimen using the Johnson-Cook model with identified parameters.

## 2 Constitutive models

There are two distinct types of constitutive model adapted to crash simulation. The work-hardening viscosity models, where plastic work-hardening is modified to take the strain rate effects into account. These models seem to be devoted to a medium range of strain rates and describe the behaviour effects briefly. The elastic-plastic viscosity models, very complete, are devoted to high strain rates and high pressures phenomena.

For the present experimental compression tests, the strain rates reached a maximum of approximately $150$ s$^{-1}$. It is for this reason that the authors have only been interested in the first approach in this study.

The work-hardening viscosity behaviour is defined by the plastic flow low [8, 9]:

$$D_{ij}^{p} = \frac{3}{2} \delta(f) \frac{\langle \dot{\bar{\sigma}} \rangle}{k(p) \dot{\bar{\sigma}}} s_{ij} \quad (1)$$

where $\delta(f)$, $p$, $k(p)$, $\dot{\bar{\sigma}}$, $s_{ij}$ and $D_{ij}^p$ represent the Kronecker symbol, the cumulative plastic strain, the work-hardening function, the equivalent von Mises stress, the deviatoric part of the stress tensor and the plastic strain rate tensor. The plastic flow law is connected to a work-hardening law strain rate dependency. Many methods can be used to introduce the strain rate effects: empirical models, additive models, multiplicative models.

In these models, the initial work-hardening function, $k(\bar{\varepsilon})$, is usually expressed as:

$$k(\bar{\varepsilon}) = \bar{\sigma} - \sigma_{y0} \quad (2)$$

where $\bar{\varepsilon} = k^{-1}(R)$, becomes $k(\bar{\varepsilon}, \dot{\bar{\varepsilon}})$. This leads to the following modification of the yield criterion:
$$f(\sigma_{ij}, R) = j(\sigma_{ij}) - (R + \sigma_{y0}) = 0 \qquad (3)$$

where R is the work-hardening internal variable, and of the work-hardening law in order to respect the thermodynamic formulation given by Lemaitre and Chaboche [8].

## 2.1 Empirical approach

Empirical models are obtained by an extensive experimental procedure and their formulation is more or less complex. A non-exhaustive list of examples of such an approach, reported amongst others by Malvern [10], Priadi et al. [11] and Zerilly and Amstrong [12], is presented in Table 1. The simplest equations such as l(a) and l(c) (Table 1), express the strain rate according to stress. Equations such as l(e) are established from the variation of the Voce rigid plastic model parameters [11]. In this case, the strain domain is not limited to a certain range. For these five models the number of parameters remains acceptable. They can be used to describe a restricted plastic strain domain. Other models are based on the dislocation mechanisms (e.g. the model developed by Zerilly and Amstrong [12]). The identification of this empirical model leads to a larger and more expensive experimental study.

<table><caption>**Table 1** Empirical strain rate flow functions</caption><thead><tr><th>N°</th><th>Name-Authors(s)</th><th>Model</th><th>Parameters</th></tr></thead><tbody><tr><td>1.a</td><td>Prandtl</td><td>ε̇ = C·sh(α·σ)</td><td>C, α</td></tr><tr><td>1.b</td><td>Sokolovsky<br>Malvern</td><td>ε̇ = γ·[σ̄ - σ<sub>y</sub> / σ<sub>y0</sub>]<sup>-1</sup></td><td>γ, σ<sub>y</sub>, σ<sub>y0</sub></td></tr><tr><td>1.c</td><td>Perzina</td><td>ε̇ = γ·[σ̄ - σ<sub>y</sub> / σ<sub>y</sub>]<sup>-1</sup></td><td>γ, σ<sub>y</sub></td></tr><tr><td>1.d</td><td>Zerilli-Amstrong</td><td>σ = C<sub>0</sub> + C<sub>2</sub>·ε<sup>n</sup>·exp(-C<sub>3</sub>·T + C<sub>4</sub>·T·lnε̇)</td><td>C<sub>0</sub>, C<sub>2</sub>, C<sub>3</sub>, C<sub>4</sub> if fcc</td></tr><tr><td>1.e</td><td>Modified<br>Krupkowsky<br>(Swift)</td><td>σ = C<sub>0</sub> + C<sub>1</sub>·ε<sup>n</sup>·[exp(-C<sub>3</sub>·T + C<sub>4</sub>·T·lnε̇) + C<sub>5</sub>·ε<sup>n</sup>]<br>σ = K<sub>ref</sub>·X<sup>a</sup>·(ε<sub>0ref</sub>·X<sup>b</sup> + ε<sup>p</sup>)<sup>n·X<sup>c</sup></sup><br>with X = ε̇/ε̇<sub>ref</sub></td><td>C<sub>0</sub>, C<sub>1</sub>, C<sub>3</sub>, C<sub>4</sub>, C<sub>5</sub> if bcc<br>K<sub>ref</sub>, ε<sub>0ref</sub>, a, b, n, c</td></tr></tbody></table>

$$^{1-1}f(\bar{f}) = 0 \text{ if } f \le 0, \langle f \rangle = f \text{ if } f > 0$$

## 2.2 Additive work-hardening viscosity

In the second approach, the strain rate effects are introduced into the work-hardening law by the means of a new work-hardening function, $k_a(\bar{\varepsilon}, \dot{\bar{\varepsilon}})$. The strain rate effects are added to the standard quasi-static with the function, $k_{ref}(\bar{\varepsilon})$.
It therefore implies that:

$$k_a(\bar{\varepsilon}, \dot{\bar{\varepsilon}}) = k_{ref}(\bar{\varepsilon}) + F(\bar{\varepsilon}, \dot{\bar{\varepsilon}}) \quad (4)$$

Various examples of these functions are given in Table 2.

<table><caption>**Table 2** Additive strain rate flow functions.</caption><thead><tr><th>N°</th><th>Name-Authors(s)</th><th>F(ε̄, ε̇)</th><th>Parameters</th></tr></thead><tbody><tr><td>2.a</td><td>Sokolovsky<br>Malvern</td><td>A · log(1 + B · ε̇)</td><td>A, B</td></tr><tr><td>2.b</td><td>Ludwick type</td><td>g(ε<sup>p</sup>) · log(ε̇/ε̇<sub>ref</sub>)</td><td>g(ε<sup>p</sup>)</td></tr><tr><td>2.c</td><td>Lindholm</td><td>(A + B · ε<sub>p</sub><sup>n</sup>) · log ε̇</td><td>A, B, n</td></tr></tbody></table>

## 2.3 Multiplicative work-hardening viscosity

Classical engineering approaches introduce strain rate effects into the work-hardening function with a new work-hardening function, $k_m(\bar{\varepsilon}, \dot{\bar{\varepsilon}})$. This function is defined as the product of the flow function $G(\bar{\varepsilon}, \dot{\bar{\varepsilon}})$ by the strain rate behaviour taken as reference:

$$k_m(\bar{\varepsilon}, \dot{\bar{\varepsilon}}) = k_{ref}(\bar{\varepsilon})G(\bar{\varepsilon}, \dot{\bar{\varepsilon}}) \quad (5)$$

This method corresponds to an adjustment of the yield surface as the strain rate increases. Various examples of such approaches are found in engineering literature and a non-exhaustive list is presented in Table 3. The simplicity of the formulation and the low number of parameters are advantageous for numerical simulations. Some of the formulae given by Symonds [13] and Johnson and Cook [3] are generally limited to a narrow plastic strain range or to materials showing the same strain rate sensitivity independently of the strain level (e.g. for an ARMCO iron, a OFHC copper or a 4340 steel under shear load and for strain rate between 10<sup>-1</sup> and 10<sup>2</sup> s<sup>-1</sup> [4]. These kinds of models introduce a homothetic effect on the work-hardening curve with respect to the strain rate. Other models, such as that of Jones [14], allow one to extend this description to a larger strain range by incorporating into the model the evolution of the strain rate influence during the deformation process.
<table><caption>**Table 3** Multiplicative strain rate flow functions.</caption><thead><tr><th>N°</th><th>Name-Authors(s)</th><th>k<sub>ref</sub>(ε̄)</th><th>G(ε̄, ˙ε̄)</th><th>Parameters</th></tr></thead><tbody><tr><td>3.a</td><td>Cowper-Symonds</td><td>analytical or<br>multi-linear<br>curve</td><td>1 + [ ˙ε̄ / D ]<sup>1/m</sup></td><td>D, m</td></tr><tr><td>3.b</td><td>Johnson-Cook</td><td>(A + B · ε<sub>p</sub><sup>n</sup>)</td><td>1 + C · ln( ˙ε̄ / ˙ε<sub>ref</sub>)<sup>-1</sup></td><td>A, B, n, C</td></tr><tr><td>3.c</td><td>Jones</td><td>analytical or<br>multi-linear<br>curve</td><td>1 + [ ε<sub>p</sub> - ε<sub>y</sub> / (D<sub>y</sub> · (ε<sub>p</sub> - ε) + D<sub>p</sub> · (ε - ε<sub>y</sub>)) ]<sup>1/m</sup></td><td>D<sub>p</sub>, D<sub>y</sub>, ε<sub>p</sub>,<br>ε<sub>y</sub>, m</td></tr></tbody></table>

# 3 Experimentation

## 3.1 Experimental set up

The lowest strain rate tests were performed on an Instron screw tension/compression electromechanical quasi-static machine, type 1195, with the following parameters: maximum force 100 kN, maximum velocity 500 mm/mn, stroke daylight 1.2 m. The highest strain rate tests were performed on a Schenck double effect hydraulic actuator piloted in displacement. It has a three level servovalve with the following parameters: maximum force in the quasi-static range 659 kN, maximum force in the dynamic range 50 kN, maximum velocity ±10 m/s.

## 3.2 Compression specimens

Small cylinders are usually used for compression [4]. Unfortunately, because of the hydraulic jack end conditions, these types of specimens were difficult to secure. Hence, two specimens of geometries described in Figure 1 were used; tubular or type I, cylindrical ‘dumbbell’ or type II.
<figure><img src="image_2.png" /><figcaption>Figure 1 Compression specimens.</figcaption></figure>

### Tubular specimen

This basic geometry offers the advantage of a more straightforward interpretation of results. However, the plastic fold location is not known before the test. This implies the use of strain gauges along the specimen's length. The following size of tube is retained for the compression tests: length 40 mm, outer and inner diameters 12 and 10 mm.

### Cylindrical ‘dumbbell’ specimen

The special interest of this specimen is to concentrate the stress on its central part and to disregard the structural effects. Strains can therefore be located more easily. The instrumentation is easier and the measures are only related to material effects. This specific geometry is based on the study of Harding *et al.* [15].

## 3.3 *Experimental results*

Table 4 summarises the experimental results for the two types of specimen and for different impact velocities, $V_{imp}$, in terms of elastic and plastic strain rates $\dot{\epsilon}^e$ and $\dot{\epsilon}^p$, Young's modulus $E$, elastic-plastic plateau length $A_p$, true yield stress $\sigma_y$, true critical stress and displacement of plastic buckling $\sigma_{CR}$ and $\delta_{CR}$ ($S$ represents the specimen area). Figures 2 and 3 show the true stress-plastic strain diagram of type I and type II specimens at different strain rates. The straight portion at the end of the curves, type I for $\dot{\epsilon}^p = 3.5s^{-1}$ and type II for $\dot{\epsilon}^p = 0.26s^{-1}$, $\dot{\epsilon}^p = 5s^{-1}$ and $\dot{\epsilon}^p = 130s^{-1}$, is due to strain gauge saturation.
<table><caption>**Table 4** Experimental results.</caption><thead><tr><th>V<sub>IMP</sub></th><th>Type</th><th>S<br>mm<sup>2</sup></th><th>ε̇<sup>e</sup><br>s<sup>-1</sup></th><th>ε̇<sup>p</sup><br>s<sup>-1</sup></th><th>E<br>MPa</th><th>A<sub>p</sub><br>%</th><th>σ<sub>y</sub><br>MPa</th><th>σ<sub>CR</sub><br>MPa</th><th>δ<sub>CR</sub><br>mm</th></tr></thead><tbody><tr><td rowspan="2">5<br>mm/mn</td><td>I</td><td>34.55</td><td>0.3 10<sup>-3</sup></td><td>1.1 10<sup>-3</sup></td><td>210 10<sup>3</sup></td><td>1.0</td><td>453</td><td>712</td><td>3.0</td></tr><tr><td>II</td><td>28.29</td><td>0.4 10<sup>-3</sup></td><td>2.7 10<sup>-3</sup></td><td>214 10<sup>3</sup></td><td>1.0</td><td>450</td><td>751.7</td><td>2.1</td></tr><tr><td rowspan="2">500<br>mm/mn</td><td>I</td><td>34.55</td><td>0.056</td><td>0.15</td><td>210 10<sup>3</sup></td><td>1.2</td><td>480</td><td>751</td><td>3.5</td></tr><tr><td>II</td><td>28.29</td><td>0.06</td><td>0.26</td><td>215 10<sup>3</sup></td><td>1.2</td><td>507.8</td><td>933</td><td>4.2</td></tr><tr><td rowspan="2">0.2<br>m/s</td><td>I</td><td>34.55</td><td>0.8</td><td>3.5</td><td>208 10<sup>3</sup></td><td>1.2</td><td>542.7</td><td>825.6</td><td>3.5</td></tr><tr><td>II</td><td>28.29</td><td>0.4</td><td>5.0</td><td>213 10<sup>3</sup></td><td>1.2</td><td>550</td><td>967.5</td><td>4</td></tr><tr><td rowspan="2">6<br>m/s</td><td>I</td><td>34.55</td><td>25</td><td>63.2</td><td>214 10<sup>3</sup></td><td>1.0</td><td>624.7</td><td>862</td><td>/</td></tr><tr><td>II</td><td>28.29</td><td>30</td><td>130</td><td>217 10<sup>3</sup></td><td>1.0</td><td>645</td><td>1085.4</td><td>/</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>**Figure 2** Type I true stress-strain diagram.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3 Type II true stress-strain diagram.</figcaption></figure>

As shown in Table 4, Young's modulus is not affected by the strain rate. The difference between Young's modulus of type I and type II specimens is less than 3%. The true yield stress increases in conjunction with the strain rate. For the two types of specimens, the true yield stress follows the same exponential progression and both curves are very close, as shown in Figure 4. The accuracy of the entire acquisition line, including strain and stress measures, is in the order of 2%. So we can conclude that there is no geometric effect on Young's modulus and yield stress results.

On the other hand, a great difference is noted for the critical stress of type I and II specimens (Figure 5). Table 5 gives an estimation of this difference for both specimens. This is because the critical stress at buckling of type I and II specimens takes place at different plastic strains (Figures 2 and 3), because of the geometric effects. To avoid these structural effects, we decided to retain as critical stresses those corresponding to an identical plastic strain. Because of the strain gauges saturation the common plastic strain that we can take is in the order of 3%. This is why all curves are cut at a 3% plastic strain. As reported in Table 6 and shown in Figure 6, the new critical stress evolution for the two specimens gets closer at the strain rate increases.
<figure><img src="image_5.png" /><figcaption>Figure 4 Yield stress evolution.</figcaption></figure>

<table><caption>Table 5 Type I and II specimens Critical stresses.</caption><thead><tr><th>σ<sub>CR</sub> (MPa)<br>Type I</th><th>σ<sub>CR</sub> MEAN<br>Type II</th><th>Error<br>%</th></tr></thead><tbody><tr><td>712</td><td>751.7</td><td>± 3.4 %</td></tr><tr><td>751</td><td>933.1</td><td>± 10.8 %</td></tr><tr><td>825.6</td><td>967.5</td><td>± 8 %</td></tr><tr><td>862</td><td>1085.4</td><td>± 11.5 %</td></tr></tbody></table>
<figure><img src="image_6.png" /><figcaption>Figure 5 Critical stress evolution.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6 Critical stress evolution for a plastic strain range of 3%.</figcaption></figure>
<table><caption>**Table 6** Type I and II specimens stresses for a plastic strain range of 3%.</caption><thead><tr><th>σ<sub>3%</sub> (MPa)<br>Type I</th><th>σ<sub>3%</sub> MEAN<br>Type II</th><th>σ<sub>3%</sub> (MPa)</th><th>Error<br>%</th></tr></thead><tbody><tr><td>657</td><td>700</td><td>678.5</td><td>± 3.17 %</td></tr><tr><td>692</td><td>735</td><td>713.5</td><td>± 3.01 %</td></tr><tr><td>760</td><td>783</td><td>771.5</td><td>± 1.49 %</td></tr><tr><td>820</td><td>820</td><td>820</td><td>± 0.00 %</td></tr></tbody></table>

# 4 Identification

The identification of the constitutive models parameters is obtained by an inverse method. This method is widely used for a large field of applications. Numerous mathematical models have been developed to describe the material behaviour. Penazzi et al. [16] present a non-exhaustive list of these models. These models require the identification of parameters for using in numerical codes. These constants are interdependent and are difficult to quantify. The solution consists in identifying these parameters with a numerical tool.

The aim in this research is to determine the constitutive models parameters. A parametric identification program is developed, based on a conjugate gradients algorithm [6, 7]. Our objective is to identify the parameters of one constitutive model which is usually employed in the current FE codes: the Johnson-Cook [3] model parameters (A, B, n, C). The principles and the progress of this program are looked at in more detail in the following paragraphs. the identification of these constants is carried out by the means of compression true stress-strain diagrams. these curves steam fro the XC48 steel experimental tests performed at various strain rates.

## 4.1 Davidon's algorithm

The Davidon's algorithm [6, 7] is one of the programs based on the conjugate gradient methods. This algorithm is split into seven main steps:

Step 0 let $z_0 \in \Re^n$, if $\nabla f(z_0)=0$, computation stop

step 1 $I=0 \quad H_0=Id^n$ (or any symmetrical matrix and defined positive)
$$g_0 = \nabla f(z_0)$$

step 2 $\underline{h}_i = -H_i g_i$

step 3 computation of $\lambda_i > 0$ as $f(z_i + \lambda_i h_i) = \min\{f(z_i + \lambda_i h_i) | \lambda_i \ge 0\}$

step 4 computation of $\nabla f(z_i + \lambda_i h_i)$

step 5 if $\nabla f(z_i + \lambda_i h_i) = 0$ then computation stops

else $z_{i+1} = z_i + \lambda_i h_i$

$g_{i+1} = \nabla f(z_{i+1})$

$\Delta g_i = g_{i+1} - g_i$

$\Delta z_i = z_{i+1} - z_i$
$$ \mathbf{H}_{i+1} = \mathbf{H}_i - \frac{\mathbf{H}_i \Delta \underline{\mathbf{g}}_i \langle \mathbf{H}_i \Delta \underline{\mathbf{g}}_i \rangle}{\langle \Delta \underline{\mathbf{g}}_i, \mathbf{H}_i \cdot \Delta \underline{\mathbf{g}}_i \rangle} + \frac{\Delta \underline{\mathbf{z}}_i \langle \Delta \underline{\mathbf{z}}_i \rangle}{\langle \Delta \underline{\mathbf{z}}_i, \Delta \underline{\mathbf{g}}_i \rangle} $$

step 6 $i+i+1$ and return to step 2

where $\underline{\mathbf{z}}$ is the identified parameters vector and $\underline{\mathbf{h}}_i$ the exploration direction vector.

$$ \text{Note: } \xi \langle \xi = \xi \xi^T \text{ and } \langle \xi, \xi \rangle = \xi^T \xi $$

## 4.2 Criterion or cost function

The criterion or cost function is the part of the program where the parameter vector $\underline{\mathbf{z}}$ intervenes. The Johnson-Cook [3] model is chosen because it is very common in engineering literature and FE codes. The criterion chosen is the quadratic sum of errors. This sum is made with every experimental point. Its analytical expression is:

$$ f(\underline{\mathbf{z}}) = \sum_{i=1}^{N_p} \frac{d_i(\underline{\mathbf{z}})^2}{\sigma_{exp}^{(i)2}} = \sum_{i=1}^{N_p} \frac{(\sigma_{exp}^{(i)} - \sigma_{cal}^{(i)}(\underline{\mathbf{z}}))^2}{\sigma_{exp}^{(i)2}} \quad (7) $$

where $\underline{\mathbf{z}}$ is the parameters vector, $d_i(\underline{\mathbf{z}})$ represents the point $i$ distance between the experimental curve and the constitutive model, $\sigma_{exp}^{(i)}$ and $\sigma_{cal}^{(i)}$ are the point $i$ experimental and identified true stresses, $N_p$ is the experimental number of points.

The main part of the conjugate gradients method concerns the $f(\underline{\mathbf{z}})$ criterion minimisation. The application of Equation (7) gives the expression of the criterion for Johnson-Cook ($f_{JC}(\underline{\mathbf{z}})$) constitutive equation:

$$ f_{JC}(\underline{\mathbf{z}}) = \sum_{i=1}^{N_p} \frac{\left( \sigma_{exp}^{(i)} - \left( A + B\varepsilon_p^{(i)} \right)^n \left( 1 + C \ln(\dot{\varepsilon}/\dot{\varepsilon}_{ref}) \right) \right)^2}{\sigma_{exp}^{(i)2}} \quad (8) $$

with $\underline{\mathbf{z}} = (A, B, n, C)$.

## 4.3 Gradient vector of the cost function

The gradient vector computation is performed with the centred derivative formulation. The expressions remain unchanged for any constitutive model. The expression of the numerical gradient vector is:

$$ \underline{\mathbf{g}} = [\nabla f(z_j)]_{j=1}^N = \left[ \frac{f(z_j + dz_j) - f(z_j - dz_j)}{2dz_j} \right]_{j=1}^N \quad (9) $$

with $N$, the number of parameters (e.g. $N = 4$ for the Johnson-Cook model), $dz_j$, the infinitesimal displacement linked to the parameter $j$.
The application of the standard derivative analytical formulation of a multi-variable function gives the theoretical gradient vector:

$$ \underline{\mathbf{g}} = \left[ \frac{\partial f(z_j)}{\partial z_j} \right]_{j=1}^{N} \qquad (10) $$

In the case of the Johnson-Cook model, the theoretical gradient vector is given by Equation (11):

$$ \underline{\mathbf{g}} = \left[ \frac{\partial f(\underline{\mathbf{z}})}{\partial z_j} \right]_{(j=A,B,n,C)} = \left[ \frac{\partial f(\underline{\mathbf{z}})}{\partial A}, \frac{\partial f(\underline{\mathbf{z}})}{\partial B}, \frac{\partial f(\underline{\mathbf{z}})}{\partial n}, \frac{\partial f(\underline{\mathbf{z}})}{\partial C} \right] \qquad (11) $$

It implies that:

$$ \underline{\mathbf{g}} = \left[ \begin{array}{l} 2(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}) \sum_{i=1}^{N_p} \frac{\left(A + B\varepsilon_p^{(i)n}\right)\left(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}}{\sigma_{\mathrm{exp}}^{(i)2}} \\ 2(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}) \sum_{i=1}^{N_p} \varepsilon_p^{(i)n} \frac{\left(A + B\varepsilon_p^{(i)n}\right)\left(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}}{\sigma_{\mathrm{exp}}^{(i)2}} \\ 2B(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}) \sum_{i=1}^{N_p} \varepsilon_p^{(i)n} \ln(\varepsilon_p^{(i)}) \frac{\left(A + B\varepsilon_p^{(i)n}\right)\left(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}}{\sigma_{\mathrm{exp}}^{(i)2}} \\ 2\mathrm{ln}\dot{\underline{\mathbf{\varepsilon}}} \sum_{i=1}^{N_p} \left(A + B\varepsilon_p^{(i)n}\right) \frac{\left(A + B\varepsilon_p^{(i)n}\right)\left(1 + \mathrm{Cln}\dot{\underline{\mathbf{\varepsilon}}}}{\sigma_{\mathrm{exp}}^{(i)2}} \end{array} \right] \qquad (12) $$

The numerical gradient is checked against theoretical results. Results of numerical and theoretical gradients computations, calculated from a 91 experimental points curve, were collected. It is clearly seen that numerical and theoretical results remain unchanged for parameters A, B and C. The influence of the infinitesimal displacement is null. On the other hand, for the exponent n, both computations, show differences as the infinitesimal displacement value increases. For a dn infinitesimal displacement of about 50%, the difference between theoretical and numerical gradients is approximately about 20%. However, because dn remains lower than 1%, error between analytical and numerical gradient has no effect on minimisation computations.

## 4.4 Termination test

As for the criterion, there are numerous formulations to stop the conjugate gradient method. When the criterion is a quadratic function, the absolute break test is satisfied when the criterion $f(\underline{\mathbf{z}})$ is equal to zero. The following criterion formulation has been chosen:
$$f(\mathbf{z}) = \sum_{i=1}^{N_p} \frac{d_i(\mathbf{z})^2}{\sigma_{\text{exp}}^{(i) 2}} = \sum_{i=1}^{N_p} \frac{(\sigma_{\text{exp}}^{(i)} - \sigma_{\text{cal}}^{(i)}(\mathbf{z}))^2}{\sigma_{\text{exp}}^{(i) 2}} \quad (13)$$

Experimental curves are obtained with inevitable measure errors. These curves intervene in the $f(\mathbf{z})$ criterion modelling and even weak errors prevent the criterion cancelling. The termination test employed for the identification program consists in computing the following relation for each experimental point:

$$\chi(i) = \left| \frac{\sigma_{\text{exp}}^{(i)} - \sigma_{\text{cal}}^{(i)}}{\sigma_{\text{exp}}^{(i)}} \right| \quad (14)$$

and for each parameter:

$$\eta^i = \left| \frac{z_i^j - z_i^{j-1}}{z_i^{j-1}} \right| \quad (15)$$

with $z_i$, the parameter under consideration, $j$, the iteration number.

The break test is satisfied when every point verifies:

$$\left| \frac{\sigma_{\text{exp}}^{(i)} - \sigma_{\text{cal}}^{(i)}}{\sigma_{\text{exp}}^{(i)}} \right| = \chi(i) \le \chi_{\text{ref}} \quad (16)$$

and when every parameter verifies:

$$\left| \frac{z_i^j - z_i^{j-1}}{z_i^{j-1}} \right| = \eta^i \le \eta_{\text{ref}} \quad (17)$$

with $\chi_{\text{ref}}$ the point accuracy, $\eta_{\text{ref}}$ the parameter accuracy.

The identification error limits are given by the Equations (14) and (16). The second set of Equations (15) and (17), gives parameter stability relation.

## 4.5 Results of identification

The previous sections have shown how experimental true stress-strain diagrams are used to identify parameters. The Johnson-Cook model parameters are determined for each kind of specimen. The initial parameter values used to identify the model were: $A_0 = 400$, $B_0 = 1000$, $n_0 = 0.4$, $C_0 = 0.001$. At least, the identification leads to one set of model parameters available for strain rates: type I $10^{-3} s^{-1} \le \dot{\epsilon}^p \le 63.2 s^{-1}$ and type II $10^{-3} s^{-1} \le \dot{\epsilon}^p \le 130 s^{-1}$.

Table 7 gives results of identification computation for the Johnson-Cook model and for both types of specimens. For each specimen and strain rate, the comparison between experimental and computational diagrams is shown in terms of minimum, $\theta_{\text{MIN}}$, maximum, $\theta_{\text{MAX}}$, mean difference, $\theta_M$, mean square deviation, $\mu_{\text{SD}}$, and percentage of points verifying Equation (16), $N_\chi$ (Table 7). For each specimen, Figures 7 and 8 show the comparison of experimental and computational curves for two strain rates and for each specimen.
<table><caption>**Table 7** Johnson-Cook model identified parameters</caption><thead><tr><th colspan="6">a - Type I</th><th colspan="6">b - Type II</th></tr><tr><th colspan="6">A=497.6 - B=2060 - n=0.6088 - C=0.01975</th><th colspan="6">A=515 - B=2468 - n=0.6435 - C=0.01911</th></tr><tr><th>ε̇<sup>P</sup><br>s<sup>-1</sup></th><th>θ<sub>MIN</sub><br>%</th><th>θ<sub>MAX</sub><br>%</th><th>θ<sub>M</sub><br>%</th><th>μ<sub>SD</sub></th><th>N<sub>χ</sub><br>%</th><th>ε̇<sup>P</sup><br>s<sup>-1</sup></th><th>θ<sub>MIN</sub><br>%</th><th>θ<sub>MAX</sub><br>%</th><th>θ<sub>M</sub><br>%</th><th>μ<sub>SD</sub></th><th>N<sub>χ</sub><br>%</th></tr></thead><tbody><tr><td>1.1 10<sup>-3</sup></td><td>0.00</td><td>3.90</td><td>1.59</td><td>0.89</td><td>100</td><td>2.7 10<sup>-3</sup></td><td>0.02</td><td>3.51</td><td>1.49</td><td>0.71</td><td>100</td></tr><tr><td>0.15</td><td>1.07</td><td>4.25</td><td>3.28</td><td>0.49</td><td>100</td><td>0.26</td><td>0.22</td><td>4.68</td><td>2.51</td><td>1.16</td><td>100</td></tr><tr><td>3.5</td><td>0.03</td><td>4.62</td><td>1.23</td><td>0.97</td><td>100</td><td>5</td><td>0.00</td><td>3.78</td><td>0.82</td><td>0.64</td><td>100</td></tr><tr><td>63.2</td><td>1.07</td><td>12.80</td><td>2.78</td><td>1.86</td><td>92.59</td><td>130</td><td>0.07</td><td>17.16</td><td>4.30</td><td>4.06</td><td>76.3</td></tr><tr><td>curve set</td><td>0.00</td><td>12.80</td><td>2.24</td><td>1.86</td><td>98.96</td><td>curve set</td><td>0.00</td><td>17.16</td><td>1.95</td><td>2.22</td><td>95.73</td></tr></tbody></table>

<figure><img src="image_8.png" /><figcaption>**Figure 7** Johnson-Cook model identification of type I specimen.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8 Johnson-Cook model identification of type II specimen.</figcaption></figure>

The Johnson-Cook model identified with the diagrams of the type II specimens is compared to the type I experimental diagrams (Table 8(a)). The same comparison is shown with the type I Johnson-Cook model and the type II experimental diagrams (Tables 8(b)). Results show that both parameter sets can be used to approximate both specimen's experimental curves ($\theta_M<4\%$). Geometrical effects are really limited and material behaviour is well identified for both specimens by the Johnson-Cook model.

<table><caption>Table 8 Johnson-Cook model results comparison.</caption><thead><tr><th colspan="6">a - Type I experimental curves<br/>/Type II model curves</th><th colspan="6">b - Type II experimental curves<br/>/Type I model curves</th></tr><tr><th colspan="6">Type I Experimental curves<br/>Type II Johnson-Cook model curves</th><th colspan="6">Type II Experimental curves<br/>Type I Johnson-Cook model curves</th></tr><tr><th>ε<sup>P</sup><br/>s<sup>-1</sup></th><th>θ<sub>MIN</sub><br/>%</th><th>θ<sub>MAX</sub><br/>%</th><th>θ<sub>M</sub><br/>%</th><th>μ<sub>SD</sub></th><th>N<sub>χ</sub><br/>%</th><th>ε<sup>P</sup><br/>s<sup>-1</sup></th><th>θ<sub>MIN</sub><br/>%</th><th>θ<sub>MAX</sub><br/>%</th><th>θ<sub>M</sub><br/>%</th><th>μ<sub>SD</sub></th><th>N<sub>χ</sub><br/>%</th></tr></thead><tbody><tr><td>1.1 10<sup>-3</sup></td><td>0.04</td><td>3.42</td><td>2.39</td><td>0.53</td><td>100</td><td>2.7 10<sup>-3</sup></td><td>0.11</td><td>6.37</td><td>4.39</td><td>1.84</td><td>51.25</td></tr><tr><td>0.15</td><td>4.74</td><td>7.75</td><td>7.02</td><td>0.40</td><td>0.5</td><td>0.26</td><td>0.05</td><td>3.25</td><td>1.58</td><td>0.88</td><td>100</td></tr><tr><td>3.5</td><td>0.08</td><td>4.36</td><td>2.33</td><td>1.02</td><td>100</td><td>5</td><td>1.77</td><td>6.84</td><td>2.84</td><td>0.69</td><td>97.65</td></tr><tr><td>63.2</td><td>0.10</td><td>9.98</td><td>1.70</td><td>1.43</td><td>95</td><td>130</td><td>0.05</td><td>19.40</td><td>5.40</td><td>4.97</td><td>55</td></tr><tr><td>curve set</td><td>0.04</td><td>9.98</td><td>3.81</td><td>2.41</td><td>70</td><td>curve set</td><td>0.05</td><td>19.40</td><td>3.75</td><td>2.74</td><td>71.5</td></tr></tbody></table>
## 5 Numerical validation

Constitutive models are used to simulate the material behaviour, for example in crash simulation. To evaluate the quality of the precedently identified parameters, a crush simulation is performed on the circular tube (type I specimen) and compared with the corresponding test. The FEM's code employed to realise this simulation is Pam-CrashTM. The specimen is completely modelled with 5346 nodes and 5280 shell elements (type 103 in the Pam-CrashTM standard). The numerical specimen is supported by a fixed rigid wall and is impacted by a second moving one with an imposed velocity of 6 m/s. Rigid walls are chosen as infinite planes with infinite mass and finite friction ($f = 0.2$). The Johnson-Cook model used to take the strain rate sensitivity of the material into account is:

$$ \sigma(\varepsilon_p, \dot{\varepsilon}) = (A + B\varepsilon_p^n)(1 + 1/p \ln \dot{\varepsilon} / \dot{\varepsilon}_{\text{ref}}) \quad (18) $$

with: $A = 497.6$ MPa, $B = 2060$ MPa, $n = 0.6088$ and $p = 1/C = 50.63$, obtained for the identification of the true stress-plastic strain diagrams of the type I specimens. The model before and after the simulation is represented on Figure 9(a) and 9(b). Simulaton leads to plastic strain and strain rates in the same range than experimentation. However, along the plastic fold, it can be locally observed a plastic strain value of 0.8. Extrapolation of stresses from 0.03 to 0.8 of plastic strain by means of the identified Johnson-Cook model leads finally to acceptable results in terms of load and local strains evolution. Figure 10 compares the experimental and simulated load-strain diagrams for a 6m/s impact test. This result shows that identification of constitutive model's parameters for this kind of steel with a constant work-hardening modulus can be performed even with a plastic strain range of about 0.03.

<figure><img src="image_10.png" /><figcaption>a- before crush</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>b- after crush</figcaption></figure>

<caption>**Figure 9(a) and (b) Circular tube FE crush simulation model.**</caption>
<figure><img src="image_12.png" /><figcaption>Figure 10 Crush simulation and experimental load-strain diagrams comparison.</figcaption></figure>

## 6 Conclusion

The characterisation of the constitutive models has been examined using an inverse method, based on a parametric identification technique. Compression tests have been performed on a XC48 steel with two kinds of specimens (tubular and cylindrical 'dumbbells'). Experimental data has been obtained for strain rates varying from $10^{-3}$ to $130 \text{ s}^{-1}$ and for a common maximum plastic strain of 0.03. A parametric identification technique, based on the conjugate gradient methods, has been developed and illustrated with one example. It was shown that there is no geometrical effects on true stress-strain diagrams. We have shown that the Johnson-Cook model with the identified parameters has a good correlation when compared to the experimental true stress-strain diagrams of compression (the differences between experimental and identified curves are lower than 3%). The identified parameters set of Johnson-Cook model are validated with the FE crush simulation of the circular tube (type I). It can be concluded that these parameters can be used in FE modelling to simulate the crash behaviour of structures loaded in compression within the same strain rate range. Crush simulation have shown that stress extrapolation from 0.03 to 0.8 of plastic strain by means of the identified model leads to satisfactory results for this kind of steel with a constant work-hardening modulus.

## Acknowledgements

The authors wish to thank Mr Jean Michel Boully for his suggestions during the experimental phase of this study, and we would like to acknowledge the support and encouragement of Mr Francis Dupriez and Mr Eric Deletombe.
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