# Modelling and simulation of thin sheet blanking using damage and rupture criteria

Christophe Poizat, Laurence Lambert-Campagne, Loïc Daridon, Saïd Ahzi, Christophe Husson, Laurent Merle

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Christophe Poizat, Laurence Lambert-Campagne, Loïc Daridon, Saïd Ahzi, Christophe Husson, et al.. Modelling and simulation of thin sheet blanking using damage and rupture criteria. International Journal of Forming Processes, 2005, 8 (1), pp.29-47. 10.3166/ijfp.8.29-47. hal-00447208

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# Modeling and Simulation of Thin Sheet Blanking using damage and rupture criteria

Christophe Poizat*,** — Laurence Campagne* — Loic Daridon* — Said Ahzi* — Christophe Husson*’*** — Laurent Merle***

* Institut de Mécanique des Fluides et des Solides, UMR 7507 CNRS
Université Louis Pasteur / UFR IPST
2, rue Boussingault – 67000 STRASBOURG – France
said.ahzi@ipst-ulp.u-strasbg.fr

** Current address: Fraunhofer IWM
Wöhlerstrasse 11, 79108 Freiburg i. Br., Germany
poizat@iwm.fraunhofer.de

*** Framatome Connectors International (FCI)
BP 50 – Rue Robert Surmont, 72402 La Ferté Bernard - France

ABSTRACT. Because of the trend in the industry to smaller sheet thickness and higher stroke rates, modeling of blanking requires the development of advanced material models. Large plastic strain, high strain rate effects, adiabatic heating and the associated temperature effects, as well as damage and rupture have to be taken into account. In this study, two material models are considered: a physically based model for plasticity (MTS model) combined with a Nucleation And Growth (NAG) model for damage and failure and a phenomenological model for both plasticity and failure (Johnson and Cook, 1985). Finite Element predictions with both models are compared with experimental results and discussed.

RÉSUMÉ. La tendance actuelle dans l'industrie conduit à découper des tôles de plus en plus minces, à des cadences de plus en plus élevées. Ceci requiert le développement de modèles avancés, capables de prendre en compte les effets des grandes déformations, des grandes vitesses de déformation et de l'endommagement jusqu'à rupture. Adiabaticité et effets de température associés sont aussi à prendre en compte. Dans cette contribution, nous considérons deux modèles : un modèle mécanistique de plasticité, le modèle MTS, combiné à un modèle de nucléation et croissance de cavités (modèle NAG) d'une part et le modèle phénoménologique de Johnson-Cook (Johnson et Cook, 1985) d'autre part. Les résultats de simulations numériques de la découpe utilisant ces deux modèles sont comparés avec l'expérience et discutés.

KEYWORDS: blanking, thin sheet, Finite Element simulation, MTS, NAG, Johnson-Cook

MOTS-CLÉS : découpe, tôle mince, simulation par éléments finis, MTS, NAG, Johnson-Cook
# 1. Introduction

Simulation of metal forming processes requires appropriate simulation tools, material models and experimental methods for the identification of the material parameters. In this contribution, we focus on the material modelling.

Several models do exist in the literature. They are mainly based on micromechanics or continuum thermodynamics, and can be characterised by the following criteria: validity domain, physical or phenomenological bases, number of parameters, complexity of the required experiments to identify the material parameters, coupled or uncoupled. In the field of forming processes, to the authors' knowledge, the more commonly used models are the Gurson model (Gurson, 1977) and its extensions (Tvergaard and Needleman, 1984; Gologanu, 1993 and 1997), the purely phenomenological Johnson-Cook model (Johnson and Cook, 1985), and thermodynamically based models (Lemaitre, 1985; Rousselier, 1987). Applications of these models to forming processes, and especially blanking, can be found for instance in (Lemiale, 2001) for the Gurson model, (Hambli, 2002) for the Gurson and Lemaitre models, (Cherouat and Saanouni, 2003) for a thermodynamically based model. The Johnson-Cook material model is mainly used in the simulation of orthogonal machining (e.g. Ng and Aspinwall, 2002), where very high strain rates are present.

All approaches have their advantages and drawbacks. Without going into details, some aspects are briefly described in this introduction. The purely phenomenological Johnson-Cook model is known for its simplicity, and its ability to model high strain rate processes with strain rate and heating effects (ABAQUS, 2001). Material parameters are identified in the literature for several materials (Johnson and Cook, 1985; Tanner et al, 1999). The damage behaviour is modelled by a critical strain rupture criterion. The damage mechanisms of void nucleation, growth and coalescence are not taken into account: there is no internal variable to describe damage and no coupling with the viscoplastic behaviour. On the contrary, models based on the Gurson approach are coupled models with damage evolution. Recent extensions to anisotropy and ellipsoidal void shape are to be found in (Benzerga, 2001) and (Gologanu, 1993 and 1997) respectively. The use of the GTN (Tvergaard and Needleman, 1984) model in the simulation of blanking is discussed in, for instance, (Lemiale, 2001; Hambli, 2001). Thermodynamically based models do not explicitly consider void growth mechanisms, lacking physical description, but are highly coupled models, developed within a rigorous framework. In the case of blanking, the thermodynamically-based Rousselier model (Rousselier, 1987), is of particular interest because of its ability of modelling damage growth under pure shear. From this short review, a trend in the modelling of plasticity and damage clearly consists of introducing more and more microstructural informations, with the aim to catch more precisely local plasticity and damage mechanisms and their influences on the forming processes.
In this contribution, using the commercially available Finite Element code ABAQUS (ABAQUS, 2001), we focus on the modelling of the plasticity and damage behaviours of thin copper sheet (thickness below 1 mm) during blanking with the help of a physically-based model, for the description of plasticity as well as damage.

Blanking is a widely used sheet metal forming operation. It involves numerous process parameters like the blanking clearance ($c=100 (D_m - D_p)/t_{sheet}$ in %, see Figure 1), the wear state of the tool, lubrication, the stroke rate, the sheet thickness... In this process, as mentioned by Kalpakjian (1997) and by Poizat et al. (2002), with a trend to smaller sheet thicknesses and higher stroke rates, the strain rates and the temperatures can now reach several $1000 s^{-1}$ and several $100°C$ in the shearing zone, respectively. Adiabatic heating and temperature effects as well as strain rate effects on plasticity and damage have hence to be taken into account, especially for thin sheets. Based on the discussion from our previous contribution (Poizat et al., 2002), where the Mechanical Threshold Stress (MTS) plasticity model (Follansbee and Kocks, 1988) was combined with a phenomenological rupture criterion, we propose here to use the Mechanical Threshold Stress (MTS) model (Follansbee and Kocks, 1988) for plasticity, combined with a Nucleation And Growth (NAG) model (Campagne et al., 2003) for damage. This model is implemented in ABAQUS/Explicit as a User Material subroutine (ABAQUS, 2001). The results obtained with this physically-based model are compared with the phenomenological Johnson-Cook model (Johnson and Cook, 1985; ABAQUS, 2001) and with experimental results.

<figure><img src="image_2.png" /><figcaption>**Figure 1.** *Blanking process and blank product*</figcaption></figure>
## 2. Material models

### 2.1. The MTS & NAG model

#### 2.1.1. Plasticity model (MTS)

The Mechanical Threshold Stress model (Follansbee and Kocks, 1988) is based on the theory of dislocation motion. It allows for a physically-based description of the material behaviour, over a large range of temperatures and strain rates. In the MTS model, a reference threshold stress, $\hat{\sigma}$, is introduced as internal state variable. The mechanical threshold stress is separated into two components, an athermal stress $\hat{\sigma}_a$ and a thermal stress $\hat{\sigma}_t$. Based on this assumption, Follansbee and Kocks (1988) derived the following expression for the yield stress $\sigma_y$:

$$ \sigma_y = \hat{\sigma}_a + \hat{\sigma}_t = \hat{\sigma}_a + (\hat{\sigma} - \hat{\sigma}_a) \left( 1 - \frac{kT \ln \left( \frac{\dot{\hat{\sigma}}_0}{\dot{\hat{\sigma}}_{eq}^p} \right)}{g_0 G b^3} \right)^{\frac{1}{p}} \quad [1] $$

In equation [1], $\dot{\hat{\sigma}}_{eq}^p$ is the equivalent plastic strain rate, $\dot{\hat{\sigma}}_0^p$ is the reference strain rate, k is the Boltzman's constant, $g_0$ is the normalized activation energy, G is the shear modulus, b is the magnitude of the Burgers vector. The constants p and q characterize the statistically averaged shape of the obstacle profile. This equation is completed with a hardening law, which takes dislocation accumulation and dynamic recovery into account. The hardening, denoted $\theta$, is given by:

$$ \theta = \frac{d\hat{\sigma}}{d\varepsilon_{eq}^p} = \theta_o (1 - F(X)) \quad [2] $$

$$ \text{with } F(X) = \frac{\tanh(2X)}{\tanh(2)} \quad \text{and} \quad X = \frac{\hat{\sigma} - \hat{\sigma}_a}{\hat{\sigma}_s - \hat{\sigma}_a} $$
$\theta_0$ is the initial strain hardening rate and $\hat{\sigma}_s$ is the saturation stress, given by:

$$ \ln\left(\frac{\dot{\epsilon}}{\dot{\epsilon}_{so}}\right) = \frac{Gb^3 A}{kT} \ln\left(\frac{\hat{\sigma}_s}{\hat{\sigma}_{so}}\right) \quad [3] $$

Here, $\dot{\epsilon}_{so}$ and A are constants and $\hat{\sigma}_{so}$ represents the saturation threshold stress for deformation at 0 K. The model also accounts for the softening due to adiabatic heating. In this case, the temperature rise $\Delta T$ is given by:

$$ \Delta T = \frac{X}{\rho C_p} \int \sigma_{ij} d\epsilon_{ij}^p \quad [4] $$

where $\sigma_{ij}$ are the components of the Cauchy stress tensor and $d\epsilon_{ij}^p$ are the components of the plastic strain increment tensor. The definitions and values of X, $\rho$, $C_p$ are given in Table 2.

The parameters and constants used in this work result from a combination of experiments and data from the literature on OFHC copper. They are listed in Tables 1 and 2. More precisely, the Young's modulus, the Poisson's coefficient and the initial yield stress are measured values. The experiments consist of quasistatic tensile tests of thin industrial copper sheets at a strain rate of $2.3 \ 10^{-4} \ s^{-1}$ and at ambient temperature. All other parameters stem from the work of Follansbee and Kocks (1988) on OFHC copper. This set of parameters is discussed in sections 3 and 4.

<table><caption>Parameters (Follansbee and Kocks, 1988)</caption><tbody><tr><td><strong>Deformation conditions</strong></td><td></td></tr><tr><td>Strain rate <i>&epsilon;&#x0307;</i></td><td>10<sup>-4</sup> to 10<sup>4</sup> s<sup>-1</sup></td></tr><tr><td>Test temperature</td><td>25 &deg;C</td></tr><tr><td>Grain size</td><td>40 &mu;m</td></tr><tr><td><strong>Normalized activation energy: g<sub>0</sub></strong></td><td>1.6</td></tr><tr><td><strong>p</strong></td><td>2/3</td></tr><tr><td><strong>q</strong></td><td>1</td></tr><tr><td><i>&epsilon;&#x0307;</i><sub>0</sub> [s<sup>-1</sup>]</td><td>10<sup>7</sup></td></tr><tr><td><strong>k/b<sup>3</sup> [MPa/K]</strong></td><td>0.823</td></tr><tr><td><strong>A</strong></td><td>0.31</td></tr><tr><td><strong>Stress at 0 K: &sigma;<sub>SO</sub> [MPa]</strong></td><td>900</td></tr><tr><td><strong>Saturation strain rate at 0 K: &epsilon;&#x0307;<sub>SO</sub> [s<sup>-1</sup>]</strong></td><td>6.2 10<sup>10</sup></td></tr><tr><td><strong>Initial strain hardening: &theta;<sub>0</sub> [MPa]</strong></td><td>2315</td></tr><tr><td><strong>Athermal stress: &sigma;&#x0302;<sub>a</sub> [MPa]</strong></td><td>40</td></tr></tbody></table>
<table><caption>**Table 1.** *MTS model parameters for OFHC copper*</caption><thead><tr><th colspan="2">Constants (measured)</th></tr></thead><tbody><tr><td>Young's modulus: E [GPa]</td><td>124</td></tr><tr><td>Poisson's coefficient: v</td><td>0.34</td></tr><tr><td>Initial yield stress: σ<sub>0</sub> [MPa]</td><td>400</td></tr><tr><td>Density: ρ [Kg/m<sup>3</sup>]</td><td>8960</td></tr><tr><td>Melting temperature: T<sub>m</sub> [K]</td><td>1356</td></tr><tr><td>Test temperature: T<sub>0</sub> [K]</td><td>300</td></tr><tr><td>Inelastic heat fraction: X</td><td>0.9</td></tr><tr><td>Specific heat: C<sub>p</sub> [J/kg K]</td><td>385</td></tr></tbody></table>

<caption>**Table 2.** *Material constants for OFHC copper*</caption>

### 2.1.2. Damage model (NAG) and rupture criterion

The damage model coupled with the MTS model is based on the physical processes of Nucleation And Growth (NAG) of a relative void volume. This approach is based on the work of Stroh (Stroh, 1957) for the statistical approach of rupture, Curran *et al.* (1987) and Hanim and Ahzi (2000) as well as and Campagne *et al.* (2003) for the modelling of nucleation due to dislocations pile-ups and Weertman (1981), Hanim and Ahzi (2000), and Campagne *et al.* (2003) for the modelling of void growth. The damage is defined in terms of a void volume fraction *f*, with a weak coupling affecting the elastic stiffness (equations 7, 8 and 9). The failure criterion is based on a critical value *f*<sub>c</sub>. In the finite element analysis, when the porosity reaches the critical value *f*<sub>c</sub>, the failure of the element occurs. For details, interested readers are referred to (Campagne, 2003) and (Campagne *et al.*, 2003). We give here the essential constitutive equations for damage. The material porosity, noted *f*, is equal to:

$$f = \frac{V_T}{1 + V_T} \qquad [5]$$

where $V_T$ is the total relative void volume (Figures 2a, 2b and 2c) due to initially existing voids ($V_0$), void nucleation ($\sum_t \Delta V_N^{new}$) and void growth ($V_{CG}$):
$$V_T = V_0 + \sum_t \Delta V_N^{new} + V_{CG} \quad [6]$$

$$\text{with} \quad V_0 = 8\pi N_0 R_0^3 \quad [6a]$$

$$\text{and} \quad \Delta V_N^{new} = 8\pi R_0^3 \dot{N} \Delta t \quad [6b]$$

where $R_0$ is the radius of the void at the beginning of the time interval and $\Delta V_N^{new}$ corresponds to the new relative void increment by nucleation during a time increment $\Delta t$.

$$\Delta V_N^{new} = 8\pi R_0^3 \dot{N}_0 \left[ 1 - \exp\left( -\omega t \left( \frac{\dot{\gamma}}{\dot{\gamma}_0} \right) \right) \right] \Delta t \quad [6c]$$

where $\dot{\gamma}$ and $\dot{\gamma}_0$ are the shear strain rate and reference shear strain rate, respectively. And, to simplify the formulation, we note $\Delta N$:

$$\Delta N = \dot{N}_0 P_D \Delta t \quad [6d]$$

$$\text{with} \quad P_D = 1 - P_B \quad [6e]$$

$$\text{and} \quad P_B = \exp\left(-\omega t \frac{\dot{\gamma}}{\dot{\gamma}_0}\right) \quad [6f]$$

where $P_D$ is associated to a probability of ductile failure and $P_B$ is associated to a probability of brittle failure due to Stroh (1957). The brittle and ductile nature of failure can be associated to the process of release or non-release of dislocations near a pile-up. We therefore assume these processes to be complementary events.

$\dot{N}_0$ is the number of potential nucleation sites per unit volume and per unit time. The relative void volume evolves by nucleation of new voids and growth of the existing ones (Figure 2c). The growth equation for the radius of a void is:

$$R = R_0 \exp\left( \frac{3\pi(1-\nu)}{16g^2G^2t^*} \int_0^t \left( \langle \sigma_{mean} - \sigma_{g0} \rangle^+ \right)^2 dt \right) \quad [6g]$$

Here, $\sigma_{mean}$ is the mean pressure. g is a constant, which is representative of the type of the growth process (details are to be found in (Weertman, 1981)), $\sigma_{g0}$ is the
threshold growth stress, t* a characteristic loading time and $<>^+$ the positive part. To calculate the relative void volume for spherical void, one should account for size distribution. Based on the relation for a size distribution proposed by Curran et al. (1987), the relative void volume by growth is obtained by integration over the entire distribution. The relative volume produced by growth, $V_G$, arises from the growth of both nucleated and existing voids. If $N_T$ denotes the total number per unit volume of pre-existing and nucleated voids, this relative volume by growth is the relative void volume of the crown (Figure 2c). It can be expressed by:

$$V_{CG} = 8\pi N_T R_0^3 \left[ \exp\left(\frac{3h\pi(1-\nu)}{16g^2G^2t^*}\int_0^t (\langle\sigma_{mean}-\sigma_{g0}\rangle^+)^2 dt \right) - 1 \right] \quad [6h]$$

$$\text{where } N_T = N_0 + \sum_t \Delta N \qquad [6i]$$

<figure><img src="image_3.png" /><figcaption>Figure 2a. Pre-existing relative void volume: $V_0$</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Figure 2b. Pre-existing relative void volume and the new relative void volume by nucleation.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 2c. Growth of initial and new relative void volume. Representation of the relative void volume of the crown.</figcaption></figure>
The creation of cavities inside a material domain modifies the mechanical properties. This can be modelled through a weak coupling. A damage parameter *d* is introduced so that the elastic modulus and shear modulus are reduced with increasing damage:

$$E_{\text{eff}} = E(1-d) \quad \text{and} \quad G_{\text{eff}} = \frac{E_{\text{eff}}}{2(1+\nu)} \qquad [7, 8]$$

Here, $E_{\text{eff}}$ is the effective Young's modulus and $G_{\text{eff}}$ is the effective shear modulus. In equation [1], G is hence replaced by $G_{\text{eff}}$.

The expression of *d* is derived from a self-coherent approach (François et al., 1991), coupled with an onset porosity. Lemaitre (1991) and Bonora (1997) have namely shown that the damage in copper starts to occur only after a specific strain level. We chose to introduce a threshold porosity $f_T$ instead of strain. With these considerations, the following expression for the damage law is proposed:

$$ \left\{ \begin{array}{ll} d = 0 & \text{if } f \in [0; f_T] \\ d = \frac{15(1-\nu)}{7-5\nu} * f_c * \left( \frac{f-f_T}{f_c-f_T} \right) & \text{if } f \in ]f_T; f_c[ \\ d = 1 & \text{if } f = f_c \end{array} \right. \qquad [9] $$

This model is implemented in the Finite Element Code ABAQUS/Explicit (ABAQUS, 2001) via a User Material Subroutine. The material parameters for the mechanistic damage model are given in Table 3. Their values lie in the physically acceptable range defined by Curran et al. (1987) and Weertman (1981). They are fitted on experimental results of the blanking process. The damage model as well as the value for the parameters are discussed in section 4 and in (Campagne, 2003).

<table><thead><tr><th colspan="2">Parameters</th></tr></thead><tbody><tr><td>Nucleation void radius R<sub>0</sub> [cm]</td><td>10<sup>-4</sup></td></tr><tr><td>Nucleation coefficient N&#x0307;<sub>0</sub> [number/cm<sup>3</sup> s]</td><td>10<sup>20</sup></td></tr><tr><td>Total number N<sub>0</sub> [number/cm<sup>3</sup>]</td><td>10<sup>4</sup></td></tr><tr><td>Frequency ω [/s]</td><td>1.0 x 10<sup>6</sup></td></tr><tr><td>Constant g</td><td>0.329</td></tr><tr><td>Hydrostatic threshold stress σ<sub>g0</sub> [MPa]</td><td>260</td></tr><tr><td>Critical relative void volume V<sub>C</sub></td><td>0.3</td></tr><tr><td>Characteristic loading time t* [s]</td><td>0.51 x 10<sup>-6</sup></td></tr></tbody></table>
<table><caption>**Table 3.** *Material parameters for the mechanistic NAG model*</caption><tr><td>Threshold porosity f<sub>r</sub></td><td>0.001</td></tr></table>

## 2.2. Johnson-Cook Model

On the contrary to the previous model, which is derived from physical considerations, a purely phenomenological model is now considered. This model is due to Johnson and Cook (1985). It is known for its simplicity and the availability of parameters for various materials of interest. It is commonly used in the simulation of highly dynamic processes, like impact or machining.

### 2.2.1. Plasticity model (JC)

Johnson and Cook (1985) proposed a purely phenomenological constitutive model for metals subjected to strains, strain rates and heating effects. In this model, the von Mises flow stress is expressed as:

$$ \sigma_y = (A + B\varepsilon^n)(1 + C \ln \frac{\dot{\varepsilon}_{eq}^p}{\dot{\varepsilon}_0}) \left(1 - \left(\frac{T - T_r}{T_m - T_r}\right)^m\right) \quad [10] $$

In equation [10], the term in the first bracket gives the dependence on strain, the second represents instantaneous strain-rate sensitivity, and the third term represents the temperature dependence of stress. T<sub>r</sub> is a reference temperature and T<sub>m</sub> is the melting temperature of the material. The five material constants are A, B, C, n and m.

### 2.2.2 Rupture

Johnson and Cook (1985) proposed a rupture criterion based on cumulative plastic strain and given by equations [11] and [12]:

$$ \varepsilon_r = \left( d_1 + d_2 \exp(d_3 \frac{\sigma_H}{\sigma_{eq}}) \right) \left( 1 + d_4 \ln \frac{\dot{\varepsilon}_{eq}^p}{\dot{\varepsilon}_0} \right) \left( 1 + d_5 \frac{T - T_r}{T_m - T_r} \right) \quad [11] $$

$$ W = \sum \frac{\Delta \varepsilon_{eq}^p}{\varepsilon_r} \quad [12] $$
In equation [11], $\sigma_H$ is the hydrostatic stress, $\sigma_{eq}$ is the von Mises equivalent stress and $\dot{\epsilon}_0 = 1 \text{ s}^{-1}$ is the reference strain rate. $d_1, d_2, d_3, d_4$ and $d_5$ are the material constants and are given in (Johnson and Cook, 1985) for OFHC copper. In equation [12], $\Delta\epsilon_{eq}^p$ is the increment of equivalent plastic strain. When $W$ reaches 1, the stresses are set to zero, simulating rupture.

### 2.2.3 Material parameters

The material data used in this contribution are given in Tables 2 and 4. They result from a combination of experimental results and data from the literature (Johnson and Cook, 1985). More precisely, in Table 4, the values of C, n, m, $T_m$ and $d_3, d_4$ and $d_5$ are the values found for OFHC copper (Johnson and Cook, 1985), whereas all other parameters are determined experimentally on the industrial material used in this contribution. The experiments consist of quasistatic tensile tests of thin industrial copper sheet at a strain rate of $2.3 \times 10^{-4} \text{ s}^{-1}$ and at ambient temperature. The sheet thickness is 0.58 mm. No tensile test at other temperatures and no dynamic test were carried out for the industrial material: this explains the choice of the authors to take the data for OFHC copper. It is thought to be a good approximation and allows for a first comparison with industrial experimental results. It is worth noting that the sign of the parameter $d_3$ has to be changed when using the Johnson-Cook model that is implemented in ABAQUS/Explicit, since the pressure is defined in the Finite Element code ABAQUS as the opposite of $\sigma_H$ (ABAQUS, 2001).

<table><caption>**Table 4. Johnson-Cook model parameters**</caption><thead><tr><th colspan="2">Parameters</th></tr></thead><tbody><tr><td>A [MPa]</td><td>440</td></tr><tr><td>B [MPa]</td><td>150</td></tr><tr><td>C</td><td>0.025</td></tr><tr><td>n</td><td>0.31</td></tr><tr><td>m</td><td>1.09</td></tr><tr><td>T<sub>m</sub> [K]</td><td>1356</td></tr><tr><td>T<sub>r</sub> [K]</td><td>300</td></tr><tr><td>ε̇<sub>0</sub> [s<sup>-1</sup>]</td><td>1</td></tr><tr><td>d<sub>1</sub></td><td>0.3</td></tr><tr><td>d<sub>2</sub></td><td>0.28</td></tr><tr><td>d<sub>3</sub></td><td>-3.03</td></tr><tr><td>d<sub>4</sub></td><td>0.014</td></tr><tr><td>d<sub>5</sub></td><td>1.12</td></tr></tbody></table>
### 3. Finite Element simulation of the blanking process

In this section, we present and discuss results obtained from Finite Element simulations of the blanking process of a thin copper sheet. The results are given for the MTS&NAG model and for the Johnson-Cook model. The sheet thickness is 0.58 mm. The punch and die radii are equal to 0.01 mm (See Figure 1). The clearance is equal to 8%. The simulation is two dimensional but not axisymmetric. For symmetry reason, just one half of the problem is simulated and shown in Figure 3b. The elements are CPE4R plain strain elements with reduced integration and hourglass control (ABAQUS, 2001, User Manual, Vol. 2, p. I.1-2). The mesh remains relatively coarse in this study (50x20 elements in the blanking area, corresponding to an element size of 30 microns). The tools are meshed with analytical rigid surface (ABAQUS, 2001). Details on the boundary conditions are given in the previous contribution of the authors (Poizat et al., 2002). The friction law is a Coulomb friction law with a coefficient of 0.02.

In Figure 3a, a typical deformed mesh obtained with the implemented MTS&NAG model is shown at rupture. The localization band, where damage occurs, is clearly seen. The roll-over, shear and fracture zones are obtained in this simulation with a high clearance (8%). In Figure 3b, the profile of the blanked part is shown for two clearances: 8%, as in Figure 3a and 0.8%. The two profiles mainly differ in the formation of a burr in the case of a large clearance. Since burrs have to be avoided in applications like connectors, small clearances should be imposed in a blanking process. Similar studies were carried out, varying the wear state of the tool or the punch velocity. The tool wear is modelled by taking two punch radii. The corresponding results tend to show that the clearance is the process parameter that mainly influences the profile of the blanked part. The punch velocity and even the wear state of the tool do not significantly modify the simulated punch profile. However, these results are to be taken with care. A quantitative prediction of the geometry of the blanked product requires more advanced adaptive remeshing techniques (Borouchaki et al., 2002) in comparison to the standard remeshing procedure of ABAQUS/Explicit, that is systematically used in this contribution (ABAQUS, 2001). No convergence was obtained with finer meshes.

In Figure 4, the relative punch force vs. relative displacement<fnref n="1" /> curves are given for a stroke rate equal to 1000 strokes/min. The numerical results, especially those obtained with the MTS&NAG model, are in fair agreement with the experimental results. This comparison with the real industrial process was not done in the previous contribution of the authors (Poizat et al., 2001). The results reported there

<footnote n="1">¹ The relative punch force is defined as the ratio between the punch force and the maximal experimental punch force; the relative punch displacement is defined as the ratio between the punch displacement and the sheet thickness</footnote>
were obtained with OFHC copper, an academic material. The load level was significantly lower than the real force level. The determination of parameter like the flow stress for the industrial material allows for first comparison between simulation and real industrial processes. In Figure 4, for both models, discrepancies with the experimental results are observed in the initial slope. This point could however be explained by sliding and clearances in the measurement set-up. In absence of rigorous material data identification of the industrial copper used in the experiments, we can not conclude on the discrepancy in the slope where softening occurs. Other reasons can explain this discrepancy. For instance, the real boundary conditions and friction behaviour at the tool-workpiece interface are not clear. The contact surface between the tool and the blanked part also clearly influence the resulting punch force. Experimental efforts towards material parameters identification, including optimisation techniques (Tanner et al, 1999), are required for a better characterisation of the material. It's worth noting that the study is a roll-over high (around 800%, see figure 4). This value is fitted to the coherent with values found in the literature for several materials (Johnson and Cook, 1985). However, this value is not experimentally for the material under consideration. Numerical results excluded: mesh size dependence and influence of the remeshing studied in details in future works.

<figure><img src="image_6.png" /><figcaption>**Figure 3a.** Deformed mesh at rupture (left) and profile (right), in the case of a sharp tool and a large clearance (8%).</figcaption></figure>

<figure><img src="image_7.png" /></figure>
<figure><img src="image_8.png" /><figcaption>**Figure 4.** Punch-force displacement curve obtained with the MTS&NAG model and the Johnson-Cook model. Comparison with experimental results on an industrial press, at a stroke rates of 1000 strokes/min.</figcaption></figure>

Figure 5 shows the same results as in Figure 4 with additionally results obtained without adiabatic heating. Since the sheet is cut in about 1 ms, the influence of adiabatic heating is significant in both the MTS&NAG and the Johnson-Cook models, and has to be taken into account. Two aspects of adiabatic heating are now discussed:

- Adiabatic heating has a stronger influence in the Johnson-Cook model as in the MTS model. The parameter m in the Johnson-Cook model was taken from OFHC copper data. Since it determines the thermal softening, an exact identification of this parameter is required for the industrial material.

- Without adiabatic heating, the slope during the softening phase for both approaches (MTS&NAG and Johnson-Cook) are identical and in better agreement with the experimental results.

In both cases, a direct measurement of the temperature field in the shearing zone is suggested. This study shows clearly that experimental works are required to determine the relative influence of adiabatic shearing versus ductile fracture on
material softening in high strain rates problems in general and in thin sheet blanking
in particular.

<figure><img src="image_9.png" /><figcaption>**Figure 5.** *Effects of adiabatic heating on the punch displacement curves*</figcaption></figure>

Independently of the comparison with experiments, it is however clear that the MTS&NAG model is more representative of the material behavior of copper than the Johnson-Cook model. The implementation of a modified version of the Johnson-Cook model as proposed in (Rule and Jones, 1998) is suggested. The set of equations becomes however more and more complex without having the physical ground of the MTS&NAG model. This point is detailed in the chapter 4, where the mechanistic NAG model is discussed. Another possibility consists of modifying the Johnson-Cook model in a more simple way. As reported in (Sing and Rao, 1997) for the prediction of forming limits curves, the hardening behavior of copper and copper alloys is better described by a Voce law ($\sigma_y = \sigma_0 - (\sigma_0 - \sigma_1)\exp(-n\epsilon)$, (Voce, 1948)) than by a Hollomon law ($\sigma_y = \sigma_0 + K\epsilon^n$), that is more appropriate for steels. The Johnson-Cook model has recently been implemented in ABAQUS by the authors in its original and in the modified forms. Numerical analyses of blanking are in progress.
4. Discussion on the damage models

The results presented in this chapter are obtained with one element tests under uniaxial loading. As in the Finite Element simulation of blanking, the element is the four-nodes plain strain element with reduced integration of ABAQUS/Explicit. These tests allow for a discussion of the main characteristics of the mechanistic Nucleation and Growth model, especially in terms of the strain rate influence.

Based on physical considerations, the MTS&NAG model allows for a detailed description of damage: the proportion of damage due to nucleation and growth is known (Figure 6) and depends on the strain rate (Figures 7 and 8). Figures 7 and 8 show that nucleation tends to dominate the damage process when the strain rate increases: the rate of nucleation as well as the relative void volume due to nucleation at rupture are higher at $1000 \text{ s}^{-1}$ than at $100 \text{ s}^{-1}$. This is in good agreement with experimental results: at such strain rates, the resulting strain at rupture is higher with increasing strain rates, which is not obvious, but in agreement with results from the literature (Johnson and Cook, 1985; Goubot, 1994, p. 193 and 194 especially). However, our rate dependence is quantitatively too high when compared with experimental results (Johnson and Cook, 1985). Experimental results are needed to show if parameters like the number of potential nucleation sites per unit volume and per unit time, $\dot{N}_0$, depend on the strain rate or if the equation giving the strain rate dependence to nucleation and/or growth has to be modified.

<figure><img src="image_10.png" /><figcaption>**Figure 6.** *Relative void volume fraction due to nucleation and growth versus strain. (one element test under uniaxial loading at $100 \text{ s}^{-1}$)*</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Figure 7.** Relative void volume due to nucleation versus strain for different strain rates (one element test under uniaxial loading at 100 s⁻¹ and 1000 s⁻¹)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Figure 8:** Relative void volume due to nucleation versus time. (one element test under uniaxial loading at 100 s⁻¹ and 1000 s⁻¹)</figcaption></figure>
**5. Conclusion**

In this contribution, a phenomenological model and a physically-based model are considered to simulate the blanking process of thin copper sheets. Both approaches are dedicated to problems involving strain rates and temperature effects. The influence of adiabatic heating is shown to significantly influence the simulated results for the punch force-displacement curve. Both models give relatively good results in terms of the punch-force displacement curve. The Johnson-Cook model is known for its simplicity. However, it does not model damage evolution and should at least be modified in the hardening law for copper and copper alloys. On the other hand, the MTS&NAG model requires significant effort towards material data identification, but is more representative of the material behavior of face centered cubic materials at small as well as high strain rates. As a consequence, a modified Johnson-Cook model can be effective in a first step, but the MTS&NAG model is helpful in providing a better insight in the blanking process, with physical backgrounds on plasticity and damage over a large range of strain rates. In the future, as discussed for impact problems (Campagne, 2003), efforts have to be made towards material parameters identification and a stronger coupling between plasticity and damage.

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