Olivier Pantalé
email: olivier.pantale@enit.fr

Pierre Tize Mha

Amèvi Tongne

Efficient implementation of non-linear flow law using neural network into the Abaqus Explicit FEM code

Keywords:

Machine learning techniques are increasingly used to predict material behavior in scientific applications and offer a significant advantage over conventional numerical methods. In this work, an Artificial Neural Network (ANN) model is used in a finite element formulation to define the flow law of a metallic material as a function of plastic strain , plastic strain rate . and temperature . First, we present the general structure of the neural network, its operation and focus on the ability of the network to deduce, without prior learning, the derivatives of the flow law with respect to the model inputs. In order to validate the robustness and accuracy of the proposed model, we compare and analyze the performance of several network architectures with respect to the analytical formulation of a Johnson-Cook behavior law for a 42CrMo4 steel. In a second part, after having selected an Artificial Neural Network architecture with 2 hidden layers, we present the implementation of this model in the Abaqus Explicit computational code in the form of a VUHARD subroutine. The predictive capability of the proposed model is then demonstrated during the numerical simulation of two test cases: the necking of a circular bar and a Taylor impact test. The results obtained show a very high capability of the ANN to replace the analytical formulation of a Johnson-Cook behavior law in a finite element code, while remaining competitive in terms of numerical simulation time compared to a classical approach.

Introduction

Numerical simulation of forming processes, machining or the behavior of structures subjected to dynamic loads and impacts requires the use of specific material behavior laws, whose parameters are identified by tests based on Taylor impacts, Hopkinson bars or Gleeble thermomechanical simulator. The behavior laws are selected according to their availability in a finite element code or the possibility, if not available, to implement them through user subroutines. In this study, our work is based on the use of the finite element code Abaqus Explicit which offers the possibility to define user behavior laws through FORTRAN subroutines VUMAT or VUHARD [START_REF] Gao | FE realization of a thermo-visco-plastic constitutive model using VUMAT in ABAQUS/Explicit program[END_REF][START_REF] Jansen Van Rensburg | Tutorial on state variable based plasticity: An Abaqus UHARD subroutine[END_REF] like the work proposed by Duc-Toan et al. [START_REF] Duc-Toan | A modified Johnson-Cook model to predict stress-strain curves of boron steel sheets at elevated and cooling temperatures[END_REF] or more recently by Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF]. The usual procedure is to select a mathematical form of the behavior law among those available in the literature (Johnson-Cook, Zerilli Armstrong, . . . ) and then, from the results of experimental tests, to identify via a regression method, the parameters of the selected law.

In most cases, the behavior of the material at high temperatures and strain rates is highly nonlinear, and the effects of many factors on the flow stress are also nonlinear, which reduces the accuracy of the prediction by the regression methods usually used and limits the field of application. In addition, the selection, development, and numerical implementation of such constitutive equations is time-consuming. Artificial intelligence techniques allow advances concerning the laws of behavior in order to allow a better identification of these laws. Thus, Versino et al. [START_REF] Versino | Data driven modeling of plastic deformation[END_REF] used a Machine Learning technique based on symbolic regression for the development of data-driven constitutive model. Obtaining a flow equation, and thus its analytical derivative, allows the use of iterative solvers that employ Jacobians (i.e., the Newton-Raphson scheme) that allow a higher order of convergence. This symbolic regression technique has also been used by other authors since, such as Bomarito et al. [START_REF] Bomarito | Development of interpretable, data-driven plasticity models with symbolic regression[END_REF], Park et al. [START_REF] Park | Multiscale constitutive model using data-driven yield function[END_REF] using constrained symbolic regression technique, or Nassr et al. [START_REF] Nassr | Developing constitutive models from EPRbased self-learning finite element analysis[END_REF] using evolutionary polynomial regression.

Given this situation, it is therefore natural to look for a method to eliminate some intermediate steps between experimental tests and numerical simulation in order to simplify the computational chain. In this perspective, recent advances in deep learning constitute a way of investigation. The basic idea is to replace the analytical formulation used to calculate the flow stress of the material as a function of the plastic strain , the plastic strain rate . and the temperature , by an Artificial Neural Network (ANN). This neural network is trained to reproduce the behavior of the considered material only from the experimental data resulting from the tests, ignoring any assumption on the analytical form of the assumed flow law. Consequently, it is no longer necessary to postulate an analytical form of the behavior law in order to implement it in a FEM code. Artificial neural networks and deep learning are becoming more and more important in today's society, and their fields of application are getting wider and wider. After a boom in the early 1990s and a decline in interest towards the end of the 20th century, neural networks are experiencing a resurgence of interest and even a huge media hype under the name of deep learning. Their use in science and physics is now widespread, notably because of the current availability of efficient tools allowing to program Artificial Neural Networks thanks to widely available libraries such as Tensorflow [START_REF] Mattmann | Machine Learning with Tensorflow[END_REF] for example. The most publicized applications of deep learning are mainly related to medical diagnosis, robotics, images and language recognition, but the applications go far beyond that and all sciences can now use these techniques, including thermomechanical numerical simulation.

Artificial neural networks can solve problems that are difficult to conceptualize using traditional computational methods. Unlike a classical approach based on a regression method, an Artificial Neural Network does not need to know the mathematical form of the model it seeks to reproduce. The Artificial Neural Network learns from the training data and can reproduce the behavior of a model from the simple knowledge of a series of input and output values with no prior assumption on their nature and their interrelations. Hornik et al. [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF] have rigorously established in 1989 that feed-forward Neural Network are a class of universal approximators, extending the work proposed twenty years before by Minsky and Papert [START_REF] Minsky | Perceptrons; an Introduction To Computational Geometry[END_REF] where they demonstrated that the simple two-layer perceptron is incapable of usefully representing or approximating functions outside a very narrow and special class. The ANN has adjustment, memorization and anticipation capabilities, and better performances than the approach based on implementing a constitutive equation. Therefore, Artificial Neural Networks can now enable novel approaches for modeling the behavior of materials and have been successfully applied in the prediction of constitutive relationships of some metals and alloys in recent years. Applicability of ANN to model path dependent plasticity has been explored and some review of the literature can be found for example in Gorji et al. [START_REF] Gorji | On the potential of recurrent neural networks for modeling path dependent plasticity[END_REF] concerning the use of Recurrent Neural Network, in Jamli et al. [START_REF] Jamli | The sustainability of neural network applications within finite element analysis in sheet metal forming: A review[END_REF] concerning their application in finite element analysis of metal forming processes, or in Jiao et al. [START_REF] Jiao | Artificial intelligence-enabled smart mechanical metamaterials: Advent and future trends[END_REF] concerning the applicability to meta-materials and their characterization. A distinction must be made between ANNbased hardening models and ANN-based constitutive models. Both approaches have been studied by many researchers over the last thirty years. Ghaboussi et al. [START_REF] Ghaboussi | Knowledge-based modeling of material behavior with neural networks[END_REF] published a pioneering paper, in which they proposed an ANN-based constitutive model for planar concrete under monotonic biaxial loading and cyclic uniaxial loading in which they successfully predicted several loading paths in the biaxial loading condition. They improved NN architecture by introducing Adaptive NN and Autoprogressive NN in [START_REF] Ghaboussi | Autoprogressive training of neural network constitutive models[END_REF][START_REF] Ghaboussi | New nested adaptive neural networks (NANN) for constitutive modeling[END_REF] where the network architecture evolves during the training phase to better learn complex stress-strain behavior of materials using a global load-deflection response. The approach adopted for this study is an ANN-based hardening model for which, the evaluation of the material flow stress calculated by the ANN is combined with a Radial Return type integration scheme.

Lin et al. [START_REF] Lin | Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel[END_REF] developed a neural network to predict the flow stress of 42CrMo4 steel in hot compression tests on a Gleeble thermomechanical device and showed a very good correlation between experimental and predicted results. An extension to the numerical implementation of this approach would have been desirable. Javadi et al. [START_REF] Javadi | Intelligent finite element method: An evolutionary approach to constitutive modeling[END_REF] used a neural network to capture the behavior of complex materials using a FE model incorporating a backpropagation neural network. Lu et al. [START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF] presented a comparative study of the modeling of an Al-Cu-Mg-Ag alloy behavior by a constitutive equation based on the Zener-Hollomon parameter and a neural network. It also shows that the model based on the ANN proposes a better prediction than the constitutive equation. Ashtiani et al. [START_REF] Ashtiani | A comparative study on the phenomenological and artificial neural network models to predict hot deformation behavior of AlCuMgPb alloy[END_REF] compared the prediction capabilities of an ANN against a conventional approach based on several behavior laws such as Johnson-Cook, Arrhenius and Strain compensated Arrhenius and concluded better efficiency and accuracy of the neural network in predicting the hot behavior of Al-Cu-Mg-Pb alloy. Ashtiani et al. [START_REF] Ashtiani | A comparative study on the phenomenological and artificial neural network models to predict hot deformation behavior of AlCuMgPb alloy[END_REF] have shown that a well-trained ANN can efficiently overcome the lacks of physics coming from analytical constitutive behaviors such as the Johnson-Cook, or the Arrhenius one. Ali et al. [START_REF] Ali | Application of artificial neural networks in micromechanics for polycrystalline metals[END_REF] presented an ANN model coupled with a rate dependent crystal plasticity finite element method to simulate the stress-strain behavior of material and its microstructure evolution in a AA6063-T6 under simple shear and tension. Stoffel et al. [START_REF] Stoffel | Artificial neural networks and intelligent finite elements in non-linear structural mechanics[END_REF][START_REF] Stoffel | Neural network based constitutive modeling of nonlinear viscoplastic structural response[END_REF] applied ANN to complicated structural deformations of shock-wave loaded plates involving both geometrical and physical non-linearities. Li et al. [START_REF] Li | Machine-learning based temperature-and ratedependent plasticity model: Application to analysis of fracture experiments on DP steel[END_REF] implemented a VUMAT subroutine for Abaqus where parameters were identified through a combination of analytical formulas and a back propagation algorithm. Recently, Huang et al. [START_REF] Huang | Constitutive models and microstructure evolution of Ti-6Al-4V alloy during the hot compressive process[END_REF] developed a neural network model to predict the flow stress and the microstructure evolution of Ti-6Al-4V alloy. It also showed the superiority of this approach over an Arrhenius behavior model, especially because the ANN can predict the flow stress in the whole range of deformation. Temporal Convolutional Networks have also been applied by Abueidda et al. [START_REF] Abueidda | Deep learning for plasticity and thermo-viscoplasticity[END_REF] to predict the historydependent responses of a class of cellular materials. Thermodynamics based ANN where also proposed by Masi et al. [START_REF] Masi | Thermodynamics-based Artificial Neural Networks for constitutive modeling[END_REF] to reduce physically inconsistencies in the predictions of the NN. They demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, using both hyper-and hypo-plasticity models. As presented by Knight et al. [START_REF] Knight | Larger GPU-accelerated brain simulations with procedural connectivity[END_REF], evolution of Neural Network Architectures is not the only way to progress, constant evolution of the hardware architecture to implement ANN will also have to be taken into account for the next future.

In Section 2, we present the main bases of the deep learning with in details the description of the structure of the neural network and the equations which govern its functioning. As we will see in Section 4 concerning the numerical implementation of the flow law, the programming of the neural network on the finite element code Abaqus Explicit requires the determination of the 3 derivatives of the flow stress with respect to the plastic strain , the plastic strain rate . and the temperature . The determination of these derivatives will be the subject of the second part of Section 2. Section 3 is devoted to a detailed presentation of the ANN learning for a Johnson-Cook type behavior law for a 42CrMo4 steel. In this section, we will show the influence of the network structure on the accuracy of the flow stress and the derivatives evaluation. Section 4 is devoted to a presentation of the numerical implementation of the neural network in the Abaqus finite element code in the form of a FORTRAN VUHARD subroutine as well as numerical test cases to validate the proposed approach. A conclusion and perspectives are finally proposed at the end.

Artificial neural network set-up

In this section, we briefly introduce the basic concepts of backward and forward propagating Artificial Neural Networks (ANNs) that are relevant to this work. The global architecture we have retained for this work is a multi-layer feed-forward network which can be seen as a universal approximator as proposed by Hornik et al. [START_REF] Hornik | Multilayer feedforward networks are universal approximators[END_REF]. The proposed neural network is used to approximate non-linear functions. The concept of neural network consists in simulating the flow of information inside the human brain by defining a set of neurons (arranged in different layers) defining functions producing results according to the inputs. These neurons are interconnected from one layer to another and continuously improve their predictive ability as the network is trained and learns a new concept [START_REF] Specht | A general regression neural network[END_REF]. Fig. 1 presents the global architecture of an ANN with multiple hidden layers where neurons are set up in different layers (from the first hidden layer to the output layer) with the following characteristics: . • The first layer, is the so-called input layer, and in our application concerning constitutive law, consists of 3 inputs. This one is not composed of neurons as they will be defined hereafter, and collects the incoming information: the three values corresponding to the plastic strain , the plastic strain rate and the temperature respectively.

• The ANN contains at least one hidden layer (in the current paper, we will consider only one and two hidden layers ANNs) containing a variable number of neurons.

• The last layer is the so-called output layer, and in our case consists of only one neuron providing the value of the von Mises flow stress .

• Neurons are not connected to other neurons of the same layer but only to the immediately preceding outputs and following inputs.

In Fig. 1 we have separated the summation ⃖ ⃗ ( ) and the activation ⃖ ⃗ ̂ ( ) parts of all neurons within the hidden layers. Concerning the notation used, note here that we are using the formalism of a vector (using an arrow over the symbol) for many quantities in the equations hereafter, even if these are not vectors in the sense of a mathematical notation. In reality, these quantities are vertical one-dimensional arrays used to store data, which visually correspond to the components of a vector. In the proposed architecture, there is no activation function associated to the neuron in the output layer, as usually done for regression problems which is our case here. The data flows from layer to layer until we obtain the output of the ANN. Every layer, except the first one, in the ANN, no matter how many neurons are in, have an input with a variable number of items and an output with a fixed number of items . In the so-called forward propagation algorithm, the output of one layer becomes the input of the next one.

Neural network governing equations 2.1.1. Input layer

Conforming to Fig. 1, the input layer is defined by ⃖ ⃗ = [ 1 , 2 , 3 ] . In the proposed application, ⃖ ⃗ has three components related to the plastic strain , the plastic strain rate . and the temperature , respectively. As it will be presented in Section 2.1.4, input variables are normalized to avoid ill-conditioning for the optimization procedure, mainly because they represent various physics with a large discrepancy in values.

Hidden layers

Any hidden layer , containing neurons, takes a weighted sum of the outputs ⃖ ⃗ of the immediately previous layer ( -1), containing neurons, given by the following equation:

( ) = ∑ =1 ( ) + ( ) (1)
where ( ) is the nodal value of the th neuron of layer , ( ) is the associated weight parameter between the th neuron of layer and the th neuron of layer ( -1) and ( ) is the associated bias of the th neuron of layer . Those weights and bias are the training parameters of the ANN that we have to adjust during the training procedure described in Section 3. Using matrix notation, Eq. ( 1) can be rewritten with the following form:

⃖ ⃗ ( ) = ( ) ⋅ ⃖ ⃗ + ⃖ ⃗ ( ) (2)
where

⃖ ⃗ ( ) = [ 1 ( ) , 2 ( ) , … , ( ) ] contains the nodal values resulting from the summation operation in layer , ⃖ ⃗ ( ) = [ 1 ( ) , 2 ( ) , … , ( ) ]
is the nodal bias of layer and ( ) is the [ × ] weight parameters matrix of layer given hereafter:

( ) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 11 ( ) 12 ( ) ⋯ 1 ( ) 21 ( ) 22 ( )
⋯ 2 ( ) ⋮ ⋮ ⋱ ⋮ 1 ( ) 2 ( ) ⋯ ( ) ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (3)
The total number of training parameters for any hidden layer is the sum of the number of weight parameters and the number of bias parameters of layer , so = ( + 1). After the summation operation defined by Eq. ( 2), each neuron in the hidden layer provides an output value ⃖ ⃗ ̂ ( ) computed from an activation function ( ) according to the following equation:

̂ ( ) = ( ) ( ) or ⃖ ⃗ ̂ ( ) = ( ) ⃖ ⃗ ( ) (4)
Many activation functions are available in literature and their choice depend mainly on the application of the ANN. In our case, it is very important that those activation functions are derivable in order to allow the computation of the derivative of the von Mises stress with regard to the plastic strain , the plastic strain rate . and the temperature . For our type of application, we have made the choice to test two among the mainly used ones:

• the Sigmoid sig( ) activation function defined by:

sig( ) = 1 1 + exp (-) , sig ′ ( ) = exp ( ) (1 + exp ( )) 2 (5)
• and the Hyperbolic tangent tanh( ) activation function defined by:

tanh( ) = exp ( ) -exp (-) exp ( ) + exp (-) , tanh ′ ( ) = 1 -tanh 2 ( ) (6)
The main guide for this choice is that the expression of their derivatives is simple, which leads to relatively compact expressions. Comparison and performance of those two activation functions will be presented in Section 3.

If there exists another hidden layer ( + 1) after the current layer, then, the output ⃖ ⃗ ̂ of layer is then used as the input ⃖ ⃗ of layer ( + 1).

Output layer

The output is computed from the values of the last hidden layer of the neural network, containing neurons, using the following equation:

= ∑ =1 ̂ ( ) + (7)
where is the bias associated to the output neuron and are the weight parameters between the last hidden layer and the output neuron . Using matrix formalism, one can rewrite this later as:

= ⃖⃖ ⃗ ⋅ ⃖ ⃗ ̂ ( ) + (8)
with

⃖⃖ ⃗ = [ 1 , 2 , … , ] .
As presented earlier, there is no activation function for the output neuron, so is directly the output of the neural network. The total number of training parameters for the output layer is + 1. For a neural network with two hidden layers having neurons on the first hidden layer and neurons on the second one, the total number of training parameters is = 4 + ( + 2) + 1.

Pre and postprocessing of data

Since the ANN are set up to treat values with a limited amplitude, it is necessary to pre-process the provided corresponding values of the plastic strain , the plastic strain rate . and the temperature in the range [0, 1] in a same manner as the one proposed by other authors [START_REF] Lin | Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel[END_REF][START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF]. Therefore, the input ⃖ ⃗ of the ANN is computed according to the constitutive flow law by:

• Since the plastic deformation rate in constitutive equations usually enhances the logarithm of the plastic strain rate, we preprocess the plastic strain rate by computing the natural logarithm of the ratio of the plastic strain rate . over the referential strain rate . 0 given by ln( . ∕ . 0 ). • Then, we normalize the variables in the range [0, 1] to avoid ill-conditioned system as presented by many other authors in the literature [START_REF] Lin | Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel[END_REF][START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF]. Therefore, the three components of the input ⃖ ⃗ are obtained from the plastic strain , the plastic strain rate . and the temperature using the following expressions:

⃖ ⃗ = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 = -[ ] [ ] -[ ] 2 = ln( . ∕ . 0 )-[ln( . ∕ . 0 )] [ln( . ∕ . 0 )] -[ln( . ∕ . 0 )] 3 = -[ ] [ ] -[ ] (9)
where [ ] and [ ] are the boundaries of the range of the corresponding field. During the training of the ANN, the von Mises stresses will also be scaled down within the range [0, 1], using the following expression:

= -[ ] [ ] -[ ] (10)
So, finally, the von Mises stress can be obtained from the output of the ANN using:

= [ ] -[ ] + [ ] (11)
The [ ] and [ ] values of plastic strain, plastic strain rate, temperature, stresses and referential strain rate . 0 used during the training phase should be recorded for later use during the implementation of the ANN in the Abaqus Explicit code. They are part of the final solution along with the weights ( ) and bias values ⃖ ⃗ ( ) of the hidden layers and the weights ⃖⃖ ⃗ and the bias of the output layer of the neural network that constitutes the training parameters of the neural network. The knowledge of these quantities after the learning phase allows then to extract the whole neural network from its formulation in Python language to a compact version, without the back propagation learning mechanism, for its implementation in FORTRAN in the Abaqus Explicit FEM code.

Loss function

In the context of optimization algorithms, the function used to evaluate the quality of a solution is called the objective function. In the application to neural networks, we try to minimize the error made by the network in the prediction of the solution. The evaluation of this error consists in measuring the difference between the predicted value computed by the NN and a reference value coming usually from experiments, or, in the present paper from an analytical equation, that we want to reach for a particular data set. There are several ways to define this error, among them, the best known is probably the average Root Mean Square Error (E RMS ) given by:

E RMS = √ √ √ √ 1 ∑ =1 ( -) 2 (12)
where is the total number of values of the training batch.

Derivatives computation

Implementation of the neural network within the radial return algorithm proposed by Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF] through a VUMAT subroutine of directly through a VUHARD one for Abaqus Explicit requires that the ANN returns not only the von Mises equivalent stress given by Eq. ( 11), but also the three derivatives ∕ , ∕ . and ∕ without having been trained to compute those derivatives. So that we do not have 4 outputs for the ANN but only one and the ability to compute those derivatives must be intrinsic. Therefore, we need to be able to compute those three derivatives using only the proposed ANN architecture. One straightforward, but not recommended, solution to this problem is to numerically compute the derivative of with respect to , . and using the following relation:

( ) = ( + ) -( ) (13)
where is a small increase ( = 10 -6 for example) applied to one of the 3 variables , . and . We need to compute 4 times a result from the ANN to compute the flow stress and approximate the three derivatives which is quite time-consuming. Another way to compute those derivatives is to compute the quantity

⃖ ⃗ ′ = [ ′ 1 , ′ 2 , ′ 3 ]
containing the 3 derivatives of the output defined by Eq. ( 8) of the ANN with respect to the input ⃖ ⃗ . This analytic computation depends on the number of hidden layers (hereafter we present the results for 1 and 2 hidden layers), and the type of activation functions used (the tanh and the sig functions). ⃖ ⃗ ′ = ∕ ⃖ ⃗ contains the derivatives of with respect to 1 , 2 and 3 respectively. Based on the chain-rule derivation, with 1 hidden layer and a tanh activation function, ⃖ ⃗ ′ takes the following form:

⃖ ⃗ ′ = (1) ⋅ ⃖⃖ ⃗ -⃖⃖ ⃗ • tanh 2 ⃖ ⃗ (1)
where ⃖ ⃗ (1) is given by Eq. ( 2) with = 1 and • the element-wise product, known as the Hadamard product, which is a binary operation that takes two matrices and of the same dimensions and produces matrix of the same dimension as operands, where each element = . The tanh operator applied on quantity ⃖ ⃗ (1) is just computed for each component of ⃖ ⃗ (1) , as presented in Fig. 1, since is not a real but only an array of reals stacked vertically. With 1 hidden layer and a sig activation function, we obtain:

⃖ ⃗ ′ = (1) ⋅ ⃖⃖ ⃗ • exp -⃖ ⃗ (1) 1 + exp -⃖ ⃗ (1) 2 (15)
When the number of hidden layers increases, the complexity of the derivative increases, so, for 2 hidden layers and a tanh activation function for both layers we obtain:

⃖ ⃗ ′ = (1) ⋅ (2) ⋅ ⃖⃖ ⃗ -⃖⃖ ⃗ • tanh 2 ⃖ ⃗ (2) • 1 -tanh 2 ⃖ ⃗ (1) (16)
Finally, for 2 hidden layers and a sig activation function for both layers we obtain:

⃖ ⃗ ′ = (1) ⋅ (2) ⋅ ⃖⃖ ⃗ • exp -⃖ ⃗ (2) 1 + exp -⃖ ⃗ (2) 2 • exp -⃖ ⃗ (1) 1 + exp -⃖ ⃗ (1) 2 (17)
Depending on the number of hidden layers and the type of activation functions used, from Eqs. ( 14) to [START_REF] Ghaboussi | New nested adaptive neural networks (NANN) for constitutive modeling[END_REF] and because of the pre and post-processing of the quantities , ,

. and defined in Section 2.1.4

by Eqs. one can finally obtain the derivatives of the flow stress with respect to , . and using the following expression:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∕ = ′ 1 [ ] -[ ] [ ] -[ ] ∕ . = ′ 2 . [ ] -[ ] [ . ] -[ . ] ∕ = ′ 3 [ ] -[ ] [ ] -[ ] (18)
.

It is important to note here, that whatever the method adopted to calculate the derivative of the output of the neural network with respect to the inputs (numerical method or formulation of the derivative of the network), the result obtained is an approximation of the derivative of the mathematical function represented by the network with respect to the inputs. Indeed, the neural network, by its construction, approximates a mathematical formulation, but, as shown by Nguyen et al. [START_REF] Nguyen-Thien | Approximation of functions and their derivatives: A neural network implementation with applications[END_REF], a neural network can also approximate the derivatives of the mathematical function it reproduces. Thus, the derivative of the neural network with respect to its inputs is correlated to the derivative of the mapped mathematical function with respect to its parameters.

Training of the ANN and performance evaluation

In order to evaluate the performance of the proposed approach, we decided to reproduce the behavior of the Johnson-Cook [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures[END_REF] flow law with an Artificial Neural Network because it is one of the most widely used flow law for the simulation of high strain rate deformation processes. It is implemented in numerous Finite Element codes such as Abaqus. Reproducing the behavior of a Johnson-Cook law by a neural network is of course not an aim of this work, but only required in order to verify numerically that the neural network is able to take into account a non-linear behavior and to have a way to measure exactly the prediction errors of the neural network. The general formulation ( , , ) is given by the following equation:

= + 1 + ln . . 0 1 - -0 -0 (19)
where . 0 is the reference strain rate, 0 and are the reference temperature and the melting temperature of the material respectively and , , , and are the five constitutive flow law parameters that we usually have to determine using for example an inverse identification procedure as the one proposed by Dalverny et al. [START_REF] Dalverny | Identification de lois constitutives et de lois de frottement adaptées aux grandes vitesses de sollicitation[END_REF] from Taylor impact tests.

Analytical expressions of the three derivatives of the Johnson-Cook flow stress with respect to , . and are given by the three following equations:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∕ = -1 1 + ln . . 0 1 - -0 -0 ∕ . = . + 1 - -0 -0 ∕ = - + -0 1 + ln . . 0 -0 -0 (20)
The usual approach to implement a new constitutive law in the Abaqus FEM code, as proposed by Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF], is to program in a VUHARD FORTRAN subroutine the evaluation of the flow stress defined by Eq. ( 19) and the derivatives of the flow stress given by Eq. (20).

Generation of the training and test data

A 42CrMo4 steel, with material parameters proposed by Sattouf et al. [START_REF] Sattouf | A methodology for the identification of constitutive and contact laws of metallic materials under High Strain Rates[END_REF] and reported in Table 1 and ∈ [START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF]500]. This later is not used during the training phase, but only during the validation presented in Section 3.3. Both data sets contains the values of the plastic strain , the plastic strain rate . , the temperature , the flow stress computed from Eq. ( 19). The second data set contains also the values of the three derivatives computed from Eq. [START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF]. It is important to remember that the neural network proposed must allow to replace the analytical formulation of the behavior law in order to be able to carry out a numerical simulation from experimental data resulting from thermomechanical tests. During these tests, we acquire the temperature, the strain, the strain rate and the stresses. But it is impossible to acquire the derivatives of the stresses with respect to the quantities , . , .

Therefore, in normal use, it is not possible to include the derivatives in the objective function for training the neural network.

Training of the artificial neural network

Training the Artificial Neural Network is finding the best set of values for all the training parameters ( ) , ⃖ ⃗ ( ) , ⃖⃖ ⃗ and defined in Section 2.1 in order to reduce the error of the ANN in computing the flow stress. The training set is used in this procedure and the ANN has been implemented using the Tensorflow Python library [START_REF] Abadi | TensorFlow: A system for large-scale machine learning[END_REF]. Training procedure is based on the use of the Adaptive Moment Estimation (ADAM) [START_REF] Kingma | Adam: A Method for Stochastic Optimization[END_REF] optimizer. We used the two activation functions presented earlier, the sig and the tanh functions, and one or two hidden layers with a variable number of neurons in the hidden layers. All models are named after their constitution, where 3-x-1-tanh refers to a one hidden layer ANN with a tanh activation function and neurons in the hidden layer and 3-x-y-1-sig refers to a two hidden layers ANN with a sig activation function in both layers and neurons in layer 1 and neurons in layer 2. All models have been trained for the same number of iterations (50 000 iterations). Training times for all models are more or less the same: around 1 hour on a Dell XPS 13 laptop.

Performance analysis of the artificial neural network

In order to illustrate the efficiency of the ANN, Fig. 2 and Table 2 reports some results obtained after training 6 different models (2 one hidden layer models and 4 two hidden layers models).

A lot more results have been obtained in this study, but only those 6 models are presented here to illustrate the tendencies. Fig. 2 shows an histogram of the average values of the E RMS defined by Eq. ( 12) evaluated during the last 5% of the training process giving an idea of the global convergence of the ADAM algorithm. In Table 2, is the number of internal parameters of the ANN. Values □ are the socalled Average Absolute Relative Error (E AAR ) given by the following expression:

□ = 1 ∑ =1 | | | | | □ -□ □ | | | | | ( 21
)
where □ is the analytical exact value obtained from Eqs. ( 19)- [START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF] and □ is the ANN predicted value of the same quantity computed by the neural network from Eqs. ( 11) and [START_REF] Lin | Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel[END_REF].

Reading the data in Table 2, the overall performance of the proposed neural networks is very good since the error on the evaluation of the flow stress is about 0.01% for the best performing network while it does not exceed 0.04% for all the tested networks. The evaluation of the derivatives is also very good since the error is about 0.2% for the best performing network while it remains below 2.0% for the ∕ term, 0.8% for the ∕ . term and 0.6% for the ∕ term. Without too much surprise, we notice that globally, the error on the evaluation of the flow stress is lower with a factor of 10 than the error on the evaluation of the derivatives. This is easily explained by the fact that the network has been trained to minimize the error on the flow stress only. Nevertheless, without ever having been optimized for computing the derivatives, we can see that the results obtained are also very good.

Fig. 3 shows the convergence of the 3-15-7-1-sig ANN model concerning the evaluation of the flow stress and the 3 derivatives with the number of iterations. From this later, one can see that the convergence of the ANN on the evaluation of is faster than convergence of the derivatives. The previously mentioned factor of 10 between the stress and the derivatives is clearly visible on this graph. If the error on the calculation of the constraint does not evolve after 10 000 iterations of the learning algorithm, the errors on the evaluation of the derivatives require a much more significant number of iterations. It is therefore necessary to continue training this type of model when the convergence criterion of the flow constraint has already been satisfied in order to allow the convergence of the derivatives. We can then be confronted with a problem of over-learning, it is therefore necessary to dimension the size of the neural network as accurately as possible with respect to the application.

Implementation of the neural network in Abaqus Explicit

In this section, we now present the numerical implementation of the proposed ANN constitutive flow law. Once the ANN network presented in Section 2 has been trained as presented in Section 3, it is time to use it for the flow stress and its three derivatives computing. The implementation is done by programming a VUHARD subroutine for the Abaqus Explicit finite element code similarly to the approach proposed by Jansen van Rensburg et al. [START_REF] Jansen Van Rensburg | Tutorial on state variable based plasticity: An Abaqus UHARD subroutine[END_REF]. This VUHARD subroutine is used inside of a Radial-Return algorithm, as illustrated in Fig. 4, to compute the flow stress and its derivative used in the expression of the two quantities ( ) and ′ ( ), thanks to the following equation [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF]:

= + . . + = √ 2 3 + 1 . + (22)
where is the consistency parameter used in the Radial-Return algorithm as defined by Simo et al. [START_REF] Simo | Computational Inelasticity[END_REF],

is the time increment, is the Taylor-Quinney coefficient defining the amount of plastic work converted into heat energy, is the specific heat coefficient and is the density of the material. So that, the ANN is used to compute the value of the flow stress and the three derivatives of the flow stress with respect to , . and involved in Eq. ( 22). This is illustrated in Fig. 4 where the yellow block in the center of the flowchart is where the ANN is used. Since this ANN is in the center of a CPU intensive loop involved for all integration points of the FEM model, this has to be optimized to reduce computing times as it will be presented further. More details on the Radial-Return algorithm can be found in Simo et al. [START_REF] Simo | Computational Inelasticity[END_REF] and more details concerning the implementation of this algorithm in Abaqus Explicit can be found in Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF], where the same approach was used, except the fact that in this paper the flow stress and its derivatives were computed analytically. Validation of the proposed implementation is done on several benchmark tests by comparing the results obtained using the ANN to native Johnson-Cook law (named Built-in hereafter) and the analytical VUHARD implementations proposed previously by Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF] (named Analytical hereafter).

Numerical implementation of the neural network

The numerical implementation of the neural network is done using a VUHARD subroutine for the Abaqus Explicit code. This is a straightforward approach to implement a new constitutive flow law in this FEM code by just implementing a FORTRAN subroutine to compute the flow stress of the material ( , . , ) according to Eq. ( 11) and its derivatives with respect to , . and defined by Eq. [START_REF] Lin | Application of neural networks to predict the elevated temperature flow behavior of a low alloy steel[END_REF]. In this approach, the main part of the Built-In constitutive law is used for time integration of the stress, for a given time increment, and the provided user subroutine is called to compute the hardening flow law.

A Python's program has been developed to extract the internal parameters of the trained neural network (the network architecture, the weights and bias values and ⃖ ⃗ of all layers, . . . ) and write automatically the FORTRAN subroutine. In the following paragraph, we detail the implementation of a 2 hidden layers neural network with a sig activation function, so that, one of the most complex model presented in this paper. of the stress and Eq. ( 17) for the derivatives. In order to optimize the numerical implementation a bit, and, as the computation of the flow stress and its derivatives share some common terms that will be stored during the computation for later use, the computation of Eqs. ( 5) and ( 17) is split into several sub-terms ⃖ ⃗ to ⃖ ⃗ . So, starting from , defining the values of ⃖ ⃗ , one can write:

= exp - ∑ (1)
- (1) ∈

[1, ], ∈ [1, 3] = 1 + ∈ [1, ] = exp - ∑ (2) ∕ -(2) ∈ [1, ], ∈ [1, ] (23)
where , and are three terms present in Eq. ( 17) corresponding to exp -⃖ ⃗ (1) , 1 + exp -⃖ ⃗ (1) and exp -⃖ ⃗ (2) respectively. Then for the computation of the derivatives, we combine the values , and to compute the whole equation ( 17) using:

= ∕(1 + ) 2 ∈ [1, ] = ∕ ∈ [1, ] = ∑ (2) ∈ [1, ], ∈ [1, ] (24)
From those definitions, one can then write the output of the neural network using the following expression:

= ∑ ∕(1 + ) + ∈ [1, ] (25)
And, the three derivatives ′ are obtained from:

′ = ∑ (1) ∈ [1, 3], ∈ [1, ] (26)
Finally, Eqs. ( 11) and ( 18) are used to compute the von Mises equivalent stress and the 3 derivatives ∕ , ∕ . and ∕ of the neural network from and ⃖ ⃗ ′ computed from Eqs. ( 25) and [START_REF] Huang | Constitutive models and microstructure evolution of Ti-6Al-4V alloy during the hot compressive process[END_REF]. As it is a straightforward implementation from Eq. ( 23) to [START_REF] Huang | Constitutive models and microstructure evolution of Ti-6Al-4V alloy during the hot compressive process[END_REF] the Python interface uses loops to explicitly write all the matrices products in a FORTRAN subroutine as illustrated in Fig. 5. This later only reports a small part of the whole FORTRAN code to illustrate how it is implemented. The reader interested in the details of this implementation can refer to the Software Heritage archive website [START_REF] Pantalé | Software Heritage archive[END_REF] giving access to the source code of this work.

The VUHARD subroutine is compiled using the GNU gfortran 9.3.0 and linked to the main Abaqus Explicit executable. All benchmarks tests have been solved using Abaqus Explicit 2021 on a Dell XPS 13 laptop running Ubuntu 20.04 64 bits with 16 GiB of Ram and one 4 core i7-10510U Intel Processor. All computations have been done using the double precision option of Abaqus, with parallel threads execution on two cores. In order to reduce the number of models presented hereafter, only the two last models presented in Table 2 are selected for the subsequent benchmark simulations.

Necking of a circular bar benchmark test

The necking of a circular bar test, already presented by Ponthot et al. [START_REF] Ponthot | Unified stress update algorithms for the numerical simulation of large deformation elasto-plastic and elasto-viscoplastic processes[END_REF], is useful to evaluate the performance of non-linear constitutive laws. An axisymmetric quarter model of the specimen, already presented in Ming et al. [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/Explicit finite element code[END_REF], is used, where dimensions of the specimen are reported in Fig. 6. We imposed a total displacement of 7 mm along the ⃖ ⃗ axis on the left side of the specimen while the radial displacement of the same edge is supposed to remain zero. On the opposite side, the axial displacement is restrained while the radial displacement is free. The mesh consists of 400 CAX4RT elements (4-node bilinear displacement and temperature, reduced integration with hourglass control element) with a refined zone of 200 elements on the right side on 1∕3 of the total length. The FEM model is a coupled temperature-displacement explicit model (this is a coupled thermal-stress analysis where the heat transfer and mechanical solutions are obtained simultaneously by an explicit coupling), and the total simulation time is set to = 0.01 s. Fig. 7 shows the von Mises stress contourplot of the deformed bar for two different models: the Built-In model (top side) and the ANN 3-15-7-1-sig model (bottom side). There is a very little difference between both models in terms of spatial distribution of stress and maximum value. The maximum stress is located into the center of the bar, so we have chosen to plot in Figs. 8 and9 the evolution of the equivalent plastic strain and the von Mises stress for the central element of the specimen (the red element in the bottom right corner in Fig. 6). As reported in Figs. 8 and9, the Built-In model, the Analytical model and both versions of the ANN model give almost the same results, except when the elongation is greater than 6 mm where the von Mises stress differs between the ANN models and the Built-in and Analytical ones as we will see further. This is also confirmed in Table 3 reporting a comparison of the four models for two values of the displacement (3.5 mm and 7 mm, named and respectively). From this later, we can see that the equivalent plastic strain , the von Mises stress and the temperature obtained with both ANN models are very close to the ones obtained by the Built-in and the Analytical models for displacement, while they differ a little for displacement. This allows us to validate the proposed approach. One interesting result concerns the values of the equivalent plastic strain and the temperatures reported in Table 3 for displacement. It can be seen that, for displacement, the values of the plastic strain reported in Table 3 are around 2.1 while the range of plastic strain used for training the model is [0, 1]. Furthermore, the maximum value of the temperature is around 587 • C while the training range is [START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF]500]. The proposed model is therefore able to extrapolate with a good accuracy some data out of the training range. It is obvious from Table 3 and Figs. 8 and9 that all models give very closed results when parameters are within the training range, while the ANN models differ a little when parameters are far out of the training range (here, with an elongation greater than 6 mm so that > 1.7). This shows that our ANN model is able, to a certain extent, and with the usual precautions insofar as neural networks are generally relatively faithful for interpolation but less so for extrapolation, to generalize a behavior out of the training range. If we want to reduce the gap between predicted and real values out of the training range, we have to enlarge the training ranges of the input variables if data is available.

Table 3 reports also the computing times and total number of increments computed after running the same model 10 times. One can see that computation time of the 3-7-4-1-sig model is equivalent to the one of the Built-in model and lower to the one of the Analytical model while there is an increase for the most complex ANN model. In the proposed approach, the ANN is used to compute the flow stress of the material within a Radial Return algorithm. It replaces the evaluation of the flow stress and its derivatives based on some analytical expressions. VUHARD implementation breaks the Built-in natural optimized algorithm used to compute the stresses by some FORTRAN subroutine call, data transfer, . . . leading to increase of CPU time (see comparison between Built-in and Analytical CPU times). This is why we cannot have a reduction of computational time with regard to the Built-in model. We therefore have to compare CPU performance of ANN models with the Analytical one. Since the Johnson-Cook analytical behavior law is quite simple, the most complex ANN model is slower than the analytical one. But, in case of more complex behavior law, such as some Modified Johnson-Cook laws proposed by Zhou et al. [START_REF] Zhou | Research on several constitutive models to predict the flow behaviour of GCr15 continuous casting bloom with heavy reduction[END_REF], this tendency will be reversed since the analytical evaluation of the derivatives needed for the radial return algorithm to work becomes quite CPU intensive. We can also conclude from the results presented in Table 3 that the 3-7-4-1-sig model is sufficient to obtain valuable results with comparable CPU times with regard to the Built-in constitutive law.

Taylor impact benchmark test

The performance of the proposed ANN subroutine will now be validated under high deformation rate with the simulation of the Taylor impact test [START_REF] Taylor | The testing of materials at high strain rates of loading[END_REF] where a cylindrical specimen is launched to impact a rigid target with a prescribed initial velocity = 287 m/s. The height of the cylinder is 32.4 mm and the radius is 3.2 mm as reported in Fig. 10. The axial displacement is restrained on the left side of the specimen while the radial displacement is free (to figure a perfect contact without friction of the projectile onto the target). A 3D quarter model of the Taylor cylindrical specimen is meshed with 4 455 C3D8T elements (8-node trilinear displacement and temperature). The total simulation time for the Taylor impact test is = 80 s. Fig. 11 shows the equivalent plastic strain contourplot of the deformed rod for two models: the Built-In model (upper side of the specimen) and the ANN-3-15-7-1-sig model (bottom side of the specimen). The distributions of the equivalent plastic strain are almost the same for both models. The maximum equivalent plastic strain is located into the center element of the model (the red element in Fig. 10) and the models give quite the same value as reported in Table 4 for , and the final dimensions of the specimen (final length) and (final diameter of the impacting face). Concerning the simulation times reported in Table 4, the same trends as those presented in Section 4.2 are noticeable in this test case. Again, the comparison of the numerical results validates the proposed approach and shows a very good level of correlation of the results.

Summary and conclusions

In this paper, an Artificial Neural Network based framework has been proposed to model the non-linear flow law ( , . , ) with its application to a 42CrMo4 steel and a constitutive behavior of type Johnson-Cook. The general architecture of the multilayer perceptron neural network was presented with a focus on the evaluation of the derivatives of the flow stress with respect to the plastic strain, the plastic strain rate and the temperature necessary to implement a VUHARD user routine in the finite element code Abaqus Explicit, without these quantities having been learned by the network according to the classical supervised training scheme. The evaluation of these derivatives for 1 or 2 hidden layers and 2 types of activation function has been presented in detail as well as the comparison of the accuracy of this evaluation with a reference solution based on the use of the Johnson-Cook flow law. The results obtained showed an excellent ability to evaluate the flow stress and a very good ability to evaluate the derivatives by the neural network. After numerical implementation of the neural network in the Abaqus code, the test cases used showed the good behavior of the proposed approach in the context of the numerical simulation of the necking of a circular bar and a Taylor impact test.

CRediT authorship contribution statement

Fig. 1 .

 1 Fig. 1. Multi-layer Artificial Neural Network architecture.

Fig. 2 .

 2 Fig. 2. Comparison of precision of various ANN.

Fig. 3 .

 3 Fig. 3. Convergence of the predicted values for the 3-15-7-1-sig ANN.

Fig. 4 .

 4 Fig. 4. Flowchart of the Radial-Return Algorithm to compute the final stresses.

Fig. 5 .

 5 Fig. 5. Partial VUHARD FORTRAN subroutine to compute the flow stress and its derivatives using the ANN for the 3-15-7-1-sig model.

Fig. 6 .

 6 Fig. 6. Numerical model for the necking of a circular bar.

Fig. 7 .

 7 Fig. 7. Von Mises stress contourplot for the necking of a circular bar for an elongation of 7 mm (top side is the Built-in flow law and bottom side is the ANN 3-15-7-1-sig flow law).

Fig. 8 .

 8 Fig. 8. Equivalent plastic strain vs. displacement for the necking of a circular bar.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Von Mises stress vs. displacement for the necking of a circular bar.

Fig. 11 .

 11 Fig. 11. Equivalent plastic strain contourplot for the Taylor impact test (upper side is the Built-in model and bottom side is the ANN 3-15-7-1-sig model).

  Pantalé: Conception and design of study, Acquisition of data, Analysis and/or interpretation of data, Writing -original draft, Writing -review & editing. Pierre Tize Mha: Acquisition of data, Writing -review & editing. Amèvi Tongne: Analysis and/or interpretation of data, Writing -review & editing.

  was selected as the material used in this study. To train and validate the ANN, we have used a Python program to generate, thanks to Eqs. (19)-[START_REF] Lu | Artificial neural network prediction to the hot compressive deformation behavior of Al-Cu-Mg-Ag heat-resistant aluminum alloy[END_REF] two distinct data sets:

	• The first data set, the training one, contains 2 520 datapoints
	defined by 70 equidistant values for rates . ∈ [1, 10, 50, 500, 5 000, 50 000] and 6 temperatures ∈ [0, 1], 6 plastic strain ∈
	[20, 100, 200, 300, 400, 500].

• The second data set, the testing set, contains 5 000 datapoints randomly generated within the ranges ∈ [0, 1],

. ∈ [1, 50 000]

Table 1

 1 Material properties of the 42CrMo4 steel[START_REF] Sattouf | Caractérisation en dynamique rapide du comportement de matériaux utilisés en aéronautique[END_REF].

	(GPa)		(MPa)	(MPa)
	206.9	0.29	806	614	0.0089	0.168	1.1
	̇ 0	0
	( -1 )	• C	• C	∕ 3	∕ • C	10 -6 ∕ • C
	1	20	1540	7830	460	12.3	0.9

Table 2

 2 Global performance analysis of the ANN during the training phase.

	Model		E RMS		( ∕ )	( ∕ ̇ )	( ∕ )
			×10 -7	%	%	%	%
	3-7-4-1-tanh	65	6.32	0.038	1.977	0.792	0.556
	3-15-1-tanh	78	3.46	0.039	1.506	0.269	0.371
	3-15-7-1-tanh	180	1.71	0.030	0.519	0.380	0.408
	3-15-1-sig	78	2.00	0.030	0.686	0.521	0.675
	3-7-4-1-sig	65	1.27	0.024	0.670	0.415	0.499
	3-15-7-1-sig	180	0.68	0.011	0.247	0.199	0.256

Table 3

 3 Comparison of results for the necking of a circular bar benchmark for a displacement of 3.5 mm (mid) and 7 mm (end).

	Model	Incr.	Time
			(s)	(MPa)	( • C)	(MPa)	( • C)
	3-7-4-1-sig	191 768 29.98 0.51 1293.81 182.01 2.16 1064.43 587.78
	3-15-7-1-sig 194 432 38.54 0.51 1293.97 181.52 2.03 1060.50 587.29
	Analytical	200 145 35.50 0.51 1293.59 182.47 2.16 1045.75 585.85
	Built-In	199 474 28.71 0.51 1293.76 180.36 2.14 1043.19 587.66

Table 4

 4 Comparison of results for the 3D Taylor impact test.

	Model	Incr.	Time
			(s)	(mm)	(mm)	( • C)
	3-7-4-1-sig	6 344	63.08	26.52	11.18	584.28	1.83
	3-15-7-1-sig	6 239	90.32	26.52	11.17	584.47	1.83
	Analytical	6 419	71.71	26.53	11.19	585.64	1.84
	Built-In	6 570	52.82	26.54	11.21	588.66	1.84

Acknowledgment

All authors approved the final version of the manuscript.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


François Ducobu
email: francois.ducobu@umons.ac.be

Anthonin Demarbaix

Olivier Pantalé

Finit Finite element modelling of the T e element modelling of the Ta ay ylor impact t lor impact test in 3D with the Coupled est in 3D with the Coupled E Eulerian-Lagr ulerian-Lagrangian method angian method

Keywords: Coupled Eulerian-Lagrangian, Finite Element, Lagrangian, Material Constitutive Model, Taylor Impact Test 1. Intr

A Abstr bstract act. . When modelling a cutting operation, the constitutive model of the machined material is one of the key parameters to obtain accurate and realistic results. Up to now, the Johnson-Cook model is still the most used, even if an increasing number of models, such as the Hyperbolic TANgent (TANH) model, were introduced last years to overcome its limitations and come closer to the actual material behaviour. Experimental tests on dedicated equipment are usually required to identify the parameters of the constitutive models. This paper introduces the Coupled Eulerian-Lagrangian (CEL) formalism to model in 3D the Taylor impact test, one of the common tests to perform that parameters identification. Indeed, one identification way involves modelling the test to determine the constitutive model parameters by comparing the experimental and the numerical samples geometries. The developed CEL model is validated against a Lagrangian reference model for a steel alloy and the Johnson-Cook constitutive model. The main goal of using the CEL method is to get rid of the elements distortion due to the high strains and strain rates during the test. Mesh dependence of the results is highlighted and a recommendation is provided on the mesh to adopt for future work. The CEL model of the 3D Taylor impact test is then extended to the use of the TANH model. The results are finally compared with that of the Johnson-Cook constitutive model. K Ke eyw ywor ords ds.

Introduction oduction

Constitutive models are one of the key aspects when modelling a cutting operation [START_REF] Arrazola | Recent advances in modelling of metal machining processes[END_REF]. They indeed directly influence the accuracy of the results: magnitude and temporal evolution of cutting forces, chip morphology and geometry, etc., justifying significant research activities in this field [START_REF] Melkote | Advances in material and friction data for modelling of metal machining[END_REF]. The severe conditions induced by the cutting process on the workpiece material makes it difficult to identify its behaviour. Empirical constitutive models are currently widely adopted. The Johnson-Cook (JC) model [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF] is the most used of them, even if it is known to have some limitations [START_REF] Melkote | Advances in material and friction data for modelling of metal machining[END_REF].

To overcome them and try to come closer to the actual material behaviour, many models have been introduced the last decade [START_REF] Ducobu | On the selection of an empirical material constitutive model for the finite element modeling of Ti6Al4V orthogonal cutting, including the segmented chip formation[END_REF]. This usually has the consequence of increasing the number of parameters to identify. This identification is also a hard task that requires dedicated equipment to reach the high levels of strains, strain rates and temperatures of the cutting process [START_REF] Melkote | Advances in material and friction data for modelling of metal machining[END_REF]. The most used in this context are the Split Hopkinson Pressure Bars [START_REF] Kolsky | An Investigation of the Mechanical Properties of Materials at very High Rates of Loading[END_REF] and the Taylor impact test [START_REF] Taylor | The testing of materials at high rates of loading[END_REF]. This later consists in launching a cylindrical specimen at a prescribed speed to impact a rigid anvil.

An identification method consists in modelling the test to determine the constitutive model parameters by comparing the experimental and the numerical samples geometries after deformation (i.e. "inverse identification" procedure).

Numerical modelling of the test is usually performed with the finite element method and the Lagrangian formulation [START_REF] Nistor | Identification of a dynamic viscoplastic flow law using a combined Levenberg-Marquardt and Monte-Carlo algorithm[END_REF]. However, elements deformation and/or distortion typically generated during the computation decreases the accuracy of the results and may significantly impact the values of the parameters. To overcome this limitation, this paper adopts the Coupled Eulerian-Lagrangian (CEL) formalism, already used for modelling high strain, strain rate and temperature problems such as a cutting operation [START_REF] Ducobu | Application of the Coupled Eulerian-Lagrangian (CEL) method to the modeling of orthogonal cutting[END_REF][START_REF] Ducobu | Finite element modelling of 3D orthogonal cutting experimental tests with the Coupled Eulerian-Lagrangian (CEL) formulation[END_REF], instead of the Lagrangian one for the development of a 3D model. The suitability of the formalism is first assessed with the JC model against a Lagrangian reference, before applying it to an improved version of the JC constitutive model.

Finit Finite element models e element models

Two 3D finite element models are developed with Abaqus/Explicit in the frame of this article: a Lagrangian model, acting as a reference, and the introduced CEL model. They both include a quarter of the actual configuration (Figure 1), with the corresponding symmetry boundary conditions, to reduce the computation time. The cylindrical sample has a length of 32.4 mm and a radius of 3.2 mm. Its prescribed initial speed is 200 m/s and its initial temperature is 20°C.

A perfect contact between the specimen and the rigid plate is assumed without friction. The simulation time is 80 μs.

This configuration is the same as Ming and Pantalé's [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF]. The mesh (Figure 1 (b)) needs to include the volume in which the material will be during the simulation. This implies that the dimensions of the mesh are larger for the CEL model than for the Lagrangian model; in the Lagrangian model, only the specimen needs to be meshed. Consequently, the mesh of the CEL model is made of a cylinder and a cone.

The cylinder is slightly larger than the undeformed part of the specimen and the cone is slightly larger than the final deformed part of the specimen. This geometry allows reducing the size of the mesh and consequently the computation time. The material of the specimen is the 42CrMo4-FP ferritic-pearlitic steel with an inelastic heat fraction coefficient of 0.9.

Its behaviour is first described by the JC constitutive model:

It requires the identification of five parameters 𝐴, 𝐵, 𝑛, 𝐶 and 𝑚. 𝜀0 is the reference strain rate. 𝑇room is the room or reference temperature and 𝑇melt is the melting temperature. The JC constitutive model is introduced in Lagrangian and CEL finite element models with the VUMAT subroutine developed by Ming and Pantalé [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF]. Material properties and JC parameters of 42CrMo4-FP steel are included in Table 1.

T Table 1. Johnson-Cook par able 1. Johnson-Cook paramet ameters and mat ers and material pr erial properties of 42CrMo4-FP st operties of 42CrMo4-FP steel [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF] eel [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF] 3. V

Validation of the CEL model alidation of the CEL model

The introduced CEL model is first validated against the Lagrangian model acting as a reference. A perfect match of the results is not expected, but they should be close enough to assess the suitability of the CEL formulation. A mesh sensitivity study is first conducted to determine the mesh configuration that should be adopted to model the 3D Taylor impact test. Four mesh densities are used: coarse (elements size of 0.8 mm), medium (elements size of 0.4 mm), fine (elements size of 0.2 mm) and very fine (elements size of 0.15 mm). In the Lagrangian model, C3D8RT elements are used, i.e. 3D linear 8-node with reduced integration and temperature degree of freedom elements. In the CEL model, Eulerian 3D linear 8-node with reduced integration and temperature degree of freedom elements, EC3D8RT, are used.

Computations (coupled temperature displacement analyses) were run on Abaqus/Explicit v.6.14-2 and Windows 10 64 bits with 8 GB of Ram. Four cores of an Intel i7-5700 HQ 3.48 GHz processor were used for the computation of each simulation with double precision. FORTRAN VUMAT subroutines were compiled with Intel Fortran Compiler 16.0.4.

Figure 2 and Figure 3 show the results of the eight models to allow comparison of the two formulations and carry out the mesh sensitivity study. As expected, high plastic strains are located at the bottom of the specimen, in the region of impact, and the highest values are in the centre. The tendency of these results is in accordance with the experimental results of a symmetric Taylor impact test for the same material but different impact speed and sample dimensions [START_REF] Nistor | Identification of a dynamic viscoplastic flow law using a combined Levenberg-Marquardt and Monte-Carlo algorithm[END_REF]. From Figure 2, it is clear that the mesh density influences the Lagrangian results: the deformed radius on the impact face increases with the mesh density. This influence is more difficult to notice on the CEL results in Figure 3. The influence of the mesh on the equivalent plastic strain values is also lower for the CEL formulation than for the Lagrangian formulation for which a strong (and unrealistic) localisation is observed. This means that the more the Lagrangian mesh is refined, the more the equivalent plastic strain increases due to distortion of the mesh with no convergence of the results. On the contrary, the equivalent plastic strain values converge when the CEL mesh is refined. Localisation of the deformation and mesh dependence of equivalent plastic strain for the Lagrangian formalism together with a comparison for the CEL formalism are shown in detail in Figure 4. Some kind of defects (looking like small indentations) are observed on the results of coarse and medium CEL meshes in the undeformed region of the specimen. They are linked to the thickness of the initial void region that is too small by comparison to the elements size; they are not spotted anymore for fine and very fine meshes. Due to the interpolation used by Abaqus to determine the contours of the area filled with material, the upper corner of the specimen is rounded. The value of the radius is directly linked to the size of the elements, explaining why it decreases when mesh density increases. Both types of artificial defects are far from the relevant region and have no influence on the results; they will be ignored in the results analysis. 2 gives the results of both models for the four different mesh sizes:

• 𝐿f and 𝑅f are the final length and radius of the specimen, respectively. Measurements of the specimen deformed geometry are performed in a lateral face of the specimen.

• 𝜀̅ max is the maximal equivalent plastic strain, 𝜎max is the maximal equivalent von Mises stress and 𝑇max is the maximal temperature.

• 𝑁elements is the number of elements of the model, 𝑁increments is the number of increments to complete the computation and 𝑡ĈPU is the normalised (by the Lagrangian model with a fine mesh) computation time of the model. differences can be noted. Regarding the mesh sensitivity study, for both models, the fine mesh should be used as it provides a good accuracy in terms of specimen geometry at the lowest computation cost. It must however be noted that, while the final lengths are very close, the final radius is larger for the Lagragian model. This is explained by the large deformation of the elements at the end of the specimen (in the impact region), decreasing the accuracy of the results. The absence of elements deformations in the CEL formulation avoids this problem and therefore provides more accurate and reliable results.

When looking at the mesh influence on the maximal equivalent plastic strain, 𝜀̅ max, the maximal equivalent von Mises stress, 𝜎max, and the maximal temperature, 𝑇max, values, convergence seems to be achieved for all but the equivalent plastic strain of the Lagrangian model. Since plastic strains are not converging, convergence of stresses and temperatures of the Lagrangian model were not expected. Convergence of stresses is linked to the shape of the flow stress, rather flat for the material conditions in that area. This implies that a large variation in plastic strain results in a small variation in stress (and so an apparent convergence). Since temperatures are computed from plastic strains and stresses, compression of element has a counterbalancing effect that can explain stabilisation of temperature values.

Regarding the CPU computation time, although the number of increments, 𝑁increments, is close for both models, the larger number of elements, 𝑁elements, and the increase in the CPU cost of one increment lead to a significant increase for the CEL model. As expected due to elements deformation with the Lagrangian formulation, time increment value decreases during computation for the Lagrangian model (from 15.8•10 -9 s at the beginning of the computation to 5.4•10 -9 s at the end of the computation for fine mesh), while it is constant in CEL (7.6•10 -9 s for fine mesh). To set the magnitude of the computation time, 221 s are needed for the Lagrangian model with the fine mesh.

Figur where 𝜎JC is computed with equation ( 1), 𝑝 and 𝑞 are additional parameters, 𝜀0 sets the strain corresponding to the peak stress and 𝑇rec is the onset temperature for strain softening. Parameters of TANH model for 42CrMo4-FP ferritic-pearlitic steel are provided in Table 3. This constitutive model has also been implemented in the models via a VUMAT subroutine developed by Ming and Pantalé [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF].

T Table 3. T able 3. TANH par ANH paramet ameters of 42CrMo4-FP st ers of 42CrMo4-FP steel [START_REF] Hor | Modelling, identification and application of phenomenological constitutive laws over a large strain rate and temperature range[END_REF] eel [START_REF] Hor | Modelling, identification and application of phenomenological constitutive laws over a large strain rate and temperature range[END_REF] Comparison of the results between JC and TANH material constitutive models (Figure 5 and Table 4), for both formulations, leads to values that are very close to each other. It can also be highlighted that results for JC and TANH are even closer for the CEL formulation, which is in accordance with the convergence of all the indicators for this formulation. The conditions of strain, strain rate and temperature in which the material is, do not trigger the modifications introduced by the TANH model. This is in accordance with previous results obtained by Ming and Pantalé [START_REF] Ming | An efficient and robust VUMAT implementation of elastoplastic constitutive laws in Abaqus/ Explicit finite element code[END_REF] and confirms that the CEL model can be used for 3D modelling of the Taylor impact test.

T Table 4. R able 4. Results with JC and T esults with JC and TANH constituti ANH constitutiv ve models f e models for Lagr or Lagrangian and CEL f angian and CEL formulations with a fine mesh (JC ormulations with a fine mesh (JC r results ar esults are the same as in T e the same as in Table 3, the able 3, they ar y are included her e included here f e for an easier comparison) or an easier comparison)

Figur Figure 5. E e 5. Equi quiv valent plastic str alent plastic strain r ain results with fine mesh f esults with fine mesh for: or: Both Johnson-Cook and Hyperbolic TANgent material constitutive models have been adopted via VUMAT subroutines.

Results showed that the CEL model delivers results close to the Lagrangian reference. Performances of the CEL formulation are better when high strains are generated thanks to the absence of elements deformation. Indeed, no plastic strain localisation, nor increase of specimen radius with mesh density were generated. In addition, the mesh convergence study showed that both geometrical and material related indicators were converging in the CEL model, contrary to the Lagrangian model. Replacing the Johnson-Cook constitutive model by the TANH model confirmed these observations. All these results show that the CEL formulation should be considered in the future as a reliable and accurate alternative to the Lagrangian formulation for inverse identification of the parameters of material constitutive models parameters from dynamic impact tests such as the Taylor impact test.

Figur Figure 1 .

 1 Figur Figure 1. Initial geometry of (a) Lagr e 1. Initial geometry of (a) Lagrangian model and (b) CEL model with their fine mesh angian model and (b) CEL model with their fine mesh Special care must be taken for the CEL model due to the coupling between the Eulerian and Lagrangian formalisms.

Figure 1 (

 1 b) shows both the initial specimen geometry in light grey and the initial void, where the specimen can deform during the simulation, in white.

Figur Figure 2 . E e 2 .

 22 Figur Figure 2. E e 2. Equi quiv valent plastic str alent plastic strain r ain results f esults for the 4 Lagr or the 4 Lagrangian models angian models (a) coarse mesh, (b) medium mesh, (c) fine mesh and (d) v (a) coarse mesh, (b) medium mesh, (c) fine mesh and (d) very fine mesh ery fine mesh Table2gives the results of both models for the four different mesh sizes:

Figur Figure 3 . E e 3 .

 33 Figur Figure 3. E e 3. Equi quiv valent plastic str alent plastic strain r ain results f esults for the 4 CEL models or the 4 CEL models (a) coarse mesh, (b) medium mesh, (c) fine mesh and (d) v (a) coarse mesh, (b) medium mesh, (c) fine mesh and (d) very fine mesh ery fine mesh They allow to validate the CEL model by comparison with the Lagrangian results: all the results are close, even if some

Figure 4 .

 4 Figur Figure 4. Detail of the mesh at the def e 4. Detail of the mesh at the deformed end of the sample, equi ormed end of the sample, equiv valent plastic str alent plastic strain: ain: Lagr Lagrangian (a) coarse mesh, (b) medium mesh, (c) fine mesh and CEL (d) coarse mesh, (e) medium mesh, (f) fine angian (a) coarse mesh, (b) medium mesh, (c) fine mesh and CEL (d) coarse mesh, (e) medium mesh, (f) fine mesh mesh TTable 2. R able 2. Results with JC constituti esults with JC constitutiv ve model (con e model (conv ver erged mesh in bold) ged mesh in bold)

  Lagrangian with JC, (b) Lagr angian with JC, (b) Lagrangian with T angian with TANH, (c) CEL with JC and (d) CEL with T ANH, (c) CEL with JC and (d) CEL with TANH ANH 5. Conclusions 5. ConclusionsThe CEL formulation has been introduced in this paper to model the 3D Taylor impact test with Abaqus/Explicit.


Keywords: path planning, environment modelling, geometry, topology, semantics, ontologies, Industry 4.0

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Path planning control using high abstraction level environment model and industrial taskoriented knowledge

Florent

  Léoty, Philippe Fillatreau, Bernard Archimède To cite this version: Florent Léoty, Philippe Fillatreau, Bernard Archimède. Path planning control using high abstraction level environment model and industrial taskoriented knowledge. 2021 IEEE International Workshop of Electronics, Control, Measurement, Signals and their application to Mechatronics (ECMSM), Jun 2021, Liberec, Czech Republic. pp.1-5, 10.1109/ECMSM51310.2021.9468850. hal-03467797

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Coarse motion planning Fine motion planning

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Hélène Welemane
email: helene.welemane@enit.fr

Sharan Raj

Rangasamy Mahendren

Olivier Dalverny

Amévi Tongne

HOMOGENIZATION BASED ANALYSIS OF THE INFLUENCE OF MICROCRACKING ON THERMAL CONDUCTION AND THERMOELASTICITY

Microcracks are known to affect the behaviour of the materials and several modelling studies are devoted to finding their effective elasticity. This paper aims to extend such investigation to thermal properties, concerning both the steady-state conduction and the thermoelasticity. Accounting simultaneously for the anisotropy induced by microcracks orientation and for their unilateral effect (different behaviour whether they are open or closed) remains a difficult task. In this work, 3D homogenization-based approaches are developed to derive closed-form estimations of the effective thermal properties of a microcracked media with arbitrarily oriented microcracks, either open or closed.

INTRODUCTION

Effects of microcracks on the elastic properties have been extensively studied through experiments and modelling approaches. In the latter case, the homogenization (up-scaling) technique appears as an effective tool to provide the overall properties from the microstructural features of the Representative Volume Element (RVE) [START_REF] Nemat-Nasser | Micromechanics: overall properties of heterogeneous materials[END_REF]. Especially for microcracks, main difficulties arise from: (i) the anisotropy induced by the oriented nature of defects, (ii) their ability to be open or closed (according to tension or compression stresses) and to influence differently the overall response of the material. Regarding the thermal steady-state conduction and thermoelastic properties of microcracked media, very few investigations of these effects have been done in the literature, even on the experimental point of view. Some micromacro modelling works have been proposed to derive effective properties of a given RVE but they consider only the open state of microcracks [START_REF] Sevostianov | On the effective properties of polycrystals with intergranular cracks[END_REF]. This work aims at exploring homogenization-based approaches to derive anisotropic thermal conductivity and thermoelasticity accounting for unilateral effect.

MICROMECHANICAL FRAMEWORK

Assuming length scale separation, let us consider a 3D RVE composed of an isotropic matrix weakened by N number of randomly distributed families of parallel microcracks. Such media exhibits a matrix-inclusion topology in which each phase (matrix, microcracks) is homogeneous, allowing the use averaging techniques. Two conditions can be imposed on the outer boundary of the RVE, i.e. uniform macroscopic temperature gradient G (for the thermal conduction problem) and uniform macroscopic strain E with constant temperature difference ∆T over the RVE (for the thermoelastic problem). Under equilibrium condition, the macroscopic quantities (G, E) correspond to the average value of their respective microscopic quantity (g, ε) over the RVE, i.e. G = g , E = ε . The microscopic and macroscopic quantities can be linked linearly as g = A • G and ε = A : E, where A and A are the second-order gradient and fourth-order strain localization tensors, respectively. Regarding the steady-state heat conduction problem, the microscopic heat flux q is given by Fourier's law:

q = -λ • g (1)
where λ is the symmetric second-order thermal conductivity tensor. On the other hand, thermoelasticity is classically described by the following linear constitutive equation:

σ = C : ε -κ ∆T ( 2
)
where σ is the microscopic stress tensor, C is the fourth-order stiffness tensor and κ = C : α is the second-order thermoelasticity tensor with α being the thermal expansion tensor.

Based on the boundary conditions, the overall heat flux Q = q and stress Σ = σ of the RVE can be expressed as:

Q = -λ hom • G and Σ = C hom : E -κ hom ∆T (3)
where λ hom , C hom , and κ hom are the effective tensors related to the thermal conductivity, stiffness and thermoelasticity of the microcracked media, respectively. Taking all simplifications into account, the effective thermal properties λ hom and κ hom are given as:

λ hom = λ m + N i=1 f c,i (λ c,i -λ m ) • A c,i and
κ hom = κ m + N i=1 f c,i κ c,i -κ m : A c,i (4)
where λ m (resp. λ c,i ) and κ m (resp. κ c,i ) are the thermal conductivity and thermoelasticity tensors of the matrix (resp. of the i th family of parallel cracks) and f c,i is the cracks volume fraction for the i th family. Estimations of localization tensors A and A can be obtained using the works of Eshelby on the single-inhomogeneity problem [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF]. In line with the recent study of [START_REF] Mahendren | Thermal conduction properties of microcracked media: Accounting for the unilateral effect[END_REF], each family of microcracks is modelled as a flat oblate ellipsoid with unit normal n i , crack density d i and aspect ratio ω i → 0. We assume both the matrix and cracks are isotropic and their properties can be given by λ r = λ r I, κ r = κ r I and C r = 3k r J + 2µ r K with r = {m; c, i} (where λ r and κ r are scalars while k r and µ r are the bulk and shear moduli). The unilateral effect is taken into account via the λ c,i and κ c,i . For open cracks, it is assumed adiabatic (for thermal conduction) and stress-free (for thermoelasticity) conditions on the microcracks lips, so λ c,i = 0, C c,i = O (k c,i = 0, µ c,i = 0) and κ c,i = 0. Inspired by the works of Deudé et al. [START_REF] Deudé | Propriétés élastiques non linéaires d'un milieu mésofissuré[END_REF], the closed cracks (frictionless contact) are considered as a fictitious isotropic material such that λ c,i = 0, κ c,i = 0 and C c,i = 3k m J (k c,i = k m , µ c,i = 0), which accounts for some heat (for conduction) and stress (for thermoelasticity) transfer continuity across the closed microcracks lips.

RESULTS

From these assumptions, closed-form expressions of the whole thermal properties (Eq (4)) are derived for dilute and Mori-Tanaka (MT) schemes and Ponte Castañeda-Willis (PCW) bound. For a single family of open microcracks, λ hom is function of d, n, λ m and κ hom is function of d, n, κ m and matrix Poisson ratio ν m . Also the material shows a damageinduced anisotropy, irrespective of the estimation. To be precise, the effective thermal properties are transversely isotropic around the axis n of the crack. As an illustration, Fig. 1 presents the generalized scalar thermal conductivity λ(v) and thermoelasticity κ(v) in any direction of unit vector v defined by: λ

(v) = v • λ hom • v and κ(v) = v • κ hom • v respectively.
While both properties are mostly degraded along the direction n normal to the crack surface, the cracks influence is stronger on thermoelastic properties for a given cracks density. We should also note the differences for both properties according to the estimations, related to the account (MT and PCW) or not (dilute) of interactions between microcracks and, in the former case, to the considered spatial distribution of microcracks (ellipsoidal for MT and spherical for PCW). On the other hand, it is demonstrated that closed cracks do not contribute to the degradation of the thermal conductivity or thermoelasticity. This result is irrespective of the estimation (dilute, MT, PCW), fictitious cracks properties or space direction, i.e. effective conductivity and thermoelasticity in any direction v recover their respective values for the virgin material: λ(v) = λ m and κ(v) = κ m , ∀ v. While elastic properties are only fully recovered in the direction n normal to closed microcracks [START_REF] Welemane | Some remarks on the damage unilateral effect modelling for microcracked materials[END_REF], we observe here a total deactivation of microcracking influence on thermal properties.

CONCLUSIONS

Using Eshelby approach, we have derived in this work several homogenization-based expressions for effective thermal properties of a microcracked media, both regarding steady-state conduction and thermoelasticity. Special attention has been paid to the unilateral effect and its consequences on the overall behaviour. This study provides some relevant information to further develop a fully coupled thermo-mechanical model with evolving damage.

Figure 1 :

 1 Figure 1: Generalized thermal conductivity λ(v) and coefficient κ(v) normalized by their initial values for a material weakened by a single family of parallel open microcracks of unit normal n (cracks density d = 0.1, matrix Poisson ratio νm = 0.15).