Olivier Goudet
email: olivier.goudet@univ-angers.fr

Mohamed Salim

Amri Sakhri

Adrien Goëffon
email: adrien.goeffon@univ-angers.fr

Frédéric Saubion
email: frederic.saubion@univ-angers.fr

Emergence of new local search algorithms with neuro-evolution

Keywords: Neuro-evolution, Local search, Black-box optimization, NK landscapes

This paper explores a novel approach aimed at overcoming existing challenges in the realm of local search algorithms. The main objective is to better manage information within these algorithms, while retaining simplicity and generality in their core components. Our goal is to equip a neural network with the same information as the basic local search and, after a training phase, use the neural network as the fundamental move component within a straightforward local search process. To assess the efficiency of this approach, we develop an experimental setup centered around NK landscape problems, offering the flexibility to adjust problem size and ruggedness. This approach offers a promising avenue for the emergence of new local search algorithms and the improvement of their problem-solving capabilities for black-box problems.

Introduction

Local search (LS) algorithms are commonly used to heuristically solve discrete optimization problems [START_REF] Hoos | Stochastic local search: Foundations and Applications[END_REF]. LS algorithms are usually composed of several components: a search space, a neighborhood relation, an evaluation function, and a selection strategy. The optimization problem instance to be solved can be fully defined by its set of feasible solutions -the decision space-and an objective function that must be optimized. A classic and direct use of local search, when applicable, is to consider the decision space as the search space, the objective function as the evaluation function, and a natural neighborhood relation, defined from an elementary transformation function (move) such as bitflip for pseudo-Boolean problems, or induced by specific operators for permutation problems [START_REF] Schiavinotto | A review of metrics on permutations for search landscape analysis[END_REF].

Starting from an initial random solution, various components collaborate to drive the search towards optimal solutions. The effectiveness of this search process depends on the complexity of the problem, including factors such as deception and other structural characteristics [START_REF] Basseur | Climbing combinatorial fitness landscapes[END_REF][START_REF] Jones | Fitness distance correlation as a measure of problem difficulty for genetic algorithms[END_REF][START_REF] Vuculescu | Human search in a fitness landscape: How to assess the difficulty of a search problem[END_REF][START_REF] Whitley | Mk landscapes, NK landscapes, MAX-kSAT: A proof that the only challenging problems are deceptive[END_REF]. The strategy to advance in the search involves selecting neighboring solutions based on their evaluations, using a wide variety of criteria. These criteria can range from simple ones, such as choosing neighbors with better or the best evaluations, to more intricate approaches involving stochastic methods. The strategy is often derived from metaheuristic frameworks based on local search such as tabu search [START_REF] Glover | Tabu search-part i[END_REF] or iterated local search [START_REF] Lourenço | Iterated local search[END_REF]. The goal is to effectively leverage the available local and partial knowledge of the landscape to identify the most promising search paths that lead to optimal solutions. Fitness landscape analysis [START_REF] Malan | A survey of advances in landscape analysis for optimisation[END_REF] provides the optimization and evolutionary computation community with theoretical and practical tools to examine search landscapes. It allows for the assessment of problem characteristics and the evaluation of the performance of search algorithms. In the context of combinatorial fitness landscapes, these are represented as graphs defined by a discrete search space and a neighborhood relation. A fundamental challenge lies in developing a search algorithm capable of navigating a fitness landscape to reach the highest possible fitness value.

In general, achieving optimal solutions through a straightforward adaptive approach is quite challenging. This difficulty arises from the complex interplay among different parts of the solution, which can lead to locally optimal solutions. These local optima cannot be escaped by intensification or exploitative move strategies. Consequently, the optimization algorithm becomes stuck in a suboptimal state. The primary concern in such optimization processes is to strike a balance between exploiting promising search areas through greedy search methods and diversifying search trajectories by temporarily exploring less promising solutions.

To overcome this challenge, researchers have developed many metaheuristics [23] and even hyperheuristic schemes [START_REF] Özcan | A comprehensive analysis of hyper-heuristics[END_REF] to mix different strategies. These approaches typically incorporate parameters that allow for precise adjustment of the trade-off between exploration and exploitation. Still, they might not perform efficiently in a black-box context, as many heuristics leverage the unique properties of the given problem to solve it efficiently. Machine learning techniques have been widely used to improve combinatorial optimization solving [START_REF] Talbi | Machine learning into metaheuristics: A survey and taxonomy[END_REF] and to address the optimal configuration of solving algorithms. An approach to algorithm design known as "programming by optimization" (PbO) was introduced by Hoos [START_REF] Hoos | Programming by optimization[END_REF]. This paradigm encourages algorithm developers to adopt and leverage extensive design possibilities that encompass a wide range of algorithmic techniques, to optimize performance for specific categories or groups of problem instances.

In various works, different machine learning approaches have been used either to consider offline adjustment, selection of parameters, or online control of the search process using reinforcement learning (RL) techniques (see [START_REF] Mamaghan | Machine learning at the service of meta-heuristics for solving combinatorial optimization problems: A state-of-the-art[END_REF] for a recent survey). The use of neural networks (NNs) in solving combinatorial optimization problems has been studied for decades [START_REF] Trafalis | Neural networks for combinatorial optimization[END_REF], starting with the early work of Hopfield [START_REF] Hopfield | Neural networks and physical systems with emergent collective computational abilities[END_REF]. Recent applications of Graph Neural Networks in the context of combinatorial optimization have been proposed to reach optimal solutions or to assist the solving algorithm in proving the optimality of a given solution (see [START_REF] Cappart | Combinatorial optimization and reasoning with graph neural networks[END_REF] for a recent survey).

In the traveling salesman problem, a GNN can be used to predict the regret associated with adding each edge to the solution to improve the computation of the fitness function of the LS algorithm [START_REF] Hudson | Graph neural network guided local search for the traveling salesperson problem[END_REF]. Note that NNs are classically used in surrogate model-based optimization [START_REF] Willmes | Comparing neural networks and kriging for fitness approximation in evolutionary optimization[END_REF]. In [START_REF] Santana | Neural networks for local search and crossover in vehicle routing: A possible overkill?[END_REF], the authors introduce a GNN into a hybrid genetic search process to solve the vehicle routing problem. The GNN is used to predict the efficiency of search operators and to select them optimally in the solving process. A deep Q-learning approach has recently been proposed to manage the different stages of an LS-based metaheuristic to solve routing and job-shop problems [START_REF] Falkner | Learning to control local search for combinatorial optimization[END_REF]. High-level solving policies can often be managed by reinforcement learning in LS processes [START_REF] Veerapen | Using local search with adaptive operator selection to solve the progressive party problem[END_REF]. In this paper, our purpose is different, as we focus on building simple search heuristics for black-box problems, rather than scheduling specific operators or parameters. In particular, we assume that the learning process cannot be based on the immediate rewards that are used in RL. This motivates our choice of neuro-evolution.

Objective of the paper This study explores the potential for emerging search strategies to overcome existing challenges. The objective is to change how information is leveraged while retaining simple and generic search components. Considering a basic hill climber algorithm to achieve a baseline search process for solving black-box binary problem instances, our aim is to benefit from machine learning techniques to get new local search heuristics that will be built from basic search information instead of choosing a priori a predefined move heuristic (e.g., always select the best neighbor). Hence, our goal is to provide a NN with the same information as a basic LS and, after training, to use the NN as the basic move component of a simple LS process. To evaluate the efficiency of our approach, we define an experimental setup based on NK landscape problems, which allows us to describe a fitness landscape whose problem size and ruggedness (determining the number of local minima) can be adjusted as parameters.

2 General framework

Pseudo-Boolean Optimization Problems and Local Search

Let us consider a finite set X ⊆ {0, 1} N of solutions to a specific problem instance. These solutions are tuples of values that must satisfy certain constraints, which may or may not be explicitly provided. We evaluate the quality of these solutions using a pseudo-Boolean objective function f obj : X → R. Therefore, a problem instance can be fully characterized by the pair (X , f obj ). In terms of solving this problem, X is referred to as the search space.

When solving an optimization problem instance with an LS algorithm, the objective is to identify a solution x ∈ X that optimizes the value of f obj (x). Since we are primarily concerned with maximization problems, let us note that any minimization problem can be reformulated as a maximization problem. In this context, an optimal solution, denoted as x * ∈ X , must satisfy the condition that for every x ∈ X , f obj (x) ≤ f obj (x * ). While exhaustive search methods, or branch and bound algorithms, guarantee the computation of an optimal solution, this is not the case with LS algorithms. However, computing optimal solutions within a reasonable time is often infeasible, leading to the use of local search algorithms within a limited budget of evaluations of f obj to approximate nearoptimal solutions.

LS algorithms operate within a structured search space, thanks to a fixedsized neighborhood function denoted as N : X → 2 X . This function assigns a set of neighboring solutions N (x) ⊆ X for each solution x ∈ X . To maintain a fundamentally generic approach to LS, we assume that N is defined using basic flip functions, f lip i : X → X , where i ∈ 1, N , and f lip i (x) is equal to x except for the i th element, which is changed from 0 to 1 or vice versa. In this case,

N (x) = {f lip i (x) | i ∈ 1, N }.
Starting from an initial solution, often selected randomly and denoted as x 0 , LS constructs a path through the search space based on neighborhood relationships. This path is typically represented as a sequence of solutions bounded by a limit (horizon) H, denoted as (x 0 , x 1 , . . . , x H ), where for each i ∈ 0, H -1 , x i+1 ∈ N (x i ). Let us denote P = X * the set of all paths (i.e., the set of all possible sequences built on X ).

In addition to the neighborhood function, this sequence of solutions is determined by a strategy, often involving the use of a fitness function f . For example, hill climbing algorithms select the next solution on the path based on a simple criterion: ∀t ∈ 0, H -1 , f (x t ) < f (x t+1 ), where f (x t ) represents the fitness evaluation of solution x t . The process of choosing the next solution on the search path is referred to as a move. We denote x t+1 = x t ⊕ f lip i the move that corresponds to x t+1 = f lip i (x t ).

Local Search as an Episodic Tasks Process

According to previous remarks, an LS process can be modeled by a sequence of actions performed on states. To be as exhaustive as possible, these states must be general enough to describe the current solution, as well as the path already explored and future possible moves. Therefore, we may define a set of states as S = P × X × 2 X . Of course, states of S cannot be managed by a practical local search algorithm, at least from a memory size point of view. We introduce the notion of observation of a state as a function o : S → Ω where Ω is a domain that corresponds to an abstraction of the real states of search to gather only useful information for the considered local search strategy.

Note that here we are in the context of episodic (discrete states) tasks with a terminal state (the end of the search fixed for instance by a maximal number of moves H). Following a reinforcement learning-based description, a local search process can be encoded by a policy π : Ω → A where A is a set of actions. Here we consider deterministic policies, i.e. for each o ∈ Ω, there exists one and only one action a ∈ A, such that π(o) = a. Note that the policy can be parameterized by a parameter vector θ. The set of all policies is Π = {π θ | θ ∈ Θ} where Θ is the parameter space.

In our context, we consider actions that are bitflips f lip i defined in Section 2.1. Hence, A ⊇ {f lip i | i ∈ 1, N }. Note that of course, other actions can be introduced to fit specific strategies.

We consider a transition function δ : X × A → X such that δ(x, a) = x ⊕ a. The transition function turns an action that is defined on the set of observations to a move in the set of solutions. Note that the transition is used to update the current state and compute the next state of the LS process. An LS run can be fully characterized by the search trajectory that it has produced from an initial starting solution.

Given an instance I = (X , f obj ), an initial solution x 0 ∈ X , a policy π θ , and a horizon H, a trajectory T (x 0 , π θ , H, I) is a sequence (x 0 , a 0 , x 1 , a 1 , . . . , x H-1 , a H-1 , x H ) such that (x 0 , . . . , x H ) ∈ P is an LS path (the multiset {x 0 , . . . , x H } will be denoted

P (T (x 0 , π θ , H, I))), ∀i ∈ 0, H -1 , a i = π θ (o(x i )) and ∀i ∈ 1, H , x i = δ(x i-1 , a i-1
). Note that the trajectory is defined with regard to solutions belonging to X , while the policy operates on the space of observations obtained from these solutions. Compared to classic reinforcement schemes, where a reward can be assigned after each action, in our context, the reward will be computed globally for a given trajectory. Note that if we considered only the best-improvement strategy, then the reward could be assigned after each move to assess that the best move has been selected. Unfortunately, such a strategy will only be suitable for simple smooth unimodal problems. Hence we translate this reward as a function

R(x 0 , π θ , H, I) = max x∈P (T (x0,π θ ,H,I)) f obj (x). Example 1.
In order to illustrate our framework, let us consider a basic hill climber (HC) that uses a simple best-improve strategy using objective function f obj as fitness function. In the following, we only consider LS processes that do not involve memory that records past decisions. Hence, a state of a HC is fully described by a current solution x ∈ X and its neighborhood N (x). An observation will abstract the state as the variation of the fitness function for each possible flip of the values of x, o(x) = (f obj (f lip 1 (x))-f obj (x), . . . , f obj (f lip N (x))-f obj (x)) ∈ R N (Ω = R N ). In a best-improve HC process, the search stops when a local optimum is reached and no improving move can be performed. We consider

A = {f lip i | i ∈ 1, N } ∪ {Id},
where Id is the identity function on X . Then, we define the policy π HC (o(x)) = argmax a∈A (f obj (δ(x, a)) -f obj (x)). Note that when a local optimum is reached, the identity action will always be selected for the remainder of the process.

Our objective is to maximize the maximum score encountered by the agent during its trajectory, and not the sum of local fitness improvements obtained during its trajectory. We are therefore not in the case of learning a Markov decision process. This is why classical reinforcement learning algorithms such as Q-learning or policy gradient are not applicable in this context. This justifies our choice of neuro-evolution where the policy parameters will be abstracted by a neural network that will be used to select the suitable action. The parameters of this neural network will be searched by means of an evolutionary algorithm according to a learning process defined below.

Policy Learning for a Set of Instances

In this paper, we focus on NK landscapes as pseudo-boolean optimization problems. The NK landscape model was introduced to describe binary fitness land-scapes [START_REF] Kauffman | The NK model of rugged fitness landscapes and its application to maturation of the immune response[END_REF]. The characteristics of these landscapes are determined by two key parameters: N , which represents the dimension (number of variables), and K (where K < N ), which indicates the average number of dependencies per variable and, in turn, influences the ruggedness of the fitness landscape. An NK problem instance is an optimization problem represented by an NK function. We use random NK functions to create optimization problem instances with adjustable search landscape characteristics. This adjustment will be achieved by varying the parameters (N, K), thus generating diverse search landscapes. Hence, we consider N K(N, K) as a distribution of instances generated by a random NK function generator.

In order to achieve our policy learning process, we must assess the performance of a policy as

F (π θ , N K(N, K), H) = E I∼N K(N,K),x0∼X [R(x 0 , π θ , H, I)]
where x 0 ∼ X stands for a uniform selection of an element in X . However, since this expectancy cannot be practically evaluated, we rely on an empirical estimator computed as an average of the score obtained by the policy π θ for a finite number q of instances I 1 , . . . , I q sampled from the distribution N K(N, K) and a finite number r of restarts x

(1) 0 , . . . , x (r) 0 drawn uniformly in X :

F (π θ , N K(N, K), H) = 1 qr q i=1 r j=1 R(x (j) 0 , π θ , H, I i ). (1)
Figure 1 highlights our general learning methodology and the connections between the LS process at the instance solving level and the policy learning task that will be achieved by a neural network presented in the next section.

(x) = (f (f lip 1 (x)) - f (x), . . . , f (f lip N (x)) -f (x)) ∈ R N .
The set of possible actions available to the different strategies always remains A = {f lip i | i ∈ 1, N }.

Neural network local search policy

We introduce a deterministic LS policy π θ : R N → A, called Neuro-LS, which uses a neural network g θ , parametrized by a vector of real numbers θ. This neural network takes as input an observation vector o ∈ R N and gives as output a vector g θ (o) of size N whose component g θ (o) i corresponds to a preference score associated to each observation o i . Then, the action a i corresponding to the highest score g θ (o) i is selected.

Permutation equivariant neural network

A desirable property of this neural network policy is to be permutation equivariant with respect to the input vector of observations, which is a property generally entailed by a local search algorithm, in order to make its behavior consistent for solving any type of instance. Formally, an LS algorithm is said to be invariant to permutations in the observations if for any permutation σ on 1, N , we have a σ(i) = π θ (o σ(1) , o σ(2) , . . . , o σ(N ) ). As an example, the basic hill climber HC defined with Example 1 in Section 2.2 has this property. In order to obtain this property for Neuro-LS, g θ must be function from R N to R N , such that for any permutation σ we have (g θ (o) σ(1) , . . . , g θ (o) σ(N ) ) = g θ (o σ(1) , . . . , o σ(N ) ). Such type of permutation equivariant neural network can be obtained by using the deep sets architecture [START_REF] Zaheer | Deep sets[END_REF].

Each layer of the proposed network combines the treatment of each observation o i associated to each of the N variables with an additional operation that performs an average of the features across the different variables. This Naveraging operation is independent by permutation of the inputs and allows to transmit some general contextual information between the N features vector.

For a vector of observation o = (o 1 , o 2 , . . . , o N ) given as input, a permutation equivariant network g θ with P hidden layers is defined as

g θ (o) = ϕ θ P • ϕ θ P -1 • • • • • ϕ θ0 (o),
where each ϕ θj is a permutation invariant function from R N ×lj to R N ×lj+1 . The l j values correspond to the layer sizes. Note that for the first layer l 0 = 1 and for the last layer l P +1 = 1. This network g θ is shown in Figure 2.

Each layer operation ϕ θj with l j input features and l j+1 output features includes a weight matrix W j ∈ R lj ×lj+1 that treats each variable information independently, a variable-mixing weight matrix Γ j ∈ R lj ×lj+1 and a bias vector β j ∈ R l . Note that the size of these matrices W j and Γ j does not depend on the size N of the observation vector, allowing the Neuro-LS strategy to adapt to pseudo-Boolean optimization problems of different sizes.

When taking as input a feature matrix v = (v 1 , . . . , v N ) of size N × l j , the matrix of weights W j processes each feature vector v i associated with each variable i independently. Then, the weight matrix Γ j processes an average vector ρ(v) computed across the different N feature vectors of size l j and given by ρ

(v) = 1 N N i=1 v i . Given v ∈ R N ×lj ,
the output of the layer ϕ θj is a matrix in R N ×lj+1 , which is the concatenation of N output vectors of size l j+1 , ϕ θj (v) = (ϕ θj (v) 1 , . . . , ϕ θj (v) N ), where for

1 ≤ i ≤ N , ϕ θj (v) i = η(β j + x i W j + ρ(v)Γ j ), (2)
with η : R lj → R lj+1 the element-wise nonlinear activation map defined by η(z) := (tanh(z 1 ), . . . , tanh(z lj )). We denote θ := {(W j , Γ j , β j )} j∈ 0,P , the set of all weight matrices and bias vectors of the neural network.

Neuro-evolution with CMA-ES

The neural network policy π θ is characterized by a set of parameters denoted as θ.

The optimization goal is to maximize the estimated score F (π θ , N K(N, K), H). This poses a stochastic black-box optimization problem within the real-valued search space R |θ| . To tackle this problem, we propose to use the covariance matrix adaptation evolution strategy (CMA-ES) [START_REF] Hansen | Completely derandomized self-adaptation in evolution strategies[END_REF], which stands out as one of the most powerful evolutionary algorithms for addressing such black box optimization problems [START_REF] Müller | Challenges in high-dimensional reinforcement learning with evolution strategies[END_REF]. The principle of CMA-ES is to iteratively test new generations of real-valued parameter vectors θ (individuals). Each new generation of parameter vectors is stochastically sampled according to a multivariate normal distribution. The mean and covariance matrix of this distribution are incrementally updated, so as to maximize the likelihood of previously successful candidate solutions. Thanks to the use of a stepwise-adapted covariance matrix, the algorithm is able to quickly detect correlations between parameters, which is an important advantage when optimizing the (many) parameters of the neural network policy. Another advantage of CMA-ES is that it relies on a ranking mechanism of the estimated scores F given by the Equation ( 1) for the different individuals of the population, rather than on their absolute values, making the algorithm more robust to stochastic noise related to the incertitude on the estimation of the performance score F with a finite number of trajectories.

Basic local search policies

We compare the neural network policy above with three basic local search policies which have been extensively studied in the literature [START_REF] Ochoa | First-improvement vs. best-improvement local optima networks of NK landscapes[END_REF][START_REF] Basseur | Hill-climbing strategies on various landscapes: an empirical comparison[END_REF]. All these strategies take as input the same vector of observation as the neural network policy and return an action a ∈ A. These three policies are two hill climbers as well as a (1, λ)-evolution strategy [START_REF] Beyer | The theory of evolution strategies[END_REF] used as a local search [START_REF] Tari | On the use of (1, λ)-evolution strategy as efficient local search mechanism for discrete optimization: a behavioral analysis[END_REF]. They are made deterministic using a pseudo-random number generator h whose seed is determined with a hash function from the current state x encountered by the LS.

Best improvement hill climber [+jump] (BHC + ). This strategy always selects the action a i = f lip i ∈ A in such a way that f (a i (x)) -f (x) is maximized, provided there is at least one action a i that strictly improves the score. If ∀i, f (a i (x)) -f (x) ≤ 0, then this strategy performs a random jump by choosing a random action a ∈ A using the pseudo-random number generator h(x).

First improvement hill climber [+jump] (FHC + ). This strategy iterates through all actions in A in random order and selects the first action a i leading to a strictly positive score improvement, i.e. such that f (a i (x)) -f (x) > 0. Similar to the BHC + strategy, if ∀i, f (a i (x)) -f (x) ≤ 0, it performs a random jump.

(1, λ)-evolution strategy ((1, λ)-ES). This strategy randomly evaluates λ actions in A and chooses the one that yields the best score, even if it results in a deteriorating move. λ is a method hyperparameter that will be calibrated to maximize the estimated score F for each type of NK landscape instance as detailed in the next section.

Computational experiments

The aim of this section is to answer two questions experimentally. The first concerns the performance of Neuro-LS compared to the baseline LS strategies presented in the last subsection for NK landscape problems of different size and ruggedness. The second is to study the emergent strategies discovered by Neuro-LS at the end of its evolutionary process. First, we discuss the experimental setting. Then, we follow the classical steps of a machine learning workflow: subsection 4.2 describes the Neuro-LS training process; it will allow us to select different emerging strategies for each type of NK landscape on validation sets, which we will then compare with the different baseline LS strategies on test sets (subsection 4.3). Finally, an in-depth analysis of the best emerging strategies discovered by Neuro-LS will be performed in subsection 4.4.

Experimental settings

In these experiments, we consider independent instances of NK-landscape problems. 12 different scenarios with N ∈ {32, 64, 128} and K ∈ {1, 2, 4, 8} are considered. For each scenario, three different sets of instances are sampled independently from the N K(N, K) distribution described in Section 2.3: a training set, a validation set and a test set. For the resolution of each instance, given a random starting point x 0 ∈ X , each LS algorithm performs a trajectory of size H = 2 × N (iterations) and returns the best solution found during this trajectory. The experiments were performed on a computer equipped with a 12th generation Intel® CoreTM i7-1265U processor and 14.8 GB of RAM.

Neuro-LS is implemented in Python 3.7 with Pytorch 1.4 library. 1 For all experiments with different values N and K, we use the same architecture of the neural network composed of two hidden layers of size 10 and 5, with a total of |θ| = 162 parameters to calibrate. To optimise the weights of the neural network, we used the CMA-ES algorithm of the pycma library [START_REF] Hansen | CMA-ES/pycma on Github[END_REF]. The multivariate normal distribution of CMA-ES is initialized with mean parameter µ (randomly sampled according to a unit normal distribution) and initial standard deviation σ init = 0.2.

Neuro-LS training phase

For each NK-landscape configuration, we run 10 different training processes of Neuro-LS with CMA-ES, with a time limit of two hours, to optimize the empirical score F defined by equation (1) computed as an average of the best fitness scores obtained for 50 trajectories (r = 5 independent random restarts for each of the q = 10 training instances). First, we find that the validation curve closely follows the training curve in average, even if it is slightly below. This reassuring consistency suggests that our model successfully generalizes the strategy learned in training instances.

Furthermore, upon comparing the validation curves of Neuro-LS and BHC + , we observe that the Neuro-LS curve progresses over generations, and eventually surpasses BHC + when evaluated on the same set of validation instances. This finding highlights that, once trained, the Neuro-LS method is able to find solutions more efficiently compared to the baseline BHC + local search for these types of instances.

In Figure 3, the minimum and maximum scores on the validation set obtained during the 10 independent neuro-evolution runs are also indicated with blue dashed lines. The figure illustrates significant variability in the results, highlighting the diversity in the performance of the emerging strategies. Nevertheless, this variability poses no issue in our context, as only the strategy with the best results on the validation set will be selected for the test phase presented in the next subsection.

Test phase

In this phase, we performed a series of evaluations to assess whether the best Neuro-LS strategies, selected based on the validation set for each configuration of NK landscape, continue to perform well in new test instances that are independently sampled from the same N K(N, K) distribution.

Table 1 summarizes the average score obtained by Neuro-LS and the three other competing methods, namely the BHC + , FHC + and (1, λ)-ES algorithms, presented in section 3.2. The strategy (1, λ)-ES has one hyperparameter λ that we calibrated in the range 1, N on training instances for each (N, K) configuration.

In order to perform a fair comparison between the different strategies, we computed an average estimated score F on the same 100 test instances. For each instance we use the same starting point to compute the trajectory produced by each LS strategy. 2In Table Table 1 shows that Neuro-LS always obtains better results for all the configurations of NK landscape, but the difference in score is only really significant when K = 4 and K = 8 for all values of N (except for N = 128 and K = 4). It means that our strategy becomes more effective than other methods when the landscape is more rugged.

This score improvement compared to the other baseline methods, obtained with the same budget of H = 2 × N iterations performed on each instance, can be attributed to a more efficient exploration of the search space as K increases (as seen in the next subsection).

Study of Neuro-LS emerging strategies

The objective here is to analyse in detail the best Neuro-LS strategies that have emerged with neuro-evolution and to understand their decision-making processes for the different types of landscape studied (smooth or rugged).

We have observed two main patterns of emerging strategies:

-For smooth landscapes, when K = 1 or K = 2, Neuro-LS learns to perform almost all the time a best improvement move, which explains why for these instances, it obtains almost the same score as the BHC + strategy (see Table 1). -For rugged landscapes, when K = 4 and K = 8, the emerging strategy is much more interesting. Figure 4 A rank of 1 on this plot indicates that Neuro-LS has chosen a best improvement move, while a rank of 64 indicates a worst deteriorating move (note that the y-axis is inverted, because it is a maximization problem). We observe on this plot that the emerging Neuro-LS strategy has three different successive operating modes:

1. Median hill climbing behavior. When the number of actions associated with a positive improvement of the score, N + a , is relatively large (∼ above N/10), Neuro-LS does not always choose the best improvement move, but instead a move with a rank approximately equal to N + a /2. This provides a good compromise between improving the score and avoiding being trapped too quickly in a local optimum. 2. Best improvement hill climbing behavior. When the number of actions corresponding to a positive delta fitness is low (∼ below N/10), Neuro-LS often chooses the best move (with rank 1) to quickly converge toward the closest local optimum. 3. Jump with worst move. When there is no more improving move, Neuro-LS does not stagnate, but instead directly chooses to perform the worst possible move (with rank 64). Even if this movement considerably deteriorates the current fitness score, it actually maximizes its long-term chances of escaping the current local optimum and continually exploring new areas of the search space. Indeed, we observe on this plot that Neuro-LS continuously improves its score with this strategy for this instance. Note that after choosing the worst possible move, it does not choose the best possible move, otherwise it would return to the same local optimum.

Conclusion

Our study explores the emergence of new local search algorithms with neuroevolution. Results on NK landscapes show that different neural network policies are learned, each adapted to the resolution of a particular landscape distribution type (smooth or rugged). Our algorithm is competitive with basic deterministic local search procedures for all the NK landscape types considered in this work. Particularly for rugged landscapes, it can achieve significantly better results with an original emerging strategy, using a worst-case improvement move, which proves very effective in the long run for escaping local optima. This study outlines avenues for future research on the automatic discovery of more advanced strategies using as input a richer set of observations to make its decision. The proposed framework could also be applied to study the emergence of strategies adapted to other types of combinatorial optimization problems.

Fig. 1 .

 1 Fig. 1. Global View of the Process

Fig. 2 .

 2 Fig. 2. Neural network policy π θ : Ω → A using a deep sets network architecture. Given a vector of observation o ∈ R N , the neural network outputs a vector g θ (o) ∈ R N . Then, the action a = argmax i∈ 1,N g θ (o)i is returned by π θ .

Figure 3

 3 displays the results of 10 independent neuro-evolution training processes for the NK landscape instances with N = 32 and K = 8. At each generation, the 10 learning instances are regenerated to avoid over-fitting, then CMA-ES samples a population of 19 individuals (19 vectors of weights θ ∈ R 162 of the neural network), and the best Neuro-LS strategy of the population on the training set is evaluated on the 10 instances of the validation set. The evolution of the average score of Neuro-LS on the training and validation sets are indicated with orange and blue lines in Figure 3. The red line is a reference score. It corresponds to the average score F obtained by the BHC + local search strategy (see Section 3.2) on the same validation set.

Fig. 3 .

 3 Fig. 3. Evolution of the average score obtained by Neuro-LS on the training and validation sets over the generations of CMA-ES.

  displays a representative example of the trajectory performed by Neuro-LS when N = 64 and K = 8. This figure shows two graphs based on data collected during the resolution of this instance. The graph on the top of this Figure shows the evolution of the fitness reached by Neuro-LS over the H = 2 × N = 128 iterations. The graph below shows at each iteration the number of available actions corresponding to an improvement of the score (in blue), and the rank of the action selected by Neuro-LS, measured in term of fitness improvement (in red).

Fig. 4 .

 4 Fig. 4. Fitness evolution curve and strategy used by Neuro-LS for the resolution of a instance with significantly rugged NK landscape (N = 64 and K = 8).

Table 1 .

 1 1, the best average result obtained among the mentioned methods for each configuration (N, K) of test instances is highlighted in bold. Results underlined indicate significant better results in average compared to all the other strategies (p-value below 0.001), measured with a Student t-test without assuming equal variance. 3 Average results on test instances obtained by different local search strategies for different NK landscape configurations.

	Instances			Methods
	N	K	BHC +	FHC +	(1, λ)-ES	Neuro-LS
	32	1	0.694	0.688	0.695	0.699
	32	2	0.713	0.709	0.717	0.721
	32	4	0.717	0.721	0.713	0.735
	32	8	0.702	0.712	0.707	0.732
	64	1	0.700	0.693	0.696	0.702
	64	2	0.712	0.709	0.712	0.715
	64	4	0.721	0.721	0.714	0.734
	64	8	0.710	0.719	0.707	0.737
	128	1	0.698	0.691	0.696	0.701
	128	2	0.712	0.711	0.710	0.713
	128	4	0.724	0.723	0.717	0.728
	128	8	0.711	0.719	0.705	0.730

The program source code, benchmark instances and result files are available at the url https://github.com/Salim-AMRI/NK_Landscape_Project.git.

For this evaluation test, we only perform one restart per instance, to avoid any dependency between the different executions that might take place on the same instance. It allows to obtain a distribution of 100 independently and identically distributed scores for each strategy and for each NK configuration.

The normality condition required for this test was first confirmed using a Shapiro statistical test on the empirical distributions of 100 iid scores obtained by each strategy.

Acknowledgment

This work was granted access to the HPC resources of IDRIS (Grant No. AD010611887R1) from GENCI. The authors would like to thank the Pays de la Loire region for its financiel support for the Deep Meta project (Etoiles Montantes en Pays de la Loire). The authors also acknowledge ANR -FRANCE (French National Research Agency) for its financial support of the COMBO project (PRC -AAPG 2023 -Axe E.2 -CE23). We are grateful to the reviewers for their comments.


F Dal

Cero Coelho

Q H Vu

D Halm

Y Nadot

Fatigue life modelling under variable amplitude multiaxial loading: comparison between fatigue criterion and incremental modelling

This study deals with fatigue of a 1045 carbon steel subjected to a cyclic tension-torsion spectrum combining inphase and out-of-phase loadings and including overload, representative of an automotive chassis loading type.

The experimental lifetimes are compared to the results of two different approaches. The first one combines a multiaxial fatigue criterion able to describe out of phase loading and a non-linear damage rule. The second one is based on an incremental mesoscale plasticity/damage model. It is shown that the criterion involving a non-linear damage rule is able to describe the experimental result for full spectrum (with overload) but the identification of the non-linearity is a function of spectrum type. The incremental approach gives better results for both spectra (with and without overloads) and does not need specific parameter identification.

Introduction

In the sector of automotive engineering, the car-to-ground connecting components undergo multiaxial cyclic loading. Usual loading corresponds to « smooth » driving on a wellmaintained road and is in the range of High Cycle Fatigue (10 6 -10 8 cycles), as well as turns or driving on cobblestones, which are however more demanding. Events in the range of Low Cycle Fatigue (1-10 4 cycles) may happen, for example impacts of potholes on wheels. The complexity of automotive fatigue loadings is taken into account by combining different types of multiaxial loadings, representative of in service stresses. In order to forecast lifetime of industrial components, reliable tools capturing the effects of multiaxiality and damage accumulation are needed. This complex topic has been addressed by researchers over the past 50 years with different types of modelling strategies, starting from very simple linear analysis up to complex nonlinear modelling, including different scales or methodologies. A short summary of major approaches is presented in this introduction, which is of course not exhaustive as many different modellings are available to address the problem, in terms of both fatigue criteria (approaches by critical planes [START_REF] Van | Sur la résistance à la fatigue des métaux[END_REF][START_REF] Findley | A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending[END_REF][START_REF] Mcdiarmid | A general criterion for high cycle multiaxial fatigue failure[END_REF][START_REF] Robert | Contribution l' tude de la fatigue multiaxiale sous sollicitations p riodiques ou al atoires[END_REF], by integrals, [START_REF] Papadopoulos | A new criterion of fatigue strength for out-of-phase bending and torsion of hard metals[END_REF][START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF][START_REF] Zenner | On the fatigue limit of ductile metals under complex multiaxial loading[END_REF], by stress invariants [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF][START_REF] Li | A computerized procedure for long-life fatigue assessment under complex multiaxial loading[END_REF][START_REF] Mamiya | Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue[END_REF][START_REF] Sines | Behaviour of metals under complex static and alternating stresses[END_REF], by energetic considerations [START_REF] Ellyin | A general fatigue theory and its application to out-ofphase cyclic loading[END_REF][START_REF] Palin-Luc | An energy based criterion for high cycle multiaxial fatigue[END_REF]) and damage accumulation (linear [START_REF] Miner | Cumulative damage in fatigue[END_REF], nonlinear [START_REF] Chaboche | Une loi diff rentielle d'endommagement de fatigue avec cumulation non linéaire[END_REF][START_REF] Chaboche | Extension and application of a nonlinear fatigue damage accumulation rule for variable amplitude loading programs[END_REF], by damage curve [START_REF] Manson | Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage[END_REF])

One of the simplest approach to solve the question of fatigue life under variable amplitude and complex multiaxial loading is to define an equivalent stress (criterion) to describe the multiaxial stresses and to perform linear accumulation damage. Most of the fatigue criteria are addressing the question of multiaxial stresses under proportional loadings with variable accuracy, especially when loadings include pure biaxiality and different load ratios. This general simple approach is not able to capture out of phase and frequency shift effects as well as nonlinear damage accumulation.

As a second step, the description of the loading complexity can be improved by integrating out of phase effect or shift frequency. Several criteria are able to deal with non-proportional loadings and already compared to experiments. The cost to be paid for this improvement leads, most of the time, to additional complexity for the criterion calculation and needs additional fatigue data. An accurate evaluation of in service fatigue life of a real component needs such level of complexity.

A third step is to improve the damage accumulation rule by taking into account nonlinear effects that are already very well-known from the experimental viewpoint but strongly material dependent. For that purpose, a nonlinear damage accumulation law is necessary and most of the time this is closely related to the material and the spectrum. It seems that no general simple approaches can deal accurately with nonlinear damage accumulation without being strongly correlated to the spectrum type. This third step of improvement corresponds to a relatively advanced method for the point of view of the design engineer and is able to solve most of the fatigue design issues but necessitates a good knowledge of the material's fatigue behavior under complex spectrum.

Another totally different approach is to link each cycle to a given level of damage. Most of the time incremental modellings are based on Thermodynamics of Irreversible Process and the load history is implicitly included in the damage internal variables so that no specific parameter for loading history is needed to describe nonlinear accumulation damage.

Advanced approaches are using scale transition in the high cycle fatigue range with local plasticity and damage at the local scale in order to describe fatigue mechanisms in the HCF range (macroscopic elasticity, plasticity and damage at the local scale). This type of approach is not so popular in the fatigue literature, probably due to a higher complexity of the identification procedure and higher computational cost for a real complex component, but should be further developed because of the direct capability of the method to describe complex loading.

While, as indicated above, one may find in the literature more or less complex approaches to predicting the in service lifetime of components subjected to high cycle fatigue, publications generally do not compare the different theories and their performance on a single material, whose behaviour has been clearly identified, and on multi-axial loading cases representative of in service use. The work presented here aims to fill this gap. It deals with a unique, widely used material, a 1045 carbon steel subjected to a specific tension-torsion loading. This spectrum combines in-phase and out-of-phase loadings, with different load ratios and includes overloadings exceeding the yield stress. In a first part, the fatigue lifetimes are measured and the damage mechanisms are observed and interpreted. The lifetimes are then compared to the results of different modelling methodologies: (1) approaches based on stress invariants [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF][START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF] are selected, owing to their reliability, their simple use and their reasonable computing time and then associated with different damage accumulation rules (Miner linear accumulation [START_REF] Miner | Cumulative damage in fatigue[END_REF], « DCA » non linear accumulation [START_REF] Manson | Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage[END_REF]), and (2) the damage model by Vu [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF]. The aim is to evaluate the respective capabilities of each of these approaches in terms of the reliability of the results.

Material and experimental conditions

In this section, the material is first presented. A 1045 carbon steel is selected because a lot of information about this material can be found in the literature. Then, the complex loading spectrum representative of stresses undergone by car-to-ground components is detailed. It comprises the effects of paved roads, curves and shocks on the wheels.

Material characteristics

The material is a 1045 carbon steel, used in many industrial applications. It has been the subject of many previous studies from the authors of the present paper [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF][START_REF] Billaudeau | Fatigue multiaxiale des matériaux à défaut: mécanismes et critère d'endurance[END_REF][START_REF] Billaudeau | Multiaxial fatigue limit for defective materials: mechanisms and experiments[END_REF][START_REF] Flaceliere | Competition between mesoplasticity and damage under HCF-Elasticity/damage shakedown concept[END_REF][START_REF] Flaceliere | Coupling between mesoplasticity and damage in high-cycle fatigue[END_REF][START_REF] Nadot | Multiaxial fatigue limit criterion for defective materials[END_REF]. The main mechanical characteristics are a Young's modulus E = 205 GPa, a monotonous yield stress R p0.2 monotonous = 350 MPa, a cyclic yield stress R p0.2 cyclic = 280 MPa and an ultimate stress R m = 580 MPa. This steel alternates ferrite and pearlite bands, the average grain size is 22 μm for ferrite and 16 μm for pearlite (a micrography can be seen in Figure 7). The chemical composition of the 1045 steel is given in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF].

The cylindrical tested specimens, whose minimum diameter is 9 mm, were machined from a round bar (diameter 80 mm). The specimen axis corresponds to that of the bar, i.e. the rolling axis. In order to observe the cracks at microscopic scale (5 -10 μm), the specimens were polished with several abrasive papers up to grade 4000. Then, the samples were tempered at 500 °C during one hour under vacuum to remove the residual stress due to the preparation stage of specimens.

Experimental procedure -Variable amplitude loading

Tests were performed on a servo-hydraulic axialtorsional machine (Instron type 1343) at room temperature and under ambient air. Applied normal stress amplitude σ a and shear stress amplitude τ a were calculated according to the following equations: , where F is the tension force, T is the torque and d is the minimum section diameter of the specimen. The replica technique [START_REF] Palin-Luc | Elastomer and resin replicas for SEM observation of metallic materials[END_REF] applied on the external specimen surface was used to monitor crack initiation and propagation. A soft dental resin is put on the specimen by the means of a mold. This allows recording and observing the surface of the specimen without disassembling and reassembling of the set-up. After metallization, the replicas were observed under SEM with low acceleration voltage. The procedure of observation starts with the last replica where the principal crack can be easily identified, and comes back to the early stages of cracking. The replica is a negative image of real surface.

Resolution of this technique is about 5 -10 μm depending on the crack opening.

The loading spectrum (Figure 1) used in tension-torsion tests is designed to represent a simplification of the real fatigue loadings undergone by in-service automotive structures.

The "Belgian Blocks (BB)" represent the effect of a paved road. The low mean stress in these blocks stands for the vehicle weight. Their recurrence is high (76% of spectrum) and their intensity low/medium. Note that the tension and torsion signals are out-of-phase, with a phase shift of 90°. The load ratio is -0.8 for tension and torsion. It represents  The "Turns (TN)" represent the effect of the curves of the road. The strong dissymmetry between the tension and torsion mean stress stands for the effect of the vehicle weight when turning. Their recurrence is medium (23% of spectrum) and their intensity low. Both signals are in-phase. The load ratio is -4.7 for tension, -0.2 for torsion.  The "Shocks (SCK)" represent the effect of a shock on the wheel (impact due to pavement, for example). Both signals are in-phase, which assumes a simultaneous impact on both wheels. The loading levels in this block may lead to local plasticity.

Their recurrence is low. The load ratio is -0.7 for tension and torsion.

The loading sequence comprises a first set composed of 5 "BB" (separated in BB1 and 2 depending on the sign of the phase shift) and followed by 12 "TN". This set is repeated 50 times. Then the specimen undergoes 2 "SCK". The whole sequence is then repeated as many times as necessary to break the specimen.

Figure 1 Spectrum for tension-torsion loadings

The spectrum reference parameter is the maximal von Mises stress in the "Shocks" block SCK; the amplitude of other blocks is defined by a ratio with respect to this reference stress level. Carrying out tests with a complex loading spectrum is a challenging task. In order to perform stable fatigue tension-torsion tests on a hydraulic device while keeping a reasonable test frequency, a preliminary study of the PID (proportional integral derivative control) is necessary. After an optimization phase, PID values have been determined to ensure a satisfactory compromise between the set point monitoring and the test stability. The theoretical spectrum levels had to be corrected according to the real mean values recorded during the tests. That is the reason why Table 1 contains the "experimental" mean and amplitude values undergone by the specimen, which are slightly different from the theoretical values. In Table 1 and in the following, the test reference refers to the theoretical value.  x is the applied tension component,  xy the shear one and σ VM the maximum Von Mises stress undergone during the "SCK" block. The test frequency for sinus loading is 3Hz for BB and 6Hz for TN and SCK.

Experimental fatigue lifetime and damage mechanisms under variable amplitude loading

The effect of "shocks" in the loading spectrum on the material lifetime is often neglected in industrial applications. Their influence can be very different according to the elastic-plastic behavior, and strongly varies between a smooth or a notched sample [START_REF] Zheng | Overload effects on fatigue behaviour and life prediction of lowcarbon steels[END_REF][START_REF] Sonsino | Interpretation of overload effects under spectrum loading o fwelded high-strength steel joints[END_REF]. The influence of the overloads is studied by analyzing the number of cycles to failure, the orientation of the critical planes and the evolution of fatigue cracks.

Fatigue lifetime under variable spectrum

For each test defined in Table 1, Table 2 provides the number of tension cycles undergone up to failure. The full spectrum sequence (Figure 1) contains 2602 tension cycles.

Four specimens have been tested with a maximum theoretical von Mises stress of 525 MPa (one of them, noted "REPLICA" has been used to observe fatigue crack initiation and growth by the means of the replica technique), two with 600 MPa and two with 480 MPa. In order to analyze the effect of shocks on the fatigue lifetime, a simplified spectrum ("NO SHOCKS" in Table 2) has been tested, in which the "SCK" cycles have been suppressed. The results are presented in a figure providing the maximal experimental Von Mises stress in "shocks" blocks vs. the number of tension cycles to failure.

Figure 2 Comparison of lifetime with or without "shocks" blocks

Although the "SCK" blocks represent only 0,1% of total spectrum, a non-negligible impact of these cycles is observed on the fatigue lifetime of specimens. Mean fatigue lifetime for "NO SHOCKS" spectrum is around 60% greater than mean fatigue lifetime for "FULL" spectrum. This confirms the harmfulness of the SCK blocks pointed out by [START_REF] Zheng | Overload effects on fatigue behaviour and life prediction of lowcarbon steels[END_REF][START_REF] Bonnen | The role of in-phase and out-ofphase overloads on the torsional fatigue of normalized SAE-1045 steel[END_REF].

Initiation and propagation mechanisms

In order to understand the influence of each type of block depicted in the previous section, damage mechanisms are observed for "FULL" and "NO SHOCKS" spectrum.

According to standard observations [START_REF] Flaceliere | Competition between mesoplasticity and damage under HCF-Elasticity/damage shakedown concept[END_REF][START_REF] Morel | Plasticity and damage heterogeneity in fatigue[END_REF][START_REF] Suresh | Fatigue of Materials[END_REF][START_REF] Verreman | High-cycle fatigue mechanisms in 1045 steel under non-proportional axial-torsional loading[END_REF], the crack propagation splits into two stages.

After the stage 1 occurring on the maximum shear planes (mode II/IIIshear stress dominated), the cracks branch into planes of maximum normal stress (stage 2mode Inormal stress dominated). Orientation of these critical planes can be calculated by the damage accumulation method for each loading case [START_REF] Karolczuk | A review of critical plane orientations in multiaxial fatigue failure criteria of metallic materials[END_REF]. Applied normal and shear stress are expressed as: 4 shows the definition of the angle α used to determine orientation of an arbitrary plane in the coordinate system associated to the specimen.

Figure 3 specimen schematization and definition of the angle α

Since cracks often initiate on the specimen surface, normal ( n ) and shear ( ns ) stress components on an arbitrary plane can be calculated under the hypothesis of plane stress as a function of the angle α (angle between the plane normal and the specimen axis):

(

The values of α ( 90 90       ) corresponding to critical (maximal) shear plane (τ ns, max ) and to critical (maximal) normal stress plane (σ n, max ) for each loading block mentioned in this work are shown in Table 3. The factor k represents the ratio of the shear and normal amplitude and is equal to 0.57 for all blocks.

Table 3 Calculated critical plane orientation

Block  xy (°) k = τ a /σ a (τ ns, max ) (°) (σ n, max ) (°)
Belgian Blocks 90 / -90 0.57 0 and 90 0 Figure 4 presents the fatigue crack after the test at the surface for the specimen quoted "525_9_REPLICA" in Table 2. The replica of the specimen surface has been observed by SEM ( Figure 5) to measure experimental orientation of the crack planes and to compare it to the values calculated in Table 3. It can be noticed that the damage state is strongly localized: only one crack seems to initiate from a polishing scratch. It then propagates until about 95% ( Figure 5-g) of total lifetime in a plane close to the one maximizing the shear stress during the "BB" cycles (α=0). Note that at this stage no explicit influence of "Turns" and "Shocks" has been observed. A similar study performed by Vu et al. [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF] showed that tensiontorsion in-phase loadings with a stress ratio k = 0.5 (like "Turns" and "Shocks") lead to multiple initiation sites. However, when localized (like "BB") and diffuse (like "Turns" and "Shocks") modes are in competition, the localized one tends to govern crack propagation. At around 95% of total lifetime, a change in the direction of the crack path is observed: the angle between the normal to the crack propagation plane and the specimen axis is about -43. This value does not correspond to any critical angle associated to the spectrum blocks (Table 3).

      sin xy xym xya xy t t t                2 cos sin 2 n t t t             

Crack growth kinetics

In spite of the three-dimensionality of the crack, the surface length is a relevant quantity to characterize the damage state during specimen lifetime [START_REF] Verreman | High-cycle fatigue mechanisms in 1045 steel under non-proportional axial-torsional loading[END_REF][START_REF] Miller | Materials Science Perspective of Metal Fatigue Resistance[END_REF][START_REF] Ohkawa | A study on fatigue crack growth under out-ofphase combined loadings[END_REF].

Figure 5 shows surface crack measures for 525_9_REPLICA specimen. In order to further characterize the crack propagation kinetics under the spectrum, Figure 6 displays the crack length (noted 2a) vs the ratio N/Nf (where N is the current number of cycles and Nf the number of cycles at failure). Different test results are drawn: the blue solid line represents the crack evolution for the full spectrum presented in Figure 1, whereas the three other lines are obtained from tests under constant amplitude (torsion, tension and a multiaxial tension-torsion loading with a stress ratio k=0.5 and a phase shift of 90°). These latter results come from [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF], who explain the different kinetics by the damage configuration:

in the case of torsion, the homogeneous and diffuse damage leads to a linear evolution, while the localized damage (observed for the other loadcases) is influenced by the microstructure heterogeneity and causes a nonlinear crack propagation evolution. The crack growth kinetics under full spectrum is highly nonlinear. A fast evolution of crack length is observed in the first 30% of lifetime, followed by a plateau until catastrophic failure. This particular behavior may be explained by the influence of the SCK blocks, which accelerate crack growth in the first stage and slow it down by plasticity in a second step. This perturbation may be correlated with the observations by [START_REF] Bonnen | The role of in-phase and out-ofphase overloads on the torsional fatigue of normalized SAE-1045 steel[END_REF] that pointed out the influence of plasticity and crack closure in crack growth kinematics. This possible role of SCK blocks on initial nonlinearity is also supported by the comparison with the multiaxial loading curve. If the effect of the TN blocks is assumed negligible, the spectrum loading is quite close to "90° out-of-phase k=0.5".

Therefore, the SCK blocks seem to be responsible for the initial strong nonlinearity.

Figure 6 Crack length vs fatigue life ratio for spectrum load compared to constant amplitude load

The red horizontal line in Figure 6 corresponds to 2a=60m. This length characterizes the required number of cycles for a crack initiated inside a ferrite grain to cross the neighboring pearlite grains, seen as the most important microstructural barrier for this material (Figure 7) [START_REF] Nasr | Fatigue initiation in C35 steel: Influence of loading and defect[END_REF]. It corresponds at the surface to a ferrite grain + 2 pearlite grains. The "crack initiation length" is independent of loading type, but the duration of this phase is different. This length takes into account micro-plasticity and propagation of micro-cracks [START_REF] Miller | The Behaviour of Short Fatigue Cracks and Their Initiation Part II -A General Summary[END_REF], and can be seen as the transition between initiation and propagation. In conclusion, this part provides clues to understand the effect of overloads on 1045 steel specimens. In spite of their limited number, the SCK blocks have a very harmful influence on the total lifetime of the specimen. The lifetime reduction is about 60%. However, these cycles do not seem to influence the critical initiation and propagation planes. The crack initiation is very rapid when the specimen is subject to the variable spectrum and its evolution is highly nonlinear.

Lifetime prediction using fatigue criterion methodology

The previous sections analyzed two spectra differing only by the presence of an overload (SCK blocks) each 2600 tension cycles. In this section, the discussion will address fatigue criterion methodology with two different damage accumulation rules: the Miner linear accumulation rule [START_REF] Miner | Cumulative damage in fatigue[END_REF] and the "Damage Curve Approach" (DCA) [START_REF] Manson | Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage[END_REF]. This latter approach is motivated by its simplicity of use and the fair correlation that can be found in general between the simulations and the experimental results. The fatigue criterion approach is widely used in industrial contexts. Its numerical processing is very simple and leads to low computer run times. However, a physically motivated criterion has to be selected to capture the main specificities of multiaxial fatigue loading. The problem encountered in this study is the very wide domain, ranging from LCF for the SCK cycles to HCF for the TN ones. In order to maintain the methodology simplicity and optimize CPU time for industrial applications, no plastic correction will be employed for SCK cycles. However, the DCA accumulation rule will be studied in order to take into account nonlinear behavior.

Multiaxial fatigue criterion

An abundant literature deals with fatigue criteria [START_REF] Van | Sur la résistance à la fatigue des métaux[END_REF][START_REF] Findley | A theory for the effect of mean stress on fatigue of metals under combined torsion and axial load or bending[END_REF][START_REF] Mcdiarmid | A general criterion for high cycle multiaxial fatigue failure[END_REF][START_REF] Robert | Contribution l' tude de la fatigue multiaxiale sous sollicitations p riodiques ou al atoires[END_REF][START_REF] Papadopoulos | A new criterion of fatigue strength for out-of-phase bending and torsion of hard metals[END_REF][START_REF] Papuga | Two new multiaxial criteria for high cycle fatigue computation[END_REF][START_REF] Zenner | On the fatigue limit of ductile metals under complex multiaxial loading[END_REF][START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF][START_REF] Li | A computerized procedure for long-life fatigue assessment under complex multiaxial loading[END_REF][START_REF] Mamiya | Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue[END_REF][START_REF] Sines | Behaviour of metals under complex static and alternating stresses[END_REF][START_REF] Ellyin | A general fatigue theory and its application to out-ofphase cyclic loading[END_REF][START_REF] Palin-Luc | An energy based criterion for high cycle multiaxial fatigue[END_REF]. In this work, the criterion recently proposed [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF] is tested by simulating the above described variable amplitude spectrum. It has been found to accurately simulate the fatigue lifetime under different loading cases (in particular non proportional [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF]), while being easy to use. It is built from the invariants of the macroscopic stress tensor (same type of approach as [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF][START_REF] Li | A computerized procedure for long-life fatigue assessment under complex multiaxial loading[END_REF][START_REF] Mamiya | Prismatic hull: a new measure of shear stress amplitude in multiaxial high cycle fatigue[END_REF][START_REF] Sines | Behaviour of metals under complex static and alternating stresses[END_REF]) and reads as:

(

Equation ( 5) defines an equivalent stress  vu . It involves not only the second invariant of the amplitude of the stress deviator J' 2 (t) but also the mean value J 2,mean of this invariant over the period. This latter term has been found to capture the phase shift effect in a simple way. The term I f is a function of the amplitude (I 1,a ) and of the mean value (I 1,m ) of the hydrostatic stress. The expression of I f depends on the type of material and can be found in [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF]. For a given loading, the maximum value of  vu is calculated during a period and compared with the value of (N), a characteristic parameter depending on the number of cycles (in order to assess the finite failure stress level). The four parameters of the criterion        can be identified from two fatigue limits (for example, fully reversed torsion and tension) and the ultimate strength. The expression ( 5) is also capable of taking into account the different effects of phase shift (increase or decrease of the fatigue limit, depending on the material). The complete identification procedure can be found in [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF]. For the 1045 steel, it leads to the values presented in Table 4.

Table 4 Fatigue criterion parameters for 1045 steel

When extension to finite lifetimes is needed, an optimal S-N curve should include experimental data from this whole range, but in the absence of such results a standard Basquin formulation, identified here on fully reversed torsion, is used as reference S-N curve. Figure 8 plots the experimental Vu equivalent stress (5) vs. the fatigue lifetime and the corresponding Basquin interpolation [START_REF] Dal Cero Coelho | Maîtrise de la tenue en fatigue des cordons de soudure[END_REF].

Figure 8 Basquin interpolation, experimental torsion R=-1

Basquin law ( 6) is identified by linear regression considering the equivalent stress as independent variable (ISO12107, [START_REF]Metallic materials -Fatigue testing -Statistical planning and analysis of data[END_REF]). An example of identification can be found in [START_REF] Statnikov | Efficiency evaluation of ultrasonic impact treatment (UIT) of welded joints in weldox 420 steel in accordance with the IIW program[END_REF]. All simulations are performed using the SN curve with 50% of survival probability and 50% of confidence interval. The identification leads to the following equation of the Basquin curve:

Damage accumulation rule

Once the fatigue criterion is selected and material parameters are identified, an accumulation rule has to be chosen in order to deal with variable amplitude loads. In this work, two usual rules are compared: the Miner linear accumulation rule [START_REF] Miner | Cumulative damage in fatigue[END_REF] and the "Damage Curve Approach" (DCA) [START_REF] Manson | Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage[END_REF]. This latter approach is motivated by its simplicity of use and the fair correlation that can be found in general between the simulations and the experimental results.

Miner's linear damage accumulation rule

Firstly, the multiaxial fatigue criterion [START_REF] Vu | Multiaxial fatigue criterion for complex loading based on stress invariants[END_REF] is combined with the classical Miner's linear damage accumulation rule [START_REF] Miner | Cumulative damage in fatigue[END_REF] to predict fatigue life of both full spectrum and « no shocks spectrum ». In order to illustrate the performance of the proposed criterion, the same analysis will be performed using the reference Crossland criterion [START_REF] Crossland | Effect of large hydrostatic pressures on the torsional fatigue strength of an alloy steel[END_REF]. full and "no shocks" spectra.

Figure 10 compares numerical and experimental results for "FULL" spectrum and "NO SHOCKS" spectrum. Several conclusions can be drawn:  As regards the prediction of fatigue lifetime for full spectrum, both approaches give similar results. The general tendency is non-conservative when compared with experimental data.  The non-conservative tendency is also observed for "NO SHOCKS" spectrum simulations. However, the proposed criterion seems to better predict the gap between fatigue lifetime of full spectrum and "NO SHOCKS" spectrum.

It is important to note that the methodology combining a fatigue criterion and the linear accumulation rule is pushed to its limits when applied to full spectrum. Indeed, the full spectrum covers a wide range of stress levels, varying from elasticity for BB and TN, and plasticity for SCK.

Damage Curve Approach

In a second step, the fatigue criterion ( 5) is associated with a nonlinear damage accumulation rule. The Miner's rule is compared with a simple non-linear cumulative damage rule, the Damage Curve Approach (DCA) [START_REF] Manson | Practical implementation of the double linear damage rule and damage curve approach for treating cumulative fatigue damage[END_REF]. This latter approach is governed by Equation [START_REF] Zenner | On the fatigue limit of ductile metals under complex multiaxial loading[END_REF], providing the damage level as a function of the current number of cycles n and the number of cycles at failure N f

D n Nf

Nf [START_REF] Zenner | On the fatigue limit of ductile metals under complex multiaxial loading[END_REF] The non-linearity is a consequence of the parameter α (the higher , the more nonlinear the accumulation), which is experimentally identified. Its value is determined by the means of tests exhibiting the nonlinearity of damage accumulation, for example uniaxial tension -torsion block loadings [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF]. Two loading sequences are carried out: (i) a given number of alternate tension cycles is applied and then the specimen is subjected to alternate torsion up to failure, (ii) a given number of alternate torsion cycles is applied and then the specimen is subjected to alternate tension up to failure. Experimental results are plotted in However, the experimental points presented in Figure 10 are very irregular and no clear tendency can be identified. This irregularity may come from a strong influence of microstructure. The identification of α is performed by creating an envelope of experimental points. The estimated lifetime using DCA is higher than the lifetime obtained from the linear accumulation of Palmgren -Miner ( ) in the case of tension -torsion ( ) and lower in the case of torsion -tension ( ). Those predicted tendencies are in agreement with several experimental results including the observations in [START_REF] Miller | Materials Science Perspective of Metal Fatigue Resistance[END_REF] for 1045 steel and in [START_REF] Chen | Fatigue life prediction of type 304 stainless steel under sequential biaxial loading[END_REF], for S45C steel. A value of α=0,6 is found to be fairly satisfactory to forecast the experimental data. Experimental To -Ta  The linear damage accumulation predicted by Miner's rule together with the Vu criterion is not sufficient to accurately represent fatigue lifetime for both "FULL" and "NO SHOCKS" spectrum, the fatigue lifetime is strongly overestimated  The nonlinear accumulation DCA approach (with α =0.6) substantially improves the simulation for "FULL" spectrum.

 The DCA approach and the Miner's rule provide very similar results in the "NO SHOCKS" case.

 From the previous point, it can be concluded that the value of α=0,6 is valid only for the "FULL" spectrum and cannot be transposed for "NO SHOCKS" spectrum. The methodology seems to be suited to loadings involving strong stress level variations.

A modification of the value of α could be a simple and tractable solution. Indeed, the evolution of damage accumulation is probably different according fatigue domains and then different values are needed for α.

It is important to note that no plastic correction is applied in order to maintain methodology simplicity and optimize CPU time. The non-linearity induced by shocks plasticity is taken into account only by DCA accumulation rule.

None of the methodologies presented in this section (fatigue criterion + linear or nonlinear damage accumulation rule) is capable of accurately simulate simultaneously the fatigue lifetimes of samples subject to a loading spectrum with and without overloads. In order to overcome this difficulty, the next section presents a more complex method: the damage evolution is simulated by a two-scale model written in the framework of the Continuum Damage Mechanics (CDM). This model has been proposed in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF], and only the main features are briefly recalled in this section. In HCF, the main source of failure of metallic components is attributed to plasticity and damage at grain scale (micro/mesoscopic scale). This argument is the starting point for many multi-scale models which can be found in the literature. For example, [START_REF] Van | Sur la résistance à la fatigue des métaux[END_REF] proposed a non-initiation criterion at microscopic scale. However, loading on structure is applied at macroscopic scale, justifying the choice of a two-scale modelling framework which is nowadays widely found in the literature [START_REF] Flaceliere | Coupling between mesoplasticity and damage in high-cycle fatigue[END_REF][START_REF] Doudard | A probabilistic two-scale model for high-cycle fatigue life predictions[END_REF][START_REF] Huyen | A critical plane fatigue model with coupled meso-plasticity and damage[END_REF][START_REF] Lemaitre | A two scale damage concept applied to fatigue[END_REF][START_REF] Monchiet | A micromechanical explanation of the mean stress effect in high cycle fatigue[END_REF]. The crack initiation process is the result of a simple slip system within one or many grains. This assumption is consistent with observations carried out on different materials [START_REF] Rasmussen | Fatigue of copper polycristals at low plastic strain amplitudes[END_REF].

The link between the stress at macroscale  and at microscale  is given by the classical Lin-Taylor localization relationship

p 2      
(where µ is the shear modulus and p  the plastic strain at the grain scale). The model is formulated in the rigorous framework of irreversible thermodynamics with internal variables for isothermal, time-independent and small deformation transformations. The evolution of internal variables is governed by: (i) one (or many) threshold function(s) answering the question "when do plasticity and damage evolve?" and (ii) one (or many) dissipation potential(s) indicating, by assumption of normality, how progress this mechanisms [START_REF] Germain | Continuum thermodynamics[END_REF]. The main ingredients of the model are briefly recalled below.

Together with the plastic strain p  , the isotropic hardening (p) and kinematic hardening (  ) variables, a scalar damage variable d is used to account for the evolution of microcracks at the grain scale. This "damage effect variable" acts on the degradation of the mechanical strength of crystal that affects directly the isotropic hardening.

Regarding the evolution laws of the plastic variables, the yield surface is based on the criterion presented in Section 4, in order to benefit from its capacity to account for the specificities of HCF loadings. The rates of the plastic variables follow the normality rule with respect to a dissipation potential different from the plastic yield surface and whose expression can be found in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF].

The damage evolution law is split into two different expressions in order to account for the initiation and propagation phases:

if d≤d p if d>d p (8)
where d p is the damage threshold of propagation phase. Moreover, the expression of directly involves, as a parameter, the hydrostatic part of the stress at the microscale in order to distinguish tension and torsion. The damage evolution laws are governed by the normality rule applied to a dissipation potential and are specified in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF]. The step-by-step integration of (8) leads to nonlinear damage accumulation. Failure occurs when the damage variable d reaches the critical value d c .

Comments on Vu et al. model and extension to account for overload

The assessment of the model briefly recalled above has been carried out for multiaxial, proportional, out-of-phase and block loadings in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF]. The obtained results prove that the model is efficient for simulating HCF lifetimes. The model is implemented in the rigorous framework of irreversible processes that allows an explicit description of the incremental evolutions of plastic and damage variables. The use of a two scale approach (mesoscopicmacroscopic) enables to capture physical processes of material degradation (plasticity, damage) at the grain scale. In addition, several proper effects of HCF are taken into account including fatigue damage accumulation below the fatigue limit, mean stress and phase shift effects. The model describes the nonlinear damage accumulation thanks to the distinctions of two damage mechanisms (tension/torsion) and two phases of crack growth (different damage evolution law for initiation and propagation). The model allows hence fairly good prediction of S-N curves for both proportional and non-proportional multiaxial loadings at constant amplitude, with and without mean stress. For variable amplitude loadings, the model proved also its efficiency in dealing with block loadings.

The model, which is dedicated to multiaxial HCF, cannot describe correctly high amplitude fatigue loading (overload) falling in low cycle fatigue (LCF) regime. In order to deal with the overloading SCK cycles of the spectrum presented in Figure 1, the plastic yield stress needs to be modified.

The plastic yield surface of the original model reads as follows:

        0 , , , 0 , 1 , 1 3 2 , 2 2 2 2 1        r r I I I J x J r x f m a f mean      (9)
where J 2 is the second invariant of the deviator of x   , J 2,mean the mean value over the period of the second invariant of the amplitude of the stress tensor and I f a function of the amplitude (I 1,a ) and of the mean value (I 1,m ) of the hydrostatic stress. The real r 0 is the initial radius of the yield surface. The tensor x is the backstress associated to the kinematic hardening variable  . The function r governs the isotropic hardening. It is supposed to tend towards an asymptotic value  r ~ when the accumulated plastic strain p increases:

    sd gp r r      exp ) exp( 1 ~ ( 10
)
where g is the isotropic hardening modulus. The parameter s is the damage sensitivity

    0 , 1 , 1 3 2 , 2 2 , r r I I I J m a f mean      (11)
During the very first loading cycles, when the accumulated plastic strain p is very low, the value of r is close to zero resulting in:

  0 , 1 , 1 3 2 , 2 2 , r I I I J m a f mean     (12)
Inequality ( 12) imposes a strong restriction on the load level and prevents from dealing with SCK overloads, when the values of mean J , 2 and f I are too high and violates [START_REF] Ellyin | A general fatigue theory and its application to out-ofphase cyclic loading[END_REF]. In order to extent the application range of the model to LCF, the expression of the yield surface is slightly modified:

        0 . , , , 0 , 1 , 1 3 2 , 2 2 2 2 1        r r A I I I J x J r x f m a f mean      (13)
with

1 1 0 0    r A eq  and   m a f mean eq I I I J , 1 , 1 3 2 , 2 2 0 ,      (14)
The parameter A can be seen as an expansion coefficient of the yield surface when switching from HCF to LCF. The Macaulay brackets <…> (<x>=x if x>0, <x>=0 otherwise) modify the value of A when the loading level suddenly increases: Table 6 summarizes the results predicted by the improved version of the model for the variable spectra mentioned in Table 2. The simulated lifetimes of the specimens subject to full spectrum are in good agreement with the experimental results and are of the same order of magnitude as those obtained by the approach "criterion + nonlinear accumulation rule"

(Figure 11). The main improvement regards the simulation of the lifetime of specimens undergoing spectra without SCK blocks. Whereas the criterion approach predicted failure after 4,000,000 cycles, the damage model provides a lifetime closer to the experimental results, moreover in a conservative way. This methodology is more complex to implement but allows to cover a wide range of lifetime.

Conclusion

The goal of this work was to compare different methodologies dedicated to the determination of fatigue lifetime of industrial components subjected to complex multiaxial spectra. For decades, an abundant literature is dealing with a wide variety of approaches, from simple fatigue criteria to damage models, via different accumulation rules. However, little work has been done to rigorously compare these different approaches on the basis of complex loads representative of real use conditions. To this end and for ease of comparison, the work presented here focused on a unique material, a 1045 carbon steel, whose samples have been subjected to a loading spectrum used in tension-torsion tests and designed to represent real fatigue loadings undergone by in-service automotive structures. This spectrum involves several sequences with different load ratios, phase shifts and maximum stress levels. In particular, it includes overloads which are responsible for local plasticity. The lifetime has been measured for different stress levels. Thanks to the observations of fracture surfaces and of crack length, a strongly non linear damage evolution has been identified. This experimental database is the starting point for the evaluation of different simulation tools to predict fatigue lifetime under complex multiaxial spectra. Because of their ease of use, two criteria built from stress invariants have been selected (Crossland and Vu) and associated to two accumulation rules (linear and nonlinear). The linear damage accumulation proves not reliable enough, even in the case of the Vu criterion which takes into account the multiaxiality character of the loading spectrum. The nonlinear accumulation DCA approach is a simple way to substantially improves the simulation.

However, the methodology consisting in the association of a fatigue criterion and a damage accumulation rule (even nonlinear) fails to accurately simulate the fatigue lifetimes of samples subject to a loading spectrum involving overloads. This issue can be overcome by simulating damage evolution by the means of the damage model proposed by Vu [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF] slightly modified in order to take into account cycles exceeding the macroscopic yield stress. This extension does not need the identification of any extra parameter compared to the original version of the model and improves its prediction. The computation time is however the price to pay for this reliability of the results: whereas the calculation "fatigue criterion + DCA" is quasi instantaneous when using a standard personal computer, the incremental approach of the damage model leads to quite high computation times (about 30 minutes for a simulation of 500 000 cycles on one material point).

Work in progress aims at extending the methodology to fully random spectra. To this end, a specific methodology has to be developed to determine the stress invariants involved in the Vu yield surface when no time period can be defined.

Figure 4

 4 Figure 4 Fatigue crack after failure / specimen 525_9_REPLICA

Figure 5

 5 Figure 5 SEM surface observation of the evolution of the main crack through lifetime (specimen 525_9_REPLICA)

Figure 7

 7 Figure 7 Definition of "initiation length" [35] a) Longitudinal direction b) Transversal direction

Figure 9

 9 Figure 9 Crossland and Vu criteria with Miner's accumulation rule prediction for

Figure 10 :

 10 Figure 10: whereas the Miner linear accumulation rule is represented by the dashed line, the test data show a nonlinear accumulation. The parameter α controls this non linearity.

Figure 10 Figure 11

 1011 Figure 10 Tension (Ta)torsion (To) block loading: experimental results and identification of parameter α

Figure 11 HCF regime with overloads 5 . 1

 1151 Figure 11 Comparison of cumulative damage rules: linear vs. non-linear

  quantity according to the local stress tensor  and the backstress tensor x at the grain scale. load must verify the following inequality:

Figure 12

 12 Figure 12 Simulated S-N curve under reversed constant amplitude loadings

Table 1 Experimental loading spectrum values: Stress amplitude and mean value per block

 1

	Test reference	Experimental value of σ VM "SCK"	Stress component	Belgian Blocks BB1 -BB2	Belgian Blocks BB3 -BB4	Turns (TN)	Shocks (SCK)
			Σ x mean [MPa]	19	25	-84	57
	600 MPa	526 MPa	Σ x amplitude [MPa] Σ xy mean [MPa]	174 11	222 18	125 45	355 19
			Σ xy amplitude [MPa]	100	129	58	169
			Σ x mean [MPa]	17	21	-72	54
	525 MPa	504 MPa	Σ x amplitude [MPa] Σ xy mean [MPa]	151 11	195 13	108 40	311 28
			Σ xy amplitude [MPa]	91	116	57	172
			Σ x mean [MPa]	13	17	-68	44
	480 MPa	463 MPa	Σ x amplitude [MPa] Σ xy mean [MPa]	140 10	179 12	102 38	288 28
			Σ xy amplitude [MPa]	83	106	49	157

Table 2 Experimental results for spectra

 2

	Specimen Reference	σvm SCK Theoretical value [MPa]	σvm SCK Experimental value [MPa]	Number of tension cycles to failure
	525_1	525	504	234 180
	525_2	525	504	353 872
	525_3	525	504	294 026
	480_5	480	463	801 416
	600_6	600	526	124 896
	600_7	600	526	140 508
	480_8	480	463	629 648
	525_9_REPLICA	525	504	265 404
	525_10_NO-SHOCKS	525	504	596 544
	525_11_NO-SHOCKS	525	504	638 560
	525_12_NO-SHOCKS	525	504	1 004 432

Max Shocks (MPa) Number of Cycles -Tension (log) FULL Spectrum NO SHOCKS Spectrum

				(2)
	1,E+05 σVM     xx t t        xm	xa	1,E+06   sin t 	1,E+07	1,E+08

where stress tensor components Σ xx (t) and Σ xy (t) are expressed as a function of mean (respectively, amplitude) normal stress Σ xm (respectively, Σ xa ) and mean (respectively, amplitude) shear stress Σ xym (respectively, Σ xya ), time t, pulsation and the phase shift angle δ xy . Figure

Table 6 Predicted results for Automobile Spectra

 6

	Specimen Reference	σvm Shock Theoretical Value [MPa]	σvm Shock Experimental Value [MPa]	Experimental number of tension cycles at failure	Predicted number tension cycles of at failure
	525_1	525	504	234 180
	525_2 525_3	525 525	504 504	353 872 294 026	276 086
	525_9_REPLICA	525	504	265 404
	600_6 600_7	600 600	526 526	124 896 140 508	119 809
	480_5 480_8	480 480	463 463	801 416 629 648	1 234 582
	525_10_NO-SHOCKS	525	504	596 544
	525_11_NO-SHOCKS	525	504	638 560	329 815
	525_12_NO-SHOCKS	525	504	1 004 432

ACKNOWLEDGMENTS

This research is funded by Renault and Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 107.02-2014.25 This work pertains to the French Government program "Investissements d'Avenir" (LABEX INTERACTIFS, reference ANR-11-LABX-0017-01).

In LCF, the yield surface radius is magnified and it is restored to its initial value when switching to HCF. In this latter case, the expression of the plastic yield surface remains the same as [START_REF] Li | A computerized procedure for long-life fatigue assessment under complex multiaxial loading[END_REF]. The value of the expansion parameter A is directly determined from the macroscopic invariants

Assessment of improved model

In this section, the improved model is tested by simulating some usual multiaxial loading cases as well as the variable amplitude spectrum (Figure 1). The model parameters for the 1045 steel mentioned in this section have been identified in [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF], and are recalled in Table 5.

The identification procedure requires three material properties (fatigue limits under fully reversed tension and torsion, ultimate strength), two S-N curves (reversed torsion, reversed tension) and one curve giving the crack length growth under reversed tension loading. In a first stage, the model is used to simulate constant amplitude loadings. Whereas [START_REF] Vu | High cycle fatigue of 1045 steel under complex loading: Mechanisms map and damage modelling[END_REF] only simulated S-N curves in the HCF regime (more than 100,000 cycles), the introduction of the expansion coefficient A≠1 allows now to extend the cycle range. The S-N curves predicted by the improved model are shown in Figure 12. It can be seen that the transition from the original version of the model (A=1) to LCF domain is quite smooth under both reversed torsion and reversed tension loadings. The yield surface modification does not introduce artificial discontinuities, the HCF curve is continued towards the lower numbers of cycles. Of course this approach would need modifications for very small number of cycles (less than 10 000) Highlights  Fatigue lifetimes of samples subjected to complex multiaxial spectrum are measured  Different simulation tools to predict fatigue lifetime are evaluated  Incremental damage model can forecast lifetime with and without overload