Garry Dorleon
email: garry.dorleon@unimes.fr.

Sylvain Rigaud

Isabelle Techer

Management of dredged marine sediments in southern France: main keys to large-scale beneficial re-use

) make some recommendations on future policy options for the sustainable management of dredged sediments.

• A sustainable approach towards managing dredged sediments requires a change of view rather than a waste; dredged materials need to be seen as valuable resources.

• Perception issues remain crucial challenges to those promoting sustainable sediments management through the dredging process and activities.

• National level support will be necessary for helping to change attitudes and to promote sustainable dredged sediments management within the existing policy framework or introducing a new policy to succeed the mission.

• An effective and favorable policy would be accomplished to achieve the sustainable utilization of dredged sediments for different applications and valuable products.

• It may be expected that sediment management may change in the future, requiring more sustainable and circular approaches to dredged sediment management.

Introduction Conclusion

The presence of anthropogenic activities throughout the globe and the harbors generate an enormous amount of dredged sediments. Various research studies [START_REF] Slimanou | Harbor Dredged Sediment as raw material in fired clay brick production: Characterization and properties[END_REF][START_REF] Yoobanpot | Multiscale laboratory investigation of the mechanical and microstructural properties of dredged sediments stabilized with cement and fly ash[END_REF][START_REF] Zheng | Framework for determining optimal strategy for sustainable remediation of contaminated sediment: A case study in Northern Taiwan[END_REF][START_REF] Bagarani | The reuse of sediments dredged from artificial reservoirs for beach nourishment: technical and economic feasibility[END_REF][START_REF] Baptist | Beneficial Use of Dredged Sediment to Enhance Salt Marsh Development by Applying A 'Mud Motor': Evaluation Based on Monitoring[END_REF][START_REF] Ulibarri | Barriers and opportunities for beneficial reuse of sediment to support coastal resilience[END_REF][START_REF] Balkaya | Beneficial use of dredged materials in geotechnical engineering[END_REF][START_REF] Houlihan | Geoenvironmental Evaluation of RCA-Stabilized Dredged Marine Sediments as Embankment Material[END_REF][START_REF] Bianchini | Sediment management in coastal infrastructures: Technoeconomic and environmental impact assessment of alternative technologies to dredging[END_REF][START_REF] Wang | Recycling dredged sediment into fill materials, partition blocks, and paving blocks: Technical and economic assessment[END_REF][START_REF] Cezarino | Diving into emerging economies bottleneck:Industry 4.0 and implications for circular economy[END_REF][START_REF] Tessier | Study of the spatial and historical distribution of sediment inorganic contamination in the Toulon bay (France)[END_REF] developed process technology to recycled contaminated dredged sediments, and it seems that there is the potential of value proposition and economic viability of the products. The integrated life cycle assessment (LCA) [START_REF] Soleimani | Environmental, economic and experimental assessment of the valorization of dredged sediment through sand substitution in concrete[END_REF], life cycle cost assessment (LCC) are also involved in economic analysis to provide insight for early stage decision-making on the valorization of the dredged sediments.

The development of technology to produce valuable materials from contaminated dredged sediments offers an opportunity to use it as an alternative to virgin resources that could save energy and optimize resource efficiency resource utilization, reduce emissions and potentially contribute to a circular economy. The choice of recycling option for dredged sediments is essential in economic, environmental, sustainability terms and the perspective of the circular economy.

4. An economic approach towards recycling dredged sediments for sustainable development

*Respect for the principle of proximity and economic aspects/realities

Managing waste with proper disposal facilities became a crucial challenge [START_REF] Lirer | Mechanical and chemical properties of composite materials made of dredged sediments in a fly-ash based geopolymer[END_REF][START_REF] Kamali | Marine dredged sediments as new materials resource for road construction[END_REF][START_REF] Norén | Integrated assessment of management strategies for metal-contaminated dredged sediments?[END_REF][START_REF] Marmin | Ocean & coastal management collaborative approach for the management of harbour-dredged sediment in the bay of seine (France)[END_REF][START_REF] Slimanou | Harbor Dredged Sediment as raw material in fired clay brick production: Characterization and properties[END_REF][START_REF] Yoobanpot | Multiscale laboratory investigation of the mechanical and microstructural properties of dredged sediments stabilized with cement and fly ash[END_REF][START_REF] Zheng | Framework for determining optimal strategy for sustainable remediation of contaminated sediment: A case study in Northern Taiwan[END_REF][START_REF] Bagarani | The reuse of sediments dredged from artificial reservoirs for beach nourishment: technical and economic feasibility[END_REF][START_REF] Baptist | Beneficial Use of Dredged Sediment to Enhance Salt Marsh Development by Applying A 'Mud Motor': Evaluation Based on Monitoring[END_REF]. Due to a shortage of land, the cost of waste dumping gradually increases. Unmanaged dumping of dredged sediments leads to damage to environmental and ecological systems. Using dredged sediments through the recycling process is a sustainable approach towards supplementing natural resources, reducing pollution and greenhouse gas (GHG), energy optimization, enhancing economic development [START_REF] Soleimani | Environmental, economic and experimental assessment of the valorization of dredged sediment through sand substitution in concrete[END_REF], improve sustainability on marine ecosystems.

Exploring the required mechanical, chemical, and physical properties of dredged sediments would lead to a most appropriate option for different applications:

The beneficial use of dredged sediments for various applications

Method: literature review on the subject 1. Production of dredged marine sediments in France and abroad

Recommendation on future policy options for the management of dredged sediments

The future policy options available with the policymakers for managing dredged sediments are to promote necessary schemes and facilities within the framework of policy and regulation by different nationals with the alliance of industry partners. The problems related to the management of dredged sediments that need to be resolved as soon as possible to promote sustainable development through dredged sediments management are the following:

Keywords

"dredging" OR "marine sediments", "management", "waste recycling", "chemical compositions", "regulatory aspect", "disposal" AND "potential recovery". ▪ Policy for limiting the use of certain unsustainable methods or technology to specific industries ▪ The particular policy required for penalties wherever applicable ▪ Unfavorable market forces ▪ Legal/policy/governance challenges at the international and national ▪ Disposal capacity and process ▪ Critical barrier on transporting of dredged sediments to a different location regarding cost, safety, physicochemical properties, environmental issues, etc. ▪ Major investments ▪ Lack of government initiative in respect of promoting mission zero waste and to make realize that dredged sediments is wealth ▪ Delay in necessary approval and responses from the government official pertaining to dealing of dredged sediments ▪ Lack of policy frameworks, especially in the context of managing dredged sediments ▪ Transparency of information systems among the government and industry partner ▪ Introduce code and standards on the secondary product from dredged sediments so that users can use the products.

▪ Facilities for transportation of enormous amount of sediments

Major bottlenecks [START_REF] Ulibarri | Barriers and opportunities for beneficial reuse of sediment to support coastal resilience[END_REF] towards managing bulk usage of the dredged sediments are as follows.

Potential applications for reuse of dredged sediments

Example below of the strategy developed for the management of dredged marine sediments from harbor Carnon (south of France)

In France, the decision on sediment suitability for sea dumping or land disposal is primarily based on chemical findings, being the ecotoxicological analyses only advised to be done on guidelines. According to the concentration limits of the required chemical parameters (such as metals, trace elements and organic micropollutants), N1 (high quality, low pollutant concentrations) and N2 (medium quality, mid pollutant concentrations) levels are legally defined. Only in the cases where any of the compounds exceeds the N2 threshold, a biological characterization of sediments is needed to define their ecotoxicity [START_REF] Ospar | Oslo and Paris Conventions[END_REF][START_REF] Harrington | Economic modelling of the management of dredged marine sediments[END_REF][START_REF] Abidi | Characterization of dredged sediments of Bouhanifia dam: potential use as a raw material[END_REF][START_REF] Tessier | Study of the spatial and historical distribution of sediment inorganic contamination in the Toulon bay (France)[END_REF].

Management options

(Based on concentration limits of some physical and chemical parameters such as particle size, metals, organic micropollutants)

Land management = Waste (ICPE thresholds)

Dredging operation and geotube dewatering, @ Carnon 2023 90% 10%

5. Bottleneck preventing bulk re-use of dredged sediments References 2 .

 2 French regulatory framework for the management of dredged sediments is complex Dredged sediments (France) Regulatory characterization ▪ Environmental Code, annex I of article L541-8 ▪ Regulation 1357/2014/EU (Waste hazard properties) ▪ Order of February 15, 2016 ▪ Order of June 30, 2020

▪

  Flexible licensing systems to process dredged sediments ▪ Safely policy ▪ Insurance scheme ▪ Startup facilities ▪ Financing facilities to set up a plant ▪ Emerging circular economic concept ▪ Approach to introduce code and standard ▪ Policy to design life cycle assessment (LCA) to reduce sediments ▪ Policy for compulsory use of sediments as secondary products wherever applicable ▪ Policy for restricting the use of natural resources

▪

  Physical, chemical and biological characterizationOther requirements▪ Determination of nitrogen and phosphorus for sediment discharged in sensitive areas; ▪ Faecal contamination to avoid impacts on shellfish, mariculture, or bathing areas. ▪ Biological characterization of the affected site in case of sediment immersion, if at least one element exceeds the N2 threshold. ▪ The particle size has to be considered in case of reuse for nourishment. ▪ Leaching test to define land storage options, ICPE thresholds


Samuel Soubeyrand
email: samuel.soubeyrand@avignon.inra.fr

Emilie Haon-Lasportes

Weak convergence of posteriors conditional on maximum pseudo-likelihood estimates and implications in ABC

Keywords: Approximate Bayesian computation, Bernstein -von Mises theorem, Weak convergence. 2010 MSC: 62F12, 62Fxx, 65C60

The weak convergence of posterior distributions conditional on maximum pseudo-likelihood estimates (MPLE) is studied and exploited to justify the use of MPLE as summary statistics in approximate Bayesian computation (ABC). Our study could be generalized by replacing the pseudo-likelihood by other estimating functions (e.g. quasi-likelihoods and contrasts).

Introduction

Approximate Bayesian computation (ABC) has been developed to make Bayesian inference with models that can be used to generate data sets but whose probability distribution of state variables is intractable [START_REF] Marin | Approximate Bayesian computational methods[END_REF]. The intractability of this distribution makes impossible the application of the exact Bayesian approach, even by using numerical algorithms.

ABC provides a sample from the parameter space that is approximately distributed under a posterior distribution of parameters conditional on sum-mary statistics. In general, this posterior does not coincide with the posterior distribution of parameters conditional on full raw data. Here, we are interested in the specific case where (some of) the summary statistics are point estimates of parameters (PEP), as in Drovandi et al. (2011), Fearnhead and[START_REF] Fearnhead | Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation[END_REF], [START_REF] Gleim | Approximate Bayesian computation with indirect summary statistics[END_REF] and [START_REF] Mengersen | Bayesian computation via empirical likelihood[END_REF].

In the classical Bayesian framework, posteriors conditional on PEP can be viewed as specific cases of posteriors conditional on partial information [START_REF] Doksum | Consistent and robust Bayes procedures for location based on partial information[END_REF][START_REF] Soubeyrand | Inference with a contrast-based posterior distribution and application in spatial statistics[END_REF]. Here, we provide new results of weak convergence when PEP are either maximum likelihood estimates (MLE) or pseudo-maximum likelihood estimates (MPLE). The case where PEP are MPLE is of specific interest because, in ABC, it may be possible to compute MPLE via simplifications of the dependence structure in the model, and to use MPLE as summary statistics.

The results of weak convergence that are provided in this note can be viewed as new extensions of the Bernstein -von Mises (BvM) theorem.

For parametric models from which independent observations are made, the BvM theorem (i) states conditions under which the posterior distribution is asymptotically normal and (ii) subsequently leads to the efficiency of Bayesian point estimators and to the convergence of Bayesian confidence sets to frequentist limit confidence sets [START_REF] Walker | On the asymptotic behaviour of posterior distributions[END_REF][START_REF] Freedman | On the Bernstein-Von Mises theorem with infinitedimensional parameters[END_REF].

Thus, the BvM theorem can be viewed as a frequentist justification of posterior distributions for the estimation of parameters. Numerous extensions of the BvM theorem have been proposed, for instance, when the model is semiparametric or nonparametric (Bickel and Kleijn, 2012;[START_REF] Bontemps | Bernstein-von Mises theorems for Gaussian regression with increasing number of regressors[END_REF]Castillo, 2012a,b;[START_REF] Castillo | Nonparametric Bernstein-von Mises theorems in Gaussian white noise[END_REF][START_REF] Rivoirard | Bernstein-von Mises theorem for linear functionals of the density[END_REF], when observations are dependent [START_REF] Borwanker | The Bernsteinvon Mises theorem for Markov processes[END_REF][START_REF] Tamaki | The Bernstein-von Mises theorem for stationary processes[END_REF], when the model is misspecified [START_REF] Kleijn | The bernstein-von-Mises theorem under misspecification[END_REF], and when the model is nonregular (e.g. when the true value of the parameter is on the boundary of the parameter space; [START_REF] Bochkina | The bernstein-von Mises theorem and nonregular models[END_REF].

Here, we extend the BvM theorem (i) when raw observations are replaced by the MLE (Lemma 1) or an MPLE (Lemma 2), and (ii) when the posterior conditional on an MPLE is approximated via ABC (Theorem 1). Using a posterior distribution (approximate or not) conditional on an MPLE, that was built by ignoring some dependences in the model, can be viewed as using a misspecified model like in [START_REF] Kleijn | The bernstein-von-Mises theorem under misspecification[END_REF].

The BvM extensions obtained in the classical Bayesian framework (Point (i) in the paragraph above) are viewed as stepping stones that lead to the BvM extension obtained in the ABC framework (Point (ii)). Advancing theory in ABC has generally no direct practical implications because assumptions that may be required to prove theorems cannot be checked for a real-life implicit stochastic model whose distribution theory is intractable.

However, showing an analytic result for a large class of theoretically tractable models may lead to conjecture that the result holds for some stochastic implicit models. Specifically, the work presented here allows us to conjecture that (i) an ABC-posterior distribution conditional on an MPLE is asymptotically normal and centered around the MPLE, and (ii) resulting point estimates and confidence sets converge to their frequentist analogues. A pseudo-likelihood is generally built by ignoring some of the dependencies in the data [START_REF] Gaetan | Modélisation et statistique spatiale[END_REF][START_REF] Gourieroux | Pseudo maximum likelihood methods: Theory[END_REF]. The vector θMP L ∈ Θ is a maximum pseudo-likelihood estimate (MPLE) of θ:

Main notations

θMP L = argmax

Posterior conditional on the MLE

The full sample posterior p(θ | D) and the posterior conditional on the MLE p(θ | θML ) exactly coincide in specific cases (e.g. when the MLE are sufficient statistics), but do not coincide in general. Our aim, in this section, is to provide an asymptotically equivalent distribution for p(θ | θML ).

Bernstein -von Mises (BvM) theorems provide, for various statistical models, the asymptotic behavior of posteriors distributions. For example, following [START_REF] Walker | On the asymptotic behaviour of posterior distributions[END_REF] and Lindley (1965, p. 130), we consider a set D = (D 1 , . . . , D n ) of n i.i.d. variables drawn from a parametric distribution with density f (• | θ) with respect to a σ-finite measure on the real line, where θ is in Θ ⊂ R q . Under this setting and additional regularity conditions, the BvM theorem establishes the asymptotic normality of the full sample posterior (Walker, 1969, Theorem 2 and conclusion): the full sample posterior density of θ is, for large n, equivalent to the normal density with mean vector equal to the MLE θML and covariance matrix equal to Ω n ( θML ) -1 :

p(θ | D) ∼ n→∞ φ θML ,Ωn( θML ) -1 (θ),
where φ µ,Σ denotes the density of the normal distribution with mean vector µ and covariance matrix Σ, and Ω n (α) is the q × q matrix with element (i, j)

equal to -∂ 2 log p(D | θ)/∂θ i ∂θ j θ=α .
To provide an asymptotically equivalent distribution for p(θ | θML ) as in BvM theorems, we assume in Lemma 1 (see below) that the MLE is asymptotically normal and consistent. For example, consider the same statistical model than above and assume that assumptions made in Lehmann and Casella (1998, Theorem 5.1 of the MLE asymptotic normality, p. 463) are satisfied. In particular, assume that data were generated with parameter vector θ. Then, the density of θML is, for large n and given θ, equivalent to the normal density with mean vector equal to the true parameter vector θ and covariance matrix equal to n -1 I(θ) -1 :

p( θML | θ) ∼ n→∞ φ θ,n -1 I(θ) -1 ( θML ).
(

) 1
where I(θ) denotes the q × q Fisher information matrix.

Lemma 1 (Asymptotic normality of the posterior conditional on the MLE).

Consider the modeling setting described in Section 2 and suppose that the MLE satisfies Equation (1) with non-singular matrix I(θ). Assume in addition that the prior π is a positive and Lipschitz function over Θ, that θ → |I(θ)| (determinant of I(θ)) and θ → x I(θ)x (for all x ∈ R q , x being the transpose of x) are Lipschitz functions over Θ, and that the constant which arises in the Lipschitz condition for θ → x I(θ)x and which is a function of x, is also a Lipschitz function over R q . Then, when n → ∞, the posterior density p(θ | θML ) conditional on the MLE is asymptotically equivalent to the density of the normal distribution with mean vector θML and covariance matrix n -1 I( θML ) -1 over a subset B n of Θ whose measure with respect to this normal density is asymptotically one in probability:

p(θ | θML ) ∼ n→∞ φ θML ,n -1 I( θML ) -1 (θ), ∀θ ∈ B n lim n→∞ Bn φ θML ,n -1 I( θML ) -1 (θ)dθ = P 1.
Thus, over the subset B n which asymptotically contains all the mass of the normal density φ θML ,n -1 I( θML ) -1 (•), the posterior conditional on the MLE is asymptotically equivalent to this normal distribution.

From a frequentist point of view, the BvM theorem (which concerns the full sample posterior p(θ | D)) is a justification of the Bayesian approach for parameter estimation since the Bayesian confidence sets asymptotically coincide with the frequentist limit confidence sets [START_REF] Freedman | On the Bernstein-Von Mises theorem with infinitedimensional parameters[END_REF]. Lemma 1 shows a similar result for the posterior conditional on the MLE p(θ | θML ). Thus, Lemma 1 can also be viewed as a justification of the use of the posterior conditional on asymptotically normal MLE for parameter estimation. Note that results similar to the one provided by Lemma 1 have already been obtained for the estimation of an univariate location parameter;

see [START_REF] Doksum | Consistent and robust Bayes procedures for location based on partial information[END_REF] and references therein.

Regarding assumptions in Lemma 1, the asymptotic normality of the MLE (Equation ( 1)) requires classical but strong assumptions (even for the simple i.i.d. case). However, the asymptotic normality of the MLE has been obtained for numerous modeling and sampling settings, even in non-i.i.d.

cases. Lemma 1 is also based on a series of Lipschitz assumptions concerning the prior π and the Fisher information matrix I(θ). These assumptions are satisfied for classical distributions (e.g. when π is uniform on a bounded domain and when (D 1 , . . . , D n ) are independent normal variables with mean µ and variance 1).

Proof. Under the assumptions of the theorem of the MLE asymptotic normality (i.e. under Equation ( 1)

), p( θML | θ) = η n (θ) + ε n (θ), where η n (θ) = φ θ,n -1 I(θ) -1 ( θML ) and ε n (θ) = n→∞ o(η n (θ)). Therefore, p(θ | θML ) = n→∞ {η n (θ) + ε n (θ)}π(θ) Θ η n (α)π(α)dα + Θ ε n (α)π(α)dα = η n (θ)π(θ)(1 + o(1)) Θ η n π + Θ ε n π .
The densities π and η n being positive over Θ and the density η n converging to the Dirac distribution at the true parameter vector θ 0 when n → ∞, Θ η n π is positive and its limit, namely π(θ 0 ), is also positive. Besides, θ → p( θML | θ) being asymptotically equivalent to η n , it also converges to the Dirac distribution at the true parameter vector θ 0 when n → ∞,

and Θ p( θML | θ)π(θ)dθ → π(θ 0 ) > 0. Therefore, Θ ε n π = Θ p( θML | θ)π(θ)dθ -Θ η n π converges to 0, Θ ε n π = o( Θ η n π) and p(θ | θML ) = n→∞ π(θ) π(θ 0 ) η n (θ)(1 + o(1)). (2)
Let B n be the ball of center θ 0 and radius r n such that r n → 0 and

r n √ n → ∞ (r n converges to zero at a lower rate than 1/ √ n). Since π is Lipshitzian (i.e. ∃A 1 < ∞, ∀θ 1 , θ 2 ∈ Θ, |π(θ 1 ) -π(θ 2 )| ≤ A 1 ||θ 1 -θ 2 ||), π(θ) = n→∞ π(θ 0 )(1 + o(1)), ∀θ ∈ B n . (3)
Let us now derive an equivalent function for η n which can be written:

η n (θ) = φ θ,n -1 I(θ) -1 ( θML ) = √ n|I(θ)| 1/2 (2π) d/2 exp - n 2 ( θML -θ) I(θ)( θML -θ) .
Using the Lipschitz condition on θ → I(θ

) (i.e. ∃A 2 < ∞, ∀θ 1 , θ 2 ∈ Θ, | |I(θ 1 )|-|I(θ 2 )| |≤ A 2 ||θ 1 -θ 2 ||), one can state that for all θ ∈ B n , |I(θ)| 1/2 = n→∞ |I( θML )| 1/2 (1 + o(1)).
Besides, the two Lipschitz conditions concerning θ → x I(θ)x can be written as follows:

∀x ∈ R q , ∃A 3 (x) < ∞, ∀θ 1 , θ 2 ∈ Θ, |x I(θ 1 )x -x I(θ 2 )x| ≤ A 3 (x)||θ 1 -θ 2 || ∃A 4 < ∞, ∀x 1 , x 2 ∈ R q , |A 3 (x 1 ) -A 3 (x 2 )| ≤ A 4 ||x 1 -x 2 ||.
These conditions imply that over

B n , |( θML -θ) I(θ)( θML -θ) -( θML - θ) I( θML )( θML -θ)| is bounded from above by 2r n (A 3 (0) + 2r n A 4 ). There- fore, ( θML -θ) I(θ)( θML -θ) = n→∞ ( θML -θ) I( θML )( θML -θ)(1 + o(1))and η n (θ) = n→∞ φ θML ,n -1 I( θML ) -1 (θ)(1 + o(1)), ∀θ ∈ B n . (4)
Using Equations (2-4), we obtain the first equation of Lemma 1:

p(θ | θML ) = n→∞ φ θML ,n -1 I( θML ) -1 (θ)(1 + o(1)), ∀θ ∈ B n .
The

∀θ ∈ R n,p , || θML -θ|| ≤ λ min {I( θML )} -1 χ 2 d (p)/n. ( 5
)
From the MLE asymptotic normality,

lim n→∞ P ( θML -θ 0 ) I(θ 0 )( θML -θ 0 ) ≤ χ 2 d (p)/n = p.
Moreover, ( θML -θ 0 ) I(θ 0 )( θML -θ 0 ) ≥ λ min {I(θ 0 )}|| θML -θ 0 ||, where λ min {I(θ 0 )} is the minimum eigenvalue of I(θ 0 ). Therefore,

lim n→∞ P || θML -θ 0 || ≤ λ min {I(θ 0 )} -1 χ 2 d (p)/n ≥ p. (6)
Since ||θ -θ 0 || ≤ || θML -θ||+|| θML -θ 0 ||, one obtains using Eq. ( 5) and ( 6),

lim n→∞ P ∀θ ∈ R n,p , ||θ -θ 0 || ≤ λ min {I( θML )} -1 + λ min {I(θ 0 )} -1 χ 2 d (p)/n ≥ p.
Since r n goes to zero more slowly than 1/ √ n, the previous inequality yields:

lim n→∞ P (∀θ ∈ R n,p , ||θ -θ 0 || ≤ r n ) ≥ p lim n→∞ P (R n,p ⊂ B n ) ≥ p lim n→∞ P Bn φ θML ,n -1 I( θML ) -1 (θ)dθ ≥ Rn,p φ θML ,n -1 I( θML ) -1 (θ)dθ ≥ p lim n→∞ P Bn φ θML ,n -1 I( θML ) -1 (θ)dθ ≥ p ≥ p.
The last inequality obtained for any p ∈ (0, 1) implies that Bn φ θML ,n -1 I( θML ) -1 (θ)dθ converges to one in probability when n → ∞, i.e.

Posterior conditional on an MPLE

Here, we propose a lemma analogous to Lemma 1 but concerning an MPLE (maximum pseudo-likelihood estimate) instead of the MLE.

Lemma 2 (Asymptotic normality of the posterior conditional on an MPLE).

Consider the modeling setting provided in Section 2. Assume that, given the vector θ under which the data D were generated, the p.d.f. of the MPLE θMP L is equivalent to the normal density with mean vector θ and covariance matrix g(n) -1 J(θ) -1 :

p( θMP L | θ) ∼ n→∞ φ θ,g(n) -1 J(θ) -1 ( θMP L ),
where g is a positive increasing function such that g(n) → ∞ and J(θ) is a positive-definite matrix. Assume in addition that the prior π is a positive and Lipschitz function over Θ, that θ → |J(θ)| and θ → x J(θ)x (for all

x ∈ R q , x being the transpose of x) are Lipschitz functions over Θ, and that the constant which arises in the Lipschitz condition for θ → x J(θ)x and which is a function of x, is also a Lipschitz function over R q . Then, when n → ∞, the posterior density p(θ | θMP L ) conditional on the MPLE is asymptotically equivalent to the density of the normal distribution with mean vector θMP L and covariance matrix g(n) -1 J( θMP L ) -1 over a subset B n of Θ whose measure with respect to this normal density is asymptoticallyone:

p(θ | θMP L ) ∼ n→∞ φ θMP L ,g(n) -1 J( θMP L ) -1 (θ), ∀θ ∈ B n lim n→∞ Bn φ θMP L ,g(n) -1 J( θMP L ) -1 (θ)dθ = 1.
Lemma 2 justifies the use of the posterior conditional on the MPLE for parameter estimation because the Bayesian confidence sets that are provided by this posterior asymptotically coincide with the frequentist limit confidence sets obtained by maximizing the pseudo-likelihood.

The asymptotic normality of the MPLE required in Lemma 2 has been obtained for various models, especially random Markov fields and spatial point processes; see Gaetan and Guyon (2008, chap. 5), [START_REF] Gourieroux | Pseudo maximum likelihood methods: Theory[END_REF], Møller and Waagepetersen (2004, chap. 9) and references therein.

It has to be noted that information is lost when MPLE are used rather than MLE and, consequently, that estimation accuracy is decreased (e.g. this has been shown for simple Markovian models using asymptotic estimation variances (Gaetan and Guyon, 2008, chap. 5)).

Proof. Follow the proof of Lemma 1 by assuming that the radius r n of the ball

B n satisfies r n g(n) → ∞ instead of r n √ n → ∞.

Approximate posterior conditional on an MPLE

Here, we derive implications of Lemma 2 in the framework of approximate Bayesian computation (ABC) when (some of) the summary statistics are MPLE (see Theorem 1 and Corollary 1). We consider the (simple) ABCrejection algorithm based on independent simulations, on a set of summary statistics and on a tolerance threshold [START_REF] Pritchard | Population growth of human Y chromosomes: a study of Y chromosome microsatellites[END_REF]:

ABC-rejection. Perform the next 3 steps for i in {1, . . . , I}, independently:

• Generate θ i from π and simulate D i from M θ i ;

• Compute summary statistics S i = s(D i ), where s is a function from D to the space S of statistics; normal density goes to one in probability and that does not depend on :

• Accept θ i if d(S i , S) ≤ ,
p (θ | θMP L ) ∼ n→∞, →0 φ θMP L ,g(n) -1 J( θMP L ) -1 (θ), ∀θ ∈ B n lim n→∞ Bn φ θMP L ,g(n) -1 J( θMP L ) -1 (θ)dθ = P 1.
As explained in the introduction, this result leads us to conjecture that, for some stochastic implicit models, (i) the ABC-posterior distribution conditional on an MPLE is asymptotically normal and centered around the MPLE, and (ii) resulting point estimates and confidence sets converge to their frequentist analogues. Corollary 1 in Appendix A provides an analogous result when the MPLE is used in conjunction with supplementary statistics.

Proof. This result is directly obtained from Lemma 2 by simply noting that the subset B n does not depend on . Indeed, following the proof of Lemma 1, pointwise in θ we have:

p (θ | θMP L ) = →0 p(θ | θMP L )(1 + o (1)) = n→∞, →0 π(θ) π(θ 0 ) η n (θ)(1 + o n (1))(1 + o (1)).
Then, B n is used to provide an equivalent of π(θ)π(θ 0 ) -1 η n (θ). This term does not depend on and, consequently, B n has not to be dependent on .

Example

The simplified example presented here illustrates the application of ABC conditional on an MPLE and a supplementary statistic. The model M θ under consideration is the following bivariate normal distribution:

N ( µ µ ) , 1 ρ ρ 1
, parameterized by the mean µ and the correlation ρ; we set θ = (µ, ρ).

Observed data D = {(D (1) k , D (2)
k ) : k = 1, . . . , n} are n = 100 vectors drawn under this normal distribution with µ = 0 and ρ = 0.5. We use a uniform prior distribution π over the rectangular domain (-3, 3) × (-1, 1). The maximum likelihood estimates of µ and ρ are the empirical mean of (D

(1) k + D (2)
k )/2 and the empirical correlation of (D

(1) k , D (2) k ), k = 1, . . . , n.
Here, we applied ABC with the two following statistics:

S = s(D) =   μMP L S 0   = 1 n n i=1   D (1) k 1{sign(D (1) k ) = sign(D (2) k )}   ,
where μMP L is an MPLE of µ that uses only partial information contained in the sample (i.e. only the first component of sampled vectors), and S 0 is a supplementary statistic that gives the mean number of vectors in the sample whose components D

k and D

(2)

k have the same sign (1{•} is the indicator function).

To assess the convergence of ABC when tends to zero, we applied ABC with varying , with I = 10 5 simulations, and with the distance d(S i , S) = (μ M P L,i -μMP L ) 2 + (S 0,i -S 0 ) 2 , where S i = (μ M P L,i , S 0,i ) is the vector of statistics computed for the simulation i. As usual in ABC-rejection, instead of fixing , we fixed the sample size τ of the posterior sample (i.e. the number of accepted parameter vectors); note that decreases when τ decreases.

The sample size τ was fixed at values ranging from 10 to 5000. For each value of τ , we computed the local posterior probability (LPP) around the true parameter vector θ = (0, 0.5) as the proportion of accepted parameter vectors in the small rectangle [-0.015, 0.015] × [0.495, 0.505] whose center is θ = (0, 0.5) and whose sides are 200 times smaller than the sides of the parameter space (-3, 3) × (-1, 1). We expect that this LPP increases with the efficiency of the inference procedure. The LPP was computed for 50000 datasets and Figure 1 shows its mean and standard deviation when τ varies.

The mean LPP around the true parameters increases when the sample size τ (and ) tends to zero; meanwhile, the dispersion of the LPP increases. This is the signature of the classical bias-variance trade-off.

To automatically select the sample size τ , we applied the procedure proposed by [START_REF] Soubeyrand | Approximate Bayesian computation with functional statistics[END_REF] where the distance between summary statistics is also optimized. In this procedure, the distance is weighted: d(S i , S; w 1 , w 2 ) = w 1 (μ M P L,i -μMP L ) 2 + w 2 (S 0,i -S 0 ) 2 , and the triplet (τ, w 1 , w 2 ) is optimized under constraints using an integrated mean square error criterion. For one of the 50000 datasets simulated above, Figure 2 shows ABC-posterior samples obtained when d is not weighted and τ is

Discussion

We provided a frequentist justification for the use of posterior distributions conditional on MPLE, both in the classical Bayesian framework and in the ABC framework. The asymptotic results presented above were obtained for a large but limited class of models satisfying regularity assumptions. In real-life studies where ABC is applied, these assumptions cannot be checked and, consequently, our asymptotic results may not hold. Therefore, it is crucial (i) to combine MPLE and supplementary summary statistics to tend to a set of sufficient summary statistics [START_REF] Joyce | Approximately sufficient statistics and Bayesian computation[END_REF], and

(ii) to apply a method for selecting, weighting or transforming the summary statistics to avoid to take into account non-relevant statistics. In Section 6, we used the weighted distance between summary statistics proposed by [START_REF] Soubeyrand | Approximate Bayesian computation with functional statistics[END_REF] where the weights are optimized with respect to an integrated mean square error. Other approaches could be applied. For example, Barnes et al. (2012), [START_REF] Joyce | Approximately sufficient statistics and Bayesian computation[END_REF] and [START_REF] Nunes | On optimal selection of summary statistics for approximate Bayesian computation[END_REF] propose a dimension reduction (that can be viewed as a binary weighting), and [START_REF] Wegmann | Efficient approximate Bayesian computation coupled with Markov chain Monte Carlo without likelihood[END_REF] proposes a PLS transformation followed by a binary weighting of the PLS axes (see also [START_REF] Blum | A comparative review of dimension reduction methods in approximate Bayesian computation[END_REF][START_REF] Fearnhead | Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation[END_REF][START_REF] Jung | Choice of summary statistic weights in approximate Bayesian computation[END_REF].

The strong implication of our approach is that an analytic work has to be made: the dependence structure of the model has to be simplified to write a tractable pseudo-likelihood and, eventually, to find an analytic expression for the maximizer. This additional work is however expected to yield relevant summary statistics directly informing (a subset of) the parameters.

In this article, we did not precisely define what is a pseudo-likelihood.

This deliberate choice is justified in the sense that the MPLE could be replaced, in Theorem 1, by any other estimates with similar normal weak convergence. Thus, our approach could be generalized by replacing the pseudo-likelihood by a quasi-likelihood or a contrast. Another extension of Theorem 1 could be derived by conditioning ABC on the score function of the pseudo-likelihood instead of the MPLE as proposed by [START_REF] Gleim | Approximate Bayesian computation with indirect summary statistics[END_REF], [START_REF] Ruli | Approximate Bayesian computation with composite score functions[END_REF] and [START_REF] Mengersen | Bayesian computation via empirical likelihood[END_REF].

Under the first set of assumptions of Corollary 1, we can also write ). Moreover, the existence of T 0 is a strong assumption that reduces the applicability of the corollary (a trivial but certainly rare example where T 0 exists is obtained when one assumes that ( θMP L , S 0 ) is a normal random vector; Indeed, in this case T 0 can be defined as T 0 = S 0 -E(S 0 | θMP L )). Nevertheless, if ( θMP L , S 0 ) is asymptotically normal, we conjecture that a result similar to Corollary 1 can be obtained.

Proof. The first part of the corollary is straightforward:

  Observed data D ∈ D are assumed to be generated under the stochastic model M θ parametrized by θ ∈ Θ with prior density π. The data space D and the parameter space Θ ⊂ R q (q ∈ N * ) are both included in multidimensional sets of real vectors. The probability distribution functions (p.d.f.) of the model and the prior are defined with respect to the Lebesgue measure. Let p(D | θ) denote the likelihood of the model and p(θ | D) = p(D | θ)π(θ)/p(D) the full sample posterior of the parameter vector θ. The vector θML ∈ Θ is the maximum likelihood estimate (MLE) of θ: θML = argmax θ∈Θ p(D | θ). The posterior of parameters conditional on the MLE is p(θ | θML ) = p( θML | θ)π(θ)/p( θML ), where p( θML | θ) is the p.d.f. of the MLE given θ. Besides, we are interested in models whose likelihoods are not tractable because of the dependence structure in the data, but for which we can build tractable pseudo-likelihoods, say p(D | θ).

  θ∈Θ p(D | θ). The posterior of parameters conditional on the MPLE is p(θ | θMP L ) = p( θMP L | θ)π(θ)/p( θMP L ), where p( θMP L | θ) is the p.d.f. of the MPLE given θ.

  2nd equation of Lemma 1 is shown below. Let p ∈ (0, 1), and consider R n,p the region consisting of the vectors θ ∈ Θ satisfying n( θMLθ) I( θML )( θML -θ) ≤ χ 2 d (p), where χ 2 d (p) is the quantile of order p of the chi-square distribution of order d (i.e. the dimension of Θ). Using the link between the normal and chi-square distributions, R n,p satisfies: Rn,p φ θML ,n -1 I( θML ) -1 (θ)dθ = p. Moreover, from a property of the Rayleigh quotient, ( θML -θ) I( θML )( θML -θ) ≥ λ min {I( θML )}|| θML -θ||, where λ min {I( θML )} is the minimum eigenvalue of I( θML ). Therefore,

  lim n→∞ Bn φ θML ,n -1 I( θML ) -1 (θ)dθ = P 1.

  where d is a distance over S and is a tolerance threshold for the distance between the observed statistics S = s(D) and the simulated ones S i .The set of accepted parameters, say Θ ,I = {θ i : d(S i , S) ≤ , i = 1, . . . , I}, forms a sample from the following posterior:p (θ | S) = B d (S, ) h(s | θ)ds π(θ) Θ B d (S, ) h(s | α)ds π(α)dα, where the ball B d (S, ) in the d-dimensional space S is the set of points from which the distance to S is less than , and h(S | θ) is the conditional probability distribution function of S given θ. When tends to zero and d is appropriate, p (θ | S) is a good approximation of p(θ | S) under regularity assumptions (see Blum (2010) and Soubeyrand et al. (2013, Appendix A)): p (θ | S) and p(θ | S) are asymptotically equivalent. However, if S is not sufficient, then p(θ | S) = p(θ | D) and information is lost by using S instead of D. Theorem 1 (Asymptotic normality of the ABC-posterior conditional on an MPLE). Consider the ABC-rejection algorithm that samples in the posterior p (θ | θMP L ) of θ conditional on the vector of summary statistics S = θMP L . Assume that when → 0, p (θ | θMP L ) converges pointwise to p(θ | θMP L ). Then, under assumptions of Lemma 2, when n → ∞ and → 0, the posterior p (θ | θMP L ) is asymptotically equivalent to the density of the normal distribution with mean vector θMP L and covariance matrix g(n) -1 J( θMP L ) -1 over a subset B n of Θ whose measure with respect to this

Figure 1 :

 1 Figure 1: Mean (solid line), pointwise 95%-confidence envelopes (dashed lines) and standard deviation (dotted line) of the local posterior probability around the true parameter vector θ = (0, 0.5) as a function of sample size τ .

Figure 2 :

 2 Figure 2: ABC-posterior samples (dots) obtained when d is not weighted and τ is fixed at 5000 (top left), 1000 (top right) and 200 (bottom left), and when d is weighted and (τ, w1, w2) is optimized (bottom right). Dashed lines are intersecting at the true value (0, 0.5) of the parameter vector θ = (µ, ρ). The grey contour line gives the smallest 95%-posterior area obtained with the classical Bayesian computation.

  that p (θ | θMP L , S 0 ) = →0 p(θ | θMP L )(1 + o (1)), where p(θ | θMP L ) is the posterior density of θ conditional on the MPLE with the modified prior p(θ | T 0 ) = p(T 0 | θ)π(θ)/p(T 0

p

  (θ | θMP L , S 0 ) = →0 p(θ | θMP L , S 0 )(1 + o (1)) = p(θ | θMP L , T 0 )(1 + o (1)) = p( θMP L , T 0 | θ)π(θ) p( θMP L , T 0 ) (1 + o (1)) = p(T 0 | θMP L , θ)p( θMP L | θ)π(θ) p(T 0 )p( θMP L ) (1 + o (1)) = p(T 0 | θ)p( θMP L | θ)π(θ) p(T 0 )p( θMP L ) (1 + o (1)) = p(T 0 | θ)p(θ | θMP L ) p(T 0 ) (1 + o (1)).The term p(θ | θMP L ) can be viewed as a prior that is based on information contained in θMP L and that is independent from data T 0 used for the final inference. The second part of the corollary corresponds to Theorem 1. Barnes, C. P., Filippi, S., Stumpf, M. P. H., Thorne, T., 2012. Considerate approaches to constructing summary statistics for abc model selection. Statistics and Computing 6, 1181-1197. Bickel, P. J., Kleijn, B. J. K., 2012. The semiparametric Bernstein-von Mises theorem. The Annals of Statistics 40, 206-237.

Acknowledgements. We thank Denis Allard, Rachid Senoussi and the reviewers for their suggestions. This work was supported by the ANR grant EMILE.

Appendix A. Combining an MPLE and supplementary statistics Corollary 1. Consider the posterior p (θ | θMP L , S 0 ) of θ conditional on the vector of summary statistics S = ( θMP L , S 0 ) in which the ABC-rejection algorithm samples. Assume that when → 0, p (θ | θMP L ) converges pointwise to p(θ | θMP L ). Assume that there exists a vector T 0 independent of θMP L such that ( θMP L , T 0 ) brings the same information on θ than S = ( θMP L , S 0 ).

Then,

where p(θ | T 0 ) is the posterior density of θ conditional on the subset of statistics T 0 given the modified prior p(θ

If in addition assumptions of Lemma 2 hold, the modified prior p(θ | θMP L ) is asymptotically equivalent to the density of the normal distribution with mean vector θMP L and covariance matrix equal to g(n) -1 J( θMP L ) -1 over a subset B n of Θ whose measure with respect to this normal density goes to one in probability and that does not depend on .


Paul Bouteiller

Complete finite-strain isotropic thermo-elasticity

Keywords:

This paper deals with finite strain isotropic thermo-elasticity without any specific Ansatz regarding the Helmholtz free energy. On the theoretical side, an Eulerian setting of isotropic thermo-elasticity is developed, based on the objective left Cauchy-Green tensor along with the Cauchy stress. The construction of the elastic model relies on a particular invariants choice of the strain measure. These invariants are built so that a succession of elementary experiments, in which the invariants evolve independently, ensures the complete identification of the Helmholtz free energy and thus of the thermo-elastic constitutive law. Expressions idealizing these experimental tests are proposed. A wide range of hyperelastic models are found to be a special case of the model proposed herein.

Introduction

Elasticity is probably one of the most extensively discussed topics in the analytical mechanics literature over the last few centuries. Since Robert Hooke's linear relationship linking the current Cauchy stress σ to the linearized strain ε, many authors studied small strain but also non-linear elasticity, the latter beeing well suited to study materials, such as polymer, which undergo large deformations. The fundamental ground of this phenomenon, which can preceed more complexe non linear events, is now a very well assessed subject, both theoretically [START_REF] Biot | Thermoelasticity and irreversible thermodynamics[END_REF][START_REF] Coleman | The thermodynamics of elastic materials with heat conduction and viscosity[END_REF][START_REF] Ogden | Non-linear elastic deformations[END_REF] and experimentally.

However, specific Ansatz are still commonly used concerning the Helmholtz free energy from which the elastic constitutive law is derived. Classical small-strain elasticity, linking the second Piola-Kirchoff tensor π to the right Green-Lagrange strain tensor E relies on a quadratic expansion of the Helmholtz free energy with respect to the small strain E. As regards finite strain elasticity, where such series expansion are no longer valid, the construction is based on a choice of the Helmholtz free energy according to invariants of the deformation tensor. Two main categories are often distinguished, depending on the state variables Email address: paul.bouteiller@cea.fr (Paul Bouteiller) Preprint submitted to Elsevier May 30, 2023 chosen. On the one hand, Neo-Hookean [START_REF] Kim | A comparison among neo-hookean model, mooneyrivlin model, and ogden model for chloroprene rubber[END_REF], Mooney-Rivlin [START_REF] Kumar | Hyperelastic mooney-rivlin model: determination and physical interpretation of material constants[END_REF], or Yeoh [START_REF] Yeoh | Some forms of the strain energy function for rubber[END_REF] models postulate a polynomial expansion of the free energy with respect to the fundamental invariants of a strain tensor. The coefficients involved in the polynomial expansion are fitted to reproduce experimental data. On the other hand, stretched based models, such as Ogden's which develop the Helmholtz free energy as a power series, retain the positive eigenvalues of the right Cauchy-Green tensor C. Ogden models have been found to be very efficient for large strain elasticity of rubber like material [START_REF] Cassels | Nonlinear elasticity: theory and applications[END_REF]. Many articles have offered reviews of these different models and their respective pros and cons [START_REF] Chagnon | Hyperelastic energy densities for soft biological tissues: a review[END_REF][START_REF] Dal | A comparative study on hyperelastic constitutive models on rubber: State of the art after 2006[END_REF][START_REF] Khaniki | A review on the nonlinear dynamics of hyperelastic structures[END_REF][START_REF] Melly | A review on material models for isotropic hyperelasticity[END_REF].

The first objective of this paper is to present a choice of a triplet of kinematically significant invariants, allowing for the rigorous identification of the Helmholtz free energy, through a succession of simple elementary experiments and without a posteriori adjustment. These invariants are chosen so that they can evolve independently.

In Section 1, we will recall the main lines of construction of the finite strain isotropic thermo-elastic behavior. In Section 2, we will present the invariants of the left Cauchy-Green deformation tensor used for the construction of the elastic model and the resulting constitutive law. Then, we will study, Section 3, the successive elementary experiments which allow the identification of the Helmholtz free energy. Physically realistic expressions for the experimental data required to build the model will then be proposed in Section 4.

Reminder of finite strain elasticity

This section briefly presents the fundamental assumption of finite strain isotropic thermo-elasticity within an Eulerian setting. The formulation of the general theory is given in the thermodynamic framework by introducing the Helmholtz free energy as a function of a finite strain measure, and by exploring both the first and the second law of thermodynamics.

Framework

Let D m 0 ∈ R 3 be a material domain corresponding to the reference configuration of a continuous medium.

Particles labeled by x 0 ∈ D m 0 are transported in the current configuration to x t ∈ D m t by a one-to-one mapping φ t . Let us note F the deformation gradient of this mapping.

Let us consider two observers (reference frames) R and R. We denote Q 0 the orthogonal transformation linking these two reference frames at the initial time and Q t the orthogonal tensor linking these same two observers in the current configuration. A second-rank tensor, Y for the observer R, equal to Ỹ for the observer R, is called invariant (or "objective Lagrangian") [START_REF] Korobeynikov | Objective tensor rates and applications in formulation of hyperelastic relations[END_REF] if:

Ỹ = Q 0 • Y • Q ⊤ 0 Q 0 ∈ SO(3); ∀t. (1)
2 Similarly, we will call "objective" any second-rank tensor X satisfying the following equality:

X = Q t • X • Q ⊤ t Q t ∈ SO(3). (2)
The invariance or objectivity property of a tensor is intrinsic and comes either from its mathematical definition or from a physically motivated assumption. The first step in the construction of a constitutive law is the choice of a deformation measure, which cannot be simply F , among two large families:

• On the one hand, strain tensors based on the product C = F ⊤ • F or its related tensors of the form

Y = 1 n (U n -1) with U = √ C (pure stretch
). These deformation tensors are often called "right strain mesure", "Lagrangian" or "invariant" because they obey the relation [START_REF] Akella | Static eos of uranium to 100 gpa pressure[END_REF]. These strain measures are used to close the mechanical problem by relating the first or second Piola-Kirchoff stress tensor P , π (which are also postulate invariant) to the displacement field u.

• On the other hand, strain measure, called "left", "Eulerian" or "objective", and based on the left Cauchy-Green deformation B = F • F ⊤ (subsequently called "Finger tensor") are more rarely considered to build elasticity models. These deformation measurements effectively link the objective Cauchy stress σ to the displacement field.

Elastic behavior

The finite-strain isotropic elastic behavior is based on the following three assertions:

1. Temperature and a strain measure are the only state variables.

2. The stress measure is a state function.

3. In all evolutions, the intrinsic dissipation is null.

The first assumption 1) forbids the presence of additional internal variables which could be the witness of phenomena such as plasticity, damage etc... But also anisotropy through a structural tensor [START_REF] Boehler | On irreducible representations for isotropic scalar functions[END_REF][START_REF] Zheng | Tensors which characterize anisotropies[END_REF]. The second assumption 2 excludes viscous effects and a possible strain rate D (D denotes the symmetric part of the spatial velocity gradient) dependency of the Cauchy stress. The last assumption 3 ensures reversibility in the sense that the entropy production other than thermal is zero. To satisfy these three requirements, most treatises devote the so-called "Lagrangian" approaches and relate the second Piola-Kirchoff tensor to the right Green-Lagrange deformation tensor. Eulerian formulations of elasticity are rarely considered (see nevertheless [START_REF] Miehe | Entropic thermoelasticity at finite strains. aspects of the formulation and numerical implementation[END_REF][START_REF] Simo | Computational inelasticity[END_REF]), especially since the Cauchy stress cannot be written as the simple derivative of a potential with respect to a strain measure.

Eulerian finite strain isotropic elasticity

Here, we choose to catch the motion with the left Cauchy-Green objective strain tensor B which we relate to the Cauchy stress σ whose physical meaning is perfectly clear.

The isotropy of the material σ(T, B)

= Q t • σ • Q ⊤ t (T, Q t • B • Q ⊤ t )
and the Rivlin-Ericksen theorem [START_REF] Ogden | Non-linear elastic deformations[END_REF] implies that the Cauchy stress necessarily expands as a quadratic polynomial of the strain measure:

σ = K B 0 1 + K B 1 B + K B 2 B 2 . (3)
The coefficients K B i are scalar functions of the temperature, but also of a triplet of B invariants. Their expressions, as a function of the free energy, can be deduced from the nullity of the intrinsic dissipation (see [START_REF] Truesdell | The non-linear field theories of mechanics[END_REF] p.295) written on the current (spatial) configuration:

Φ = -ρ ψm + Ṫ s m + σ : D = 0. (4)
For an isotropic material, the Helmholtz massic free energy is an isotropic function [START_REF] Boehler | On irreducible representations for isotropic scalar functions[END_REF][START_REF] Smith | On isotropic functions of symmetric tensors, skew-symmetric tensors and vectors[END_REF] 3) so that there exists f B ψ such that:

ψ m (T, B) = ψ m (T, Q t • B • Q ⊤ t ), ∀Q t ∈ SO(
ψ m (T, B) = f B ψ (T, B I , B II , B III ), (5)
where the fundamental invariants (B I , B II , B III ) denote the coefficients of the characteristic polynomial of B (main invariant). Note that any other triplet of invariants bijectively related to (B I , B II , B III ), is legitimate.

Thus, the "principal" invariants defined by B i = 1 i TrB i are often used to construct elastic laws as their partial derivatives with respect to the strain measure are straightforward. In the following, we will retain invariants whose kinematic meaning is clear.

According to the fundamental invariants (B I , B II , B III ), the isotropic finite strain thermo-elastic law reads, (see Appendix A but also [START_REF] Garrigues | Comportement élastique[END_REF][START_REF] Simo | Computational inelasticity[END_REF]):

σ = 2ρ 0 √ B III B III ∂ BIII f B ψ 1 + (∂ BI f B ψ + B I ∂ BII f B ψ )B -∂ BII f B ψ B 2 . (6)

Elastic model based on volumetric split

Most materials exhibit completely different volumetric and isochoric responses so that we make use of the isochoric-volumetric commutative split of the gradient F [START_REF] Flory | Thermodynamic relations for high elastic materials[END_REF][START_REF] Simo | Variational and projection methods for the volume constraint in finite deformation elasto-plasticity[END_REF]:

F = J 1 3 1 • J -1 3 F = J 1 3 1 • F , (7)
⇒ B = (J 2 3 1) • (J -2 3 B) = B sph • B. ( 8
)
B sph is a spherical tensor whose only invariant is the volume expansion J:

J = det F = √ det B = B III . (9)
The isochoric Finger tensor B satisfies det B = 1 and is therefore represented by two invariants. We note, without any ambiguity, B I and B II the two fundamental invariants of the isochoric Finger tensor.

These invariants are trivially related to those of B by:

B I = J -2 3 B I = B -1 3 III B I and B II = J -4 3 B II = B -2 3 III B II . (10)
One can find a function f B ψ of the invariants (T, B I , B II , B III ) coinciding with the Helmholtz massic free energy:

ψ m (T, B) = f B ψ (T, B I , B II , B III ) = f B ψ (T, B I , B II , B III ). (11)
The partial derivatives of the new function f B ψ are then obtained by simple algebraic calculations. The elastic law ( 6) is then transformed to give (see Appendix B):

σ = 2ρ 0 √ B III B III ∂ BIII f B ψ 1 + ∂ BI f B ψ DevB -∂ BII f B ψ Dev B 2 -Tr B B , σ = 2ρ 0 √ B III B III ∂ BIII f B ψ 1 + ∂ BI f B ψ B 1 3 III DevB - ∂ BII f B ψ B 2 3 III Dev B 2 -Tr (B) B . (12)
DevX = X -1 3 TrX 1 refers to the deviatoric (trace-less) part of a second-order tensor. [START_REF] Doll | On the development of volumetric strain energy functions[END_REF] emphasizes the deep link between the isochoric strain and the deviatoric part of the stress.

Construction of the thermo-elastic model

The invariants used so far are natural, but inefficients for the identification of the Helmholtz free energy, as they evolve simultaneously in classical experimental tests. The corner-stone of the subsequent section is the choice of independent invariants associated with simple elementary experiments, allowing them to evolve one after the other.

State variables

First invariant of the isochoric Finger tensor. We introduce a first invariant γ s of the isochoric Finger tensor B, defined by:

γ s = B I -3 = B I B 1 3 III -3, (13)
γ s is bijectively related to the maximum distortion δ max (see [START_REF] Garrigues | Cinématique des milieux continus[END_REF]):

γ s = √ 3 (δ max ) 2 3 -1 with δ max = max Q∈SO(3) 1 |det (Q • u t , Q • u ′ t , Q • u ′′ t )| , (14)
where (u t , u ′ t , u ′′ t ) denotes an initially orthogonal material direction triplet (at t = 0, det (u 0 , u ′ 0 , u ′′ 0 ) = 1).

The maximum distortion δ max reflects the maximum decrease of the solid angle formed by three initially orthogonal material directions. Moreover, in an isochoric simple shear test, defined by the transformation field (see Figure 1):

x t = φ(x 0 , t) = J 1 3 (x 0 + γY e x ), (15)
simple algebra yield γ s = γ, so that γ s equalized the motion magnitude.

Set γ ⊥ = 0 and e = 1 in (D.3) shows that the two fundamentals invariants of ( 16) stay equal B II = B I .

We define the invariant γ t , identically zero in any spherical evolution and in a simple shear test, by:

γ t = B I -3 -B II -3 = γ s - B II J 4 3 -3, (17)
γ t is always well defined (i.e. the terms under the roots are always positive) by concavity of the logarithm.

A closed form of its variations' domain is given in Appendix C.

Constitutive law with the new invariants set

The list of reduced state variables retained is thus (T, J, γ s , γ t ) (9), ( 13), [START_REF] Garrigues | Comportements inélastiques[END_REF]. The relations are inversed in:

J = B 1 2 III ; γ s = B I -3; γ t = γ s -B II -3 ⇔ B III = J 2 ; B I = γ 2 s + 3; B II = (γ s -γ t ) 2 + 3; (18)
The generic elastic law [START_REF] Doll | On the development of volumetric strain energy functions[END_REF], expressed in terms of the principal invariants (T, B I , B II , B III ) is transformed by applying the chain rule:

σ = ρ 0 ∂ J fψ 1 + 1 γ s J 5 3 ∂ γs fψ + ∂ γt fψ DevB + ∂ γt fψ (γ s -γ t )J 7 3 Dev B 2 -Tr(B)B . (19)
This is the most general isotropic thermo-elastic law, formulated with respect to the variables (T, J, γ s , γ t ).

Postulating that the free energy does not depend on the γ t invariant, i.e. ∂ γt fψ = 0, reduces [START_REF] Khaniki | A review on the nonlinear dynamics of hyperelastic structures[END_REF] to the elastic law used by [START_REF] Miehe | Entropic thermoelasticity at finite strains. aspects of the formulation and numerical implementation[END_REF][START_REF] Pamin | Gradient-enhanced large strain thermoplasticity with automatic linearization and localization simulations[END_REF][START_REF] Simo | Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation[END_REF][START_REF] Simo | Computational inelasticity[END_REF]. It should be noted, however, that in these papers, the expression of the free energy is presented as a hypothesis and is not deduced from experimentally assessed stresses.

These models, being limited to the triplet (T, J, γ s ), choose not to distinguish energetically two different deformations which would share the same maximum distortion δ max and thus the same invariant γ s .

Experimental identification

The thermodynamic construction presented herein follows the generic method presented by Guarrigues [START_REF] Garrigues | Comportement élastique[END_REF][START_REF] Garrigues | Comportements inélastiques[END_REF].

Let fψ denote the free energy variation within the non-isothermal thermo-elastic process from a reference configuration, E 0 = (T 0 , 1, 0, 0) for which the free energy is set to zero, to an arbitrary configuration given by the state variables (T, J, γ s , γ t ):

ψ m = fψ (T, J, γ s , γ t ). (20)
To define the Helmholtz massic free energy at any point (T, J, γ s , γ t ) of the state space, we define a particular path E 0 = (T 0 , 1, 0, 0)

P (1)
-→ E 1 = (T, 1, 0, 0)

P (2)
-→ E 2 = (T, J, 0, 0)

P (3)
-→ E 3 = (T, J, γ s , 0)

P (4) -→ E t = (T, J, γ s , γ t )
such that fψ is additively found, given the energy variation in every path:

ψ m = g (1) (T ) + g (2) (T, J) + g (3) (T, J, γ s ) + g (4) (T, J, γ s , γ t ). (21)
g (1) (T ) is the Helmholtz massic free energy variation within P (1) (strain-locked heating):

Ṫ ̸ = 0; J = 1; J = 0; γ s = 0; γs = 0; γ t = 0; γt = 0; (22)
g (2) (T, J) is the Helmholtz massic free energy variation within P (2) (isothermal spherical motion):

Ṫ = 0; J ̸ = 0; γ s = 0; γs = 0; γ t = 0; γt = 0; (23)
g (3) (T, J, γ s ) is the Helmholtz massic free energy variation within P (3) (isochoric simple shear test):

Ṫ = 0; J = 0; γs ̸ = 0; γ t = 0; γt = 0; (24)
g (4) (T, J, γ s , γ t ) is the Helmholtz massic free energy variation within P (4) (isothermal isochoric isotrace motion):

Ṫ = 0; J = 0; γs = 0; γt ̸ = 0; (25)
with the initial conditions:

g (1) (T 0 ) = 0; g (2) (T, 1) = 0, ∀T ; g (3) (T, J, 0) = 0, ∀(T, J); g (4) (T, J, γ s , 0) = 0, ∀(T, J, γ s );

which leads to cancel the following partial derivatives:

∂ T g (2) (T, 1) = 0; ∂ T g (3) (T, J, 0) = ∂ J g (3) (T, J, 0) = 0 ∀ T, J; ∂ T g (4) (T, J, γ s , 0) = ∂ J g (4) (T, J, γ s , 0) = ∂ γs g (4) (T, J, γ s , 0) = 0 ∀ T, J, γ s . (27)
The general form of the massic entropy is given by the Helmholtz relation:

s m = -∂ T fψ = -∂ T g (1) -∂ T g (2) -∂ T g (3) -∂ T g (4) . ( 28
)
The massic internal energy is deduced from the definition of the Helmholtz free energy: 4) ).

e m = fψ + T s m = g (1) + g (2) + g (3) + g (4) -T (∂ T g (1) + ∂ T g (2) + ∂ T g (3) ∂ T g (
(29)

Elementary evolutions

The identification of the four unknown functions g (1) (T ), g (2) (T, J), g (3) (T, J, γ s ) and g (4) (T, J, γ s , γ t ) comes down to experimental measures in a few experiments carried under the following ideal experimental conditions:

1) body forces are negligible;

2) state variables (T, J, γ s , γ t ) are uniform across the specimen.

3) experimental data are evaluated in a quasi-static setting so that the kinetic energy can be neglected;

g (1) (T ) will be determined by a simple energy balance whereas direct integration of experimental stress data shall clarify the rest of the Helmholtz free energy.

Strain-locked pure heating

The first evolution P (1) is a pure heating, strain locked, experiment: B = 1, J = 1, J = 0, γ s = 0, γs = 0, γ t = 0, γt = 0. We measure the algebraic massic heat q m (1) exp (T ) supplied to the system to go from T 0 to T . The conservation of energy between the states E 0 and E 1 is written (quasi-static evolution: ∆e m c = 0, no deformation: w m ext = 0):

e m 1 -e m 0 0 = q m (1) exp (T ), (30)
which leads to the differential equation in g (1) from (29):

g (1) (T ) -T (∂ T g (1) (T ) + ∂ T g (2) (T, 1) + ∂ T g (3) (T, 1, 0)) + ∂ T g (4) (T, 1, 0, 0)) = q m (1) exp (T ), ⇔ g (1) (T ) -T ∂ T g (1) (T ) = q m (1) exp (T ) see (27), ⇔ g (1) (T ) = -T T T0 q m (1) exp ( T ) T 2 d T . (31)
This experimental measurement is very difficult to perform, especially when the samples tend to contract (usually during cooling). We will see later, Section 3.1.5, that this ideal experiment can be replaced by a free thermal expansion followed by an isothermal spherical motion.

Spherical motion

In the second path P (2) , the deformation is purely spherical (B = J 2 3 1, γ s = 0, γs = 0, γ t = 0, γt = 0)

and isothermal ( Ṫ = 0). The elastic law [START_REF] Khaniki | A review on the nonlinear dynamics of hyperelastic structures[END_REF] proves that stress tensor must also be spherical:

σ (2) = ρ 0 ∂ J fψ 1, (32)
and is thus completely characterized by its mean stress.

In this experiment, we measure the pressure p

exp (T, J) which is the opposite of the uniform normal stress applied on the boundary if the hypothesis 2) p.8 holds:

p (2) exp (T, J) = - Trσ (2) 3 = -ρ 0 ∂ J g (2) (T, J) see (27). ( 33
)
The second contribution to the Helmholtz massic free energy variation then follows naturally.

g (2) (T, J) = - 1 ρ 0 J 1 p (2) exp (T, J)d J. (34)
In the shock-wave mechanics community, the determination of this pressure with respect to the volumetric change (or equivalently to the mass density ρ) and the temperature, is often referred to as the "equation of state (EOS)" [START_REF] Akella | Static eos of uranium to 100 gpa pressure[END_REF].

Isochoric iso-T planar simple shear test

To perform an isochoric isothermal planar simple shear [START_REF] Boni | Application of the plane simple shear test for determination of the plastic behaviour of solid polymers at large strains[END_REF] in the (e x , e y ) plane with an initial volume expansion J, we impose the following position field on the particles:

x t = J 1 3 (x 0 + γY e x ), (35)
where x 0 = Xe x + Y e y + Ze z is the reference position of a particle. The Finger tensor B = F • F ⊤ in the orthonormal basis (e x , e y , e z ) for this isochoric motion have already been given [START_REF] Garrigues | Comportement élastique[END_REF]. The strain tensor is uniform in the specimen ∂ x0 B = 0 so that γ s is uniform as well and satisfies:

γ s = B I -3 = γ 2 = ±γ.
Stress tensor in the P (3) evolution. The general thermo-elastic law reads:

σ = ρ 0 ∂ J fψ 1 + 1 γ s J 5 3 ∂ γs fψ + ∂ γt fψ DevB + ∂ γt fψ (γ s -γ t )J 7 3 Dev B 2 -Tr(B)B . ( 36
)
Given the equalities γ t = 0, ∂ γs g (4) (T, J, γ s , 0) = 0, simple calculations show that the tangential stress

τ (3) exp (T, J, γ s ) = σ (3)
12 within this evolution is directly related to ∂ γs g (3) by:

τ (3) exp = ρ 0 J ∂ γs g (3) (T, J, γ s ). ( 37
)
integration the value of the function g (3) :

g (3) = J ρ 0 γs 0 τ (3) exp (T, J, γ)dγ. (38)
3.1.4. Iso-γ s iso-T isochoric double shear test P (4) is an isothermal, isochoric, iso-γ s experiment. Several tests grants this triple condition. For the sake of simplicity, we introduce herein the "double shear test", eventhough this experiment does not scan the whole domain of variation of γ t . In the Appendix D, we will present a wider two-parameter test family, which include the present "double shear experiment", and which browse the whole domain for the variable

γ t .
Isochoric double shear test. Consider the motion defined by the following transformation (see also Figure 2):

x t = J 1 3 (x 0 + γ 2 s -γ 2 ⊥ Y e x + γ ⊥ Ze y ), (39)
where x 0 = Xe x + Y e y + Ze z is the reference position of a particle, J is the volume dilatation prior from path P (2) and constant in path P (3) and P (4) , γ s is the slip prior from path P (3) and constant in path P (4) . The components of the Finger tensor B = F • F ⊤ in the orthonormal basis {e x , e y , e z } for this isochoric motion are:

F = J 1 3      1 γ 2 s -γ 2 ⊥ 0 0 1 γ ⊥ 0 0 1      ⇒ B =      1 + γ 2 s -γ 2 ⊥ γ 2 s -γ 2 ⊥ 0 γ 2 s -γ 2 ⊥ 1 + γ 2 ⊥ γ ⊥ 0 γ ⊥ 1      ⇒ B I = γ 2 s + 3. ( 40
)
This test continuously shifts the deformation from a simple shear test in the (e x , e y ) plane to the same state in the (e y , e z ) plane while keeping the first invariant γ s constant.

Stress tensor in the double shear test. Injecting the strain measure deduced from (39) in [START_REF] Yeoh | Some forms of the strain energy function for rubber[END_REF], one can see that the out of plane shear stress σ

13 is directly related to the derivative of the Helmholtz massic free energy with 10 respect to γ t . Indeed, B 2 [START_REF] Flory | Thermodynamic relations for high elastic materials[END_REF] given by (D.2) and [START_REF] Yeoh | Some forms of the strain energy function for rubber[END_REF] proves that the tangential stress τ

(4) exp (T, J, γ s , γ t ) = σ (4) 13
in this evolution equalized:

τ (4) exp (T, J, γ s , γ ⊥ ) = ρ 0 J(γ s -γ t ) ∂ γt g (4) (T, J, γ s , γ t )γ ⊥ γ 2 s -γ 2 ⊥ with γ t = γ s -γ 2 s + γ 2 ⊥ (γ 2 s -γ 2 ⊥ ) see (D.9), τ (4) exp (T, J, γ s , γ t ) = ρ 0 J(γ s -γ t ) ∂ γt g (4) (T, J, γ s , γ t ) (γ s -γ t ) 2 -γ 2 s . (41)
The expression for the massic free energy g (4) is obtained once more by integrating the experimental data:

g (4) (T, J, γ s , γ t ) = J ρ 0 γt 0 γ s -τ (γ s -τ ) 2 -γ 2 s τ (4) exp (T, J, γ s , τ )dτ. ( 42
)
When γ ⊥ evolves in the interval 0,

γ s √ 2 , the invariant γ t takes its values in γ t ∈ γ s 1 -1 + γ 2 s 4 , 0
(see Appendix D).
Note that this out-of-plane shear-stress σ (4)

13 is purely "non-linear" in the sense that it arises from the quadratic term of the strain measure, therefore, it cannot be predicted by any "linearized" versions of the elastic law or by any model neglecting the contribution of the second strain invariant B II .

The "double shear experiment" is not a standard laboratory test but can be performed using an efficient hexapod machine (Stewart machine [START_REF] Dalemat | Une experimentation reussie pour l'identification de la reponse mecanique sans loi de comportement: Approche data-driven appliquee aux membranes elastomeres[END_REF][START_REF] Stewart | A platform with six degrees of freedom[END_REF]).

3.1.5. Free thermal expansion (P (5) )

A direct measurement of the massic heat q m (1) exp in the first path P (1) (strain-locked heating) is rather difficult. This experiment can be replaced by a free-stress thermal expansion (p

exp (T, J

exp (T )) = 0, ∀T ), followed by an isothermal spherical deformation which brings the volume expansion J

exp (T ) back to 1.

Writing the conservation of energy for this evolution between the initial state (T 0 , 1, 0, 0) and the freely expanded state (T, J

exp , 0, 0), it follows: e m 5 = q m (5) exp (T ). Moreover, the massic internal energy in the freely expanded state (T, J

exp , 0, 0) is also given by ( 29):

e m 5 =q m (1) exp (T ) - 1 ρ 0 J (5) exp 1 p (2) exp (T, J) -T ∂ T p (2) exp (T, J)d J, (43)
⇒ q m (1) exp (T ) = q m (5) exp (T ) + 1 ρ 0 J (5) exp 1 p (2) exp (T, J) -T ∂ T p (2) exp (T, J)d J. (44)
(44) proves that given the experimental pressure p 2 exp (T, J) and the free-stress expansion massic heat q m (5) exp (T ), one may determine q m (1)

exp (T ). The function [START_REF] Simo | Computational inelasticity[END_REF] is then finally written:

g (1) (T ) = -T T T0 q m (5) exp ( T ) T 2 d T - T ρ 0 T T0 J (5) exp ( T ) 1 p (2) exp ( T , J) -T ∂ T p (2) exp ( T , J))d J d T . (45)

Synthesis

The complete identification of the finite strain thermo-elastic model relies on four experimental data:

1) q m (1)
exp (T ): massic heat exchanged in the first path P (1) (strain-locked heating or cooling), with the condition:

q m (1) exp (T 0 ) = 0. (46) 2) p (2)
exp (T, J): pressure in the spherical evolution P (2) (isothermal spherical deformation), with the condition:

p (2) exp (T, 1) = p (1) exp (T ), ∀T. (47)
3) τ

exp (T, J, γ s ): in-plane shear stress in the isochoric planar shear test P (3) , along with the condition:

τ (3) exp (T, J, 0) = 0, ∀T ∀J. (48) 4) τ (4)
exp (T, J, γ s , γ t ): out-of-plane tangential stress in the fourth path P (4) , with the condition:

τ (4) exp (T, J, γ s , 0) = 0, ∀T ∀J ∀γ s . (49)
g (1) , g (2) , g (3) , g (4) which determine the state functions are expressed in terms of these measurements:

g (1) (T ) = -T T T0 q m (1) exp ( T ) T 2 d T or (45), (50)
g (2) (T, J) = - 1 ρ 0 J 1 p (2) exp (T, J)d J, (51)
g (3) (T, J, γ s ) = J ρ 0 γs 0 τ (3) exp (T, J, γ)dγ, (52)
g (4) (T, J, γ s , γ t ) = J ρ 0 γt 0 γ s -τ (γ s -τ ) 2 -γ 2 s τ (4) exp (T, J, γ s , τ )dτ. (53)
The internal massic energy is assumed to be a state function, as to ensure (29) takes a simpler expression:

e m = q m (1) exp + g (2)
-T ∂ T g (2) + g (3) -T ∂ T g (3) + g (4) -T ∂ T g (4) .

For the numerical assessment of the model, it is not necessary to compute explicitly g (1) because this function does not appear neither in the constitutive law, nor in the internal energy.

Volumetric/isochoric splitting assumption

As already mentioned, hyperelastic materials exhibit radically different volume and shear behavior. This explains a well-accepted splitting of the Helmholtz free energy with respect to the isochoric/volumetric part of the strain tensor (see [START_REF] Miehe | Entropic thermoelasticity at finite strains. aspects of the formulation and numerical implementation[END_REF][START_REF] Sansour | On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy[END_REF]). In this section, we emphasize that this partition comes down to a very natural assertion: the pressure remains unchanged in isochoric evolutions. With the previous results, the pressure in any state (T, J, γ s , γ t ) is:

- Trσ 3 = -ρ 0 ∂ J g (2) (T, J) + g (3) (T, J, γ s ) + g (4) (T, J, γ s , γ t ) = p (2) exp (T, J) + γs 0 τ (3) exp (T, J, γ s ) -J∂ J τ (3) exp (T, J, γ s )dγ (55)
- γt 0 γ s -τ (γ s -τ ) 2 -γ 2 s τ (4) exp + J∂ J τ (4) exp (T, J, γ s , τ )dτ.
The pressure in any state is a function of the tangential stresses τ

(3) exp , τ (4)
exp measured during the paths P (3) , P (4) . If one performs the P (3) experiment from the initial state (p

exp (T 0 , 1) = 0, no prior volume expansion), the Cauchy stress tensor generated is not deviatoric.

Hypothesis Optional Simplification. In isochoric deformation, the pressure does not vary. Both conditions ∂ γt Trσ = 0, ∂ γs Trσ = 0 and (55) lead to the differential equations:

τ (4) exp + J∂ J τ (4) exp = 0 ⇒ τ (4) exp (T, J, γ s , γ t ) = τ (4) exp (T, γ s , γ t ) J , τ (3) exp + J∂ J τ (3) exp = 0 ⇒ τ (3) exp (T, J, γ s ) = τ (3) exp (T, γ s ) J . (56)
Inserting these expressions into the general formula ( 52)-( 53), (56) obviously split the Helmholtz free energy into an isochoric/volumetric contributions:

ψ m (T, J, γ s , γ t ) = g (1) (T ) + g (2) (T, J) + g (3) (T, γ s ) + g (4) (T, γ s , γ t ).

(

) 57
This assumption is certainly valid in a wide range of solicitation (e.g is consistent with the classical linearized elasticity), but must be somehow validated experimentally. Note that the complete model ( 50)-(53) can take into account the volumetric/deviatoric coupling which is relevant for high velocity impact [START_REF] Scheidler | On the coupling of pressure and deviatoric stress in hyperelastic materials[END_REF]. Indeed, extreme strain-rate experiments can induced finite elastic deformations before plasticity occurs, so that pressure-dependent shear modulus are requiered (see for instance the SCG model [START_REF] Banerjee | An evaluation of plastic flow stress models for the simulation of high-temperature and high-strain-rate deformation of metals[END_REF][START_REF] Steinberg | A constitutive model for metals applicable at high-strain rate[END_REF]).

Time derivative of the massic internal energy

The massic internal energy takes a rather elegant form:

e m = q m (5) exp (T ) - 1 ρ 0 J J (5) exp (T ) (p (2) exp -T ∂ T p (2) exp )(T, J)d J + 1 ρ 0 γs 0 τ (3) exp -T ∂ T τ (3) exp (T, γs )dγ s + 1 ρ 0 γt 0 γ s -τ (γ s -τ ) 2 -γ 2 s τ (4) exp -T ∂ T τ (4) exp (T, γ s , τ )dτ. ( 58
)
The time derivative of the internal energy, essential for the numerical implementation, is obtained by evaluating the partial derivatives of the massic internal energy with respect to its four variables (see Appendix E for details):

de m dt = ∂e m ∂T Ṫ + ∂e m ∂J J + ∂e m ∂γ s γs + ∂e m ∂γ t γt . ( 59
)

Idealizations of elementary paths

The expressions (50)-(53) yields the "exact" thermo-elastic behavior of an isotropic material and lies upon four experimental results. In this section, we will propose physically consistent expressions for the experimental stresses. We will see that some rather simple forms reduce our model to a few hyperelastic models classically found in the literature.

Pressure assumption

Hypothesis Experimental pressure p

(2)

exp . We assume the following form for the pressure p

(2)
exp in any isothermal spherical deformation:

p (2) exp (T, J) = p (1) exp (T ) -κ(T ) ln J, ( 60
)
where p

(1) exp (T ) is the pressure from the path P (1) κ(T ) is a temperature-dependent bulk modulus.

The pressure (60) is reasonable (note for instance lim

J→∞ p (2) exp = -∞, lim J→0 p (2) exp = ∞)
, but predicts a finite strain energy under infinite compression J → 0. For further details about the volumetric part of the Helmholtz free energy from which the pressure is derived, we refer to [START_REF] Doll | On the development of volumetric strain energy functions[END_REF] and references therein. A more reasonable approximation of the experimental stress would be:

p (2) exp = p (1) exp (T ) - κ(T ) 2 J - 1 J , (61)
⇒ g (2) (T, J) = -(J -1)p

(1)

exp (T ) ρ 0 + κ(T ) 2 1 2 J 2 -1 -ln J . ( 62
)
It's worth noticing that (62) coincides with that proposed by [START_REF] Simo | Computational inelasticity[END_REF] which is a variant of [START_REF] Ciarlet | Mathematical elasticity: Three-dimensional elasticity[END_REF].

For the sake of calculus simplicity, we will however keep the pressure given by (60) for the following, noting that the differences between (60) and (61) are very small for any volumic expansion up to 200%.

Both model comes down to the so called linearized elasticity when J ≈ 1 as ln J ≈

1 2 J - 1 J ≈ Trε.
g (2) is obtained using both (60) and (51):

g (2) (T, J) = - (J -1)p (1) exp (T ) ρ 0 + κ(T ) ρ 0 (J ln(J) + 1 -J) . ( 63
)
If we replace the first path P (1) by a free expansion P (5) (in which one save q m (5) exp (T ) and J

(5) exp (T ))

followed by an isothermal spherical compression to bring J back to 1, p

exp (T ) is obtained by evaluating (60) in (T, J

exp ). Remembering that the stress is zero in a freely expanded test, it comes:

p (1) exp (T ) = κ(T ) ln J (5) exp (T ) . (64)
Therefore, by reinjecting this expression (64) into the general formula (60), we deduce the expression of

p (2)
exp :

p (2) exp (T, J) = -κ(T ) ln J J (5) exp (T ) . ( 65
)
Hypothesis Free-stress volume expansion J

(5) exp (T ). One can set arbitrary for J

(5) exp (T ):

J (5) exp (T ) = 1 + β(T -T 0 ), (66)
where β refers to a free-stress expansion coefficient.

Inserting both the experimental pressure (64) and the free-stress volume expansion (66) in (63), g (2) (T, J) now yields:

g (2) (T, J) = κ(T ) ρ 0 J ln J 1 + β(T -T 0 ) + 1 -J + ln (1 + β(T -T 0 )) . ( 67
)

Massic heat q m (1) exp

As a reminder, the first path P (1) can be substituted by a free thermal expansion followed by isothermal compression. We then obtain:

q m (1) exp (T ) = q m (5) exp (T ) + 1 ρ 0 J (5) exp 1 p (2) exp (T, J) -T ∂ T p (2) exp (T, J)d J [see (44)]. ( 68
)
The conservation of energy in a free expansion is then written according to (65):

q m (5) exp (T ) = q m (1) exp (T ) + 1 ρ 0 T κ(T ) ∂ T J
(5) exp

J

(5) exp J (5) exp -1 + (κ(T ) -T κ ′ (T )) 1 + ln(J (5) exp ) -J (5) exp .

We can choose a simple function of q m (5)

exp (T ) or q m (1)
exp (T ) the other being determined by (69).

It is noteworthy that the idealizations of experimental curves presented in this section are only arbitrary examples that have no theoretical justification.

Hypothesis Massic heat q m (5)

exp (T ).

A physically reasonable approximation to q m (5) exp (T ) may be, for example:

q m (5) exp (T ) = C p (T -T 0 ), (70)
where C p refers to a specific heat capacity in free expansion.

Calculation of g (3)

Hypothesis In-plane shear stress τ (3) exp . We assume that a temperature dependent simple shear modulus µ(T ) linearly links the shear stress with respect to the shear parameter γ:

τ (3) exp (T, J, γ) = µ(T )γ. (71)
The shear stress (71) leads to a quadratic massic free energy with respect to γ s that is also independent of the volume expansion J:

g (3) (T, γ s ) = µ(T )γ 2 s 2ρ 0 . (72)
This Helmholtz massic free energy has for example been postulated in [START_REF] Simo | Associative coupled thermoplasticity at finite strains: Formulation, numerical analysis and implementation[END_REF]. We now choose to enhance the model considering the contribution of the second invariant B II through γ t .

4.4. Calculation of g (4) The out-of-plane shear stress is a purely non-linear effect and its interpretation in terms of a "classical" shear or compressibility coefficient is therefore impossible.

Hypothesis Out-of-plane shear stress τ (4) exp . We postulate that the shear stress τ (4) exp in the fourth path 4) is given by:

P (
τ (4) exp (T, γ s , τ ) = α(T ) (γ s -τ ) 2 -γ 2 s , (73)
α(T ) refers to a non-linear shear modulus. We further recall that the invariant γ t is always negative during the double shear test so that the square root involved in (73) is well-defined. Using the expression (73) g (4) now reads:

g (4) (T, γ s , γ t ) = α(T ) ρ 0 γt 0 (γ s -τ )dτ = α(T ) 2ρ 0 γ 2 s -(γ s -γ t ) 2 . (74) 4.5

. Idealized elastic law

With the idealizations of the experimental stresses (60), (71), (73) the g (•) functions were explicitly calculated (63), (72), (74). The derivation of the Helmholtz free energy yields the current Cauchy stress:

σ = ρ 0 ∂ J fψ 1 + 1 γ s J 5 3 (µ(T )γ s + α(T ) (γ s -γ t )) DevB + α(T ) J 7 3 Dev B 2 -Tr(B)B . (75)
The invariant γ t is upper bounded by γ s whereas the positivity of the second invariant J B 2 implies the lower bound:

J B 2 = 3 2 ∥DevB∥ ≥ 0 ⇒ γ t ≥ γ s 1 - 1 + γ 2 s 4 . (76)
Finally lim γs→0 γ t γ s = 0 so that (75) does not diverge when γ s → 0. As we will see in the following section, the expression (75) is algebraically equivalent to a standard Mooney-Rivlin model.

Analogy with the Mooney-Rivlin model

If we retain the idealizations (71), (73), the dependence of the massic free energy with respect to the isochoric invariants is written:

g (3) (T, γ s ) + g (4) (T, γ s , γ t ) = 1 2ρ 0 µ(T )γ 2 s + α(T ) γ 2 s -(γ s -γ t ) 2 = 1 2ρ 0 µ(T ) B I -3 + α(T ) B I -3 -B II -3 . (77)
The right and left Cauchy-Green strain tensors C and B share their invariants B i = C i , as C and B are related through the rotation tensor R arising from the polar decomposition of the transformation gradient

F = R • U = V • R.
Therefore the free energy (77) coincides exactly with that of a Mooney-Rivlin model [START_REF] Kumar | Hyperelastic mooney-rivlin model: determination and physical interpretation of material constants[END_REF]:

ψ m M -R = g 1 (T ) + g 2 (T, J) + C 01 ρ 0 B I -3 + C 10 ρ 0 B II -3 , (78)
with the parameters

C 01 = 1 2 (µ(T ) + α(T )), C 10 = -α(T ) 2 .
If we assume that the shear stress along the P (4) path is identically zero, the massic free energy becomes independent of the γ t invariant and we then observe that the Helmholtz massic free energy evolves linearly with B I -3, which corresponds to the Neo-Hookean model

ψ m N -H = g 1 (T )+g 2 (T, J)+ µ 2ρ 0 B I -3 .
Consequently our thermodynamic approach includes these two models which result from the experimental idealizations (71) (73).

More generally, if one looks at the in-plane shear stress curve τ

exp as a function of the slip parameter γ, it is certain that a polynomial expansion fitting experimental data can be found. By symmetry, this polynomial expansion must necessarily contain only even powers:

τ (3) exp (T, γ) = i µ i γ 2i+1 . (79)
(79) then generates a generalized Neo-Hookean model (power series of the free energy with respect to the "B I -3" invariant).

Moreover, if we postulate by the same reasoning that the out-of-plane shear stress τ (4) exp develops in the form:

τ (3) exp (T, γ) = i µ i γ 2i+1 and τ (4) exp (T, γ s , τ ) = i α i (γ s -τ ) 2 -γ 2 s (γ s -τ ) 2i . (80)
Then one found the following Helmholtz massic free energy:

ψ m = 1 ρ 0 i µ i (T ) 2i + 2 B I -3 i + α i (T ) 2i + 2 B I -3 i -B II -3 i + g (1) (T ) + g (2) (T, J), ψ m = 1 ρ 0 i C i0 B I -3 i + C 0i B II -3 i + g (1) (T ) + g (2) (T, J) C ij = 0 if i ̸ = j. (81)
That is, a generalized Mooney-Rivlin material [START_REF] Kumar | Hyperelastic mooney-rivlin model: determination and physical interpretation of material constants[END_REF] .

In the general case, the integral expressions given in section 3.2 yield the exact finite strain isotropic thermo-elastic constitutive law.

Conclusion

We have developed a complete finite-strain isotropic thermo-elastic model without any assumption concerning the form of the Helmholtz free energy. The identification of the latter relies on the successive realization of four elementary experiments, and more precisely the measurement of: a massic heat, a pressure, and two shear stresses. Experiments must be carried many times in order to scan all the accessible values of the state variables e.g. the double shear test P (4) must be performed for a whole collection of temperature T , volume expansion J, and slip γ s .

Some idealizations of the experimental stresses have been proposed. They can be replaced by any realistic curves, or even be given by the upscaling results of a large scale molecular dynamic simulation.

The well known Mooney-Rivlin model, and consequently the Neo-Hookean, are special cases of our methodology. They are derived from an assumption concerning the shear stress along the P (3) and P (4) paths.

The present work could admit several natural extensions among which:

• The description of an anisotropic medium by the introduction of structural tensors {N • t } representing the actual directions of anisotropy. Enhancing the Helmholtz free energy with these new unknowns necessarily yields cross-invariant effects, reflecting the current orientation of the deformation with respect to the anisotropic directions.

• A finite strain plasticity model with the addition of an objective plastic strain tensor B p in the arguments' list of the Helmholtz free energy. Taking this tensor into account adds 6 state variables.

Three of them are specific to the plastic strain B p (2 if one further assumes that the plastic evolution is isochoric), three others orient the principal directions orthogonal trihedron of B and B p .

• The study of large-amplitude shock waves and more particularly shock tails. Indeed, after a large plastic phase which is out of the scope of this study, the elastic release wave on the shock tail can be considered as large strain. A proper coupling between both, an equation of state driving the pressure, and the deviatoric elastic law presented herein is a very interesting perspective.

III ∂ BI f B ψ , ∂ BII f B ψ = ∂ BII f B ψ ∂ BII B II = B -2 3 III ∂ BII f B ψ , ∂ BIII f B ψ = ∂ BI f B ψ ∂ BIII B I + ∂ BII f B ψ ∂ BIII B II + ∂ BIII f B ψ , ∂ BIII f B ψ = -∂ BI f B ψ B I 3B III -∂ BII f B ψ 2B II 3B III + ∂ BIII f B ψ . (B.1)
And the elastic law (6) now reads:

σ = 2ρ 0 √ B III B III ∂ BIII f B ψ -∂ BI f B ψ B I 3 -∂ BII f B ψ 2B II 3 1 + B -1 3 III ∂ BI f B ψ + B 1 3 III B I B -2 3 III ∂ BII f B ψ B -B -2 3 III ∂ BII f B ψ B 2 , σ = 2ρ 0 √ B III B III ∂ BIII f B ψ -∂ BI f B ψ B I 3 -∂ BII f B ψ 2B II 3 1 + ∂ BI f B ψ + B I ∂ BII f B ψ B -∂ BII f B ψ B 2 . (B.2)
This formula is more elegantly written using the following relations, which hold for any tensor of order 2: Let (λ i ) i∈ 1,3 be the principal stretches of the isochoric Finger tensor B. We further assume that the invariant γ s remains fixed. The following system is necessarily verified by the triplet (λ i ) i∈ 1,3 :

X II = 1 2 X 2 I -TrX 2 ⇒ - 2 3 X II 1 + X I X -X 2 = - 1 3 X 2 I -TrX 2 1 + X I X -X 2 , = -Dev X 2 -Tr (X) X . (B.3) σ = 2ρ 0 √ B III B III ∂ BIII f B ψ 1 + ∂ BI f B ψ DevB -∂ BII f B ψ Dev B 2 -Tr B B , σ = 2ρ 0 √ B III B III ∂ BIII f B ψ 1 + ∂ BI f B ψ B 1 3 III DevB - ∂ BII f B ψ B 2
λ 2 1 + λ 2 2 + λ 2 3 = γ 2 s + 3 = B I , (C.1) λ 2 1 λ 2 2 + λ 2 2 λ 2 3 + λ 2 1 λ 2 3 = B II , (C.2) λ 1 λ 2 λ 3 = 1. (C.3)
By combining the equations (C.1) and (C.3), we deduce:

λ 4 2 + λ 2 1 -B I λ 2 2 + 1 λ 2 1 = 0. (C.4) (C.4
) has a positive or zero discriminant:

∆ = λ 2 1 -B I 2 - 4 λ 2 1 ≥ 0 ⇒ λ 2 2± = B I -λ 2 1 ± λ 2 1 -B I - 2 λ 1 λ 2 1 -B I + 2 λ 1 2 . (C.5)
The positivity of the term under the square root implies that (C.4) has real solutions if and only if the eigenvalue λ 1 evolves in the interval:

λ ∈ [X 2 , X 0 ] with X k = 3 √ α cos 1 3 arccos -(α) -3 2 + 2kπ 3 with α = B I 3 ≥ 0. (C.6)
We can then express γ t as a function of the unique variable λ 1 :

B II (λ 1 ) = λ 2 1 λ 2 2± + 1 λ 2 1 + 1 λ 2 2± = λ 2 1 B I -λ 4 1 + 1 λ 2 1 ⇒ γ t (λ 1 ) = γ s -B II (λ 1 ) -3 (C.7)
Fixing γ s successively, we plot the evolution domain of γ t (see Figure C.3a) and compare it to the set that can be scanned by the double shear test: Therefore, we present in the following Appendix D a more general motion which scan the whole domain of variation for γ t . We define as "isochoric double shear-traction test" the motion given by the following gradient:

γ t ∈ γ s 1 -1 + γ 2 s 4 , 0
F = J 1 3       1 √ e γ 0 0 1 √ e γ ⊥ 0 0 e       ⇒ B =       1 e + γ 2 γ √ e 0 1 e + γ 2 ⊥ γ ⊥ e Sym e 2       . (D.1)
The "double shear test" mentioned in Section 3.1.4 corresponds to e = 1. We also denote as "simple tensileshear test" the case γ ⊥ = 0.

The square of the Finger isochoric strain tensor is equal to:

B 2 =        1 e + γ 2 2 + γ 2 e γ √ e 2 e + γ 2 + γ 2 ⊥ γγ ⊥ √ e γ 2 e + (γ ⊥ e) 2 + 1 e + γ 2 ⊥ 2 γ ⊥ e 1 e + γ 2 ⊥ + e 2 Sym (γ ⊥ e) 2 + e 4        . (D.2)
It's worth noticing a non-zero B 2 13 deformation which induces a purely non-linear stress. This out-of plane shear stress is relevant for identifying the dependence of the free energy on the second invariant γ t .

A simple computation of the characteristic polynomial of the isochoric Finger tensor shows that the fundamental invariants are equal to:

B I = γ 2 + γ 2 ⊥ + e 2 + 2 e and B II = γ 2 γ 2 ⊥ + e 2 + 2e + 1 e 2 + γ 2 ⊥ e 2 . (D.3)
The conservation of the parameter γ s is ensured if at each instant, the following equality holds:

B I = γ 2 + γ 2 ⊥ + e 2 + 2 e = γ 2 s + 3 with e(0) = 1, γ(0) = γ s , γ ⊥ (0) = 0. (D.4)
The motion has two independent parameters: a shear γ ⊥ and a traction magnitude e. The value of γ is adjusted so as to guarantee at each instant the equality (D.4).

We then deduce, by simple calculations, the value of the invariant γ t : The identification of the constitutive law with respect to the second invariant γ t Section 3.1.4 was built upon the double shear test. The reason is quite simple: the single shear tensile test, although scanning the whole domain for γ t , does not allow to simply identify the dependence on γ t because the non-linear stress σ 13 vanishes. Therefore, measuring either σ 12 or σ 13 will couple g (4) to the other functions. The most efficient test, but also the hardest to perform experimentally, shall set a small shear strain γ ⊥ and then perform a tensile test by varying e. Hence σ 13 would be non-zero and one could characterize fψ for γ t ≥ 0. The partial derivative with respect to the volume expansion J thanks to the volumetric/isochoric split:

γ t (e, γ ⊥ ) = γ s -(γ 2 ⊥ + e 2 ) B I -γ 2 ⊥ -e 2 -
∂e m ∂J = - 1 ρ 0 p (2)
exp (T, J) -T ∂ T p (2) exp (T, J) . The partial derivative with respect to γ s and γ t are written as:

∂e m ∂γ s = 1 ρ 0 τ (3) exp (T, γ s ) -T ∂ T τ (3

Figure 1 :

 1 Figure 1: Isochoric planar simple shear kinematics

Figure 2 :

 2 Figure 2: Double shear kinematic

3 IIIDev B 2 -(B. 4 )

 324 Tr (B) B . Appendix C. Evolution domain of γ t

. (C. 8 )

 8 Some values of the shear parameter γ t are not accessible through the double shear test (see Figure C.3b).

Analytical

  Figure C.3: Value of the second invariant γt for various γs

B I -γ 2 ⊥ -e 2 - 2 e ≥ 0 + 1 e 2 e 2 e

 22022 and e > 0. (D.6)The existence conditions (D.6) define a sub-domain of R 2 . At γ ⊥ ≤ γ s fixed, the range of variation of e is analytic and simply given by e ∈ [e 2 , e 0 ] with: values of the parameter γ t that can be scanned through the simple shear tensile test ((D.9) with γ ⊥ = 0) are:γ t (e, 0) = γ s -e 2 B I -e 2 -2 e + 2e γ t (e, 0) = γ s -e 2 B I -e 4 + 1 ∈ [e 2 (0), e 0 (0)], e k (0) given by (D.7). (D.8)Whereas for the double-shear test ((D.9) with e = 1):γ t (1, γ ⊥ ) = γ s -(γ 2 ⊥ + 1) B I -γ 2 ⊥ -3 + γ 2 ⊥ = γ s -γ 2 s + γ 2 ⊥ (γ 2 s -γ2⊥ ). (D.9) We represent in figure D.4 the values of γ t covered by the double shear tensile test with γ s = 3. The red line corresponds to the single tensile-shear test (γ ⊥ = 0), while the orange line corresponds to the double shear test (e = 1). The widest γ t interval is scanned when γ ⊥ = 0 (see D.4), i.e., during a single tensile-shear test. We further represent the γ t that can be scanned by both the tensile-shear (Figure (D.5b)) and double shear (Figure D.5a) tests for various γ s . We note that the double-shear test only explores negative values of γ t while the tensile-shear test scans the whole domain (see Figure D.6 or more simply the equality between (C.7) and (D.8) as e = λ 1 ).

Figure D. 4 :(γ s -τ ) 2 -γ 2 s T ∂ 2 Tτ ( 4 )

 42224 Figure D.4: Value of γt, γs = 3 fixed, for different values of e and γ ⊥

Figure D. 6 :

 6 Figure D.6: Upper and lower bound of γt

  ) exp (T, γ s ) +

	1 ρ 0 γ ∂e m γt 0 ∂ γs ∂γ t ρ 0 = 1 γ

s -τ (γ s -τ ) 2 -γ 2 s τ (4) exp -T ∂ T τ (4) exp (T, γ s , τ ) dτ, s -γ t (γ s -γ t ) 2 -γ 2 s τ (4) exp -T ∂ T τ (4) exp (T, γ s , γ t ). (E.3)

Acknowledgement

The author would like to thank his colleagues from the CEA for their fruitful remarks but also Pr. Jean Guarrigues whose rigorous work gave rise to this article.

Appendix A. Justification of the finite strain isotropic elastic law

The definition of the Finger tensor is B = F • F ⊤ , hence its time derivative reads:

where D (resp W ) denotes the symmetric (resp skew-symmetric) part of the spatial velocity gradient.

Using the combined product

, we get successively, ∀n ∈ Z:

Orthogonality of the antisymmetric and symmetric tensors for the doubly contracted product has been used to simplify (A.2).

The time derivatives of the coefficients of the characteristic polynomials of B are (see also [START_REF] Garrigues | Algèbre et analyse tensorielles pour l'étude des milieux continus[END_REF][START_REF] Garrigues | Cinématique des milieux continus[END_REF]):

The nullity of intrinsic dissipation (4), is written using the coincident isotropic function ( 5):

Injecting both (A.3), (A.2) in (A.4) and using the fact that σ and f B ψ are state functions so that they do not depend on D, the nullity of (A.4) ∀T ∀D yields the elastic law [START_REF] Cassels | Nonlinear elasticity: theory and applications[END_REF].

Appendix B. Thermo-elastic law with respect to isochoric invariants

The partial derivatives of the new function f B ψ defined by [START_REF] Dalemat | Une experimentation reussie pour l'identification de la reponse mecanique sans loi de comportement: Approche data-driven appliquee aux membranes elastomeres[END_REF] are simple algebraic calculations: