Fen Xiu

Xuetao Li

Wenjie Zhang

Fan He

Xixiang Ying

Didier Stien

A new alkaloid from Portulaca oleracea L. and its antiacetylcholinesterase activity

Keywords: Portulaca oleracea L., alkaloid, NMR, anticholinesterase activity

A new alkaloid, (10E, 12E)-9-ureidooctadeca-10, 12-dienoic acid, named oleraurea (1) and 10 known compounds, p-hydroxybenzaldehyde (2), p-hydroxybenzoic acid (3), p-hydroxyacetophenone (4), benzamide (5), (E)-p-coumaramide (6), (E)-ferulamide (7), soyalkaloid A (8), βcarboline-3-carboxylic acid (9), 2, 3, 4, 9-tetrahydro-1H-pyrido [3, 4-b] indole-3-carboxylic acid (10), (1S, 3S)-1-methyl-1, 2, 3, 4-tetrahydroβ-carboline-3-carboxylic acid (11) were obtained from Portulaca oleracea L., in which, compounds 4, 5, 8-11 were isolated from the plant for the first time. The structure of the compound 1 was identified using spectroscopic methods including 1D and 2D NMR, HR-ESI-TOFMS. The compounds 1, 5-11 presented anticholinesterase activities, but the P. oleracea extract (POE) presented very low anticholinesterase activity.

Introduction

Portulaca oleracea L. is succulent annual herb belonging to the family Portulacaceae, order Caryophyllales in phytochemical taxonomy [START_REF] Lei | Separation and identification of four new compounds with antibacterial activity from Portulaca oleracea L[END_REF], which is widely distributed in the tropical and subtropical areas in the world [START_REF] Zhou | Portulaca oleracea L.: a review of phytochemistry and pharmacological effects[END_REF]. P. oleracea is name ' a Chi Xian' because i s lea es seem like ee h of ho se an as e likes ama an h, also, name 'Chan hou Cai' because of its cold resistance, tenacious vitality and prolonging lifespan. oleracea is e a e as he plan of 'homolo y of foo an me icine' ian e al. 2014) because of its safety for daily consumption as a vegetable [START_REF] Yen | Evaluation of the cytotoxicity, mutagenicity and antimutagenicity of emerging edible plants[END_REF]) and its many health effects, i.e. anti-inflammatory [START_REF] Li | Three novel alkaloids from Portulaca oleracea L. and their anti-inflammatory effects[END_REF], antioxidant [START_REF] Yang | Phenolic alkaloids as a new class of antioxidants in Portulaca oleracea[END_REF], antimicrobia [START_REF] Nayaka | Antibacterial attributes of apigenin, isolated from Portulaca oleracea L[END_REF], antidiabetics [START_REF] Li | Preparation and antidiabetic activity of polysaccharide from Portulaca oleracea L[END_REF], anti-tumor [START_REF] Zhao | Antitumor activity of Portulaca oleracea L. polysaccharides against cervical carcinoma in vitro and in vivo[END_REF], and the effects should mainly attribute to the constituents in P. oleracea, such as alkaloids [START_REF] Li | Three novel alkaloids from Portulaca oleracea L. and their anti-inflammatory effects[END_REF][START_REF] Xu | A novel alkaloid from Portulaca oleracea L[END_REF][START_REF] Li | Two new similar alkaloids from Portulaca oleracea L[END_REF][START_REF] Zhao | A new lactam alkaloid from Portulaca oleracea L. and its cytotoxity[END_REF], flavonoids [START_REF] Xu | Determination of flavonoids in Portulaca oleracea L. by capillary electrophoresis with electrochemical detection[END_REF], terpenoids [START_REF] Xin | Two novel triterpenoids from Portulaca oleracea L[END_REF], organic acids [START_REF] Wang | Isolating and identifying organic acids from Portulaca oleracea and determining their anti-cyanobacterial activity[END_REF], phenolic compounds [START_REF] Erkan | Antioxidant activity and phenolic compounds of fractions from Portulaca oleracea L[END_REF], cerebrosides [START_REF] Zheng | Portulacerebroside A induces apoptosis via activation of the mitochondrial death pathway in human liver cancer HCCLM3 cells[END_REF].

Concerning alkaloids have a wide range of structural diversity and biological activities, for examples, some alkaloids are anticholinesterase (AChE) inhibitors, which a e he useful pa hway fo cu in he Alzheime 's isease AD Richa e al. 2010), in this paper we investigated the alkaloids in P. oleracea extract (POE) and evaluated their AChE inhibitory activities. A new alkaloid, (10E, 12E)-9-ureidooctadeca-10, 12-dienoic acid, named oleraurea, together with 10 known compounds were finally obtained (Figure 1). In addition, the anticholinesterase activities of the 11 compounds and the POE were studied by Ellman's colo ime ic me ho wi h some mo ifica ions. The compoun s 1, 8-11 presented higher anticholinesterase activities among the pure compounds with IC 50 between 62.22-67.65 μM, conversely, the POE with IC 50 of 257.09 ± 5.90 μM nearly presented no anticholinesterase activity, and eserine with IC 50 of 29.74 ± 0.14 μM was to be the positive control in the study.

Results and discussion

Structure elucidation

Compound 1 was a white power, which presented a dark spot at UV 254 nm and a blue fluoresce at UV 365 nm. The UV (MeOH) λmax: 226 nm (Figure S4) and IR: 3442, 1650, 1606 cm -1 (Figure S5). S1) (Figure S7) and DEPT spectrum (Figure S8) revealed 19 carbon resonances, which include one methyl (δ C 13.9), eleven methylenes (δ C 21.9, 24.6, 25.3, 28.4, 28.6, 28.7, 28.8, 30.8, 31.9, 34.1, 35.2), one methine group (δ C 50.2), four olefinic methines (δ C 128.6, 129.9, 133.3, 134.0) NOESY spectrum showed that the four olefinic carbons belong to two double bonds and were respectively assigned to C-10, 11, 12 and 13, the intensive signals leading the adjacent methylene protons shifting to downfield at 31.9

(position 14) and methine protons at 50.2 (position 9), furthermore, the coupling constant (14.4 Hz) confirmed the trans-configuration of two double bonds. Thus, the structure of compound 1 was elucidated as (10E, 12E)-9-ureidooctadeca-10, 12-dienoic acid, named oleraurea.

By comparison of the 1 H and 13 C NMR spectral data with literature, the 10 known compounds (2-11) were identified, respectively, as p-hydroxybenzaldehyde (2) [START_REF] Kim | NMR analysis of lignins in CAD-deficient plants. Part 1. Incorporation of hydroxycinnamaldehydes and hydroxybenzaldehydes into lignins[END_REF], p-hydroxybenzoic acid (3) [START_REF] Wang | Isolation and structural elucidation of two new glycosides from sage (Salvia officinalis L.)[END_REF], p-hydroxyacetophenone (4) [START_REF] Liu | Phenolic constituents from Chinese a on's bloo an i s ul a iole spec um cha ac e is ic[END_REF], benzamide (5) [START_REF] Cheng | Secondary metabolites of Tin Tailings Streptomyces AE21985[END_REF], (E)-p-coumaramide (6) [START_REF] Kokubun | Amides and an alkaloid from Portulaca oleracea[END_REF], (E)-ferulamide (7) [START_REF] Kokubun | Amides and an alkaloid from Portulaca oleracea[END_REF], soyalkaloid A (8) [START_REF] Wang | New alkaloids from green vegetable soybeans and their inhibitory activities on the proliferation of concanavalin a-activated lymphocytes[END_REF], β-carboline-3-carboxylic acid (9) [START_REF] Tran | Isolation and total synthesis of stolonines A-C, unique taurine amides from the Australian marine tunicate Cnemidocarpa stolonifera[END_REF], 2, 3, 4, 9-tetrahydro-1H-pyrido [3,4-b]indole-3-carboxylic acid (10) [START_REF] Goda | Constituents in watercress: inhibitors of histamine release from RBL-2H3 cells induced by antigen stimulation[END_REF]), (1S,3S)-1-methyl-1,2,3,4-tetrahydro-β-carboline-3-carboxylic acid (11) [START_REF] Brossi | Alkaloids in mammalian tissues. 3. Condensations of L-tryptophan and L-5-hydroxytryptophan with formaldehyde and acetaldehyde[END_REF].

Anticholinesterase activity

AD is the most common form of neurodegenerative disease in the aged and the patient have loss of memory, cognitive abilities and obstacle in behavior [START_REF] Mohamed | Design, synthesis and structure-activity relationship (SAR) studies of 2,4-disubstituted pyrimidine derivatives: dual activity as cholinesterase and Aβaggregation inhibitors[END_REF]. Currently, the more accepted AD pathology is the cholinergic hypothesis and the effective treatment strategy is to inhibit the activity of AChE and improve the level of acetylcholine in the brain [START_REF] Kurt | Synthesis, antioxidant and anticholinesterase activities of novel coumarylthiazole derivatives[END_REF]. Our experiment herein was devoted to search for the effective cholinesterase inhibitors. The anticholinesterase activities of the 11 compounds and the POE were determined using a microplate assay, and IC 50 values of the of the literature (Boga, et al. 2011). IC 50 value of positive control eserine was 29.74 ± 0.14 μM.

Experimental

General experimental procedures

The further separation and purity of compounds 1-11 used a Shimadzu Nexera X2 UHPLC LC-30A system (Shimadzu, Kyoto, Japan) with a UV

Plant materials and reagents

The air-dried aerial parts P. oleracea was collected from Shijiazhuang, Hebei, were used for column chromatography (CC). TLC was performed on TLC plates pre-coated with silica gel GF254 (Qingdao Marine Chemical Co., Qingdao, China).

Extraction and isolation

A sample (150 kg) of the air-dried aerial parts P. oleracea was extracted with 50% ethanol twice under reflux for 2 h each time. The ethanol POE was concentrated under reduced pressure to yield a crude POE (21 kg) and then subjected to chromatography on a silica-gel (80-100 mesh) column eluted sequentially with ethyl acetate and ethanol (1:0, 4:1, 2:1, v/v). The part of 2:1 was provided (1 kg) after concentrated, then the POE was divided into two equal parts, adsorbed on a polyamide resin (80-100 mesh, Taizhou Luqiao Sijia Biochemical Plastic Factory, Zhejiang, China) column eluted with equal distilled water, 30, 50 and 70% ethanol, respectively. The fractions of 30% ethanol (150 g) were evaporated and applied to a silica-gel (200-300 mesh, approximately 2.5 kg) column (8 × 120 cm), eluting sequentially withi isocratic ethyl acetate and methanol (5:1, 2:1, 1:2, 1:3, v/v) for further purification to obtain eluates 1-4. The resulting Fr. 1 (ethyl acetate and methanol 5:1, 14 g) was applied to 20-40 μm ODS-C 18 (150 g) column (φ3 × 70 cm) under medium pressure and eluted with MeOH-H 2 O (20:80, 40:60, 60:40, 80:20, 100:0, v/v)

Acetylcholinesterase activity assay

Ace ylcholines e ase ac i i y was es e by Ellman's colo ime ic me ho [START_REF] Ellman | A new and rapid colorimetric determination of acetylcholinesterase activity[END_REF]) with some modifications. In the 96-well plates, 140 μL PBS

Conclusion

In the study, a new alkaloid, (10E, 12E)-9-ureidooctadeca-10, 12-dienoic acid, along with 10 known compounds were isolated from P. oleracea, in which, compounds 4, 5, 8, 9, 10, 11 were isolated from the plant for the first time, and compounds 1, 5, 6, 7, 8, 9, 10, 11 exhibited higher anticholinesterase activities than that of the POE, suggested that the natural compounds in the P. oleracea presented a better pharmacological activities than that the extract, and therefore it is interesting to isolate more natural bioactive compounds from P.

oleracea in the further study.

Disclosure statement

No potential conflict of interest was reported by the authors.

Figure 1 .

 1 Figure 1. Chemical structures of compounds isolated from P. oleracea

  spectrophotometric detector (SPD-20A), a vacuum degasser (DGU-20A), a solvent delivery pump (LC-30AD) and Lab Station software (Shimadzu). The analytical column was a Diamonsil C 18 column (150 mm × 4.6 mm i.d., 5 μm) (Dikma Technologies, Bejing, China). The UV spectrum was detected using a HITACHI U-3010 spectrophotometer (Hitachi Ltd, Tokyo, Japan). The IR spectrum was recorded on an IR 200 spectrophotometer (Thermo Electron Corporation, Waltham, MA). The optical rotation was detected on an autopol I antomatic polarmeter (Rudolph Research Analytical, Hackettstown, NJ). The 1 H (1D, 2D COSY) and 13 C (1D, 2D HMBC) NMR spectra were determined using an AVANCE 500 MHz instrument (Bruker Corporation, Switzerland) in MeOD and DMSO, using TMS as an internal standard. An Agilent 1290 UHPLC system coupled to an Agilent 6520 quadrupole-time of flight mass spectrometer (Agilent, Waldbronn, Germany) was carried out to reference UHPLC-Q-TOF/MS spectra. The acetylcholinesterase inhibitory assay was performed in 96-well microplates and determinedusing an Emax precision microplate reader (Nanjing defe experiment equipment co. LTD).

  (0.1 M, pH = 8.0, containing 0.1 mol/L NaH 2 PO 4 and Na 2 HPO 4 ), 10 μL DTNB (15 mmol/L), 15 μL AChE (0.2 U/mL) and 20 μL of a serially diluted solution of the tested samples were mixed and incubated for 10 min, then added with 10 μL ATCI (15 mmol/L) and incubated at 20 °C for 10 min. The absorbance of each well was determined at 405 nm using a microplate reader. In the blank group, the tested sample was replaced by methanol. The percentage inhibition was calculated according to the following formula: inhibition activity (%) = [(blank group-test group)/blank group] × 100%. Anticholinesterase activities were evaluated by the value of IC 50 .

  5-11 isolated from P. oleracea ranged 1-35 μM exhibited different degree of activities against the AChE with IC 50 values of 62.22 ± 1.02-121.77 ± 0.72 μM far less than that of POE with IC 50 of 257.09 ± 5.90 μM, meaning that the POE presented very low anticholinesterase activity which was similar with the results

  China in June 2014. The plant material was kindly identified by Professor Dr.

	Xixiang Ying. Voucher specimens (No. 20140312) were deposited at School of
	Pharmacy, Liaoning University of Traditional Chinese Medicine. Methanol and
	acetonitrile were all of HPLC grade purchased from Damao Chemical Reagent
	Plant (Tianjin, China). All other reagents were analytical grade provided by
	Jinfeng Chemical Factory (Tianjin, China) and the water was WAHAHA purified
	water (Shenyang, China). Sodium dihydrogen phosphate and disodium
	hydrogen phosphate were purchased from Sinopharm Chemical Reagent Co.,
	Ltd. Eserine was provided by Biochempartner. 5, 5-dithiobis-2-nitrobenzoic acid
	(DTNB) was purchased from Shanghai Jinsui Bio-Technology Co., Ltd.

Acetylthiocholine iodide (ATCI) and AChE were provided by Dalian Meilun Biotech Co., Ltd. Polyamide and silica gel

(Qingdao Marine Chemical Co., Qingdao, China)

. Sephadex

 Marlborough, MA)

Funding

This work was supported by the National Natural Science Foundation of China [grant number 81573546].


Djaffar Boussaa
email: boussaa@lma.cnrs-mrs.fr

Effects of superimposed eigenstrains on the overall thermoelastic moduli of composites

Keywords: coefficient of thermal expansion, heat capacity, residual stress, temperature dependent material

This study investigates the effects that an initial local eigenstrain field, when superimposed on the thermal eigenstrain field, has on the overall thermal expansion coefficients and heat capacities of thermoelastic composites. The study can also be seen as an investigation into how a local residual stress field affects these overall moduli, as initial eigenstrains are generally a source of residual stresses. The approach taken is thermodynamic. Expressions that include the superimposed eigenstrain field are developed for the overall moduli within the framework of small strain thermoelasticity with temperature dependent materials. These expressions, which are written in terms of the concentration tensors and residual fields (stress and strain fields given rise to by the eigenstrains under zero overall stress and strain, respectively), contain correction terms that are absent in the expressions developed within linear thermoelasticity. Taking into account the temperature dependence of the constituent moduli is shown to be essential to capture the effects of the superimposed eigenstrain field. A Ti-6Al-4V/ZrO2 composite is investigated for which the correction terms are found to be negligible for the heat capacities but significant for the thermal expansion coefficients. This suggests that, for applications with large temperature changes, using the linear-thermoelasticity-based expressions can affect the accuracy of the estimates of the overall moduli, and therefore the accuracy of thermostructural analyses of composite structures. The proposed expressions can be of use to estimate the overall thermoelastic moduli in contexts in which the strains remain small, temperature changes are large, and superimposed eigenstrains may be present.

Introduction

This study investigates how the thermal expansion coefficients and heat capacities of thermoelastic composites are affected by the presence of an initial local eigenstrain field superimposed on the thermal eigenstrain field. Eigenstrains being generally a source of residual stresses, this study can also be seen as 1 an investigation into how local residual stresses affect the overall thermoelastic moduli of composites.

A broader framework than linear thermoelasticity is needed to explore the effects of the superimposed eigenstrain field on the overall thermoelastic moduli. Linear thermoelasticity assumes that the Helmholtz potential is quadratic in strain and temperature [START_REF] Lubarda | On thermodynamic potentials in linear thermoelasticity[END_REF]. This assumptions implies, among other things, temperature independent elasticities, which turns out to be too restrictive for the purposes of this study. One such broader framework, and the one used here, is that of small strain thermoelasticity with temperature dependent materials [START_REF] Kovalenko | Osnovy termouprugosti[END_REF]. This framework is used in many studies on components having to withstand large temperature changes, including ceramic components under thermal shocks [START_REF] Han | Thermal shock resistance of ceramics with temperature-dependent material properties at elevated temperature[END_REF][START_REF] Cheng | Effects of mechanical boundary conditions on thermal shock resistance of ultra-high temperature ceramics[END_REF]. For applications of this kind, considering the temperature dependence of the materials is essential in the processes of materials selection [START_REF] De Faoite | Regression analysis of temperature-dependent mechanical and thermal properties of dielectric technical ceramics[END_REF], structural analysis [START_REF] Ching | Thermal stress analysis of functionally graded composites with temperature-dependent material properties[END_REF], and structural optimization [START_REF] Bobaru | Designing optimal volume fractions for functionally graded materials with temperature-dependent material properties[END_REF][START_REF] Boussaa | Optimization of temperature-dependent functionally graded material bodies[END_REF].

Being more general, the framework accounts for more phenomena than does linear thermoelasticity. Thus, within the broader framework, [START_REF] Nadeau | Effective thermal expansion of heterogeneous materials with application to low temperature environments[END_REF] showed theoretically that the presence of a superimposed local eigenstrain field can induce anisotropy in the overall thermal expansion response in an otherwise macroscopically isotropic composite of isotropic constituents. They also drew a parallel between their findings and a "most surprising" thermal expansion anisotropy experimentally observed in a polycrystal, one explanation for which could be the presence of residual stresses [START_REF] Finlayson | The low-temperature thermal expansion of polycrystalline V 3 Ge[END_REF][START_REF] Gibbs | Anisotropic thermal expansion of polycrystalline V25.75at%Si[END_REF].

Several approaches have been developed in small strain thermoelasticity to estimate the overall thermoelastic moduli for composites with temperaturedependent constituents. A first approach relies on numerical homogenization. E.g., the finite-element method was used to estimate the overall elasticities and thermal expansion coefficients of silicon nitride [START_REF] Wippler | Homogenization of the thermoelastic properties of silicon nitride[END_REF]. A second approach is analytical and yields the overall moduli in terms of the strain and stress concentration tensors and residual fields. One variant of this approach consists in combining the stress-strain relation with the concentration relations [START_REF] Benveniste | On a correspondence between mechanical and thermal effects in two-phase composites[END_REF][START_REF] Bahei-El-Din | Isothermal fatigue behavior of Sigma/TIMETAL 21S laminates, Part II: Modeling and numerical analysis[END_REF][START_REF] Nadeau | Effective thermal expansion of heterogeneous materials with application to low temperature environments[END_REF]Dvorak, 2012, pp. 250-252). A second variant consists in combining thermodynamic potentials with the concentration relations [START_REF] Boussaa | Effective thermoelastic properties of composites with temperaturedependent constituents[END_REF]. This second variant has the advantage of providing not only the overall elasticities and thermal expansion coefficients but also the overall heat capacities.

This paper adopts and extends this second variant to accommodate the presence of the superimposed eigenstrain field, and assesses the effects of that presence on the overall thermal expansion coefficients as well as on the overall heat capacities.

The paper is organized as follows. Section 2 provides notation and the definitions of the material moduli of interest here. Section 3 develops expressions for the Gibbs and Helmholtz potentials of a homogeneous material in which an initial eigenstrain is superimposed on the thermal eigenstrain. Section 4 recalls the concentration relations and obtains the macroscopic Gibbs and Helmholtz potentials of a heterogeneous material with an initial superimposed eigenstrain field. Section 5 derives the overall moduli from the macroscopic thermody-namic potentials. Section 6 provides Levin-type formulas for two-phase composites. Section 7 discusses a numerical example illustrating the application of the developed framework to estimate the overall moduli of a high temperature composite with a superimposed eigenstrain field. Finally, Section 8 gives some concluding remarks.

Thermoelastic material moduli of interest

The material moduli of interest here are the isothermal elasticity tensor, L; the isothermal compliance tensor, M ; the coefficient of thermal expansion (CTE) tensor, α; the stress-temperature tensor, β; the heat capacity per unit reference volume at constant strain, C ; and the heat capacity per unit reference volume at constant stress, C σ . These moduli are defined as follows:

L = ∂σ ∂ T , M = ∂ ∂σ T , (1)
α = ∂ ∂T σ , β = ∂σ ∂T , (2)
C = T ∂η ∂T , C σ = T ∂η ∂T σ , (3)
where σ is the stress tensor, is the small strain tensor, T is the absolute temperature, and η is the entropy. The moduli thus defined depend a priori on the two independent variables used to describe the thermodynamic state of the material, which here are taken to be either stress and temperature or strain and temperature. When stress and temperature are used as independent variables, the dual variables, strain and entropy, are given by

= - ∂g ∂σ T , η = - ∂g ∂T σ , ( 4
)
where g is the Gibbs potential per unit reference volume. Similarly, when strain and temperature are used as independent variables, the dual variables, stress and entropy, are given by

σ = ∂f ∂ T , η = - ∂f ∂T , ( 5
)
where f is the Helmholtz potential per unit reference volume.

From the above definitions, it follows that the moduli enjoy the usual symmetries.

The symbols f 0 , g 0 and η 0 will be used to denote the values of f , g, and η at some reference state, respectively. The reference state referred to will be specified at each use. The dot "•" will denote the usual inner product between second-order tensors: For any two second-order tensors X and Y , X • Y = X ij Y ij , with summation over repeated indices.

The overall or effective moduli will be denoted with superscript "eff".

3 Gibbs and Helmholtz potentials of a homogeneous thermoelastic material with a superimposed eigenstrain

In [START_REF] Boussaa | Effective thermoelastic properties of composites with temperaturedependent constituents[END_REF], which deals with the case without superimposed eigenstrain, the starting point for developing expressions for the Gibbs and Helmholtz potentials is the postulation of appropriate forms for them. Here, it is more straightforward to start with the postulation of constitutive relations, and then integrate them to obtain expressions for the potentials. The constitutive relations are

= M (T )σ + m(T ), (6) m(T ) = m th (T ) + m * , ( 7
)
where m is the total eigenstrain, m th is the thermal eigenstrain, and m * is the superimposed eigenstrain, which is assumed not to depend on T and whose physical origin is irrelevant to the analysis. The linear stress-strain relationship ( 6), the additive decomposition (7), and the compliance tensor dependence on the temperature but not on the superimposed eigenstrain are the core assumptions of the constitutive model. These are common assumptions in the framework of small strains.

Gibbs potential

In view of the stress-strain relation (6), integrating (4-1) gives

g(σ, T ) = - 1 2 σ • M (T )σ -m(T ) • σ + φ(T ), ( 8
)
where φ is a function of the temperature alone. From (3-2), (4-2) and ( 8), one can relate φ to C σ as follows:

C σ (σ, T ) = T 1 2 σ • d 2 M dT 2 σ + d 2 m dT 2 • σ - d 2 φ dT 2 . ( 9
)
Writing this equation for σ = 0, dividing both sides by T , and integrating the resulting equation twice with respect to temperature gives

φ(T ) = - T T0 ξ T0 C σ (0, ν) dν ν dξ -(T -T 0 ) η 0 + g 0 , (10)
where T 0 is an arbitrary temperature, and η 0 and g 0 are the entropy and the Gibbs potential in the state characterized by σ = 0 and T = T 0 , respectively. Substituting (10) back into (8) gives the desired expression for the Gibbs potential:

g(σ, T ) = - 1 2 σ • M (T ) σ -m(T ) • σ - T T0 ξ T0 C σ (0, ν) dν ν dξ -(T -T 0 ) η 0 + g 0 . (11)
The following remarks about the thermoelastic constitutive model characterized by this potential are in order:

1. The superimposed eigenstrain does not affect the compliance (by assumption).

2. By (2-1) and (4-2), the CTE tensor is given by

α (σ, T ) = dM dT σ + dm th dT .
This equation shows that the CTE tensor is independent of the superimposed eigenstrain, m * , and that

m th (T ) = T T ref α(0, ν) dν,
where T ref is the temperature for which the thermal eigenstrain vanishes (i.e., m th (T ref ) = 0). The quantity α(0, T ) is the coefficient of free thermal expansion (CFTE) tensor. It is related to α (σ, T ) through

α (σ, T ) = dM dT σ + α (0, T ) . (12)
3. By (3-2), the heat capacity at constant stress is

C σ (σ, T ) = C σ (0, T ) + T 1 2 σ • d 2 M dT 2 σ + d 2 m th d 2 T 2 • σ . ( 13
)
This shows that the heat capacity at constant stress also is independent of the superimposed eigenstrain, m * .

4. The choice of σ = 0 as a reference stress in deriving ( 11) is one of convenience, as it leads to a compact expression for the Gibbs potential. Any choice of this reference stress, however, is equally valid. Writing (9) for σ = σ 0 , where σ 0 is arbitrary, and proceeding as was done for σ = 0 gives

g(σ, T ) = - 1 2 σ • M (T )σ -m(T ) • σ + 1 2 σ 0 • M (T )σ 0 + m(T ) • σ 0 - T T0 ξ T0 C σ (σ 0 , ν) dν ν dξ -(T -T 0 ) η 0 + g 0 , (14)
where η 0 and g 0 are here the entropy and the Gibbs potential in the state characterized by σ = σ 0 and T = T 0 , respectively.

5. In summary, within the present thermoelastic framework, the material data needed to specify the Gibbs potential of a homogeneous material, and thus its state equations and moduli at any state, are the following: M (T ), the value of the compliance tensor as a function of temperature; m th , the value of the thermal eigenstrain as a function of temperature; m * , the value of the superimposed eigenstrain; C σ (σ 0 , T ), the value the heat capacity at constant stress as a function of temperature at an arbitrary given value of the stress σ 0 ; and η 0 and g 0 , the values of the entropy and Gibbs potential at a reference state (σ 0 , T 0 ), respectively. The quantities η 0 and g 0 do not enter the expressions for the moduli.

Helmholtz potential

The Helmholtz potential, f , is obtained by Legendre transforming expression (11) for the Gibbs potential with respect to stress, that is,

f = g + σ • . ( 15
)
Inverting the stress-strain relation (6) and combining the resulting equation with (15) and definition (3-1) of C gives

f ( , T ) = 1 2 • L(T ) + l(T ) • - T T0 ξ T0 C (0, ν) dν ν dξ -(T -T 0 ) η 0 + f 0 , (16)
where η 0 and f 0 are here the entropy and the Helmholtz potential in the state characterized by = 0 and T = T 0 , respectively, and the total eigenstress, l, is defined by

l(T )= -L(T )m(T ). (17)
The state equations associated with the Helmholtz potential (16) read

σ = L + l, (18)
η = - 1 2 • dL dT - dl dT • + T T0 C (0, ν) dν ν + η 0 . ( 19
)
The following remarks complement those in the previous subsection:

1. From the state equation for stress (18), the stress-temperature tensor β ( , T ) is given by

β ( , T ) = dL dT + dl dT .
In view of definition (17) of l, the stress-temperature tensor β ( , T ) depends on the superimposed eigenstrain, unless L is independent of T .

2. The state equation for entropy (19) implies that

C ( , T ) = C (0, T ) -T 1 2 • d 2 L dT 2 + d 2 l dT 2 • . (20)
Again, unless L is independent of, or varies linearly with, T , the heat capacity C ( , T ) depends on the superimposed eigenstrain.

3. The reference strain = 0 used to express f is again a choice of convenience. The expression for the Helmholtz potential for an arbitrary reference state ( = 0 , T = T 0 ) is given by

f ( , T ) = 1 2 • L(T ) + l(T ) • - 1 2 0 • L(T ) 0 -l(T ) • 0 - T T0 ξ T0 C ( 0 , ν) dν ν dξ -(T -T 0 ) η 0 + f 0 , (21)
where η 0 and g 0 are here the entropy and the Helmholtz potential in the reference state ( = 0 , T = T 0 ), respectively.

4 Gibbs and Helmholtz potentials of a heterogeneous thermoelastic material with a superimposed eigenstrain field

Assumptions and concentration relations

A representative volume element (RVE) of a heterogeneous material is considered. Its constituents are assumed to show (i) a thermoelastic behavior governed by the Gibbs potential ( 11) or ( 14), or equivalently by the Helmholtz potential ( 16) or ( 21), and the associated state equations, and (ii) perfect bonding and thermal contact at their interfaces. Within the RVE, (iii) the superimposed eigenstrain field is assumed to be given, (iv) the temperature is assumed to be uniform, and (v) the reference temperature for zero thermal strain is the same for all constituents.

If the RVE is considered under controlled macroscopic stress σ and temperature T , then the local stress σ(x) can be written as

σ (x) = B (x; T ) σ + b (x; T ) . ( 22
)
In this equation, the term Bσ is the stress that develops in the RVE under σ alone, and the fourth-order tensor B, which is independent of m * , is the stress concentration tensor at temperature T . The residual stress b is the stress that develops in the RVE under the total eigenstrain field m and zero overall stress conditions. By linearity, b can be decomposed into two terms,

b (x; T ) = b th (x; T ) + b * (x; T ) ,
the first of which represents the stress developed under m th (x; T ) alone while the second represents the stress developed under m * (x) alone. Although the stress b * (x; T ) arises from the temperature-independent superimposed eigenstrain, m * , it generally depends on the temperature, the elasticity tensors of the constituents being temperature dependent. In the case of temperature independent constituent elasticities and CFTEs, m is affine in T , and so is b, by linearity again. If either the elasticities or CFTEs or both depend on temperature, then b is, in general, non affine in T .

If the RVE is considered under controlled macroscopic strain and temperature T , then the local strain (x) can be written as

(x) = A (x; T ) + a (x; T ) . ( 23
)
In this equation, the term A is the strain that develops in the RVE under alone, and the fourth-order tensor A, which is independent of m * , is the strain concentration tensor at temperature T . The residual strain a is the strain that develops in the RVE under the eigenstrains and zero overall strain conditions. By linearity again, a also can be decomposed into two terms,

a (x; T ) = a th (x; T ) + a * (x; T ) ,
the first of which represents the strain developed under m th (x; T ) alone while the second represents the strain developed under m * (x) alone. Although the strain a * (x; T ) arises from eigenstrain m * , it depends on the temperature because of the temperature dependence of the constituent elasticities. The terminology "residual stress" and "residual strain" is adopted from (Dvorak, 2012, p. 54).

Macroscopic Gibbs potential

The macroscopic Gibbs potential density G is defined as

G (σ, T ) = g = 1 |V | V g (x; σ, T ) d V.
Here, g (x; σ, T ) is of the form (11) (or 14) where M , m, C σ , η 0 , and g 0 , and T 0 are made to depend on position, and σ is the stress that develops in the RVE under the combined effects of σ and m, V is an RVE and |V | its volume, and • is the average over V . Substituting expression (11) for g and the stress concentration relation ( 22) into the foregoing equation gives

G (σ, T ) = - 1 2 σ • B M B σ -B (M b + m) • σ - 1 2 b • M b + m • b - T T0 ξ T0 C σ (0, ν) dν ν dξ -(T -T 0 ) η 0 + g 0 . ( 24
)
The derivation of this equation uses classical properties of the stress concentration tensor B and residual stress b, as can be found in [START_REF] Boussaa | Effective thermoelastic properties of composites with temperaturedependent constituents[END_REF].

If the reference state is (σ 0 (x), T 0 (x)), then the expression for G takes the following form:

G (σ, T ) = - 1 2 σ • B M B σ -B (M b + m) • σ - 1 2 b • M b + m • b + 1 2 σ 0 • M σ 0 + σ 0 • m - T T0 ξ T0 C σ (σ 0 , ν) dν ν dξ -(T -T 0 ) η 0 + g 0 . ( 25
)

Macroscopic Helmholtz potential

Similarly to the macroscopic Gibbs potential, the macroscopic Helmholtz potential density F is defined as

F ( , T ) = f = 1 |V | V f (x; , T ) d V.
Here, f (x; σ, T ) is of the form ( 16) where L, m, C , η 0 , f 0 , and T 0 are made to depend on position, and is the strain that develops in the RVE under the combined effects of and m. Substituting expression (16) for f and the strain concentration relation (23) into the foregoing equation gives

F ( , T ) = 1 2 • A LA + A (La + l) • + 1 2 a • La + l • a - T T0 ξ T0 C (0, ν) dν ν dξ -(T -T 0 ) η 0 + f 0 . (26)
Again, the derivation of this equation uses classical properties of tensors A and a, as can be found in [START_REF] Boussaa | Effective thermoelastic properties of composites with temperaturedependent constituents[END_REF]. If a state ( 0 (x), T 0 (x)) is taken as the reference state instead of (0, T 0 (x)), then the expression for F becomes

F ( , T ) = 1 2 • A LA + A L (a -m) • + 1 2 a • La -a • Lm - 1 2 0 • L 0 -0 • Lm 9 - T T0 ξ T0 C ( 0 , ν) dν ν dξ -(T -T 0 ) η 0 + f 0 . ( 27
)
5 Overall moduli

Values deduced from the macroscopic Gibbs potential

The macroscopic Gibbs potential ( 24) is that of a thermoelastic material. The corresponding macroscopic strain = (∂G/∂σ) T takes the form

= M eff σ + m eff , (28)
where the effective compliance tensor, M eff , and the total eigenstrain, m eff , are given by

M eff = B M B = M B = B M , (29)
and

m eff = B (M b + m) = M b + m = B m . (30)
From equation ( 28) and definition (2-1) of the CTE tensor, the effective CTE tensor α eff (σ, T ) = (∂ /∂T ) σ is found to be given by

α eff (σ, T ) = dM eff dT σ + d B m dT . ( 31
)
The effective CFTE tensor,

α eff (0, T ) = d B m dT , (32)
can be additively decomposed as

α eff (0, T ) = d dT B T T0 α(0, T ) + dB dT m * = B α(0, T ) + dB dT m th + dB dT m * . ( 33
)
The first term on the right-hand side represents the estimate one obtains if one neglects the temperature dependence of the elasticity tensors of the constituents, whether a superimposed eigenstrain is present or not. The second term is a correction term accounting for the temperature dependence of the elasticity tensors of the constituents in the absence of any superimposed eigenstrain. The third term captures the effects of the superimposed eigenstrain. From the form of this term, it follows that if the elasticity tensors of the constituents depend on temperature, then (i) the superimposed eigenstrain enters the expression of the effective CFTE tensor and (ii) the effective CFTE tensor depends on temperature, even if the CFTE tensors of the constituents are temperature independent. This confirms that a framework broader than linear thermoelasticity, in which the elasticity tensors are assumed temperature-independent, is needed to capture the effects of the superimposed eigenstrain.

The macroscopic entropy η (σ, T ) = -(∂G/∂T ) σ may be written as

η = 1 2 σ • dM eff dT σ + dm eff dT • σ + 1 2 d dT b • m + T T0 C σ (0, ν) ν dν + η 0 , (34)
so that the effective heat capacity at constant stress C eff σ (σ, T ) = T (∂η/∂T ) σ is given by

C eff σ (σ, T ) = C σ (0, T ) + 1 2 T d 2 dT 2 b • m + T 1 2 σ • d 2 M eff dT 2 σ + d 2 m eff dT 2 • σ . ( 35
)
For a reference stress σ 0 (x), the same reasoning leads to

C eff σ (σ, T ) = C σ (σ 0 , T ) + 1 2 T d 2 dT 2 b • m -T 1 2 σ 0 • d 2 M dT 2 σ 0 + σ 0 • d 2 m dT 2 + T 1 2 σ • d 2 M eff dT 2 σ + d 2 m eff dT 2 • σ . ( 36
)
Within linear thermoelasticity, the elasticity tensors and CFTEs of the constituents are temperature independent, so that B is temperature independent and m, b, and m eff are affine in T . This and equation (35) (or (36)) imply that the superimposed eigenstrain has no effect on C eff σ (σ, T ) within that framework. The conclusions below follow from expressions (29), ( 30), ( 32), ( 33), (35), and (36) for the overall moduli. A sentence of the form "the superimposed eigenstrain affects some overall modulus" means that the superimposed eigenstrain occurs in the expression for that modulus, but nothing is implied about the significance of the effects of the superimposed eigenstrain on the modulus.

• The elasticity tensors of the constituents being assumed to be independent of the superimposed eigenstrain, the stress concentration tensor B is unaffected by the superimposed eigenstrain, and so are the effective elasticity tensor.

• The superimposed eigenstrain affects the effective total eigenstrain.

• The superimposed eigenstrain, in general, affects the effective CFTE tensor. Its effect is made possible by the temperature dependence of the elasticity tensors of the constituents. If the elasticity tensors of the constituents are temperature independent, then the superimposed eigenstrain has no effect on the effective CFTE tensor even if the CFTE tensors of the constituents depend on temperature.

• The superimposed eigenstrain, in general, induces anisotropy in both the free thermal expansion response and CFTE tensor in an otherwise isotropic composite (see the third term on the right-hand-side of ( 33)).

• In the limiting case of equal constituent CFTE tensors, the presence of a superimposed eigenstrain makes it possible for the effective CFTE tensor to differ from the common value of the constituent CFTE tensors. In particular, the composite can be thermally deformable even when the constituents are thermally undeformable, provided that the elasticity tensors of the constituents are temperature dependent. By (33), the effective thermal expansion of a composite with thermally undeformable constituents is (dB /dT ) m * .

• The superimposed eigenstrain, in general, affects the effective heat capacity at constant stress, C σ . Unlike in the case of the effective CFTEs, the temperature independence of the elasticity tensors of the constituents is not a sufficient condition for the superimposed eigenstrain to disappear from the expression of C σ . Assume that the elasticity tensors of the constituents do not depend on temperature but that their CFTEs do. Then, b is, in general, not affine in T , so that the term d 2 b • m * /dT 2 in ( 35) or (36) does not vanish. For the same reason, the superimposed eigenstrain does not, in general, disappear from the expression of C σ when the elasticity tensors of the constituents depend on temperature, even if the CFTE tensors of the constituents do not depend on temperature.

• The superimposed eigenstrain has no effect on the effective moduli if it is uniform.

Values deduced from the macroscopic Helmholtz potential

Following the same steps as for the macroscopic Gibbs potential, one can show that the effective stiffness tensor, L eff , the effective eigenstress, l eff , and the effective stress-temperature tensor, β eff (σ, T ) , are given by

L eff = A LA = LA = A L , l eff = A (La + l) = A l = La + l , β eff ( , T ) = dL eff dT + d A l dT ,
and that the effective heat capacity at constant strain C eff ( , T ) is given by either

C eff ( , T ) = C (0, T ) - 1 2 T d 2 dT 2 a • l -T • d 2 L eff dT 2 + d 2 l eff dT 2 • , or C eff ( , T ) = C ( 0 , T ) - 1 2 T d 2 dT 2 a • l + T 1 2 0 • d 2 L dT 2 0 + 0 • d 2 l dT 2 -T • d 2 L eff dT 2 + d 2 l eff dT 2 • ,
depending on the choice of the reference strain.

6 Case of the two-phase composite

This section considers two-phase composites, for which Levin-type formulas are given. It is assumed that the superimposed eigenstrain is uniform over each phase. Let m

(1) * and m

(2) * denote the values of the superimposed eigenstrains in phases 1 and 2, respectively.

If the stress and temperature are used as independent variables, the following Levin-type formulas can be obtained by using the usual arguments:

m eff = m + M eff -M [M ] -1 [m] , α eff (0, T ) = α (0, T ) + d dT M eff -M [M ] -1 [m] , C eff σ (0, T ) = C σ (0, T ) + 1 2 T d 2 dT 2 [M ] -1 [m] • M eff -M [M ] -1 [m] ,
where

[m] = T T0 [α(0, T )] dt + [m * ]
The square brackets denote the difference in the bracketed quantity between phase 1 and phase 2, e.g., [M ] = M (1) -M (2) , [m] = m (1) -m (2) . These formulas show that α eff (0, T ) and C eff σ (0, T ) depend on the superimposed eigenstrains m

(1) * and m

(2) * only through their difference. If the strain and temperature are used as independent variables, formulas analogous to the above can be obtained by using the same type of arguments. E.g.,

l eff = l + L eff -L [L] -1 [l] , β eff (0, T ) = β (0, T ) + d dT L eff -L [L] -1 [l] , C eff (0, T ) = C (0, T ) - 1 2 T d 2 dT 2 [L] -1 [l] • L eff -L [L] -1 [l] .
7 Numerical example: Titanium alloy matrix with zirconium oxide inclusions This section discusses a numerical example related to high temperature applications. The example data are drawn from the literature on functionally graded materials [START_REF] Tanigawa | Optimization of material composition to minimize thermal stresses in nonhomogeneous plate subjected to unsteady heat supply[END_REF]; [START_REF] Ootao | Neural network optimization of material composition of a functionally graded material plate at arbitrary temperature range and temperature rise[END_REF]Ootao et al. ( , 1999a,b),b); [START_REF] Wang | Transient one-dimensional heat conduction problems solved by finite element[END_REF]; Ching and Chen ( 2007)). For the proper modeling of these materials at high-temperatures, consideration of the effects of the local residual stresses and those of the material temperature dependence is regarded as critical [START_REF] Birman | Modeling and analysis of functionally graded materials and structures[END_REF]).

Below, the magnitude of a (symmetric) tensor quantity will refer to the absolute value of its eigenvalue with largest absolute value, and the order of magnitude of a (symmetric) tensor quantity will refer to the order of magnitude of its eigenvalue with largest absolute value.

Example data

Temperature range

The effective moduli are estimated over the temperature range of 300 to 1300 K. It is the same as that used by [START_REF] Tanigawa | Optimization of material composition to minimize thermal stresses in nonhomogeneous plate subjected to unsteady heat supply[END_REF] in some of their analyses involving the composite considered here.

Constituents

The composite material considered is two-phase, consisting of ZrO 2 inclusions embedded in a Ti-6Al-4V matrix. Both constituents are assumed to be isotropic. The materials' moduli are represented over the temperature range by the best fit expressions given in Table 1.

The coefficient of free linear thermal expansion of Ti-6Al-4V, which is defined piecewise, increases with temperature and plateaus above around 1100 K. As interpolated above, it is not T -differentiable at T = 1100 K. In the results below, the plateau is made to begin at the temperature at which the coefficient reaches its maximum, that is, at 1033.46 K (instead of 1100 K) and its value is fixed at 10.303 × 10 -6 K -1 (instead of 10.291 × 10 -6 K -1 ). This slight modification, which negligibly affects the predicted effective moduli, removes the artificial discontinuity introduced by the interpolation.

Figures 1234sum up the data for the constituents. In these figures, G, K, α, and C p generically denote shear modulus, bulk modulus, coefficient of free linear thermal expansion, and heat capacity per unit reference volume at constant pressure, respectively. For ZrO 2 , the shear and bulk moduli are both two (at higher temperatures) to three (at lower temperatures) orders of magnitude larger (in absolute value) than their respective T -derivatives, which are in turn three orders of magnitude larger than their respective T -derivatives. For Ti-6Al-4V, the shear and bulk moduli and the T -derivative of the bulk modulus are each three orders of magnitude larger than their respective T -derivatives, whereas the T -derivative of the shear modulus is four orders of magnitude larger than its T -derivative.

Aspect ratio and composition

The ZrO 2 inclusions are assumed to be spherical and randomly distributed within the matrix, so that the overall behavior of the composite is isotropic in the absence of superimposed eigenstrain. Their volume fraction is fixed at 20%.

Superimposed eigenstrain field

The superimposed eigenstrain field used in the simulations, assumed to be phasewise uniform, is such that

m (m) * -m (i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ) , (37)
where superscripts m and i refer to the matrix phase and the inclusion phase, respectively.

Reference temperature for thermal expansions and reference pressure for heat capacities

The reference temperature for zero thermal eigenstrain is taken as T ref = 300 K. The CTEs given above are assumed to be the CFTEs of the constituents. In view of the orders of magnitude of the eigenvalues of the compliance tensor T -derivatives discussed below, one can check that the CFTEs and the CTEs of the constituents, which are related by ( 12), are very close at the atmospheric pressure (σ 0 ≈ 10 5 i Pa, i being the second-order identity tensor). The constituent heat capacities given above are assumed to be the constituent heat capacities at atmospheric pressure. Below, the effective heat capacity C eff σ is also estimated at σ = σ 0 ,

C eff σ (σ 0 , T ) = C σ (σ 0 , T ) + 1 2 T d 2 dT 2 b • m + T 2 σ 0 • d 2 m eff dT 2 - d 2 m dT 2 + T 2 σ 0 • d 2 M eff dT 2 - d 2 M dT 2 σ 0 . ( 38
)
7.2 Effective moduli

Micromechanical model

The Mori-Tanka model is used to estimate the effective moduli of the composite.

In the case of two-phase inclusion-matrix composites, the equations defining the model include the following:

B (m) = c (m) I + c (i) L (i) (I -P [L]) -1 M (m) -1 , b (m) = B (m) -I [M ] -1 [m] ,
where B (m) and b (m) denote the Mori-Tanaka estimates to the averages over the matrix phase of B and b, respectively; c (m) and c (i) are the volume fractions of the matrix and the inclusions, respectively; I is the symmetric fourth-order identity tensor;

[L] = L (m) -L (i) ; [M ] = M (m) -M (i) ; [m] = m (m) -m (i)
; and P is the Hill polarization tensor.

Computing the effective moduli requires the first and second order T -derivatives of the above and similar quantities. Numerical differentiation is one option, which was discussed by Dvorak (2012, p. 252). Analytical differentiation offers no difficulty in principle. In practice, however, the computation of the first and second T -derivatives of P is analytically impractical in the general case. A sense of the difficulty can be gained by consulting [START_REF] Suvorov | Rate form of the Eshelby and Hill tensors[END_REF] where analytical expressions were derived for the first-order T -derivative of P in the cases of inclusions with aspect ratios of 0, 1 and infinity. For this study, a Mathematica package was developed that estimates the effective moduli for arbitrary shape-factor inclusions in an isotropic matrix. For this, techniques akin to automatic differentiation were implemented by using Mathematica's command OptimizeExpression in the Experimental package.

Effective elasticities

The eigenvalues and their first and second T -derivatives as functions of temperature for stress concentration factor B (m) and effective compliance tensor M eff are given in figures referred to in the discussion to follow.

Effective thermal expansion

The first (respectively second, third) term on the right-hand side of decomposition (33) of the effective CFTE tensor will subsequently be referred to as Term 1 (respectively 2, 3).

For the composite under consideration, Terms 1 and 2 are isotropic, because tensors B, m th and their T -derivatives are isotropic. Term 3, in contrast, is anisotropic.

As expected, the higher the temperature, the more apparent the combined effects of the material temperature dependence and the presence of the superimposed eigenstrains. Thus, below 760 K, the magnitudes of Terms 2 and 3 are both less than 1% of that of Term 1, while, at 1190 K, both magnitudes exceed 5% of that of Term 1. At 1300 K, these magnitudes are respectively larger than 14% and 11% of the magnitude of Term 1. The sum of Terms 2 and 3 accounts for 20% of component 22 of the effective CFTE tensor with superimposed eigenstrain (Figure 5). Also, the anisotropy induced by the superimposed eigenstrain increases with increasing temperature: It is imperceptible for small to moderate temperature changes, becoming evident in the upper part of the temperature range (Figure 6).

Term 3 dominates Term 2 for a good portion of the temperature range. The magnitude of Term 2, which is zero at the lower end of the temperature range and increases with increasing temperature, reaches the value of the smallest eigenvalue of Term 3 at around 500 K, that of the intermediate eigenvalue at around 700 K, and that of the largest eigenvalue at about 1100 K. Above 1100 K, the magnitude of Term 2 exceeds that of Term 3 (Figure 7).

The sharp increase of Terms 2 and 3 starting around 900 K can be traced to a sharp increase in the magnitude of dB (m) /dT around that temperature (Figure 8-b). The sharp increase of dB (m) /dT can in turn be traced to the relative loss of stiffness of the inclusions with increasing temperature (Figures 1, 2 and8). To illustrate, the magnitude of dM ZrO2 /dT increases by more than two orders of magnitude over the temperature range (Figure 9-a), whereas the magnitude of dM Ti-6Al-4V /dT increases by less than one order of magnitude (Figure 9-b).

Effective heat capacity at constant pressure per unit reference volume

The effective heat capacities predicted by the present approach are very close to those predicted by the law of mixtures. The heat capacities are thus negligibly affected by the consideration of all the correction terms introduced by the thermoelastic coupling, let alone by the superimposed eigenstrains. This can be checked by examining the orders of magnitude of the various terms occurring in expression (38) for C σ :

• The order-of-magnitude of term C σ (σ 0 , T ) is 10 6 J • m -3 • K -1 (Figure 4).

• The order-of-magnitude of term

(T /2)d 2 b • m /dT 2 is 10 3 J • m -3 • K -1 .
• The order-of-magnitude of term

(T /2)σ 0 • d 2 m eff /dT 2 -d 2 m/dT 2 is 1 J • m -3 • K -1 .
• The order-of-magnitude of term (T /2)

σ 0 • d 2 M eff /dT 2 -d 2 M /dT 2 σ 0 is 10 -2 J • m -3 • K -1 .
These orders of magnitude can be obtained by noting that the orders of magnitude of the individual quantities entering these terms are as follows. The order of magnitude of b is 10 8 Pa, that of db/dT is 10 5 Pa • K -1 , and that of d 2 b/dT 2 is 10 3 Pa•K -2 (Figure 10). The maximum order of magnitude of terms m is 10 -2 , that of terms dm/dT is 10 -5 K -1 , and that of terms d 2 m/dT 2 is 10 -8 K -2 (Figures 3 and6). The maximum order of magnitude of T is 10 3 K. The order of magnitude of the second T -derivatives of the compliances varies markedly with temperature. The order of magnitude of the second T -derivative of M ZrO2 increases from 10 -17 Pa -1 • K -2 at 300 K to 10 -14 Pa -1 • K -2 at 1300 K (Figure 9-a); that of M Ti-6Al-4V , from 10 -17 Pa -1 • K -2 at 300 K to 10 -16 Pa -1 • K -2 at 1300 K (Figure 9-b); that of M eff , from 10 -17 Pa -1 • K -2 at 300 K to 10 -15 Pa -1 • K -2 at 1300 K (Figure 9-c).

The conclusion that the effective heat capacities of the composite predicted by the present approach and those predicted by the law of mixtures are similar is expected to hold true for other structural materials at room and higher temperatures, at which the heat capacity per unit volume has an order of magnitude of 10 6 J • m -3 • K -1 [START_REF] Hatta | Heat capacity per unit volume[END_REF]. Such a conclusion, however, may not hold in the case of low temperatures, in which the orders of magnitude of the heat capacities are much lower.

Conclusions

The effects that the presence of a eigenstrain field, when superimposed on the thermal eigenstrain, has on the overall thermoelastic moduli of composites were quantified within the framework of small strain thermoelasticity with temperature dependent materials. Expressions for the Gibbs and Helmholtz potentials of a homogeneous thermoelastic material with a superimposed eigenstrain were first developed. Combining these potentials with the concentration relations then yielded the Gibbs and Helmholtz potentials of the heterogeneous thermoelastic material with a superimposed eigenstrain field, from which expressions for the effective CTE tensor, stress-temperature tensor, and heat capacities were finally obtained. A numerical example was thoroughly investigated. Besides the effective moduli, the orders of magnitude of the various correction terms induced by the temperature dependence of the constituent moduli and the presence of the superimposed eigenstrain were discussed.

The paper provides a framework for modeling thermoelastic composites undergoing small thermoelastic strains and subjected to large temperature changes and possibly to superimposed eigenstrains.

Eigenvalues of the compliance tensors (curves labeled 1 and 2), those of their respective first (curves labeled 1 and2 ) andsecond (curves labeled 1 and2 ) m (i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ). The curves labeled 1 correspond to Term 1 (i.e., B α ); the curves labeled to 2 (i.e., (dB/dT ) m th ); the curves labeled 3, to Term 3 (i.e., (dB/dT ) m * ). The gray curves correspond to the sum of the three terms, that is, to the non-zero components of α eff (0, T ). (i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ). The curves labeled 1 (respectively 2, 3) represent components 11 (respectively 22, 33). The gray curves correspond to the case of no superimposed eigenstrain field, in which case b (i) and its first and second T -derivatives are isotropic. In subfigure (c), the black curves are ordered as in subfigure (b).

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :

 12345 Figure 1: Constituent shear moduli (a) and respective first (b) and second (c) T -derivatives as functions of temperature.

Figure 6 :

 6 Figure 6: Non-zero components (a) and their T -derivatives (b) as functions of temperature for the effective CFTE tensor in the presence of a superimposed eigenstrain field m * such that m (m) * -m (i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ). The curves labeled 1 (respectively 2, 3) represent component α eff 11 (0, T ) (a) and component dα eff 11 /dT (0, T ) (b) (respectively α eff 22 (0, T ) (a) and dα eff 22 /dT (0, T ) (b), α eff33 (0, T ) (a) and dα eff 33 /dT (0, T ) (b)) of α eff (0, T ). The gray curves, which are hardly distinguishable from the curves labeled 3, correspond to the value of the coefficient of free thermal expansion (a) and that of its T -derivative in the case of no superimposed eigenstrain.

Figure 7 :Figure 8 :

 78 Figure 7: Comparison of the non zero components of Term 2 (i.e., (dB/dT ) m th ) and Term 3 (i.e., (dB/dT ) m * ) of effective CFTE tensor in the of a superimposed eigenstrain field m * such that m (m) *m(i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ).The curve labeled 2 represents the diagonal term of Term 2, which is isotropic. The curve labeled 3 11 represents component 11 of Term 3, and so on. The components of Term 2 are zero at 300 K. Those of Term 3 are relatively small (around 10 -8 ) but non-zero at 300 K.

Figure 9 :

 9 Figure9: Eigenvalues of the compliance tensors (curves labeled 1 and 2), those of their respective first (curves labeled 1 and 2 ) and second (curves labeled 1 and 2 ) T -derivatives as functions of temperature for ZrO (a), Ti6Al4V (b), and the composite (c). The curves labeled 1, 1 , and 1 correspond to the eigenvalues of multiplicity 5.

Figure 10 :

 10 Figure 10: The three vanishing components (a) and their first (b) and second (c) -derivatives as functions of temperature for b (i) in the presence of a superimposed eigenstrain field m * such that m (m) * -m

List of Figures

(i) * = 10 -3 (e 1 ⊗ e 1 + 2e 2 ⊗ e 2 ). The curves labeled 1 correspond to Term 1 (i.e., B α ); the curves labeled 2, to Term 2 (i.e., (dB/dT ) m th ); the curves labeled 3, to Term 3 (i.e., (dB/dT ) m * ). The gray curves correspond to the sum of the three terms, that is, to the non-zero components of α eff (0, T ). List of Tables 1 Material properties of the constituents as functions of the absolute temperature [START_REF] Tanigawa | Optimization of material composition to minimize thermal stresses in nonhomogeneous plate subjected to unsteady heat supply[END_REF] 1: Material properties of the constituents as functions of the absolute temperature [START_REF] Tanigawa | Optimization of material composition to minimize thermal stresses in nonhomogeneous plate subjected to unsteady heat supply[END_REF]. Heat capacity refers to the heat capacity per unit reference volume at constant pressure. The quantity α entering the expression for the mass density of a material is its linear CTE.


Raviella Zgheib

Sylvain Chaillou

Naim Ouaini

Douglas N Rutledge

Didier Stien

Amine Kassouf

Marc El Beyrouthy
email: marcelbeyrouthy@usek.edu.lb

Investigation of Origanum libanoticum Boiss. essential oils chemical polymorphism by independent components analysis (ICA)

Keywords: Origanum libanoticum Boiss., Essential oil, GC/MS, Independent components analysis, Harvesting date, Geographical location

The essential oils obtained from Origanum libanoticum Boiss., a plant endemic to Lebanon, were analyzed by GC/MS. Seventy compounds were identified, covering till 99.8% of the total oil composition. All samples were p-cymene and/or β-caryophyllene chemotype, with variable percentage of other compounds such as α-pinene, myrcene, α-phellandrene, limonene, etc. Compared to traditional drying method, lyophilized samples provided the highest essential oil (EO) yields and yields were higher at flowering stage (Chouwen: 0.33% in 2013 and 0.32% in 2014; Qartaba: 0.27% in 2013 and 0.37% in 2014). According to independent components analysis (ICA), date and site of harvest, altitude and drying technique had no effect on the variation of O. libanoticum EO chemical composition. An annual variation of EOs composition was observed since a particular variation in some major components concentration was revealed monthly and annually between 2013 and 2014.

There are at least 250 000 species of vascular plants existing worldwide. Only a small number of species has been investigated both phytochemically and pharmacologically [START_REF] Hostettmann | Strategy for the biological and chemical evaluation of plant extracts[END_REF]. Lebanon occupies only 0.007 % of the world's land surface area and is a home to 2612 plant species compared to only 1600 species in Great-Britain which is 30 times more extended, and, 4200 species in mainland France, a country that is 55 times bigger than Lebanon [START_REF] Mouterde | Nouvelle flore du Liban et de la Syrie. Tomes I[END_REF][START_REF] Mouterde | Nouvelle flore du Liban et de la Syrie[END_REF][START_REF] Mouterde | Nouvelle flore du Liban et de la Syrie[END_REF]. The vegetation of Lebanon has the exceptionally high species-area ratio of 0.25 species per km 2 , compared with a ratio of 0.0081 per km 2 for South Africa, 0.0044 for Brazil, 0.0021 for Egypt, 0.022 for Jordan, 0.015 for Spain, 0.017 for Syria and 0.011 for Turkey [START_REF]Fourth National Report of Lebanon to the Convention on Biological Diversity[END_REF]. Relative to its size, Lebanon is recognized as a center of plant diversity and one of the Mediterranean region's mini-hotspots, sheltering high biodiversity richness including many rare and endemic plant species [START_REF] Sattout | Botanical assessment, chemical profiling, and conservation status of Lebanese endemics: Guidelines for national conservation policies and adaptation strategy[END_REF]. According to national floras, dating back to the early 1930s and 1960s, the country is shelter to almost 311 endemic species found in Lebanon, Syria and Palestine. A high percentage of endemic plant species (3.5%) is observed in Lebanon: among the 2612 wild species, 92 are endemics [START_REF] El Beyrouthy | Plants used as remedies antirheumatic and antineuralgic in the traditional medicine of Lebanon[END_REF]. The genus Origanum (Labiatae) consists of 43 species belonging to 10 sections. It also has 18 natural hybrids [START_REF] Kintzios | Oregano: The genera Origanum and Lippia[END_REF]. Origanum libanoticum Boiss., known under the vernacular name "za'atar lebnen", a wild cascading ornamental plant, endemic to Lebanon, is widespread throughout Lebanon, at various altitudes. It blooms from June till October on tall stems with cascading sprays of pink bracts and flowers.

Environmental factors were proven to affect the development of Origanum plant species. Many examples showed that the chemical composition of their essential oils (EOs) varies with geographical location, harvesting date, climate factors, soil composition, plant part, age, seasonality, vegetative stage, and so on [START_REF] Angioni | Chemical composition, seasonal variability, and antifungal activity of Lavandula stoechas L. ssp. stoechas essential oils from stem/leaves and flowers[END_REF][START_REF] Chalcha | Morphological and chemical studies of Origanum clones: Origanum vulgare L. ssp. vulgare[END_REF][START_REF] D'antuono | Variability of essential oil content and composition of Origanum vulgare L. populations from a north Mediterranean area (Liguria region, northern Italy)[END_REF][START_REF] Masotti | Seasonal and phenological variations of the essential oil from the narrow endemic species Artemisia molinieri and its biological activities[END_REF][START_REF] Mockute | The essential oil of Origanum vulgare L. ssp. vulgare growing wild in Vilnius district (Lithuania)[END_REF][START_REF] Werker | The essential oils and glandular hairs in different chemotypes of Origanum vulgare L[END_REF]. The influence of those factors on the accumulation of distinct terpene metabolites in Origanum species defines its chemotypes.

We aimed, in this work, to investigate the yield and the chemical composition of the EO isolated from O. libanoticum and to seek for a possible chemical variability of its EO, for the first time, in relation to seasonal variations, drying method, geographical location and soil composition. We should point out that, no published work has presented a detailed characterization such as what is discussed in this paper. To fulfill this objective, sampling has covered two regions corresponding to two different altitudes: Qartaba (Jbeil District/Mount Lebanon Governorate) located at 1250 m and Chouwen (Keserwan District/Mount Lebanon Governorate) at 600 m.

The EO content increased from January to July (flowering stage) then decreased after flowering. The flowering stage was characterized by the highest EO yields. The highest yields were: 0.33% in 2013 (0.2 mL in 60.35 g of the dried herb) and 0.32% in 2014 (0.4 mL in 124.3g of the dried herb) for samples harvested from Chouwen and dried by lyophilization; 0.27% in 2013 (0.2 mL in 75.1 g of the dried herb) and 0.37% in 2014 (0.2 mL in 54.52 g of the dried herb) for samples harvested from Qartaba and dried by lyophilization. One-way ANOVA demonstrated the relation between EO yield and date of harvest (p < 0.05). Clearly the EO yield was influenced by vegetative stages and climate factors. Our results are in full agreement with most of previous works who reported that Origanum sp. EO content depends on the collection month and that the flowering stage increased the EO yield [START_REF] Kizil | Effect of different developing stages on some agronomical characteristics and essential oil composition of oregano (Origanum onites)[END_REF][START_REF] Zgheib | Chemometric tools to highlight the variability of the chemical composition and yield of Lebanese Origanum syriacum L. essential oil[END_REF].

Student's t-test showed that the drying method of plant material influenced the EO yield (p < 0.05): lyophilized samples provided the highest yield values (Chouwen: 0.33% in 2013 and 0.32% in 2014; Qartaba: 0.27% in 2013 and 0.37% in 2014) (Figure 1). Table 1 summarizes the altitudes of the two geographical locations and the main chemical parameters of the two soils. O. libanoticum harvested from Qartaba and Chouwen naturally grows in sandy soils. A statistically non-significant difference was noticed between altitude and oil content (p > 0.05). Region of harvest, elevation and temperature differences between the 2 sites probably did not influence the EO yield of the plants growing in the same edaphic conditions (similar soil characteristics). In total, 70 components were identified in the volatile oil of O. libanoticum plants collected from Qartaba and Chouwen, accounting for 99.8% of the total oil composition. Tables 2 and3 gather the compositions of the EOs of the 53 samples of oregano collected during 2013 and 2014. The compounds were grouped into five classes: monoterpene hydrocarbons, oxygenated monoterpenes, sesquiterpenes hydrocarbons, oxygenated sesquiterpenes and others.

With reference to their EO composition, Origanum taxa may be characterized by the dominant occurrence of the following compounds:

-Monoterpene alcohol (linalool, α-terpineol, terpinen- [START_REF] Hafi | Chemical Composition and Antimicrobial Activity of Origanum libanoticum, Origanum ehrenbergii and Origanum syriacum Growing Wild in Lebanon[END_REF], thymol methyl ether (32.8%), δcadinene (9.7%) and γ-terpinene (9.6%) [START_REF] Arnold | Composition of the essential oil of three different species of Origanum in the eastern Mediterranean[END_REF], and, hexadecanoic acid (11.3%), thymol methyl ether (9.8%), (E)-β-caryophyllene (7.7%) and linalool (6.5%) [START_REF] Marrelli | Composition, antibacterial, antioxidant and antiproliferative activities of essential oils from three Origanum species growing wild in Lebanon and Greece[END_REF], our results analyzing 53 samples, indicate that it is possible to attribute O. libanoticum growing wild in Qartaba and Chouwen to the p-cymene and/or β-caryophyllene chemotype. It was also noticed that the percentage of the dominance of p-cymene and β-caryophyllene varied from one month to another and between years.

Qualitative and quantitative analysis of the volatile oils extracted from 53 samples, harvested from the two Lebanese sites, throughout two years (2013 and 2014), was performed by GC/MS. The chemical constituents were submitted to multivariate analysis (ICA or independent components analysis), in order to study their intraspecific variation.

ICA-DA was applied on the original matrix X (peak areas) concatenated with the matrix Y (predefined groups) corresponding to year of harvest (two years vectors), extracted seven ICs. Thus, an ICA with seven ICs was applied to the matrix X. IC1 and IC6 were found to be informative since clear discrimations were observed. In IC1, samples harvested in 2013 (1) were characterized by the lowest proportions compared to the ones harvested in 2014 (2), which presented higher proportions (Figure 2). The source signal of IC1, in Figure 3, showed that β-caryophyllene is mainly at the origin of this discrimination. IC1 highlighted that samples harvested in 2014 contained more of β-caryophyllene, thymol methyl ether, germacrene D, bicyclogermacrene and δ-cadinene than the samples harvested in 2013. IC6 was also able to discriminate the samples (Figure 4). Looking at the extracted source signal in Figure 5, it can be seen that this discrimination is mainly due to β-sesquiphellandrene, thymol methyl ether, terpinen-4-ol, spathulenol, α-copaene and α-cubebene.

Samples harvested in 2013 (1) contained less of βsesquiphellandrene, terpinen-4-ol, spathulenol, α-copaene and αcubebene, and, more of thymol methyl ether than the samples harvested in 2014 [START_REF] Mouterde | Nouvelle flore du Liban et de la Syrie. Tomes I[END_REF]. (1) Samples harvested in 2013

(2) Samples harvested in 2014 Six ICs were extracted by ICA-DA applied on the matrix X of the peak areas and the matrix Y which presented by the two drying methods vectors. The six extracted independent components illustrated an overlap of the proportions, and showed that there is no discrimination between the samples traditionally dried and lyophilized. This result demonstrated that the two methods of harvest selected in this work, presented the same effect on the quantity and the quality of the O. libanoticum EO.

Discrimination was not detected between the different months of the year. An overlap of proportions is shown in the nine extracted independent components. So, date of harvest and vegetative stages had no effect on the variation of O. libanoticum EO chemical composition.

Two vectors (predefined groups) representing the two geographical locations were concatenated with the original matrix of peak areas. The application of ICA-DA allowed the determination of eight ICs as the optimal number. Samples presented superimposed proportions in the eight ICs which implied that the chemical composition of the EOs was not influenced by the site of harvest. O. libanoticum harvested from Qartaba and Chouwen naturally grows in sandy soils. Therefore, probably, neither the climatic factors associated to altitudes (such as temperature) nor the edaphic factors which are similar in the 2 sites, had an influence on the EO chemical composition (Table 1).

In summary, wild populations of O. libanotictum collected throughout 2013 and 2014 from Qartaba and Chouwen produced EO characterized by the prevalence of p-cymene and βcaryophyllene. In addition, we demonstrated that the yield of EO varied according to the date of harvest and the drying method.

Flowering phase was the most productive period. A comparison of the chemical composition of the EO of O. libanoticum at all stages of development throughout the 2 considered years revealed differences. However, ICA revealed that the chemical composition of the EOs was not influenced by the harvesting site, altitudes, drying techniques and date of harvest. It is worth noting that the combination of more than one factor (environmental conditions, temperature, light, humidity, ecological conditions, plant adaptation, etc.) could explain the variation observed in the EO main compounds concentration between the two years investigated.

The investigations at chemotypes level and at the intraspecific chemical polymorphism are needed because biological activity of EOs depends on their chemical composition, the major compounds functional groups (alcohols, phenols, etc.) and the possible synergistic effects between the components. In this context, the value of an EO is its totum [START_REF] Lahlou | Methods to study the phytochemistry and bioactivity of essential oils[END_REF][START_REF] Hussain | Chemical composition, antioxidant and antimicrobial activities of basil (Ocimum basilicum) essential oils depends on seasonal variations[END_REF]. EO chemical composition varies in quality and quantity according to environmental and agronomic factors [START_REF] Karousou | Essential oil composition is related to the natural habitats: Coridothymus capitatus and Satureja thymbra in NATURA 2000 sites of Crete[END_REF][START_REF] Rota | Antimicrobial activity and chemical composition of Thymus vulgaris, Thymus zygis and Thymus hyemalis essential oils[END_REF]. However, at variance with Al Hafi et al. libanoticum EO and the variability affecting its chemical composition, which is observed from one month to another in our results, any determination of the EO biological activity must take into consideration, not only the chemical signature, but also its chemical variability. Therefore, larger scale and more long-term studies will allow room for validation and determination of the EO bioactivity. This will lead the research to an optimal harvest of the plant; a harvest adapted to the user's needs. Further studies conducted over several years on seasonality, genetics and growing conditions of wild and cultivated O. libanoticum should be performed in order to better understand the annual chemical variability observed. Furthermore, standardization of cultivation and propagation methods leading to efficient production and preservation of this plant is needed. Moreover, investigations of nutrient doses on quantitative and qualitative production of O. libanoticum EO compounds should be considered in future studies (each nutrient used in nutrient solution for the cultivation) in order to determine the optimal nutrient doses ensuring higher concentrations of the major EO compounds. Future investigations will focus the research on the biological activities of O. libanoticum EOs. Drying method (L: lyophilization; D: traditional drying). Harvest number (1: harvest at the beginning of the month; 2: harvest in the middle of the month).

Retention index (Ri a : Retention index calculated on a HP-5MS column; Ri b : Retention index on a HP-5MS column from literature [START_REF] Zgheib | Chemometric tools to highlight the variability of the chemical composition and yield of Lebanese Origanum syriacum L. essential oil[END_REF]; Ri c : Retention index calculated on a HP Innowax column; Ri d : Retention index on a HP Innowax column from literature [START_REF] De Falco | Chemical Composition and Biological Activity of Essential Oils of Origanum vulgare L. subsp. vulgare L. under Different Growth Conditions[END_REF][START_REF] Khoury | Report on the Medicinal Use of Eleven Lamiaceae Species in Lebanon and Rationalization of Their Antimicrobial Potential by Examination of the Chemical Composition and Antimicrobial Activity of Their Essential Oils[END_REF]). t: trace (less than 0.05.%). Blanks: not detected.

Yield evaluation:

The EO content was calculated as percentage of the dry mass. EO yield, expressed in mL/g, was calculated by measuring the volume of oil extracted per weight of dried/lyophilized plant material.

Soil analysis: Soil samples were collected from each site at a depth of 0-30 cm. Each sample was dried at room temperature, sieved at 2 mm and quartered before being analyzed at the Lebanese Agricultural Research Institute (LARI) Lebanon. Soil samples were taken during June 2014 at the time of flowering period of plants.

Essential oils analyses GC analyses:

The GC analysis was performed on a Thermo Electron Corporation apparatus fitted with a flame ionization detector (FID), a non-polar HP-5MS (5% Phenyl Methyl Siloxane) capillary column (30 m × 0.25 mm i.d., film thickness 0.25 μm) and a polar fused-silica HP Innowax capillary column (polyethylene glycol, 50 m × 0.20 mm i.d., film thickness 0.20 μm) in order to confirm the identification of the components. Helium was the carrier gas (0.8 mL/minute). The column temperature was initially set to 35°C before being gradually increased to 85°C at 5°C/minute, held for 20 minutes at 85°C, raised to 300°C at 10°C/minute and finally held for 5 minutes at 300°C. Diluted 1 µL samples (1/100, v/v in pentane) were manually injected at 250°C and in the splitless mode. Flame ionisation detection (FID) was performed at 310˚C.

GC/MS analyses:

The GC/MS analyses were performed using an Agilent gas chromatograph 6890 coupled with Mass Detector 5975. The 7683 B auto sampler injected 1 µL of each oil sample diluted in pentane (1/100 v/v). A fused-silica capillary column HP-5MS (30 m × 0.25 mm i.d., film thickness 0.25 μm) and a fused-silica HP Innowax polyethylene glycol capillary column (50 m × 0.20 mm i.d., film thickness 0.20 μm) were used in order to confirm the identification of the components. Helium was the carrier gas (0.8 mL/minute). The oven temperature program was identical to that described in 2.4.1. The mass spectra were recorded at 70 eV with an ion source temperature of 310°C and a transfer line heated to 320°C. The acquisition was recorded in full scan mode (50 -400 amu).

Qualitative and quantitative analyses: Most constituents were identified by comparing mass spectra on both columns with those provided in the NIST (National Institute of Standards and Technology, Gaithersburg, MD, USA) and Wiley 275 libraries, our home-made library constructed with pure compounds and EOs of known composition (such as the EO of Rosmarinus officinalis L. from Phytosun Aroms, Plélo, France) and those from literature [START_REF] Adams | Identification of Essential Oil Components by Gas Chromatography/Mass Spectrometry[END_REF][START_REF] Jennings | Qualitative analysis of flavour and fragrance volatiles by glass capillary gas chromatography[END_REF]. A further identification was achieved by comparing their retention indices (RI) on both polar and apolar columns with those from literature [START_REF] Jennings | Qualitative analysis of flavour and fragrance volatiles by glass capillary gas chromatography[END_REF][START_REF] Davies | Gas chromatographic retention indices of monoterpenes and sesquiterpenes on methyl silicon and carbowax 20M phases[END_REF] or with those of standard compounds available in our laboratories, obtained from Sigma-Aldrich (Germany). Retention indices were determined in relation to a homologous series of n-alkanes (C 7 to C 25 ) analyzed under the same GC/MS operating conditions. Relative proportions of oil constituents were calculated from the GC peak areas corrected using the Probabilistic Quotient Normalization (PQN), using the median of all the chromatograms as reference signal for the correction [START_REF] Kohl | State-of-the art data normalization methods improve NMR-based metabolomics analysis[END_REF].

Independent components analysis (ICA): theoretical background:

ICA is one of the most widely used blind source separation (BSS) techniques for revealing hidden factors that underlie sets of variables, measurements, or signals [START_REF] Hyvärinen | Independent component analysis[END_REF]. The goal of this statistical method is to extract the underlying source signals from a set of mixed signals, assuming that the original underlying source signals are mutually independently distributed. ICA looks for a linear transformation of non-Gaussian data that maximizes the statistical independence of the components and decomposes the measured overlapping signals of mixtures into a set of independent components (ICs) that are as independent as possible [START_REF] Comon | Independent component analysis, a new concept[END_REF][START_REF] Hyvärinen | Independent component analysis: Algorithms and applications[END_REF].

Maximizing independence means maximizing the non-Gaussianity of the signals.

The general model of ICA can be described as: ICA consists in estimating both matrices A and S knowing only X.

X = A
In this study the Joint Approximate Diagonalization of Eigenmatrices (JADE) algorithm was used in order to calculate a demixing matrix W which approximates A -1 . Among all the available algorithms, the JADE algorithm has proven to provide the most consistently reproducible and reasonable solutions [START_REF] Rutledge | Independent components analysis with the JADE algorithm[END_REF].

The component signals (source signals: the rows of the matrix S) are recovered from the measured mixed signals (peak areas of the chromatograms: the rows of the matrix X) by: S = W . X [START_REF] Bouveresse | Identification of significant factors by an extension of ANOVA-PCA based on multi-block analysis[END_REF][START_REF] Kassouf | Rapid discrimination of plastic packaging materials using MIR spectroscopy coupled with independent components analysis (ICA)[END_REF] The proportions, A, can then be calculated by projecting X onto S: A = X . S T (S S T ) -1

The ICA-DA procedure was used to determine the optimal numbers of independent components (ICs) to use in the ICA decomposition.

In this procedure, a matrix Y is created formed by vectors representing predefined groups corresponding to the different levels of the studied factors: geographical location of harvest (two sites implying two predefined groups), time of harvest (twelve months implying twelve predefined groups) and drying methods (two methods implying two predefined groups). The matrix Y is concatenated with the original matrix X of the peak areas in order to create a new matrix Z on which ICA is applied with from 1 to n max ICs. The number of ICs giving maximum discrimination of samples, based on the predefined groups, is then selected as the optimal number of ICs. The source signals extracted from the concatenated matrices contain contributions from both the MS variables and from the scaled groups matrix, so the corresponding proportions (scores) reflect not only the effect of the MS data, X, but also that of the predefined groups, Y. It is therefore necessary to recalculate the proportions using only the original matrix X, in order to determine whether the MS variables selected by ICA-DA are able alone to discriminate the samples into the predefined groups.

Figure 1 :

 1 Figure 1: Monthly yields of essential oil (2013-2014) extracted from O. libanoticum harvested in Qartaba and Chouwen for lyophilized and traditionally dried plants.

Figure 2 :( 1 )( 2 )Figure 3 :

 2123 Figure 2: Pproportions on IC1 for ICA-DA based on the year of harvest. (1) Samples harvested in 2013 (2) Samples harvested in 2014

Figure 4 :

 4 Figure 4: Representation of proportions on IC6 for ICA-DA based on the year of harvest.

Figure 5 :

 5 Figure 5: Source signal of IC6 for ICA-DA based on the year of harvest. These discriminations highlighted the variation of EO chemical composition according to the year of harvest. The concentration of the main components (β-caryophyllene and p-cymene) defining O. libanoticum chemotype varied between 2013 and 2014. These variations were not observed in other Origanum species such as O. syriacum [16] and, were not only found among samples of different regions but also among samples of the same region with different months during 2 years reflecting the intraspecific variability.

  (2016) [START_REF] Hafi | Chemical Composition and Antimicrobial Activity of Origanum libanoticum, Origanum ehrenbergii and Origanum syriacum Growing Wild in Lebanon[END_REF],[START_REF] Arnold | Composition of the essential oil of three different species of Origanum in the eastern Mediterranean[END_REF] [START_REF] Arnold | Composition of the essential oil of three different species of Origanum in the eastern Mediterranean[END_REF] and[START_REF] Marrelli | Composition, antibacterial, antioxidant and antiproliferative activities of essential oils from three Origanum species growing wild in Lebanon and Greece[END_REF] [START_REF] Marrelli | Composition, antibacterial, antioxidant and antiproliferative activities of essential oils from three Origanum species growing wild in Lebanon and Greece[END_REF] results, our samples were p-cymene and/or β-caryophyllene chemotype. Given the intraspecific polymorphism characterizing O.

  . S Where X (n × m) is the matrix of observed signals that contains n measured signals of m variables each, A (n × d) is a mixing matrix of unknown source signal proportions, S (d × m) is the matrix containing the unknown source signals (the ICs).

Table 1 :

 1 Altitude and soil parameters results at two Lebanese localities.

		Qartaba	Chouwen
	Altitude (m)	1250	600
	pH	7.94	7.98
	Organic Matter (%)	2.83	1.93
	P2O5 (mg/kg)	12.73	11.1
	K2O (mg/kg)	142.06	88.59
	N (%)	0.22	0.228

Table 2 :

 2 Chemical composition of the essential oil of O. libanoticum harvested from Chouwen.

						Harvesting date	Apr -13	Jun -13	Jul-13	Jul-13	Aug -13	Aug -13	Sep -13	Sep -13	Oct -13	Dec-13
						Drying method	L1	L1	L1	L2	L1	L2	L1	L2	L1	L1
	N°	Ri a		Ri b		Volatile compound
	1	924	924	1035	1035	α-thujene	0.9		0.6	1.5	0.3	0.6	0.8	1.6
	2	933	932	1076	1076	α-pinene	2.8	2.7	1.6	2.1	0.9	1.8	2.3	4.3	1.0	0.9
	3	947	946	1076	1076	camphene								0.3
	4	973	969	1132	1132	sabinene	0.6		0.4	0.6	0.9	0.2	0.5	0.5	0.3	1.2
	5	980	974	1118	1118	β-pinene	0.9		0.6	0.5	0.5		0.9	1.1
	6	991	988	1174	1174	myrcene	0.7	1.4	1.6	2.2	1.9	1.6	1.9	3.2	1.3	4.2
	7	1003	1002	1188	1188	α-phellandrene		6.4	0.3	0.3			0.3	0.6
	8	1009	1008	1157	1159	δ-3-carene				0.2			0.2	0.4
	9	1014	1014	1188	1189	α-terpinene	1.0	1.8	1.2	2.3	0.7	1.0	1.6	2.6	0.7
	10	1024	1020	1280	1278	p-cymene	15.5	17.7	14.7	33.1	7.5	11.8	19.8	33.2	8.2	9.8
	11	1025	1024	1203	1203	limonene	2.5	4.4	3.5	4.4	1.9	3.0	4.6	7.7	2.1	2.3
	12	1027	1026	1213	1213	1,8-cineole	4.7	6.7	2.4	2.4	1.1	2.2	3.0	5.4	1.4	1.1
	13	1043	1045	1269	1269	Z-β-ocimene
	14	1057	1054	1255	1256	γ-terpinene	3.7	6.1	5.0	7.5	2.7	3.6	6.0	8.9	2.7	2.9
	15	1086	1086	1265	1265	α-terpinolene
	16	1098	1095	1553	1553	linalool			0.7		0.4		0.2
	17	1174	1174	1611	1611	terpinen-4-ol					0.5			0.4
	18	1201	1186	1706	1706	α-terpineol			2.1	0.8	3.0	1.5	1.3	3.4	3.1	3.3
	19	1238	1232	1607	1609	thymol methyl ether	2.6	t	5.9	4.2	5.9	6.7	6.7	4.0	8.0	8.3
	20	1294	1289	2198	2198	thymol			1.3
	21	1300	1298	2239	2239	carvacrol								0.6
	22	1350	1345	1466	1466	α-cubebene	0.7	2.9	1.7	1.1	2.7	1.6	1.5	0.7	1.7	1.8
	23	1378	1374	1497	1497	α-copaene	0.5		1.2	1.3	2.8	1.8	1.0	0.4
	24	1379	1379	1765	1765	geranyl acetate	3.7	4.3	3.1	2.6	2.6	4.3	2.4	1.4	8.8	6.3
	25	1386	1387	1535	1535	β-bourbonene	1.6		1.1	0.9	1.3	2.2	1.2	1.0	2.9	3.3
	26	1389	1387	1547	1547	β-cubebene			0.4	0.2	0.4	0.4	0.2		0.4
	27	1391	1389	1600	1600	β-elemene			0.3	0.2	0.3	0.4	0.2		0.3
	28	1411	1410	1568	1568	α-cedrene	1.2	1.1	0.9	0.8	1.0	1.4	0.8	0.4	2.2	1.4
	29	1416	1417	1612	1612	β-caryophyllene	10.2	18.3	16.4	12.1	19.7	18.6	13.0	6.4	14.9	23.1
	30	1432	1431	1612	1612	β-gurjunene	0.5	1.5	0.7	0.4	0.9	1.0	0.5	0.2	0.9	0.8
	31	1438	1439	1628	1628	aromadendrene	0.2		1.0	0.3	0.6	0.4	0.4		0.4
	32	1454	1452	1668	1670	α-humulene	1.7	2.9	2.4	1.5	2.6	2.6	1.6	0.9	2.1	2.7
	33	1457	1454	1689	1689	E-β-farnesene	0.3	1.9		0.2		0.4	0.2		0.6
						epi-
	34	1462	1469	1734	1734	bicyclosesquiphella	0.7		1.4	0.3	1.0	0.5	0.5	0.4	0.6
						ndrene
	35	1478	1478	1704	1704	γ-muurolene		6.7			0.2		0.2	1.5
	36	1480	1484	1726	1726	germacrene D	4.1	4.5	7.8	3.9	9.2	8.6	4.4		7.9	6.1
	37	1483	1486	1784	1784	α-curcumene	0.3

Table 3 :

 3 Chemical composition of the essential oil of O. libanoticum harvested from Qartaba. In order to cover all the vegetative stages for plants development, aerial parts of O. libanoticum were collected twice monthly throughout 2013 and 2014 from March 1 st , 2013 and continued until December 1 st , 2014

						Harvesting date	Jul-13	Aug-13	Sep-13			Oct-13	Jul-14	Aug-14	Sep-14	Oct-14
						Drying method	L1	L1	D1	L1	D1	L1	D1	L1	L1	L1	L1
	N°	Ri a		Ri b		Volatile compound
	1	924	924	1035	1035	α-thujene		1.0	1.4		1.6	1.7	0.9				1.4
	2	933	932	1076	1076	α-pinene	0.9	4.0	5.2	1.3	5.9	6.4	3.5	0.4			2.1
	3	947	946	1076	1076	camphene		0.4				0.7
	4	973	969	1132	1132	sabinene	1.1	0.3	0.4		0.5	0.5	0.3	1.0			1.0
	5	980	974	1118	1118	β-pinene	1.5	0.9	1.2		1.5	2.0	1.2		2.0	0.6	1.0
	6	991	988	1174	1174	myrcene	1.3	2.2	3.4	t	3.8	3.7	2.2	0.8	1.2	0.4	2.1
	7	1003	1002	1188	1188	α-phellandrene	8.2	0.8	2.1		2.1	1.4	0.7				1.5
	8	1009	1008	1157	1159	δ-3-carene	0.5	0.4	1.1		0.8	0.6
	9	1014	1014	1188	1189	α-terpinene	2.5	2.1	3.2	2.4	3.6	3.3	1.9	0.5	0.4		2.6
	10	1024	1020	1280	1278	p-cymene	24.8	25.2	40.9	41.5	43. 1	40. 9	23.5	4.1	2.3	2.9	28.6
	11	1025	1024	1203	1203	limonene	6.2	6.9	11.8	6.5	12. 5	11. 3	6.7	1.0	1.0	0.3	6.7

a : Retention index calculated on a HP-5MS column; Ri b : Retention index on a HP-5MS column from literature

[START_REF] Zgheib | Chemometric tools to highlight the variability of the chemical composition and yield of Lebanese Origanum syriacum L. essential oil[END_REF]

; Ri c : Retention index calculated on a HP Innowax column; Ri d : Retention index on a HP Innowax column from literature

[START_REF] De Falco | Chemical Composition and Biological Activity of Essential Oils of Origanum vulgare L. subsp. vulgare L. under Different Growth Conditions[END_REF][START_REF] Khoury | Report on the Medicinal Use of Eleven Lamiaceae Species in Lebanon and Rationalization of Their Antimicrobial Potential by Examination of the Chemical Composition and Antimicrobial Activity of Their Essential Oils[END_REF]

). t: trace (less than 0.05.%). Blanks: not detected.

Experimental

Plant material and essential oil extraction:

20082008 caryophyllen e oxide

Acknowledgments -We are grateful for the support from CNRS Lebanon and USEK.