The extension of the finite strain expansion of the Mie-Griineisen equation (taking into account the elastic constants in the reference configuration up to the fourth-order) is used to derive the temperature dependence of the volumetric compression and of the second-order elastic adiabatic moduli of cubic solids. Numerical results are given for germanium and silicon and compared to experimental data. FinalIy, the second pressure derivative of these constants is estimated a t zero pressure and 300 K.L'extension au domaine des dkformations finies du modele d'kquation d'ktat de Mie-Gruneisen (prenant en compte les constantes klastiques dans la configuration de rkfkrence jusqu'au quatrikme ordre inclus) permet d'obtenir les variations, en fonction de la tempkrature, du taux de compression volumique et des modules klastiques adiabatiques du second ordre pour les solides de symktrie cubique. Des applications numkriques sont donnkes pour le germanium et le silicium et comparkes aux mesures expkrimentales. Finalement, nous donnons une estimation des dkrivkes secondes des modules effectifs par rapport B la pression, B pression nulle et 300 K.
The fourth-order anharmonic finite strain theory leads to a model for the temperature and pressure dependences of effective elastic moduli, for non-cubic crystals, involving the elastic constants up to the fourth-order. These relationships are used to give an estimation of some combinations of fourth-order elastic constants, using a mean deformation in the fourth-order term and fitting the experimental data at low temperatures. The fourth-order variations of the effective adiabatic elastic moduli with temperature at zero pressure can then be determined. A comparative study of the results with experimental data is given for three hexagonal metals-cadmium, magnesium and zinc-which shows a good agreement between theory and experiment over a large range of temperatures. This development uses both the Lagrangian and Eulerian strain measures previously introduced.
Expressions for the linear adiabatic compressibilities of an at least orthorhombic crystal and their pressure derivatives are derived in terms of the second-, third-, and fourth-order elastic constants within a finite-strain theory which includes thermal effects according to the quasi-harmonic approximation of lattice dynamics. The expressions are derived in terms of the Lagrangian strain tensor and the frame-indifferent analogue of the Eulerian strain tensor. Numerical applications are given for three hexagonal metals: cadmium, magnesium, and zinc, namely the temperature variations of the linear adiabatic compressib es and their pressure derivatives, including recent estimations of the combinations of fourth-order elastic constants appearing in the present work. When possible, the comparison with experimental data shows a rather good agreement with the compressibilities calculated within the present theory using the Green-Lagrange strain measure of the deformation. About the variations of the first pressure derivatives of the adiabatic compressibilities with temperature, the present work provides us with theoretical values in the lack of experimental data.Les coefficients lineaires de compressibilite adiabatique et leurs derivees par rapport a la pression sont exprimes, pour un cristal de symetrie au moins orthorhombique, en fonction des constantes elastiques des deuxiime, troisikme et quatrieme ordres dans le cadre d'une theorie des dkformations finies qui inclut les effects thermiques grice a la thtorie de la dynamique des reseaux cristallins developpee dans I'approximation quasi-harmonique. Deux tenseurs de deformation sont utilises: le tenseur de GreenLagrange et fe tenseur de deformation, invariant par changement de referentiel, analogue a u tenseur d'Almansi-Euler. Des applications numeriques sont donnees pour trois metaux hexagonaux: le cadmium, le magnesium et le zinc, a savoir les variations avec la temperature des coefficients lintaires de compressibilite adiabatique et de leurs derivees par rapport a la pression. Ces calculs utilisent de recentes estimations des combinaisons de constantes tlastiques du quatritme ordre qui figurent dans cette etude. Lorsque cela est possible, nos resultats sont compares aux valeurs experimentales, comparaison qui met en evidence un accord satisfaisant entre les compressibilites calculees d'apres la presente theorie avec le tenseur de deformation de Green-Lagrange et les valeurs mesurees experimentalement. En ce qui concerne les variations, avec la temperature, des dtrivees premieres par rapport a la pression des coefficients lineaires de compressibilite adiabatique, le present article permet de donner des valeurs theoriques en l'absence de resultats experimentaux.
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