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Comptes Rendus. Mécanique

Polycrystal thermo-elasticity revisited: theory and applications
Comptes Rendus. Mécanique, Tome 348 (2020) no. 10-11, pp. 877-891.

Article du numéro thématique : Contributions in mechanics of materials

The self-consistent (SC) theory is the most commonly used mean-field homogenization method to estimate the mechanical response behavior of polycrystals based on the knowledge of the properties and orientation distribution of constituent single-crystal grains. The original elastic SC method can be extended to thermo-elasticity by adding a stress-free strain to an elastic constitutive relation that expresses stress as a linear function of strain. With the addition of this independent term, the problem remains linear. Although the thermo-elastic self-consistent (TESC) model has important theoretical implications for the development of self-consistent homogenization of non-linear polycrystals, in this paper, we focus on TESC applications to actual thermo-elastic problems involving non-cubic (i.e. thermally anisotropic) materials. To achieve this aim, we provide a thorough description of the TESC theory, which is followed by illustrative examples involving cooling of polycrystalline non-cubic metals. The TESC model allows studying the effect of crystallographic texture and single-crystal elastic and thermal anisotropy on the effective thermo-elastic response of the aggregate and on the internal stresses that develop at the local level.

Première publication : 2020-11-18
Publié le : 2021-01-13
DOI : https://doi.org/10.5802/crmeca.18
Mots clés : Homogenization, Self-consistent methods, Thermo-elasticity, Polycrystals, Anisotropy, Metals
     author = {Carlos N. Tom\'e and Ricardo A. Lebensohn},
     title = {Polycrystal thermo-elasticity revisited: theory and applications},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {877--891},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {348},
     number = {10-11},
     year = {2020},
     doi = {10.5802/crmeca.18},
     language = {en},
     url = {comptes-rendus.academie-sciences.fr/mecanique/item/CRMECA_2020__348_10-11_877_0/}
Carlos N. Tomé; Ricardo A. Lebensohn. Polycrystal thermo-elasticity revisited: theory and applications. Comptes Rendus. Mécanique, Tome 348 (2020) no. 10-11, pp. 877-891. doi : 10.5802/crmeca.18. https://comptes-rendus.academie-sciences.fr/mecanique/item/CRMECA_2020__348_10-11_877_0/

[1] A. V. Hershey The elasticity of an isotropic aggregate of anisotropic cubic crystals, J. Appl. Mech.-Trans. ASME, Volume 21 (1954), p. 236 | Zbl 0059.17604

[2] E. Kröner Berechnung der elastischen Konstanten des Vielkristalls aus den Konstanten des Einkristalls, Z. Phys., Volume 15 (1958), p. 504 | Article

[3] R. Hill Continuum micro-mechanics of elastoplastic polycrystals, J. Mech. Phys. Solids, Volume 13 (1965), p. 89 | Article | Zbl 0127.15302

[4] R. Zeller; P. H. Dederichs Elastic constants of polycrystals, Phys. Status Solidi (b), Volume 55 (1973), p. 831 | Article

[5] C. N. Tomé; R. A. Lebensohn ViscoPlastic Self-Consistent (VPSC) code (Los Alamos National Laboratory)

[6] L. J. Walpole On the overall elastic moduli of composite materials, J. Mech. Phys. Solids, Volume 17 (1969), p. 235 | Article | Zbl 0177.53204

[7] J. R. Willis Variational and related methods for the overall properties of composites, Adv. Appl. Mech., Volume 21 (1981), p. 1 | Article | MR 706965 | Zbl 0476.73053

[8] V. A. Buryachenko Multiparticle effective field and related methods in micromechanics of random structure composites, Math. Mech. Solids, Volume 6 (2001), p. 577 | Article | Zbl 1128.74301

[9] G. W. Milton The Theory of Composites, Cambridge University Press, 2002 | Zbl 0993.74002

[10] C. N. Tomé; N. Christodoulou; P. A. Turner; M. A. Miller; C. H. Woo; J. Root; T. M. Holden Role of internal stresses in the transient of irradiation growth of Zircaloy-2, J. Nucl. Mater., Volume 227 (1996), p. 237 | Article

[11] R. Hill The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids, Volume 15 (1967), p. 79 | Article

[12] T. Mura Micromechanics of Defects in Solids, Martinus-Nijhoff Publishers, Dordrecht, The Netherlands, 1987

[13] J. D. Eshelby The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. R. Soc. Lond. A, Volume 241 (1957), p. 366 | MR 87326 | Zbl 0079.39606

[14] M. Berveiller; O. Fassi-Fehri; A. Hihi The problem of two plastic and heterogeneous inclusions in an anisotropic medium, Int. J. Eng. Sci., Volume 25 (1987), p. 691 | Article | Zbl 0612.73010

[15] M. Bobeth; G. Diener Static elastic and thermoelastic fluctuations in multiphase composites, J. Mech. Phys. Solids, Volume 35 (1987), p. 37 | Article | Zbl 0601.73001

[16] W. Kreher Residual stresses and stored elastic energy of composites and polycrystals, J. Mech. Phys. Solids, Volume 38 (1990), p. 115 | Article | Zbl 0701.73008

[17] Y. Liu Macroscopic behavior, fluctuations and texture evolution in viscoplastic polycrystals (2003) (Ph. D. Thesis)

[18] R. Brenner; O. Castelnau; L. Badea Mechanical field fluctuations in polycrystals estimated by homogenization techniques:, Proc. R. Soc. Lond. A, Volume 460 (2004), p. 3589 | Article | MR 2166303 | Zbl 1104.74054

[19] R. A. Lebensohn; C. N. Tomé; P. P. Castañeda Self-consistent modeling of the mechanical behavior of viscoplastic polycrystals incorporating intragranular field fluctuations, Phil. Mag., Volume 87 (2007), p. 4287 | Article

[20] G. Simmons; H. Wang Single Crystal Elastic Constants and Calculated Aggregate Properties: A Handbook, MIT Press, 1971

[21] S. R. MacEwen; C. N. Tomé; J. Faber Jr. Residual stresses in annealed Zircaloy, Acta Metall., Volume 37 (1989), p. 979 | Article

[22] O. Castelnau; D. K. Blackman; R. A. Lebensohn; P. P. Castañeda Micromechanical modelling of the viscoplastic behavior of olivine, J. Geophys. Res. B, Volume 113 (2008), p. B09202 | Article

[23] P. Gordon A high temperature precision X-ray camera. Some measurements of the thermal coefficients of expansion of Beryllium, J. Appl. Phys., Volume 20 (1949), p. 908 | Article

[24] D. W. Brown; T. A. Sisneros; B. Clausen; S. Abeln; M. A. M. Bourke; B. G. Smith; M. L. Steinzig; C. N. Tomé; S. C. Vogel Development of intergranular thermal residual stresses in Beryllium during cooling from processing temperatures, Acta Mater., Volume 57 (2009), p. 972 | Article

[25] E. S. Fisher Temperature dependence of the elastic moduli in α-uranium single crystals, J. Nucl. Mater., Volume 18 (1966), p. 39 | Article

[26] L. T. Lloyd; C. S. Barrett Thermal expansion of α-uranium, J. Nucl. Mater., Volume 18 (1966), p. 55 | Article

[27] P. A. Turner; C. N. Tomé A study of residual stresses in Zircaloy-2 with rod texture, Acta Metall. Mater., Volume 42 (1994), p. 4143 | Article

[28] Y. Jeong; C. N. Tomé Extension of the visco-plastic self-consistent model to account for elasto-visco-plastic behavior using a perturbed visco-plastic approach, Modell. Simul. Mater. Sci. Eng., Volume 27 (2019), 085013 | Article

[29] M. Zecevic; R. A. Lebensohn New robust self-consistent homogenization schemes of elasto-viscoplastic polycrystals, Int. J. Solids Struct., Volume 202 (2020), p. 434 | Article