Sage Journals: Discover world-class research

Abstract

Several families of elastic anisotropies have been introduced by Saint Venant (Saint Venent, B. (de) (1863). Sur la distribution des élasticitiés autour de chaque point d'un solide ou d'un milieu de contexture quelconque, particulièrement lorsqu'il est amorphe sams être isotrope, Journal de Math. Pures et Appliquées, Tome VIII (2^éme série) pp. 257–430) for which the polar diagram of elastic parameters in different directions of the material (indicator surface) is ellipsoidal. These families cover a large variety of models introduced in recent years for damaged materials or as effective moduli of heterogeneous materials. Ellipsoidal anisotropy has also been used as a guideline in phenomenological modeling of materials. Then a question that naturally arises is to know in which conditions the assumption that some indicator surfaces are ellipsoidal allows one to entirely determine the elastic constants. This question has not been rigorously studied in the literature. In this study, first, several basic classes of ellipsoidal anisotropy are presented. Then the problem of the determination of elastic parameters from indicator surfaces is discussed in several basic cases that can occur in phenomenological modeling. Finally, the compatibility between the assumption of ellipsoidal form for different indicator surfaces is discussed. In particular, it is shown that if the indicator surfaces of ⁴ √E(n) and of ^-4 √c(n) (where E(n) and c(n) are, respectively, the Young's modulus and the elastic coefficient in the direction n) are ellipsoidal, then the two ellipsoids have necessarily the same principal axes, and the material in this case is orthotropic.

Keywords

linear elasticity anisotropy indicator surface ellipsoidal anisotropy amorphous materials damaged materials

Get full access to this article

View all access options for this article.

References

Alliche, A. (2004). Damage Model for Fatigue Loading of Concrete , Int. J. Fatigue, 26: 915–921 .

Benitez, F.G. and Rosakis, A.J. (1987). Three-dimensional Elastostatics of a Layer and a Layered Medium , J. Elasticity, 18: 3–50 .

Boehler, J.-P. (1975). Contribution théorique et expérimentale à l'étude des milieux plastiques anisotropes, Thèse d'état Institut de Meécanique de Grenoble, France.

Boehler, J.-P. (1982). Comportement mécanique des solides anisotropes, Colloque Euromech 115, Villard-de-Lans, Juin 1979, Editions CNRS, Paris.

Cazzani, A. and Marco, Rovati (2003). Extrema of Young's Modulus for Cubic and Transversely Isotropic Solids , Int. J. Solids and Structures, 40: 1713–1744 .

Cazzani, A. and Marco, Rovati (2005). Extrema of Young's Modulus for Elastic Solids with Tetragonal Symmetry , Int. J. Solids and Structures (in press).

Chiarelli, A.S. , Shao, J.F. and Hoteit, N. (2003). Modeling of Elastoplastic Damage Behaviour of a Claystone , Int. J. Plasticity, 19: 23–45 .

Cowin, S.C. and Mehrabadi, M.M. (1987). On the Identification of Material Symmetry for Anisotropic Elastic Materials , Quart. J. Mech. Appl. Math., 40: 451–476 .

Cowin, S.C. and Mehrabadi, M.M. (1995). Anisotropic Symmetries of Linear Elasticity , Appl. Mech. Rev., 48: 247–285 .

10.

Daley, P.F. and Hron, F. (1979). Reflection and Transmission Coefficients for Seismic Waves in Ellipsoidally Anisotropic Media , Geophysics, 44: 27–38 .

11.

Dragon, A. , Halm, D. and Désoyer, Th . (2000). Anisotropic Damage in Quasi-brittle Solids: Modelling Computational Issues and Applications, Comput . Methods Appl. Mech. Engrg., 183: 331–352 .

12.

Halm, D. and Dragon, A. (1988). An Anisotropic Model of Damage and Frictional Sliding for Brittle Materials , Eur. J. Mech., A/Solids, 17(3): 439–460 .

13.

Hé, Q.C. and Curnier, A. (1995). A More Fundamental Approach to Damaged Elastic Stress– Strain Relations , Int. J. Solids Structures, 32(10): 1433–1457 .

14.

Hefny, A.M. and Lo, K.L. (1999). Analytical Solutions for Stresses and Displacements around Tunnels Driven in Cross-anisotropic Rocks , Int. J. Numer. Anal. Meth. Geomech., 23: 161–177 .

15.

Homand, F. , Morel, E. , Henry, J.-P. , Cuxac, P. and Hammade, E. (1993). Characterization of the Moduli of Elasticity of an Anisotropic Rock using Dynamic and Static Methods , Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 30(5): 527–535 .

16.

Kachanov, M. (1992). Effective Elastic Properties of Cracked Solids: Critical Review of Some Concepts , Appl. Mech. Review, 45(8): 304–335 .

17.

Lekhnitskii, S.G. (1963). Theory of Elasticity of an Anisotropic Elastic Body, Holden Day Series in Mathematical Physics, Holden-Day, San-Francisco .

18.

Louis, L. , Robion, Ph . and David, Ch . (2004). A Single Method for Inversion of Anisotropic Data Sets with Application to Structural Studies , J. Structural Geology, 26: 2065–2072 .

19.

Milgrom, M. and Shtrikman, S. (1992). The Energy of Inclusions in Linear Media, Exact Shape Independent Relations , J. Mech. Phys. Solids, 40(5): 927–937 .

20.

Milton, G.W. (2002). The Theory of Composites, Cambridge University Press , pp. 145–147.

21.

Min, K.B. and Jing, L. (2003). Numerical Determination of the Equivalent Elastic Compliance Tensor for Fractured Rock Masses using the Distinct Element Method , Int. J. Rock Mech. Min. Sci., 40: 795–816 .

22.

Pan, E. and Amadei, B. (1996). Water Reservoir Loading of Long Anisotropic Valleys with Irregular Topographies , Appl. Math. Modelling, 20: 909–924 .

23.

Pan, Y.-C. and Chou, T.-W. (1976). Point Force Solution for an Infinite Transversely Isotropic Solid, Transactions of the ASME , Journal of Applied Mechanics, pp 608–612 .

24.

Peres Rodrigues, F. and Aires-Barros, L. (1970). Anisotropy of Endogenetic Rocks-correlation between Micropetrographic Index, Ultimate Strength and Modulus of Elasticity Ellipsoids , Proc. 2nd Congress of the ISRM, Belgrad, pp. 1–23 .

25.

Pouya, A. 2000. A Transformation of the Problem of Linear Elastic Structure for Application to Inclusion Problem and to Green Functions, Comptes-Rendus de l'Académie des Sciences de Paris, t. 328, Série II b, pp. 437–443.

26.

Pouya A. and Reiffsteck, Ph . (2003). Analytical Solutions for Foundations on Anisotropic Elastic Soils , In: Proceedings of International Symposium FONDSUP, Presses des Ponts et Chaussées, Paris.

27.

Pouya A. and Zaoui, A. (2005). A Transformation of Elastic Boundary Value Problems with Application to Anisotropic Behavior , I. J. Solids Structures (in press).

28.

Rongved, L. (1955). Force Interior to One of Two Joined Semi-infinite Solids , Proc. 2nd Midwestern Conf. Solid Mechanics, pp. 1–13 .

29.

Saint Venant, B. (de) (1863). Sur la distribution des élasticités autour de chaque point d'un solide ou d'un milieu de contexture quelconque, particulièrement lorsqu'il est amorphe sans être isotrope , Journal de Math. Pures et Appliqueées, Tome VIII (2éme série) pp. 257–430 .

30.

Sirotine, Y. and Chaskolskaia, M. (1984). Fondements de la physique des cristaux, Editions MIR, Moscou.

31.

Stroh, A.N. (1958). Dislocations and Cracks in Anisotropic Elasticity , Phil. Mag., 3: 625–646 .

32.

Takemura, T. , Golshani, A. , Oda, M. and Suzuki, K. (2003). Preferred Orientations of Open Microcracks in Granite and their Relation with Anisotropic Elasticity , Int. J. Rock Mech. & Mininig Sci., 40: 443–454 .

33.

Thomsen, L. (1986). Weak Elastic Anisotropy , Geophysics, 51(10): 1954–1966 .

34.

Ting, T.C.T. (1996). Anisotropic elasticity, Oxford University Press, Oxford .

35.

Zheng, Q.S. 1997. A Unified Invariant Description of Micromechanically-based Effective Elastic Properties for Two-dimensional Damaged Solids , Mechanics of materials, 25: 273–289 .

Ellipsoidal Anisotropies in Linear Elasticity: Extension of Saint Venant's Work to Phenomenological Modeling of Materials

Abstract

Keywords

Get full access to this article

References