Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Youngs modulus EðnÞ, embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which EðnÞ attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Youngs modulus on direction n are given as well.

[1]  L. Walpole,et al.  Fourth-rank tensors of the thirty-two crystal classes: multiplication tables , 1984, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[2]  S. G. Lekhnit︠s︡kiĭ.  Theory of elasticity of an anisotropic body , 1981 .

[3]  Anthony Kelly,et al.  Crystallography and crystal defects , 1970 .

[4]  J. Rychlewski,et al.  Generalized proper states for anisotropic elastic materials , 2001 .

[5]  J. Rychlewski.  A qualitative approach to Hooke's tensors. Part I , 2000 .

[6]  M. Hayes,et al.  On young's modulus for anisotropic media , 1995 .

[7]  A. Maradudin,et al.  An Introduction To Applied Anisotropic Elasticity , 1961 .

[8]  E. Schmid,et al.  Kristallplastizität mit besonderer Berücksichtigung der Metalle , 1935 .

[9]  Maurizio Vianello,et al.  Symmetry classes for elasticity tensors , 1996 .

[10]  C. Zener,et al.  Élasticité et anélasticité des métaux , 1955 .

[11]  P. Pedersen.  On optimal orientation of orthotropic materials , 1989 .

[12]  M. Gurtin.  The Linear Theory of Elasticity , 1973 .

[13]  Stephen C. Cowin,et al.  EIGENTENSORS OF LINEAR ANISOTROPIC ELASTIC MATERIALS , 1990 .

[14]  J. Nye.  Physical Properties of Crystals: Their Representation by Tensors and Matrices , 1957 .

[15]  Michael Hayes,et al.  On the Extreme Values of Young’s Modulus, the Shear Modulus, and Poisson’s Ratio for Cubic Materials , 1998 .

[16]  J. Rychlewski,et al.  On Hooke's law☆ , 1984 .

[17]  H. B. Huntington.  The Elastic Constants of Crystals , 1958 .

[18]  F. Fedorov.  Theory of Elastic Waves in Crystals , 1968 .

[19]  Walter Noll,et al.  Material symmetry and thermostatic inequalities in finite elastic deformations , 1964 .

[20]  Robert Bruce Lindsay,et al.  Physical Properties of Crystals , 1957 .

[21]  A. Love.  A treatise on the mathematical theory of elasticity , 1892 .

[22]  G. F. Smith,et al.  THE STRAIN-ENERGY FUNCTION FOR ANISOTROPIC ELASTIC MATERIALS , 1958 .

[23]  E. Goens.  Über eine verbesserte Apparatur zur statischen Bestimmung des Drillungsmoduls von Kristallstäben und ihre Anwendung auf Zink‐Einkristalle , 1933 .

[24]  Stephen C. Cowin,et al.  A new proof that the number of linear elastic symmetries is eight , 2001 .

[25]  R. Bechmann,et al.  Numerical data and functional relationships in science and technology , 1969 .

[26]  J. Nadeau,et al.  On optimal zeroth-order bounds with application to Hashin–Shtrikman bounds and anisotropy parameters , 2001 .

[27]  H. Wenk,et al.  Texture and Anisotropy , 2004 .

[28]  G. Piero,et al.  On the completeness of the crystallographic symmetries in the description of the symmetries of the elastic tensor , 1991 .

[29]  J. W. Edington.  Interpretation of transmission electron micrographs , 1975 .

[30]  Henry E. Bass,et al.  Handbook of Elastic Properties of Solids, Liquids, and Gases , 2004 .

[31]  J. C. Nadeau,et al.  Invariant Tensor-to-Matrix Mappings for Evaluation of Tensorial Expressions , 1998 .