On the elastic moduli and compliances of transversely isotropic and orthotropic materials

The relationships between the elastic moduli and compliances of transversely isotropic and orthotropic materials, which correspond to different appealing sets of linearly independent fourth-order base tensors used to cast the elastic moduli and compliances tensors, are derived by performing explicit inversions of the involved fourth-order tensors. The deduced sets of elastic constants are related to each other and to common engineering constants expressed in the Voigt notation with respect to the coordinate axes aligned along the directions orthogonal to the planes of material symmetry. The results are applied to a transversely isotropic monocrystalline zinc and an orthotropic human femural bone.

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

[2]  P. Theocaris,et al.  Spectral decomposition of the compliance fourth-rank tensor for orthotropic materials , 2000 .

[3]  J. P. Boehler,et al.  Representations for Isotropic and Anisotropic Non-Polynomial Tensor Functions , 1987 .

[4]  Stephen C. Cowin,et al.  Anisotropic Symmetries of Linear Elasticity , 1995 .

[5]  S. White,et al.  Stress analysis of fiber-reinforced composite materials , 1997 .

[6]  Vlado A. Lubarda,et al.  An analysis of large-strain damage elastoplasticity , 1994 .

[7]  T. Ting.  INVARIANTS OF ANISOTROPIC ELASTIC CONSTANTS , 1987 .

[8]  Vlado A. Lubarda,et al.  On the mechanics of solids with a growing mass , 2002 .

[9]  S. Nigam,et al.  Invariant Elastic Constants for Crystals , 1968, July 1.

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

[11]  C. Gans,et al.  Biomechanics: Motion, Flow, Stress, and Growth , 1990 .

[12]  J. O. Thompson.  HOOKE'S LAW. , 1926, Science.

[13]  George Z. Voyiadjis,et al.  Advances in Damage Mechanics: Metals and Metal Matrix Composites With an Introduction to Fabric Tensors , 1999 .

[14]  S. Sutcliffe.  Spectral Decomposition of the Elasticity Tensor , 1992 .

[15]  A. Drozdov.  Mechanics of Viscoelastic Solids , 1998 .

[16]  Q. Zheng.  Theory of Representations for Tensor Functions—A Unified Invariant Approach to Constitutive Equations , 1994 .

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

[18]  M. Epstein,et al.  Cardiovascular Solid Mechanics: Cells, Tissues, and Organs , 2002 .

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

[20]  E. Dill,et al.  Theory of Elasticity of an Anisotropic Elastic Body , 1964 .

[21]  Q. He,et al.  Decomposition of elasticity tensors and tensors that are structurally invariant in three dimensions , 2006 .

[22]  Robert J. Asaro,et al.  Mechanics of Solids and Materials , 2006 .

[23]  A. Spencer,et al.  The Formulation of Constitutive Equation for Anisotropic Solids , 1982 .

[24]  J. Betten.  Invariants of Fourth-Order Tensors , 1987 .

[25]  E. Morozov,et al.  Mechanics and analysis of composite materials , 2001 .

[26]  Adam Morecki.  Biomechanics of Motion , 1980 .

[27]  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.

[28]  I. Kunin,et al.  An algebra of tensor operators and its applications to elasticity , 1981 .

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

[30]  T. Ting.  Anisotropic Elastic Constants That are Structurally Invariant , 2000 .

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

[32]  M. Meyers,et al.  On the negative poisson ratio in monocrystalline zinc , 1999 .