Abstract
In this chapter the general concepts of anisotropy are introduced, starting from the basic question: what is anisotropy? The mathematical consequences of anisotropy on algebraic operators are introduced along with the concept of geometrical symmetries. Different examples of anisotropic physical phenomena are given, and a brief account of the anisotropy of crystals is proposed. The Chapter ends recalling some fundamental laws of elasticity, useful in the following.
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Notes
- 1.
In old texts, anisotropy is sometimes called æolotropy, like in the classical book of Love (1944); this term comes from ancient Greek \(\alpha \grave{\iota } \acute{o} \lambda o \varsigma \), that means changeful.
- 2.
Following common standard rules, we will indicate by \(\mathscr {E}\) the natural Euclidean space, by \(\mathscr {V}\) the vector space of translations on \(\mathscr {E}\), or generally speaking any other vector space, when not differently indicated, by Lin the manifold of second-order tensors and by \(\mathbb {L}\)in that of fourth-rank tensors. Normally, elements of \(\mathscr {E}\), points, will be denoted by italic lowercase letters, e.g. p, elements of \(\mathscr {V}\), vectors, by bold lowercase letters, e.g. \(\mathbf {v}\), elements of Lin, second-rank tensors, by bold uppercase letters, e.g. \(\mathbf {L}\), and elements of \(\mathbb {L}in\), fourth-rank tensors, by special characters, for instance like \(\mathbb {E}\).
- 3.
An orthogonal tensor U is a tensor that preserves the inner product (and hence angles):
$$ \mathbf {Uv}_1\cdot \mathbf {Uv}_2=\mathbf {v}_1\cdot \mathbf {v}_2\ \forall \mathbf {v}_1,\mathbf {v}_2\in \mathscr {V}; $$choosing \(\mathbf {v}_1=\mathbf {v}_2\) we see that an orthogonal tensor preserves not only angles but also lengths: that is why any rigid transformation of the space can be represented uniquely by an orthogonal tensor. From the above definition, it follows immediately that
$$ \mathbf {UU}^\top =\mathbf {U}^\top \mathbf {U}=\mathbf {I}, $$and finally, by the uniqueness of the inverse tensor,
$$ \mathbf {U}^{-1}=\mathbf {U}^\top , $$while by the Binet’s theorem on the determinant of a product of tensors we get
$$ \det \mathbf {U}=\pm 1. $$ - 4.
An orthonormal basis \(\mathscr {B}=\{\mathbf {e}_1,\mathbf {e}_2,\mathbf {e}_3\}\) is said to be positively oriented if
$$ \mathbf {e}_1\times \mathbf {e}_2\cdot \mathbf {e}_3=+1. $$Because of the identity, see e.g. (Gurtin 1981, p. 8),
$$ \mathbf {Lv}_1\times \mathbf {Lv}_2\cdot \mathbf {Lv}_3=\det \mathbf {L}\ \mathbf {v}_1\times \mathbf {v}_2\cdot \mathbf {v}_3\ \ \forall \mathbf {v}_1,\mathbf {v}_2,\mathbf {v}_3\in \mathscr {V}\ \mathrm {and}\ \mathbf {L}\in \mathrm {Lin}, $$we see that only orthogonal tensors with positive determinant preserve the space orientation and hence can represent rigid rotations. Because an inversion of the space orientation corresponds to change the orientation of one of the vectors of the triad, reflexions are represented by orthogonal tensors with negative determinant.
- 5.
Unless explicitly indicated, we will use the Einstein’s convention on dummy indexes for denoting summation.
- 6.
Generally speaking, the comma as subscript will be used to denote differentiation: \(u_{i,j}=\frac{\partial u_i}{\partial x_j}\).
- 7.
\(\forall \mathbf {a},\mathbf {b},\mathbf {c}\in \mathscr {V}\), the dyad \((\mathbf {a}\otimes \mathbf {b})\) is the second-rank tensor defined by the operation \((\mathbf {a}\otimes \mathbf {b})\mathbf {c}:=\mathbf {b}\cdot \mathbf {c}\ \mathbf {a}\). Given an orthonormal basis \(\mathscr {B}=\{\mathbf {e}_1,\mathbf {e}_2,\mathbf {e}_3\}\) of \(\mathscr {V}\), any second-rank tensor \(\mathbf {L}\) can be expressed as
$$ \mathbf {L}=L_{ij}\mathbf {e}_i\otimes \mathbf {e}_j, $$where the Cartesian components \(L_{ij}\) of \(\mathbf {L}\) are
$$ L_{ij}=\mathbf {e}_i\cdot \mathbf {Le}_j. $$
References
M.E. Gurtin, An Introduction to Continuum Mechanics (Academic Press Inc., New York, 1981)
S.G. Lekhnitskii, Theory of elasticity of an anisotropic elastic body, 1950. (English translation by P. Fern, Holden-Day, San Francisco, 1963)
W.J. Lewis, Treatise on Cystallography (Cambridge University Press, Cambridge, 1899)
A.E.H. Love, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944)
H.A. Miers, Mineralogy (Oxford University Press, Oxford, 1902)
F. Neumann, Vorlesungen über die Theorie der Elasticität (B. G. Teubner, Leipzig, 1885)
J.F. Nye, Physical Properties of Crystals (Clarendon Press, Oxford, 1957)
O. Rand, V. Rovenski, Analytical Methods in Anisotropic Elasticity (Birkhäuser, Boston, 2005)
A.M. Schoenflies, Krystallsysteme und Krystallstructur (B. G. Teubner, Leipzig, 1891)
W. Voigt, Lehrbuch der Kristallphysik (B. G. Teubner, Leipzig, 1910)
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Vannucci, P. (2018). Basic Concepts on Anisotropy. In: Anisotropic Elasticity . Lecture Notes in Applied and Computational Mechanics, vol 85. Springer, Singapore. https://doi.org/10.1007/978-981-10-5439-6_1
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DOI: https://doi.org/10.1007/978-981-10-5439-6_1
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