Anisotropy

Anisotropic solids also are common in nature and technology. Examples are single crystals; polycrystals in which the grains are not completely random in their crystallographic orientation but have a “texture,” typically owing to some plastic or creep flow process that has left a preferred grain orientation; fibrous biological materials such as wood or bone; and composite materials that, on a microscale, either have the structure of reinforcing fibres in a matrix, with fibres oriented in a single direction or in multiple directions (e.g., to ensure strength along more than a single direction), or have the structure of a lamination of thin layers of separate materials. In the most general case, the application of any of the six components of stress induces all six components of strain, and there is no shortage of elastic constants. There would seem to be 6 × 6 = 36 in the most general case, but, as a consequence of the laws of thermodynamics, the maximum number of independent elastic constants is 21 (compared with 2 for isotropic solids). In many cases of practical interest, symmetry considerations reduce the number to far below 21. For example, crystals of cubic symmetry, such as rock salt (NaCl); face-centred cubic metals, such as aluminum, copper, or gold; body-centred cubic metals, such as iron at low temperatures or tungsten; and such nonmetals as diamond, germanium, or silicon have only three independent elastic constants. Solids with a special direction, and with identical properties along any direction perpendicular to that direction, are called transversely isotropic; they have five independent elastic constants. Examples are provided by fibre-reinforced composite materials, with fibres that are randomly emplaced but aligned in a single direction in an isotropic or transversely isotropic matrix, and by single crystals of hexagonal close packing such as zinc.

General linear elastic stress-strain relations have the formwhere the coefficients C_ijkl are known as the tensor elastic moduli. Because the ε_kl are symmetric, one may choose C_ijkl = C_ijlk, and, because the σ_ij are symmetric, C_ijkl = C_jikl. Hence the 3 × 3 × 3 × 3 = 81 components of C_ijkl reduce to the 6 × 6 = 36 mentioned. In cases of temperature change, the ε_ij above is replaced by ε_ij − ε_ij^thermal, where ε_ij^thermal = α_ij(θ − θ₀) and α_ij is the set of thermal strain coefficients, with α_ij = α_ji. An alternative matrix notation is sometimes employed, especially in the literature on single crystals. That approach introduces 6-element columns of stress and strain {σ} and {ε}, defined so that the columns, when transposed (superscript T) or laid out as rows, are {σ}^T = (σ₁₁, σ₂₂, σ₃₃, σ₁₂, σ₂₃, σ₃₁) and {ε}^T = (ε₁₁, ε₂₂, ε₃₃, 2ε₁₂, 2ε₂₃, 2ε₃₁). These forms assure that the scalar {σ}^T{dε} ≡ tr([σ][dε]) is an increment of stress working per unit volume. The stress-strain relations are then written {σ} = [c]{ε}, where [c] is the 6 × 6 matrix of elastic moduli. Thus, c₁₃ = C₁₁₃₃, c₁₅ = C₁₁₂₃, c₄₄ = C₁₂₁₂, and so on.