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When a seismic wave passes through a rock, the wave imposes stresses and strains on the rock and on the individual mineral grains that compose the rock. In Earth’s mantle, these stresses and strains are much smaller than those that arise from compression and heating of the rock to pressures and temperatures in the mantle. The mineral grains or crystals react to compression and heating as well as to seismic waves mainly by reversible elastic deformation. For each crystal grain in the rock, the small stresses and strains imposed by a seismic wave can be assumed to obey a linear relationship between the stress tensor σ_ij and the infinitesimal strain tensor e_ij (Haussühl, 2007; Nye, 1985):

with the components of the elastic stiffness tensor c_ijkl and the elastic compliance tensor s_ijkl of the crystal.

Large strains that arise from compression and thermal expansion can be described by the Eulerian finite‐strain tensor E_ij (Birch, 1947; Davies, 1974; Stixrude & Lithgow‐Bertelloni, 2005). For hydrostatic compression and heating, the resulting finite strain is dominated by volume strain and can be approximated by an isotropic tensor with E_ij = − fδ_ij where f = [(V₀/V)^2/3 − 1]/2. V₀/V is the ratio of the volume V₀ of the crystal at the reference state to the volume V in the compressed and hot state. In principle, the definition of the reference state is arbitrary. From an experimental point of view, it is convenient to define the reference state to be the state of the mineral at ambient pressure and temperature, i.e., P₀ = 1 × 10^–4 GPa and T₀ = 298 K. Based on an expansion of the Helmholtz free energy F in finite strain, physical properties, including pressure and elastic stiffnesses, can be described as a function of volume and temperature (Birch, 1947; Davies, 1974; Thomsen, 1972a). Throughout this chapter, I will use the self‐consistent formalism presented by Stixrude and Lithgow‐Bertelloni (2005) that combines finite‐train theory with a quasi‐harmonic Debye model for thermal contributions to calculate pressure and elastic properties (Ita & Stixrude, 1992; Stixrude & Lithgow‐Bertelloni, 2005).

For a truncation of the Helmholtz free energy after the fourth‐order term in finite strain, the components of the isentropic elastic stiffness tensor are given by (Davies, 1974; Stixrude & Lithgow‐Bertelloni, 2005):

where δijkl = −δ_ijδkl −δilδjk −δjlδik. The parameters of the cold part (lines 1–5) of this expression are the components of the isothermal elastic stiffness tensor c_ijkl0 at the reference state and their first and second pressure derivatives and , respectively. They combine to the isothermal bulk modulus K₀ at the reference state as:

In an analogous way, the first and second pressure derivatives of the components of the elastic stiffness tensor combine to the first and second pressure derivatives of the bulk modulus and , respectively. The number of independent components of the elastic stiffness tensor depends on the crystal symmetry of the mineral and ranges from 21 for monoclinic (lowest) symmetry to 2 for an elastically isotropic material (Haussühl, 2007; Nye, 1985). Many mantle minerals, including olivine, wadsleyite, and bridgmanite, display orthorhombic crystal symmetry with 9 independent components of the elasticity tensors. For minerals with cubic crystal symmetry, such as ringwoodite, most garnets, and ferropericlase, the number of independent components is reduced to 3.

The thermal contribution (lines 6–7) includes the changes in internal energy Δ_THU and in the product of isochoric heat capacity C_V and temperature T, Δ_TH(C_VT ), that result from heating the mineral at constant volume V from the reference temperature T₀ to the temperature T. The internal energy U is computed from a Debye model (Ita & Stixrude, 1992) based on an expansion of the Debye temperature θ in finite strain (Stixrude & Lithgow‐Bertelloni, 2005). The Grüneisen tensor is then defined as:

and the strain derivative of the Grüneisen tensor as:

The Grüneisen tensor and the tensor η_ijkl are functions of finite strain themselves that are parameterized using their values at ambient conditions, i.e., γ_ij0 and η_ijkl0 (Stixrude & Lithgow‐Bertelloni, 2005).

Seismic waves travel with wavelengths on the order of kilometers that by far exceed the grain sizes of mantle rocks. Seismic waves thus probe the elastic response of a collection or an aggregate of mineral grains. In the simplest case of a monomineralic aggregate, all grains are crystals of the same mineral. The crystals can be randomly oriented or aligned to form a crystallographic preferred orientation (CPO). CPO can result from deformation or from crystallization of mineral grains under the action of an anisotropic external field (Karato, 2008). In general, crystals are elastically anisotropic, and any preferred orientation of crystals will partly transfer the elastic anisotropy of individual crystals to the aggregate (Mainprice, 2015; Mainprice et al., 2000).

When combining the elastic properties of individual crystals to those of the aggregate, we further have to take into account that stresses and strains may not be distributed homogeneously throughout the aggregate as some crystals may behave stiffer and deform less than others. The effective stiffness of a crystal depends on the orientation of the crystal with respect to the imposed stress and strain fields. Crystals may also be clamped between neighboring grains and be forced into a strain state that may not correspond to a state of unconstrained mechanical equilibrium with the imposed external stress field. Approximations and bounds to the complex elastic behavior of crystalline aggregates have been reviewed by Watt et al. (1976). Here, I adopt the simplest bounding scheme based on the Voigt and Reuss bounds (Reuss, 1929; Voigt, 1928; Watt et al., 1976) and assume a random orientation of crystals.

The Voigt bound is based on the assumption that the strain is homogeneous throughout the aggregate (Reuss, 1929; Watt et al., 1976). Regardless of its orientation, each crystal in the aggregate is deformed according to the external strain field imposed, for example, by a seismic wave. Because the resulting stresses are calculated from the strains using the elastic stiffnesses, we need to average the components of the elastic stiffness tensor c_ijkl over all orientations. For a monomineralic aggregate, this leads to the following expressions for the isotropic bulk modulus K and the isotropic shear modulus G (Hill, 1952; Watt et al., 1976):

The Reuss bound assumes that the imposed external stress field is homogeneous throughout the aggregate (Reuss, 1929; Watt et al., 1976). In this case, the strains are calculated from the stresses using the components of the elastic compliance tensor s_ijkl. The isotropic bulk and shear moduli of a monomineralic aggregate of randomly oriented crystals are then (Hill, 1952; Watt et al., 1976):

Most mantle rocks are composed of several mineral phases, each of which has different elastic properties. For a polymineralic rock composed of different mineral phases with volume fractions v_i, the bulk and shear moduli M_i of individual mineral phases are then combined to the Voigt bound as (Watt et al., 1976):

and to the Reuss bound as (Watt et al., 1976):

While the Voigt and Reuss averages provide bounds to the elastic behavior of a crystalline aggregate, the Voigt‐Reuss‐Hill averages are often used as an approximation to the actual elastic response (Chung & Buessem, 1967; Hill, 1952; Thomsen, 1972b):

Note that the calculation of the Voigt and Reuss bounds in their strictest sense would require the elastic stiffness and compliance tensors of all minerals in the rock at the respective pressure and temperature. For many minerals, however, bulk and shear moduli have so far only been determined on polycrystalline aggregates and are themselves averages over the elastic properties of many individual crystals. For polycrystalline aggregates, the changes of the aggregate bulk and shear moduli of a mineral that result from compression and heating can be described by combining the finite‐strain expressions for individual components c_ijkl to finite‐strain expressions for the isotropic bulk and shear moduli (Birch, 1947; Davies, 1974; Stixrude & Lithgow‐Bertelloni, 2005). Under the assumption of isotropic finite strain, the expressions would resemble those for the Voigt bound. In analogy to the elastic moduli, the components of the tensor η_ijkl are combined to the isotropic parameters η_V and η_S with respect to volume and shear strain, respectively (Stixrude & Lithgow‐Bertelloni, 2005). When determined on a polycrystalline sample, however, aggregate moduli are not readily identified with either of the bounds or with a simple average as the elastic response of the aggregate falls somewhere in between the Voigt and Reuss bounds and may even fall outside the bounds if a CPO is present.

Once the elastic properties of individual minerals have been combined to bulk and shear moduli of an isotropic polymineralic rock, the propagation velocities v of longitudinal waves (P) and transverse or shear waves (S) can be calculated by:

The density ρ of the rock is computed by combining the molar masses and volumes of all minerals according to their abundances. Note that wave velocities discussed throughout this chapter are based on the elastic response of minerals and rocks only and do not include corrections for anelastic behavior that might affect wave velocities at seismic frequencies and at high temperatures (Jackson, 2015; Karato, 1993).