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Use the same real elastic stiffness tensor for both fields, including its minor and major symmetries, and let $t_i^a={\hat{\sigma }}_{ij}^a{n}_j$, with outward unit boundary normal $n_j$. No complex conjugation is needed for the initial bilinear identity. DefineThe divergence has two force-balance terms and two cross-gradient terms. The latter cancel becausethis follows by exchanging the two strain-gradient factors using major symmetry and the minor pair symmetries. The frequency-domain equations then giveIntegrating and applying the divergence theorem provesThis is the Betti identity for elastodynamic fields. Spatially varying stiffness is allowed because its derivatives remain inside the divergence. The original PDF's constitutive contraction is $c_{ijpq}{\partial }_p{\hat{u}}_q$; the TeX's repeated displacement field index is erroneous.

Take a positive comparison elastic stiffness tensor $C^0$, excess energies $F_r(e)=W_r(e)-e:C^{0}e/2$, and a phasewise constant elastic polarization. The lower conjugate inequality majorizes the local energy by a quadratic-plus-linear function. Minimizing that function over compatible strains gives the displayed bound, where $A_{rs}=\int \int ({\chi }_r-c_r)\Gamma ({\chi }_s-c_s)$. Stationarity in ${\tau }^r$ gives $c_r{e}^r+{\sum }_s{A}_{rs}{\tau }^s=c_{r}E$, with $e^r=D{F}_{r\ast }({\tau }^r)$. Without this stationarity, dual strains must not be identified with the comparison solution's phase means.

A localized zero-resultant seismic moment tensor represents the leading force-dipole part of an elastic wave source through the displayed equivalent body force. For a prescribed displacement jump $b_i$ across a surface with normal $n_j$, its moment density is $c_{ijkl}{b}_k{n}_l$, where $c$ is the elastic stiffness tensor. Spatial integration gives the point-source moment when the source is small compared with the wavelengths.

A small planar crack of area $S$ opening normally by $d(t)$ has volume change $V=Sd$ and the displayed seismic moment tensor. Its principal dipoles are $\lambda V,\lambda V,(\lambda +2\mu )V$: two act in the crack plane and the largest acts normally for positive opening and ordinary positive Lamé parameters. The definition follows by contracting the isotropic elastic stiffness tensor with two copies of the crack normal.

A rotation through $\pi$ about the third coordinate axis preserves the strain components $e_{11},e_{22},e_{33},e_{12}$ and reverses $e_{13},e_{23}$. A symmetry-invariant elastic stiffness tensor cannot couple the two parity spaces. Its independent symmetric blocks therefore contain $4\cdot 5/2+2\cdot 3/2=13$ coefficients. A twofold axis alone does not impose invariance under every rotation about that axis.

The Voigt elastic approximation assumes uniform strain. Differentiating its averaged strain energy density gives the averaged local stress. For linear elasticity the predicted elastic stiffness tensor is $C_V=⟨C⟩$. The uniform strain is an admissible affine displacement trial field, so this gives an upper elastic energy bound for convex local energy.