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Small displacements in an isotropic elastic solid satisfy a vector wave equation governed by density and the two Lamé moduli.

In a homogeneous isotropic elastic body without body forces, static equilibrium of the displacement field is

For the one-dimensional reduction of isotropic linear elasticity, nondegeneracy means that the effective longitudinal modulus $\lambda +2\mu$ is nonzero. The static Navier-Cauchy equation then forces the displacement to be affine.

Average the instantaneous elastic-wave energy flux over one temporal period at fixed position. For complex amplitudes,Write the evanescent P displacement amplitude asFor an isotropic linear elastic material with Lamé parameters $\lambda ,\mu$, its stress and velocity amplitudes areApart from a real common factor,which is purely imaginary. Its real part vanishes, and henceThe converted P-wave is a reactive near-boundary field and transports no time-averaged acoustic energy in the normal direction.

Poisson's ratio is minus the transverse strain divided by longitudinal strain under uniaxial loading in isotropic linear elasticity.

The second moment of area of a cross-section about an axis is $I=\int y^{2}dA$, where $y$ is the perpendicular distance from that axis. Its dimensions are length to the fourth power. For a circular section of radius $a$, polar integration gives $I={\int }_0^a{r}^{3}dr{\int }_0^{2\pi }{\sin }^2\theta d\theta =\pi a^4/4$. A slender rod with Young's modulus $E$ has filament bending modulus $B=EI$: the axial strain caused by curvature $\kappa$ is $-y\kappa$, and integrating the linear elasticity energy density $E{y}^2{\kappa }^2/2$ over the section gives $B{\kappa }^2/2$ per unit length.

Young's modulus is the ratio of axial stress to axial strain in uniaxial linear elasticity. For an isotropic plate, the bending rigidity is $E{t}^3/[12(1-{\nu }^2)]$, with thickness $t$ and Poisson's ratio $\nu$.