Linear elasticity

The Euler-Bernoulli beam equation models transverse linear bending waves in a slender beam. The normalized equation has

c_{b} = 1

and may be reduced by the Schrodinger factorization of the elastic beam equation. Prescribing displacement and curvature at an endpoint supplies two boundary traces for its fourth spatial derivative.

Schrodinger factorization of the elastic beam equation ( $u_{t} = - v_{xx}, v_{t} = u_{xx}$ )

Writing

q = u + i v

in the free Schrodinger equation gives

u_{t} = - v_{xx}

and

v_{t} = u_{xx}

, hence

u_{tt} + u_{xxxx} = 0

. To encode initial velocity

u_{1}

, choose the decaying primitive

v_{0} (x) = - \int_{x}^{\infty} (s - x) u_{1} (s) d s

. A prescribed endpoint curvature becomes the time derivative of the imaginary boundary trace. This turns the normalized Euler-Bernoulli beam equation into a complex second-order boundary problem.

Poisson's ratio ( $ν$ )

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

Young's modulus ( $E$ )

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 - ν^{2})]

, with thickness

t

and Poisson's ratio

ν

Fracture toughness ( $K_{c}$ )

Fracture toughness measures a material's resistance to crack advance. In a thin elastic plate model, the threshold for advance can be represented by a prescribed front curvature

κ_{f}

. Such a curvature parameter is not numerically interchangeable with a bulk fracture-toughness parameter without specifying the fracture model.

Elastic plate

An elastic plate is a thin elastic body whose leading deformation is bending of its middle surface.

Flexural wave ( $ω^{2} = (B / m) k^{4}$ )

For a thin plate with areal mass

m

and bending stiffness

B

, small free transverse disturbances satisfy

m η_{tt} + B \nabla^{4} η = 0

. A plane wave of tangential wavenumber

k

therefore satisfies

ω^{2} = (B / m) k^{4} .

The frequency is quadratic in wavenumber, so these waves are dispersive. Matching this free-wave relation cancels the plate's inertial and bending response in an acoustic transmission problem.

Flexural-gravity wave ( $ω^{2} = (g k + β k^{5}) / (1 + α k)$ )

A flexural-gravity wave bends a floating elastic plate while displacing the underlying water. For a uniform plate with areal mass

m

and bending stiffness

D

above inviscid deep water of mass density

ρ_{w}

, put

α = m / ρ_{w}

and

β = D / ρ_{w}

. Combining the plate equation with the water's kinematic boundary condition gives the displayed dispersion relation. Both gravitational acceleration and plate bending restore displacement; water and plate inertia resist acceleration. It assumes small amplitude, no prestress and negligible shear deformation of the plate.

Group-velocity minimum of a flexural-gravity wave ( $c_{g} = d ω / d k$ )

The group velocity of a flexural-gravity wave diverges on the long-wave gravity branch and on the short-wave bending branch, and therefore has a positive minimum between them when the bending stiffness is positive. This minimum bounds the propagation speed of narrow wave packets within the ideal model. It does not by itself determine a wind-generation threshold: steady forcing requires matching phase velocity, while dissipation and coupling determine growth.

Viscous peeling of an elastic plate ( $U_{p} \sim B h_{f}^{3} / (μ l_{p}^{5})$ )

A fluid advancing under an elastic plate can be resisted mainly by lubrication theory flow near its front. For front gap

h_{f}

and peeling length

l_{p}

, the elastic pressure gradient is of order

B h_{f} / l_{p}^{5}

, while the gap volume flux is of order

B h_{f}^{4} / (μ l_{p}^{5})

. Mass balance therefore gives peeling speed

U_{p} \sim B h_{f}^{3} / (μ l_{p}^{5})

Axisymmetric clamped-plate deflection ( $h = p (R^{2} - r^{2})^{2} / (64 B)$ )

A circular elastic plate with bending stiffness

B

, constant excess pressure

p

and clamped boundary conditions at radius

R

obeys

B \nabla_{r}^{4} h = p

, where

\nabla_{r}^{2} = r^{- 1} \partial_{r} (r \partial_{r})

. Its regular solution is

h = p (R^{2} - r^{2})^{2} / (64 B)

. Consequently its enclosed volume is

V = π p R^{6} / (192 B)

, and its edge curvature is

h_{rr} (R) = p R^{2} / (8 B) = 24 V / (π R^{4})

Bending stiffness ( $B$ )

Bending stiffness is the coefficient relating plate curvature to bending moment. In linear thin-plate theory a transverse displacement

ζ

produces a restoring load

B \nabla^{4} ζ

Nondegenerate one-dimensional elastic material

For the one-dimensional reduction of isotropic linear elasticity, nondegeneracy means that the effective longitudinal modulus

λ + 2 μ

is nonzero. The static Navier-Cauchy equation then forces the displacement to be affine.

Displacement field

A displacement field

u (x)

gives the change in position of each material point from a reference configuration.

Displacement gradient tensor

The displacement gradient has components

(\nabla u)_{ij} = \partial_{i} u_{j}

and records local changes of the displacement field.

Lamé parameter

The two Lamé parameters

λ

and

μ

determine the linear isotropic relation between stress and strain;

μ

is the shear modulus.

Isotropic linear-elastic energy density

For the two-dimensional convention

\nabla u = (\partial_{i} u_{j})

, the energy density

W = \frac{μ}{2} \nabla u : \nabla u^{T} + \frac{λ + μ}{2} (\nabla \cdot u)^{2}

produces the static isotropic elastic equations by variation.

Navier-Cauchy equation

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

μ \nabla^{2} u + (λ + μ) \nabla (\nabla \cdot u) = 0.

Uniform extension of a one-dimensional elastic body

For a one-dimensional body with endpoints displaced by zero and

Δ

, the static Navier-Cauchy equation reduces to

u^{''} = 0

and gives the uniform strain solution

u (x) = Δ x / L

Elastic wave in an isotropic solid

For density

ρ

and Lamé parameters

λ, μ

, the displacement satisfies

ρ u_{tt} = (λ + μ) \nabla (\nabla \cdot u) + μ \nabla^{2} u .

Its longitudinal and transverse wave speeds are

c_{P} = \frac{λ + 2 μ}{ρ}, c_{S} = \frac{μ}{ρ} .

(2)

Shear-horizontal wave (SH wave)

A shear-horizontal wave propagating in the

x

direction has displacement perpendicular to the sagittal

x y

plane, for example

u = w (x, y, t) e_{z}

. It is divergence-free and obeys the scalar shear-wave equation

w_{tt} = c_{S}^{2} (w_{xx} + w_{yy})

Guided shear-horizontal mode (SH mode)

Between parallel boundaries, a guided shear-horizontal mode has a discrete transverse wavenumber

q_{n}

and dispersion relation

ω_{n}^{2} = c_{S}^{2} (k^{2} + q_{n}^{2})

. Its cutoff frequency is

c_{S} q_{n}

Mode conversion at a planar elastic interface

An obliquely incident in-plane P- or SV-wave generally produces reflected and transmitted P- and SV-waves. Continuity of displacement and traction supplies four scalar conditions for their four amplitudes, while frequency and tangential wavenumber are shared.

Snell law for elastic and acoustic waves

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds

c_{j}

and angles

θ_{j}

from the normal satisfy

\frac{s i n θ _{j}}{c _{j}} = constant .

Acoustic impedance ( $Z = ρ c$ )

The acoustic impedance of a fluid is

Z = ρ c

. For a progressive plane acoustic wave, pressure amplitude

P

and velocity amplitude

U

in its propagation direction satisfy

P = Z U

. At an oblique planar interface, zero reflection requires a matching condition involving the normal impedances

Z / cos θ

or, equivalently for displacement amplitudes,

Z cos θ

Normal acoustic impedance ( $Z_{n}$ )

The normal acoustic impedance of a progressive acoustic wave at a planar boundary is the pressure amplitude divided by the normal-velocity amplitude. For propagation angle

θ

from the normal,

Z_{n} = \frac{ρ c}{c o s θ} = \frac{ρ ω}{k _{n}} .

The distinction from the acoustic impedance

ρ c

matters at oblique incidence: interface reflection depends on the ratio of normal impedances. For an evanescent outgoing wave

k_{n}

is imaginary and the normal impedance is reactive, giving no mean normal acoustic energy flux.

Surface acoustic impedance ( $Z = P / V$ )

At an acoustic boundary, the surface acoustic impedance is the pressure amplitude divided by the velocity amplitude directed into that boundary. With upward surface velocity

- V

and harmonic convention

e^{iω t}

, it is

Z = P / V

. Reflection compares this impedance with the incident fluid's normal acoustic impedance. The velocity orientation and harmonic convention must be fixed before interpreting signs of inertial, elastic, and radiation terms.

Passive acoustic impedance ( $Re Z \geq 0$ )

With pressure amplitude

P = Z V

and velocity

V

directed into a boundary, the mean absorbed acoustic energy flux is

\frac{1}{2} Re (P V^{*}) = \frac{1}{2} Re Z ∣ V ∣^{2}

. A passive boundary cannot supply net energy, so

Re Z \geq 0

at a real frequency. Its reflection from a propagating acoustic plane wave has magnitude at most one. A pole continued to a complex incidence angle or frequency must be interpreted separately from real-frequency passive scattering.

Acoustic transmission through a uniform layer

A layer of impedance

Z_{1}

, wavenumber

k_{1}

, and thickness

L

between identical media of impedance

Z_{0}

has pressure-amplitude transmission coefficient

∣ T ∣ = [cos^{2} (k_{1} L) + \frac{1}{4} (\frac{Z _{1}}{Z _{0}} + \frac{Z _{0}}{Z _{1}})^{2} sin^{2} (k_{1} L)]^{- 1/2} .

It transmits perfectly at the layer resonances

k_{1} L = nπ

, irrespective of the impedance mismatch.

Displacement reflection from an interface between two inviscid elastic liquids

For a P-wave incident at angle

θ

from a liquid

(ρ, c, λ)

onto

(ρ^{'}, c^{'}, λ^{'})

, with transmitted angle

θ^{'}

obeying Snell law for elastic and acoustic waves,

R = \frac{λ ^{'} s i n 2 θ - λ s i n 2 θ ^{'}}{λ ^{'} s i n 2 θ + λ s i n 2 θ ^{'}} .