Elasticity

Elasticity studies reversible deformation and the stresses produced by it.

An elastic filament is a slender elastic body modeled by a centerline and bending rigidities. Its elastic energy penalizes curvature. In a small-deflection representation $(x,y(x),z(x))$, the bending energy is quadratic in $y^{\prime \prime }$ and $z^{\prime \prime }$. Compression can cause Euler buckling of an elastic filament.

Adding $\frac{1}{2}\int \sigma (x)(h^{\prime }{)}^{2}dx$ to the quadratic bending energy of an elastic filament contributes $-(\sigma h^{\prime }{)}^{\prime }$ to its variational derivative. The endpoint term is $A{h}^{\prime \prime }{\eta }^{\prime }+(\sigma h^{\prime }-A{h}^{\prime \prime \prime })\eta$. If the real filament tension vanishes at both ends, the usual self-adjoint endpoint conditions for filament bending remain valid. A complete eigenfunction expansion diagonalizes the energy even when the eigenfunctions have no explicit formula. Thermal equipartition theorem arguments require positive eigenvalues after fixing any kernel; strong compression can violate this requirement.

For a positive quadratic bending operator $K$ with a real orthonormal basis of eigenfunctions, the equipartition theorem gives $C(x,y)=k_{B}T{\sum }_n{W}_n(x)W_n(y)/{\mu }_n$, where $K{W}_n={\mu }_n{W}_n$. This is the inverse-operator Green function multiplied by Boltzmann constant and temperature. Unconstrained zero-energy filament modes prevent a normalizable canonical ensemble, and negative modes signal an unstable quadratic model.

For both ends clamped and $K=A{\partial }_x^4$, the diagonal is $C(x,x)=k_{B}T{x}^3(L-x{)}^3/(3A{L}^3)$. This differs from a clamped-free tip variance. The center variance is $k_{B}T{L}^3/(192A)$.

A zero-energy mode lies in the kernel of the quadratic elastic filament operator. For pure bending, zero curvature makes the displacement affine. Free-free endpoints therefore leave translation and tilt modes, while slope-constrained, force-free endpoints leave only translation. Both clamped-clamped and hinged-hinged endpoints remove these modes. An unrestricted modal amplitude has constant Boltzmann distribution weight, so no normalizable canonical ensemble or finite displacement variance exists. Fixing the rigid degrees of freedom allows the equipartition theorem on the positive complement.

A normal mode of a uniformly bending elastic filament fixed in position and slope at both ends satisfies $W^{\prime \prime \prime \prime }=k^{4}W$. With $\beta =kL>0$, the allowed wavenumbers solve $\cos \beta \cosh \beta =1$. The first root is $4.730040745$ and ${\beta }_n=(n+\frac{1}{2})\pi +O(e^{-(n+1/2)\pi })$ for $n\ge 1$. The apparent root at zero is not an eigenfunction: the zero-eigenvalue cubic polynomial satisfying all four clamped boundary conditions is identically zero. This spectrum differs from that of a filament clamped only at one end.

For the elastic filament operator $K=A{\partial }_x^4$, two integration by parts operations give boundary form $A[\overline{u}{v}^{\prime \prime \prime }-\overline{u^{\prime }}{v}^{\prime \prime }+\overline{u^{\prime \prime }}{v}^{\prime }-\overline{u^{\prime \prime \prime }}v]$. Four elementary homogeneous choices at each endpoint are $h=h^{\prime }=0$ (clamped), $h=h^{\prime \prime }=0$ (hinged), $h^{\prime \prime }=h^{\prime \prime \prime }=0$ (free), and $h^{\prime }=h^{\prime \prime \prime }=0$ (slope-constrained and transverse-force-free). Each defines a self-adjoint operator on a finite interval when imposed at both ends. More general real Robin boundary conditions also give self-adjointness; the four choices are not an exhaustive classification of all boundary subspaces.

For a clamped elastic filament of filament bending modulus $A$, a transverse tip force $f$ gives $h(x)=f{x}^2(3L-x)/(6A)$ in the small-slope model. Thus $h(L)/f=L^3/(3A)$. This differs from the effective tip stiffness under a distributed load. The zero-load Gaussian response reproduces the thermal bending fluctuations of a clamped filament, $⟨h(L{)}^2⟩=k_{B}T{L}^3/(3A)$ for one transverse coordinate.

Elastic energy is the recoverable energy stored by deforming an elastic material. In the small-deflection model of an elastic filament, the bending contribution is a positive quadratic functional of the curvatures, such as $\frac{1}{2}\int (A{y}^{\prime \prime 2}+B{z}^{\prime \prime 2})dx$. Work by applied loads must also be included when minimizing the total potential energy.

Linear elasticity approximates strain and stress to first order in a small displacement field.

An elastic membrane is a thin deformable sheet whose transverse restoring force is supplied by in-plane tension. For uniform tension $T$ and mass per area $m$, a two-dimensional small-displacement model has $m{\eta }_{tt}=T{\eta }_{xx}-[p]$, where the pressure jump is defined from the lower fluid to the upper fluid. Fluid loading modifies its traveling modes through the acoustic wave on a tensioned massive membrane.

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.

Writing $q=u+iv$ 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 $v_0(x)=-{\int }_x^{\infty }(s-x)u_1(s)ds$. 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.

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.

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

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 $\nu$.

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 ${\kappa }_f$. Such a curvature parameter is not numerically interchangeable with a bulk fracture-toughness parameter without specifying the fracture model.

Hooke's law states that the restoring force or stress is proportional to a sufficiently small displacement or strain. For a one-dimensional spring, $f=kx$ in magnitude.

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

For a thin plate with areal mass $m$ and bending stiffness $B$, small free transverse disturbances satisfy $m{\eta }_{tt}+B{\nabla }^4\eta =0$. A plane wave of tangential wavenumber $k$ therefore satisfiesThe 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.

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 ${\rho }_w$, put $\alpha =m/{\rho }_w$ and $\beta =D/{\rho }_w$. 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.

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.

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 $B{h}_f^4/(\mu l_p^5)$. Mass balance therefore gives peeling speed $U_p\sim B{h}_f^3/(\mu l_p^5)$.

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 $h=p(R^2-r^2{)}^2/(64B)$. Consequently its enclosed volume is $V=\pi p{R}^6/(192B)$, and its edge curvature is $h_{rr}(R)=p{R}^2/(8B)=24V/(\pi R^4)$.

Bending stiffness is the coefficient relating plate curvature to bending moment. In linear thin-plate theory a transverse displacement $\zeta$ produces a restoring load $B{\nabla }^4\zeta$.

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.

A displacement field $\mathbf{u}(\mathbf{x})$ gives the change in position of each material point from a reference configuration.

The displacement gradient has components $(\nabla \mathbf{u}{)}_{ij}={\partial }_i{u}_j$ and records local changes of the displacement field.

The two Lamé parameters $\lambda$ and $\mu$ determine the linear isotropic relation between stress and strain; $\mu$ is the shear modulus.

For the two-dimensional convention $\nabla \mathbf{u}=({\partial }_i{u}_j)$, the energy densityproduces the static isotropic elastic equations by variation.

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

For a one-dimensional body with endpoints displaced by zero and $\Delta$, the static Navier-Cauchy equation reduces to $u^{\prime \prime }=0$ and gives the uniform strain solution $u(x)=\Delta x/L$.

For density $\rho$ and Lamé parameters $\lambda ,\mu$, the displacement satisfiesIts longitudinal and transverse wave speeds are

A shear-horizontal wave propagating in the $x$ direction has displacement perpendicular to the sagittal $xy$ plane, for example $\mathbf{u}=w(x,y,t){\mathbf{e}}_z$. It is divergence-free and obeys the scalar shear-wave equation $w_{tt}=c_S^2(w_{xx}+w_{yy})$.

Between parallel boundaries, a guided shear-horizontal mode has a discrete transverse wavenumber $q_n$ and dispersion relation ${\omega }_n^2=c_S^2(k^2+q_n^2)$. Its cutoff frequency is $c_S{q}_n$.

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.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy

The acoustic impedance of a fluid is $Z=\rho c$. For a progressive plane acoustic wave, pressure amplitude $P$ and velocity amplitude $U$ in its propagation direction satisfy $P=ZU$. At an oblique planar interface, zero reflection requires a matching condition involving the normal impedances $Z/\cos \theta$ or, equivalently for displacement amplitudes, $Z\cos \theta$.

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 $\theta$ from the normal,The distinction from the acoustic impedance $\rho c$ 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.

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 $e^{i\omega t}$, 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.

With pressure amplitude $P=ZV$ and velocity $V$ directed into a boundary, the mean absorbed acoustic energy flux is $\frac{1}{2}\mathrm{Re}(P{V}^{\ast })=\frac{1}{2}\mathrm{Re}Z|V{|}^2$. A passive boundary cannot supply net energy, so $\mathrm{Re}Z\ge 0$ 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.

A layer of impedance $Z_1$, wavenumber $k_1$, and thickness $L$ between identical media of impedance $Z_0$ has pressure-amplitude transmission coefficientIt transmits perfectly at the layer resonances $k_{1}L=n\pi$, irrespective of the impedance mismatch.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.

For displacement velocity $\dot{u}$ and stress $\sigma$, the instantaneous elastic-energy flux isFor complex harmonic amplitudes, its average over one temporal period is

An incident SV-wave generally reflects as both an SV-wave and a P-wave. A rigid boundary determines their amplitudes by requiring both displacement components to vanish, while phase matching preserves frequency and tangential wavenumber.

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

When the converted P-wave is evanescent, its amplitude decays away from the boundary and the reflected SV amplitude has modulus one. The P field stores reactive energy near the boundary but carries zero mean normal power.

At a perfectly rigid plane, every component of the displacement field vanishes.

A wavevector points in the direction of phase propagation; its magnitude is the wavenumber and its scalar product with position gives the spatial phase.

For a slowly varying local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, wave-packet rays obey

The WKB method seeks a rapidly oscillating field in the formand determines the phase and slowly varying amplitude order by order in $\epsilon$.

The leading WKB phase obeys an eikonal equation. For a local dispersion relation $\omega =\Omega (k;x,t)$ and $k=\nabla \theta$, it is

Along a ray $\mathbf{x}(t)$, the derivative of a field $f(\mathbf{x},t)$ is

An evanescent wave has an imaginary component of its wavevector and therefore decays exponentially in that direction instead of transporting energy away as a propagating wave.