Elasticity and continuum mechanics

Stress, strain, elastic moduli, tension, plate and membrane mechanics, wave speeds.

An elastic solid is a continuum that stores energy when deformed and recovers its shape when released. The constitutive law tying the deformation to the internal forces is Hooke’s law in 3-D; the elastic moduli are its coefficients. The basilar membrane is an elastic membrane; the ossicles are nearly-rigid bones; water has a non-zero bulk modulus that controls the speed of sound in liquids.

Stress — force per unit area, acting on a plane

A force acting across a surface inside a continuum, normalised by the area of that surface, is a stress. Because the surface has an orientation, stress is a tensor, not a scalar. At any interior point, the force per unit area that the material on one side of a plane exerts on the material on the other is

\mathbf{t}(\hat{\mathbf{n}}) \;=\; \boldsymbol{\sigma}\, \hat{\mathbf{n}},

where $\hat{\mathbf{n}}$ is the unit normal of the plane and $\boldsymbol{\sigma}$ is the symmetric Cauchy stress tensor with components $\sigma_{ij}$ .

σ₁₁ = 2.00 σ₂₂ = 1.00 σ₁₂ (shear) = 0.50 Plane normal angle θ = 30°

The stress tensor σ acts on a plane with normal n̂ to give the traction vector t = σ·n̂ — the force per unit area that the material on the +n̂ side exerts on the material on the −n̂ side. The traction has a normal component (pointing along n̂, the "pressure-like" part) and a shear component (perpendicular to n̂, the "drag-like" part). Rotating the plane changes how σ apportions itself between the two.

Slide the plane-normal angle: the same stress tensor produces different traction vectors on differently-oriented planes. The diagonal components $\sigma_{ii}$ are normal stresses (tension if positive, compression if negative); off-diagonal $\sigma_{ij}$ are shear stresses. For a fluid at rest the only non-zero components are diagonal: $\sigma_{ij} = -p\, \delta_{ij}$ .

Strain — deformation of a material element

When a body deforms, each material point $\mathbf{r}$ moves to $\mathbf{r} + \mathbf{u}(\mathbf{r})$ where $\mathbf{u}$ is the displacement field. For small deformations, the relevant kinematic object is the symmetric strain tensor

\varepsilon_{ij} \;=\; \tfrac12\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right).

ε₁₁ (stretch in x) = 0.150 ε₂₂ (stretch in y) = -0.080 ε₁₂ (shear) = 0.050

Each grid line maps under the displacement field u = ε·r. Diagonal terms ε₁₁, ε₂₂ stretch the grid in the corresponding direction. Off-diagonal ε₁₂ shears it — turning squares into parallelograms. The strain tensor records all of this with six independent components (in 3-D); positive trace is dilation, traceless part is pure shear.

Diagonal $\varepsilon_{ii}$ measure local stretching along the $i$ -axis; off-diagonal $\varepsilon_{ij}$ ( $i \ne j$ ) are half the shear strain $\gamma_{ij}$ . The trace $\varepsilon_{kk} = \nabla\cdot\mathbf{u}$ is the local volumetric strain — the fractional change in volume.

Hooke’s law in 3-D and the elastic moduli

For a linear, isotropic, elastic solid, the stress is linear in the strain. The constitutive law has two independent constants:

\sigma_{ij} \;=\; \lambda\, \varepsilon_{kk}\, \delta_{ij} \;+\; 2\mu\, \varepsilon_{ij},

where $\lambda$ and $\mu$ are the Lamé parameters. $\mu$ is the shear modulus (also called $G$ ); $\lambda$ has no direct physical name. The same two constants are repackaged in several equivalent forms:

Young’s modulus $E$ — ratio of uniaxial stress to uniaxial strain when the lateral faces are free.
Shear modulus $G = \mu$ .
Bulk modulus $K$ — ratio of pressure to fractional volume change.
Poisson’s ratio $\nu$ — ratio of lateral contraction to axial extension.

Any two determine the others:

G \;=\; \frac{E}{2(1+\nu)}, \qquad K \;=\; \frac{E}{3(1-2\nu)}, \qquad \lambda \;=\; K - \tfrac23 G.

For most solids $\nu$ is in the range $0.2\!-\!0.35$ . Two extreme limits matter: $\nu \to 0.5$ (incompressible, $K \to \infty$ , the operative limit for most biological tissues and water in shear) and $\nu \to -1$ (auxetic, engineered metamaterials only).

σ_xx = 0.050 σ_yy = 0.000 τ_xy = 0.000 Young modulus E = 1.00 Poisson ratio ν = 0.30

Tension in x alone (σ_xx > 0, σ_yy = 0) elongates the block in x and *contracts* it in y by Poisson's ratio ν. A pure shear stress τ_xy tilts the block without changing its volume. At ν = 0.5 the material is incompressible (volumetric strain = 0 for any deviatoric stress), and the bulk modulus K diverges — the limit relevant for water and most biological tissues.

Slide the three stress components and watch the unit block deform. Tension in one direction contracts the perpendicular direction by Poisson’s ratio. Shear stress tilts the block without changing its volume.

Elastic wave speeds in a solid

Apply Newton’s second law to a small element of an elastic continuum: the net force per unit volume is the divergence of the stress tensor, and $\rho \ddot{\mathbf{u}} = \nabla\cdot\boldsymbol{\sigma}$ . Substituting Hooke’s law gives the Navier–Cauchy equation:

\rho\, \ddot{\mathbf{u}} \;=\; (\lambda + \mu)\, \nabla(\nabla\cdot\mathbf{u}) \;+\; \mu\, \nabla^2 \mathbf{u}.

Decomposing $\mathbf{u}$ into longitudinal (curl-free) and transverse (divergence-free) parts gives two independent wave equations with two characteristic speeds:

c_P \;=\; \sqrt{\frac{\lambda + 2\mu}{\rho}} \;=\; \sqrt{\frac{K + 4G/3}{\rho}}, \qquad c_S \;=\; \sqrt{\frac{\mu}{\rho}} \;=\; \sqrt{\frac{G}{\rho}}.

K (bulk)1.50

G (shear)0.60

c_P/c_S1.96

Bulk modulus K = 1.50 Shear modulus G = 0.60

The P-wave (top) is compressional: particles oscillate *along* the propagation direction; the medium alternately compresses and rarefies. The S-wave (bottom) is shear: particles oscillate *perpendicular* to the propagation direction. P-waves are always faster (involve both K and G); S-waves involve only G — they don't exist in fluids (where G = 0). The c_P / c_S ratio is always > √2 and diverges as the material approaches incompressibility.

The P-wave (primary, longitudinal) is compressional and involves both $K$ and $G$ ; particles oscillate along the propagation direction. The S-wave (secondary, shear) involves only $G$ ; particles oscillate perpendicular. P-waves are always faster than S-waves. Fluids have $G = 0$ — no shear elasticity — so they cannot transmit S-waves, only acoustic P-waves with speed $c = \sqrt{K/\rho}$ .

For steel ( $E \approx 200\,\text{GPa}$ , $\nu \approx 0.3$ , $\rho \approx 7800\,\text{kg/m}^3$ ): $c_P \approx 5900\,\text{m/s}$ , $c_S \approx 3200\,\text{m/s}$ . For seismology this $c_P/c_S$ ratio is the basic diagnostic: the P-S delay at a seismograph tells you the distance to the earthquake.

A 1-D string — tension as restoring stiffness

The string is the simplest elastic system: a 1-D continuum whose transverse deflection $y(x, t)$ has

\rho_\text{lin}\, \frac{\partial^2 y}{\partial t^2} \;=\; T\, \frac{\partial^2 y}{\partial x^2}, \qquad c \;=\; \sqrt{\frac{T}{\rho_\text{lin}}}.

Tension T = 1.00 Linear density μ_lin = 1.00

A transverse pulse on a taut string propagates at c = √(T/μ_lin). Raising the tension quadruples-the-stiffness — and the speed doubles. Doubling the mass density slows the wave by √2. The wave equation u_tt = c² u_xx is exactly what you get from applying Newton's second law to an element of the string with tension restoring its curvature.

$T$ is the tension (force, not stress); $\rho_\text{lin}$ is the linear mass density. The Sound book’s chapter 3 lessons develop this equation from a discrete chain of masses; here the same algebra applies to a stretched string of any continuum material.

Plate and membrane mechanics

The basilar membrane is treated in the Hearing book as a thin elastic structure. There are two limiting regimes:

Membrane: bending stiffness negligible, restoring force comes entirely from in-plane tension. Drumhead. Transverse modes look like 2-D string analogues with linear dispersion $\omega = ck$ .
Plate: bending stiffness dominant. Flexural waves are dispersive: $\omega \propto k^2$ .

Highlight:

A membrane stretches under in-plane tension; the restoring force is tension, and the dispersion is *linear* (ω = ck): all wavelengths travel at the same speed. A plate resists bending via a stiffness D ∝ Eh³; the dispersion is *quadratic* (ω ∝ k²): short wavelengths travel faster than long ones. The basilar membrane is closer to a plate than a membrane, and its quadratic dispersion is what makes the cochlear traveling-wave place-map work.

The basilar membrane sits closer to the plate end of this spectrum — its quadratic dispersion is the operative reason the cochlear traveling-wave place-map exists.

Viscoelasticity

Real biological tissues are neither perfectly elastic nor perfectly viscous. They are viscoelastic: the stress depends on both the strain and its rate. Two simple models build intuition:

Maxwell (spring in series with dashpot): under sustained stress, strain grows linearly (creep). Stress relaxes exponentially under sustained strain.
Kelvin–Voigt (spring in parallel with dashpot): instantaneous stiffness on loading; long-time strain approaches an elastic plateau.

Both models predict the form of frequency-domain mechanical impedance $Z(\omega) = b + i(\omega m - k/\omega)$ that the cochlea chapter uses to define $\kappa^2(x, \omega)$ . The $b$ is the dashpot, the $k$ the spring, the $m$ the inertial loading from the cochlear fluid.

The principle of virtual work

A variational reformulation: the equilibrium configuration of an elastic body is the one that minimises the total potential energy

\Pi[\mathbf{u}] \;=\; \int_V \tfrac12 \sigma_{ij} \varepsilon_{ij}\, dV \;-\; \int_V f_i u_i\, dV \;-\; \int_S t_i u_i\, dS,

over all kinematically admissible $\mathbf{u}$ . The Euler–Lagrange equation is the Navier–Cauchy equation above. This is the operative formulation for finite-element methods.

⏳ The history — Robert Hooke's anagram and the slow disclosure of linear elasticity

Robert Hooke published his law of elasticity in 1660 as an anagram: ceiiinosssttuv. The convention was a way to establish priority without revealing the discovery to rivals; he did not give the solution publicly until 1678, in De Potentia Restitutiva (“Of Spring”). The solved anagram reads ut tensio sic vis — “as the extension, so the force.” It was the first quantitative statement of a constitutive law.

The 3-D generalisation took 150 more years. Augustin-Louis Cauchy in 1822 introduced the stress tensor and gave the first systematic theory of continuum mechanics. Gabriel Lamé developed the modern elastic-modulus algebra in the 1850s. The two-parameter (E, ν) representation of an isotropic linear elastic solid crystallised in the engineering literature only in the early twentieth century.

What is remarkable is that the same linearity Hooke posited for a single spring — extension proportional to force — survives to the 3-D continuum case for small deformations. It is the first-order Taylor expansion of any smooth stress–strain relation around the unstressed state.

For the cross-book applications — basilar-membrane mechanics, ossicular impedance, speed of sound in solids vs fluids, viscoelastic cochlear damping — see the key examples sub-page.