Elasticity, Stress and Continuum Mechanics: /20 Summary

Context and scope

the supplied physics reference, Applications of Classical Physics — Chapters 11–12 Elastostatics · Elastodynamics

/20 Summary

• Elasticity = small-deformation response of solids to applied forces, modeled by linear constitutive law (Hooke).
• Two primary tensors:
  • Strain tensor $e_{ij}$ (geometric: how the body deforms)
  • Stress tensor $T_{ij}$ (mechanical: forces transmitted across surfaces — already from Part I)
• Constitutive bridge — Hooke's law (isotropic linear elasticity): $$T_{ij} = -K\Theta\,\delta_{ij} - 2\mu\,\sigma_{ij}$$ with bulk modulus $K$ (resists compression), shear modulus $\mu$ (resists shape change), $\Theta = e_{kk}$ (dilatation), $\sigma_{ij}$ (deviatoric strain).
• Just two scalars ($K, \mu$ or equivalently Young's $E$ + Poisson $\nu$) characterize an isotropic elastic solid.
• Two wave types in an elastic continuum:
  • P-waves (longitudinal, compressional) — speed $c_P = \sqrt{(K + 4\mu/3)/\rho}$
  • S-waves (transverse, shear) — speed $c_S = \sqrt{\mu/\rho}$
• Plus surface waves (Rayleigh, Love) at free/layered boundaries — dominant in earthquakes.
• Applications drive intuition: stretching a rod, bending a beam, torsion of a shaft, buckling, plates & shells, seismic waves, Earth's free oscillations, planetary interiors.

Master Map

mindmap
  root((Elasticity))
    Ch.11 Elastostatics
      Displacement u
      Strain tensor e_ij
        Expansion Θ
        Shear σ_ij
        Rotation Ω
      Stress T_ij (Part I)
      Hooke isotropic
      Moduli
        Bulk K
        Shear μ
        Young E
        Poisson ν
        Lamé λ μ
      Energy density
      Equilibrium ∂T = -f
      Applications
        Rod stretch
        Beam bending
        Cantilever
        Torsion
        Buckling
        Plates
        Shells
      Saint-Venant
      Dislocations
      Fracture (Griffith)
    Ch.12 Elastodynamics
      Wave eq for u
      P-waves
      S-waves
      Energy flux
      Reflection / refraction
      Surface waves
        Rayleigh
        Love
        Stoneley
      Normal modes
        Rod
        Sphere
        Earth (1S0 etc.)
      Seismology
      Quake source mech

Elastostatics

Kinematics — The Displacement Field

Deformation: each material point at $\mathbf{x}$ in the undeformed (reference) state moves to $\mathbf{x} + \mathbf{u}(\mathbf{x})$.

Displacement vector $\mathbf{u}(\mathbf{x})$ — small (linear theory: $|\partial_j u_i| \ll 1$).

Strain Tensor — The Symmetric Part of $\partial \mathbf{u}$

Expand $\partial_j u_i$ into symmetric + antisymmetric parts:

$$\partial_j u_i = e_{ij} + \Omega_{ij}, \qquad e_{ij} = \tfrac{1}{2}(\partial_i u_j + \partial_j u_i), \quad \Omega_{ij} = \tfrac{1}{2}(\partial_j u_i - \partial_i u_j)$$

• Strain $e_{ij}$ (symmetric, 6 components): true deformation.
• Rotation $\Omega_{ij}$ (antisymmetric, 3 components): rigid rotation, no deformation. Encodes infinitesimal rotation vector $\Omega_k = \tfrac{1}{2}\epsilon_{kij}\Omega_{ij}$.

Decomposition of Strain

$$\boxed{e_{ij} = \underbrace{\tfrac{1}{3}\Theta\,\delta_{ij}}_{\text{isotropic expansion}} + \underbrace{\sigma_{ij}}_{\text{pure shear, traceless}}}$$

with dilatation $\Theta \equiv e_{kk} = \nabla\cdot\mathbf{u}$ (fractional volume change) and shear strain $\sigma_{ij} = e_{ij} - \tfrac{1}{3}\Theta\delta_{ij}$.

Compatibility Conditions

For an arbitrary symmetric $e_{ij}(\mathbf{x})$, a displacement field $\mathbf{u}$ giving rise to it exists only if compatibility is satisfied:

$$\epsilon_{ikl}\epsilon_{jmn}\partial_k\partial_m e_{ln} = 0$$

— 6 equations, only 3 independent. (Trivially satisfied if $\mathbf{u}$ is given; constraint when solving directly in terms of $e_{ij}$.)

Hooke's Law — The Constitutive Relation

For an isotropic linear elastic solid at small strain:

$$\boxed{T_{ij} = -K\,\Theta\,\delta_{ij} - 2\mu\,\sigma_{ij}}$$

(B&T sign convention: positive $T_{ii}$ = tension. Some textbooks use opposite.) Equivalently using Lamé parameters $(\lambda, \mu)$:

$$T_{ij} = -\lambda\,\Theta\,\delta_{ij} - 2\mu\, e_{ij}, \quad \lambda = K - \tfrac{2}{3}\mu$$

Inverted (strain in terms of stress):

$$e_{ij} = \frac{1+\nu}{E}T_{ij} - \frac{\nu}{E}T_{kk}\delta_{ij}$$

The Elastic Moduli — All Equivalent for Isotropic Solids

Conversion Identities

Pick any two and the rest follow. Most-used pairs: $(K, \mu)$ or $(E, \nu)$.

$$E = \frac{9 K \mu}{3K + \mu}, \qquad \nu = \frac{3K - 2\mu}{2(3K + \mu)}$$

$$K = \frac{E}{3(1-2\nu)}, \qquad \mu = \frac{E}{2(1+\nu)}$$

$$\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}, \qquad c_P^2/c_S^2 = \frac{K + 4\mu/3}{\mu} = \frac{2(1-\nu)}{1-2\nu}$$

Bounds (thermodynamic stability): $$K > 0, \quad \mu > 0 \quad \Longleftrightarrow \quad -1 < \nu < \tfrac{1}{2}, \quad E > 0$$

Special cases:

• $\nu = 1/2$ ⇒ incompressible ($K\to\infty$). Approached by rubber.
• $\nu = 0$ ⇒ no lateral strain. Cork, ~ some foams.
• $\nu < 0$ ⇒ auxetic materials (rare, engineered foams, certain crystals).
• $\nu = 1/4$ ("Poisson solid") — most metals approximate this; gives $c_P/c_S = \sqrt{3}$.

Typical Values

Elastic Strain Energy

For a linear-elastic isotropic solid:

$$\boxed{w_{\text{elast}} = \tfrac{1}{2} T_{ij} e_{ij} = \tfrac{1}{2} K \Theta^2 + \mu\, \sigma_{ij}\sigma_{ij}}$$

per unit volume. Decomposes cleanly into bulk (compression) + shear (distortion) energy — orthogonal contributions because $\delta_{ij}\sigma_{ij} = 0$.

Properties:

• Quadratic in strain ⇒ Hooke's law is the linearization of a quadratic energy.
• $w \ge 0$ if and only if $K, \mu > 0$ (stability bound).
• Total elastic potential energy: $U = \int w\, dV$.

Thermodynamic interpretation: $w$ is the free energy at fixed entropy (adiabatic moduli) or at fixed temperature (isothermal moduli) — they differ slightly because of thermal expansion (next section).

Equation of Elastostatic Equilibrium

From Cauchy (Part I): $\partial_j T_{ij} + f_i^{\text{ext}} = \rho \ddot u_i$. Static case ⇒ $\ddot u_i = 0$:

$$\boxed{\partial_j T_{ij} + f_i^{\text{ext}} = 0}$$

Substitute Hooke's law:

$$\mu \nabla^2 u_i + (K + \tfrac{\mu}{3})\,\partial_i(\nabla\cdot\mathbf{u}) + f_i^{\text{ext}} = 0$$

(in B&T's sign convention; check carefully, since some books write with $(\lambda + \mu)$.)

Equivalent forms:

$$\mu \nabla^2 \mathbf{u} + (\lambda + \mu)\,\nabla(\nabla\cdot\mathbf{u}) = -\mathbf{f} \quad \text{(Navier-Cauchy)}$$

Boundary conditions:

1. Traction (Neumann): $T_{ij} n_j = \tau_i^{\text{applied}}$ on $\partial V$ (surface stress prescribed)
2. Displacement (Dirichlet): $u_i = u_i^{\text{applied}}$ on $\partial V$
3. Mixed: different conditions on different parts of $\partial V$

Saint-Venant's Principle

The detailed distribution of an applied surface load matters only within a distance ~ (load region size). Beyond that, only the net force, torque, and stress moments matter.

⇒ For long beams, only net loading is needed — local details (point-of-application etc.) wash out.

Canonical Static Problems

Uniaxial Stretch of a Rod

Apply uniform tension $T_{11} = \sigma$, others zero. Solve $e_{ij} = $ inverse Hooke:

$$e_{11} = \sigma/E, \quad e_{22} = e_{33} = -\nu\sigma/E$$

⇒ A rod under tension thins (if $\nu>0$). For $\nu = 1/2$ (incompressible), no volume change.

Pure Bending of a Beam (Euler-Bernoulli)

Beam along $x$-axis; bending moment $M(x)$ about $y$-axis.

Assumptions: plane sections remain plane and perpendicular to the deformed centerline; small deflections.

Curvature: $$\kappa(x) = \frac{1}{R} = \frac{d^2 w}{dx^2}$$ where $w(x)$ is the transverse deflection.

Constitutive relation:

$$\boxed{M(x) = E I\, \kappa(x)}$$

Second moment of area $I = \int_A z^2\, dA$ (about neutral axis):

Beam bending equation (combine static equilibrium + $M = E I \kappa$):

$$\boxed{E I\, \frac{d^4 w}{dx^4} = q(x)}$$

with $q(x)$ = distributed transverse load per unit length.

Boundary conditions for each end:

• Clamped: $w = 0$, $w' = 0$
• Simply supported: $w = 0$, $M = E I w'' = 0$
• Free: $M = 0$, $V = -E I w''' = 0$
• Pin: $w = 0$ only (rotation free)

Common Beam Solutions

Torsion of a Circular Shaft

For a circular rod (radius $a$, length $L$) under torque $\tau$:

$$\tau = \frac{\mu J\, \theta}{L}, \quad J = \int_A r^2\, dA = \frac{\pi a^4}{2}$$

with $\theta$ = total twist angle and $J$ = polar moment of area.

Shear stress at radius $r$: $\sigma = \mu r\,d\theta/dx$, linear in $r$, max at outer surface.

Stored energy: $U = \tfrac{1}{2}\tau\theta = \tau^2 L/(2\mu J)$.

For non-circular cross-sections: torsion induces warping of cross-section; use Saint-Venant torsion theory (stress function).

Buckling — Loss of Stability under Compression

A slender rod under axial compression $P$ buckles when $P$ exceeds Euler's critical load:

$$\boxed{P_{cr} = \frac{\pi^2 E I}{L_{\text{eff}}^2}}$$

Mechanism: above $P_{cr}$, the straight configuration becomes unstable; the rod adopts a sinusoidal deflection $w \propto \sin(\pi x/L)$. Bifurcation — second-order phase-transition analogy.

Important: $P_{cr}$ depends on $L^{-2}$, so doubling length cuts critical load by 4. Long thin columns fail by buckling long before stress reaches yield.

Thin Plates (Kirchhoff-Love)

For a flat plate of thickness $h$ (small), midplane deflection $w(x,y)$:

$$D\nabla^4 w = q(x, y)$$

with flexural rigidity $D = E h^3 / [12(1-\nu^2)]$. (Plate is stiffer than a beam by factor $1/(1-\nu^2)$ because lateral contraction is constrained.)

Plate boundary conditions richer than beam; principal modes for circular/rectangular plates have closed-form solutions.

Thin Shells

Curved thin structures (cylindrical, spherical) — predominantly membrane stress (in-plane), much stiffer than flat plates. Failure modes include buckling.

Spherical pressure vessel (radius $R$, thickness $h$, internal pressure $p$):

• Membrane stress $\sigma = pR/(2h)$ (uniform, biaxial).
• Useful for tanks, gas cylinders. Why pressure vessels are spherical.

Cylindrical pressure vessel: hoop stress $\sigma_\theta = pR/h$; axial $\sigma_z = pR/(2h)$. Hoop is the larger ⇒ failure usually splits longitudinally (witness: hot dogs splitting along their length).

Thermoelasticity

Temperature change $\Delta T$ induces isotropic expansion:

$$e_{ij}^{\text{thermal}} = \alpha\,\Delta T\,\delta_{ij}$$

with $\alpha$ = linear thermal expansion coefficient (~$10^{-5}$ K⁻¹ for metals, $10^{-7}$ for fused silica).

Modified constitutive law:

$$T_{ij} = \lambda\,(\Theta - 3\alpha\Delta T)\,\delta_{ij} + 2\mu\,(e_{ij} - \alpha\Delta T\delta_{ij}) \quad \text{(sign per B\&T)}$$

Consequences: constrained heated bodies develop thermal stress — railway tracks buckle, glass shatters from thermal shock, bimetallic strips bend.

Adiabatic vs isothermal moduli:

• Difference is small (~ $0.5$%) for normal metals.
• $C_p - C_v = TV \alpha^2 K$ (general thermo relation).

Defects and Failure

Dislocations

• Line defects in crystals — Burgers vector $\mathbf{b}$ characterizes them.
• Edge dislocation: extra half-plane of atoms.
• Screw dislocation: helical lattice distortion.
• Motion of dislocations → plastic deformation (irreversible, beyond linear elasticity).
• Yield stress in pure crystals is ~ $\mu/30$, far below theoretical $\mu/2\pi$ — dislocations make slip easy.

Fracture — Griffith Criterion

• Crack length $a$, surface energy $\gamma$, applied stress $\sigma$.
• Energy balance: elastic energy released = surface energy created.

$$\boxed{\sigma_c = \sqrt{\frac{2 E \gamma}{\pi a}}}$$

A material with crack of size $a$ fails at $\sigma_c \propto 1/\sqrt{a}$. Crack sharpness amplifies stress; pre-existing flaws determine real-world strength.

Stress intensity factor: $K_I = \sigma\sqrt{\pi a}$. Fracture occurs when $K_I > K_{IC}$ (material toughness).

Elastodynamics

The Elastic Wave Equation

Cauchy momentum equation with Hooke's law:

$$\rho \frac{\partial^2 u_i}{\partial t^2} = \partial_j T_{ij} = (\lambda + \mu)\,\partial_i(\nabla\cdot\mathbf{u}) + \mu\,\nabla^2 u_i$$

Or in vector form:

$$\boxed{\rho \ddot{\mathbf{u}} = (\lambda + 2\mu)\,\nabla(\nabla\cdot\mathbf{u}) - \mu\,\nabla\times(\nabla\times\mathbf{u})}$$

using $\nabla^2\mathbf{u} = \nabla(\nabla\cdot\mathbf{u}) - \nabla\times(\nabla\times\mathbf{u})$.

Helmholtz Decomposition — P and S Waves

Any vector field decomposes uniquely:

$$\mathbf{u} = \nabla\phi + \nabla\times\boldsymbol\psi$$

with scalar potential $\phi$ and vector potential $\boldsymbol\psi$ (gauge $\nabla\cdot\boldsymbol\psi = 0$).

Substitute into the wave equation; the two parts decouple:

$$\boxed{\ddot\phi = c_P^2\,\nabla^2 \phi, \qquad \ddot{\boldsymbol\psi} = c_S^2\,\nabla^2\boldsymbol\psi}$$

Ratio: $c_P/c_S = \sqrt{2(1-\nu)/(1-2\nu)}$. For $\nu = 1/4$, $c_P/c_S = \sqrt{3} \approx 1.73$.

Why P-waves are "primary": they travel faster, so on a seismogram they arrive first. The lag $\Delta t = (1/c_S - 1/c_P) \times d$ allows distance estimation.

Fluids/liquids: $\mu = 0$ ⇒ no S-waves. Earth's outer core is liquid — S-waves don't propagate through it. This is how we know.

Energy in Elastic Waves

Energy density:

$$u = \tfrac{1}{2}\rho |\dot{\mathbf{u}}|^2 + w_{\text{elast}}$$

(kinetic + elastic potential).

Energy flux (analog of Poynting vector):

$$F_i = -T_{ij}\dot u_j$$

For plane P-wave: $|F| = \tfrac{1}{2}\rho c_P A^2 \omega^2$ (with displacement amplitude $A$).

Acoustic impedance $Z = \rho c$ controls reflection/transmission at interfaces.

Reflection and Refraction at Interfaces

At a planar interface between media 1 and 2, an incident wave gives rise to multiple reflected and transmitted waves:

• Incident P → reflected P, reflected S, transmitted P, transmitted S (mode conversion)
• Incident S → reflected S, reflected P, transmitted S, transmitted P

Snell's law applies separately to each: $\sin\theta/c = $ constant across interface, with $c$ = appropriate wave speed.

Critical angles: when $\sin\theta_i > c_i/c_j$ for some outgoing mode, that mode becomes evanescent (no transmission). Wave totally reflects (with mode conversion).

Free surface (top of Earth): S → S + P (reflection only). P → P + S. Boundary condition: $T_{ij} n_j = 0$ on free surface.

Surface Waves

When the boundary supports trapped modes localized near the surface:

Rayleigh Waves

Combination of P and SV at a free surface, exponentially decaying away from the surface.

Properties:

• Speed $c_R < c_S < c_P$; for Poisson solid ($\nu = 1/4$): $c_R \approx 0.9194\, c_S$.
• Retrograde elliptical particle motion near surface.
• Penetration depth ~ wavelength.
• Most damaging earthquake waves at large distances because they're 2-D (decay $1/\sqrt{r}$, not $1/r$).

Love Waves

SH (horizontally polarized shear) waves trapped in a slow surface layer over a faster substrate.

Properties:

• Dispersive — different frequencies travel at different speeds (unlike P/S in homogeneous media).
• Speed $c_S^{\text{layer}} < c_L < c_S^{\text{substrate}}$.
• Horizontal, perpendicular to propagation.
• Cause "side-to-side" ground motion — often most destructive to buildings.

Stoneley Waves

At solid-solid or solid-liquid interfaces, between two media.

flowchart TD
    A[Earthquake source] --> B[P-wave arrives first]
    A --> C[S-wave next]
    A --> D[Surface waves: Rayleigh + Love]
    D --> E[Largest amplitude<br/>most damage]
    B --> F[Outer core blocks S<br/>shadow zone 105-140°]

Free Oscillations of Finite Bodies

A finite body (rod, sphere, Earth) has discrete normal modes indexed by integer quantum numbers.

Thin Rod (Longitudinal)

Boundary conditions: free ends ⇒ $\partial_x u = 0$. Modes:

$$\omega_n = n\pi c_{\text{bar}}/L, \quad c_{\text{bar}} = \sqrt{E/\rho}$$

Note: $c_{\text{bar}} \ne c_P$. Thin rod allows lateral contraction (Poisson), so effective stiffness is $E$ not $K + 4\mu/3$.

Sphere — Spheroidal and Toroidal Modes

For a solid elastic sphere (e.g., Earth idealized), modes split into two classes:

Labels: $l$ = angular degree (spherical harmonic), $n$ = radial overtone.

Earth's Free Oscillations (Seismology)

Excited by very large earthquakes (Mw 8+); decay over days. Spectrum tells us density and elastic moduli vs depth in Earth — inverse problem.

Earthquake Source Mechanism

Faulting and Moment Tensor

Earthquakes = sudden slip on a fault plane. Modeled as a double couple (no net force, no net torque).

Seismic moment: $$M_0 = \mu\, A\, D$$ where $A$ = fault area, $D$ = average slip, $\mu$ = shear modulus of rocks at the source.

Moment magnitude (Hanks-Kanamori): $$M_w = \tfrac{2}{3}\log_{10}(M_0/\text{N·m}) - 6.07$$

Each unit of $M_w$ = $\sim 32\times$ in energy. Largest recorded: Mw 9.5 (Chile 1960).

Far-field radiation

Each mode radiates with a characteristic focal mechanism ("beach ball" diagram) — 4-lobed P-wave amplitude pattern from a double couple.

Practical Applications

Workflow / Process

flowchart TD
    A["Elasticity problem"] --> B["Static or dynamic?"]

    B -->|Static| C["Elastostatics\n∂T = -f"]
    B -->|Dynamic| D["Wave equation\nρ ü = ∂T"]

    C --> E["Geometry?"]

    E -->|"Rod uniaxial"| F["T = E e\nPoisson contraction"]
    E -->|"Beam bending"| G["EI w'''' = q\nApply BCs"]
    E -->|"Torsion"| H["τ = μ J θ / L"]
    E -->|"Buckling risk"| I["Compare P to π²EI / L²"]
    E -->|"Plate"| J["D ∇⁴ w = q"]

    D --> K["Decompose fields:\n∇·u and ∇×u"]

    K --> L["P-wave:\nc_P = √((λ + 2μ)/ρ)"]
    K --> M["S-wave:\nc_S = √(μ/ρ)"]

    L --> N["Bulk or surface?"]
    M --> N

    N -->|"Surface"| O["Rayleigh / Love waves"]
    N -->|"Free body"| P["Normal modes"]

    C --> Q["Boundary conditions:\ntraction, displacement, mixed"]
    D --> Q

Comparison Tables

Strain Decomposition

Elastic Moduli Relations

Bulk Wave Modes

Surface Waves

Stress Concentration / Failure

Common Mistakes

• ❌ Confusing engineering shear $\gamma_{ij}$ with tensor shear $e_{ij}$. Engineers use $\gamma_{ij} = 2 e_{ij}$ for $i\ne j$. Factors of 2 contaminate Hooke's-law formulas.
• ❌ Mixing sign conventions for $T_{ij}$. B&T: tension positive (so $T_{11}>0$ pulls outward); other texts negate. Recheck before mixing sources.
• ❌ Using only one elastic modulus. Hooke's law needs two. $E$ alone is incomplete (lateral effects vanish only if $\nu=0$).
• ❌ Treating $\lambda$ and $\mu$ as independent of $E$ and $\nu$. They're not; just a different basis.
• ❌ Forgetting Poisson contraction. Squeeze in one direction, bulge in others — easy to ignore in derivations.
• ❌ Using $c_P = \sqrt{E/\rho}$ as the P-wave speed. That's the thin-rod speed $c_{\text{bar}}$, which allows lateral motion. The bulk P-wave is $\sqrt{(K + 4\mu/3)/\rho}$.
• ❌ Beam bending: confusing $I$ (second moment of area) with mass moment of inertia. Different objects: $I_{\text{area}} = \int z^2\,dA$ (Pa·m²·m² = N·m²/E), not $\int z^2\rho dV$.
• ❌ Applying Euler-Bernoulli to short, deep beams. Need Timoshenko (shear deformation) when $L/h$ is small.
• ❌ Treating buckling as a stress problem. It's a stability problem. The rod can buckle far below yield stress.
• ❌ Using bulk wave equations for surface phenomena. Surface waves need careful boundary-condition treatment.
• ❌ Forgetting mode conversion at interfaces. Incident P generates reflected S; incident S generates reflected P. Both must be included for energy conservation.
• ❌ Assuming isotropy for a crystal. Crystals require $c_{ijkl}$ tensor (up to 21 independent components, fewer with symmetry).
• ❌ Plate $\propto h^3$, beam $\propto h^3$ confusion. They look similar but with different constants because of biaxial stress in plates ($1/(1-\nu^2)$ stiffening).
• ❌ Ignoring fluid-loaded surfaces. Underwater acoustics, biological tissue — fluid loading modifies surface-wave dispersion strongly.

Expert Insights

All of elasticity is just Hooke's law + Newton's law + tensor algebra. The complexity comes from boundary conditions, not the underlying physics.

Two moduli are the minimum for an isotropic linear solid. Geophysics likes $(K, \mu)$ or $(c_P, c_S)$; engineering likes $(E, \nu)$. Translation matters when reading across fields.

Poisson's ratio is the most counterintuitive elastic constant. $\nu=1/2$ ⇒ rubber/incompressible, $\nu=0$ ⇒ no lateral coupling, $\nu<0$ ⇒ auxetic (rare). Most familiar materials sit at $\nu\approx 0.2$–0.35.

Saint-Venant is a deep statement — it says elasticity is a low-pass filter: the rapidly-varying high-frequency surface load is smoothed within ~$L_{\text{loading}}$ inside the body.

Bending dominates because $I\propto h^3$ for given mass per length. A beam twice as tall (same width) bends 8× less — engineering miracle of I-beams.

Buckling failures are spectacular. Sudden, often catastrophic. Designers add safety factors of 3–5 over $P_{cr}$ because imperfections amplify the sensitivity.

The shear modulus is what distinguishes solids from fluids. Both have $K>0$; only solids have $\mu>0$. That's why fluids can't support S-waves, and why Earth's outer core was identified as liquid.

Mode conversion is the source of complexity in seismograms — the recorded ground motion is a superposition of P, S, Rayleigh, Love, multiple reflections, and refractions from the geological structure.

The ratio $c_P/c_S$ tells you Poisson's ratio of the medium. Seismologists use travel-time differences to infer composition: $c_P/c_S \approx 1.73$ for crustal rock, much higher for fluid-saturated regions.

Free oscillations of Earth are the planet's equivalent of atomic spectroscopy. $_0S_2$ at 54 min was first detected unambiguously after the 1960 Chile earthquake.

Griffith's $\sigma_c\propto 1/\sqrt{a}$ is the most universal fracture law. Long cracks are weak; sharp cracks are weaker. Tempered glass survives by minimizing surface flaws and inducing compressive surface stress.

The Burgers vector of a dislocation is conserved along the line. Dislocations can end on surfaces or grain boundaries, never in bulk. Their motion is the microscopic origin of plasticity, hence work hardening, creep, fatigue.

Elastic anisotropy can be huge in crystals. Diamond varies only ~10%, but quartz, graphite, and other layered/aligned structures vary by factors of several. SAW (surface acoustic wave) devices exploit anisotropy precisely.

Thermoelastic damping — converting strain energy to heat via thermal-expansion-induced flow — sets the intrinsic $Q$ factor of MEMS resonators. Fundamental noise source in many precision instruments.

Plate tectonics is elastic strain release on a planetary scale. Stress accumulates in locked fault zones over centuries; releases in seconds during earthquakes. The slow → fast transition is what makes prediction so hard.

Troubleshooting

Cheatsheet

=== STRAIN ===
e_ij = (1/2)(∂_i u_j + ∂_j u_i)         symmetric, 6 components
Ω_ij = (1/2)(∂_j u_i − ∂_i u_j)         antisymmetric, rotation
Θ = e_kk = ∇·u                          dilatation
σ_ij = e_ij − Θ δ_ij/3                  deviatoric (traceless)

=== HOOKE'S LAW (B&T sign convention) ===
T_ij = −K Θ δ_ij − 2μ σ_ij
     = −λ Θ δ_ij − 2μ e_ij           (λ = K − 2μ/3)
e_ij = [(1+ν)T_ij − ν T_kk δ_ij]/E    (inverted)

=== MODULI ===
K   bulk            μ   shear (=G)
E   Young           ν   Poisson (−1 < ν < 1/2)
λ = K − 2μ/3

E = 9Kμ/(3K+μ)
ν = (3K−2μ)/[2(3K+μ)]
K = E/[3(1−2ν)]
μ = E/[2(1+ν)]
λ = Eν/[(1+ν)(1−2ν)]

c_P² = (K + 4μ/3)/ρ = (λ+2μ)/ρ
c_S² = μ/ρ
c_P/c_S = √[2(1−ν)/(1−2ν)]   = √3 for ν=1/4
c_bar = √(E/ρ)               (thin-rod longitudinal)

=== ENERGY ===
w = (1/2) T_ij e_ij = (1/2) K Θ² + μ σ_ij σ_ij

=== EQUILIBRIUM ===
Static:    ∂_j T_ij + f_i = 0
Dynamic:   ρ ü_i = ∂_j T_ij + f_i
Navier-Cauchy:
  (λ+2μ)∇(∇·u) − μ ∇×(∇×u) + f = ρ ü

=== UNIAXIAL ROD ===
ΔL/L = σ/E
Δr/r = −νσ/E
ΔV/V = (1−2ν)σ/E

=== BEAM BENDING ===
M = EI κ = EI w''
EI w'''' = q(x)
I (rect b×h)  = bh³/12
I (circ a)    = π a⁴/4
Cantilever, point load P:   δ_max = PL³/(3EI)
Simply supp., point P:      δ_max = PL³/(48EI)
Simply supp., dist. q:      δ_max = 5qL⁴/(384EI)

=== TORSION (circular) ===
τ = μJθ/L,   J = πa⁴/2
σ_shear(r) = μr dθ/dx,   max at r = a

=== BUCKLING ===
P_cr = π² EI / L_eff²
  pin-pin   L_eff = L
  clamp-free L_eff = 2L
  clamp-clamp L_eff = L/2
  clamp-pin L_eff = 0.7 L

=== PLATE (Kirchhoff-Love) ===
D ∇⁴ w = q
D = E h³ / [12(1−ν²)]

=== SHELL (pressure vessel, radius R, thickness h) ===
Sphere:   σ = pR/(2h)
Cylinder: σ_hoop = pR/h, σ_axial = pR/(2h)

=== FRACTURE / FAILURE ===
Griffith:  σ_c = √(2Eγ/πa)
Stress intensity: K_I = σ√(πa)
Fracture when K_I > K_IC

=== ELASTODYNAMICS WAVES ===
ρ ü = (λ+2μ) ∇(∇·u) − μ ∇×(∇×u)
Helmholtz: u = ∇φ + ∇×ψ
  φ:  ∂²φ/∂t² = c_P² ∇²φ
  ψ:  ∂²ψ/∂t² = c_S² ∇²ψ

Energy flux: F_i = −T_ij u̇_j
Impedance:   Z = ρc

Rayleigh wave: c_R ≈ 0.9194 c_S  (ν=1/4)
Love wave: dispersive, c_S^layer < c_L < c_S^sub

=== EARTHQUAKE / SOURCE ===
Seismic moment: M_0 = μ A D
Magnitude:      M_w = (2/3)log₁₀(M_0/N·m) − 6.07

=== TYPICAL VALUES (metals) ===
E ~ 70–200 GPa,  ν ~ 0.25–0.35
ρ ~ 2700–8000 kg/m³
c_P ~ 5–6 km/s,  c_S ~ 3 km/s (rock)
α ~ 10⁻⁵ /K (thermal expansion)