Elasticity, Stress and Continuum Mechanics: Final Takeaways

Glossary

• Anisotropic — Direction-dependent elastic response; needs $c_{ijkl}$ (4-tensor).
• Auxetic — $\nu < 0$ material; expands transversely when stretched.
• Beam — 1-D elastic structure; bending characterized by $EI$.
• Bulk modulus ($K$) — Resistance to uniform compression: $\Delta P = -K\,\Delta V/V$.
• Buckling — Loss of stability of slender compressed structure; first occurs at Euler's $P_{cr}$.
• Burgers vector — Lattice mismatch around a dislocation; conserved along its line.
• Cantilever — Beam clamped at one end, free at other.
• Compatibility conditions — Integrability constraint relating strain components to a displacement field.
• Compliance — Inverse of stiffness; $e_{ij} = S_{ijkl}T_{kl}$.
• Constitutive relation — Stress-strain relationship for a material (Hooke for linear elastic).
• Deviatoric strain ($\sigma_{ij}$) — Traceless part of strain; shape change only.
• Dilatation ($\Theta$) — $e_{kk} = \nabla\cdot\mathbf{u}$; fractional volume change.
• Dislocation — Linear lattice defect; carrier of plastic deformation.
• Displacement field ($\mathbf{u}$) — Vector field giving the move of each material point.
• Elastic limit — Stress beyond which deformation is no longer reversible (plasticity sets in).
• Elastodynamics — Time-dependent linear elasticity; supports P, S, surface waves.
• Elastostatics — Time-independent linear elasticity.
• Euler-Bernoulli beam — Slender beam theory; ignores shear deformation.
• Flexural rigidity — $EI$ for beam, $D = Eh^3/[12(1-\nu^2)]$ for plate.
• Fracture toughness ($K_{IC}$) — Critical stress intensity for crack propagation.
• Free oscillation — Normal mode of finite body.
• Griffith criterion — Fracture stress $\propto 1/\sqrt{a}$ for crack length $a$.
• Hooke's law — Linear stress-strain relation: $T_{ij} = -\lambda\Theta\delta_{ij} - 2\mu e_{ij}$.
• Isotropic — Direction-independent material response.
• Lamé parameters ($\lambda, \mu$) — Common parametrization of isotropic Hooke's law.
• Linear elasticity — Small-strain limit where stress is linear in strain.
• Love wave — Horizontally polarized shear surface wave in layered media.
• Mode conversion — At interface, incident P generates reflected S (and vice versa).
• Moment of inertia (second moment of area) — $I = \int z^2\,dA$ for beam.
• Navier-Cauchy equation — Elasticity in terms of displacement: $\mu\nabla^2\mathbf{u} + (\lambda+\mu)\nabla(\nabla\cdot\mathbf{u}) + \mathbf{f} = \rho\ddot{\mathbf{u}}$.
• Neutral axis — Plane in a bent beam where strain is zero.
• P-wave — Primary, compressional, longitudinal wave; faster than S.
• Plane strain — Zero strain in one direction (thick body).
• Plane stress — Zero stress in one direction (thin plate).
• Plasticity — Irreversible deformation beyond yield; outside linear-elastic theory.
• Poisson's ratio ($\nu$) — Ratio of transverse contraction to axial extension.
• Polar moment of area — $J = \int r^2\,dA = I_x + I_y$ (perpendicular-axis theorem); for torsion.
• Rayleigh wave — Free-surface elastic wave; mix of P + SV decaying with depth.
• Saint-Venant's principle — Distant fields depend only on net force/moment of load distribution.
• Seismic moment ($M_0$) — $\mu A D$; measure of earthquake size.
• Shear modulus ($\mu$) — Resistance to shape change; ratio of shear stress to shear strain.
• Spheroidal mode ($_nS_l$) — Normal mode of sphere with radial component.
• Stoneley wave — Interface wave between two solids or solid/fluid.
• Strain tensor ($e_{ij}$) — Symmetric gradient of displacement.
• Stress concentration — Local amplification of stress near holes, notches, cracks.
• Stress intensity factor ($K_I$) — Pre-factor of crack-tip stress singularity.
• Surface acoustic wave (SAW) — Elastic surface wave used in RF filters, sensors.
• Surface energy ($\gamma$) — Energy per unit area of new surface; controls fracture.
• Tensor stiffness ($c_{ijkl}$) — 4-tensor for anisotropic linear elasticity.
• Thermal expansion coefficient ($\alpha$) — $de/dT$ at zero stress.
• Toroidal mode ($_nT_l$) — Purely tangential normal mode of sphere.
• Torsion — Twisting deformation of a rod about its axis.
• Wave impedance — $Z = \rho c$; controls reflection at interfaces.
• Yield stress — Stress above which permanent (plastic) deformation begins.
• Young's modulus ($E$) — Stress / strain for uniaxial loading.

Final Takeaways

1. Elasticity = Hooke + Newton. Two equations and a constitutive law generate the whole theory.
2. Two moduli (any two of $K, \mu, E, \nu, \lambda$) fully specify an isotropic linear elastic solid. All conversions are bidirectional.
3. Strain is symmetric gradient of displacement; rotation is antisymmetric. Don't conflate them.
4. The shear modulus is what makes a solid a solid. No shear modulus = fluid, no S-waves.
5. Energy decomposes into bulk + shear orthogonally. $w = \tfrac{1}{2}K\Theta^2 + \mu\sigma_{ij}\sigma_{ij}$.
6. Beam bending scales as $h^3$; the I-beam exists for this reason. Move material away from the neutral axis to maximize $I$ per kg.
7. Buckling is a stability problem, not a strength problem. Slender columns fail at $P_{cr} \propto 1/L^2$ regardless of yield.
8. Saint-Venant's principle is why simple boundary models work. Detailed loading washes out within a characteristic length.
9. Helmholtz decomposes elastic waves cleanly into P and S. They propagate independently in a homogeneous isotropic medium.
10. Surface waves dominate at distance. 2-D geometric spreading ($1/\sqrt{r}$) means Rayleigh/Love carry the energy farthest — most earthquake damage.
11. Earth's free oscillations probe its interior. Spectroscopy on a planetary scale.
12. Fracture is a competition between elastic energy release and surface energy creation. Griffith $\sigma_c \propto 1/\sqrt{a}$ is universal.
13. Defects (dislocations) make real materials weaker than theory by ~ 100× — but also make them ductile.
14. Elastodynamic ideas reappear later: acoustic waves in fluids (Part V), MHD waves (Part VI), gravitational waves (Part VII). Linear-wave propagation is one mathematics.

Next: Part V — Fluid Dynamics. Continuum mechanics with $\mu = 0$ at zeroth order, but viscosity, vorticity, turbulence, magnetohydrodynamics enter. Seven chapters; the biggest Part in the book.