Classical Mechanics · Advanced Topics

Continuum Mechanics & Elasticity

Continuum mechanics describes the deformation of materials under applied forces. Elasticity theory — the mechanics of reversible deformation — governs everything from the vibration of guitar strings to earthquake seismic waves to the bending of DNA molecules.

Prerequisites

Newton's laws (Ch. 3) \cdot Vectors and calculus (Ch. 21-22) \cdot Linear algebra (Ch. LA) \cdot Fluid

mechanics (Ch. FM)

Learning Goals

Define the linearised strain tensor and identify diagonal (stretch) vs. off-diagonal (shear) components.
$State the isotropic Hooke's law in terms of Lamé constants and relate \lambda and \mu to Young's m$ odulus, Poisson ratio, shear modulus, and bulk modulus.
Derive the speeds of P-waves and S-waves from the Navier equation and explain why S-waves cannot propagate in a fluid.
Calculate the tip deflection of a cantilever beam using beam-bending theory and determine its fundamental natural frequency.
Apply the Griffith fracture criterion to compute the critical crack length for a given material and applied stress.

CE.1 Strain and Stress Tensors

A continuous body deforms when forces are applied. The displacement fieldu(r) maps each material point r to its new position r + u. The strain tensormeasures local deformation:

\varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i}\right) \qquad \text{(linearized strain tensor)}

(CE.1)

The diagonal components ε_ii represent stretching/compression; off-diagonal ε_ij (i≠j) represent shear. The dilatation (fractional volume change) is the trace: θ = ε_ii = ∇·u.

The stress tensor σ_ij is the i-th component of force per unit area on a surface with normal ĵ. The symmetry σ_ij = σ_ji follows from torque balance on an infinitesimal volume element.

Equilibrium condition (Newton's 2nd for a volume element):

\frac{\partial\sigma_{ij}}{\partial x_j}+f_i=\rho\frac{\partial^2u_i}{\partial t^2} \qquad \text{(Cauchy equation of motion)}

(CE.2)

where f_i is the body force density (e.g., gravity ρg).

CE.2 Hooke's Law for Solids

For a linear elastic (Hookean) material, stress is proportional to strain:

\sigma_{ij}=C_{ijkl}\varepsilon_{kl} \qquad \text{(generalized Hooke's law)}

(CE.3)

The stiffness tensor C_ijkl has 81 components, reduced to 21 by symmetries. For an isotropic material (elastic properties same in all directions):

\sigma_{ij}=\lambda\theta\delta_{ij}+2\mu\varepsilon_{ij} \qquad \text{(isotropic elastic solid, Lamé constants }\lambda,\mu\text{)}

(CE.4)

The two Lamé constants (λ, μ) relate to the familiar moduli:

Young's modulus E = μ(3λ+2μ)/(λ+μ) — uniaxial stress/strain ratio.Poisson ratio ν = λ/(2(λ+μ)) — ratio of transverse to axial strain.Shear modulus G = μ — shear stress/strain ratio.Bulk modulus K = λ + 2μ/3 — hydrostatic pressure/volume change.

Definition CE.1 — Common Elastic Moduli

For steel: E \approx 200 GPa, \nu \approx 0.29, G \approx 78 GPa. For rubber: E \approx 0.01–0.1 GPa, \nu \approx 0.50 (near

ly incompressible, K ≫ G). For water (fluid): G = 0 (no shear resistance), K = 2.2 GPa. Fo

r bone: E \approx 20 GPa (along axis), highly anisotropic. The constraint -1 < \nu < \frac{1}{2} follows fro

m thermodynamic stability (positive-definite elastic energy). Materials with \nu < 0 (auxeti

c) expand when stretched — exist in some foams and metamaterials.

CE.3 Elastic Waves

Substituting Hooke's law into the Cauchy equation gives the Navier equation:

(\lambda+\mu)\nabla(\nabla\cdot\mathbf{u})+\mu\nabla^2\mathbf{u}=\rho\frac{\partial^2\mathbf{u}}{\partial t^2} \qquad \text{(Navier equation)}

(CE.5)

This supports two types of waves. Decompose u = ∇φ + ∇×H (Helmholtz):

P-waves (compressional, primary): longitudinal displacement, ∇×u = 0. Faster: v_P = √((λ+2μ)/ρ) = √(K+4G/3)/√ρ.

S-waves (shear, secondary): transverse displacement, ∇·u = 0. Slower: v_S = √(μ/ρ) = √G/√ρ.

For the Earth's crust: v_P ≈ 6 km/s, v_S ≈ 3.5 km/s. For the Earth's liquid outer core: μ = 0 (fluid) → v_S = 0 (S-waves don't propagate), which is how we know the outer core is liquid. The ratio v_P/v_S = √((λ+2μ)/μ) ≥ √2 (for ν ≥ 0).

Example CE.1 — Bending of a Cantilever Beam

A beam of length L, width b, height h, Young's modulus E, is clamped at one end with force F at the free end. Find the deflection curve and tip deflection.

Setup:

Bending moment at position x from clamped end: M(x) = F(L - x). Beam bending equation: EI

d^{2}y/dx^{2} = M(x) where I = bh^{3}/12 is the second moment of area

Integrate:

EI y'' = F(L-x). EI y' = F(Lx - x^{2}/2) + C_{1}. At clamped end: y'(0) = 0 \to C_{1} = 0. EI y = F(L

x^{2}/2 - x^{3}/6) + C_{2}. At clamped end: y(0) = 0 \to C_{2} = 0

Tip deflection:

y(L) = FL^{3}/(3EI) = FL^{3}/(3E \times bh^{3}/12) = 4FL^{3}/(Ebh^{3}). For a steel beam (E = 200 GPa) 1 m \times 0

05 m \times 0.01 m, F = 100 N: \delta = 4\times100\times1^{3}/(200\times10^{9}\times0.05\times10^{-6}) = 400/10^{4} = 0.04 m = 4 cm

Natural frequency:

The fundamental mode of a cantilever: f_{1} = (1.875)^{2}/(2\piL^{2}) \sqrt(EI/\rhoA) where A = bh. For same

beam: f_{1} \approx 35 Hz. Heavier loads lower f_{1} — used to measure mass in cantilever MEMS sensors

CE.4 Seismic Waves and Geophysics

Earthquakes generate elastic waves that propagate through the Earth. The moment magnitude M_w relates to the seismic moment M₀ = G A d (G = shear modulus, A = fault area, d = average slip):

M_w=\frac{2}{3}\log_{10}(M_0)-10.7 \qquad \text{(moment magnitude scale)}

(CE.6)

Each unit of M_w is a factor of 10 in seismic moment (factor 31.6 in energy). M_w = 9.0 (Tohoku 2011): M₀ ≈ 3.5×10²² N·m, energy ∼10¹⁸ J ≈ 500 million Hiroshima bombs.

Surface waves(Love and Rayleigh) propagate along the Earth's surface. Rayleigh waves have a retrograde elliptical motion (coupled P and SV), speed ≈ 0.92 v_S. Seismic reflection/refraction is used to map Earth's interior — the same principles as ultrasound imaging.

CE.5 Elastic Energy and Fracture

The elastic strain energy density:

U=\frac{1}{2}\sigma_{ij}\varepsilon_{ij}=\frac{\sigma^2}{2E}\ \text{(uniaxial)} \qquad \text{(elastic energy density)}

(CE.7)

Griffith fracture criterion: a crack of length 2a in a plate under stress σ propagates when the strain energy release rate G_I = K_I²/E exceeds the fracture toughness G_Ic:

K_I=\sigma\sqrt{\pi a} \qquad \text{(stress intensity factor, mode I)}

(CE.8)

Critical crack length: a_c = K_Ic²/(πσ²). For glass (K_Ic = 0.7 MPa√m) under σ = 70 MPa: a_c = 0.49/(π×4900) ≈ 32 μm — scratches this size cause failure! For steel (K_Ic = 50 MPa√m): a_c ≈ 16 cm — much more defect-tolerant.

Definition CE.2 — Common Traps

Stress and strain are tensors: direction and orientation matter, not just magnitude.
Linear elasticity is a small-deformation theory: large rotations or strains require nonlinear measures.
Young's modulus is not the only stiffness: shear, bulk, and Poisson response matter for 3D loading.
Fluids cannot support static shear: $set \mu = 0 for ideal fluids, so S-waves vanish$
Fracture is defect-controlled: small cracks can dominate failure even when average stress is modest.

Exercises — CE.1–CE.5 Continuum Mechanics

Write the strain tensor for a pure shear deformation u_{x} = \gammay. Find the corresponding stre

ss and principal strains for an isotropic material.

Straightforward

The Earth's crust has P-wave speed 6 km/s, S-wave speed 3.5 km/s, and density 2700 kg/m^{3}

. Find the Young's modulus E (GPa).

GPa

Intermediate

A steel railway rail i

s clamped between two rigid supports at 20°C. Find the compressive stress (MPa) at 60°C (\Delta

perature would it yield?

MPa

Intermediate

4.Explain why Rayleigh waves are dispersive in a layered Earth but not in a homogeneous half-space. How is this dispersion used in seismic tomography?

Challenging

Key Takeaways

$Strain tensor: \varepsilon_ij = \frac{1}{2}(\partialu_i/\partialx_j + \partialu_j/\partialx_i). Diagonal = stretch, off-diagonal = shear$ .
$Isotropic Hooke's law: \sigma_ij = \lambda\theta\delta_ij + 2\mu\varepsilon_ij. Two Lamé constants describe all elastic beh$ avior.
$P-waves (longitudinal): v_{P} = \sqrt((\lambda+2\mu)/\rho). S-waves (transverse): v_{S} = \sqrt(\mu/\rho). Always v_{P}$ > $v_{S}$
$Cantilever tip deflection: \delta = FL^{3}/(3EI). Stiffness scales as h^{3} — height is critical$ .
$Griffith fracture: K_{I} = \sigma\sqrt(\pia). Critical crack size a_{c} = K_{Ic}^{2}/(\pi\sigma^{2}$ .
Seismic waves map Earth's interior. Liquid outer core detected by absence of S-waves.