Stress, Strain, and the Equation of Motion

ESS 314 Geophysics · University of Washington

Week 1, Lecture 3 · April 7, 2026

Marine Denolle

By the end of this lecture…

  • [LO-3.1] Define the stress and strain tensors; identify normal vs. shear components
  • [LO-3.2] Relate the four elastic moduli (EE, KK, μ\mu, ν\nu) to deformation geometries
  • [LO-3.3] Write isotropic Hooke's law using Lamé parameters λ\lambda and μ\mu
  • [LO-3.4] Derive the equation of motion; identify VP=(λ+2μ)/ρV_P = \sqrt{(\lambda+2\mu)/\rho}
  • [LO-3.5] Evaluate the assumptions of linear elastic theory

Why Does the Ground Shake?

A Cascadia M9 earthquake will reach Seattle in ~90 seconds as elastic waves traveling through rock

The wave speed — and the shaking intensity — depend on the elastic properties of every rock layer the wave passes through

Today: building the physics that connects rock stiffness to wave speed

Elastic Deformation: The Key Assumption

Elastic = material returns to its original shape after stress is removed

Linear elastic (Hookean) = stress ∝ strain

Seismic strains are ~10⁻⁷ — far inside the Hookean regime. Fault zones, magma chambers, and the deep Earth are exceptions.

alt text: stress-strain curve with blue linear Hookean region, amber nonlinear zone, orange plastic zone, and dashed green unloading path showing permanent strain after the elastic limit
Figure 3.1. Seismic waves operate in the blue (linear elastic) region only. Python-generated — assets/scripts/fig_stress_strain_curve.py

Two Modes of Elastic Deformation

Volumetric (dilatational) strain θ\theta

  • Change in volume, no change in shape
  • Resisted by bulk modulus KK
  • → P-waves

Shear strain εij\varepsilon_{ij} (i≠ji \neq j)

  • Change in shape, no change in volume
  • Resisted by shear modulus μ\mu
  • → S-waves

Fluids: μ=0\mu = 0 → no resistance to shear → S-waves CANNOT travel in fluids

The Stress Tensor

σ=(σxxσxyσxzσyxσyyσyzσzxσzyσzz)\boldsymbol{\sigma} = \begin{pmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{pmatrix}

  • Diagonal = normal stresses (compression / tension)
  • Off-diagonal = shear stresses
  • Symmetric (σij=σji\sigma_{ij} = \sigma_{ji}) → 6 independent components
  • Force = Stress × Area: Fx=σxx AxF_x = \sigma_{xx}\,A_x — stress is force per unit area on a surface

alt text: two 3D diagrams showing force components on a surface element decomposing into one normal and two shear forces, and a unit cube with blue normal stress arrows and vermilion shear stress arrows on each visible face
Figure 3.2. Normal stresses (blue) and shear stresses (vermilion) on a unit cube. Python-generated — assets/scripts/fig_stress_tensor.py

Three Modes of Strain

alt text: three-panel figure showing panel a with a cylinder compressed axially with labels h and delta-h and longitudinal strain formula, panel b with a cube compressed uniformly by pressure arrows from all sides with volumetric strain formula, panel c with a rectangle sheared into a parallelogram by a horizontal force with shear angle psi and shear strain formula
Figure 3.3. (a) Longitudinal ε_xx = Δh/h. (b) Volumetric θ = ΔV/V. (c) Shear γ = tan ψ. P-waves involve (a)+(b); S-waves involve (c). Python-generated — assets/scripts/fig_strain_types.py

The Strain Tensor

xx = coordinate (fixed, meters) · u(x)u(x) = displacement (how far that material point moved)

Strain = symmetric part of the displacement gradient:

εij=12(∂ui∂xj+∂uj∂xi)\varepsilon_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)

  • Diagonal (i=ji=j): extension / compression — modes (a) longitudinal and (b) volumetric above
  • Off-diagonal (i≠ji\neq j): angular distortion — mode (c) shear above
  • Factor of ½ excludes rigid-body rotation

Dilatation (volume change):

θ=εxx+εyy+εzz=∇⋅u\theta = \varepsilon_{xx} + \varepsilon_{yy} + \varepsilon_{zz} = \nabla\cdot\mathbf{u}

Four Elastic Moduli

alt text: four-panel figure showing Young's modulus E with a cylinder compressed axially, shear modulus mu with a block sheared into a parallelogram, bulk modulus K with a cube compressed uniformly, and Poisson ratio nu with a cylinder compressed axially that bulges laterally, each panel labeled with defining formula in a green equation box
Figure 3.4. E (axial stiffness), μ (shear stiffness), K (bulk stiffness), ν (lateral/axial ratio). Python-generated — assets/scripts/fig_elastic_moduli.py

Elastic Moduli: Relationships

Any two moduli specify all others. Seismology uses Lamé parameters λ\lambda, μ\mu:

λ=K−23μ=νE(1+ν)(1−2ν)\lambda = K - \tfrac{2}{3}\mu = \frac{\nu E}{(1+\nu)(1-2\nu)}

μ=E2(1+ν)(shear modulus = rigidity)\mu = \frac{E}{2(1+\nu)} \quad\text{(shear modulus = rigidity)}

Key conversions needed for seismology:

  • VPV_P and VSV_S → λ\lambda, μ\mu, ρ\rho → EE, KK, ν\nu
  • Typical crustal granite: λ≈30\lambda \approx 30 GPa, μ≈25\mu \approx 25 GPa, ρ≈2700\rho \approx 2700 kg/m³

Isotropic Hooke's Law

For a homogeneous, isotropic, linear elastic solid:

σij=λ δij θ+2μ εij\sigma_{ij} = \lambda\,\delta_{ij}\,\theta + 2\mu\,\varepsilon_{ij}

Term 1 (λδijθ\lambda\delta_{ij}\theta): volume change drives normal stresses in ALL directions — the coupling term

Term 2 (2μεij2\mu\varepsilon_{ij}): direct resistance to any strain (normal and shear)

Two parameters (λ\lambda, μ\mu) because isotropy collapses 21 stiffness components to 2

Units: Pa · dimensionless + Pa · dimensionless = Pa = [stress] ✓

Force Balance on a Continuum Element

Apply Force = Stress × Area → Newton's F=maF = ma on an infinitesimal element of density ρ\rho.

alt text: two-panel figure. Left panel shows an element between x and x-plus-dx with face area A_x and displacement arrows u and u-plus-du. Right panel shows the same element with a vermilion force arrow F_x entering from the left face and F_x-plus-dF_x leaving the right face, with equation boxes showing net force equals partial of sigma over x times dx, Newton's law, and the final equation of motion
Figure 3.5. Net force = stress gradient × volume. Divide by A_x dx → equation of motion. Python-generated — assets/scripts/fig_force_balance.py

The Equation of Motion → Wave Equation

Step 1 — Net force on element:

dFx=Ax ∂σxx∂x dxdF_x = A_x\,\frac{\partial\sigma_{xx}}{\partial x}\,dx

Step 2 — Newton's 2nd law (F=maF = ma):

ρ ∂2u∂t2=∂σxx∂x\rho\,\frac{\partial^2 u}{\partial t^2} = \frac{\partial\sigma_{xx}}{\partial x}

Step 3 — Substitute Hooke's law (σxx=(λ+2μ)∂u/∂x\sigma_{xx} = (\lambda+2\mu)\partial u/\partial x):

ρ ∂2u∂t2=(λ+2μ) ∂2u∂x2\rho\,\frac{\partial^2 u}{\partial t^2} = (\lambda + 2\mu)\,\frac{\partial^2 u}{\partial x^2}

⇒VP=λ+2μρ\Rightarrow \quad V_P = \sqrt{\frac{\lambda+2\mu}{\rho}}

Two Wave Speeds from One Equation

Wave Equation Speed
P (compressional) ρ u¨=(λ+2μ) u′′\rho\,\ddot{u} = (\lambda+2\mu)\,u'' VP=(λ+2μ)/ρV_P = \sqrt{(\lambda+2\mu)/\rho}
S (shear) ρ u¨=μ u′′\rho\,\ddot{u} = \mu\,u'' VS=μ/ρV_S = \sqrt{\mu/\rho}

Since λ≥0\lambda \geq 0: λ+2μ>μ\lambda + 2\mu > \mu → VP>VSV_P > V_S always

Units: Pa/(kg/m3)=(kg/m⋅s2)/(kg/m3)=m/s\sqrt{\text{Pa}/(\text{kg/m}^3)} = \sqrt{(\text{kg/m·s}^2)/(\text{kg/m}^3)} = \text{m/s} ✓

Stiffer rock → faster waves. Denser rock → slower waves.
Their ratio sets the speed — not either quantity alone.

Worked Example: Granite

λ=30\lambda = 30 GPa, μ=25\mu = 25 GPa, ρ=2700\rho = 2700 kg/m³

VP=(30+50)×1092700=2.96×107≈5443 m/sV_P = \sqrt{\frac{(30+50)\times 10^9}{2700}} = \sqrt{2.96\times 10^7} \approx 5443 \text{ m/s}

VS=25×1092700≈3043 m/sV_S = \sqrt{\frac{25\times 10^9}{2700}} \approx 3043 \text{ m/s}

VPVS=8025=3.2≈1.79⇔ν≈0.27\frac{V_P}{V_S} = \sqrt{\frac{80}{25}} = \sqrt{3.2} \approx 1.79 \quad\Leftrightarrow\quad \nu \approx 0.27

Characteristic of upper-crustal granite ✓

Seismic Velocities: Typical Values

alt text: horizontal bar chart with material names on the vertical axis and P-wave velocity in meters per second on the horizontal axis from 0 to 8000. Dark blue bars for crystalline rocks span 2000 to 6500 m/s, sky blue bars for unconsolidated sediments span 60 to 2000 m/s, green bars for fluids cluster near 1200 to 1540 m/s, amber bars for engineering materials like steel and aluminum are between 5800 and 6400 m/s. A vertical dotted reference line marks 1480 m/s for water.
Figure 3.7. V_P spans nearly two orders of magnitude across Earth materials. Dry sand is ~100× slower than granite. Python-generated — assets/scripts/fig_seismic_velocities.py

The V_P / V_S Ratio as a Fluid Diagnostic

For ν=0.25\nu = 0.25 (typical crust): VP/VS=3≈1.73V_P/V_S = \sqrt{3} \approx 1.73

As ν→0.5\nu \to 0.5 (fluid saturation): VP/VS→∞V_P/V_S \to \infty

Seattle Basin example:

  • VP≈1800V_P \approx 1800 m/s, VS≈300V_S \approx 300 m/s
  • VP/VS=6.0V_P/V_S = 6.0, ν≈0.49\nu \approx 0.49
  • → water-saturated sediment

High VP/VSV_P/V_S = fluid. Low VP/VSV_P/V_S = dry rock or gas sand.
This is the single most useful seismic diagnostic in exploration and hazard.

AI Prompt Lab

Try this after class:

"Is V_P = sqrt(E/rho) or V_P = sqrt((λ+2μ)/rho) for seismic P-waves?"

Both can be correct — but in different contexts. Evaluate whether the AI explains:

  • E/ρ\sqrt{E/\rho}: slender rod, uniaxial stress, free lateral expansion
  • (λ+2μ)/ρ\sqrt{(\lambda+2\mu)/\rho}: 3D bulk wave, constrained lateral deformation
  • The conversion: λ+2μ=E(1−ν)/[(1+ν)(1−2ν)]\lambda + 2\mu = E(1-\nu)/[(1+\nu)(1-2\nu)]

If the AI gives only one answer without qualification → it has oversimplified.

Concept Check

  1. A rock has λ=50\lambda = 50 GPa, μ=30\mu = 30 GPa, ρ=3100\rho = 3100 kg/m³. Calculate VPV_P, VSV_S, and ν\nu. Show unit checks.

  2. A sediment has VP=1500V_P = 1500 m/s and VS=50V_S = 50 m/s. Calculate Poisson's ratio. What does this tell you about the physical state of the sediment?

  3. The equation of motion was derived assuming the material is homogeneous and isotropic. Name one real-Earth situation where each assumption fails, and describe what new physics is needed.

Summary

Concept Key Result
Elastic deformation Hookean (linear elastic), small strains
Stress tensor 6 independent components (σij=σji\sigma_{ij} = \sigma_{ji})
Strain tensor εij=12(∂jui+∂iuj)\varepsilon_{ij} = \frac{1}{2}(\partial_j u_i + \partial_i u_j)
Hooke's law σij=λδijθ+2μεij\sigma_{ij} = \lambda\delta_{ij}\theta + 2\mu\varepsilon_{ij}
Equation of motion ρu¨=(λ+2μ)u′′\rho\ddot{u} = (\lambda+2\mu)u'' or μu′′\mu u''
P-wave speed VP=(λ+2μ)/ρV_P = \sqrt{(\lambda+2\mu)/\rho}
S-wave speed VS=μ/ρV_S = \sqrt{\mu/\rho}

Next lecture: Wave types (P, S, Rayleigh, Love) and Snell's law

Source: USGS / Wikimedia Commons — Public Domain