Stress and Strain: Elastic Properties of Earth Materials

Stress and Strain: Elastic Properties of Earth Materials#

Duration: 30 minutes Learning Objectives:

  • Derive Lamé parameters from seismic velocities

  • Apply 2D stress-strain constitutive relationships

  • Interpret strain measurements from geodetic observations

  • Use eigenvalue decomposition to find principal stresses

1. Introduction: Why Stress and Strain Matter in Seismology#

Seismic waves propagate through Earth because rocks are elastic materials that deform under stress and recover their shape. Understanding the relationship between:

  • Stress (force per unit area, Pa) — what causes deformation

  • Strain (dimensionless fractional deformation) — the response

…allows us to connect seismic wave velocities to material properties and interpret earthquake-related ground deformation.

2. Geometry of Forces in a Continuum: Traction and Stress#

Before introducing elastic constants or constitutive laws, we must understand how forces act inside a continuous medium. Stress is not defined on volumes or points in isolation, but on oriented surfaces embedded within the material.

2.1 Traction: Force Acting on an Oriented Surface#

Consider an infinitesimal surface element of area ΔA inside a solid. This surface is characterized by a unit normal vector $\( \hat{\mathbf{n}} \)$ which defines its orientation.

The traction vector (\mathbf{t}(\hat{\mathbf{n}})) is defined as the force per unit area exerted across that surface: $\( \mathbf{t}(\hat{\mathbf{n}}) = \lim_{\Delta A \to 0} \frac{\Delta \mathbf{F}}{\Delta A} \)$

Key properties:

  • Traction is a vector quantity

  • Traction depends on surface orientation

  • For the opposite face, \(\mathbf{t}(-\hat{\mathbf{n}}) = -\mathbf{t}(\hat{\mathbf{n}})\)

This orientation dependence is fundamental: the same material point experiences different tractions on differently oriented planes.

2.2 Decomposition of Traction: Normal and Shear Stress#

The traction vector can be decomposed into:

Normal stress: $\( t_N = \mathbf{t} \cdot \hat{\mathbf{n}} \)$

Shear traction: $\( \mathbf{t}_S = \mathbf{t} - t_N \hat{\mathbf{n}} \)$

By convention in seismology:

  • Extension is positive

  • Compression is negative

2.3 Stress Tensor as a Linear Mapping#

Experiments and force balance arguments (Cauchy’s theorem) show that the traction vector varies linearly with the surface normal. This motivates the definition of the Cauchy stress tensor \(\boldsymbol{\sigma}\):

\[ \boxed{ \mathbf{t}(\hat{\mathbf{n}}) = \boldsymbol{\sigma}\,\hat{\mathbf{n}} } \]

In Cartesian coordinates: $$ \boldsymbol{\sigma} =

(1)#\[\begin{pmatrix} \sigma_{xx} & \sigma_{xy} & \sigma_{xz} \\ \sigma_{yx} & \sigma_{yy} & \sigma_{yz} \\ \sigma_{zx} & \sigma_{zy} & \sigma_{zz} \end{pmatrix}\]
\[ \begin{align}\begin{aligned}Interpretation:\\Force balance (no net torque) requires: \end{aligned}\end{align} \]

\sigma_{ij} = \sigma_{ji} $$ so the stress tensor is symmetric and has six independent components.

2.4 Worked Example: Stress Acting on an Inclined Plane (2-D)#

Consider a two-dimensional stress tensor: $$ \boldsymbol{\sigma} =

(2)#\[\begin{pmatrix} \sigma_{xx} & \sigma_{xy} \\ \sigma_{xy} & \sigma_{yy} \end{pmatrix}\]
\[Let a fault plane be oriented such that its unit normal is: \]

\hat{\mathbf{n}} = (\cos\theta, \sin\theta) $$

The traction acting across the fault is: $\( \mathbf{t} = \boldsymbol{\sigma}\hat{\mathbf{n}} \)$

The normal and shear stresses on the fault are obtained by projection: $\( t_N = \mathbf{t} \cdot \hat{\mathbf{n}}, \quad t_S = \mathbf{t} \cdot \hat{\mathbf{f}} \)\( where \)\hat{\mathbf{f}}$ is a unit vector parallel to the fault.

This geometric procedure—stress → traction → normal/shear decomposition—is central to fault mechanics, Coulomb stress, and earthquake triggering.

3. Geometry of Deformation: Strain vs. Motion#

Stress describes internal forces. Strain describes internal deformation. The two are related, but conceptually distinct.

3.1 Displacement vs. Strain#

The displacement field \(\mathbf{u}(\mathbf{x})\) describes how points move: $\( \mathbf{u} = (u_x, u_y, u_z) \)$

Strain measures relative deformation, not absolute motion. It is defined from spatial gradients of displacement.

Example: A 100 m long bar stretched uniformly to 101 m $\( \epsilon = \frac{\Delta L}{L} = 0.01 \)$

3.2 Infinitesimal Strain Tensor#

The displacement gradient tensor can be decomposed into:

The infinitesimal strain tensor is: $\( \boxed{ \epsilon_{ij} = \tfrac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right) } \)$

  • Only strain (not rigid rotation) produces stress

3.3 Physical Interpretation of Strain Components#

  • \(\epsilon_{xy}\): shear strain (change in right angles)

Important geometric insight:

Even “pure extension” produces shear strain when viewed in a rotated coordinate system.

This is why principal strains and eigenvectors play a central role in seismology.

3.4 Volumetric Strain and Rotation#

The dilatation (relative volume change) is given by: $\( \epsilon_v = \text{tr}(\boldsymbol{\epsilon}) = \epsilon_{xx} + \epsilon_{yy} + \epsilon_{zz} = \nabla \cdot \mathbf{u} \)$

The antisymmetric part of the displacement gradient corresponds to rigid-body rotation and does not contribute to stress.#

Preview: Why This Geometry Matters#

  • Seismic waves, fault slip, and moment tensors all follow directly from these definitions

With this geometric foundation established, we can now introduce constitutive relations that link stress and strain in elastic materials.

4. Elastic Constants and Seismic Velocities#

4.1 The Lamé Parameters#

For isotropic, linearly elastic materials, two parameters fully describe elastic behavior:

  • μ (mu): Shear modulus (Pa)

These relate to seismic wave velocities:

\[v_p = \sqrt{\frac{\lambda + 2\mu}{\rho}}\]
\[v_s = \sqrt{\frac{\mu}{\rho}}\]

  • \(\rho\) = density (kg/m³)

4.2 Deriving Lamé Parameters from Velocities#

Starting from the S-wave equation:

\[\mu = \rho v_s^2\]

Substituting into the P-wave equation and solving for λ:

\[\lambda = \rho v_p^2 - 2\mu = \rho v_p^2 - 2\rho v_s^2\]
\[\boxed{\lambda = \rho(v_p^2 - 2v_s^2)}\]

  • \(\lambda = 2700 \times (6000^2 - 2 \times 3500^2) = 3.08 \times 10^{10}\) Pa = 30.8 GPa

Physical Insight: Typical crustal rocks have μ ≈ 30 GPa, comparable to structural steel!

5. The Stress-Strain Relationship (Hooke’s Law)#

5.1 General 3D Form#

For small deformations, stress and strain are linearly related:

\[\sigma_{ij} = \lambda \epsilon_{kk} \delta_{ij} + 2\mu \epsilon_{ij}\]

  • \(\delta_{ij}\) = Kronecker delta (1 if i=j, 0 otherwise)

5.2 Simplification to 2D (Horizontal Deformation)#

For horizontal plane strain (neglecting vertical stress), we work with 2×2 tensors:

\[\sigma_{11} = \lambda(\epsilon_{11} + \epsilon_{22}) + 2\mu\epsilon_{11}\]
\[\sigma_{22} = \lambda(\epsilon_{11} + \epsilon_{22}) + 2\mu\epsilon_{22}\]
\[\sigma_{12} = 2\mu\epsilon_{12} = \sigma_{21}\]

  • 12 = shear component

6. Physical Meaning of Strain Measurements#

6.1 Geodetic Strain: GPS/GNSS Networks#

Modern seismology uses Global Navigation Satellite Systems (GPS/GNSS) to measure ground displacements with millimeter precision:

  • Convert to strain rates using spatial gradients: \(\epsilon_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)\)

6.2 Typical Strain Values#

  • Tectonic strain rates: ~\(10^{-7}\) to \(10^{-8}\) per year

    • Example: 0.1 μstrain/yr = \(0.1 \times 10^{-6}\)/yr

    • Over 1000 years: ~\(10^{-4}\) total strain

  • Earthquake-induced strain: ~\(10^{-6}\) to \(10^{-4}\) (sudden)

    • Example: 1992 Landers M7.3 earthquake caused strains of \(-0.26 \times 10^{-6}\) (compression) to \(+0.92 \times 10^{-6}\) (extension) 80 km away

  • A 1 m rock sample changes by 1 μm

6.3 InSAR: Imaging Earthquakes from Space#

  • Creates 2D strain maps across entire fault zones

7. Principal Stresses: The Eigenvalue Problem#

7.1 Why Eigenvalues?#

  • These are the principal stress directions

Mathematically, we seek eigenvectors \(\mathbf{v}\) such that:

\[\sigma_{ij} v_j = \lambda_{\text{eig}} v_i\]

or in matrix form: \(\mathbf{\sigma}\mathbf{v} = \lambda_{\text{eig}}\mathbf{v}\)

7.2 Physical Meaning#

  • Eigenvalues \(\lambda_1, \lambda_2\): magnitudes of principal stresses (Pa)

    • \(\lambda_1 > \lambda_2\): maximum and minimum principal stresses

  • Eigenvectors \(\mathbf{v}_1, \mathbf{v}_2\): orientations of principal stresses

    • Express as azimuth (angle from North, clockwise)

7.3 Example: Landers Earthquake Stress Field#

From measured strain at Pinon Flat: $\(\epsilon = \begin{pmatrix} -0.26 & -0.69 \\ -0.69 & 0.92 \end{pmatrix} \times 10^{-6}\)$

Calculate stress using \(\sigma = \lambda \text{tr}(\epsilon) \mathbf{I} + 2\mu \epsilon\): $\(\sigma \approx \begin{pmatrix} -9.5 & -45.6 \\ -45.6 & 50.3 \end{pmatrix} \text{ kPa}\)$

  • Orientation: \(\sigma_1\) at ~N30°E (NE-SW extension, typical for Landers)

7.4 Computing Eigenvalues in Python#

import numpy as np
from numpy import linalg as la

# Define 2D stress tensor
sigma = np.array([[s11, s12],
                  [s12, s22]])

# Compute eigenvalues and eigenvectors
eigenvalues, eigenvectors = la.eig(sigma)

# Convert eigenvector to azimuth (degrees clockwise from North)
azimuth = np.degrees(np.arctan2(eigenvectors[0, 0], eigenvectors[1, 0]))

8. Applications to Earthquake Cycle#

  • Interseismic Strain Accumulation

  • When stress exceeds fault strength (~10-100 MPa), rupture occurs

  • Coseismic Stress Changes

  • Can trigger aftershocks where stress increased

  • 6.3 Postseismic Relaxation

  • Measured by continuous GPS networks

9. Key Takeaways#

  1. Lamé parameters (λ, μ) connect seismic velocities to elastic response:

    • \(\mu = \rho v_s^2\)

    • \(\lambda = \rho(v_p^2 - 2v_s^2)\)

  2. Stress-strain relationship (Hooke’s Law):

    • \(\sigma_{ij} = \lambda \epsilon_{kk}\delta_{ij} + 2\mu\epsilon_{ij}\)

  3. Geodetic strain from GPS/InSAR:

    • Tectonic: ~\(10^{-7}\)/yr

    • Earthquake: ~\(10^{-6}\) (sudden)

  4. Principal stresses from eigenvalue decomposition:

    • Magnitudes: eigenvalues (Pa)

    • Orientations: eigenvectors (azimuth)

  5. Earthquake cycle interpretation:

    • Interseismic accumulation → coseismic release → postseismic relaxation

10. References & Further Reading#

  • 1992 Landers Earthquake: Wald & Heaton (1994), BSSA

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