Chapter 2 Lecture: Stress and Strain - EPS 164 Seismology Course Materials

Purpose¶

To describe how the Earth deforms and how forces are transmitted through it, we need a framework for:

Forces inside materials → stress
Deformation of materials → strain

These two quantities are linked through constitutive relationships, which ultimately control:

Seismic wave propagation
Faulting and earthquakes
Rock deformation

In this chapter, we build that framework.

Learning Objectives¶

By the end of this lecture, you should be able to:

Explain how stress describes internal forces in the Earth and how it acts on planes of arbitrary orientation.
Describe the stress tensor as a mathematical representation of internal forces and interpret its physical meaning.
Understand the concept of principal stresses and why they provide a natural coordinate system for analyzing stress.
Distinguish between hydrostatic and deviatoric stress, and explain their roles in deformation and faulting.
Explain how strain describes deformation and how it is derived from spatial variations in displacement.
Distinguish between deformation and rigid-body rotation, and interpret the physical meaning of strain components.
Understand how stress and strain are related through elasticity, and recognize the assumptions of linear elastic behavior.
Relate elastic properties of materials to seismic wave propagation, including the origin of P- and S-wave speeds.

1. The Stress Tensor¶

1.1 Traction¶

Imagine cutting an infinitesimal plane inside a material. Forces act across that plane.

The force per unit area is called the traction:

\mathbf{t}(\hat{\mathbf{n}}) = (t_x, t_y, t_z)

(1)

where $\hat{\mathbf{n}}$ is the unit normal to the plane.

We can always decompose traction into:

Normal stress → perpendicular to the plane
Shear stress → parallel to the plane

Key symmetry:

\mathbf{t}(-\hat{\mathbf{n}}) = -\mathbf{t}(\hat{\mathbf{n}})

(2)

👉 This just says: flip the plane, and the force reverses.

Special case (fluids):

Fluids cannot support shear stress, so:

\mathbf{t} = -P \hat{\mathbf{n}}

(3)

Only normal forces (pressure) exist.

1.2 Stress Tensor¶

Instead of describing traction separately for every possible plane, we define a single object:

The stress tensor:

\boldsymbol{\tau} = \begin{bmatrix} \tau_{xx} & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \tau_{yy} & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \tau_{zz} \end{bmatrix}

(4)

Interpretation:

$\tau_{ij}$ = force in direction $i$ acting on a plane normal to $j$

The key idea is:

\mathbf{t}(\hat{\mathbf{n}}) = \boldsymbol{\tau} \hat{\mathbf{n}}

(5)

👉 The stress tensor maps plane orientation → traction.

1.3 Why is Stress Symmetric?¶

Physics requires that a tiny volume cannot spontaneously rotate.

This leads to:

\tau_{ij} = \tau_{ji}

(6)

So:

Only 6 independent components
Not 9

👉 This is a big simplification.

1.4 Principal Stresses¶

There exist special directions where traction has no shear component:

\boldsymbol{\tau} \hat{\mathbf{n}} = \lambda \hat{\mathbf{n}}

(7)

This is an eigenvalue problem:

(\boldsymbol{\tau} - \lambda I)\hat{\mathbf{n}} = 0

(8)

$\lambda$ → principal stresses
$\hat{\mathbf{n}}$ → principal directions

👉 In these directions, stress is purely normal.

1.5 Principal Coordinate System¶

If we rotate into the principal directions:

\boldsymbol{\tau}_R = \begin{bmatrix} \tau_1 & 0 & 0 \\ 0 & \tau_2 & 0 \\ 0 & 0 & \tau_3 \end{bmatrix}

(9)

👉 No shear stresses remain.

This is often the most physically meaningful coordinate system.

1.6 Maximum Shear Stress¶

The largest shear stress occurs on planes at 45° to the principal axes:

\tau_{\max} = \frac{\tau_1 - \tau_3}{2}

(10)

👉 This is directly related to failure and faulting.

1.7 Hydrostatic vs Deviatoric Stress¶

We can split stress into:

Hydrostatic (mean) stress:¶

\tau_m = \frac{\tau_1 + \tau_2 + \tau_3}{3}

(11)

Causes volume change only

Deviatoric stress:¶

\boldsymbol{\tau}_D = \boldsymbol{\tau} - \tau_m I

(12)

Causes shape change

Key takeaway:

Earthquakes are driven by deviatoric stress, not hydrostatic pressure.

2. The Strain Tensor¶

2.1 Displacement Field¶

We describe motion using displacement:

\mathbf{u}(\mathbf{r}_0, t) = \mathbf{r} - \mathbf{r}_0

(13)

Tracks how points move
This is a Lagrangian description

2.2 From Displacement → Strain¶

Strain describes how displacement varies in space:

J = \nabla \mathbf{u}

(14)

This is the displacement gradient tensor.

2.3 Strain Tensor¶

We split deformation into:

J = \mathbf{e} + \boldsymbol{\omega}

(15)

$\mathbf{e}$ → strain (deformation)
$\boldsymbol{\omega}$ → rotation (rigid motion)

Strain is:

e_{ij} = \frac{1}{2} \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)

(16)

👉 Only the symmetric part changes shape.

2.4 Physical Meaning¶

Diagonal terms → stretching or compression
Off-diagonal terms → shear

👉 Strain tells you how the material deforms, not where it moves.

2.5 Volume Change (Dilatation)¶

\Delta = \nabla \cdot \mathbf{u} = \text{tr}(\mathbf{e})

(17)

$\Delta > 0$ → expansion
$\Delta < 0$ → compression

2.6 Principal Strains¶

Strain behaves just like stress:

\mathbf{e} \hat{\mathbf{n}} = \lambda \hat{\mathbf{n}}

(18)

👉 There are directions with pure extension/compression and no shear.

2.7 Connection to Seismic Waves¶

P-waves:
- Produce volume change
- $\nabla \cdot \mathbf{u} \neq 0$
S-waves:
- Produce shear deformation
- No volume change

2.8 Typical Strain Magnitudes¶

\sim 10^{-6}

(19)

👉 Very small → justifies linear elasticity.

3. Stress–Strain Relationship (Elasticity)¶

To connect forces and deformation, we need a constitutive law.

3.1 General Linear Form¶

\tau_{ij} = c_{ijkl} e_{kl}

(20)

Maps strain → stress
Fully general (anisotropic materials)

3.2 Isotropic Case¶

For most Earth materials (first approximation):

\tau_{ij} = \lambda \delta_{ij} e_{kk} + 2\mu e_{ij}

(21)

Only two parameters:

$\lambda$
$\mu$ (shear modulus)

3.3 Physical Interpretation¶

$\lambda$ → controls volume response
$\mu$ → controls shear resistance

👉 $\mu$ is especially important—it controls whether S-waves exist.

3.4 Seismic Velocities¶

Elasticity directly determines wave speeds:

\alpha = \sqrt{\frac{\lambda + 2\mu}{\rho}}, \quad \beta = \sqrt{\frac{\mu}{\rho}}

(22)

P-waves depend on compression + shear
S-waves depend only on shear

Summary¶

Stress describes internal forces
Strain describes deformation
Elasticity links the two

This framework underpins everything we do in seismology: wave propagation, earthquake mechanics, and Earth structure.