8.3 Stress and strain in the earthquake source region

Earthquakes occur when stress builds up in rocks until they break. This section explores how stress and strain interact in the source region, setting the stage for rupture.

Understanding stress components, strain types, and fault mechanics is crucial for grasping earthquake processes. We'll look at how these factors combine to trigger seismic events and shape their characteristics.

Stress Components

Stress Tensor and Principal Stresses

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• Stress tensor represents the state of stress at a point in a material
• Consists of nine components describing forces acting on infinitesimal cube faces
• Principal stresses define maximum and minimum normal stresses
• Three principal stresses (σ1, σ2, σ3) act perpendicular to principal planes
• Principal stresses determined by solving characteristic equation of stress tensor
• Eigenvalues of stress tensor correspond to magnitudes of principal stresses
• Eigenvectors indicate directions of principal stresses

Shear and Normal Stress

• Shear stress acts parallel to surface, causing sliding or deformation
• Normal stress acts perpendicular to surface, causing compression or tension
• Shear stress crucial in fault mechanics and earthquake generation
• Normal stress influences friction and fault strength
• Relationship between shear and normal stress determines fault stability
• Stress state on a plane described by combination of shear and normal stresses
• Stress resolution used to calculate shear and normal stress on arbitrary planes

Strain and Deformation

Elastic Strain and Material Behavior

• Strain measures relative displacement between particles in a material
• Elastic strain involves reversible deformation under applied stress
• Characterized by linear relationship between stress and strain (Hooke's Law)
• Elastic moduli describe material's resistance to deformation (Young's modulus, shear modulus)
• Poisson's ratio relates lateral strain to axial strain in elastic materials
• Elastic strain energy stored in material during deformation
• Release of elastic strain energy contributes to earthquake generation

Types of Strain and Deformation Mechanisms

• Volumetric strain involves changes in material volume
• Shear strain results in shape change without volume change
• Plastic strain occurs when stress exceeds elastic limit, causing permanent deformation
• Brittle deformation characterized by sudden failure and fracturing
• Ductile deformation involves continuous, plastic flow without fracturing
• Strain rate affects material behavior and deformation mechanisms
• Time-dependent strain includes creep and stress relaxation phenomena

Fault Mechanics

Coulomb Failure Criterion and Mohr Circle

• Coulomb failure criterion defines conditions for shear failure on a plane
• Expressed as $τ = C + μσn$, where τ is shear stress, C is cohesion, μ is friction coefficient, σn is normal stress
• Mohr circle graphically represents stress state on all possible planes
• Radius of Mohr circle indicates maximum shear stress
• Failure occurs when Mohr circle touches or exceeds failure envelope
• Failure envelope determined by material properties and stress conditions
• Mohr circle analysis used to predict fault orientation and slip direction

Stress Drop and Fault Strength

• Stress drop measures stress release during earthquake rupture
• Calculated as difference between initial and final shear stress on fault
• Typical earthquake stress drops range from 1 to 10 MPa
• Stress drop influences earthquake ground motion and seismic energy release
• Fault strength determined by frictional properties and effective normal stress
• Static fault strength resists initial slip, while dynamic strength controls ongoing slip
• Fault weakening mechanisms (thermal pressurization, lubrication) reduce fault strength during rupture
• Stress accumulation and fault strength evolution control earthquake recurrence intervals