Quantitative Structural Analysis:
Where does it start?
David D. Pollard, Stanford University
Stephen J. Martel, Univ. of Hawaii
Structural Geology and Tectonics Forum
Golden, CO – June 16 to 18, 2014
Objectives
• Introduce a new way to teach undergraduate
structural geology
– Make the foundational concepts accessible
– Integrate teaching, research and practice
– Unite all aspects of structural geology
– Prepare students for professional work
• Recruit help for testing the new textbook
There are many good textbooks
Change may not be easy…
Why is a new book needed?
• Suppose you asked your students this after
completing the structure course :
What are the foundational concepts that
underlie all tectonic processes and their
structural products?
• Would truly meaningful answers emerge?
A meaningful answer
• Identifies the glue to integrate field work, lab
testing, modeling, and theory
• Connects geometry, kinematics, constitutive
laws, stress states, and boundary conditions
• Informs students how to study the natural
complexity of rock deformation
• Ties structural geology to related disciplines
(e.g. geophysics, rock mechanics, engineering
geology, civil engineering)
Foundational concepts
• Conservation of
– mass
– momentum
– energy
The conservation laws underlie all tectonic
processes and their structural products.
Here, we focus on mass and momentum
Cauchy’s first law of motion
• Quantifies conservation of mass & momentum
Dv i  ji


  gi
Dt
x j
• This law plays a central role in field work, lab
experiments, and modeling
• Tectonic processes are conservative
ma  F
• The only equation of this talk…
Independent variables: Coordinates
Dv i  ji


  gi
Dt
x j
Referential
Descriptions of motion
Spatial
Independent variable: Time
Dv i  ji


  gi
Dt
x j
Relative age
Geochronology
Rate
Dependent variables: Velocity, Stress
• Stress analysis and kinematic analysis do not
stand alone, but are inextricably linked by the
equation of motion
Dv i  ji


  gi
Dt
x j
• These variables are associated as the ‘effect’
and ‘cause’ of deformation
Accommodates the natural complexity
of geologic structures
• Deformation varies spatially
Dv i  ji


  gi
Dt
x j
• Deformation varies temporally
Dv i  ji


  gi
Dt
x j
• Partial derivatives are essential
Boundary conditions at the outcrop
It looks like these curved faults
opened when they slipped.
We should use ‘contact’
boundary conditions.
Photo of Dave and Steve, Bear Creek,
Balloon overhead with Cauchy’s eq.
Dv i  ji


  gi
Dt
x j
Universal value
Dv i  ji


  gi
Dt
x j
• All relevant constitutive properties of rock
– brittle elastic, ductile plastic, viscous, …
• All relevant rates of deformation
– quasi-static to dynamic
• All relevant magnitudes of strain
– infinitesimal to finite
• All relevant length scales
– nm to crustal
• All relevant time scales
– ms to Ma
Putting the equations of motion to use
• Too many dependent variables…
vi (3)
ij (6)
Dv i  ji


  gi
Dt
x j
• Choose a constitutive law and reduce the
number of variables
• Choose the appropriate kinematic relations
Example 1
• Hooke’s Law for linear elasticity and small
strain kinematics
• Navier’s equations of motion for solid
mechanics
• Applications: fractures, faults, dikes…
– Elastic brittle deformation
Sheeting joints
Martel , S.J., 2011, Geophys. Res. Ltrs., v. 38, p. L20303 (photo by Greg Stock)
Example 2
• Stoke’s Law for linear viscosity and rate of
deformation kinematics
• Navier-Stokes equations of motion for fluid
mechanics
• Applications: folding, magmatic intrusions,
salt tectonics…
Buckle folds
Hudleston, P.J. & Treagus, S.H., 2010, J. Structural Geology, v. 32, p. 2042
Example 3
• Von Mises yield criterion
• FEM analysis for elasto-plasticity
• Applications: folding, shear zones, fabrics…
– elastic-ductile deformation
Localized mylonitic foliation
Nevitt, J.M., Pollard, D.D., & Warren, J.M., 2014, J. Structural Geology, v. 60, p. 55-69
Summary
• Cauchy’s equations of motion provide a much
needed universal and foundational concept
for undergrad students of structural geology.
• Teaching from this foundation builds on the
pre-requisite courses in the undergrad
curriculum, particularly calculus and physics.
Summary
• Teaching from this foundation makes it clear
why one should choose a constitutive law and
not divorce kinematic and dynamic variables.
• Teaching from this foundation makes it clear
that deformation varies in space and time,
and provides the tools (partial derivatives) for
analyzing the relevant field quantities.
Will you help us?
• If you would like to be a member of the
testing team for the new book, please contact:
dpollard@stanford.edu
Testing constitutive laws
Nevitt, J.M., PhD Thesis, 2014
Testing constitutive laws
Von Mises
Drucker-Prager
Power-law creep
Constrained viscoplastic
Viscoplastic w/ relaxation
Viscoplastic w/ more relaxation
Nevitt, J.M., PhD Thesis, 2014
Earthquake faulting
1999 Hector mine earthquake (Mw 7.1), southern California
InSAR data
Elastic model
Maerten, F., et. al., 2005, B.S.S.A., v. 95, p. 1654
Volcanic eruption
Curtain of fire, Kilauea