# Quantitative Structural Analysis: Where does it start?

David Pollard, Stanford University
Stephen Martel, University of Hawaii

We address the question: what approach should one adopt to analyze problems of structural geology from a quantitative point of view? We focus here on physical processes of deformation and flow, which are key ingredients necessary to understand the development of most geologic structures, including fractures, faults, fabrics, folds, and intrusions. Underpinning these processes are the conservation laws for mass and momentum. These laws are an attractive starting place for quantitative structural analyses, because they address general and fundamental relationships, while avoiding the need to specify particular mechanical behaviors such as elasticity, plasticity, or viscosity. These laws should be part of the conversation as one starts to analyze structures, whether standing on an outcrop, preparing an analogue experiment, loading rock into a triaxial rig, or contemplating an FEM code. In the context of continuum mechanics, these laws become the Equation of Continuity and Cauchy's First and Second Laws of Motion. In this form, stress and the kinematic quantities (velocity, rate of deformation, strain rate) are linked clearly and inextricably, indicating that one should approach geologic structures using both to guarantee the analysis is consistent with physical reality. Furthermore, the Laws of Motion encourage one to acknowledge that deformation rarely is homogeneous, and they provide the tools for analyzing spatial variations. Because Cauchy's Laws offer too few equations for the unknowns, the next step is to choose a constitutive law. Although laboratory data guide us, in most cases the ambiguities of nature necessitate treating the constitutive laws as 'trial balloons', to see what might satisfy the geologic constraints. We offer a case study of a ductile shear zone to exemplify some benefits and shortcomings of this approach. The analysis also requires geometric data, usually taken from field measurements, and often lacking in 3D coverage. Finally, boundary conditions in the form of tractions, displacements, or velocities are required. Often these are poorly constrained, so a range of conditions must be evaluated. The resulting stress, displacement, and velocity fields then are compared to field data and the procedure is repeated with new field data and laboratory insights leading to a sequence of quantitative analyses that successively provide an improved understanding. We suggest that a first course in structural geology should employ this approach in order to integrate research and teaching. We are preparing a new textbook for undergraduate students using this methodology, and welcome new members to our beta-test team.