1. CMG Research: Mathematical Modeling of the Dynamics of Multi-scale Phenomena During Folding and Fracturing of Sedimentary Rocks
Ronaldo I. Borja, Craig D. Foster, and Pablo F. Sanz
Department of Civil and Environmental Engineering
Stanford University

2. Outline of Presentation
Ronnie’s Presentation (40 minutes):
General kinematics of deformation
Constitutive model - plasticity and damage models
Finite element model
Boundary conditions, interface elements
Pablo’s Presentation (10 minutes):
Three-invariant plasticity models
Craig’s Presentation (10 minutes):
Element enhancements for strain localization analysis

3. Kinematics of Deformation
Features that must be captured:
Folding - change in geometry
Fracturing - large strain at crack tip
Finite rotation of fracture zone
Finite deformation formulations:
Lagrangian formulation - appropriate for solids
Eulerian formulation - appropriate for fluids
Both formulations can handle geometric nonlinearity,
large strain, finite rotation, and frame-invariant
constitutive models.

4. Lagrangian versus Eulerian Formulations
We follow the motion of a material point
Constitutive model applies to a material point
Finite element mesh could be severely distorted
Re-meshing possible for severely distorted meshes
Eulerian:
We describe the motion of a fixed point in space
Point in space occupied by different material points
Finite element mesh will not be distorted
ALE (Arbitrary Lagrangian-Eulerian)

5. Lagrangian Formulation
Why this formulation is OK for the CMG project:
Constitutive response applies to a material point
Accommodates coupled plasticity-damage models
We do not expect the mesh to be severely distorted
Captures rotation of fracture as rock folds
Facilitates specification of boundary conditions
Finite element implementation is well developed
Finite element implementation:
Integration performed over the undeformed mesh
Constitutive model uses Kirchhoff or Cauchy stress

6. Illustration of Lagrangian Formulation
Graphics 1

7. Illustration of Lagrangian Formulation

8. Constitutive Model Features to capture: Brittle response
Quasi-brittle response
Ductile response
Plasticity
Damage
Localized deformation
Framework for continuum model:
Coupled elastoplastic-damage formulation
One unifying model that captures all required features
Framework for post-localized model:
Finite element with embedded strong discontinuity