The complexity of such problems stems from the inherent material nonlinearity of the solid phase of an inelastic porous medium (such as soil, rock, or concrete skeleton, porous metal, foam, or extracellular matrix of a soft biologically tissue), the coupled mechanical behavior of the solid and fluid phases, and the possibility of large strains and motions encountered in the boundary value problem. As a first course on computational mechanics for inelastic porous media, attention will be focussed on small deformations while providing a basis for advanced computational studies in large deformation and failure of these materials.

The course will cover general constitutive modeling for solids (plasticity, visco-plasticity, visco-elasticity, ...), Finite Element (FE) implementation of constitutive models, and FE implementation of coupled solid-fluid mechanical governing equations for inelastic porous media (cf. http://www.olemiss.edu/sciencenet/poronet/ ). Specifics of the numerical integration and FE implementation will be taught in the context of a commercially-available FE software program (see syllabus). Governing equations for biphasic porous media (solid skeleton and pore fluid) will be derived and expressed in weak form for implementation by the FE method. Steady state and transient conditions will be considered.

Course Objective: To develop the mathematical language, numerical skills, and thought process to formulate, numerically implement, and use nonlinear constitutive models and poromechanics for finite element analysis of inelastic porous media.

1. Overview of Nonlinear FE method and Newton-Raphson solution (1 week)
2. Constitutive modeling and FE implementation:
• 1D plasticity and viscoplasticity (3 weeks): theory (thermodynamics, yield criteria, Kuhn-Tucker conditions, evolution of internal variables for isotropic and kinematic hardening/softening, uniqueness); and numerical integration in time
• 3D deviatoric (isochoric) plasticity: (2.5 weeks) deviatoric J2 plasticity for metals and soils in locally undrained condition; numerical integration (return mapping algorithm) and consistent tangent operator
• 3D pressure-sensitive plasticity: (2 weeks) Overview of plastic mechanical behavior of geomaterials, Mohr-Coulomb, Drucker-Prager, cap, critical state (e.g., Cam-Clay), and threeinvariant models; return mapping/numerical integration; consistent tangent
• Constitutive modeling for other materials: (2.5 weeks) metal damage-plasticity; viscoelasticity for rubber, polymers, etc.; address student interest
3. Balance laws and FE implementation for porous media using mixture theory:
• Introduction to physics of flow through deformable biphasic porous media (1 week): bridging pore-space/particulate scale to continuum scale through concept of volume fraction; conservation laws; Darcy's law
• Analytical formulation and FE analysis of solid-fluid mixture with and without inertiaterms (4 weeks): coupled strong and weak forms; Galerkin approximation; matrix FE equations; time integration (stability and accuracy) for hyperbolic and parabolic systems; wave propagation in porous media

Required