Continuum mechanics of two-phase porous media (5p)

Instructor: Ragnar Larsson, 7725267, ragnar.larsson@me.chalmers.se

General purpose and contents

The area of multiphase materials modelling is a well established and growing field in the mechanical scientific community. There has been a tremendous development in recent years including the conceptual theoretical core of multiphase materials modelling, the development of computational methodologies as well as experimental procedures. The applications concern biomechanics, modelling of structural foams, composites modelling, soils, road mechanics etc. Specific related issues concern modelling of: solid-fluid interaction, compressible-incompressible fluids/solids including phenomena like consolidation, compaction, erosion, growth, wetting, drying etc.

The main purpose of this course is to give an up-to-date account of the fundamental continuum mechanical principles pertinent to the theory of porous materials considered as mixture of two constituents. The course will provide a framework for the modelling of a “solid” porous material with compressible and incompressible fluid phases. As to constitutive modelling, we restrict to hyper-elasticity and the ordinary Darcy model describing the interaction between the constituents. Computational procedures associated with the nonlinear response of the coupled two-phase material will be emphasized. The lecture notes [1] focus on the general description of kinematics and material models for FE-modeling of large deformation problems.

Organisation of lectures

The course material is defined by [1] plus additional literature references given during the course.
Start: Mon 25/9, 10.00, Materialtekniks seminarierum.
A tentative course outline is as follows:

1. Introduction and applications of the porous media theory, Course outline, [1]:

o The concept of a two-phase mixture: Volume fractions, Effective mass, Effective velocities, Homogenized stress

2. A homogenized theory of porous media

o Kinematics of a two-phase continuum

o Conservation of mass:

§ One-phase material, Two-phase material, Mass balance of fluid phase in terms of relative velocity, Mass balance in terms of internal mass supply, Mass balance - final result

3. a) Conservation of momentum changes and b) energy - isothermal case
• a) Total format, Individual phases and transfer of momentum change between phases
• b) Total formulation, Individual phases, Energy equation in localized format, Assumption about ideal viscous fluid and the effective stress of Terzaghi
4. Conservation of energy (cont’d) and Entropy inequality
• General approach (effective free energy), Localization, Effective drag (or interaction) force
5. Constitutive relations
• Effective stress response, Solid-fluid interaction, Solid densification - reduction of a three phase model, Gas densification - the ideal gas law
6. Summary - Balance relations for different types of porous media
• Classical incompressible solid-fluid medium, Compressible solid-fluid medium, Compressible solid-gas medium
• Restriction to small deformations - Compressible solid-fluid medium
7. Assignment: specific model, cont’d
• Computational aspects
• Modelling of effective solid phase (Hyper-elasticity), Darcy interaction, Issue of incompressibility, boundary value problem
• Discretization, Set of non-linear FE equations, Solution of coupled problem (monolithic/staggered solution techniques)
8. Summary of the course

Course work and examination

Pertinent assignments, involving “derivation of continuum mechanical relations” and “computer implementation of a specific model”, are given. Completed course work gives 5 credit points.

Literature

[1] R. Larsson, Lecture notes.

Lecture notes and overheads

The lecture notes and overheads used will be available in electronic form, cf. the pdf files,

Overheads: Lecture 1, Lecture 2, Lecture 3, Lecture 4, Lecture 5, Lecture 6.

Lecture notes

Course material from course on Computational Continuum Mechanics, 1997, RLa.