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Click to add title Ragnar Larsson Division of Material and Computational Mechanics Department of Applied Mechanics Chalmers University of Technology S-412 96 Göteborg, Sweden 6/9 2012

Continuum mechanics of two-phase porous media

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General purpose and contents • Main purpose = up-to-date account of the fundamental continuum

mechanical principles of theory of porous materials

• Idea: Provide a framework for modelling of “solid” porous material with “fluid” phase.

• Consider a mixture of two constituents: Solid + fluid, gas (compressible and incompressible)

• Focus on continuum mechanical framework

• Lectures focus on general description of continuum mechanical relationships (kinematics, momentum, mass, energy etc.)

• Emphasize computational procedures associated with nonlinear response of coupled two-phase material.

• Restrict to hyper-elasticity • Ordinary Darcy modelling of interaction between the constituents.

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

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

• 2. A homogenized theory of porous media – Kinematics of a two-phase continuum – 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

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Organization of lectures, cont’d

• 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

– Modelling of effective solid phase (Hyper-elasticity), Darcy interaction, Issue of incompressibility,

– boundary value problem – Computational aspects – Discretization, Set of non-linear FE equations, Solution of coupled problem – (monolithic/staggered solution techniques)

• 8. Summary of the course – Dugga

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Applications of the porous media theory

• Geomechanics - Fluid saturated soils – Consolidation problems - Settlements – Stability of natural slopes

• Composites manufacturing – Wet-out of fibre composite - Solid fibre network and (wet)

fluid resin • Crash safety

– Modeling of foams (closed - open cells) - interaction with fluid phase

• Other topics: – Drying, Erosion – Biomechanics: fluid saturated brain tissue ...

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Time-schedule

• One lecture per week. • Dates agreed on:

– 1. To 6/9, 15-17 – 2. Ti 18/9, 13-15 – 3. Mo 24/9, 10-12 – 4. On 3/10, 10-12 – 5. Mo 8/10, 10-12 – 6. On 17/10, 10-12 – 7. Mo 29/10, 10-12

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• Course work and examination – “Theory questions” involving derivation of continuum mechanical relations

related to the modeling of porous materials. – An assigment including computational aspects, and “computer

implementation”, of a chosen specific porous media model is given. – Completed course work gives 7.5 credit points.

• Literature – [1] R. Larsson, Continuum mechanics of two-phase porous media, Lecture notes

(2006). – The lecture notes and overheads will be available in electronic form. Note!

Course material defined by [1]; Note! Additional literature references given during the course.

– The following reference literature is also proposed: – R. de Boer, Theory of Porous Media - Highlights in the Historical Development

and Current State, Springer Verlag Berlin - Heidelberg - New York, (2000) – O. Coussy, Mechanics of Porous Media, John Wiley & Sons, Chichester, (1995)

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• Related courses

• Note! Orientation towards “continuum mechanics” – Continuum mechanics, M Ekh (ongoing). – Computational Finite Strain Hyper-Elasto Plasticty Theory.

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Lecture 1: The concept of a two-phase mixture - Relation to micromechanics

Theory questions 1. Define and discuss the concept of volume fractions in relation to the micro-

constituents of a two-phase mixture of solid and fluid phases related to an RVE of the body.

2. Define the effective mass in terms of intrinsic and bulk densities of the phases from equivalence of mass. Discuss the issue of (in)compressibility of the basis of this discussion.

3. Define the effective (representative) velocities of the solid and fluid phases related their micromechanical variations across an RVE. Discuss also the issue of “direct averaging”.

4. Based on the quite general result of stress homogenization of a one-phase material, generalize the result to the two-phase situation. Discuss the partial stresses and their relation to intrinsic stresses and micro-stress fields.

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Volume fractions

Note! main aim formulate homogenized theory of two-phase mixture => Refrain from detailed consideration of the micromechanics of the constituents

Consider the constituents homogenized w.r.t. volume fractions an RVE with volume V

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Effective mass

principle of “mass equivalence” to the RVE (with volume V) => total mass M of RVE:

p.s.s.= “principle of scale separation"

intrinsic macroscopic densities

Relation to local micromechanical densities

Introduce the bulk (or saturated) density:

Note intrinsic/bulk densities!

Issue of incompressibility =>

(w.r.t. ref. configurations)

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Effective velocities

Consider equivalence of momentum P produced by “micro-” and “effective fields” within RVE =>

Relationship between micro- and effective fields from "equivalence":

Note! Effective velocity fields are basically mean properties of the momentum of the respective constituents!

Assume: independent (or uncorrelated)

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Homogenized stress From homogenization theory (micromechanics of solid materials, cf. course by KR, FL)=>

Consider total stress as volumetric mean value RVE (with the volume V) micro-mechanical variation of stress field

Generalize this result to two-phase mixture: Idea

=> Micromechanical motivation for homogenized partial stresses of the phases partial stresses = homogenized stress response of phases.

Relation to intrinsic homogenized stresses: consider

Example: consider assumption of ideal fluid => intrinsic stress response defined fluid pressure p