Thermo-Hydromechanical and Chemical Coupling in Geomaterials and Applications: Proceedings of the 3rd International Symposium GeoProc'2008

GeoProc2008 collects the proceedings of the International Conference on Coupled T-H-M-C (thermal, hydraulic, mechanical, chemical) Processes in Geosystems.

• Language: English
• Publisher: Wiley
• Release date: May 10, 2013
• ISBN: 9781118623664

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Table of Contents

Preface

Keynote Lectures

Physical Mechanics of In-Pore Phase Transition

1. Introduction

2. Phase Transition within a Porous Solid

3. Unsaturated Poroelasticity

4. Drying shrinkage and freezing expansion

5. Conclusion

6. Acknowledgements

7. References

Localized Failure in Brittle Rock

1. Introduction

2. Background

3. True triaxial data

4. Predictions of failure angle

5. Comparison with data

6. Discussion

7. Conclusions

8. Acknowledgments

9. References

Coupled Analysis of Chemo-Mechanical Processes

1. Introduction

2. General description of the formulation

3. Example of application of the THMC formulation

4. A chemo-mechanical model for expansive clays

5. Conclusions

6. Acknowledgements

7. References

Drilling Into the San Andreas Fault

1. Introduction

2. Frictional strength and heat flow

3. Fault stress state and pore pressure distribution

4. Discussion

5. Acknowledgment

6. References

Section 1. Fundamentals of Mechanics of Porous Media

A Numerical Model for CO2 Wells Ageing through Water/Supercritical CO2/Cement Interactions

1. Introduction

2. Selected experimental features to be reproduced by the reaction-transport model

3. Modelling of the Portland cement-CO2 fluids interactions

4. Results

5. Conclusion

6. References

Study on Shear Stress-Strain Model for Unsaturated Soil

1. Introduction

2. Hyperbolic Shear Stress-strain Model and Its Deficiency

3. New Stress-strain Model Establishing for Unsaturated Soil

4. Mathematical Property of the New Model

5. Comparisons with Laboratory Tests

6. Conclusions

7. References

How Lead Affects the Hydraulic and Microscopic Properties of a Smectite

1. Introduction

2. Material and methods

3. Results

4. Physico-chemical study

5. Conclusion

6. References

Study of Settlements in a Granular Medium by a Probabilistic Approach

1. Introduction

2. Probabilistic model

3. Formulation of the diffusion equation and modelling

4. Equation of diffusion and displacements in a granular medium

5. Parametric study, influence of the diffusion coefficient D

6. Conclusion

7. References

Mechanics of a Soil, a Dynamically Coupled Solid-Water Gas System.

1. Introduction

2. Soil model

3. Dynamic coupling

4. Incremental approach

5. Conclusion

6. References

Mechanics of a Soil, a Dynamically Coupled Solid-Water Gas System.

1. Introduction

2. General stress-strain components

3. Water-gas mixture

4. Isotropic undrained

5. Conclusion

6. References

Mechanics of a Soil, a Dynamically Coupled Solid-Water Gas System

1. Introduction

2. Isotropic compression

3. Shear

4. Branch of linear shear

5. Triaxial test

6. Conclusion

7. References

Simulation and Contrastive Analysis of Typical Pollutant Transporting

1. Introduction

2. Test design

3. Tests process and the result analysis

4. Conclusions

5. Acknowledgement

6. References

Section 2. Experimental Characterization of Coupled T-H-M-C Processes in Porous Media

Gas Retention Phenomenon in Dry or Partially-Saturated Concrete: Permeability Assessment

1. Introduction

2. Theory

3. Experimental methodology

4. Results and Discussion

5. Conclusion and further work

6. References

Simultaneous Measurement of Expansion and Water Humidity Sorption on Montmorillonitic Clays

1. Introduction

2. Materials and methods

3. Results and discussion

4. Conclusions

5. Acknowledgements

6. References

Effect of Temperature on Migration of Gas and Brine in Compacted Salt

1. Introduction

2. Experimental Approach

3. Numerical Approach

4. Results and Discussion

5. Conclusion

6. References

The effect of Wetting Conditions on the Mechanical Strength of Chalk

1. Introduction

2. Test Materials, Fluids and Methods

3. Results and discussion

4. Conclusions

5. References

Induced Geometry in Chalk during Hydrochloric Acid Stimulation

1. Introduction

2. Theory and experimental set-up

3. Results and discussion

4. Conclusion

5. References

An Experimental Investigation of the Evolution of Rock Poromechanical Properties Associated with Chemical Alteration Processes

1. Introduction

2. Evolution of petrophysical properties induced by chemical alteration

3. Geomechanical characterization

4. Evolution of geomechanical properties induced by chemical alteration

5. Conclusions

6. Acknowledgements

7. References

Electrokinetic Treatment of a Natural Silt in Saturated and Unsaturated Conditions

1. Introduction

2. Material, testing equipment and procedures

3. Experimental programme

4. Conclusions

5. Acknowledgements

6. References

Normal Stress-Induced Permeability Reduction of a Fracture in a Large Granite Cylinder

1. Introduction

2. Theoretical aspects

3. Experimental results

4. Conclusion

5. Acknowledgements

6. References

Experimental Study of the Water Permeability of a Partially Saturated Argillite

1. Introduction

2. Bure argillite

3. Experimental set-up and method

4. Experimental results

5. Conclusion

6. Acknowledgements

7. References

Application of the Maturity Concept for the Prediction of Restrained Autogenous Shrinkage of Cement Pastes

1. Introduction

2. Background: the maturity concept

3. Experimental program

4. Results and discussions

5. Conclusions

6. References

Laboratory Experiments on Thermal Effects on Clay Rocks

1. Introduction

2. Thermal expansion and contraction

3. Pore-water pressure response to heating

4. Temperature influence on deformation

5. Conclusions

6. Acknowledgements

7. References

Mechanical Compaction of Porous Sandstone: an Experimental Study using Acoustic Emission (AE) Monitoring

1. Introduction

2. Materials and Methods

3. Results and discussion

4. Conclusions

5. References

An Analysis of the Pulse Test and the Light of Residual Hydraulic Potentials

1. Introduction

2. Governing equations

3. Numerical results

4. Conclusion

5. Acknowledgements

6. References

Section 3. Constitutive Models for T-H-M-C Coupling and Multi-scale Approaches

Formulating Material Properties in Coupled Hydro-Mechanical Modeling

1. Introduction

2. Numerical model

3. Non-linear elastic compressibility applied to a consolidation problem

4. Elasto-plastic material behavior in a triaxial test

5. Conclusion

6. References

Partially Coupled Fluid Flow Modeling for Stress Sensitive Naturally Fractured Reservoirs

1. Introduction

2. Review of literature

3. Methodology

4. Results/Discussion

5. Conclusion

6. References

7. Figures

Poromechanical Modeling of Hygric Shrinkage and Crystallization Swelling in Layered Porous Materials

1. Introduction

2. Experimental work

3. Modeling

4. Results

5. Conclusions

6. References

Fan-shaped Model of Clay Swelling Process

1. Introduction

2. Fan-shaped model of clay swelling process

3. Model of filtration in clays

4. Conclusion

5. References

Early Age Autogenous Deformations of Cement-Based Materials

1. Introduction

2. Experimental study

3. Micromechanics modelling

4. Analysis and discussion of the experimental and numerical results

5. Conclusions

6. References

Identification of the Hydro-Mechanical in-Situ Properties of Tournemire Argillite from Mine-by-test Experiment

1. Introduction

2. Theoretical model

3. Model setup

4. Materials parameters

5. Comparison of results

6. Influence of rock mass viscoplastic behaviour

7. Conclusion

8. References

Model of Coupled Thermo-Hydraulic Transport in Bentonite Based on Mobile and Immobile Water Phase

1. Introduction

2. Governing physical principles

3. Material parameters and functions

4. Numerical code implementation

5. Simulation of the UPC heating experiment

6. Acknowledgements

7. References

Orthotropic Anisotropic Damage Coupled Modeling of Saturated Porous Rock

1. Introduction

2. Continuous damage model

3. Calibration of model parameters

4. Hydraulic mechanical coupled damage

5. Conclusion

6. References

Numerical Evaluation of Effective Transport Properties of Random Cell Models: Two-Point Probability Approach

1. Introduction

2. Formulation of the numerical technique

3. Microstructure generation

4. Determination of the two-point probability function

5. Evaluation of effective properties and the appropriate sample size

6. Conclusions

7. References

Section 4. Numerical Modeling of T-H-M-C Processes

Numerical Analysis of the Desaturation Process at the Argillaceous Toumemire Site (France)

1. Introduction

2. Numerical Model

3. Model Setup

4. Simulation

5. Sensitivity study

6. Comparison with measurements

7. Conclusion

8. References

Numerical Study of the Influence of Fractures on the EDZ around a Nuclear Waste Emplacement Drift

1. Introduction

2. Modelling of near field model domain

3. Modelling steps

4. Concluding remarks

5. Acknowledgements

6. References

Modeling THM Processes in Rocks with the Aid of Parallel Computing

1. Introduction

2. THM processes and their finite element analysis

3. Parallel finite element analysis

4. An example of parallel TM computations

5. Mixed finite element modelling of Darcy flow

6. Conclusions

7. References

Influence of Excavation of Disposal Tunnel on the Near-Field Coupled Thermal, Hydraulic and Mechanical Phenomena

1. Introduction

2. Description of the numerical code

3. Calibration of bentonite and rock property

4. THM analysis of the near field

5. Summary

6. References

The Probabilistic Method: An Efficient Tool to Take into Account the Parameters Variability of Modeling for Durability Design Process

1. Introduction

2. The chloride ingress

3. Variability of the model parameters

4. Calculation of probability of failure

5. Practical application

6. Conclusion

7. References

The Influence of Fractures in the Wall-Block Model Domain in the EDZ using an EPCA Code

1. Introduction

2. Constitutive relation of cell element and HM coupling model

3. Fracture representation in numerical modelling

4. Linear elastic response

5. Elasto-plastic analysis

6. Concluding remarks

7. Acknowledgements

8. References

Simulations of the Thermo-Hydro-Mechanical Behavior of an Annular Reinforced Concrete Structure Heated up to 200°C

1. Introduction

2. Description of the MAQBETH mock-up

3. THM governing equations

4. Numerical Results

5. Conclusions

6. Acknowledgments

7. References

Hydraulic Modeling of Unsaturated Zones Around Three Openings at the Argillaceous Tournemire Site (France)

1. Introduction

2. Description of code

3. Model setup

4. Modeling results

5. Conclusions

6. References

Modeling of Non-Isothermal THM Coupled Processes In Multi-Phase Porous Media

1. Introduction

2. Equations of the problems

3. Numerical scheme

4. Simulation of FEBEX type repository

5. Conclusions

6. References

Scale and Stress Effects on Permeability Tensor of Fractured Rocks with Correlated Fracture Length and Aperture

1. Introduction

2. Deformability of the single fractures with correlation between aperture and trace length

3. Numerical study on permeability REV using UDEC

4. Results

5. Discussion and Conclusion

6. Acknowledgment

7. References

3D Fully Coupled Multiphase Modeling of Ekofisk Reservoir

1. Introduction

2. The complex mechanical behaviour of chalk

3. The PASACHALK constitutive model

4. Numerical 3D model of the Ekofisk field

5. Conclusion

6. Acknowledgements

7. References

Evolution of Permeability in Siliceous Rocks by Dissolution and Precipitation Under Hydrothermal Conditions

1. Introduction

2. Conceptual model

3. Numerical simulations

4. Conclusions

5. Acknowledgements

6. References

Are Uncertainties on the Spatial Distribution of Rock Properties Influential in Coupled Reservoir/Geomechanical Modehng?

1. Overview of the problem

2. Description of the case study

3. Discussion of results

4. Conclusion and further work

5. Acknowledgments

6. References

Thermo-Hydro-Mechanical Behavior of Concrete at High Temperature

1. Introduction

2. Thermo-hydro-mechanical model

3. Numerical simulation

4. Conclusions

5. References

Development of Loads in a Shaft Foundation in Salt Rock due to Seasonal Temperature Changes

1. Introduction

2. Layout of the shaft

3. TM modeling-theory: verification of the determining processes

4. TM modeling of a shaft in rock salt

5. Modeling results

6. Conclusions

7. Acknowledgements

8. References

An Analytical Model to Calculate the Stress Field Induced by a Thin Axisymmetric Producing Reservoir

1. Introduction

2. Mathematical model outline

3. Stress distributions induced by depletion

4. Conclusions

5. References

Time and Chemical Effects on Rock Sample Failure

1. Introduction

2. Subcritcal crack growth and time dependent failure model

3. Laboratory experiments to determine subcritical crack growth parameters

4. Numerical modeling of time dependent failure

5. Discussion and conclusions

6. Acknowledgment

7. References

Effects of Pore Pressure on Failure Process and Acoustic Emissions of Rock Specimen with Pre-existing Random Imperfections

1. Introduction

2. Constitutive relation and model

3. Numerical results and discussions

4. Conclusions

5. References

Modeling the Three-dimensional Hydraulic Performance of a Prototype Repository System within Fractured Crystalline Rock

1. Introduction

2. Hydraulic modeling

3. Prototype repository project in Äspö

4. Conclusions

5. Acknowledgements

6. References

Numerical Simulation of Laboratory Coupled Shear-Flow Tests for Rock Fractures

1. Introduction

2. Experimental study – laboratory coupled shear-flow tests

3. Numerical simulations

4. Results

5. Conclusions

6. References

Section 5 - T-H-M-C Processes in Durability Mechanics of Concrete and Structures

Ultra High Performance Fibre Reinforced Concrete Activation Energy

1. Introduction

2. Material

3. Influence of curing conditions on compressive strength and degree of hydration

4. Prediction of UHPFRC compressive strength under different curing temperatures

5. Activation energy of the tested UHPFRC

6. Conclusions

7. References

Modeling of Mechanical Behavior of Steel Fibre-Reinforced Concrete in a Chemical Evolution Context

1. Introduction

2. Experiments

3. Presentation of the model

4. Model fitting

4. Conclusion

5. Acknowledgments

6. References

A Model for Hydration-Drying Interactions in the Concrete Cover

1. Introduction

2. Presentation of the hydration model

3. Modelling of drying-hydration interactions

4. Results and discussion

5. Conclusions

6. References

Performance Assessment of a Mortar Added with High Calcareous Filler Amounts

1. Introduction

2. Experimental methodology

3. Results and discussion

3.2. Compressive strength and Young's modulus

4. Conclusion

5. Acknowledgments

6. References

Modeling of Isothermal Drying Process in Cementitious Materials

1. Introduction

2. Isothermal drying modelling

3. Modelling analysis

4. Conclusions

5. References

Separation of Damage Mechanisms in Concrete at High Temperature

1. Introduction

2. Modeling

3. Identification of damage mechanisms

4. Conclusions

5. Bibliography

Experimental Analysis of Concrete Structures Affected by DEF

1. Background and motivations

2. Experimental programme

3. Original techniques and specific devices

4. Conclusions and perspectives

5. Acknowledgements

6. References

Percolation and Early Age Behavior of Concrete

1. Introduction

2. Percolation threshold and hydration model

3. Computation Methods

4. Results Obtained for the Entire Microstructure

5. Percolation Algorithm

6. Results Obtained for the Entire Microstructure

7. Conclusions

8. References

How can a Crack Opening be Extracted from a Continuous Damage Finite Element Computation? Application for the Estimation of Permeability

1. Introduction

2. Non-local damage approach

3. Non-local strong discontinuity

4. Numerical comparison

5. Validation

6. Conclusions

7. References

Effect of Carbonation on the Hydro-Mechanical Properties of Portland Cement

1. Introduction

2. Experimental methods

3. Results and discussion

4. Conclusion

5. Acknowledgement

6. References

Assessing the Long-Term Behavior of a Radioactive Waste Disposal Tunnel with a Damage Model Incorporating Chemical Degradation Effects

1. Introduction

2. Modelling of chemical degradation

3. Modelling of near field

4. Examination Results

5. Conclusions

6. References

Thermo-Hydro-Mechanical Behavior of a Petroleum Cement Paste: Chemical Degradation Effects

1. Introduction

2. Experimental investigation

3. Presentation of results

4. Conclusions

5. Acknowledgements

6. References

Experimental Study of Water Desorption and Shrinkage in Mortars and Cement Pastes

1. Introduction

2. Desorption curves and shrinkage

3. Experimental set-up

4. Results

5. Conclusions

6. References

Section 6. T-H-M-C in Engineering Applications and In-situ Investigations

Hydro-Mechanical Response of the Tournemire Argillite to the Excavation of Underground Openings: Unsaturated Zones and Mine-by-test Experiment

1. Introduction

2. Tournemire site and its hydro-mechanical characteristics

3. Mine-by-test experiment

4. Unsaturated zones experiment

5. Assessment of disturbed zones

6. Conclusion

7. References

The Belgian Supercontainer Concept for Radioactive Waste Geological Disposal

1. Introduction

2. Presentation of the simulations

3. Thermal calculation

4. Thermo-Hydric calculation

5. Conclusions

6. Acknowledgements

7. References

Comparative Simulation Study on THM-induced Changes in Hydrological Properties of Fractured Rock near Nuclear Waste Repositories

1. Introduction

2. Basic THM responses

3. Potential for Fracture-Shear-Enhanced Permeability

4. Estimate of Permeability Change by Fracture Normal Stress

5. Results and Discussion on the Impact of Permeability Changes

6. Conclusions

7. References

Long-term Response of Near-Field BMT Models around a Deposition Hole by BEM

1. Introduction

2. Near-field BMT model

3. Modeling methodology

4. Modeling results

5. Conclusions

6. Acknowledgments

7. References

Assessment of Modeling Approaches for Analysis of Coupled THMC Processes in the EDZ of Geological Nuclear Waste Repositories

1. Introduction

2. Model Setup

3. Approaches for Modeling of EDZ Evolution

4. Simulation Results for EDZ Evolution

5. Discussion

6. References

Hydro-Mechanical Modeling of Seepage in Gotvand Dam Foundation

1. Introduction

2. Geology of Aghajari formation

3. Geotechnical investigation

4. Numerical modeling

5. Results and discussion

6. references

Atomized Rainfall Effect on Stability of Coupling Hydraulic-Mechanical Unsaturated Rock Slope

1. Introduction

2. Unsaturated fluid-solid coupling governing function

3. Computation model

4. Conclusions

5. References

Index of Authors

First published in Great Britain and the United States in 2008 by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

© ISTE Ltd, 2008

The rights of Jian-Fu Shao and Nicolas Burlion to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

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Library of Congress Cataloging-in-Publication Data

GeoProc'2008 (2008 : Lille, France)

  Thermo-hydromechanical and chemical coupling in geomaterials and applications : proceedings of the 3rd international symposium GeoProc'2008 / edited by Jian-Fu Shao, Nicolas Burlion.

      p. cm.

  Includes bibliographical references.

  ISBN 978-1-84821-043-1

 1. Rock mechanics--Mathematical models-Congresses. 2. Soil mechanics--Mathematical models-

Congresses. I. Shao, Jian-Fu. II. Burlion, Nicolas. III. Title.

  TA706.G473 2008

  624.1'51--dc22

2008016125

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British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-043-1

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Local Organization Committee

Jian-Fu Shao Chairman

Nicolas Burlion Co-Chairman

Thomas Rougelot Secretary

D. Kondo

F. Skoczylas

International Scientific Committee

E. Alonso (Spain)

J. Carmeliet (Belgium)

A.H.-D. Cheng (USA)

K. T. Chau (Hong Kong)

O. Coussy (France)

E. Detoumay (USA)

L. Dormieux (France)

M. Dusseault (Canada)

D. Elsworth (USA)

X.-T. Feng (China)

A. Gens (Spain)

R. M. Holt (Norway)

Y. Jiang (Japan)

L. Jing (Sweden)

P. Landais (France)

C.F. Lee (Hong Kong, China)

C.I. Lee (Korea)

R.W. Lewis (UK)

J. Liu (Australia)

D. Lydzba (Poland)

G. Meschke (Germany)

I. Neretnieks (Sweden)

H. Niitsuma(Japan)

Y. Ohnishi (Japan)

A. Onaisi (France)

S. Pietruzczak (Canada)

G. Pijaudier-Cabot (France)

J.-C Rougiers (USA)

J. Rudnicki (USA)

B. A. Schrefler (Italy)

P. Selvadurai (Canada)

O. Stephansson (Sweden)

K. Su (France)

H. Thomas (UK)

M. Wallner (Germany)

J. Wang (China)

S.-J. Wang (China)

H. Xie (China)

W. Xu(China)

Q.Yang (China)

J. H. Yin (Hong Kong)

J. Zhao (Switzerland)

R. Zimmerman (Sweden)

Preface

The international conference GeoProc’2008 is the third one of a series of GeoProc, which started in Stockholm in 2003 and continued in Nanjing in 2006. The objective of these conferences is to provide an international forum of exchanges and discussions on the recent advances in experimental investigations, fundamental developments and numerical modelling of thermo-hydromechanical and chemical (THM-C) couplings in geo-systems. Indeed, realistic and robust modelling of THM-C coupling phenomena are fundamental for many engineering applications, such as standard geotechnical engineering and natural risks analyses and management, geological storage and disposal of radioactive and toxic waste, exploration of complex reservoir in the oil industry, sequestration of acid gas as CO2, durability analyses of infrastructures in civil engineering, etc.

Compared to the previous conferences, GeoProc’2008 extends to the THM-C modelling in cement-based materials and structures and their interactions with geological system. The main topics of GeoProc’2008 cover the following topics:

  Fundamentals of mechanics of porous media

  Experimental characterization of coupled T-H-M-C processes in porous media

  Constitutive models for T-H-M-C coupling

  Numerical modelling of T-H-M-C processes

  Multi-scale approaches

  In situ investigation

  T-H-M-C processes in durability mechanics of concrete and structures

 T-H-M-C in engineering applications (radioactive waste disposal, geothermal reservoirs, oil and gas engineering, geological systems, geotechnical and environmental engineering, hydraulic and hydropower engineering, natural hazards and environments, mining engineering)

This book contains 4 keynote lectures and 70 scientific papers written by authors coming from 23 countries. All the contributions have been reviewed by the members of the GeoProc’2008 scientific committee. The proceedings include many papers written by experts recognised internationally in their respective fields. These provide a wealth of information, which would be useful not only for researchers but also for practising engineers.

Special thanks go to all members of the scientific committee for their valuable supports to this conference. The efforts of all the contributing authors and all the persons who assisted in organizing the conferences are gratefully acknowledged.

Nicolas BURLION

Jian-Fu SHAO

Co-Chairs of GeoProc’2008

Keynote

Lectures

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Physical Mechanics of In-Pore Phase Transition

O. Coussy¹

¹ Université Paris-Est,

Ecole des Ponts

Institut Navier

 6-8 Av. Blaise Pascal - Cité Descartes

F 77455 Marne-la-Vallée Cedex 2

France

[e-mail]

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ABSTRACT: In this paper we show how the mechanics of confined phase transition within a deformable porous solid can be addressed in a unique framework, whatever the phase transition considered, either the liquid-gas transition involved in the drying of porous materials or the liquid-solid transition involved in their freezing. Indeed, owing to stability considerations a hydrostatic stress is shown to ultimately prevail within the solid crystal phase so that the latter behaves like a compressible elastic fluid as long as only in-pore phase transition is involved. The extension of saturated poroelasticity to unsaturated conditions allows us to work out appropriate constitutive equations to capture the deformation resulting from in-pore phase transition within an elastic porous solid, while the use of homogenization schemes provides estimates of the unsaturated poroelastic these constitutive equations involve. The prediction of the drying shrinkage or that of the deformation due to cryosuction during freezing reveals the significant effect of the pore size distribution, since the intensity of both the deformation and the elastic energy that the solid matrix can store strongly depends upon the homogenization scheme.

KEY WORDS: poroelasticity, pore size distribution, homogenization, drying shrinkage, crystallization.

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1. Introduction

The understanding of the mechanical behaviour of porous materials upon confined phase transition relates to various issues: cement-based materials in civil engineering (Baroghel-Bouny et al., 1999), woods in the building industry (Santos 00), plants in botanic (Kozlowski et al., 2002), soils in soil science (Chertkov 02), gels in physical chemistry (Smith et al., 2004), vegetables in foods engineering (Ratti 94), tissues in biomechanics (Gusnard et al., 1977), etc. In civil engineering various concerns are attached to confined phase transitions. The drying shrinkage of materials can induce cracks and enhance the penetration of aggressive agents in concrete structures (Bazant et al., 1982). The crystallization of sea-salts induced by successive imbibition-drying cycles is recognized as being an important weathering phenomenon in dry environments close to the sea (Evans 69). It often leads to serious deterioration in porous sedimentary rocks used for building in coastal areas (Cardell et al., 2003, Fassina 00). Besides, the delayed formation of ettringite crystals, which may be encountered in cement-based materials after hardening (Taylor et al., 2001), can seriously damage concrete structures. Furthermore, the durability of water-infiltrated materials subjected to frost action is a major concern in cold climates (Pigeon et al., 1995). Ice formation in concrete is the cause of billions of euros in damage undergone by concrete structures, even in temperate regions: in 1980 the supervision of French civil engineering works concluded that most damage experienced by buildings, bridges, etc. was originated by frost action (LCPC 03). A better understanding of the mechanics of the liquid-gas phase transition and the liquid-solid phase transition in confined conditions within deformable porous materials could help to improve the resistance of building materials in environmental conditions and, thereby, reduce maintenance and repair costs.

The mechanism of drying shrinkage induced by isothermal evaporation is well known. When a porous material is subjected to an outer relative humidity smaller than its initial inner relative humidity, the thermodynamic vapour imbalance enforces the porous material to exchange water vapour with the outer atmosphere, so that the outer relative humidity progressively settles within the material. In turn the liquid water simultaneously evaporates in order to maintain the vapour-liquid equilibrium. This causes the decrease of the liquid saturation degree. The shrinkage of the porous material finally results from the liquid de-pressure induced at the gas-liquid water interface by the de-saturation process. While the drying kinetics is governed by transport phenomena (Mainguy et al., 2001), the asymptotic drying shrinkage is governed by the outer relative humidity only, since, asymptotically, the air pressure recovers the atmospheric pressure value. Through the years the macroscopic modelling of the drying shrinkage has been addressed this way by many authors (Bazant et al., 1982, Coussy et al., 1998).

At first sight it might be thought that the mechanics of freezing porous materials is similar to that of a sealed water-filled bottle subjected to the frost action, where the pressure build up and the consecutive failure result only from the undrained 9% expansion of liquid water solidifying within a single large pore. Unfortunately that is not that straightforward. Indeed, cement pastes can still slightly expand if the saturating liquid water is replaced by benzene, which, unlike water, contracts when solidifying (Beaudoin et al., 1974). One of the main causes of this unexpected expansion is the cryosuction process, which drives liquid water from the remaining solution to the crystallized sites (Everett 61). In contrast, water-saturated porous materials can exhibit a contraction if air-voids are present (Piltner et al., 2000). These observations bring to light that the mechanics of confined crystallization within a porous solid does involve the material micro-structural properties, and particularly the pore size distribution, so that a global approach in the context of the mechanics of porous solids has to be worked out (Coussy 05, Coussy et al., 2007, Coussy et al., 2008a). Although drying and crystallization are two quite distinct phenomena, the main purpose of this paper is to show how the physical mechanics of confined phase transitions occurring within a porous solid can be addressed in a unique framework.

2. Phase Transition within a Porous Solid

2.1. Phase equilibrium laws

Since the celebrated works of Gibbs (Gibbs 1899) it is well known that the coexistence of two phases of the same substance requires their specific chemical potential to be equal. Owing to the very definition of the specific (per mass unit) chemical potential μJ related to any phase J, the free energy supply to the phase during the formation of the new mass dm is μJdm. Conversely the loss in free energy undergone by the phase K, during the extraction of the same mass dm, is −μJdm. If we assume that the J-K phase transition occurs with no dissipation, the free energy balance must be zero, resulting in the equality of the specific chemical potential related to each phase in the form

(1)

If the two coexisting phases J and K are the liquid phase, assumed to be poorly compressible, and the vapour phase, assumed to be an ideal gas, the equilibrium relation (1) provides the celebrated Kelvin’s law, reading

(2)

where patm, hR and ρJ are respectively the atmospheric pressure, the relative humidity and the volumetric mass of phase J, while T and r = R/M are respectively the absolute temperature and the ratio of the ideal gases constant R per the liquid molar mass M.

When the phase coexisting with the liquid is the solid phase (index J = C, with C for Crystals), the derivation of the phase equilibrium relation needs further attention because of the non-hydrostatic stress state that a stressed solid is apt to support at rest, and of the elastic energy it can store. Indeed the law governing the liquid-solid equilibrium is still a matter of debate (Sereka et al., 2004). The key point is the determination of the expression related to the chemical potential μJ of an elastic solid phase J, either solid or liquid. As previously defined the free energy supply μJdm associated with the formation of the mass dm within the phase J can be split in two terms according to

(3)

In (3) the first term AJdm, where AJ stands for the specific Helmholtz free energy, is the free energy supplied to the phase J by advection, owing to the additional mass dm that the phase will ultimately contain at the end of the formation process. The second term dW accounts for the additional work to be done against the already existing phase J to make room for the infinitesimal mass in formation within the phase. In order to determine dW let then the index 0 refer to an undeformed (and unstressed) reference state. For instance, the mass dm occupies the volume dV0 = dm/ρJO in the undeformed reference state, whereas it occupies the volume dV = dm/ρJ in the deformed current state. The distinction between the reference undeformed state and the current deformed state allows us to split the work dW in two terms according to

(4)

where dW0→dVo accounts for the work done against the already existing phase J to make room for the undeformed volume dV0 = dm/ρJ0 that the mass dm would occupy prior to any deformation; dWdVo→dV accounts for the additional work done against the phase J in order that the volume dV0 deforms into the volume dV that the mass dm will finally occupy in the current strain state of phase J. The work dWdVo→dV is thereby simply equal to the opposite of the strain work undergone by the mass occupying initially the volume dVo and transforming into the volume dV. Let then σJ and sJ be respectively the mean stress and the deviatoric stress tensor. Let also εJ and eJ be respectively the current volumetric strain and the current deviatoric strain tensor related to phase J with respect to the reference state. In infinitesimal transformations both

(5)

and the norm of eJ are much smaller than 1. Owing to their definitions, works dW0→dVo and dWdVo→dV can then be expressed in the respective forms

(6)

Collecting the results (3)-(6) we finally get μJ in the form

(7)

In linear elasticity AJ is the sum of a linear form and a quadratic form of its arguments T, 1/ρJ and eJ /ρJO, while σJ and sJ are linear functions of the variations of the latter. As a result μJ is the sum of a linear form and of a quadratic form of its natural arguments T, σJ and sJ. In infinitesimal transformations the quadratic terms related to the stress components are second order terms with regard to the linear terms. Besides, when addressing the liquid-solid phase equilibrium, it is convenient to adopt for the reference state of both phases the melting temperature Tm related to the phase equilibrium which is achieved when both phases are at atmospheric pressure in the absence of any shear. For an isotropic solid (J = C), omitting the subscript J for the stress components, neglecting the quadratic terms both with regard to the stress contribution and with regard to the relative temperature variation (Tm−T)/Tm, this choice for the reference state finally provides

(8)

where sJ is the specific entropy related to phase J. At the melting point, where Tm−T and σJ + patm are zero, the liquid-solid phase equilibrium relation (1) requires the reference chemical potentials to be equal, resulting in μC0 = μL0. When the temperature drops below the melting temperature Tm, combining (1) and (8) for J = L or C and making the approximation

(9)

we finally find out that the liquid phase and the solid phase can co-exist provided that they are pressurized according to

(10)

where ΔSm is the volumetric melting entropy defined by

(11)

when the solid crystal phase is subjected to a hydrostatic stress state, so that σC = −pC (10) reduces to the standard Thomson law (Markov 03), reading

(12)

2.2. In-pore phase transition

The previous section has analysed the general conditions governing a phase transition. This section addresses the case where the phase transition occurs within a rigid porous solid, leaving to the next section the case of a deformable porous solid. First considering the liquid-vapour phase transition, dry air is assumed to freely penetrate the porous volume where the phase transition occurs. At any time the porous volume is therefore filled up by the liquid and by the gas (index J = G) consisting of dry air and of vapour in equilibrium with the liquid. Let then ϕ0 and ϕJ be respectively the initial porosity and the current partial porosity related to the phase J, and let SL and SG be the liquid and gas saturations accounting for the fractions of the porous volume occupied by the liquid and by the gas. Owing to the assumed rigidity of the porous solid, we have

(13)

Let in addition pJ be the pressure related to phase J, and let F be the free energy of the porous solid once the bulk phases L and G have been removed. Assuming no hysteretic capillary effect the first law and the second law of the thermodynamics combine to the isothermal incremental free energy balance (Coussy 04)

(14)

Since the bulk liquid and gas phases have been removed, and that the porous solid is assumed to be undeformable, the free energy F reduces to the surface energy of the interfaces between the phases and the solid matrix forming the solid part of the porous solid. Denoting by U the surface energy per unit of porous volume this allows us to write

(15)

Substituting (13) and (15) in (14) we get

(16)

which shows that the liquid saturation SL is a state function of the capillary pressure pG− pL. We write

(17)

where ϖ is the so-called capillary curve. The macroscopic capillary curve can receive a simple microscopic interpretation at the pore scale. At that scale the mechanical equilibrium of the current gas-liquid interface is governed by Laplace’s law. As a result, assuming a zero wetting angle and assuming the liquid to be the wetting fluid, we write

(18)

where γGL is the energy per unit of surface of the gas-liquid interface, whose r is the mean curvature radius. Standard porosimetry then provides the cumulated porous volume fraction S(r) of pores having a pore entry radius smaller than r, or, equivalently, the cumulated porous volume fraction 1 − S(r) of pores having a pore entry radius greater than r. This is illustrated in the left side of Fig. 1 for a cement paste. For a given value of the capillary pressure pG − pL, the pores with an entry radius smaller than that given by (18) will still remained filled by the liquid while the other pores with larger entry radius will be invaded by the gas. As a consequence we write

(19)

Figure 1. Left hand: cumulative pore volume fraction 1−S(r) of pore shaving a pore entry radius greater than r for a typical cement paste from porosimetry data extracted from literature (Cheng-yi et al., 1985). Right hand: cooling ΔT = (T − Tm) and relative humidity hR plotted against the liquid saturation SL as predicted by (24) and (21) for the experimental data of the left hand figure

Combining (18) and (19) we get

(20)

which provides an explicit determination of the capillary curve. If in addition we assume that the gas remains constantly at the atmospheric pressure patm of Kelvin’s law (2) and (20) finally combine to give

(21)

which provides the liquid saturation SL as a function of the current relative humidity hR.

Figure 2. Solid crystal block in a cylindrical pore for two distinct stress states allowed at equilibrium for the same cooling: a) the end face is flat and the stress state is non-hydrostatic; b) the end face is spherical and the stress state is hydrostatic. However it can be shown that the flat interface is unstable and will inexorably transform into the curved interface

Similarly to the relation linking the liquid saturation SL to the current relative humidity hR we found for the liquid-vapour transition, it is tempting to look for an analogous relation linking the liquid saturation SL to the current cooling Tm− T for the liquid-solid phase transition. To that purpose we first replace (18) by

(22)

where γCL is the interface energy of the liquid-solid interface and r its curvature radius as illustrated in right hand of Fig. 2. As result, instead of (21) we now find

(23)

which, combined with Thomson’s law (12), provides the liquid saturation SL as a function of the current cooling Tm− T in the form

(24)

However, instead of the hydrostatic stress state and the curved solid-liquid interface represented in the right hand of Fig. 2, the phase equilibrium law (10), when associated with the same cooling, allows the stress state to be non-hydrostatic and the interface to be flat, as represented in the left hand of Fig. 2. The key relation (24) would then no longer be valid. Fortunately it can be shown (Coussy et al., 2008c) that the non-hydrostatic stress state associated with the flat interface is not stable. Indeed, for the same cooling the free Helmholtz energy associated with a flat interface is larger than the one associated with a curved interface because of the additional shear contribution to the elastic energy in the former case. As a result the flat solid-liquid interface corresponding to a non-hydrostatic stress state will inexorably transform into a curved interface corresponding to a hydrostatic stress state, so that (24) finally applies. Based on relations (21) and (24) we have just derived, the right hand of Fig. 1 shows the liquid saturation SL as a function of both the relative humidity hR and the cooling Tm− T when the liquid is water and for the cement paste corresponding to the experimental data reported in the right hand of Fig. 1.

3. Unsaturated Poroelasticity

According to the analysis of the previous section, two phases of the same substance which co-exist in a porous space are not subjected to the same pore pressure. Accordingly the prediction of drying shrinkage of elastic porous solids, or that of the deformation due to their freezing, requires the extension of saturated poroelasticity to unsaturated conditions where the internal solid walls delimiting two separate parts of the porous space are not subjected to the same pressure. This is the topic of this section.

3.1. State equations of unsaturated poroelasticity

Let us consider a porous solid which is now deformable and whose porous space is saturated both by a liquid phase (index L) and by a phase either gaseous or solid (index J = G or C). Noting σij and εij the overall stress and strain components, the free energy balance (14) extends in the form

(25)

where σijdεij accounts for the strain work. Since the porous space now deforms, instead of (13) we now write

(26)

where φJ accounts for the change in the partial porosity ϕJ due the deformation only. Note that saturation SJ can be coined as a Lagrangian saturation related to phase J since it refers to the undeformable configuration. More precisely, starting from fully liquid saturated conditions, ϕ0SL is the fraction of the porous volume in the non deformed configuration whose solid walls will be still wetted by the liquid in the current deformed configuration (Coussy 05, Coussy 07).

With regard to the undeformable porous solid the free energy F of the deformable system obtained by removing the bulk phases L and J now splits into the surface energy ϕ0U associated with the interfaces, and the elastic energy ψS stored in the deformable solid matrix. Accordingly expression (15) transforms into

(27)

Substitution of (26) and (27) into (25) provides

(28)

The change in U is mainly due to the creation of new interfaces between the phases during the invasion process by the phase J, U being slightly affected by the deformation when SL is held constant. According to the analysis of the previous section U can be still considered as a function of the liquid saturation SL only. Conversely, if we assume infinitesimal elastic deformations of the porous solid, its elastic energy ψS will be slightly affected by the variation dSL of the liquid saturation. As a result, (28) allows us to conclude that (15) will still hold, while the free energy balance related to the deformable porous solid obtained by removing the interfaces is

(29)

from which we derive

(30)

Letting WS = σijεij + φL pL + φJ pJ - ψS be the Legendre transform of ψS with regard to both φL and φJ we alternatively get

(31)

In the context of both infinitesimal deformation and linear isotropic poroelasticity the expression of the elastic energy of the solid matrix, WS = ψS, is

(32)

where σ = σkk/3 and sij are respectively the mean stress and the components of the deviatoric stress tensor. Letting in addition ε = εkk be the volumetric strain and substituting (32) in (31) we finally get

(33)

(34)

(35)

K and G are therefore identified as the bulk modulus and the shear modulus of the dry porous solid which can be recorded in the absence of pore pressures.

When pL = pJ we must retrieve the saturated case so that we have (Coussy 91)

(36)

where b and N are the poroelastic properties of the porous solid with uniform pore pressure, while kS is the bulk modulus of the solid matrix assumed to be homogeneous. Using mesoscopic-macroscopic considerations (Coussy 91, Coussy et al., 2008) or more refined up scaling methods (Dormieux et al., 2006), it can be further shown that

(37)

Provided that kS is known the previous relations are independent of the porous solid considered. In contrast separate expressions of the poroelastic properties K, G, bJ and NJK as functions of the porosity ϕ0 and saturation SJ require specific pieces of information on the porous solid considered. This is addressed in the next section.

3.2. Estimates of the unsaturated poroelastic properties

3.2.1. Pore iso-deformation

The first relation of (36) shows that bL and bJ are not independent and allows us to introduce a Bishop-like parameter (see (Bishop et al. 1963)), depending on the liquid saturation SL and such that

(38)

The explicit determination of function (SL) needs new pieces of information. Using up scaling procedures, the Bishop-like parameter can be shown to reduce to the liquid saturation SL provided that the deformation localisation tensor is the same for all pores (Chateau et al., 2002). In fact, without the need of having recourse to such sophisticated procedures, this result can be simply recovered by assuming that all pores undergo the same volumetric deformation (Coussy 07). This assumption amounts to writing

(39)

Substitution of (39) in (34) and (35) then provides

(40)

Substituting (40) in the first of relations (36), we finally get the above mentioned identification, that is

(41)

Because of relations (37) and (41), it can be then shown that the porous volumes occupied by the liquid L and the phase K would still deform in the same manner provided that they are subjected to the same pressure.

3.2.2. Mori-Tanaka scheme

When the pore iso-deformation assumption is relevant, we are left with the determination of the expressions of the bulk modulus K and the shear modulus G as functions of the porosity ϕ0. This determination can be achieved by using up scaling procedures. It is no question here to enter the details of these procedures, the reader being referred to (Dormieux et al., 2006) for a comprehensive application of micromechanics to porous materials. In this section we will restrict ourselves to recall well known results associated with the Mori-Tanaka scheme.

Since the overall volumetric strain ε is the average of the volumetric strain, letting εS be the volumetric strain of the solid matrix we first write

(42)

The variation of volume φKV of a spherical void of initial volume ϕ0SKV, which is embedded within an elastic matrix with k and g as bulk and shear moduli, while subjected to the pore pressure pK, can be expressed in the form

(43)

where ε0 is the volumetric strain prescribed at infinity. The homogenization schemes differ from each other by the choice of the embedding medium having the elastic properties k and g. The Mori-Tanaka scheme consists of choosing the solid matrix as the embedding medium, that is choosing k = and g = , resulting in ε0 = εS. The self-consistent scheme consists of choosing the porous solid whose we look for poroelastic properties as the embedding medium, that is choosing k = K and g = G. In the following we will restrict to the former choice so that (43) becomes

(44)

Substitution of (44) for successively K = L and K = J into (42) provides εS in the form

(45)

In turn, substituting (45) in (43) we recover the constitutive equations (34) and (35) of unsaturated poroelasticity and the associated relations (36)-(37), but now with the benefit of the new relations

(46)

The determination of the relations providing the missing relation involving the shear modulus G is by far less straightforward since it corresponds to prescribe at infinity the deviatoric strain instead of the volumetric strain. We will restrict here to recall the final result (see for instance (Dormieux et al., 2006))

(47)

3.2.3. Beyond the pore iso-deformation assumption

The comparison of the two first relations in (46) with (41) shows that the pore iso-deformation holds when adopting the Mori-Tanaka scheme. This is due to the fact that all pores see the same embedding medium irrespective of their size, with thereby no associated scale effect. To roughly account for this scale effect, it can be first considered that there are only two sizes of pores referred to by subscript J(= G or C) for the largest pores occupied by the non wetting gas or the solid crystal phase, and by subscript L for the smallest pores occupied by the wetting liquid, in conformity with the analysis we did in the first section. This rough scale separation, which is actually quite approximate since the smallest pores occupied by the gas or the solid crystals have a size comparable with the largest pores occupied by the liquid, leads us to develop a two-scale scheme where we write

(48)

where κJ is the bulk modulus of the porous solid matrix consisting of the original solid matrix and of the smaller pores forming the porous volume at pressure pL. The porosity ϕ0J of this porous solid matrix is the ratio of the porous volume at pressure pL to the overall volume from which we remove the porous volume at pressure pJ. Accordingly we write

(49)

Owing to the general relation (37) the other poroelastic properties are then derived from the sequence

(50)

The assessment of the poroelastic properties needs the determination of κJ, which can be achieved by using a specific homogenization scheme. However the two-scale approach does not affect the results we already derived, when using the Mori-Tanaka scheme. Indeed it can be easily checked that the pore iso-deformation condition bJ = bSJ is preserved when using (50) and the expression of κJ associated with the Mori-Tanaka scheme and obtained by replacing K and ϕ0 respectively by κJ and ϕ0J in the second relation in (46).

Figure 3. Sketch out of the differential homogenization scheme for determining the unsaturated poroelastic properties

A more refined homogenization scheme is the differential homogenization scheme sketched out in Fig. 3. Whatever the value of r the differential scheme consists of considering only the pores occupying the fraction ϕ0dS(r − dr) as forming the porous space related to the porous solid that the larger pores occupying the volume fraction ϕ0dS(r) see as embedding medium. Similarly to (49) the porosity df0 of the embedding medium is then given by

(51)

In view of the explicit determination of the unsaturated poroelastic properties further calculations let then first rewrite (46) and (47) in the form

(52)

Applying (52), where we replace κ, G, kS, gS and ϕ0 by respectively κ, γ, κ + dκ, + dγ and df0, and retaining only the terms of main order with regard to dκ, dγ and dS we finally get

(53)

Let then a be defined by

(54)

so that

(55)

From (53)-(55) we derive

(56)

which can be integrated in the form

(57)

From (53) and (55) we get

(58)

Integrating the left hand side between kS and κJ and the right hand side between αS and α(SJ)=4γJ/3κJ, we successively obtain

(59)

and

(60)

Finally, since b = 1-K/kS (48), (50) and (59) combine to

(61)

Although no simple expression can be obtained for defined by (38), expression (61) of Biot coefficients related to the differential scheme shows that relation (41) no more holds. Indeed the pores do not undergo the same volumetric deformation owing to the coupling between of the deformation of pores having distinct sizes. The remaining poroelastic properties NJK are then provided by substituting (59) and (61) into their expressions (50).

Figure 4. Bishop-like parameter x and poroelastic coupling properties gS/NJK plotted against SL for Mori-Tanaka and differential schemes

In Fig. 4, adopting the value 0.3 for both the Poisson coefficient vS and the porosity ϕ0, we plotted the Bishop-like parameter and the various poroelastic coupling properties gS/NJK against the liquid saturation SL for both the Mori-Tanaka scheme and the differential scheme. In the differential scheme the larger pores deform more than the smaller ones so that exhibits a lower value than the liquid saturation SL which is the value associated with pore iso-deformation. Furthermore, in the absence of pore iso-deformation the poroelastic coupling properties gS/NJK are not symmetric with regard to the line SL/2 properties gS/NLL and gS/NLJ reaching their maximum value for values larger than SL = 1/2 with the opposite for gS/NJJ.

4. Drying shrinkage and freezing expansion

When an initially liquid-saturated porous solid is subjected to surroundings of decreasing relative humidity, it shrinks. Indeed, as captured by (21) the pore entry radius, and consequently the liquid saturation SL, have to adjust to the current relative humidity in order that the confined liquid water to remain in thermodynamic equilibrium with the current vapour pressure imposed by the current relative humidity. This is accompanied with a depressurization of the remaining liquid water, which in turn provokes a drying shrinkage. When an initially liquid-saturated porous solid is subjected to a decreasing temperature of the surroundings, it generally expands. As indicated in the introduction, experiments performed on benzene-saturated cement pastes have even shown that the expansion still occurs even though the saturating liquid contracts when solidifying (Beaudoin et al., 1974). This apparent paradoxical expansion can now be explained as follows. As the temperature T decreases below the melting point Tm, according to (12) the crystal pressure pC has to increase within the already frozen part of the porous volume whose extension is governed by (24). This crystal pressure increase is achieved by the further freezing of some extra liquid water entering the already frozen zone after having been driven from the remaining liquid solution. This is the so-called cryosuction effect. The purpose of this section is to show how both the drying shrinkage and the freezing expansion caused solely by the cryosuction effect can be both addressed by unsaturated poromechanics.

Figure 5. Normalized drying shrinkage (left) and elastic energy (right) for the Mori-Tanaka and differential schemes and the pore size distribution reported in Fig. 1

Consider then a stress-free drying process so that σ = 0, starting from a reference initial state where the porous material is saturated (SL = 1) the pore pressure is atmospheric (pL = patm) and the relative humidity is 100%. With regard to a zero pore pressure state, the deformation ε0 related to the drying initial state is provided by substituting the initial conditions in (33) with bL = b and bJ = 0 so that

(62)

When the relative humidity is lowered below 100% and the gas pressure is maintained at atmospheric pressure patm a drying shrinkage of intensity is observed. Its intensity is obtained by substituting σ = 0, (2) and (38) in (33) and using (62). With the help of the same equations the intensity of the extra elastic energy Wdrying = W(ε) − W(ε0), which is stored during the drying process, can be derived from (32). We get

(63)

Consider now a freezing process with the same initial conditions, but the liquid now being the phase maintained at atmospheric pressure. Letting Thomson’s law (12) and the crystal pressure pC play the role of respectively Kelvin’s law (2) and the gas pressure pG in the previous development which has lead to (63), we similarly get

(64)

It becomes then quite similar to analyse the drying shrinkage or the volumetric deformation induced by cryosuction Indeed, using a specific homogenization scheme the poroelastic properties b, K, and NLL or NCC are then known as a function of the porosity ϕ0, the current liquid saturation SL and the matrix elastic properties kS and gS. In addition the current liquid saturation SL is known as a function of the pore size distribution and either the current relative humidity hR or the cooling Tm−T through (21) or (24), according to the phase transition considered.

For the cement paste corresponding to the experimental data reported Fig. 1, we plotted in Fig. 5 the volumetric deformation εdrying and the elastic energy Wdrying against SL for both the Mori-Tanaka scheme and the differential scheme. The deformation induced by the phase transition exhibits a peak which is achieved when the increase of the difference of pressure becomes compensated by the decrease of the extent of the still wetted porous volume and the associated decrease of the Bishop-like parameter . As shown in Fig. 5 the drying shrinkage is the most significant for the differential scheme because the latter accounts for the softening of the material induced by the difference of deformation of pores having a different size. Similarly the elastic energy stored in the solid matrix during the drying process is more significant for the differential scheme. As a result the difference of deformation of pores having a different size can significantly effect the strength of a porous material subjected to drying if its fracture is brittle and ultimately governed by a threshold in the elastic energy that the solid matrix can store.

5. Conclusion

In this paper we have shown that the mechanics of confined phase transition within a deformable porous solid can be addressed in a unique framework whatever the phase transition considered, either liquid-gas in the case of the drying of porous materials or liquid-solid in the case of their freezing. Indeed, owing to stability considerations a hydrostatic stress will ultimately prevail within the solid crystal phase so that the latter behaves like a compressible elastic fluid as long as only in-pore phase transition is involved. The extension of saturated poroelasticity to unsaturated conditions has allowed us to work out appropriate constitutive equations to capture the deformation resulting from in-pore phase transition within an elastic porous solid, while the use of homogenization schemes has provided estimates of the unsaturated poroelastic these constitutive equations involve. The prediction of the deformation induced by the phase transition has then shown the significant effect of the pore size distribution, since the intensity of the deformation as well as that of the elastic energy that the solid matrix can store strongly depends upon the homogenization scheme. As developed in (Coussy 06) this analysis can be easily extended to address the drying-induced crystallization of sea salts, which combines the two phase transitions explored in this paper. However the improvement in the accuracy of the analysis needs further refinements since it is must be recalled that it is based upon a rough scale separation between the sizes of the pore volumes invaded by the two phases. In addition the progressive cracking of the solid matrix under the internal stresses generated by the drying process might significantly alter the conclusions drawn from a poroelastic reversible analysis. Further research is thereby needed to assess the effects of the size of the pores upon the ultimate strength of porous materials subjected to confined phase transitions, using the tools provided by microporomechanics (Dormieux et al., 2006). In addition, in the analysis of the drying and the freezing of non cohesive porous materials like soils, use of the extension of unsaturated poroelasticity to unsaturated poroplasticity will certainly prove to be fruitful (Coussy et al., 2008b). Besides, the results obtained in this paper have been derived by using a macroscopic approach to in-pore phase transition. This rather simple general approach needs a further confirmation by performing more advanced molecular simulations applied to crystallization processes (Denoyel et al., 2002), which will allow us to account for specific pore geometries and realistic intermolecular interactions related to the solidifying substance and the surrounding solid matrix.

6. Acknowledgements

The author acknowledges the support of the ATILH-CNRS research program Porosity, Transport, Strength.

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