A phase field framework for capillary-induced fracture in unsaturated porous media: Drying-induced vs. hydraulic cracking
Abstract
This work introduces a unified mathematical framework to replicate both desiccation-induced and hydraulic fracturing in low-permeable unsaturated porous materials observed in experiments. The unsaturated porous medium is considered as a three-phase solid–liquid–gas effective medium of which each constituent occupies a fraction of the representative elementary volume. As such, an energy-minimization-based phase-field model (PFM) is formulated along with the Biot’s poroelasticity theory to replicate the sub-critical crack growth in the brittle regime. Unlike hydraulic fracturing where the excess pore liquid pressure plays an important role at the onset and propagation of cracks, desiccation cracks are mainly driven by deformation induced by water retention. Therefore, the wettability of the solid skeleton may affect the evolution of the capillary pressure (suction) and change the path-dependent responses of the porous media. This air–water–solid interaction may either hinder or enhance the cracking occurrence. This difference of capillary effect on crack growth during wetting and drying is replicated by introducing retention-sensitive degradation mechanisms in our phase field fracture approach. To replicate the hydraulic behaviors of the pore space inside the host matrix and that of the cracks, the path-dependent changes of the intrinsic permeability due to crack growth and porosity changes are introduced to model themore »
- Authors:
-
- Columbia Univ., New York, NY (United States); RWTH Aachen Univ. (Germany)
- Columbia Univ., New York, NY (United States)
- Publication Date:
- Research Org.:
- Columbia Univ., New York, NY (United States)
- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE); National Science Foundation (NSF); US Army Research Office (ARO); US Air Force Office of Scientific Research (AFOSR)
- OSTI Identifier:
- 1801239
- Alternate Identifier(s):
- OSTI ID: 1580796
- Grant/Contract Number:
- NE0008534; W911NF-18-2-0306; FA9550-17-1-0169; CMMI-1846875
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Computer Methods in Applied Mechanics and Engineering
- Additional Journal Information:
- Journal Volume: 359; Journal Issue: C; Journal ID: ISSN 0045-7825
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 42 ENGINEERING; engineering; mathematics; mechanics; unsaturated porous media; desiccation cracking; hydraulic fracturing; phase field fracture
Citation Formats
Heider, Yousef, and Sun, WaiChing. A phase field framework for capillary-induced fracture in unsaturated porous media: Drying-induced vs. hydraulic cracking. United States: N. p., 2019.
Web. doi:10.1016/j.cma.2019.112647.
Heider, Yousef, & Sun, WaiChing. A phase field framework for capillary-induced fracture in unsaturated porous media: Drying-induced vs. hydraulic cracking. United States. https://doi.org/10.1016/j.cma.2019.112647
Heider, Yousef, and Sun, WaiChing. Wed .
"A phase field framework for capillary-induced fracture in unsaturated porous media: Drying-induced vs. hydraulic cracking". United States. https://doi.org/10.1016/j.cma.2019.112647. https://www.osti.gov/servlets/purl/1801239.
@article{osti_1801239,
title = {A phase field framework for capillary-induced fracture in unsaturated porous media: Drying-induced vs. hydraulic cracking},
author = {Heider, Yousef and Sun, WaiChing},
abstractNote = {This work introduces a unified mathematical framework to replicate both desiccation-induced and hydraulic fracturing in low-permeable unsaturated porous materials observed in experiments. The unsaturated porous medium is considered as a three-phase solid–liquid–gas effective medium of which each constituent occupies a fraction of the representative elementary volume. As such, an energy-minimization-based phase-field model (PFM) is formulated along with the Biot’s poroelasticity theory to replicate the sub-critical crack growth in the brittle regime. Unlike hydraulic fracturing where the excess pore liquid pressure plays an important role at the onset and propagation of cracks, desiccation cracks are mainly driven by deformation induced by water retention. Therefore, the wettability of the solid skeleton may affect the evolution of the capillary pressure (suction) and change the path-dependent responses of the porous media. This air–water–solid interaction may either hinder or enhance the cracking occurrence. This difference of capillary effect on crack growth during wetting and drying is replicated by introducing retention-sensitive degradation mechanisms in our phase field fracture approach. To replicate the hydraulic behaviors of the pore space inside the host matrix and that of the cracks, the path-dependent changes of the intrinsic permeability due to crack growth and porosity changes are introduced to model the flow conduit in open and closed cracks. Numerical examples of drying-induced and hydraulic fracturing demonstrate the capability of the proposed model to capture different fracture patterns, which qualitatively agrees with the fracture mechanisms of related experiments documented in the literature.},
doi = {10.1016/j.cma.2019.112647},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = C,
volume = 359,
place = {United States},
year = {Wed Oct 30 00:00:00 EDT 2019},
month = {Wed Oct 30 00:00:00 EDT 2019}
}
Web of Science
Works referenced in this record:
Modeling of hydraulic fracturing using a porous-media phase-field approach with reference to experimental data
journal, October 2018
- Heider, Yousef; Reiche, Sönke; Siebert, Philipp
- Engineering Fracture Mechanics, Vol. 202
A unified variational eigen-erosion framework for interacting brittle fractures and compaction bands in fluid-infiltrating porous media
journal, May 2017
- Wang, Kun; Sun, WaiChing
- Computer Methods in Applied Mechanics and Engineering, Vol. 318
A Phase-Field Method for Propagating Fluid-Filled Fractures Coupled to a Surrounding Porous Medium
journal, January 2015
- Mikelić, Andro; Wheeler, Mary F.; Wick, Thomas
- Multiscale Modeling & Simulation, Vol. 13, Issue 1
Advances in modelling hysteretic water retention curve in deformable soils
journal, November 2008
- Nuth, Mathieu; Laloui, Lyesse
- Computers and Geotechnics, Vol. 35, Issue 6
Deformation and localization analysis of partially saturated soil
journal, July 2004
- Ehlers, W.; Graf, T.; Ammann, M.
- Computer Methods in Applied Mechanics and Engineering, Vol. 193, Issue 27-29
Phase field modeling of fracture in multi-physics problems. Part I. Balance of crack surface and failure criteria for brittle crack propagation in thermo-elastic solids
journal, September 2015
- Miehe, Christian; Schänzel, Lisa-Marie; Ulmer, Heike
- Computer Methods in Applied Mechanics and Engineering, Vol. 294
Iterative coupling of flow, geomechanics and adaptive phase-field fracture including level-set crack width approaches
journal, April 2017
- Lee, Sanghyun; Wheeler, Mary F.; Wick, Thomas
- Journal of Computational and Applied Mathematics, Vol. 314
Phase-field modeling of fracture in variably saturated porous media
journal, August 2017
- Cajuhi, T.; Sanavia, L.; De Lorenzis, L.
- Computational Mechanics, Vol. 61, Issue 3
Modeling 3-D desiccation soil crack networks using a mesh fragmentation technique
journal, October 2014
- Sánchez, Marcelo; Manzoli, Osvaldo L.; Guimarães, Leonardo J. N.
- Computers and Geotechnics, Vol. 62
A fully coupled dynamic model for two-phase fluid flow in deformable porous media
journal, March 2001
- Schrefler, Bernhard A.; Scotta, Roberto
- Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issue 24-25
A phase-field modeling approach of hydraulic fracture in saturated porous media
journal, March 2017
- Heider, Y.; Markert, B.
- Mechanics Research Communications, Vol. 80
Phase field model of fluid-driven fracture in elastic media: Immersed-fracture formulation and validation with analytical solutions: PHASE FIELD MODEL FLUID-DRIVEN FRACTURE
journal, April 2017
- Santillán, David; Juanes, Ruben; Cueto-Felgueroso, Luis
- Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 4
Mathematical framework for unsaturated flow in the finite deformation range: FRAMEWORK FOR UNSATURATED FLOW IN THE FINITE DEFORMATION RANGE
journal, December 2013
- Song, Xiaoyu; Borja, Ronaldo I.
- International Journal for Numerical Methods in Engineering, Vol. 97, Issue 9
A continuum phase field model for fracture
journal, December 2010
- Kuhn, Charlotte; Müller, Ralf
- Engineering Fracture Mechanics, Vol. 77, Issue 18
Diffusive fluid‐driven brittle fracture in saturated porous media
journal, December 2018
- Heider, Yousef
- PAMM, Vol. 18, Issue 1
A Variational Approach to the Numerical Simulation of Hydraulic Fracturing
conference, April 2013
- Bourdin, Blaise; Chukwudozie, Chukwudi P.; Yoshioka, Keita
- SPE Annual Technical Conference and Exhibition
Determination of fracture storativity in hard rocks using high-pressure injection testing
journal, October 1998
- Rutqvist, Jonny; Noorishad, Jahan; Tsang, Chin-Fu
- Water Resources Research, Vol. 34, Issue 10
A microstructurally based effective stress for unsaturated soils
journal, December 2010
- Alonso, E. E.; Pereira, J. -M.; Vaunat, J.
- Géotechnique, Vol. 60, Issue 12
A quasi-static phase-field approach to pressurized fractures
journal, April 2015
- Mikelić, Andro; Wheeler, Mary F.; Wick, Thomas
- Nonlinearity, Vol. 28, Issue 5
A variational phase-field model for hydraulic fracturing in porous media
journal, April 2019
- Chukwudozie, Chukwudi; Bourdin, Blaise; Yoshioka, Keita
- Computer Methods in Applied Mechanics and Engineering, Vol. 347
A phase-field approach embedded in the Theory of Porous Media for the description of dynamic hydraulic fracturing
journal, March 2017
- Ehlers, Wolfgang; Luo, Chenyi
- Computer Methods in Applied Mechanics and Engineering, Vol. 315
The Effective Stress Principle: Incremental or Finite Form?
journal, November 1996
- Schrefler, Bernhard A.; Gawin, Dariusz
- International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 20, Issue 11
Mechanics of continuous porous media
journal, January 1980
- Prévost, Jean H.
- International Journal of Engineering Science, Vol. 18, Issue 6
Revisiting brittle fracture as an energy minimization problem
journal, August 1998
- Francfort, G. A.; Marigo, J. -J.
- Journal of the Mechanics and Physics of Solids, Vol. 46, Issue 8
An updated Lagrangian LBM–DEM–FEM coupling model for dual-permeability fissured porous media with embedded discontinuities
journal, February 2019
- Wang, Kun; Sun, WaiChing
- Computer Methods in Applied Mechanics and Engineering, Vol. 344
A phase-field approach embedded in the Theory of Porous Media for the description of dynamic hydraulic fracturing, Part II: The crack-opening indicator
journal, November 2018
- Ehlers, Wolfgang; Luo, Chenyi
- Computer Methods in Applied Mechanics and Engineering, Vol. 341
Optimal approximations by piecewise smooth functions and associated variational problems
journal, July 1989
- Mumford, David; Shah, Jayant
- Communications on Pure and Applied Mathematics, Vol. 42, Issue 5
Thermodynamically consistent phase-field models of fracture: Variational principles and multi-field FE implementations
journal, August 2010
- Miehe, C.; Welschinger, F.; Hofacker, M.
- International Journal for Numerical Methods in Engineering, Vol. 83, Issue 10
Damage, gradient of damage and principle of virtual power
journal, March 1996
- Frémond, Michel; Nedjar, Boumediene
- International Journal of Solids and Structures, Vol. 33, Issue 8
Inf–sup conditions for twofold saddle point problems
journal, March 2011
- Howell, Jason S.; Walkington, Noel J.
- Numerische Mathematik, Vol. 118, Issue 4
Dynamics of unsaturated poroelastic solids at finite strain: DYNAMICS OF UNSATURATED POROELASTIC SOLIDS AT FINITE STRAIN
journal, August 2011
- Uzuoka, Ryosuke; Borja, Ronaldo I.
- International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 36, Issue 13
Permeability due to the Increase of Damage in Concrete: From Diffuse to Localized Damage Distributions
journal, September 2009
- Pijaudier-Cabot, Gilles; Dufour, Frédéric; Choinska, Marta
- Journal of Engineering Mechanics, Vol. 135, Issue 9
Phase-field modeling of a fluid-driven fracture in a poroelastic medium
journal, November 2015
- Mikelić, A.; Wheeler, M. F.; Wick, T.
- Computational Geosciences, Vol. 19, Issue 6
Hydromechanical Modeling of Unsaturated Flow in Double Porosity Media
journal, December 2016
- Choo, Jinhyun; White, Joshua A.; Borja, Ronaldo I.
- International Journal of Geomechanics, Vol. 16, Issue 6
Crack patterns obtained by unidirectional drying of a colloidal suspension in a capillary tube: experiments and numerical simulations using a two-dimensional variational approach
journal, March 2013
- Maurini, C.; Bourdin, B.; Gauthier, G.
- International Journal of Fracture, Vol. 184, Issue 1-2
Partitioned analysis of coupled mechanical systems
journal, March 2001
- Felippa, Carlos A.; Park, K. C.; Farhat, Charbel
- Computer Methods in Applied Mechanics and Engineering, Vol. 190, Issue 24-25
Fracture of a model cohesive granular material
journal, January 2017
- Schmeink, Alexander; Goehring, Lucas; Hemmerle, Arnaud
- Soft Matter, Vol. 13, Issue 5
Mechanics of Hydraulic Fractures
journal, January 2016
- Detournay, Emmanuel
- Annual Review of Fluid Mechanics, Vol. 48, Issue 1
Branching of hydraulic cracks enabling permeability of gas or oil shale with closed natural fractures
journal, January 2019
- Rahimi-Aghdam, Saeed; Chau, Viet-Tuan; Lee, Hyunjin
- Proceedings of the National Academy of Sciences, Vol. 116, Issue 5
Desiccation cracking of soils
journal, September 2009
- Peron, Hervé; Laloui, Lyesse; Hueckel, Tomasz
- European Journal of Environmental and Civil Engineering, Vol. 13, Issue 7-8
On the effective stress in unsaturated porous continua with double porosity
journal, August 2009
- Borja, Ronaldo I.; Koliji, Azad
- Journal of the Mechanics and Physics of Solids, Vol. 57, Issue 8
A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening
journal, June 1979
- Allen, Samuel M.; Cahn, John W.
- Acta Metallurgica, Vol. 27, Issue 6
Free Energy of a Nonuniform System. I. Interfacial Free Energy
journal, February 1958
- Cahn, John W.; Hilliard, John E.
- The Journal of Chemical Physics, Vol. 28, Issue 2
Fundamentals of desiccation cracking of fine-grained soils: experimental characterisation and mechanisms identification
journal, October 2009
- Peron, H.; Hueckel, T.; Laloui, L.
- Canadian Geotechnical Journal, Vol. 46, Issue 10
A semi-implicit discrete-continuum coupling method for porous media based on the effective stress principle at finite strain
journal, June 2016
- Wang, Kun; Sun, WaiChing
- Computer Methods in Applied Mechanics and Engineering, Vol. 304
Experiments versus theory for the initiation and propagation of radial hydraulic fractures in low-permeability materials: INITIATION OF HYDRAULIC FRACTURES
journal, February 2017
- Lecampion, B.; Desroches, J.; Jeffrey, R. G.
- Journal of Geophysical Research: Solid Earth, Vol. 122, Issue 2
A phase field model for rate-independent crack propagation: Robust algorithmic implementation based on operator splits
journal, November 2010
- Miehe, Christian; Hofacker, Martina; Welschinger, Fabian
- Computer Methods in Applied Mechanics and Engineering, Vol. 199, Issue 45-48
Multiscale method for characterization of porous microstructures and their impact on macroscopic effective permeability
journal, May 2011
- Sun, W. C.; Andrade, J. E.; Rudnicki, J. W.
- International Journal for Numerical Methods in Engineering, Vol. 88, Issue 12
Salient factors controlling desiccation cracking of clay in laboratory experiments
journal, January 2013
- Costa, S.; Kodikara, J.; Shannon, B.
- Géotechnique, Vol. 63, Issue 1
A comparative molecular dynamics-phase-field modeling approach to brittle fracture
journal, December 2016
- Patil, Sandeep P.; Heider, Yousef; Hernandez Padilla, Carlos Alberto
- Computer Methods in Applied Mechanics and Engineering, Vol. 312
The Variational Approach to Fracture
journal, March 2008
- Bourdin, Blaise; Francfort, Gilles A.; Marigo, Jean-Jacques
- Journal of Elasticity, Vol. 91, Issue 1-3
Phase-field modeling of ductile fracture at finite strains: A robust variational-based numerical implementation of a gradient-extended theory by micromorphic regularization: PHASE FIELD MODELING OF DUCTILE FRACTURE
journal, February 2017
- Miehe, Christian; Aldakheel, Fadi; Teichtmeister, Stephan
- International Journal for Numerical Methods in Engineering, Vol. 111, Issue 9
A numerical manifold method model for analyzing fully coupled hydro-mechanical processes in porous rock masses with discrete fractures
journal, April 2017
- Hu, Mengsu; Rutqvist, Jonny; Wang, Yuan
- Advances in Water Resources, Vol. 102
A formulation for an unsaturated porous medium undergoing large inelastic strains
journal, March 2002
- Sanavia, L.; Schrefler, B. A.; Steinmann, P.
- Computational Mechanics, Vol. 28, Issue 2
Numerical experiments in revisited brittle fracture
journal, April 2000
- Bourdin, B.; Francfort, G. A.; Marigo, J-J.
- Journal of the Mechanics and Physics of Solids, Vol. 48, Issue 4
A stabilized finite element formulation for monolithic thermo-hydro-mechanical simulations at finite strain: STABILIZED FEM FOR THERMO-HYDRO-MECHANICS AT FINITE STRAIN
journal, April 2015
- Sun, WaiChing
- International Journal for Numerical Methods in Engineering, Vol. 103, Issue 11
Formation of drying crack patterns in soils: a deterministic approach
journal, August 2012
- Peron, Hervé; Laloui, Lyesse; Hu, Liang-Bo
- Acta Geotechnica, Vol. 8, Issue 2
Phase field modeling of fracture in multi-physics problems. Part III. Crack driving forces in hydro-poro-elasticity and hydraulic fracturing of fluid-saturated porous media
journal, June 2016
- Miehe, Christian; Mauthe, Steffen
- Computer Methods in Applied Mechanics and Engineering, Vol. 304
Hydraulic fracturing and its peculiarities
journal, May 2014
- Secchi, Stefano; Schrefler, Bernhard A.
- Asia Pacific Journal on Computational Engineering, Vol. 1, Issue 1
Automatic integration of Euler-Lagrange equations with constraints
journal, May 1985
- Gear, C. W.; Leimkuhler, B.; Gupta, G. K.
- Journal of Computational and Applied Mathematics, Vol. 12-13
A two-scale approach for fluid flow in fractured porous media
journal, December 2006
- Réthoré, Julien; Borst, René de; Abellan, Marie-Angèle
- International Journal for Numerical Methods in Engineering, Vol. 71, Issue 7
Desiccation and cracking behavior of three compacted landfill liner soils
journal, June 2000
- Yesiller, N.; Miller, C. J.; Inci, G.
- Engineering Geology, Vol. 57, Issue 1-2
Soil desiccation cracks as a suction–contraction process
journal, December 2017
- Cordero, J. A.; Useche, G.; Prat, P. C.
- Géotechnique Letters, Vol. 7, Issue 4
A survey of selected coupled multifield problems in computational mechanics
journal, April 2013
- Markert, Bernd
- Journal of Coupled Systems and Multiscale Dynamics, Vol. 1, Issue 1
Phase Field Model of Hydraulic Fracturing in Poroelastic Media: Fracture Propagation, Arrest, and Branching Under Fluid Injection and Extraction
journal, March 2018
- Santillán, David; Juanes, Ruben; Cueto-Felgueroso, Luis
- Journal of Geophysical Research: Solid Earth, Vol. 123, Issue 3
A stabilized assumed deformation gradient finite element formulation for strongly coupled poromechanical simulations at finite strain: STABILIZED F-BAR FINITE ELEMENT FORMULATION FOR POROMECHANICS
journal, January 2013
- Sun, WaiChing; Ostien, Jakob T.; Salinger, Andrew G.
- International Journal for Numerical and Analytical Methods in Geomechanics
Preferential Mode of gas invasion in sediments: Grain-scale mechanistic model of coupled multiphase fluid flow and sediment mechanics
journal, January 2009
- Jain, A. K.; Juanes, R.
- Journal of Geophysical Research, Vol. 114, Issue B8
Validity of Cubic Law for fluid flow in a deformable rock fracture
journal, December 1980
- Witherspoon, P. A.; Wang, J. S. Y.; Iwai, K.
- Water Resources Research, Vol. 16, Issue 6
Pressure and fluid-driven fracture propagation in porous media using an adaptive finite element phase field model
journal, June 2016
- Lee, Sanghyun; Wheeler, Mary F.; Wick, Thomas
- Computer Methods in Applied Mechanics and Engineering, Vol. 305
Numerical modelling of desiccation cracking of clayey soil using a cohesive fracture method
journal, May 2017
- Vo, Thi Dong; Pouya, Amade; Hemmati, Sahar
- Computers and Geotechnics, Vol. 85
Modelling the deterioration of the near surface caused by drying induced cracking
journal, September 2017
- Stirling, Ross A.; Glendinning, Stephanie; Davie, Colin T.
- Applied Clay Science, Vol. 146
Minimization principles for the coupled problem of Darcy–Biot-type fluid transport in porous media linked to phase field modeling of fracture
journal, September 2015
- Miehe, Christian; Mauthe, Steffen; Teichtmeister, Stephan
- Journal of the Mechanics and Physics of Solids, Vol. 82
A review on phase-field models of brittle fracture and a new fast hybrid formulation
journal, December 2014
- Ambati, Marreddy; Gerasimov, Tymofiy; De Lorenzis, Laura
- Computational Mechanics, Vol. 55, Issue 2
Cracking and damage from crystallization in pores: Coupled chemo-hydro-mechanics and phase-field modeling
journal, June 2018
- Choo, Jinhyun; Sun, WaiChing
- Computer Methods in Applied Mechanics and Engineering, Vol. 335
Computational thermo-hydro-mechanics for multiphase freezing and thawing porous media in the finite deformation range
journal, May 2017
- Na, SeonHong; Sun, WaiChing
- Computer Methods in Applied Mechanics and Engineering, Vol. 318
Phase-field modeling of proppant-filled fractures in a poroelastic medium
journal, December 2016
- Lee, Sanghyun; Mikelić, Andro; Wheeler, Mary F.
- Computer Methods in Applied Mechanics and Engineering, Vol. 312
Porous media fracturing dynamics: stepwise crack advancement and fluid pressure oscillations
journal, February 2018
- Cao, Toan D.; Hussain, Fazle; Schrefler, Bernhard A.
- Journal of the Mechanics and Physics of Solids, Vol. 111
Laboratory experiments on desiccation cracking of thin soil layers
journal, December 2006
- Nahlawi, H.; Kodikara, J. K.
- Geotechnical and Geological Engineering, Vol. 24, Issue 6
Phase-field modeling of hydraulic fracture
journal, November 2016
- Wilson, Zachary A.; Landis, Chad M.
- Journal of the Mechanics and Physics of Solids, Vol. 96
A new monolithic Newton-multigrid-based FEM solution scheme for large strain dynamic poroelasticity problems: A new monolithic Newton-multigrid-based FEM solution scheme for large strain dynamic poroelasticity problems
journal, July 2016
- Obaid, A.; Turek, St.; Heider, Y.
- International Journal for Numerical Methods in Engineering, Vol. 109, Issue 8
An Experimental Study of Cracking Induced by Desiccation
journal, February 1994
- Groisman, A.; Kaplan, E.
- Europhysics Letters (EPL), Vol. 25, Issue 6
Experimental and numerical analysis of desiccation of a mining waste
journal, June 2007
- Rodríguez, R.; Sánchez, M.; Ledesma, A.
- Canadian Geotechnical Journal, Vol. 44, Issue 6
A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils1
journal, January 1980
- van Genuchten, M. Th.
- Soil Science Society of America Journal, Vol. 44, Issue 5
Fracture-like borehole breakouts in high-porosity sandstone: Are they caused by compaction bands?
journal, January 2001
- Haimson, B. C.
- Physics and Chemistry of the Earth, Part A: Solid Earth and Geodesy, Vol. 26, Issue 1-2
On the capillary stress tensor in wet granular materials
journal, July 2009
- Scholtès, L.; Hicher, P. -Y.; Nicot, F.
- International Journal for Numerical and Analytical Methods in Geomechanics, Vol. 33, Issue 10
A phase-field method for computational modeling of interfacial damage interacting with crack propagation in realistic microstructures obtained by microtomography
journal, December 2016
- Nguyen, T. T.; Yvonnet, J.; Zhu, Q. -Z.
- Computer Methods in Applied Mechanics and Engineering, Vol. 312
An introduction to phase-field modeling of microstructure evolution
journal, June 2008
- Moelans, Nele; Blanpain, Bart; Wollants, Patrick
- Calphad, Vol. 32, Issue 2
A idealized framework for the analysis of cohesive soils undergoing desiccation
journal, August 1997
- Konrad, J. -M; Ayad, R.
- Canadian Geotechnical Journal, Vol. 34, Issue 4
Modeling the hydro-mechanical responses of strip and circular punch loadings on water-saturated collapsible geomaterials
journal, November 2013
- Sun, WaiChing; Chen, Qiushi; Ostien, Jakob T.
- Acta Geotechnica, Vol. 9, Issue 5
Analysis of Stresses and Strains Near the End of a Crack Traversing a Plate
journal, September 1957
- Irwin, G. R.
- Journal of Applied Mechanics, Vol. 24, Issue 3