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January  2021, 14(1): 41-55. doi: 10.3934/dcdss.2020303

Cahn-Hilliard equation with capillarity in actual deforming configurations

Tomáš Roubíček 1,2,

1. 

Mathematical Institute, Math.-Phys. Faculty, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic
2. 

Institute of Thermomechanics,Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic

Dedicated to Alexander Mielke on the occasion of his sixtieth birthday.

Received  March 2019 Revised  August 2019 Published  January 2021 Early access  March 2020

Fund Project: * This research has been supported from the grants 17-04301S (regarding the focus on the dissipative evolution of internal variables), 19-04956S (regarding the focus on the dynamic and nonlinear behaviour), and 19-29646L (especially regarding the focus on the large strains in materials science) of Czech Science Foundation, and from the FWF grant I 4052 N3 with BMBWF through the OeAD-WTZ project CZ04/2019, and also from the institutional support RVO: 61388998 (ČR).

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity (i.e. concentration gradient) term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Faedo-Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at an actual configuration allow for similar models and analysis.

Keywords: Poro-elastodynamics, large strains, 3rd-grade nonsimple materials, Darcy/Fick flow, Cahn-Larché system, Galerkin approximation, weak solutions, existence.
Mathematics Subject Classification: 35Q74, 37N15, 65M60, 74A30, 74G25, 74H20, 76S05.
Citation: Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
References:
[1]

E. K. Agiasofitou and M. Lazar, Conservation and balance laws in linear elasticity of grade three, J. Elasticity, 94 (2009), 69-85.  doi: 10.1007/s10659-008-9185-x.

[2]

S. M. Allen and J. W. Cahn, Ground state structures in ordered binary alloys with second neighbor interactions, Acta Metall., 20 (1972), 423-433.  doi: 10.1016/0001-6160(72)90037-5.

[3]

L. Anand, A Cahn-Hilliard-type theory for species diffusion coupled with large elastic-plastic deformations, J. Mech. Phys. Solids, 60 (2012), 1983-2002.  doi: 10.1016/j.jmps.2012.08.001.

[4]

P. Areias, E. Samaniego and T. Rabczuk, A staggered approach for the coupling of Cahn-Hilliard type diffusion and finite strain elasticity, Comput. Mech., 57 (2016), 339-351.  doi: 10.1007/s00466-015-1235-1.

[5]

A. Bedford, Hamilton's Principle in Continuum Mechanics, Pitman, Boston, 1985. doi: 10.13140/2.1.1603.4887.

[6]

E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65.  doi: 10.1016/S0167-2789(02)00373-1.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[8]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807.

[9]

H. Dal and C. Miehe, Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains, Comput. Mech., 55 (2015), 303-325.  doi: 10.1007/s00466-014-1102-5.

[10]

C. Di Leo, E. Rejovitzky and L. Anand, A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials, J. Mech. Phys. Solids, 70 (2014), 1-29.  doi: 10.1016/j.jmps.2014.05.001.

[11]

F. P. Duda, A. C. Souza and E. Fried, A theory for species migration in a finitely strained solid with application to polymer network swelling, J. Mech. Phys. Solids, 58 (2010), 515-529.  doi: 10.1016/j.jmps.2010.01.009.

[12]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Arch. Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.

[13]

H. Garcke, On Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.  doi: 10.1017/S0308210500002419.

[14]

S. Govindjee and J. C. Simo, Coupled stress-diffusion: Case II, J. Mech. Phys. Solids, 41 (1993), 863-887.  doi: 10.1016/0022-5096(93)90003-X.

[15]

T. J. Healey and S. Krömer, Injective weak solutions in second-gradient nonlinear elasticity, ESAIM: Control Optim. Calc. Var., 15 (2009), 863-871.  doi: 10.1051/cocv:2008050.

[16]

C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes, Springer Spektrum, Wiesbaden, 2014. doi: 10.1007/978-3-658-05252-2.

[17]

C. Hesch, A. J. Gil, R. Ortigosa, M. Dittmann, C. Bilgen and et al., A framework for polyconvex large strain phase-field methods to fracture, Comput. Methods Appl. Mech. Engrg., 317 (2017), 649-683.  doi: 10.1016/j.cma.2016.12.035.

[18]

W. Hong and X. Wang, A phase-field model for systems with coupled large deformation and mass transport, J. Mech. Phys. Solids, 61 (2013), 1281-1294.  doi: 10.1016/j.jmps.2013.03.001.

[19]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fuides si lón tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. Sci. Exactes Nat., 6 (1901), 1-24. 

[20]

S. Krömer and T. Roubíček, Quasistatic viscoelasticity with self-contact at large strains, preprint, arXiv: 1904.02423, 2019.

[21]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. doi: 10.1007/978-3-030-02065-1.

[22]

F. C. Larché and J. W. Cahn, The effect of self–stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.  doi: 10.1016/0001-6160(82)90023-2.

[23]

V. I. Levitas, Phase field approach to martensitic phase transformations with large strains and interface stresses, J. Mech. Phys. Solids, 70 (2014), 154-189.  doi: 10.1016/j.jmps.2014.05.013.

[24]

C. Miehe, S. Mauthe and H. Ulmer, Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn-Hilliard-type and standard diffusion in elastic solids, Internat. J. Numer. Meth. Engrg., 99 (2014), 737-762.  doi: 10.1002/nme.4700.

[25]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.

[26]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[27]

A. Mielke and T. Roubíček, Rate-Independent Systems – Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.

[28]

A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, preprint, arXiv: 1903.11094, 2019. To appear: Arch. Ration. Mech. Anal..

[29]

R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Internat. J. Solids Structures, 1 (1965), 417-438.  doi: 10.1016/0020-7683(65)90006-5.

[30]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), 11pp. doi: 10.1007/s00526-017-1212-y.

[31]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191.  doi: 10.3934/dcds.2006.15.1169.

[32]

I. Pawłow and W. M. Zajaczkowski, Weak solutions to 3-D Cahn-Hilliard system in elastic solids, Topol. Methods Nonlinear Anal., 32 (2008), 347-377. 

[33]

T. Roubíček, Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids, ZAMM Z. Angew. Math. Mech., 97 (2017), 990-1002.  doi: 10.1002/zamm.201600269.

[34]

T. Roubíček and U. Stefanelli, Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modeling, SIAM J. Appl. Math., 78 (2018), 2597-2625.  doi: 10.1137/17M1137656.

[35]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Z. Angew. Math. Phys., 69 (2018), Art. no. 55, 34pp. doi: 10.1007/s00033-018-0932-y.

[36]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.

[37]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.

[38]

T. Waffenschmidt, C. Polindara, A. Menzel and S. Blanco, A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials, Comput. Methods Appl. Mech. Engrg., 268 (2014), 801-842.  doi: 10.1016/j.cma.2013.10.013.

[39]

V. V. Yashin and A. C. Balazs, Theoretical and computational modeling of self-oscillating polymer gels, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2672951.

[40]

V. V. Yashin, S. Suzuki, R. Yoshida and A. C. Balazs, Controlling the dynamic behavior of heterogeneous self-oscillating gels, J. Mater. Chem., 22 (2012). doi: 10.1039/c2jm32065g.

show all references

References:
[1]

E. K. Agiasofitou and M. Lazar, Conservation and balance laws in linear elasticity of grade three, J. Elasticity, 94 (2009), 69-85.  doi: 10.1007/s10659-008-9185-x.

[2]

S. M. Allen and J. W. Cahn, Ground state structures in ordered binary alloys with second neighbor interactions, Acta Metall., 20 (1972), 423-433.  doi: 10.1016/0001-6160(72)90037-5.

[3]

L. Anand, A Cahn-Hilliard-type theory for species diffusion coupled with large elastic-plastic deformations, J. Mech. Phys. Solids, 60 (2012), 1983-2002.  doi: 10.1016/j.jmps.2012.08.001.

[4]

P. Areias, E. Samaniego and T. Rabczuk, A staggered approach for the coupling of Cahn-Hilliard type diffusion and finite strain elasticity, Comput. Mech., 57 (2016), 339-351.  doi: 10.1007/s00466-015-1235-1.

[5]

A. Bedford, Hamilton's Principle in Continuum Mechanics, Pitman, Boston, 1985. doi: 10.13140/2.1.1603.4887.

[6]

E. Bonetti, P. Colli, W. Dreyer, G. Gilardi, G. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65.  doi: 10.1016/S0167-2789(02)00373-1.

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267. 

[8]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807.

[9]

H. Dal and C. Miehe, Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains, Comput. Mech., 55 (2015), 303-325.  doi: 10.1007/s00466-014-1102-5.

[10]

C. Di Leo, E. Rejovitzky and L. Anand, A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials, J. Mech. Phys. Solids, 70 (2014), 1-29.  doi: 10.1016/j.jmps.2014.05.001.

[11]

F. P. Duda, A. C. Souza and E. Fried, A theory for species migration in a finitely strained solid with application to polymer network swelling, J. Mech. Phys. Solids, 58 (2010), 515-529.  doi: 10.1016/j.jmps.2010.01.009.

[12]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Arch. Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.

[13]

H. Garcke, On Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.  doi: 10.1017/S0308210500002419.

[14]

S. Govindjee and J. C. Simo, Coupled stress-diffusion: Case II, J. Mech. Phys. Solids, 41 (1993), 863-887.  doi: 10.1016/0022-5096(93)90003-X.

[15]

T. J. Healey and S. Krömer, Injective weak solutions in second-gradient nonlinear elasticity, ESAIM: Control Optim. Calc. Var., 15 (2009), 863-871.  doi: 10.1051/cocv:2008050.

[16]

C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes, Springer Spektrum, Wiesbaden, 2014. doi: 10.1007/978-3-658-05252-2.

[17]

C. Hesch, A. J. Gil, R. Ortigosa, M. Dittmann, C. Bilgen and et al., A framework for polyconvex large strain phase-field methods to fracture, Comput. Methods Appl. Mech. Engrg., 317 (2017), 649-683.  doi: 10.1016/j.cma.2016.12.035.

[18]

W. Hong and X. Wang, A phase-field model for systems with coupled large deformation and mass transport, J. Mech. Phys. Solids, 61 (2013), 1281-1294.  doi: 10.1016/j.jmps.2013.03.001.

[19]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fuides si lón tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. Sci. Exactes Nat., 6 (1901), 1-24. 

[20]

S. Krömer and T. Roubíček, Quasistatic viscoelasticity with self-contact at large strains, preprint, arXiv: 1904.02423, 2019.

[21]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. doi: 10.1007/978-3-030-02065-1.

[22]

F. C. Larché and J. W. Cahn, The effect of self–stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.  doi: 10.1016/0001-6160(82)90023-2.

[23]

V. I. Levitas, Phase field approach to martensitic phase transformations with large strains and interface stresses, J. Mech. Phys. Solids, 70 (2014), 154-189.  doi: 10.1016/j.jmps.2014.05.013.

[24]

C. Miehe, S. Mauthe and H. Ulmer, Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn-Hilliard-type and standard diffusion in elastic solids, Internat. J. Numer. Meth. Engrg., 99 (2014), 737-762.  doi: 10.1002/nme.4700.

[25]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.

[26]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.

[27]

A. Mielke and T. Roubíček, Rate-Independent Systems – Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.

[28]

A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, preprint, arXiv: 1903.11094, 2019. To appear: Arch. Ration. Mech. Anal..

[29]

R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Internat. J. Solids Structures, 1 (1965), 417-438.  doi: 10.1016/0020-7683(65)90006-5.

[30]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), 11pp. doi: 10.1007/s00526-017-1212-y.

[31]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191.  doi: 10.3934/dcds.2006.15.1169.

[32]

I. Pawłow and W. M. Zajaczkowski, Weak solutions to 3-D Cahn-Hilliard system in elastic solids, Topol. Methods Nonlinear Anal., 32 (2008), 347-377. 

[33]

T. Roubíček, Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids, ZAMM Z. Angew. Math. Mech., 97 (2017), 990-1002.  doi: 10.1002/zamm.201600269.

[34]

T. Roubíček and U. Stefanelli, Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modeling, SIAM J. Appl. Math., 78 (2018), 2597-2625.  doi: 10.1137/17M1137656.

[35]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Z. Angew. Math. Phys., 69 (2018), Art. no. 55, 34pp. doi: 10.1007/s00033-018-0932-y.

[36]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.

[37]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.

[38]

T. Waffenschmidt, C. Polindara, A. Menzel and S. Blanco, A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials, Comput. Methods Appl. Mech. Engrg., 268 (2014), 801-842.  doi: 10.1016/j.cma.2013.10.013.

[39]

V. V. Yashin and A. C. Balazs, Theoretical and computational modeling of self-oscillating polymer gels, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2672951.

[40]

V. V. Yashin, S. Suzuki, R. Yoshida and A. C. Balazs, Controlling the dynamic behavior of heterogeneous self-oscillating gels, J. Mater. Chem., 22 (2012). doi: 10.1039/c2jm32065g.

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