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April  2020, 25(4): 1415-1437. doi: 10.3934/dcdsb.2019234

Dynamics of charged elastic bodies under diffusion at large strains

Tomáš Roubíček 1,2,, and  Giuseppe Tomassetti 3,

1. 

Mathematical Institute, Charles University, Sokolovská 83, 18675 Praha 8, Czech Republic
2. 

Institute of Thermomechanics, Czech Acad. Sci., Dolejškova 5, 18200 Praha 8, Czech Republic
3. 

Università degli Studi Roma Tre, Dipartimento di Ingegneria, Via Vito Volterra 62, 00146 Roma, Italy

* Corresponding author: Tomáš Roubíček

Received  August 2018 Revised  July 2019 Published  November 2019

Fund Project: The authors are thankful to an anonymous referee for many comments to the model and to Dr. Giuseppe Zurlo for a discussion on the concept and applicability of the ideal dielectric model. This research was partly supported through the grants 17-04301S (as far as dissipative evolution concerns) and 19-04956S (as far as dynamic and nonlinear behaviour concerns) of the Czech Science Foundation, through the institutional project RVO: 61388998 (ČR), and the Grant of Excellence Departments, MIUR-Italy (Art.1, commi 314-337, Legge 232/2016), as well as through INdAMGNFM.

We present a model for the dynamics of elastic or poroelastic bodies with monopolar repulsive long-range (electrostatic) interactions at large strains. Our model respects (only) locally the non-self-interpenetration condition but can cope with possible global self-interpenetration, yielding thus a certain justification of most of engineering calculations which ignore these effects in the analysis of elastic structures. These models necessarily combines Lagrangian (material) description with Eulerian (actual) evolving configuration evolving in time. Dynamical problems are studied by adopting the concept of nonlocal nonsimple materials, applying the change of variables formula for Lipschitz-continuous mappings, and relying on a positivity of determinant of deformation gradient thanks to a result by Healey and Krömer.

Keywords: Elastodynamics, poroelasticity, nonsimple materials, long-range interactions.
Mathematics Subject Classification: 35Q60, 35Q74, 65M60, 74A30, 74F15, 76S99, 78A30.
Citation: Tomáš Roubíček, Giuseppe Tomassetti. Dynamics of charged elastic bodies under diffusion at large strains. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1415-1437. doi: 10.3934/dcdsb.2019234
References:
[1]

J. M. Ball, Some open problems in elasticity., In: P. Newton, P. Holmes, A. Weinstein (eds): Geometry, Mechanics, and Dynamics, Springer, New York, (2002), 3–59. doi: 10.1007/0-387-21791-6_1.  Google Scholar

[2]

J. Benziger, E. Chia, J. F. Moxley and I. G. Kevrekidis, The dynamic response of PEM fuel cells to changes in load, Chemical Engineering Science, 60 (2005), 1743-1759.  doi: 10.1016/j.ces.2004.10.033.  Google Scholar

[3]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.  doi: 10.1063/1.1712886.  Google Scholar

[4]

M. A. Biot, Theory of finite deformations of porous solids, Indiana Univ. Math. J., 21 (1971/72), 597-620.  doi: 10.1512/iumj.1972.21.21048.  Google Scholar

[5]

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.  Google Scholar

[6]

A. DeSimone and P. Podio-Guidugli, Pointwise balances and the construction of stress fields in dielectrics, Math. Mod. Meth. Appl. Sci., 7 (1997), 477-485.  doi: 10.1142/S0218202597000268.  Google Scholar

[7]

L. Dorfmann and R. W. Ogden, Nonlinear Theory of Electroelastic and Magnetoelastic Interactions, Springer, New York, 2014. doi: 10.1007/978-1-4614-9596-3.  Google Scholar

[8]

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.  Google Scholar

[9]

A. C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.  Google Scholar

[10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[11]

M. Foss, W. J. Hrusa and V. J. Mizel, The Lavrentiev gap phenomenon in nonlinear elasticity, Arch. Rat. Mech. Anal., 167 (2003), 337-365.  doi: 10.1007/s00205-003-0249-6.  Google Scholar

[12]

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.  Google Scholar

[13]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[14]

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

[15]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[16]

M. Jirásek, Nonlocal theories in continuum mechanics, Acta Polytechnica, 44 (2004), 16-34.   Google Scholar

[17]

M. Kružík, U. Stefanelli and J. Zeman, Existence results for incompressible magnetoelasticity, Disc. Cont. Dynam. Systems, 35 (2015), 2615-2623.  doi: 10.3934/dcds.2015.35.2615.  Google Scholar

[18]

M. Kružík and T. Roubíček, Mathematical Methods in Contiuum Mechanics of Solids, Springer, Switzerland, 2019. Google Scholar

[19]

P. W. Majsztrik, A. B. Bocarsly and J. B. Benziger, Viscoelastic response of Nafion. Effects of temperature and hydration on tensile creep, Macromolecules, 41 (2008), 9849-9862.  doi: 10.1021/ma801811m.  Google Scholar

[20]

M. Marcus and V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc., 79 (1973), 790-795.  doi: 10.1090/S0002-9904-1973-13319-1.  Google Scholar

[21]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), Art114, 11 pp. doi: 10.1007/s00526-017-1212-y.  Google Scholar

[22]

K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802.  Google Scholar

[23]

R. C. Rogers, Nonlocal variational problems in nonlinear electromagneto-elastotatics, SIAM J. Math. Analysis, 19 (1988), 1329-1347.  doi: 10.1137/0519097.  Google Scholar

[24]

T. Roubíček, An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat, Disc. Cont. Dynam. Syst. S, 10 (2017), 867-893.  doi: 10.3934/dcdss.2017044.  Google Scholar

[25]

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

[26]

T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their Thermodynamics and analysis, Arch. Rat. Mech. Analysis, 210 (2013), 1-43.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[27]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Zeit. Angew. Mat. Phys., 69 (2018), Art 55, 34 pp. doi: 10.1007/s00033-018-0932-y.  Google Scholar

[28]

M. Šilhavý, A variational approach to nonlinear electro-magneto-elasticity: Convexity conditions and existence theorems, Math. Mech. Solids, 23 (2018), 907-928.  doi: 10.1177/1081286517696536.  Google Scholar

[29]

R. A. Toupin., The elastic dielectric., J. Rat. Mech. Analysis, 5 (1956), 849-915.  doi: 10.1512/iumj.1956.5.55033.  Google Scholar

[30]

R. A. Toupin, A dynamical theory of elastic dielectrics, Int. J. Eng Sci., 1 (1963), 101-126.  doi: 10.1016/0020-7225(63)90027-2.  Google Scholar

[31]

X. H. Zhao and Z. G. Suo, Method to analyze electromechanical stability of dielectric elastomers, Appl. Phys. Lett., 91 (2007), 061921.  doi: 10.1063/1.2768641.  Google Scholar

[32]

G. Zurlo, M. Destrade and T. Q. Lu, Fine tuning the electro-mechanical response of dielectric elastomers, Appl. Phys. Lett., 113 (2018), 162902.  doi: 10.1063/1.5053643.  Google Scholar

show all references

References:
[1]

J. M. Ball, Some open problems in elasticity., In: P. Newton, P. Holmes, A. Weinstein (eds): Geometry, Mechanics, and Dynamics, Springer, New York, (2002), 3–59. doi: 10.1007/0-387-21791-6_1.  Google Scholar

[2]

J. Benziger, E. Chia, J. F. Moxley and I. G. Kevrekidis, The dynamic response of PEM fuel cells to changes in load, Chemical Engineering Science, 60 (2005), 1743-1759.  doi: 10.1016/j.ces.2004.10.033.  Google Scholar

[3]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164.  doi: 10.1063/1.1712886.  Google Scholar

[4]

M. A. Biot, Theory of finite deformations of porous solids, Indiana Univ. Math. J., 21 (1971/72), 597-620.  doi: 10.1512/iumj.1972.21.21048.  Google Scholar

[5]

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.  Google Scholar

[6]

A. DeSimone and P. Podio-Guidugli, Pointwise balances and the construction of stress fields in dielectrics, Math. Mod. Meth. Appl. Sci., 7 (1997), 477-485.  doi: 10.1142/S0218202597000268.  Google Scholar

[7]

L. Dorfmann and R. W. Ogden, Nonlinear Theory of Electroelastic and Magnetoelastic Interactions, Springer, New York, 2014. doi: 10.1007/978-1-4614-9596-3.  Google Scholar

[8]

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.  Google Scholar

[9]

A. C. Eringen, Nonlocal Continuum Field Theories, Springer-Verlag, New York, 2002.  Google Scholar

[10] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Revised Edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2015.   Google Scholar
[11]

M. Foss, W. J. Hrusa and V. J. Mizel, The Lavrentiev gap phenomenon in nonlinear elasticity, Arch. Rat. Mech. Anal., 167 (2003), 337-365.  doi: 10.1007/s00205-003-0249-6.  Google Scholar

[12]

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.  Google Scholar

[13]

M. E. Gurtin, Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance, Physica D, 92 (1996), 178-192.  doi: 10.1016/0167-2789(95)00173-5.  Google Scholar

[14]

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

[15]

H. Federer, Geometric Measure Theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.  Google Scholar

[16]

M. Jirásek, Nonlocal theories in continuum mechanics, Acta Polytechnica, 44 (2004), 16-34.   Google Scholar

[17]

M. Kružík, U. Stefanelli and J. Zeman, Existence results for incompressible magnetoelasticity, Disc. Cont. Dynam. Systems, 35 (2015), 2615-2623.  doi: 10.3934/dcds.2015.35.2615.  Google Scholar

[18]

M. Kružík and T. Roubíček, Mathematical Methods in Contiuum Mechanics of Solids, Springer, Switzerland, 2019. Google Scholar

[19]

P. W. Majsztrik, A. B. Bocarsly and J. B. Benziger, Viscoelastic response of Nafion. Effects of temperature and hydration on tensile creep, Macromolecules, 41 (2008), 9849-9862.  doi: 10.1021/ma801811m.  Google Scholar

[20]

M. Marcus and V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc., 79 (1973), 790-795.  doi: 10.1090/S0002-9904-1973-13319-1.  Google Scholar

[21]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), Art114, 11 pp. doi: 10.1007/s00526-017-1212-y.  Google Scholar

[22]

K. Promislow and B. Wetton, PEM fuel cells: A mathematical overview, SIAM J. Appl. Math., 70 (2009), 369-409.  doi: 10.1137/080720802.  Google Scholar

[23]

R. C. Rogers, Nonlocal variational problems in nonlinear electromagneto-elastotatics, SIAM J. Math. Analysis, 19 (1988), 1329-1347.  doi: 10.1137/0519097.  Google Scholar

[24]

T. Roubíček, An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat, Disc. Cont. Dynam. Syst. S, 10 (2017), 867-893.  doi: 10.3934/dcdss.2017044.  Google Scholar

[25]

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

[26]

T. Roubíček and G. Tomassetti, Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their Thermodynamics and analysis, Arch. Rat. Mech. Analysis, 210 (2013), 1-43.  doi: 10.1007/s00205-013-0648-2.  Google Scholar

[27]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Zeit. Angew. Mat. Phys., 69 (2018), Art 55, 34 pp. doi: 10.1007/s00033-018-0932-y.  Google Scholar

[28]

M. Šilhavý, A variational approach to nonlinear electro-magneto-elasticity: Convexity conditions and existence theorems, Math. Mech. Solids, 23 (2018), 907-928.  doi: 10.1177/1081286517696536.  Google Scholar

[29]

R. A. Toupin., The elastic dielectric., J. Rat. Mech. Analysis, 5 (1956), 849-915.  doi: 10.1512/iumj.1956.5.55033.  Google Scholar

[30]

R. A. Toupin, A dynamical theory of elastic dielectrics, Int. J. Eng Sci., 1 (1963), 101-126.  doi: 10.1016/0020-7225(63)90027-2.  Google Scholar

[31]

X. H. Zhao and Z. G. Suo, Method to analyze electromechanical stability of dielectric elastomers, Appl. Phys. Lett., 91 (2007), 061921.  doi: 10.1063/1.2768641.  Google Scholar

[32]

G. Zurlo, M. Destrade and T. Q. Lu, Fine tuning the electro-mechanical response of dielectric elastomers, Appl. Phys. Lett., 113 (2018), 162902.  doi: 10.1063/1.5053643.  Google Scholar

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