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

Dynamics of charged elastic bodies under diffusion at large strains

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  April 2020 Early access  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.

Citation: Tomáš Roubíček, Giuseppe Tomassetti. Dynamics of charged elastic bodies under diffusion at large strains. Discrete and 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.

[2]

J. BenzigerE. ChiaJ. 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.

[3]

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

[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.

[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.

[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.

[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.

[8]

F. P. DudaA. 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.

[9]

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

[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. 
[11]

M. FossW. 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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. W. MajsztrikA. 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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

G. ZurloM. 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.

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.

[2]

J. BenzigerE. ChiaJ. 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.

[3]

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

[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.

[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.

[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.

[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.

[8]

F. P. DudaA. 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.

[9]

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

[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. 
[11]

M. FossW. 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.

[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.

[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.

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[19]

P. W. MajsztrikA. 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.

[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.

[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.

[22]

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

[23]

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

[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.

[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.

[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.

[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.

[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.

[29]

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

[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.

[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.

[32]

G. ZurloM. 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.

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