Dynamics of charged elastic bodies under diffusion at large strains

Giuseppe Tomassetti; Tomas Roubicek

REVIEW 3 major objections 4 minor 32 references

Dynamics of charged elastic bodies under diffusion at large strains

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves existence of weak solutions for the dynamics of charged elastic and poroelastic bodies at large strains, allowing global self-penetration.

desk verdict Solid extension of the authors' program to dynamic long-range self-interactions; Proposition 2 has a missing term in a key identity that needs fixing, but the paper deserves referee time. read the letter →

arxiv 1908.01811 v1 pith:FGKVBW4W submitted 2019-08-05 math.AP cond-mat.mtrl-scimath-phmath.MP

Tomas Roubicek , Giuseppe Tomassetti This is my paper · ORCID

classification math.APcond-mat.mtrl-scimath-phmath.MP MSC 35Q6035Q7465M6074A3074F1576S9978A30

keywords elastodynamicsporoelasticitynonsimplematerialslong-rangeinteractionselectrostaticself-interactionsweaksolutionsGalerkinapproximationlargestrains

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves existence of weak solutions for a dynamical model of an elastic body carrying electric charge and undergoing arbitrarily large strains. The central difficulty is that the electrostatic potential lives on the current, deformed configuration while the mechanics is formulated on the reference configuration, and the charge density involves the inverse deformation, which need not be globally injective in dynamics. The authors resolve this by admitting only local non-self-penetration, using Federer's change-of-variables formula to define the charge density through the preimage of the deformation, and adding a nonlocal second-gradient energy whose singular kernel forces deformations into a fractional Sobolev space. A theorem of Healey and Kroemer then gives a uniform positive lower bound on the determinant of the deformation gradient, which makes the change of variables legitimate and controls the pullback of the electric force. The main results, Propositions 1 and 2, show that Galerkin approximations exist on the whole time interval and their weak* limits solve the coupled system, including the diffusion of the charge in the poroelastic extension.

What carries the argument

The central machinery is the nonlocal nonsimple energy (1)-(2) with a singular kernel satisfying (30), combined with the Healey-Kroemer determinant bound in the form (40): any deformation with bounded energy has det∇χ uniformly bounded below by a positive constant. This gives Lipschitz regularity and local invertibility, so Federer's change-of-variables formula (7) defines the Eulerian charge density (6) as a sum over the preimage points, and the force term q∇φ∘χ can be rewritten via (33) in a form that passes to the limit under only weak convergence of ∇²χ. The p-regularized Poisson equation (9) with p > d ensures the potential converges in the space of continuous functions on compact sets, which is needed for the composition φ∘χ.

What would settle it

A numerical or analytical example satisfying (29a)-(c), (30), and (31) in which the Galerkin solutions develop a point or time where det∇χ_k(t,x) tends to 0 while the kinetic plus stored energy remains bounded would refute the determinant bound (42) on which Proposition 1 rests; equivalently, exhibiting a sequence χ_k with bounded Φ(χ_k) but no uniform positive lower bound on det∇χ_k would falsify the Healey-Kroemer form (40) used here.

Watch

Extended reading notes

Core claim

Proposition 1 establishes that the dynamical system (21)-(23), coupling elastodynamics with a p-regularized Poisson equation for the electrostatic potential, admits a weak solution in the sense of Definition 3.1. Proposition 2 extends this to the poroelastic case where the charge density q is an evolving diffusant, governed by the electrochemical potential inclusion (63a) and a Biot-type diffusion equation (63b,c), with the quadruple (χ, φ, q, μ) satisfying (34a), (34b), (68), and (69). The argument is constructive: nested Galerkin subspaces produce approximate solutions on the full time interval, uniform estimates (41a)-(41b) and (70) yield weakly* converging subsequences, and the determinant lower bound (42) supplies the strong convergence needed to pass to the limit in the terms containing ∇$χ^{{-T}}$ and φ∘χ.

Load-bearing premise

Everything rests on the assertion that any deformation with bounded energy keeps the determinant of its deformation gradient uniformly positive; if a bounded-energy motion could drive the determinant to zero, the charge-density formula and the entire weak formulation would lose meaning.

Editorial extensions

If this is right

For the purely elastic model of Section 2, a weak solution exists on any time interval for the initial-boundary-value problem, so the model can serve as a rigorous basis for engineering computations that ignore global self-penetration.
Including the charged diffusant gives weak solutions of the coupled elastodynamics-diffusion system, so Darcy, Fick, and drift responses such as (74)-(75) are obtained within a large-strain existence theory.
The Galerkin construction is explicit and the determinant bound rules out the Lavrentiev-type singularity on the approximation level, so the singular energy barrier at detF→0+ needs no additional regularization.
For the diffusion extension, the energy balance (66) holds, combining the conservative mechanical-electrostatic balance with the dissipative diffusion rate, which is exactly the structure the a priori estimates exploit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the determinant bound (40) holds beyond the assumptions made here, the same change-of-variables device could handle attractive monopolar interactions such as gravitation, but the paper explicitly leaves ε1 = 0 out, so Proposition 1 does not cover self-gravitating bodies.
The p-regularization of the Poisson equation is a technical device; the authors note as an open problem whether the standard electrostatic setting can be recovered, and a testable extension would be to attempt the same limit passage with p = 2 under stronger assumed regularity of the deformation.
The model's allowance of global self-penetration while retaining local injectivity suggests that charge conservation for overlapping deformation images is encoded in the multiplicity sum (6); one could test this by computing the electrostatic energy of a deformation that folds a charged body onto itself and comparing it with the unfolded configuration.
The multi-component version in Remark 3 merges fixed dopant charges with mobile diffusant charges, giving a direct route to polymer-electrolyte fuel cell and semiconductor device models, though the proof would need adaptation to the cross-diffusion mobility tensor.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

This is a real working paper, not a gesture. What's new: it extends Roubíček and Tomassetti's previous work to dynamical problems with monopolar long-range self-interactions, where global invertibility of the deformation is not assumed. The trick is to use the nonlocal nonsimple regularization to get H^{2+γ} regularity, then Healey-Kroemer to keep det∇χ uniformly positive, and Federer's area formula to make sense of the pullback charge density under non-injective Lipschitz deformations. That combination is legitimate and the modeling discussion is honest about the ideal-dielectric assumption and p-regularization being limitations. I read the paper as doing what it claims: existence of weak solutions for the p-regularized system, with a serious Galerkin construction.

Where I'd push back. Proposition 1's proof is terse in places. The strong almost-everywhere convergence φ_k(t)→φ(t) in C(B) at (57) is asserted without a full argument; you only have φ_k bounded in L∞(I;W^{1,p}(R^d)) with p>d, and the equation gives some uniform continuity in time only in weaker norms. It's probably repairable by using the equation and the strong convergence of χ_k, but as printed it's a gap in the presentation.

Proposition 2 has a more concrete problem. Identity (72) is wrong: differentiating (63a) gives an extra ∇χ^T(∇φ∘χ) term when solving for ∇q. This is the step that yields the L²(I;H¹(Ω)) bound for q_k and the strong W^{1,1} convergence of q_k(t), both needed for the limit passage in the diffusive case. The missing term looks controllable — ∇φ_k is bounded in L∞(I;L^p) with p>d and ∇χ_k in L∞(I;L∞), so the product lands in L²(Q), and strong convergence should follow from the available convergences — but the paper doesn't give that argument. So as written, Proposition 2's proof has a load-bearing gap, not just a typo.

The negative claims are fine: no fitted parameters, no circularity, and the self-citations are to the authors' own earlier work that establishes the framework. The p-regularization is presented as a modeling compromise, which it is.

Bottom line: Proposition 1 is likely sound after some filling-in; Proposition 2 needs a corrected identity and the extra estimate. This is a subfield-level result but it's the kind of thing people doing numerical large-strain electromechanics will want to cite. I'd send it to a serious referee and ask them to check the diffusive section carefully, but I wouldn't desk-reject.

Referee Report

3 major / 4 minor

Summary. The paper proposes a continuum model for the dynamics of charged elastic (and poroelastic) bodies at large strains, allowing for possible global self-interpenetration while retaining local injectivity. The key modeling ingredients are a nonlocal nonsimple free energy with singular kernel, a p-regularized nonlinear Poisson equation for the electrostatic potential, and Federer's area formula to define the Eulerian charge density through a set-valued inverse deformation. The analytical core consists of two Galerkin-based existence theorems: Proposition 1 establishes weak solutions to the elastodynamics system with fixed charge, and Proposition 2 extends this to the case of a diffusing charge driven by an electrochemical potential. The proofs rely crucially on the Healey-Kroemer determinant lower bound, on strong compactness of the deformation in C^1 obtained from the nonlocal second-gradient regularization, and on strong convergence of the scalar potential in C(B) for a.e. time.

Significance. If the technical gaps are repaired, the paper is a valuable contribution to the mathematical theory of electro-elastodynamics at large strains. Its use of Federer's change-of-variable formula to handle non-injective deformations is a genuine novelty with respect to earlier work, and the constructive Galerkin approach with explicit a priori estimates is appropriate for the hyperbolic-elliptic structure. The paper honestly states its limitations (monopolar interactions only, ideal-dielectric assumption, p-regularization instead of standard electrostatics), and it does not fit parameters or assume the desired conclusion. The main theorems are plausible and the overall framework is likely to be useful for further work on coupled electro-chemo-mechanics, but the proofs contain load-bearing gaps that need to be addressed.

major comments (3)

[§4, Eq. (72)] Equation (72) is not the correct derivative of (63a). Since µ_k ∈ ∂_qϕ(∇χ_k,q_k)+φ_k(χ_k), spatial differentiation gives ∇µ_k = ∂²_Fqϕ(∇χ_k,q_k)∇²χ_k + ∂²_qqϕ(∇χ_k,q_k)∇q_k + ∇χ_k^T(∇φ_k∘χ_k). The last term is omitted in (72). This is load-bearing: (72) is the only step that yields the L²(I;H¹(Ω)) estimate for q_k in (70a), and the subsequent strong convergence ∇q_k(t)→∇q(t) in L¹ is used in the passage to the limit after (72). The missing term is likely controllable, since ∇φ_k is bounded in L∞(I;L^p(R^d)) with p>d and ∇χ_k is bounded in L∞(I;L∞(Ω)), so the product lies in L²(Q); however this argument is not given. As written, the proof of Proposition 2 is incomplete.
[§3, proof of Proposition 1, around (57)] The assertion that φ_k(t)→φ(t) strongly in C(B) for a.a. t∈(0,T) and every closed ball B is not justified in the text. The preceding compactness statements give only (51b) and the pointwise strong convergence (56) for χ_k(t). To obtain (57), one must show that a single subsequence works for all t outside a null set, using the continuity of the solution map of the monotone equation (58) as the right-hand side converges in W^{1,p}(R^d)^*, together with uniqueness of φ(t). This is a standard argument, but it should be written out explicitly because the limit passage in the composed terms (61a)–(61c) depends on it.
[§3, Eq. (42)] The successive-continuation argument leading to the uniform determinant lower bound det∇χ_k ≥ ε on the whole time interval is only sketched. The energy identity (43) and the Healey-Kroemer implication (40) are invoked before the global a priori estimate is established, so the proof should explain how the maximal time of existence is shown to be T: on any interval where the local Galerkin solution satisfies the energy bound, (40) prevents the determinant from approaching zero and hence allows continuation. Without this bootstrapping detail, the uniform bound (42) that underpins the change-of-variable formula and the control of ∇χ_k^{-⊤} is not fully justified.

minor comments (4)

[§3, before Eq. (50)] In the displayed inequality following (48), the right-hand side is missing a dt: the term ∫_0^τ ‖χ̇_k(t)‖_{L²} should read ∫_0^τ ‖χ̇_k(t)‖_{L²} dt, and the displayed estimate should also account for the term ‖χ_k(t)‖²_{H^{2+γ}} dt that appears in the following line.
[§3, Eq. (41a)] The notation ‖‖‖ 1/det(∇χ_k) ‖‖‖ contains a typographical artifact; it should presumably be the standard L∞(Q) norm.
[Definition 4.1, Eq. (68)] The test function for the scalar mass-balance equation is written w∈H¹(Q;R^d), but w is scalar (it multiplies the scalar quantities M∇µ and q); it should likely be H¹(Q) or W¹,¹(Q) with w(T)=0.
[Throughout] There are several typographical and stylistic slips, such as "Lagrangean" instead of "Lagrangian", "an result", and inconsistent uses of upright and italic fonts for Eulerian and Lagrangian quantities. These should be corrected in the revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the existence proofs rest on external Healey–Kroemer and Federer results, and the paper's self-citations are motivational or corroborative rather than reductions.

full rationale

The claimed derivations are Galerkin existence proofs for (21) and for (21)+(63). The inputs are the constitutive assumptions (29), the kernel bound (30), the p-regularized Poisson problem (9), the external determinant lower bound (39)–(40) from Healey and Kroemer [14], and Federer's change-of-variables formula (7), used to rewrite the Eulerian charge density (6). None of these inputs contains the target weak-solution existence statement, and no parameter is fitted to data and then renamed as a prediction. The self-citations ([27], [18], [24], [25], [26]) are not the load-bearing argument: [27] is cited for motivation and for the earlier model without long-range interactions, while [18, Sect. 2.5] is a pointer for the recorded modification of the external Healey–Kroemer theorem, not a substitute for it. The proof gap at (72), where differentiating (63a) also produces the term ∇χ^T(∇φ∘χ), is a correctness issue in the L^2(I;H^1(Ω)) estimate for q_k; it is not a circular reduction, since the missing term is not the conclusion of Proposition 2 and the estimate is not obtained by assuming the theorem. The model's p-regularization is an explicit ad hoc modeling choice, openly described as an open problem to remove, so it is an assumption rather than a hidden identification of input and output.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

No fitted material parameters appear. The parameters listed above are analytical and regularization parameters chosen by hand, not calibrated to data. No new physical entities are introduced.

free parameters (3)

p = p>d
Exponent in the p-Laplacian term ε1|∇φ|^{p-2}∇φ; chosen ad hoc to obtain W^{1,p}(R^d) potentials and compact embedding to C(B). Standard electrostatics p=2 is not covered in d≥2.
γ = γ>d/2−1
Fractional smoothness exponent in the nonlocal kernel (30); chosen to place H^{2+γ}(Ω) compactly in C^1(Ω) and to satisfy the Healey-Kroemer threshold.
ε1 = unspecified positive constant
Coefficient of the |∇φ|^p term in (11); without it the electrostatic energy lacks the coercivity used in the proof. It is not determined by data.

assumptions (7)

standard math Healey-Kroemer injectivity criterion (Theorem 3.1 of [14]) with energy level set Φ≤K implies det∇χ≥ε on Ω
External result, invoked at (39)-(40) to obtain uniform positive determinant bounds for Galerkin solutions.
standard math Federer change-of-variables and area formula, Eq. (7)-(8)
Used to define the spatial charge density q under non-injective Lipschitz deformations in Eq. (6).
domain assumption Potential energy blow-up (29c): ϕ(F)→+∞ sufficiently fast as detF→0+
Constitutive modeling assumption that guarantees det∇χ stays away from zero via (40).
domain assumption Nonlocal nonsimple elastic energy (1)-(2) with singular kernel (30) giving H^{2+γ} regularity
Regularization that yields Lipschitz deformations and excludes the Lavrentiev phenomenon; central to the proof.
domain assumption Ideal-dielectric split of the free energy into mechanical and electrostatic parts
Limits electromechanical coupling; stated in Section 2 and discussed in Section 5.
ad hoc to paper p-regularized Poisson equation (9)-(11) with p>d replacing standard electrostatics
Introduced for technical convergence; the authors explicitly leave the standard setting p=2 open.
domain assumption Regularity and loading assumptions: q∈W^{1,1}(Ω), qext∈L^1(R^d) time-independent, data (31), and in Prop. 2 uniform convexity and smoothness of ϕ(F,·)
Needed for the Galerkin construction and limit passages; stated in Propositions 1 and 2.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Dynamics of charged elastic bodies under diffusion at large strains." pith.science (2026). https://pith.science/paper/FGKVBW4W

@misc{pith2026190801811,
  author       = {Pith},
  title        = {Pith review of: Dynamics of charged elastic bodies under diffusion at large strains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FGKVBW4W}},
  note         = {Machine review of arXiv:1908.01811}
}

read the original abstract

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 combine 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 the positivity of the determinant of the deformation gradient thanks to a result by Healey and Kroemer.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

32 extracted references · 32 canonical work pages

[1]

J.M. Ball. Some open problems in elasticity. In: Geometry, mechanics, and dynam- ics. Springer, New York, NY, p. 3-59, 2002

work page 2002
[2]

Benziger, E

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

work page 2005
[3]

M.A. Biot. General theory of three-dimensional consolidation. J. Appl. Phys. , 12 (1941), 155–164

work page 1941
[4]

Biot and G

M.A. Biot and G. Temple. Theory of ﬁnite deformations of porous solids. Indiana Univ. Math. J. , 21 (1972), 597–620. 23

work page 1972
[5]

Ciarlet and J

P.G. Ciarlet and J. Neˇ cas. Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal., 97 (1987), 171–188

work page 1987
[6]

DeSimone and P

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

work page 1997
[7]

Dorfmann and R.W

L. Dorfmann and R.W. Ogden, Nonlinear theory of electroelastic and magnetoelastic interactions. Springer, 2014

work page 2014
[8]

Duda, A.C

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

work page 2010

Show all 32 references

[9]

A.C. Eringen. Nonlocal continuum ﬁeld theories. Springer, New York, 2002

2002
[10]

Evans and R.F

L.C. Evans and R.F. Gariepy. Measure theory and ﬁne properties of functions, Revised Edition. CRC Press, 2015

2015
[11]

Foss, W.J

M. Foss, W.J. Hrusa, and V.J. Mizel. The Lavrentiev gap phenomenon in nonlinear elasticity. Arch. Rat. Mech. Anal., 167 (2003), 337–365

2003
[12]

Govindjee and J.C

S. Govindjee and J.C. Simo: Coupled stress-diﬀusion: case II. J. Mech. Phys. Solids, 41 (1993), 863–887

1993
[13]

M.E. Gurtin. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D, 92 (1996), 178–192

1996
[14]

Healey and S

T.J. Healey and S. Kr¨ omer. Injective weak solutions in second-gradient nonlinear elasticity. ESAIM: Control, Optim. & Cal. Var. , 15 (2009), 863–871

2009
[15]

H. Federer. Geometric Measure Theory, Springer, Berlin, 1969

1969
[16]

Jir´ asek

M. Jir´ asek. Nonlocal theories in continuum mechanics. Acta Polytechnica, 44 (2004),16–34

2004
[17]

Kruˇ z´ ık, U

M. Kruˇ z´ ık, U. Stefanelli, and J. Zeman. Existence results for incompressible mag- netoelasticity. Disc. Cont. Dynam. Systems , 35 (2015), 2615–2623

2015
[18]

Kruˇ z´ ık and T

M. Kruˇ z´ ık and T. Roub´ ıˇ cek.Mathematical Methods in Contiuum Mechanics of Solids. Springer, Cham/Heidelberg, 2019, in print

2019
[19]

Majsztrik, A.B

P.W. Majsztrik, A.B. Bocarsly, and Jay B. Benziger. Viscoelastic response of Naﬁon. Eﬀects of temperature and hydration on tensile creep. Macromolecules 41 (2008), 9849–9862

2008
[20]

Marcus and V.J

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), 790–796

1973
[21]

Palmer and T.J

A.Z. Palmer and T.J. Healey. Injectivity and self-contact in second-gradient nonlin- ear elasticity. Calc. Var., P.D.E. 56 (2017), 114. 24

2017
[22]

Promislow and B

K. Promislow and B. Wetton. PEM fuel cells: a mathematical overview. newblock SIAM J. Appl. Math. , 70 (2009), 369–409

2009
[23]

R.C. Rogers. Nonlocal variational problems in nonlinear electromagneto- elastotatics. SIAM J. Math. Analysis , 19 (1988), 1329–1347

1988
[24]

Roub´ ıˇ cek

T. Roub´ ıˇ cek. 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

2017
[25]

Roub´ ıˇ cek

T. Roub´ ıˇ cek. Variational methods for steady-state Darcy/Fick ﬂow in swollen and poroelastic solids. Zeit. angew. Math. Mech. , 97 (2017), 990–1002

2017
[26]

Roub´ ıˇ cek and G

T. Roub´ ıˇ cek and G. Tomassetti. Phase transformations in electrically conductive ferromagnetic shape-memory alloys, their Thermodynamics and analysis. Arch. Rat. Mech. Analysis, 210 (2013), 1–43

2013
[27]

Roub´ ıˇ cek and G

T. Roub´ ıˇ cek 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), 55

2018
[28]

ˇSilhav´ y

M. ˇSilhav´ y. A variational approach to nonlinear electro-magneto-elasticity: Convex- ity conditions and existence theorems. Math. Mech. Solids , 23 (2018), 907–928

2018
[29]

R.A. Toupin. The elastic dielectric. J. Rat. Mech. Analysis , 5 (1956), 849–915

1956
[30]

R.A. Toupin. A dynamical theory of elastic dielectrics. Int. J. Eng Sci. , 1 (1963), 101–126

1963
[31]

Zhao and Z

X. Zhao and Z. Suo. Method to analyze electromechanical stability of dielectric elastomers. Appl. Phys. Lett. , 91 (2007), 061921

2007
[32]

Zurlo, M

G. Zurlo, M. Destrade, and T. Lu. Fine tuning the electro-mechanical response of dielectric elastomers. Appl. Phys. Lett. , 113 (2018), 162902. 25

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

J.M. Ball. Some open problems in elasticity. In: Geometry, mechanics, and dynam- ics. Springer, New York, NY, p. 3-59, 2002

work page 2002

[2] [2]

Benziger, E

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

work page 2005

[3] [3]

M.A. Biot. General theory of three-dimensional consolidation. J. Appl. Phys. , 12 (1941), 155–164

work page 1941

[4] [4]

Biot and G

M.A. Biot and G. Temple. Theory of ﬁnite deformations of porous solids. Indiana Univ. Math. J. , 21 (1972), 597–620. 23

work page 1972

[5] [5]

Ciarlet and J

P.G. Ciarlet and J. Neˇ cas. Injectivity and self-contact in nonlinear elasticity. Arch. Rational Mech. Anal., 97 (1987), 171–188

work page 1987

[6] [6]

DeSimone and P

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

work page 1997

[7] [7]

Dorfmann and R.W

L. Dorfmann and R.W. Ogden, Nonlinear theory of electroelastic and magnetoelastic interactions. Springer, 2014

work page 2014

[8] [8]

Duda, A.C

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

work page 2010