Skip to main content

From Statistical Polymer Physics to Nonlinear Elasticity

Abstract

A polymer-chain network is a collection of interconnected polymer-chains, made themselves of the repetition of a single pattern called a monomer. Our first main result establishes that, for a class of models for polymer-chain networks, the thermodynamic limit in the canonical ensemble yields a hyperelastic model in continuum mechanics. In particular, the discrete Helmholtz free energy of the network converges to the infimum of a continuum integral functional (of an energy density depending only on the local deformation gradient) and the discrete Gibbs measure converges (in the sense of a large deviation principle) to a measure supported on minimizers of the integral functional. Our second main result establishes the small temperature limit of the obtained continuum model (provided the discrete Hamiltonian is itself independent of the temperature), and shows that it coincides with the \(\Gamma \)-limit of the discrete Hamiltonian, thus showing that thermodynamic and small temperature limits commute. We eventually apply these general results to a standard model of polymer physics from which we derive nonlinear elasticity. We moreover show that taking the \(\Gamma \)-limit of the Hamiltonian is a good approximation of the thermodynamic limit at finite temperature in the regime of large number of monomers per polymer-chain (which turns out to play the role of an effective inverse temperature in the analysis).

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    Uniqueness fails when points are not in general position, which we rule out by condition (v) of this definition.

References

  1. 1.

    Adams, S., Kotecký, R., Müller, S.: Strict convexity of the surface tension for non-convex potentials. Preprint. arXiv:1606.09541 (2016)

  2. 2.

    Akcoglu, U., Krengel, M.A.: Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323, 53–67, 1981

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Alicandro, R., Cicalese, M.: A general integral representation result for continuum limits of discrete energies with superlinear growth. SIAM J. Math. Anal. 36(1), 1–37, 2004

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alicandro, R., Cicalese, M., Gloria, A.: Integral representation results for energies defined on stochastic lattices and application to nonlinear elasticity. Arch. Ration. Mech. Anal. 200(3), 881–943, 2011

    MathSciNet  Article  Google Scholar 

  5. 5.

    Arruda, E.M., Boyce, M.C.: A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids41, 389–412, 1993

    ADS  Article  Google Scholar 

  6. 6.

    Arruda, E.M., Boyce, M.C.: Constitutive models of rubber elasticity: a review. Rubber Chem. Technol. 72, 504–523, 2000

    Google Scholar 

  7. 7.

    Bastide, J., Herz, J., Boué, F.: Loss of affineness in gels and melts. J. Phys. Fr. 46, 1967–1979, 1985

    Article  Google Scholar 

  8. 8.

    Blanc, X., Le Bris, C., Lions, P.L.: The energy of some microscopic stochastic lattices. Arch. Ration. Mech. Anal. 184(2), 303–339, 2007

    MathSciNet  Article  Google Scholar 

  9. 9.

    Blanc, X., Lewin, M.: Existence of the thermodynamic limit for disordered quantum Coulomb systems. J. Math. Phys. 53(9), 095209, 2012

    ADS  MathSciNet  Article  Google Scholar 

  10. 10.

    Böl, M., Reese, S.: Finite element modelling of rubber-like materials—a comparison between simulation and experiment. J. Mater. Sci. 40, 5933–5939, 2005

    ADS  Article  Google Scholar 

  11. 11.

    Böl, M., Reese, S.: Finite element modelling of rubber-like polymers based on chain statistics. Int. J. Sol. Struct. 43, 2–26, 2006

    Article  Google Scholar 

  12. 12.

    Braides, A.: \(\Gamma \)-Convergence for Beginners, Volume 22 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford 2002

  13. 13.

    Braides, A., Defranceschi, A.: Homogenization of Multiple Integrals. Oxford University Press, Oxford 1998

    MATH  Google Scholar 

  14. 14.

    Braides, A., Gelli, M.S.: Non-local variational limits of discrete systems. Commun. Contemp. Math. 2(2), 286–297, 2002

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Braides, A., Gelli, M.S.: Continuum limits of discrete systems without convexity hypotheses. Math. Mech. Solids7(1), 41–66, 2002

    MathSciNet  Article  Google Scholar 

  16. 16.

    Braides, A., Gelli, M.S.: Limits of discrete systems with long-range interactions. J. Convex Anal. 9(2), 363–399, 2002

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Braides, A., Gelli, M.S.: The passage from discrete to continuous variational problems: a nonlinear homogenization process. Nonlinear Homogenization and Its Applications to Composites, Polycrystals and Smart Materials, NATO Science Series II Mathematics and Physical Chemistry, 170, Kluwer, Dordrecht, 45–63, 2004

  18. 18.

    Braun, J., Schmidt, B.: Existence and convergence of solutions of the boundary value problem in atomistic and continuum nonlinear elasticity theory. Calc. Var. Partial Differ. Equ. 55(125), 55–125, 2016

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Braun, J., Schmidt, B.: On the passage from atomistic systems to nonlinear elasticity theory for general multi-body potentials with p-growth. Netw. Heterog. Media8(4), 879–912, 2013

    MathSciNet  Article  Google Scholar 

  20. 20.

    Ciarlet, P.G.: Three-Dimensional Elasticity. Studies in Mathematics and its Applications, 20. North-Holland Publishing Co., Amsterdam 1988

    Google Scholar 

  21. 21.

    Cotar, C., Deuschel, J.D., Müller, S.: Strict convexity of the free energy for a class of non-convex gradient models. Commun. Math. Phys. 286, 359–376, 2009

    ADS  MathSciNet  Article  Google Scholar 

  22. 22.

    Cotar, C., Deuschel, J.D.: Decay of covariances, uniqueness of ergodic component and scaling limit for a class of gradient systems with non-convex potential. Ann. Inst. Henri Poincaré Probab. Stat. 48, 819–853, 2012

    ADS  MathSciNet  Article  Google Scholar 

  23. 23.

    Dal Maso, G.: An introduction to \(\Gamma \)-convergence. Progress in Nonlinear Differential Equations and their Applications, 8. Birkhäuser Boston Inc., Boston, MA, 1993

  24. 24.

    Dembo, A., Zeitouni, O.: Large Deviation Techniques and Applications. Jones and Bartlett, Boston 1993

    MATH  Google Scholar 

  25. 25.

    Deuschel, J.D., Giacomin, G., Ioffe, D.: Large deviations and concentration properties for \(\nabla \varphi \) interface models. Probab. Theory Relat. Fields117, 49–111, 2000

    MathSciNet  Article  Google Scholar 

  26. 26.

    Duerinckx, M., Gloria, A.: Stochastic homogenization of nonconvex unbounded integral functionals with convex growth. Arch. Ration. Mech. Anal. 221, 1511–1584, 2016

    MathSciNet  Article  Google Scholar 

  27. 27.

    Flory, P.J.: Statistical Mechanics of Chain Molecules. Interscience Publishers, New York 1969

    Book  Google Scholar 

  28. 28.

    Flory, P.J.: Network topology and the theory of rubber elasticity. Br. Polym. J. 17(2), 96–102, 1985

    Article  Google Scholar 

  29. 29.

    Francfort, G., Giacomini, A., Lopez-Pamies, O.: Fracture with healing: a first step towards a new view of cavitation. Anal. PDE12(2), 417–447, 2019

    MathSciNet  Article  Google Scholar 

  30. 30.

    Gloria, A., Le Tallec, P., Vidrascu, M.: Foundation, analysis, and numerical investigation of a variational network-based model for rubber. Contin. Mech. Thermodyn. 26, 1–31, 2014

    ADS  MathSciNet  Article  Google Scholar 

  31. 31.

    Gloria, A., Le Tallec, P., Lequeux, F., Vidrascu, M.: In preparation

  32. 32.

    Gloria, A., Penrose, M.D.: Random parking, Euclidean functionals, and rubber elasticity. Commun. Math. Phys. 321(1), 1–31, 2013

    ADS  MathSciNet  Article  Google Scholar 

  33. 33.

    Heinrich, G., Kaliske, M.: Theoretical and numerical formulation of a molecular based constitutive tube-model of rubber elasticity. Comput. Theor. Polym. Sci. 7(3–4), 227–241, 1997

    Article  Google Scholar 

  34. 34.

    Kotecký, R., Luckhaus, S.: Nonlinear elastic free energies and gradient Young–Gibbs measures. Commun. Math. Phys. 326, 887–917, 2014

    ADS  MathSciNet  Article  Google Scholar 

  35. 35.

    Kuhn, W., Grün, F.: Beziehung zwischen elastische Konstanten und Dehnungsdoppelberechnung Eigenschaften hochpolymerer Stoffe. Kolloid-Z. 101, 248–271, 1942

    Article  Google Scholar 

  36. 36.

    Kumar, A., Francfort, G., Lopez-Pamies, O.: Fracture and healing of elastomers: a phase-transition theory and numerical implementation. J. Mech. Phys. Solids112, 523–551, 2018

    ADS  MathSciNet  Article  Google Scholar 

  37. 37.

    Masurel, R.J., Cantournet, S., Dequidt, A., Long, D.R., Montes, H., Lequeux, F.: Role of dynamical heterogeneities on the viscoelastic spectrum of polymers: a stochastic continuum mechanics model. Macromolecules48, 6690–6702, 2015

    ADS  Article  Google Scholar 

  38. 38.

    Miehe, C., Göktepe, S., Lulei, F.: A micro–macro approach to rubber-like materials—part I: the non-affine micro-sphere model of rubber elasticity. J. Mech. Phys. Solids52, 2617–2660, 2004

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Møller, J.: Lectures on Random Voronoi Tessellations. Lecture Notes in Statistics, vol. 87. Springer, New York 1994

    Book  Google Scholar 

  40. 40.

    Müller, S., Spector, S.J.: An existence theory for nonlinear elasticity that allows for cavitation. Arch. Rational Mech. Anal. 131(1), 1–66, 1995

    ADS  MathSciNet  Article  Google Scholar 

  41. 41.

    Neukamm, S., Schäffner, M.: Quantitative homogenization in nonlinear elasticity for small loads. Arch. Ration. Mech. Anal. 230(1), 343–396, 2018

    MathSciNet  Article  Google Scholar 

  42. 42.

    Rubinstein, M., Panyukov, S.: Elasticity of polymer networks. Macromolecules35, 6670–6686, 2002

    ADS  Article  Google Scholar 

  43. 43.

    Rubinstein, M., Colby, R.H.: Polymer physics. Oxford at the Clarendon Press, Oxford 2003

    Google Scholar 

  44. 44.

    Tkachuk, M., Linder, C.: The maximal advance path constraint for the homogenization of materials with random network microstructure. Philos. Mag. 2012

  45. 45.

    Treloar, L.R.G.: The Physics of Rubber Elasticity. Oxford at the Clarendon Press, Oxford 1949

    Google Scholar 

Download references

Acknowledgements

AG warmly thanks François Lequeux and Michael Rubinstein for inspiring discussions on polymer physics. AG and MR acknowledge the financial support from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410). The work of MC was supported by the DFG Collaborative Research Center TRR 109, “Discretization in Geometry and Dynamics”.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Matthias Ruf.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Communicated by A. Garroni

Appendix A

Appendix A

In this appendix we collect and prove some of the results we used in the paper. We start with the technical proof of the interpolation inequality.

Proof of Proposition 1

We set \(\beta =1\) to reduce the notation. Given \(\delta >0\) and \(N\in {\mathbb {N}}\), for \(i\in \{1,\dots ,N+1\}\) we introduce the open sets

$$\begin{aligned} O_i=\left\{ x\in O:\;{\mathrm{dist}}(x,\partial O)>(i+1)\frac{\delta }{2N}\right\} . \end{aligned}$$

Then the stripes \(S_i:=O_{i-1}\backslash \overline{O_{i+2}}\) fulfill \(S_i\cap S_j=\emptyset \) whenever \(|i-j|>2\). Thus for every \(u:O_{\varepsilon }^{\mathcal {L}}\rightarrow {\mathbb {R}}^n\) we obtain by averaging

$$\begin{aligned} \frac{1}{N}\sum _{i=1}^{N}H_{\varepsilon }(S_i,u)\leqq \frac{3}{N}H_{\varepsilon }(O,u), \end{aligned}$$

so that we can decompose the set \({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )=\bigcup _{i=1}^N{\mathcal {P}}_{i,\varepsilon }\) (we omit the dependence on O and \(\kappa \)), where

$$\begin{aligned} {\mathcal {P}}_{i,\varepsilon }=\left\{ u\in {\mathcal {N}}_p(v,O,\varepsilon ,\kappa ):\;H_{\varepsilon }(S_i,u)\leqq \frac{3}{N}H_{\varepsilon }(O,u)\right\} . \end{aligned}$$

Let \(\theta _i:O\rightarrow [0,1]\) be the Lipschitz-continuous cut-off function defined by

$$\begin{aligned} \theta _i(z)=\min \left\{ \max \left\{ \frac{2N}{\delta }{\mathrm{dist}}(z,\partial O)-(i+1),0\right\} ,1\right\} , \end{aligned}$$

so that \(\theta _i\equiv 1\) on \(\overline{O_{i+1}}\), \(\theta _i\equiv 0\) on \(O\backslash O_{i}\) and its Lipschitz constant can be bounded by \(\mathrm {Lip}(\theta _i)\leqq \frac{2N}{\delta }\). We then define an interpolation between functions \(u,\psi :O_{\varepsilon }^{\mathcal {L}}\rightarrow {\mathbb {R}}^n\) as

$$\begin{aligned} T_{i,\varepsilon }(u,\psi )(x)=\theta _i(\varepsilon x)u(x)+(1-\theta _i(\varepsilon x))\psi (x). \end{aligned}$$

Observe that if \(u\in {\mathcal {P}}_{i,\varepsilon }\) as well as \(\varphi \in {\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\) and \(\psi \in {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )\), by the Minkowski inequality we have

$$\begin{aligned} \varepsilon ^{\frac{d}{p}}\Vert v_{\varepsilon }-\varepsilon T_{i,\varepsilon }(u,\psi )\Vert _{\ell ^p_{\varepsilon }(O)}\leqq & {} 2\kappa |O|^{\frac{1}{p}+\frac{1}{d}}+\varepsilon ^{\frac{d}{p}}\Vert \varepsilon \psi -\varepsilon \varphi \Vert _{\ell ^p_{\varepsilon }(O\backslash \overline{O_{i+1}})}\\\leqq & {} 2\kappa |O|^{\frac{1}{p}+\frac{1}{d}}+C\varepsilon |O|^{\frac{1}{p}}, \end{aligned}$$

so that \(T_{i,\varepsilon }(u,\psi )\in {\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\) for \(\varepsilon \) small enough.

For technical reasons the interpolations will not suffice to prove the estimates. For every i let us choose \(t_i\in [\frac{1}{4},\frac{3}{4}]\) such that, setting \(S_i^t=\{x\in O:\;\theta _i(x)=t\}\), the coarea formula implies

$$\begin{aligned} \frac{1}{2}{\mathcal {H}}^{d-1}( S_i^{t_i})\leqq \int _{\frac{1}{4}}^{\frac{3}{4}}{\mathcal {H}}^{d-1}(S_i^t)\,\mathrm {d}t\leqq \int _0^1{\mathcal {H}}^{d-1}( S_i^t)\,\mathrm {d}t=\int _{O}|\nabla \theta _i|\leqq \frac{2N}{\delta }|O_i\backslash {O_{i+1}}|.\nonumber \\ \end{aligned}$$
(A.1)

We set \(S_i^*=\{x\in O:\theta _i(x)<t_i\}\). Note that for \(\delta \) small enough (depending only on O), we have \(S_i^*\in {\mathcal {A}}^R(D)\) (see for instance [32, Lemma 2.2]). Let us introduce the product set

$$\begin{aligned} {\mathcal {U}}_{\varepsilon }^i(M):=\big ({\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )\big )\times {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon ), \end{aligned}$$

as well as the integral

$$\begin{aligned} e^i_{\varepsilon }(M):=\bigg (\int _{{\mathcal {U}}_{\varepsilon }^i(M)}\exp \Big (- H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))-c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\Big )\,\mathrm {d}u\,\mathrm {d}\psi \bigg ), \end{aligned}$$

where \(c_0>0\) is a small constant such that \(c_0|\xi |^p\leqq f(\cdot ,\xi )+c_0^{-1}\) (cf. Hypothesis 1). This integral quantity will be the main ingredient to prove the interpolation inequality. We split the remaining argument into several steps.

Step 1. Energy bounds for the interpolation.

To bound the energy of \(T_{i,\varepsilon }(u,\psi )\), we use the pointwise inequality

$$\begin{aligned} |\psi (x)-\psi (y)|^p\leqq & {} C|(\psi -\varphi )(x)-(\psi -\varphi )(y)|^p+C|\varphi (x)-\varphi (y)|^p\\\leqq & {} C+C|\varphi (x)-\varphi (y)|^p, \end{aligned}$$

which is valid for all \(x,y\in (O\backslash \overline{O_{i+1}})_{\varepsilon }\). Combined with the two-sided growth condition in Hypothesis 1 we infer that

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))&\leqq H_{\varepsilon }(O_{i+1},u)+H_{\varepsilon }(O\backslash \overline{O_i},\psi )+H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi ))\nonumber \\&\leqq H_{\varepsilon }(O_{i+1},u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi )), \end{aligned}$$
(A.2)

where \(O^{\delta }\) is defined in the statement of Proposition 1. In order to estimate the last term on the right hand side we use the formula

$$\begin{aligned} T_{i,\varepsilon }(u,\psi )(x)-T_{i,\varepsilon }(u,\psi )(y)&=\big (\theta _i(\varepsilon x)-\theta _i(\varepsilon y)\big )\big (u(x)-\psi (x)\big ) \\&\quad +\theta _i(\varepsilon y)\big (u(x)-u(y)\big )\\&\quad +(1-\theta _i(\varepsilon y))\big (\psi (x)-\psi (y)\big ) \end{aligned}$$

and the bound on the Lipschitz constant of \(\theta _i\) to estimate the energy on the interpolation stripe via

$$\begin{aligned}&H_{\varepsilon }(S_i,T_{i,\varepsilon }(u,\psi )) \nonumber \\&\quad \leqq C\Vert \nabla _{{\mathbb {B}}}T_{i,\varepsilon }(u,\psi )\Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+C|(S_i)^{{\mathcal {L}}}_{\varepsilon }|\nonumber \\&\quad \leqq C\left( \Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+\Vert \nabla _{{\mathbb {B}}}\psi \Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+\frac{(N\varepsilon )^p}{\delta ^p}\Vert u-\psi \Vert ^p_{\ell ^p_{\varepsilon }(S_i)}+|(S_i)^{{\mathcal {L}}}_{\varepsilon }|\right) \nonumber \\&\quad \leqq \frac{C}{N}H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{1+\frac{p}{d}}\varepsilon ^{-d}, \end{aligned}$$
(A.3)

where we have used again that the degree of each vertex is equibounded and that, after suitable extension, \(\psi \in {\mathcal {N}}_p(v,O,\varepsilon ,2\kappa )\) for \(\varepsilon \) small enough. Combining (A.2) and (A.3) we infer that

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))\leqq & {} H_{\varepsilon }(O_{i+1},u)+\frac{C}{N}H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }| \nonumber \\&+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{\frac{p}{d}}|O_{\varepsilon }^{\mathcal {L}}|. \end{aligned}$$
(A.4)

Step 2. Lower bound for \(e^i_{\varepsilon }(M)\).

In order to prove a lower bound for the integral, first note that due to Hypothesis 1 and the definition of \(S_i^*\)

$$\begin{aligned} c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\leqq H_{\varepsilon }(O\setminus O_{i+1},u)+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|, \end{aligned}$$

so that, up to increasing C, we can add this inequality to (A.4) and obtain the estimate

$$\begin{aligned} H_{\varepsilon }(O,T_{i,\varepsilon }(u,\psi ))+c_0\Vert \nabla _{{\mathbb {B}}}u\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\leqq & {} \left( 1+\frac{C}{N}\right) H_{\varepsilon }(O,u)+CH_{\varepsilon }(O^{\delta },\varphi )\\&+C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{CN^p}{\delta ^p}\kappa ^p|O|^{\frac{p}{d}}|O_{\varepsilon }^{\mathcal {L}}| \end{aligned}$$

Rearranging the terms we obtain by Fubini’s Theorem that

$$\begin{aligned} e_{\varepsilon }^i(M)&\geqq \exp \bigg (-C \Big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}| \nonumber \\&\quad + H_{\varepsilon }(O^{\delta },\varphi )\Big )\bigg )\int _{{\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )}\mathrm {d}\psi \nonumber \\&\quad \times \int _{{\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )}\exp (- H_{\varepsilon }(O,u))\,\mathrm {d}u\nonumber \\ \geqq&\exp \bigg (-C \Big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\Big )\bigg )\nonumber \\&\quad \times Z_{\varepsilon ,O}({\mathcal {P}}_{i,\varepsilon }\cap {\mathcal {S}}_M(O,\varepsilon )) \end{aligned}$$
(A.5)

where we used that the measure of \({\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )\) can be bounded from below by \(\exp (-C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\).

Step 3. Upper bound for \(e^i_{\varepsilon }(M)\) and conclusion.

To estimate \(e^i_{\varepsilon }(M)\) from above, similar to [34] we perform a suitable change of variables. Define \(\Phi _{i,\varepsilon }:{\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\times {\mathcal {N}}_{\infty }(\varphi ,O\backslash \overline{O_{i+1}},\varepsilon )\rightarrow {\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\times {\mathcal {N}}_p(u,O\backslash \overline{O_{i+1}},\varepsilon ,3\kappa )\) by

$$\begin{aligned} \Phi _{i,\varepsilon }(u,\psi )(x)= {\left\{ \begin{array}{ll} (T_{i,\varepsilon }(u,\psi )(x),\psi (x)) &{}\text{ if } \theta _i(\varepsilon x)\geqq t_i,\\ (T_{i,\varepsilon }(u,\psi )(x),u(x)) &{}\text{ if } \theta _i(\varepsilon x)<t_i. \end{array}\right. } \end{aligned}$$

Note that for \(\varepsilon \) small enough \(\Phi _{i,\varepsilon }\) is well-defined and bijective onto its range \({\mathcal {R}}(\Phi _{i,\varepsilon })\). For the idea how to calculate the Jacobian, we refer to the proof of Proposition 3. As \(t_i\in [\frac{1}{4},\frac{3}{4}]\), it holds that

$$\begin{aligned} |\det (D\Phi _{i,\varepsilon }(u,\psi ))|^{-1}= & {} \left( \prod _{x:\theta _i(\varepsilon x)\geqq t_i}|\theta _i(\varepsilon x)|^n\prod _{x:\theta _i(\varepsilon x)<t_i}|1-\theta _i(\varepsilon x)|^n\right) ^{-1}\\\leqq & {} \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|). \end{aligned}$$

Setting \((g,h)=\Phi _{i,\varepsilon }(u,\psi )\), by construction of the interpolation we have

$$\begin{aligned} \begin{aligned} g&\in {\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ), \\ h&=(h_1,h_2)\in {\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )\\&\quad \times \underbrace{\{h:(S_i^*)_{\varepsilon }\rightarrow {\mathbb {R}}^n:\;\Vert h-\varepsilon ^{-1}v_{\varepsilon }\Vert _{\infty }\leqq C\kappa |O_{\varepsilon }|^{\frac{1}{p}+\frac{1}{d}}\}}_{=:R_{i,\varepsilon }}. \end{aligned} \end{aligned}$$

As the measure of the set \({\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )\) can be bounded by \(\exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\), the above change of variables and Fubini’s Theorem imply

$$\begin{aligned} e_{\varepsilon }^i(M)&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{{\mathcal {R}}(\Phi _{i,\varepsilon })}\exp \Big (- H_{\varepsilon }(g,O)-c_0\Vert \nabla _{{\mathbb {B}}}h\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)}\Big )\,\mathrm {d}g\,\mathrm {d}h\nonumber \\&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{{\mathcal {N}}_{\infty }(\varphi ,O\backslash (\overline{O_{i+1}}\cup S_i^*),\varepsilon )}\mathrm {d}h_1\int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2\nonumber \\&\quad \times Z({\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\nonumber \\&\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|)\int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^t)})\,\mathrm {d}h_2\; \nonumber \\&\quad \times Z_{\varepsilon ,O}({\mathcal {N}}_p(u,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi )). \end{aligned}$$
(A.6)

In order to bound the integral on the right hand side, we apply Lemma 3.3 to the graph \(G_{S_i^*,\varepsilon }\) with \(\alpha =c_0\) and \(\gamma =C\kappa |O_{\varepsilon }|^{\frac{1}{p}+\frac{1}{d}}\) and infer

$$\begin{aligned} \int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2&\leqq \Big (C\kappa ^n |O_{\varepsilon }|^{\frac{n}{p}+\frac{n}{d}}\Big )^{N_{i,\varepsilon }}C^{|(S_i^*)^{{\mathcal {L}}}_{\varepsilon }|-N_{i,\varepsilon }}, \end{aligned}$$

where we denoted by \(N_{i,\varepsilon }\) the number of connected components of the graph \(G_{S_i^*,\varepsilon }\). For \(\varepsilon \) small enough (possibly depending on \(N,\delta \)), by Remark 7, (3.3) and the fact that \(S_i^*\in {\mathcal {A}}^R(D)\) we can bound the number of components via

$$\begin{aligned} N_{i,\varepsilon }\leqq \#\{x\in O_{\varepsilon }^{\mathcal {L}}:\;{\mathrm{dist}}(x,\partial (S_i^*)_{\varepsilon })\leqq C_0\}\leqq C\varepsilon ^{1-d}({\mathcal {H}}^{d-1}(S_i^{t_i})+{\mathcal {H}}^{d-1}(\partial O)). \end{aligned}$$

In particular, for \(N,\delta \) and \(\kappa >0\) fixed, due to (A.1) there exists \(\varepsilon _0\) such that for all \(\varepsilon <\varepsilon _0\)

$$\begin{aligned} \int _{R_{i,\varepsilon }}\exp (-c_0\Vert \nabla _{\mathbb {B}}h_2\Vert ^p_{\ell ^p_{\varepsilon }(S_i^*)})\,\mathrm {d}h_2\leqq \exp (C|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|). \end{aligned}$$

Plugging this bound into (A.6) and comparing with (A.5) yields

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {P}}_{\varepsilon }^i\cap {\mathcal {S}}_M(O,\varepsilon )) \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi )) \\&\quad \times \exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ) \end{aligned}$$

Summing this inequality over i, by the definition of the sets \({\mathcal {P}}_{i,\varepsilon }\) we infer that

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\cap {\mathcal {S}}_M(O,\varepsilon )) \leqq \sum _{i=1}^N Z_{\varepsilon ,O}({\mathcal {P}}_{\varepsilon }^i\cap {\mathcal {S}}_M(O,\varepsilon ))\nonumber \\&\quad \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\nonumber \\&\qquad \times N\exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}|O_{\varepsilon }^{\mathcal {L}}|+\frac{M}{N}|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ) \end{aligned}$$
(A.7)

Now, choosing

$$\begin{aligned} M=2\left( \frac{1}{|O_{\varepsilon }^{\mathcal {L}}|}\log \Big (Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))\Big )+{\overline{C}}+\frac{\log (2)}{|O_{\varepsilon }^{\mathcal {L}}|}\right) , \end{aligned}$$

where \({\overline{C}}\) is the constant of Lemma 3.4, we obtain by the same Lemma and Remark 8 that, for any \(\kappa >0\) fixed and all \(\varepsilon \) small enough,

$$\begin{aligned} Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\backslash {\mathcal {S}}_M(O,\varepsilon ))\leqq \frac{1}{2}Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )). \end{aligned}$$

Thus (A.7) and the definition of M yield the final estimate

$$\begin{aligned}&Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))\\&\quad \leqq 2Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa )\cap {\mathcal {S}}_M(O,\varepsilon )) \\&\quad \leqq Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,3\kappa )\cap {\mathcal {B}}_{\varepsilon }(O,\varphi ))\,Z_{\varepsilon ,O}({\mathcal {N}}_p(v,O,\varepsilon ,\kappa ))^{\frac{C}{N}}\nonumber \\&\qquad \times 2N\exp \Big (C\big (|(O^{\delta })^{{\mathcal {L}}}_{\varepsilon }|+\big (\frac{(N\kappa |O|^{\frac{1}{d}})^p}{\delta ^p}+\frac{C}{N}\big )|O_{\varepsilon }^{\mathcal {L}}|+ H_{\varepsilon }(O^{\delta },\varphi )\big )\Big ). \end{aligned}$$

\(\square \)

Remark 14

Note that the restriction on \(\delta \) in the interpolation inequality comes only from the requirement that tubular neighbourhoods of the boundary have again Lipschitz boundary. In particular, if \(\delta \) satisfies the condition for a set \(O\subset {\mathbb {R}}^d\), then \(\delta ^{\prime }=\delta \rho \) satisfies the condition for all sets of the form \(O^{\prime }=z+\rho O\). Applying this fact to the family of cubes \(Q(z,\rho )\) with \(z\in D\) and \(\rho >0\), we obtain that there exists \(\delta _0>0\) such that for all \(\delta <\delta _0\), all \(N\in {\mathbb {N}}\) and all \(\kappa >0\) it holds that

$$\begin{aligned} \frac{N-C}{N}{\mathcal {F}}_{\kappa }^-(Q(z,\rho ), {\overline{\varphi }}_\Lambda )&\geqq {{\overline{W}}}(\Lambda )-C(1+|\Lambda |^p)\frac{|Q(z,\rho )^{\delta \rho }|}{|Q(z,\rho )|}\\&\quad -C\left( \frac{(N\kappa |O(z,\rho )|^{\frac{1}{d}})^p}{(\delta \rho )^p}+\frac{1}{N}\right) \\&\geqq {{\overline{W}}}(\Lambda )-C\left( (1+|\Lambda |^p)\delta +\frac{(N\kappa )^p}{\delta ^p}+\frac{1}{N}\right) \end{aligned}$$

\({\mathbb {P}}\)-almost surely. Here we used Lemma 4.1, Proposition 3 and (3.1) in order to pass to the limit as \(\varepsilon \rightarrow 0\) in the interpolation inequality almost surely. Note that the last bound is independent of \(\rho \) and z.

Lemma A.1

Let \(p\in (1,+\infty )\). For all \(u,v\in {\mathbb {R}}^n\) it holds that

$$\begin{aligned} |u-v|^p+|u+v|^p\leqq \max \{2^{p-1},2\}(|u|^p+|v|^p). \end{aligned}$$

Proof of Lemma A.1

For \(p\geqq 2\) the claimed estimate follows from Clarkson’s inequality. If \(p<2\), then \((x_1^p+x_2^p)^{\frac{1}{p}}\geqq (x_1^2+x_2^2)^{\frac{1}{2}}\) for all \(x_1,x_2\geqq 0\). Moreover, with elementary analysis one can show that \((x_1^p+x_2^p)^{\frac{1}{p}}\leqq 2^{\frac{1}{p}-\frac{1}{2}}(x_1^2+x_2^2)^{\frac{1}{2}}\). Applying these two inequalities first with \(x_1=|u-v|\) and \(x_2=|u+v|\) and then with \(x_1=|u|\) and \(x_2=|v|\) we obtain

$$\begin{aligned} (|u-v|^p+|u+v|^p)^{\frac{1}{p}}\leqq & {} 2^{\frac{1}{p}-\frac{1}{2}}(|u-v|^2+|u+v|^2)^{\frac{1}{2}}=2^{\frac{1}{p}}(|u|^2+|v|^2)^{\frac{1}{2}}\\\leqq & {} 2^{\frac{1}{p}}(|u|^p+|v|^p)^{\frac{1}{p}}. \end{aligned}$$

\(\square \)

Lemma A.2

Let \(p\in (1,+\infty )\). Then there exists a constant \(c_{p}\) such that the Hausdorff measure of the sphere \(S^{n-1}_{p}=\{y\in {\mathbb {R}}^n:\;|y|_p=1\}\) fulfills

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_{p})\geqq \left( \frac{c_{p}}{n}\right) ^{\frac{n}{p}}. \end{aligned}$$

Proof of Lemma A.2

Note that \(S^{n-1}_{p}\) is a compact smooth \((n-1)\)-dimensional manifold. Hence we can characterize its Hausdorff measure by its Minkowski content. To be more precise, it holds that

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_{p})=\lim _{\varepsilon \rightarrow 0}\frac{{\mathcal {H}}^{n}(S^{n-1}_{p}+B_{\varepsilon }(0))}{2\varepsilon }, \end{aligned}$$
(A.8)

where the factor 2 comes from the Lebesgue measure of the 1D unit ball \([-1,1]\). Note however that \(B_{\varepsilon }(0)\) is a ball with respect to the Euclidean metric on \({\mathbb {R}}^n\). We now give a lower bound for the nominator on the right hand side of (A.8). To this end, set \(c_{n,p}=\max \{1,n^{\frac{1}{2}-\frac{1}{p}}\}\). Then, for \(y\ne 0\), we have

$$\begin{aligned} \left| y-\frac{y}{|y|_p}\right| _2\leqq ||y|_p-1|\frac{|y|_2}{|y|_p}\leqq ||y|_p-1|c_{n,p}, \end{aligned}$$

where we used that by definition \(|y|_2\leqq c_{n,p}|y|_p\) for all \(y\in {\mathbb {R}}^n\). We conclude that

$$\begin{aligned} \{y\in {\mathbb {R}}^n:\;1-c^{-1}_{n,p}\varepsilon<|y|_p<1+c^{-1}_{n,p}\varepsilon \}\subset S^{n-1}_p+B_{\varepsilon }(0). \end{aligned}$$

Hence we deduce from (A.8) and the well-know formula for the volume of p-norm balls that

$$\begin{aligned} {\mathcal {H}}^{n-1}(S^{n-1}_p)&\geqq \liminf _{\varepsilon \rightarrow 0}\frac{{\mathcal {H}}^n(\{|y|_p<1+c_{n,p}^{-1}\varepsilon \})-{\mathcal {H}}^n(\{|y|_p<1-c^{-1}_{n,p}\varepsilon \})}{2\varepsilon }\\&=\frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}\lim _{\varepsilon \rightarrow 0}\frac{(1+c^{-1}_{n,p}\varepsilon )^n-(1-c^{-1}_{n,p}\varepsilon )^n}{2\varepsilon }\\&=\frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}nc^{-1}_{n,p}\geqq \frac{(2\Gamma (\frac{1}{p}+1))^n}{\Gamma (\frac{n}{p}+1)}n^{\frac{1}{2}}. \end{aligned}$$

We conclude the proof using Stirling’s formula in the form of the upper bound

$$\begin{aligned} \Gamma \left( \frac{n}{p}+1\right) \leqq \left( \frac{2\pi n}{p}\right) ^{\frac{1}{2}}\left( \frac{n}{pe}\right) ^{\frac{n}{p}}\exp (p/12). \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cicalese, M., Gloria, A. & Ruf, M. From Statistical Polymer Physics to Nonlinear Elasticity. Arch Rational Mech Anal 236, 1127–1215 (2020). https://doi.org/10.1007/s00205-019-01487-1

Download citation