February  2017, 10(1): 55-73. doi: 10.3934/dcdss.2017003

A note on $3$d-$1$d dimension reduction with differential constraints

Fakultät für Mathematik, Universität Regensburg, 93040 Regensburg, Germany1

Received  April 2015 Revised  June 2015 Published  December 2016

Starting from three-dimensional variational models with energies subject to a general type of PDE constraint, we use Γ-convergence methods to derive reduced limit models for thin strings by letting the diameter of the cross section tend to zero. A combination of dimension reduction with homogenization techniques allows for addressing the case of thin strings with fine heterogeneities in the form of periodically oscillating structures. Finally, applications of the results in the classical gradient case, corresponding to nonlinear elasticity with Cosserat vectors, as well as in micromagnetics are discussed.

Citation: Carolin Kreisbeck. A note on $3$d-$1$d dimension reduction with differential constraints. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 55-73. doi: 10.3934/dcdss.2017003
References:
[1]

E. AcerbiG. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[3]

M. BaíaM. ChermisiJ. Matias and P. M. Santos, Lower semicontinuity and relaxation of signed functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity, Calc. Var. Partial Differential Equations, 47 (2013), 465-498.  doi: 10.1007/s00526-012-0524-1.  Google Scholar

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G. BouchittéI. Fonseca and M. L. Mascarenhas, Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.  doi: 10.1023/B:ELAS.0000029996.20973.92.  Google Scholar

[5]

G. BouchittéI. Fonseca and M. L. Mascarenhas, The Cosserat vector in membrane theory: A variational approach, J. Convex Anal., 16 (2009), 351-365.   Google Scholar

[6]

A. Braides, $Γ$-convergence for Beginners volume 22 of Oxford Lecture Series in Mathematics and its Applications Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[7]

A. BraidesI. Fonseca and G. Leoni, $\mathcal{A}$-quasiconvexity: Relaxation and homogenization, ESAIM Control Optim. Calc. Var., 5 (2000), 539-577.  doi: 10.1051/cocv:2000121.  Google Scholar

[8]

W. Brown, Micromagnetics John Wiley and Sons, New York, 1963. Google Scholar

[9]

B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals volume 922. Springer, 1982.  Google Scholar

[10]

G. Dal Maso, An Introduction to $Γ$-convergence volume 8 of Progress in nonlinear differential equations and their applications Birkhäuser Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[11]

E. Davoli, Thin-walled beams with a cross-section of arbitrary geometry: Derivation of linear theories starting from 3D nonlinear elasticity, Adv. Calc. Var., 6 (2013), 33-91.  doi: 10.1515/acv-2011-0003.  Google Scholar

[12]

I. Fonseca and S. Krömer, Multiple integrals under differential constraints: Two-scale convergence and homogenization, Indiana Univ. Math. J., 59 (2010), 427-457.  doi: 10.1512/iumj.2010.59.4249.  Google Scholar

[13]

I. FonsecaG. Leoni and S. Müller, $\mathcal{A}$-quasiconvexity: weak-star convergence and the gap, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 209-236.  doi: 10.1016/j.anihpc.2003.01.003.  Google Scholar

[14]

I. Fonseca and S. Müller, $\mathcal{A}$-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal., 30 (1999), 1355-1390.  doi: 10.1137/S0036141098339885.  Google Scholar

[15]

I. Fonseca and E. Zappale, Multiscale relaxation of convex functionals, J. Convex Anal., 10 (2003), 325-350.   Google Scholar

[16]

T. KatoM. MitreaG. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 7 (2000), 643-650.  doi: 10.4310/MRL.2000.v7.n5.a10.  Google Scholar

[17]

J. Kräamer, S. Krömer, K. Martin and G. Pathó, $\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals, Preprint, arXiv: 1401.6358, 2014. Google Scholar

[18]

C. Kreisbeck, Another approach to the thin-film $Γ$-limit of the micromagnetic free energy in the regime of small samples, Quart. Appl. Math., 71 (2013), 201-213.  doi: 10.1090/S0033-569X-2012-01323-5.  Google Scholar

[19]

C. Kreisbeck and F. Rindler, Thin-film limits of functionals on $\mathcal{A}$-free vector fields, Indiana Univ. Math. J., 64 (2015), 1383-1423.  doi: 10.1512/iumj.2015.64.5653.  Google Scholar

[20]

C. Kreisbeck and S. Krömer, Heterogeneous thin films: Combining homogenization and dimension reduction with directors, SIAM J. Math. Anal., 48 (2016), 785-820.  doi: 10.1137/15M1032557.  Google Scholar

[21]

S. Krömer, Private notes 2012. Google Scholar

[22]

S. Krömer, Dimension reduction for functionals on solenoidal vector fields, ESAIM Control Optim. Calc. Var., 18 (2012), 259-276.  doi: 10.1051/cocv/2010051.  Google Scholar

[23]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of continuous media, In Course of Theoretical Physics volume 8. Pergamon Press, New York, 1984.  Google Scholar

[24]

H. Le Dret and N. Meunier, Modeling heterogeneous martensitic wires, Math. Models Methods Appl. Sci., 15 (2005), 375-406.  doi: 10.1142/S0218202505000406.  Google Scholar

[25]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578.   Google Scholar

[26]

I. P. Lizorkin, (Lp; Lq)-multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 152 (1963), 808-811(Engl. trans. Sov. Math. Dokl., 4 (1963), 1420-1424.  Google Scholar

[27]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152.  doi: 10.1007/BF02417888.  Google Scholar

[28]

M. G. Mora and S. Müller, A nonlinear model for inextensible rods as a low energy $Γ$-limit of three-dimensional nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 271-293.  doi: 10.1016/S0294-1449(03)00044-1.  Google Scholar

[29]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.  doi: 10.1007/s00526-006-0039-8.  Google Scholar

[30]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212.  doi: 10.1007/BF00284506.  Google Scholar

[31]

F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 489-507.   Google Scholar

[32]

F. Murat, Compacité par compensation: Condition nécessaire et suffisante de continuité faible sous une hypothése de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 69-102.   Google Scholar

[33]

S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal., 206 (2012), 645-706.  doi: 10.1007/s00205-012-0539-y.  Google Scholar

[34]

L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by $Γ$-convergence, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1037-1070.  doi: 10.1017/S0308210507000194.  Google Scholar

[35]

L. Tartar, Compensated compactness and applications to partial differential equations, In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. Ⅳ, volume 39 of Res. Notes in Math., pages 136-212. Pitman, 1979.  Google Scholar

show all references

References:
[1]

E. AcerbiG. Buttazzo and D. Percivale, A variational definition of the strain energy for an elastic string, J. Elasticity, 25 (1991), 137-148.  doi: 10.1007/BF00042462.  Google Scholar

[2]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[3]

M. BaíaM. ChermisiJ. Matias and P. M. Santos, Lower semicontinuity and relaxation of signed functionals with linear growth in the context of $\mathcal{A}$-quasiconvexity, Calc. Var. Partial Differential Equations, 47 (2013), 465-498.  doi: 10.1007/s00526-012-0524-1.  Google Scholar

[4]

G. BouchittéI. Fonseca and M. L. Mascarenhas, Bending moment in membrane theory, J. Elasticity, 73 (2003), 75-99.  doi: 10.1023/B:ELAS.0000029996.20973.92.  Google Scholar

[5]

G. BouchittéI. Fonseca and M. L. Mascarenhas, The Cosserat vector in membrane theory: A variational approach, J. Convex Anal., 16 (2009), 351-365.   Google Scholar

[6]

A. Braides, $Γ$-convergence for Beginners volume 22 of Oxford Lecture Series in Mathematics and its Applications Oxford University Press, Oxford, 2002. doi: 10.1093/acprof:oso/9780198507840.001.0001.  Google Scholar

[7]

A. BraidesI. Fonseca and G. Leoni, $\mathcal{A}$-quasiconvexity: Relaxation and homogenization, ESAIM Control Optim. Calc. Var., 5 (2000), 539-577.  doi: 10.1051/cocv:2000121.  Google Scholar

[8]

W. Brown, Micromagnetics John Wiley and Sons, New York, 1963. Google Scholar

[9]

B. Dacorogna, Weak Continuity and Weak Lower Semicontinuity for Nonlinear Functionals volume 922. Springer, 1982.  Google Scholar

[10]

G. Dal Maso, An Introduction to $Γ$-convergence volume 8 of Progress in nonlinear differential equations and their applications Birkhäuser Boston, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[11]

E. Davoli, Thin-walled beams with a cross-section of arbitrary geometry: Derivation of linear theories starting from 3D nonlinear elasticity, Adv. Calc. Var., 6 (2013), 33-91.  doi: 10.1515/acv-2011-0003.  Google Scholar

[12]

I. Fonseca and S. Krömer, Multiple integrals under differential constraints: Two-scale convergence and homogenization, Indiana Univ. Math. J., 59 (2010), 427-457.  doi: 10.1512/iumj.2010.59.4249.  Google Scholar

[13]

I. FonsecaG. Leoni and S. Müller, $\mathcal{A}$-quasiconvexity: weak-star convergence and the gap, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 209-236.  doi: 10.1016/j.anihpc.2003.01.003.  Google Scholar

[14]

I. Fonseca and S. Müller, $\mathcal{A}$-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal., 30 (1999), 1355-1390.  doi: 10.1137/S0036141098339885.  Google Scholar

[15]

I. Fonseca and E. Zappale, Multiscale relaxation of convex functionals, J. Convex Anal., 10 (2003), 325-350.   Google Scholar

[16]

T. KatoM. MitreaG. Ponce and M. Taylor, Extension and representation of divergence-free vector fields on bounded domains, Math. Res. Lett., 7 (2000), 643-650.  doi: 10.4310/MRL.2000.v7.n5.a10.  Google Scholar

[17]

J. Kräamer, S. Krömer, K. Martin and G. Pathó, $\mathcal{A}$-quasiconvexity and weak lower semicontinuity of integral functionals, Preprint, arXiv: 1401.6358, 2014. Google Scholar

[18]

C. Kreisbeck, Another approach to the thin-film $Γ$-limit of the micromagnetic free energy in the regime of small samples, Quart. Appl. Math., 71 (2013), 201-213.  doi: 10.1090/S0033-569X-2012-01323-5.  Google Scholar

[19]

C. Kreisbeck and F. Rindler, Thin-film limits of functionals on $\mathcal{A}$-free vector fields, Indiana Univ. Math. J., 64 (2015), 1383-1423.  doi: 10.1512/iumj.2015.64.5653.  Google Scholar

[20]

C. Kreisbeck and S. Krömer, Heterogeneous thin films: Combining homogenization and dimension reduction with directors, SIAM J. Math. Anal., 48 (2016), 785-820.  doi: 10.1137/15M1032557.  Google Scholar

[21]

S. Krömer, Private notes 2012. Google Scholar

[22]

S. Krömer, Dimension reduction for functionals on solenoidal vector fields, ESAIM Control Optim. Calc. Var., 18 (2012), 259-276.  doi: 10.1051/cocv/2010051.  Google Scholar

[23]

L. D. Landau, E. M. Lifshitz and L. P. Pitaevskii, Electrodynamics of continuous media, In Course of Theoretical Physics volume 8. Pergamon Press, New York, 1984.  Google Scholar

[24]

H. Le Dret and N. Meunier, Modeling heterogeneous martensitic wires, Math. Models Methods Appl. Sci., 15 (2005), 375-406.  doi: 10.1142/S0218202505000406.  Google Scholar

[25]

H. Le Dret and A. Raoult, The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity, J. Math. Pures Appl. (9), 74 (1995), 549-578.   Google Scholar

[26]

I. P. Lizorkin, (Lp; Lq)-multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR, 152 (1963), 808-811(Engl. trans. Sov. Math. Dokl., 4 (1963), 1420-1424.  Google Scholar

[27]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl. (4), 117 (1978), 139-152.  doi: 10.1007/BF02417888.  Google Scholar

[28]

M. G. Mora and S. Müller, A nonlinear model for inextensible rods as a low energy $Γ$-limit of three-dimensional nonlinear elasticity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 271-293.  doi: 10.1016/S0294-1449(03)00044-1.  Google Scholar

[29]

M. G. Mora and S. Müller, Derivation of a rod theory for multiphase materials, Calc. Var. Partial Differential Equations, 28 (2007), 161-178.  doi: 10.1007/s00526-006-0039-8.  Google Scholar

[30]

S. Müller, Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. Rational Mech. Anal., 99 (1987), 189-212.  doi: 10.1007/BF00284506.  Google Scholar

[31]

F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (1978), 489-507.   Google Scholar

[32]

F. Murat, Compacité par compensation: Condition nécessaire et suffisante de continuité faible sous une hypothése de rang constant, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 8 (1981), 69-102.   Google Scholar

[33]

S. Neukamm, Rigorous derivation of a homogenized bending-torsion theory for inextensible rods from three-dimensional elasticity, Arch. Ration. Mech. Anal., 206 (2012), 645-706.  doi: 10.1007/s00205-012-0539-y.  Google Scholar

[34]

L. Scardia, Asymptotic models for curved rods derived from nonlinear elasticity by $Γ$-convergence, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 1037-1070.  doi: 10.1017/S0308210507000194.  Google Scholar

[35]

L. Tartar, Compensated compactness and applications to partial differential equations, In Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. Ⅳ, volume 39 of Res. Notes in Math., pages 136-212. Pitman, 1979.  Google Scholar

Figure 1.  Transformation of $\Omega_\varepsilon $ into $\Omega_1$ and rescaling of the differential constraints.
Figure 2.  Examples of heterogeneous strings. a) Fibered heterogeneities ($f$ constant in $y_d$); b) Fine material layers ($f$ constant in $y'$).
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