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 and 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.

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

S. Krömer, Private notes 2012.

[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.

[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.

[24]

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

[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. 

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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. 

[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.

[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.

[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.

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.

[2]

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

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[15]

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

[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.

[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.

[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.

[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.

[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.

[21]

S. Krömer, Private notes 2012.

[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.

[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.

[24]

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

[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. 

[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.

[27]

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

[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.

[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.

[30]

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

[31]

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

[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. 

[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.

[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.

[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.

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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