Estimates for elliptic systems in a narrow region arising from composite materials
Authors:
Hongjie Ju, Haigang Li and Longjuan Xu
Journal:
Quart. Appl. Math. 77 (2019), 177-199
MSC (2010):
Primary 35B65, 74B05, 35J47, 35Q74
DOI:
https://doi.org/10.1090/qam/1518
Published electronically:
September 5, 2018
MathSciNet review:
3897923
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Additional Information
Abstract: In this paper, we establish the pointwise upper and lower bounds of the gradients of solutions to a class of elliptic systems, including linear systems of elasticity, in a general narrow region and in all dimensions. This problem arises from the study of damage analysis of high-contrast composite materials. Our results show that the damage may initiate from the narrowest place.
References
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- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 162050, DOI https://doi.org/10.1002/cpa.3160170104
- Habib Ammari, Eric Bonnetier, Faouzi Triki, and Michael Vogelius, Elliptic estimates in composite media with smooth inclusions: an integral equation approach, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 453–495 (English, with English and French summaries). MR 3346176, DOI https://doi.org/10.24033/asens.2249
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Kihyun Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal. 208 (2013), no. 1, 275–304. MR 3021549, DOI https://doi.org/10.1007/s00205-012-0590-8
- Habib Ammari, George Dassios, Hyeonbae Kang, and Mikyoung Lim, Estimates for the electric field in the presence of adjacent perfectly conducting spheres, Quart. Appl. Math. 65 (2007), no. 2, 339–355. MR 2330561, DOI https://doi.org/10.1090/S0033-569X-07-01034-1
- Habib Ammari, Hyeonbae Kang, and Mikyoung Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. MR 2178063, DOI https://doi.org/10.1007/s00208-004-0626-y
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Jungwook Lee, and Mikyoung Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324 (English, with English and French summaries). MR 2384571, DOI https://doi.org/10.1016/j.matpur.2007.07.005
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Habib Zribi, Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions, J. Differential Equations 247 (2009), no. 11, 2897–2912. MR 2569851, DOI https://doi.org/10.1016/j.jde.2009.08.005
- Ivo Babuška, Börje Andersson, Paul J. Smith, and Klas Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1-4, 27–77. MR 1685902, DOI https://doi.org/10.1016/S0045-7825%2898%2900225-4
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. MR 2506075, DOI https://doi.org/10.1007/s00205-008-0159-8
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. MR 2754076, DOI https://doi.org/10.1080/03605300903564000
- Jiguang Bao, Hongjie Ju, and Haigang Li, Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients, Adv. Math. 314 (2017), 583–629. MR 3658726, DOI https://doi.org/10.1016/j.aim.2017.05.004
- JiGuang Bao, HaiGang Li, and YanYan Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients, Arch. Ration. Mech. Anal. 215 (2015), no. 1, 307–351. MR 3296149, DOI https://doi.org/10.1007/s00205-014-0779-0
- JiGuang Bao, HaiGang Li, and YanYan Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two, Adv. Math. 305 (2017), 298–338. MR 3570137, DOI https://doi.org/10.1016/j.aim.2016.09.023
- Eric Bonnetier and Faouzi Triki, Asymptotics in the presence of inclusions of small volume for a conduction equation: a case with a non-smooth reference potential, Imaging microstructures, Contemp. Math., vol. 494, Amer. Math. Soc., Providence, RI, 2009, pp. 95–111. MR 2581768, DOI https://doi.org/10.1090/conm/494/09645
- Eric Bonnetier and Faouzi Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal. 209 (2013), no. 2, 541–567. MR 3056617, DOI https://doi.org/10.1007/s00205-013-0636-6
- Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481, DOI https://doi.org/10.1137/S0036141098333980
- B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech. 51 (1984), 733-735.
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9) 99 (2013), no. 2, 234–249. MR 3007847, DOI https://doi.org/10.1016/j.matpur.2012.06.013
- Hyeonbae Kang, Hyundae Lee, and KiHyun Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann. 363 (2015), no. 3-4, 1281–1306. MR 3412359, DOI https://doi.org/10.1007/s00208-015-1203-2
- J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech. 60 (1993), 1054-1056.
- J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 3 (1963), 991-993.
- Haigang Li, Yanyan Li, Ellen Shiting Bao, and Biao Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), no. 3, 589–596. MR 3237564, DOI https://doi.org/10.1090/S0033-569X-2014-01339-0
- Haigang Li and Longjuan Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal. 49 (2017), no. 4, 3125–3142. MR 3686796, DOI https://doi.org/10.1137/16M1067858
- Yanyan Li and Louis Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. Dedicated to the memory of Jürgen K. Moser. MR 1990481, DOI https://doi.org/10.1002/cpa.10079
- Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI https://doi.org/10.1007/s002050000082
- Mikyoung Lim and Kihyun Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1287–1315. MR 2581974, DOI https://doi.org/10.1080/03605300903079579
- X. Markenscoff, Stress amplification in vanishingly small geometries, Comput. Mech. 19 (1996), 77-83.
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
- Kihyun Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730. MR 2300307, DOI https://doi.org/10.1137/060648817
- KiHyun Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. MR 2476915, DOI https://doi.org/10.1016/j.jmaa.2008.09.057
References
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727. MR 0125307, DOI https://doi.org/10.1002/cpa.3160120405
- S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35–92. MR 0162050, DOI https://doi.org/10.1002/cpa.3160170104
- Habib Ammari, Eric Bonnetier, Faouzi Triki, and Michael Vogelius, Elliptic estimates in composite media with smooth inclusions: an integral equation approach, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015), no. 2, 453–495 (English, with English and French summaries). MR 3346176, DOI https://doi.org/10.24033/asens.2249
- Habib Ammari, Giulio Ciraolo, Hyeonbae Kang, Hyundae Lee, and Kihyun Yun, Spectral analysis of the Neumann-Poincaré operator and characterization of the stress concentration in anti-plane elasticity, Arch. Ration. Mech. Anal. 208 (2013), no. 1, 275–304. MR 3021549, DOI https://doi.org/10.1007/s00205-012-0590-8
- Habib Ammari, George Dassios, Hyeonbae Kang, and Mikyoung Lim, Estimates for the electric field in the presence of adjacent perfectly conducting spheres, Quart. Appl. Math. 65 (2007), no. 2, 339–355. MR 2330561, DOI https://doi.org/10.1090/S0033-569X-07-01034-1
- Habib Ammari, Hyeonbae Kang, and Mikyoung Lim, Gradient estimates for solutions to the conductivity problem, Math. Ann. 332 (2005), no. 2, 277–286. MR 2178063, DOI https://doi.org/10.1007/s00208-004-0626-y
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Jungwook Lee, and Mikyoung Lim, Optimal estimates for the electric field in two dimensions, J. Math. Pures Appl. (9) 88 (2007), no. 4, 307–324 (English, with English and French summaries). MR 2384571, DOI https://doi.org/10.1016/j.matpur.2007.07.005
- Habib Ammari, Hyeonbae Kang, Hyundae Lee, Mikyoung Lim, and Habib Zribi, Decomposition theorems and fine estimates for electrical fields in the presence of closely located circular inclusions, J. Differential Equations 247 (2009), no. 11, 2897–2912. MR 2569851, DOI https://doi.org/10.1016/j.jde.2009.08.005
- Ivo Babuška, Börje Andersson, Paul J. Smith, and Klas Levin, Damage analysis of fiber composites. I. Statistical analysis on fiber scale, Comput. Methods Appl. Mech. Engrg. 172 (1999), no. 1-4, 27–77. MR 1685902, DOI https://doi.org/10.1016/S0045-7825%2898%2900225-4
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect conductivity problem, Arch. Ration. Mech. Anal. 193 (2009), no. 1, 195–226. MR 2506075, DOI https://doi.org/10.1007/s00205-008-0159-8
- Ellen Shiting Bao, Yan Yan Li, and Biao Yin, Gradient estimates for the perfect and insulated conductivity problems with multiple inclusions, Comm. Partial Differential Equations 35 (2010), no. 11, 1982–2006. MR 2754076, DOI https://doi.org/10.1080/03605300903564000
- Jiguang Bao, Hongjie Ju, and Haigang Li, Optimal boundary gradient estimates for Lamé systems with partially infinite coefficients, Adv. Math. 314 (2017), 583–629. MR 3658726, DOI https://doi.org/10.1016/j.aim.2017.05.004
- JiGuang Bao, HaiGang Li, and YanYan Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients, Arch. Ration. Mech. Anal. 215 (2015), no. 1, 307–351. MR 3296149, DOI https://doi.org/10.1007/s00205-014-0779-0
- JiGuang Bao, HaiGang Li, and YanYan Li, Gradient estimates for solutions of the Lamé system with partially infinite coefficients in dimensions greater than two, Adv. Math. 305 (2017), 298–338. MR 3570137, DOI https://doi.org/10.1016/j.aim.2016.09.023
- Eric Bonnetier and Faouzi Triki, Asymptotics in the presence of inclusions of small volume for a conduction equation: a case with a non-smooth reference potential, Imaging microstructures, Contemp. Math., vol. 494, Amer. Math. Soc., Providence, RI, 2009, pp. 95–111. MR 2581768, DOI https://doi.org/10.1090/conm/494/09645
- Eric Bonnetier and Faouzi Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D, Arch. Ration. Mech. Anal. 209 (2013), no. 2, 541–567. MR 3056617, DOI https://doi.org/10.1007/s00205-013-0636-6
- Eric Bonnetier and Michael Vogelius, An elliptic regularity result for a composite medium with “touching” fibers of circular cross-section, SIAM J. Math. Anal. 31 (2000), no. 3, 651–677. MR 1745481, DOI https://doi.org/10.1137/S0036141098333980
- B. Budiansky and G. F. Carrier, High shear stresses in stiff fiber composites, J. Appl. Mech. 51 (1984), 733-735.
- Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Asymptotics and computation of the solution to the conductivity equation in the presence of adjacent inclusions with extreme conductivities, J. Math. Pures Appl. (9) 99 (2013), no. 2, 234–249. MR 3007847, DOI https://doi.org/10.1016/j.matpur.2012.06.013
- Hyeonbae Kang, Hyundae Lee, and KiHyun Yun, Optimal estimates and asymptotics for the stress concentration between closely located stiff inclusions, Math. Ann. 363 (2015), no. 3-4, 1281–1306. MR 3412359, DOI https://doi.org/10.1007/s00208-015-1203-2
- J. B. Keller, Stresses in narrow regions, Trans. ASME J. Appl. Mech. 60 (1993), 1054-1056.
- J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 3 (1963), 991-993.
- Haigang Li, Yanyan Li, Ellen Shiting Bao, and Biao Yin, Derivative estimates of solutions of elliptic systems in narrow regions, Quart. Appl. Math. 72 (2014), no. 3, 589–596. MR 3237564, DOI https://doi.org/10.1090/S0033-569X-2014-01339-0
- Haigang Li and Longjuan Xu, Optimal estimates for the perfect conductivity problem with inclusions close to the boundary, SIAM J. Math. Anal. 49 (2017), no. 4, 3125–3142. MR 3686796, DOI https://doi.org/10.1137/16M1067858
- Yanyan Li and Louis Nirenberg, Estimates for elliptic systems from composite material, Comm. Pure Appl. Math. 56 (2003), no. 7, 892–925. Dedicated to the memory of Jürgen K. Moser. MR 1990481, DOI https://doi.org/10.1002/cpa.10079
- Yan Yan Li and Michael Vogelius, Gradient estimates for solutions to divergence form elliptic equations with discontinuous coefficients, Arch. Ration. Mech. Anal. 153 (2000), no. 2, 91–151. MR 1770682, DOI https://doi.org/10.1007/s002050000082
- Mikyoung Lim and Kihyun Yun, Blow-up of electric fields between closely spaced spherical perfect conductors, Comm. Partial Differential Equations 34 (2009), no. 10-12, 1287–1315. MR 2581974, DOI https://doi.org/10.1080/03605300903079579
- X. Markenscoff, Stress amplification in vanishingly small geometries, Comput. Mech. 19 (1996), 77-83.
- O. A. Oleĭnik, A. S. Shamaev, and G. A. Yosifian, Mathematical problems in elasticity and homogenization, Studies in Mathematics and its Applications, vol. 26, North-Holland Publishing Co., Amsterdam, 1992. MR 1195131
- Kihyun Yun, Estimates for electric fields blown up between closely adjacent conductors with arbitrary shape, SIAM J. Appl. Math. 67 (2007), no. 3, 714–730. MR 2300307, DOI https://doi.org/10.1137/060648817
- KiHyun Yun, Optimal bound on high stresses occurring between stiff fibers with arbitrary shaped cross-sections, J. Math. Anal. Appl. 350 (2009), no. 1, 306–312. MR 2476915, DOI https://doi.org/10.1016/j.jmaa.2008.09.057
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Additional Information
Hongjie Ju
Affiliation:
School of Sciences, Beijing University of Posts and Telecommunications, Beijing 100876, People’s Republic of China
Email:
hjju@bupt.edu.cn
Haigang Li
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
MR Author ID:
928017
Email:
hgli@bnu.edu.cn
Longjuan Xu
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
MR Author ID:
1225839
Email:
ljxu@mail.bnu.edu.cn
Received by editor(s):
August 9, 2017
Received by editor(s) in revised form:
July 3, 2018
Published electronically:
September 5, 2018
Additional Notes:
The first author was partially supported by NSFC (11301034) (11471050).
The second author was partially supported by NSFC (11571042) (11371060) (11631002), Fok Ying Tung Education Foundation (151003). Corresponding author.
Article copyright:
© Copyright 2018
Brown University
