Singularities of the stress concentration in the presence of $C^{1,\alpha }$-inclusions with core-shell geometry
Authors:
Xia Hao and Zhiwen Zhao
Journal:
Quart. Appl. Math. 81 (2023), 203-243
MSC (2020):
Primary 78A48, 35Q74; Secondary 35B44, 35C20, 35J25
DOI:
https://doi.org/10.1090/qam/1634
Published electronically:
September 28, 2022
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Abstract: In high-contrast composites, if an inclusion is in close proximity to the matrix boundary, then the stress, which is represented by the gradient of a solution to the Lamé systems of linear elasticity, may exhibit the singularities with respect to the distance $\varepsilon$ between them. In this paper, we establish the asymptotic formulas of the stress concentration for core-shell geometry with $C^{1,\alpha }$ boundaries in all dimensions by precisely capturing all the blow-up factor matrices, as the distance $\varepsilon$ between interfacial boundaries of a core and a surrounding shell goes to zero. Further, a direct application of these blow-up factor matrices gives the optimal gradient estimates.
References
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- 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 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 10.1007/s00205-014-0779-0
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- 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 10.1080/03605300903564000
- 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 10.1137/S0036141098333980
- Yu Chen and Haigang Li, Estimates and asymptotics for the stress concentration between closely spaced stiff $C ^{1,\gamma }$ inclusions in linear elasticity, J. Funct. Anal. 281 (2021), no. 2, Paper No. 109038, 63. MR 4242965, DOI 10.1016/j.jfa.2021.109038
- Giulio Ciraolo and Angela Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl. (9) 127 (2019), 268–298 (English, with English and French summaries). MR 3960144, DOI 10.1016/j.matpur.2018.09.006
- Giulio Ciraolo and Angela Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differential Equations 266 (2019), no. 9, 6149–6178. MR 3912777, DOI 10.1016/j.jde.2018.10.041
- Hongjie Dong and Haigang Li, Optimal estimates for the conductivity problem by Green’s function method, Arch. Ration. Mech. Anal. 231 (2019), no. 3, 1427–1453. MR 3902466, DOI 10.1007/s00205-018-1301-x
- Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math. 58 (2005), no. 1, 84–106. MR 2136971, DOI 10.1093/qjmamj/hbh022
- Yuliya Gorb and Alexei Novikov, Blow-up of solutions to a $p$-Laplace equation, Multiscale Model. Simul. 10 (2012), no. 3, 727–743. MR 3022019, DOI 10.1137/110857167
- Yuliya Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul. 13 (2015), no. 4, 1312–1326. MR 3418221, DOI 10.1137/140982076
- M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Springer Science Business Media, 2013.
- D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 1998.
- Xia Hao and Zhiwen Zhao, The asymptotics for the perfect conductivity problem with stiff $C^{1,\alpha }$-inclusions, J. Math. Anal. Appl. 501 (2021), no. 2, Paper No. 125201, 27. MR 4239007, DOI 10.1016/j.jmaa.2021.125201
- 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 10.1016/j.matpur.2012.06.013
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math. 74 (2014), no. 1, 125–146. MR 3162415, DOI 10.1137/130922434
- 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 10.1007/s00208-015-1203-2
- Hyeonbae Kang and Sanghyeon Yu, Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity, Arch. Ration. Mech. Anal. 232 (2019), no. 1, 121–196. MR 3916973, DOI 10.1007/s00205-018-1318-1
- Hyeonbae Kang and Sanghyeon Yu, A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 22, 13. MR 4048331, DOI 10.1007/s00526-019-1692-z
- Hyeonbae Kang and KiHyun Yun, Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions, J. Differential Equations 266 (2019), no. 8, 5064–5094. MR 3912742, DOI 10.1016/j.jde.2018.10.018
- Junbeom Kim and Mikyoung Lim, Electric field concentration in the presence of an inclusion with eccentric core-shell geometry, Math. Ann. 373 (2019), no. 1-2, 517–551. MR 3968879, DOI 10.1007/s00208-018-1688-6
- 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 10.1090/S0033-569X-2014-01339-0
- Haigang Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal. 52 (2020), no. 4, 3350–3375. MR 4126320, DOI 10.1137/19M1282623
- HaiGang Li, YanYan Li, and ZhuoLun Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul. 17 (2019), no. 3, 899–925. MR 3977105, DOI 10.1137/18M1214329
- Haigang Li and Zhiwen Zhao, Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of $m$-convex hard inclusions, SIAM J. Math. Anal. 52 (2020), no. 4, 3777–3817. MR 4134031, DOI 10.1137/19M1306038
- 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 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 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 10.1080/03605300903079579
- V. G. Maz′ya, A. B. Movchan, and M. J. Nieves, Uniform asymptotic formulae for Green’s tensors in elastic singularly perturbed domains, Asymptot. Anal. 52 (2007), no. 3-4, 173–206. MR 2339953
- C. X. Miao and Z. W. Zhao, Singular analysis of the stress concentration in the narrow regions between the inclusions and the matrix boundary, arXiv:2109.04394.
- 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 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 10.1016/j.jmaa.2008.09.057
- Zhiwen Zhao and Xia Hao, Asymptotics for the concentrated field between closely located hard inclusions in all dimensions, Commun. Pure Appl. Anal. 20 (2021), no. 6, 2379–2398. MR 4274389, DOI 10.3934/cpaa.2021086
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 125307, DOI 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 162050, DOI 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 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 10.1007/s00205-012-0590-8
- 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 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 10.1016/j.matpur.2007.07.005
- 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 10.1016/S0045-7825(98)00225-4
- 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 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 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 10.1016/j.aim.2016.09.023
- 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 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 10.1080/03605300903564000
- 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 10.1137/S0036141098333980
- Yu Chen and Haigang Li, Estimates and asymptotics for the stress concentration between closely spaced stiff $C ^{1,\gamma }$ inclusions in linear elasticity, J. Funct. Anal. 281 (2021), no. 2, Paper No. 109038, 63. MR 4242965, DOI 10.1016/j.jfa.2021.109038
- Giulio Ciraolo and Angela Sciammetta, Gradient estimates for the perfect conductivity problem in anisotropic media, J. Math. Pures Appl. (9) 127 (2019), 268–298 (English, with English and French summaries). MR 3960144, DOI 10.1016/j.matpur.2018.09.006
- Giulio Ciraolo and Angela Sciammetta, Stress concentration for closely located inclusions in nonlinear perfect conductivity problems, J. Differential Equations 266 (2019), no. 9, 6149–6178. MR 3912777, DOI 10.1016/j.jde.2018.10.041
- Hongjie Dong and Haigang Li, Optimal estimates for the conductivity problem by Green’s function method, Arch. Ration. Mech. Anal. 231 (2019), no. 3, 1427–1453. MR 3902466, DOI 10.1007/s00205-018-1301-x
- Y. Gorb and L. Berlyand, Asymptotics of the effective conductivity of composites with closely spaced inclusions of optimal shape, Quart. J. Mech. Appl. Math. 58 (2005), no. 1, 84–106. MR 2136971, DOI 10.1093/qjmamj/hbh022
- Yuliya Gorb and Alexei Novikov, Blow-up of solutions to a $p$-Laplace equation, Multiscale Model. Simul. 10 (2012), no. 3, 727–743. MR 3022019, DOI 10.1137/110857167
- Yuliya Gorb, Singular behavior of electric field of high-contrast concentrated composites, Multiscale Model. Simul. 13 (2015), no. 4, 1312–1326. MR 3418221, DOI 10.1137/140982076
- M. Giaquinta and L. Martinazzi, An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs, Springer Science Business Media, 2013.
- D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order, Springer, 1998.
- Xia Hao and Zhiwen Zhao, The asymptotics for the perfect conductivity problem with stiff $C^{1,\alpha }$-inclusions, J. Math. Anal. Appl. 501 (2021), no. 2, Paper No. 125201, 27. MR 4239007, DOI 10.1016/j.jmaa.2021.125201
- 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 10.1016/j.matpur.2012.06.013
- Hyeonbae Kang, Mikyoung Lim, and KiHyun Yun, Characterization of the electric field concentration between two adjacent spherical perfect conductors, SIAM J. Appl. Math. 74 (2014), no. 1, 125–146. MR 3162415, DOI 10.1137/130922434
- 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 10.1007/s00208-015-1203-2
- Hyeonbae Kang and Sanghyeon Yu, Quantitative characterization of stress concentration in the presence of closely spaced hard inclusions in two-dimensional linear elasticity, Arch. Ration. Mech. Anal. 232 (2019), no. 1, 121–196. MR 3916973, DOI 10.1007/s00205-018-1318-1
- Hyeonbae Kang and Sanghyeon Yu, A proof of the Flaherty-Keller formula on the effective property of densely packed elastic composites, Calc. Var. Partial Differential Equations 59 (2020), no. 1, Paper No. 22, 13. MR 4048331, DOI 10.1007/s00526-019-1692-z
- Hyeonbae Kang and KiHyun Yun, Optimal estimates of the field enhancement in presence of a bow-tie structure of perfectly conducting inclusions in two dimensions, J. Differential Equations 266 (2019), no. 8, 5064–5094. MR 3912742, DOI 10.1016/j.jde.2018.10.018
- Junbeom Kim and Mikyoung Lim, Electric field concentration in the presence of an inclusion with eccentric core-shell geometry, Math. Ann. 373 (2019), no. 1-2, 517–551. MR 3968879, DOI 10.1007/s00208-018-1688-6
- 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 10.1090/S0033-569X-2014-01339-0
- Haigang Li, Asymptotics for the electric field concentration in the perfect conductivity problem, SIAM J. Math. Anal. 52 (2020), no. 4, 3350–3375. MR 4126320, DOI 10.1137/19M1282623
- HaiGang Li, YanYan Li, and ZhuoLun Yang, Asymptotics of the gradient of solutions to the perfect conductivity problem, Multiscale Model. Simul. 17 (2019), no. 3, 899–925. MR 3977105, DOI 10.1137/18M1214329
- Haigang Li and Zhiwen Zhao, Boundary blow-up analysis of gradient estimates for Lamé systems in the presence of $m$-convex hard inclusions, SIAM J. Math. Anal. 52 (2020), no. 4, 3777–3817. MR 4134031, DOI 10.1137/19M1306038
- 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 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 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 10.1080/03605300903079579
- V. G. Maz′ya, A. B. Movchan, and M. J. Nieves, Uniform asymptotic formulae for Green’s tensors in elastic singularly perturbed domains, Asymptot. Anal. 52 (2007), no. 3-4, 173–206. MR 2339953
- C. X. Miao and Z. W. Zhao, Singular analysis of the stress concentration in the narrow regions between the inclusions and the matrix boundary, arXiv:2109.04394.
- 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 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 10.1016/j.jmaa.2008.09.057
- Zhiwen Zhao and Xia Hao, Asymptotics for the concentrated field between closely located hard inclusions in all dimensions, Commun. Pure Appl. Anal. 20 (2021), no. 6, 2379–2398. MR 4274389, DOI 10.3934/cpaa.2021086
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Additional Information
Xia Hao
Affiliation:
Department of Mathematics Science, Hebei Normal University, Shijiazhuang, Hebei 050024, People’s Republic of China
ORCID:
0000-0003-1911-5118
Email:
xiahao0915@163.com
Zhiwen Zhao
Affiliation:
Beijing Computational Science Research Center, Beijing 100193, People’s Republic of China
ORCID:
0000-0002-9708-1198
Email:
zwzhao365@163.com
Received by editor(s):
August 3, 2022
Received by editor(s) in revised form:
September 4, 2022
Published electronically:
September 28, 2022
Additional Notes:
Zhiwen Zhao is the corresponding author. The second author was partially supported by CPSF (2021M700358).
Article copyright:
© Copyright 2022
Brown University
