Upper and lower bounds for stress concentration in linear elasticity when 𝐶^{1,𝛼} inclusions are close to boundary

Chen, Yu; Hao, Xia; Xu, Longjuan

doi:10.1090/qam/1621

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Upper and lower bounds for stress concentration in linear elasticity when $C^{1, \alpha }$ inclusions are close to boundary

Authors: Yu Chen, Xia Hao and Longjuan Xu
Journal: Quart. Appl. Math. 80 (2022), 607-639
MSC (2020): Primary 35J47, 35B44, 35J25
DOI: https://doi.org/10.1090/qam/1621
Published electronically: March 21, 2022
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Abstract: We establish boundary gradient estimates for both Lamé systems with partially infinite coefficients and perfect conductivity problem. The inclusion and the matrix domain are both assumed to be of $C^{1, \alpha }$, weaker than $C^{2, \alpha }$ assumptions in the previous work by Bao-Ju-Li [Adv. Math. 314 (2017), pp. 583–629]. When the inclusion is located close to the boundary of matrix domain, we give the specific examples of boundary data to obtain the lower bound gradient estimates in all dimensions, which guarantee the blow-up occurs and indicate that the blow-up rates of the gradients with respect to the distance between the interfacial surfaces are optimal.

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
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, 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, 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 10.1090/S0033-569X-07-01034-1
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
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
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
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 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 10.1137/S0036141098333980
Yu Chen, Haigang Li, and Longjuan Xu, Optimal gradient estimates for the perfect conductivity problem with $C^{1, \alpha }$ inclusions, Ann. Inst. H. Poincaré C Anal. Non Linéaire 38 (2021), no. 4, 953–979. MR 4266231, DOI 10.1016/j.anihpc.2020.09.009
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
Hongjie Dong, Gradient estimates for parabolic and elliptic systems from linear laminates, Arch. Ration. Mech. Anal. 205 (2012), no. 1, 119–149. MR 2927619, DOI 10.1007/s00205-012-0501-z
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
Hongjie Dong and Hong Zhang, On an elliptic equation arising from composite materials, Arch. Ration. Mech. Anal. 222 (2016), no. 1, 47–89. MR 3519966, DOI 10.1007/s00205-016-0996-9
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
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
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
Y. Y. Hou and H. G. Li, The convexity of inclusions and gradient’s concentration for the Lamé systems with partially infinite coefficients, arXiv: 1802.01412v1, 2018.
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, 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 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
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
J. B. Keller, Conductivity of a medium containing a dense array of perfectly conducting spheres or cylinders or nonconducting cylinders, J. Appl. Phys. 34 (1963), 991–993.
Haigang Li, Lower bounds of gradient’s blow-up for the Lamé system with partially infinite coefficients, J. Math. Pures Appl. (9) 149 (2021), 98–134 (English, with English and French summaries). MR 4238998, DOI 10.1016/j.matpur.2020.09.007
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
H. G. Li and Y. Li, An extension of Flaherty-Keller formula for density packed m-convex inclusion, arXiv:1912.13261v1, 2019
HaiGang Li and YanYan Li, Gradient estimates for parabolic systems from composite material, Sci. China Math. 60 (2017), no. 11, 2011–2052. MR 3714566, DOI 10.1007/s11425-017-9153-0
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, 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 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 10.1137/16M1067858
Haigang Li, Fang Wang, and Longjuan Xu, Characterization of electric fields between two spherical perfect conductors with general radii in 3D, J. Differential Equations 267 (2019), no. 11, 6644–6690. MR 4001067, DOI 10.1016/j.jde.2019.07.007
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
Y. Y. Li and Z. L. Yang, Gradient estimates of solutions to the insulated conductivity problem in dimension greater than two, arXiv:2012.14056, 2020.
M. Lim and S. Yu, Stress concentration for two nearly touching circular holes, arXiv:1705.10400v1, 2017.
Mikyoung Lim and Sanghyeon Yu, Asymptotics of the solution to the conductivity equation in the presence of adjacent circular inclusions with finite conductivities, J. Math. Anal. Appl. 421 (2015), no. 1, 131–156. MR 3250470, DOI 10.1016/j.jmaa.2014.07.002
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, N. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains, Springer Science and Business Media, 2000.
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
KiHyun Yun, An optimal estimate for electric fields on the shortest line segment between two spherical insulators in three dimensions, J. Differential Equations 261 (2016), no. 1, 148–188. MR 3487255, DOI 10.1016/j.jde.2016.03.005