doi: 10.3934/nhm.2021023
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Asymptotic analysis of an elastic material reinforced with thin fractal strips

Laboratory of Mathematics and Applications, Abdelmalek Essaâdi University, FST Tangier, B.P. 416 Tangier, Morocco

* Corresponding author: M. El Jarroudi

Received  February 2021 Revised  August 2021 Early access November 2021

We study the asymptotic behavior of a three-dimensional elastic material reinforced with highly contrasted thin vertical strips constructed on horizontal iterated Sierpinski gasket curves. We use $ \Gamma $-convergence methods in order to study the asymptotic behavior of the composite as the thickness of the strips vanishes, their Lamé constants tend to infinity, and the sequence of the iterated curves converges to the Sierpinski gasket in the Hausdorff metric. We derive the effective energy of the composite. This energy contains new degrees of freedom implying a nonlocal effect associated with thin boundary layer phenomena taking place near the fractal strips and a singular energy term supported on the Sierpinski gasket.

Citation: Mustapha El Jarroudi, Youness Filali, Aadil Lahrouz, Mustapha Er-Riani, Adel Settati. Asymptotic analysis of an elastic material reinforced with thin fractal strips. Networks & Heterogeneous Media, doi: 10.3934/nhm.2021023
References:
[1]

J. E. Adkins, Finite plane deformations of thin elastic sheets reinforced with inextensible cords, Philos. Trans. R. Soc. London A, 249 (1956), 125-150.  doi: 10.1098/rsta.1956.0017.  Google Scholar

[2]

J. E. Adkins, Cylindrically symmetrical deformations of incompressible elastic materials reinforced with inextensible cords, J. Ration. Mech. Anal., 5 (1956), 189-202.  doi: 10.1512/iumj.1956.5.55005.  Google Scholar

[3]

J. E. Adkins, A three-dimensional problem for highly elastic materials subject to constraints, Q. J. Mech. Appl. Math., 11 (1958), 88-97.  doi: 10.1093/qjmam/11.1.88.  Google Scholar

[4]

J. E. Adkins and R. S. Rivlin, Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords, Philos. Trans. R. Soc. London A, 248 (1955), 201-223.  doi: 10.1098/rsta.1955.0014.  Google Scholar

[5]

H. Attouch, Variational Convergence for Functions and Operators, Appl. Math. Series. London, Pitman, 1984.  Google Scholar

[6]

M. Bellieud and G. Bouchitté, Homogenization of a soft elastic material reinforced by fibers, Asymptotic Anal, 32 (2002), 153-183.   Google Scholar

[7]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3. Springer, New York, 2006. doi: 10.1007/978-0-387-31256-9.  Google Scholar

[8]

B. E. Breckner and C. Varga, Elliptic problems on the Sierpinski gasket, Topics in Mathematical Analysis and Applications. Springer Optim. Appl., 94 (2014), 119-173.  doi: 10.1007/978-3-319-06554-0_6.  Google Scholar

[9]

D. Caillerie, The effect of a thin inclusion of high rigidity in an elastic body, Math. Methods Appl. Sci., 2 (1980), 251-270.   Google Scholar

[10]

M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals,, Arch. Ration. Mech. Anal., 170 (2003), 211-245.  doi: 10.1007/s00205-003-0272-7.  Google Scholar

[11]

J. R. Cannon and G. H. Meyer, On diffusion in a fractured medium, Siam J. Appl. Math., 20 (1971), 434-448.  doi: 10.1137/0120047.  Google Scholar

[12]

R. CapitanelliM. R. Lancia and M. A. Vivaldi, Insulating layers of fractal type, Differ. Integ. Equs, 26 (2013), 1055-1076.   Google Scholar

[13]

R. Capitanelli and M. A. Vivaldi, Reinforcement problems for variational inequalities on fractal sets, Calc. Var. Partial Differential Equations, 54 (2015), 2751-2783.  doi: 10.1007/s00526-015-0882-6.  Google Scholar

[14]

R. Capitanelli and M. A. Vivaldi, Dynamical quasi-filling fractal layers, SIAM J. Math. Anal., 48 (2016), 3931-3961.  doi: 10.1137/15M1043893.  Google Scholar

[15]

M. CefaloM. R. Lancia and H. Liang, Heat-flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential Integral Equations, 26 (2013), 1027-1054.   Google Scholar

[16]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, PNLDEA 8, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[17]

M. EI Jarroudi, Homogenization of a nonlinear elastic fibre-reinforced composite: A second gradient nonlinear elastic material, J. Math. Anal. Appl., 403 (2013), 487-505.  doi: 10.1016/j.jmaa.2013.02.042.  Google Scholar

[18]

M. El Jarroudi, Homogenization of an elastic material reinforced with thin rigid von Kármán ribbons, Math. Mech. Solids, 24 (2019), 1965-1991.  doi: 10.1177/1081286518810757.  Google Scholar

[19]

M. El Jarroudi, Homogenization of a quasilinear elliptic problem in a fractal-reinforced structure, SeMA, (2021). doi: 10.1007/s40324-021-00250-5.  Google Scholar

[20]

M. El Jarroudi and M. Er-Riani, Homogenization of elastic materials containing self-similar rigid micro-inclusions, Contin. Mech. Thermodyn., 31 (2019), 457-474.  doi: 10.1007/s00161-018-0700-4.  Google Scholar

[21]

M. El JarroudiM. Er-RianiA. Lahrouz and A. Settati, Homogenization of elastic materials reinforced by rigid notched fibres, Appl. Anal., 97 (2018), 705-738.  doi: 10.1080/00036811.2017.1285015.  Google Scholar

[22]

K. Falconer, Techniques in Fractal Geometry, J. Wiley and Sons, Chichester, 1997.  Google Scholar

[23]

U. R. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen, 23 (2004), 115-137.  doi: 10.4171/ZAA/1190.  Google Scholar

[24]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110889741.  Google Scholar

[25]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket,, Potential Anal., 1 (1992), 1-35.  doi: 10.1007/BF00249784.  Google Scholar

[26]

P. H Hung and E. Sanchez-Palencia, Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47 (1974), 284-309.  doi: 10.1016/0022-247X(74)90023-7.  Google Scholar

[27]

A. Jonsson and H. Wallin, Boundary value problems and Brownian motion on fractals, Chaos Solitons Fractals, 8 (1997), 191-205.  doi: 10.1016/S0960-0779(96)00048-3.  Google Scholar

[28]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia. Math., 112 (1995), 285-300.   Google Scholar

[29]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[30]

S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357.  doi: 10.1007/BF02096647.  Google Scholar

[31]

M. R. LanciaU. Mosco and M. A. Vivaldi, Homogenization for conductive thin layers of pre-fractal type, J. Math. Anal. Appl., 347 (2008), 354-369.  doi: 10.1016/j.jmaa.2008.06.011.  Google Scholar

[32]

M. R. LanciaM. Cefalo and G. DellÁcqua, Numerical approximation of transmission problems across Koch-type highly conductive layers, Appl. Math. Comput., 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033.  Google Scholar

[33]

Y. Le Jean, Measures associées á une forme de Dirichlet, Appl., Bull. Soc. Math., 106 (1978), 61-112.   Google Scholar

[34]

M. Lobo and E. Perez, Boundary homogenization of certain elliptic problems for cylindrical bodies, Bull Sci Math., 116 (1992), 399-426.   Google Scholar

[35]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[36]

U. Mosco, Variational fractals,, Ann. Scuola Norm. Sup. Pisa, Special Volume in Memory of E. De Giorgi, 25 (1997), 683-712.   Google Scholar

[37]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., Amer. Math. Soc., 54 (1998), 301-323.  doi: 10.1090/psapm/054/1492702.  Google Scholar

[38]

U. Mosco, Energy functionals on certain fractal structures, J. Conv. Anal., 9 (2002), 581-600.   Google Scholar

[39]

U. Mosco and M. A. Vivaldi, An example of fractal singular homogenization, Georgian Math. J., 14 (2007), 169-193.  doi: 10.1515/GMJ.2007.169.  Google Scholar

[40]

U. Mosco and M. A. Vivaldi, Fractal reinforcement of elastic membranes, Arch. Ration. Mech. Anal., 194 (2009), 49-74.  doi: 10.1007/s00205-008-0145-1.  Google Scholar

[41]

U. Mosco and M. A. Vivaldi, Thin fractal fibers, Math. Meth. Appl. Sci., 36 (2013), 2048-2068.  doi: 10.1002/mma.1621.  Google Scholar

[42]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials,, J. Math. Pures Appl., 103 (2015), 1198-1227.  doi: 10.1016/j.matpur.2014.10.010.  Google Scholar

[43]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[44]

R. S. Rivlin, Plane strain of a net formed by inextensible cords, J. Rational Mech. Anal., 4 (1955), 951-974.  doi: 10.1512/iumj.1955.4.54037.  Google Scholar

[45]

R. S. Rivlin, The deformation of a membrane formed by inextensible cords, Arch. Rational Mech. Anal., 2 (1958), 447-476.  doi: 10.1007/BF00277942.  Google Scholar

[46]

R. S. Strichartz and C. Wong, The p-Laplacian on the Sierpinski gasket, Nonlinearity, 17 (2004), 595-616.  doi: 10.1088/0951-7715/17/2/014.  Google Scholar

show all references

References:
[1]

J. E. Adkins, Finite plane deformations of thin elastic sheets reinforced with inextensible cords, Philos. Trans. R. Soc. London A, 249 (1956), 125-150.  doi: 10.1098/rsta.1956.0017.  Google Scholar

[2]

J. E. Adkins, Cylindrically symmetrical deformations of incompressible elastic materials reinforced with inextensible cords, J. Ration. Mech. Anal., 5 (1956), 189-202.  doi: 10.1512/iumj.1956.5.55005.  Google Scholar

[3]

J. E. Adkins, A three-dimensional problem for highly elastic materials subject to constraints, Q. J. Mech. Appl. Math., 11 (1958), 88-97.  doi: 10.1093/qjmam/11.1.88.  Google Scholar

[4]

J. E. Adkins and R. S. Rivlin, Large elastic deformations of isotropic materials X. Reinforcement by inextensible cords, Philos. Trans. R. Soc. London A, 248 (1955), 201-223.  doi: 10.1098/rsta.1955.0014.  Google Scholar

[5]

H. Attouch, Variational Convergence for Functions and Operators, Appl. Math. Series. London, Pitman, 1984.  Google Scholar

[6]

M. Bellieud and G. Bouchitté, Homogenization of a soft elastic material reinforced by fibers, Asymptotic Anal, 32 (2002), 153-183.   Google Scholar

[7]

J. M. Borwein and A. S. Lewis, Convex Analysis and Nonlinear Optimization. Theory and Examples, 2$^{nd}$ edition, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 3. Springer, New York, 2006. doi: 10.1007/978-0-387-31256-9.  Google Scholar

[8]

B. E. Breckner and C. Varga, Elliptic problems on the Sierpinski gasket, Topics in Mathematical Analysis and Applications. Springer Optim. Appl., 94 (2014), 119-173.  doi: 10.1007/978-3-319-06554-0_6.  Google Scholar

[9]

D. Caillerie, The effect of a thin inclusion of high rigidity in an elastic body, Math. Methods Appl. Sci., 2 (1980), 251-270.   Google Scholar

[10]

M. Camar-Eddine and P. Seppecher, Determination of the closure of the set of elasticity functionals,, Arch. Ration. Mech. Anal., 170 (2003), 211-245.  doi: 10.1007/s00205-003-0272-7.  Google Scholar

[11]

J. R. Cannon and G. H. Meyer, On diffusion in a fractured medium, Siam J. Appl. Math., 20 (1971), 434-448.  doi: 10.1137/0120047.  Google Scholar

[12]

R. CapitanelliM. R. Lancia and M. A. Vivaldi, Insulating layers of fractal type, Differ. Integ. Equs, 26 (2013), 1055-1076.   Google Scholar

[13]

R. Capitanelli and M. A. Vivaldi, Reinforcement problems for variational inequalities on fractal sets, Calc. Var. Partial Differential Equations, 54 (2015), 2751-2783.  doi: 10.1007/s00526-015-0882-6.  Google Scholar

[14]

R. Capitanelli and M. A. Vivaldi, Dynamical quasi-filling fractal layers, SIAM J. Math. Anal., 48 (2016), 3931-3961.  doi: 10.1137/15M1043893.  Google Scholar

[15]

M. CefaloM. R. Lancia and H. Liang, Heat-flow problems across fractal mixtures: Regularity results of the solutions and numerical approximation, Differential Integral Equations, 26 (2013), 1027-1054.   Google Scholar

[16]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, PNLDEA 8, Birkhäuser, Basel, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[17]

M. EI Jarroudi, Homogenization of a nonlinear elastic fibre-reinforced composite: A second gradient nonlinear elastic material, J. Math. Anal. Appl., 403 (2013), 487-505.  doi: 10.1016/j.jmaa.2013.02.042.  Google Scholar

[18]

M. El Jarroudi, Homogenization of an elastic material reinforced with thin rigid von Kármán ribbons, Math. Mech. Solids, 24 (2019), 1965-1991.  doi: 10.1177/1081286518810757.  Google Scholar

[19]

M. El Jarroudi, Homogenization of a quasilinear elliptic problem in a fractal-reinforced structure, SeMA, (2021). doi: 10.1007/s40324-021-00250-5.  Google Scholar

[20]

M. El Jarroudi and M. Er-Riani, Homogenization of elastic materials containing self-similar rigid micro-inclusions, Contin. Mech. Thermodyn., 31 (2019), 457-474.  doi: 10.1007/s00161-018-0700-4.  Google Scholar

[21]

M. El JarroudiM. Er-RianiA. Lahrouz and A. Settati, Homogenization of elastic materials reinforced by rigid notched fibres, Appl. Anal., 97 (2018), 705-738.  doi: 10.1080/00036811.2017.1285015.  Google Scholar

[22]

K. Falconer, Techniques in Fractal Geometry, J. Wiley and Sons, Chichester, 1997.  Google Scholar

[23]

U. R. Freiberg and M. R. Lancia, Energy form on a closed fractal curve, Z. Anal. Anwendungen, 23 (2004), 115-137.  doi: 10.4171/ZAA/1190.  Google Scholar

[24]

M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. doi: 10.1515/9783110889741.  Google Scholar

[25]

M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket,, Potential Anal., 1 (1992), 1-35.  doi: 10.1007/BF00249784.  Google Scholar

[26]

P. H Hung and E. Sanchez-Palencia, Phénomènes de transmission à travers des couches minces de conductivité élevée, J. Math. Anal. Appl., 47 (1974), 284-309.  doi: 10.1016/0022-247X(74)90023-7.  Google Scholar

[27]

A. Jonsson and H. Wallin, Boundary value problems and Brownian motion on fractals, Chaos Solitons Fractals, 8 (1997), 191-205.  doi: 10.1016/S0960-0779(96)00048-3.  Google Scholar

[28]

A. Jonsson and H. Wallin, The dual of Besov spaces on fractals, Studia. Math., 112 (1995), 285-300.   Google Scholar

[29]

T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag New York, Inc., New York 1966. doi: 10.1007/978-3-642-66282-9.  Google Scholar

[30]

S. M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys., 153 (1993), 339-357.  doi: 10.1007/BF02096647.  Google Scholar

[31]

M. R. LanciaU. Mosco and M. A. Vivaldi, Homogenization for conductive thin layers of pre-fractal type, J. Math. Anal. Appl., 347 (2008), 354-369.  doi: 10.1016/j.jmaa.2008.06.011.  Google Scholar

[32]

M. R. LanciaM. Cefalo and G. DellÁcqua, Numerical approximation of transmission problems across Koch-type highly conductive layers, Appl. Math. Comput., 218 (2012), 5453-5473.  doi: 10.1016/j.amc.2011.11.033.  Google Scholar

[33]

Y. Le Jean, Measures associées á une forme de Dirichlet, Appl., Bull. Soc. Math., 106 (1978), 61-112.   Google Scholar

[34]

M. Lobo and E. Perez, Boundary homogenization of certain elliptic problems for cylindrical bodies, Bull Sci Math., 116 (1992), 399-426.   Google Scholar

[35]

U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal., 123 (1994), 368-421.  doi: 10.1006/jfan.1994.1093.  Google Scholar

[36]

U. Mosco, Variational fractals,, Ann. Scuola Norm. Sup. Pisa, Special Volume in Memory of E. De Giorgi, 25 (1997), 683-712.   Google Scholar

[37]

U. Mosco, Lagrangian metrics on fractals, Proc. Symp. Appl. Math., Amer. Math. Soc., 54 (1998), 301-323.  doi: 10.1090/psapm/054/1492702.  Google Scholar

[38]

U. Mosco, Energy functionals on certain fractal structures, J. Conv. Anal., 9 (2002), 581-600.   Google Scholar

[39]

U. Mosco and M. A. Vivaldi, An example of fractal singular homogenization, Georgian Math. J., 14 (2007), 169-193.  doi: 10.1515/GMJ.2007.169.  Google Scholar

[40]

U. Mosco and M. A. Vivaldi, Fractal reinforcement of elastic membranes, Arch. Ration. Mech. Anal., 194 (2009), 49-74.  doi: 10.1007/s00205-008-0145-1.  Google Scholar

[41]

U. Mosco and M. A. Vivaldi, Thin fractal fibers, Math. Meth. Appl. Sci., 36 (2013), 2048-2068.  doi: 10.1002/mma.1621.  Google Scholar

[42]

U. Mosco and M. A. Vivaldi, Layered fractal fibers and potentials,, J. Math. Pures Appl., 103 (2015), 1198-1227.  doi: 10.1016/j.matpur.2014.10.010.  Google Scholar

[43]

O. A. Oleinik, A. S. Shamaev and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, Studies in Mathematics and its Applications, 26. North-Holland Publishing Co., Amsterdam, 1992.  Google Scholar

[44]

R. S. Rivlin, Plane strain of a net formed by inextensible cords, J. Rational Mech. Anal., 4 (1955), 951-974.  doi: 10.1512/iumj.1955.4.54037.  Google Scholar

[45]

R. S. Rivlin, The deformation of a membrane formed by inextensible cords, Arch. Rational Mech. Anal., 2 (1958), 447-476.  doi: 10.1007/BF00277942.  Google Scholar

[46]

R. S. Strichartz and C. Wong, The p-Laplacian on the Sierpinski gasket, Nonlinearity, 17 (2004), 595-616.  doi: 10.1088/0951-7715/17/2/014.  Google Scholar

Figure 1.  The graph $ \Sigma _{h} $ for $ h = 0,1,2,3 $
Figure 2.  The network $ \left\{ \mathcal{T}_{m}\right\} _{m\in \mathbb{N}} $ where $ \sigma _{\alpha 3}|_{\Sigma \times \left\{ 0^{+}\right\} } $ is the outward normal stress on $ \Sigma \cap \partial \mathcal{T}_{m} $ and $ -\sigma _{\alpha 3}|_{\Sigma \times \left\{ 0^{-}\right\} } $ is the inward normal stress
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