February  2022, 17(1): 47-72. doi: 10.3934/nhm.2021023

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 Published  February 2022 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 and Heterogeneous Media, 2022, 17 (1) : 47-72. 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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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. 

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

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

[6]

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

[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.

[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.

[9]

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

[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.

[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.

[12]

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

[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.

[14]

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

[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. 

[16]

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

[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.

[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.

[19]

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

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[29]

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

[30]

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

[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.

[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.

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

[38]

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

[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.

[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.

[41]

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

[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.

[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.

[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.

[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.

[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.

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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