Research article Special Issues

Homogenisation of high-contrast brittle materials

  • Received: 13 October 2019 Accepted: 26 November 2019 Published: 31 December 2019
  • This paper is an overview on some recent results concerning the variational analysis of static fracture in the so-called high-contrast brittle composite materials. The paper is divided into two main parts. The first part is devoted to establish a compactness result for a general class of free-discontinuity functionals with degenerate (or high-contrast) integrands. The second part is focussed on some specific examples which show that the degeneracy of the integrands may lead to non-standard limit effects, which are specific to this high-contrast setting.

    Citation: Caterina Ida Zeppieri. Homogenisation of high-contrast brittle materials[J]. Mathematics in Engineering, 2020, 2(1): 174-202. doi: 10.3934/mine.2020009

    Related Papers:

  • This paper is an overview on some recent results concerning the variational analysis of static fracture in the so-called high-contrast brittle composite materials. The paper is divided into two main parts. The first part is devoted to establish a compactness result for a general class of free-discontinuity functionals with degenerate (or high-contrast) integrands. The second part is focussed on some specific examples which show that the degeneracy of the integrands may lead to non-standard limit effects, which are specific to this high-contrast setting.


    加载中


    [1] Acerbi E, Chiadó Piat V, Dal Maso G, et al. (1992) An extension theorem from connected sets, and homogenization in general periodic domains. Nonlinear Anal Theor 18: 481-496. doi: 10.1016/0362-546X(92)90015-7
    [2] Ambrosio L, Fusco N, Pallara D (2000) Functions of Bounded Variations and Free Discontinuity Problems. Oxford: Clarendon Press.
    [3] Barchiesi M (2018) Toughening by crack deflection in the homogenization of brittle composites with soft inclusions. Arch Ration Mech Anal 227: 749-766. doi: 10.1007/s00205-017-1173-5
    [4] Barchiesi M, Dal Maso G (2009) Homogenization of fiber reinforced brittle materials: The extremal cases. SIAM J Math Anal 41: 1874-1889. doi: 10.1137/080744372
    [5] Barchiesi M, Focardi M (2011) Homogenization of the Neumann problem in perforated domains: An alternative approach. Calc Var Partial Dif 42: 257-288. doi: 10.1007/s00526-010-0387-2
    [6] Barchiesi M, Lazzaroni G, Zeppieri CI (2016) A bridging mechanism in the homogenization of brittle composites with soft inclusions. SIAM J Math Anal 48: 1178-1209. doi: 10.1137/15M1007343
    [7] Bouchitté G, Fonseca I, Leoni G, et al. (2002) A global method for relaxation in W1,p and in SBVp. Arch Ration Mech Anal 165: 187-242. doi: 10.1007/s00205-002-0220-y
    [8] Braides A, Defranceschi A (1998) Homogenization of Multiple Integrals. New York: Oxford University Press.
    [9] Braides A, Defranceschi A, Vitali E (1996) Homogenization of free discontinuity problems. Arch Ration Mech Anal 135: 297-356. doi: 10.1007/BF02198476
    [10] Braides A, Garroni A (1995) Homogenization of periodic nonlinear media with stiff and soft inclusions. Math Mod Meth Appl Sci 5: 543-564. doi: 10.1142/S0218202595000322
    [11] Braides A, Solci M (2013) Multi-scale free-discontinuity problems with soft inclusions. Boll Unione Mat Ital 1: 29-51.
    [12] Cagnetti F, Dal Maso G, Scardia L, et al. (2019) Γ-convergence of free-discontinuity problems. Ann Inst H Poincaré Anal Non Linéaire 36: 1035-1079. doi: 10.1016/j.anihpc.2018.11.003
    [13] Cagnetti F, Dal Maso G, Scardia L, et al. (2019) Homogenisation of stochastic free-discontinuity problems. Arch Ration Mech Anal 233: 935-974. doi: 10.1007/s00205-019-01372-x
    [14] Cagnetti F, Scardia L (2011) An extension theorem in SBV and an application to the homogenization of the Mumford-Shah functional in perforated domains. J Math Pur Appl 95: 349-381. doi: 10.1016/j.matpur.2010.03.002
    [15] Dal Maso G (1993) An Introduction to Γ-Convergence. Boston: Birkhäuser.
    [16] Dal Maso G, Zeppieri CI (2010) Homogenization of fiber reinforced brittle materials: The intermediate case. Adv Calc Var 3: 345-370.
    [17] Focardi M, Gelli MS, Ponsiglione M (2009) Fracture mechanics in perforated domains: A variational model for brittle porous media. Math Mod Meth Appl Sci 19: 2065-2100. doi: 10.1142/S0218202509004042
    [18] Giacomini A, Ponsiglione M (2006) A Γ-convergence approach to stability of unilateral minimality properties. Arch Ration Mech Anal 180: 399-447. doi: 10.1007/s00205-005-0392-3
    [19] Pellet X, Scardia L, Zeppieri CI (2019) Homogenization of high-contrast Mumford-Shah energies. SIAM J Math Anal 51: 1696-1729. doi: 10.1137/18M1189804
    [20] Scardia L (2008) Damage as Γ-limit of microfractures in anti-plane linearized elasticity. Math Mod Meth Appl Sci 18: 1703-1740. doi: 10.1142/S0218202508003170
    [21] Scardia L (2010) Damage as the Γ-limit of microfractures in linearized elasticity under the non-interpenetration constraint. Adv Calc Var 3: 423-458.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2496) PDF downloads(365) Cited by(2)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog