In crack problems, the elastic system is described on the domain $\Omega $ with a crack $\Sigma $, where $\Omega =D\setminus \Sigma $ by a domain $D$ with the Lipschitz property and a surface (curve in 2-d case) $\Sigma $ with the edge $\partial \Sigma $. The boundary of $\Omega $ has not the Lipschitz property.

The definition of the GJ-integral (2.5) changes slightly. $\Omega $ is devided into domains $\Omega ^+, \Omega ^-$ with the Lipschitz property and the surface (line) $\Pi $ such that $\Sigma \subset \Pi $ and $D=\Omega ^+\cup \Omega ^-\cup (\Pi \cap D)$. In crack problem, if $\omega \subset D$, (2.11) is written as \begin{eqnarray*} \int _{\omega \cap \Omega }\nabla _x\widehat{W}(u)\cdot \mu \, dx &=&\int _{\partial (\omega \cap \Omega ^+)}\widehat{W}(u)^+(\mu \cdot n^+)\,ds\\ &&+\int _{\partial (\omega \cap \Omega ^-)}\widehat{W}(u)^-(\mu \cdot n^-)\,ds,\\ &&-\int _{\omega \cap \Omega }\widehat{W}(u)\textrm{div}\mu \, dx,\\ &=&\int _{\partial \omega }\widehat{W}(u)(\mu \cdot n)\,ds+ \int _{\omega \cap \Sigma }[\! [\widehat{W}(u)]\! ](\mu \cdot n^+)\,ds,\\ &&-\int _{\omega \cap \Omega }\widehat{W}(u)\textrm{div}\mu \, dx. \end{eqnarray*} since $W(u)^+=W(u)^-$ on $\Pi \setminus \Sigma $, where $\widehat{W}(u)^±$ is the trace of $\widehat{W}(u)|_{\Omega ^±}$ on $\partial (\omega \cap \Omega ^{±})$, $n^±$ the outward unit normal to $\Omega ^±$ respectively, and $[\! [\widehat{W}(u)]\! ]$ stands for $\widehat{W}(u)^+-\widehat{W}(u)^-$ on $\Sigma $. Also the first term in the right-hand side of (2.12) is written as $$ \int _{\partial (\omega \cap \Omega )}\widehat{T}(u)(\nabla u\cdot \mu )ds =\int _{\partial \omega }\widehat{T}(u)(\nabla u\cdot \mu )ds +\int _{\omega \cap \Sigma }[\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]ds$$ where $[\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]=n_j^+ \left ([\partial \widehat{W}(u)/\partial \zeta _{ij}]^+(\nabla u_i\cdot \mu )^+-[\partial \widehat{W}(u)/\partial \zeta _{ij}]^-(\nabla u_i\cdot \mu )^-\right )$. In the crack problem, for $\omega \subset D$, (2.5b) is defined by \begin{eqnarray} P_{\omega }(u,\mu )& :=&\int _{\partial \omega }\left \lbrace \widehat{W}(u)(\mu \cdot n)-\widehat{T}(u)(\nabla u\cdot \mu )\right \} \,ds, \notag \\ &&+\int _{\omega \cap \Sigma }\left \{[\! [\widehat{W}(u)]\! ](\mu \cdot n)- [\! [\widehat{T}(u)(\nabla u\cdot \mu )]\! ]\right \} ds. \tag{2.15} \end{eqnarray} If $\mu $ is the vector $\mu _C$ obtained from the crack extension, then $\mu _C\cdot n=0$ on $\omega \cap \Sigma $ and $\widehat{T}(u)=0$ on $\omega \cap \Sigma $ by stress free on the crack surface so the last term in ((2.15) ) vanishes.

Information about the page: The current position is painted circle in the diagram below. Blue is the main MaKR and orange is a duplicate for MaKR's public use, where dashed line means the connection to the private area The dashed lines are only connections to main MaKR.

Top

top

[Adams] R.A.Adams, Sobolev spaces, Academic Press, 1975.
[A-P06] G. Allaire and O. Pantz, Structural optimization with FreeFem++, Struct. Multidiscip. Opt, 32 (2006), 173--181.
[Al07] G. Allaire, Conception optimale de structures, Springer, 2007.
[Az94] H. Azegami, Solution to domain optimization problems, Trans. Japan Soc. Mech. Engrs. Series A, 60, No.574 (1994), 1479--1486. (in Japanese)
[A-W96] H. Azegami and Z. Wu, Domain optimization analysis in linear elastic problems: Approach using traction method, JSME Inter. J. Series A, 39 (1996), 272--278.
[Az17] H. Azegami. Solution of shape optimization problem and its application to product design, Mathematical Analysis of Continuum Mechanics and Industrial Applications, Springer, 2017, 83--98.
[B-S04] M.P. Bends{\o }e and O. Sigmund, Topology optimization: theory, methods, and applications, Springer, 2004.
[Bu04] H.D. Bui, Fracture mechanics -- Inverse problems and solutions, Springer, 2006.
[Ch67] G.P. Cherepanov, On crack propagation in continuous media, Prikl. Math. Mekh., 31 (1967), 476--493.
[Cir88] P.G. Ciarlet, Mathematical elasticity: Three-dimensional elasticity, North-Holland, 1988.
[Co85] R. Correa and A. Seeger, Directional derivative of a minimax function. Nonlinear Anal., 9(1985), 13--22.
[D-Z88] M.C. Delfour and J.-P. Zolésio, Shape sensitivity analysis via min max differentiability, SIAM J. Control and Optim., 26(1988), 834--862.
[D-D81] Ph. Destuynder and M. Djaoua, Sur une interprétation de l'intégrale de Rice en théorie de la rupture fragile. Math. Meth. in Appl. Sci., 3 (1981), 70--87.
[E-G04] A. Em and J.-L. Guermond, Theory and practice of finite elements, Springer, 2004.
[Es56] J.D. Eshelby, The Continuum theory of lattice defects, Solid State Physics, 3 (1956), 79--144.
[F-O78] D. Fujiwara and S. Ozawa, The Hadamard variational formula for the Green functions of some normal elliptic boundary value problems, Proc. Japan Acad., 54 (1978), 215--220.
[G-S52] P.R. Garabedian and M. Schiffer, Convexity of domain functionals, J.Anal.Math., 2 (1952), 281--368.
[Gr21] A.A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. London, Series A 221 (1921), 163--198.
[Gr24] A.A. Griffith, The theory of rupture, Proc. 1st.Intern. Congr. Appl. Mech., Delft (1924) 55--63.
[Gr85] P. Grisvard, Elliptic problems in nonsmooth domains, Pitman, 1985.
[Gr92] P. Grisvard, Singularities in boundary value problems, Springer, 1992.
[Had68] J. Hadamard, Mémoire sur un problème d'analyse relatif à l'équilibre des plaques élastiques encastrées, Mémoire des savants étragers, 33 (1907), 515--629.
[Hau86] E.J. Haug, K.K. Choi and V. Komkov, Design sensitivity analysis of structural systems, Academic Press, 1986.
[ffempp] F. Hecht, New development in freefem++. J. Numer. Math. 20 (2012), 251--265. 65Y15, (FreeFem++ URL:http://www.freefem.org)
[Kato] T. Kato, Perturbation theory for linear operators, Springer, 1980.
[K-W06] M. Kimura. and I. Wakano, New mathematical approach to the energy release rate in crack extension, Trans. Japan Soc. Indust. Appl. Math., 16(2006) 345--358. (in Japanese) \bibitem {K-W11} M. Kimura and I. Wakano, Shape derivative of potential energy and energy release rate in rracture mechanics, J. Math-for-industry, 3A (2011), 21--31.
[Kne05] D. Knees, Regularity results for quasilinear elliptic systems of power-law growth in nonsmooth domains: boundary, transmission and crack problems. PhD thesis, Universität Stuttgart, 2005. http://elib.uni-stuttgart.de/opus/volltexte/2005/2191/.
[Ko06] V.A. Kovtunenko, Primal-dual methods of shape sensitivity analysis for curvilinear cracks with nonpenetration, IMA Jour. Appl. Math. 71 (2006), 635--657.
[K-O18] V.A. Kovtunenko and K. Ohtsuka, Shape differentiability of Lagrangians and application to stokes problem, SIAM J. Control Optim. 56 (2018), 3668--3684.
[M-P01] B. Mohammadi and O. Pironneau, Applied shape optimization for fluids. Oxford University Press, 2001.
[Na94] S.Nazarov and B.A.Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, de Gruyter Expositions in Mathematics 13. Walter de Gruyter \& Co., 1994.
[Nec67] J. Nečas, Direct methods in the theory of elliptic equations, Springer, 2012. Translated from ``Méthodes directes en théorie des équations elliptiques, 1967, Masson''.
[Noe18] E. Noether, Invariante variationsprobleme, göttinger nachrichten, Mathematisch-Physikalische Klasse (1918), 235--257.
[N-S13] A.A. Novotny and J. Sokolowski, Topological derivatives in shape optimization, Springer, 2013.
[Oh81] K. Ohtsuka, Generalized J-integral and three dimensional fracture mechanics I, Hiroshima Math. J., 11(1981), 21--52.
[Oh85] K. Ohtsuka, Generalized J-integral and its applications. I. -- Basic theory, Japan J. Appl. Math., 2 (1985), 329--350.
[O-K00] K. Ohtsuka and A. Khludnev, Generalized J-integral method for sensitivity analysis of static shape design, Control \& Cybernetics, 29 (2000), 513--533.
[Oh02] K. Ohtsuka, Comparison of criteria on the direction of crack extension, J. Comput. Appl. Math., 149 (2002), 335--339.
[Oh02-2] K. Ohtsuka, Theoretical and numerical analysis on 3-dimensional brittle fracture, Mathematical Modeling and Numerical Simulation in Continuum Mechanics, Springer, 2002, 233--251.
[Oh09] K. Ohtsuka, Criterion for stable/unstable quasi-static crack extension by extended griffith energy balance theory, Theor. Appl. Mech. Japan, 57 (2009), 25--32.
[Oh12] K. Ohtsuka, Shape optimization for partial differential equations/system with mixed boundary conditions, RIMS K\^oky\^uroku 1791 (2012), 172--181.
[OT-K12] K. Ohtsuka and M. Kimura, Differentiability of potential energies with a parameter and shape sensitivity analysis for nonlinear case: the p-Poisson problem, Japan J. Indust. Appl. Math., 29 (2012), 23--35.
[Oh14] K. Ohtsuka and T. Takaishi, Finite element anaysis using mathematical programming language FreeFem++, Kyoritsu Shuppan, 2014. (in Japanese)
[Oh17] K. Ohtsuka, Shape optimization by GJ-integral: Localization method for composite material, Mathematical Analysis of Continuum Mechanics and Industrial Applications, Springer, 2017, 73--109.
[Oh18] K. Ohtsuka, Shape optimization by Generalized J-integral in Poisson's equation with a mixed boundary condition, Mathematical Analysis of Continuum Mechanics and Industrial Applications II, Springer, 2018, 73--83.
[Pr10] A.N. Pressley, Elementary differential geometry, Springer, 2010.
[Ri68] J.R. Rice, A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35(1968), 379--386.
[Ri68-2] J.R. Rice, Mathematical analysis in the mechanics of fracture, Fracture Volume II, Academic Press, 1968, 191--311.
[Pi84] O. Pironneau, Optimal shape design for elliptic systems, Springer-Verlag, 1984.
[Sa99] J.A. Samareh, A survey of shape parameterization techniques, NASA Report CP-1999-209136 (1999), 333--343.
[Sc91] B.-W. Schulze, Pseudo-differential operators on manifolds with singularities, North-Holland, 1991.
[Sok92] J. Sokolowski and J.-P. Zolesio, Introduction to shape optimization, Springer, 1992.
[St14] K. Sturm, On shape optimization with non-linear partial differential equations, Doctoral thesis, Technische Universiltät of Berlin, 2014. https://d-nb.info/106856959X/34
[Sumi] Y. Sumi, Mathematical and computational analyses of cracking formation, Springer, 2014.
[Zei/2B] E. Zeidler. Nonlinear functional analysis and its applications II/B, Springer, 1990.
[Z-S73] O.C. Zienkiewicz and J.S. Campbell, Shape optimization and sequential linear programming, Optimum Structural Design, Wiley, 1973, 109--126.

top