refs.bib

% References for Dissertation



@article{bib:Fri.Jam.Mor.Mue.,
title = {Derivation of nonlinear bending theory for shells from three-dimensional nonlinear elasticity by Gamma-convergence},
journal = {Comptes Rendus Mathematique},
volume = {336},
number = {8},
pages = {697-702},
year = {2003},
issn = {1631-073X},
%doi = {https://doi.org/10.1016/S1631-073X(03)00028-1},
%url = {https://www.sciencedirect.com/science/article/pii/S1631073X03000281},
author = {Gero Friesecke and Richard D James and Maria {Giovanna Mora} and Stefan Müller},
abstract = {We show that the nonlinear bending theory of shells arises as a Γ-limit of three-dimensional nonlinear elasticity. To cite this article: G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).
Résumé
Nous montrons que la théorie non linéaire des coques en flexion émerge comme Γ-limite de la théorie de l'élasticité tridimensionelle. Pour citer cet article : G. Friesecke et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).},
label = {bib:Fri.Jam.Mor.Mue.}
}
\bibitem{fjmm03}
G.~Friesecke, R.~D. James, M.~G. Mora, and S.~M{\"u}ller. Derivation of nonlinear bending theory for shells from
  three-dimensional nonlinear elasticity by gamma-convergence.
\textit{Comptes Rendus Mathematique}, Volume 336, Issue 8, 15 April 2003, Pages  697--702.
\label{bib:Fri.Jam.Mor.Mue.}

@article{bib:Almroth,
author = {Almroth, B. O.},
title = {Postbuckling behaviour of axially compressed circular cylinders},
journal = {AIAA J},
volume = {1},
pages = {627--633},
year = {1963},
label = {bib:Almroth}
}


@techreport{bib:Bud.Hut.,
author = {Budiansky, B. and Hutchinson, J.},
title = {A survey of some buckling problems},
institution = {NASA},
type = {Technical Report},
number = {CR-66071},
month = {2},
year = {1966},
label = {bib:Bud.Hut.}
}

@article{bib:Bushnell,
    author = {Bushnell, D.},
    title = {Buckling of shell-pitfall of designers},
    journal = {AIAA J},
    volume = {19},
    number = {9},
    pages = {1183--1226},
    year = {1981},
    label = {bib:Bushnell}
}



@Inbook{bib:Calladine,
author="Calladine, C.R.",
editor="Durban, D.
and Givoli, D.
and Simmonds, J. G.",
title="A Shell-buckling Paradox Resolved",
bookTitle="Advances in the Mechanics of Plates and Shells: The Avinoam Libai Anniversary Volume",
year="2002",
publisher="Springer Netherlands",
address="Dordrecht",
pages="119--134",
%isbn="978-0-306-46954-1",
%doi="10.1007/0-306-46954-5_8",
%url="https://doi.org/10.1007/0-306-46954-5_8",
label={bib:Calladine}
}


@incollection{bib:Dog.Kli.Zim.Ode.Ara.,
author = {Richard Degenhardt and Alexander Kling and Rolf Zimmermann and Falk Odermann and F.C. de Araújo},
title = {Dealing with Imperfection Sensitivity of Composite Structures Prone to Buckling},
booktitle = {Advances in Computational Stability Analysis},
publisher = {IntechOpen},
address = {Rijeka},
year = {2012},
editor = {Safa Bozkurt Coskun},
chapter = {1},
%doi = {10.5772/45810},
%url = {https://doi.org/10.5772/45810},
label = {bib:Dog.Kli.Zim.Ode.Ara.}
}


@article{bib:Fri.Jam.Mue.1,
author = {Friesecke, Gero and James, Richard D. and Müller, Stefan},
title = {A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity},
journal = {Comm. on Pure and Appl Math.},
volume = {55},
number = {11},
pages = {1461-1506},
doi = {https://doi.org/10.1002/cpa.10048},
%url = {https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.10048},
%eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1002/cpa.10048},
label = {bib:Fri.Jam.Mue.1}
}



@article{bib:Fri.Jam.Mue.2,
  author    = {G. Friesecke and R. D. James and S. M{\"u}ller},
  title     = {A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence},
  journal   = {Arch. Ration. Mech. Anal.},
  volume    = {180},
  number    = {2},
  pages     = {183-236},
  year      = {2006},
  label     = {bib:Fri.Jam.Mue.2}
}
\bibitem{} G. Friesecke, R. D. James, and S. M\"uller. A hierarchy of plate models derived from
nonlinear elasticity by gamma-convergence. \textit{Arch. Ration. Mech. Anal.}, 180(2):183-236, 2006.
\label{bib:Fri.Jam.Mue.2}



@article{bib:Gor.Eva.,
  author    = {D. J. Gorman and R. M. Evan-Iwanowski},
  title     = {An analytical and experimental investigation of the effects of large prebuckling deformations on the buckling of clamped thin-walled circular cylindrical shells subjected to axial loading and internal pressure},
  journal   = {Developments in Theoretical and Applied Mechanics},
  volume    = {4},
  pages     = {415-426},
  year      = {1970},
  label     = {bib:Gor.Eva.}
}



@article{bib:Gra.Har.1,
  author    = {Y. Grabovsky and D. Harutyunyan},
  title     = {Exact scaling exponents in Korn and Korn-type inequalities for cylindrical shells},
  journal   = {SIAM Journal on Mathematical Analysis},
  volume    = {46},
  number    = {5},
  pages     = {3277--3295},
  year      = {2014},
  label     = {bib:Gra.Har.1}
}

@article{bib:Gra.Har.2,
  author    = {Y. Grabovsky and D. Harutyunyan},
  title     = {Scaling instability of the buckling load in axially compressed circular cylindrical shells},
  journal   = {Journal of Nonlinear Science},
  volume    = {26},
  pages     = {83--119},
  year      = {2016},
  label     = {bib:Gra.Har.2}
}


@article{bib:Gra.Har.3,
  author    = {Y. Grabovsky and D. Harutyunyan},
  title     = {Rigurous derivation of the formula for the buckling load in axially compressed circular cylindrical shells},
  journal   = {Journal of Elasticity},
  volume    = {120},
  pages     = {249--276},
  year      = {2015},
  label     = {bib:Gra.Har.3}
}


@article{bib:Gra.Har.4,
  author    = {Y. Grabovsky and D. Harutyunyan},
  title     = {Korn inequalities for shells with zero Gaussian curvature},
  journal   = {Ann. d'Inst. Henri Poincare (C), Nonl. Anal.},
  volume    = {35},
  number    = {1},
  pages     = {267--282},
  year      = {2018},
  label     = {bib:Gra.Har.4}
}


@article{bib:Gra.Tru.,
  author    = {Y. Grabovsky and L. Truskinovsky},
  title     = {The flip side of buckling},
  journal   = {Cont. Mech. Thermodyn.},
  volume    = {19},
  number    = {3-4},
  pages     = {211--243},
  year      = {2007},
  label     = {bib:Gra.Tru.}
}


@book{bib:Gur.Fri.Ana.,
  author    = {M. Gurtin and E. Fried and L. Anand},
  title     = {The Mechanics and Thermodynamics of Continua},
  publisher = {Cambridge University Press},
  year      = {2010},
  label     = {bib:Gur.Fri.Ana.}
}
\bibitem{Gurtin} 
M.~Gurtin, E.~Fried, and L.~Anand. \textit{The and Thermodynamics of Continua.} Cambridge University Press, 2010.
\label{bib:Gur.Fri.Ana.}

@article{bib:Harutyunyan.1,
  author    = {D. Harutyunyan},
  title     = {On the Korn interpolation and second inequalities in thin domains},
  journal   = {SIAM J. Math. Anal.},
  volume    = {50},
  number    = {5},
  pages     = {4964--4982},
  year      = {2018},
  label     = {bib:Harutyunyan.1}
}


@article{bib:Harutyunyan.2,
  author    = {D. Harutyunyan},
  title     = {Gaussian curvature as an identifier of shell rigidity},
  journal   = {Archive for Rational Mechanics and Analysis},
  volume    = {226},
  number    = {2},
  pages     = {743--766},
  year      = {2017},
  label     = {bib:Harutyunyan.2}
}


@misc{bib:Harutyunyan.3,
  author    = {D. Harutyunyan},
  title     = {A hint on the localization of the buckling deformation at vanishing curvature points on thin elliptic shells},
  journal   = {Journal of Elasticity},
  year      = {2022},
  volume    = {252},
  pages     = {61-77},
  label     = {bib:Harutyunyan.3}
}

@article{bib:Hor.Lor.Pel.,
  author    = {J. Hor{\'a}k and G. J. Lord and M. A. Peletier},
  title     = {Cylinder buckling: the mountain pass as an organizing center},
  journal   = {SIAM J. Appl. Math.},
  volume    = {66},
  number    = {5},
  pages     = {1793--1824},
  year      = {2006},
  label     = {bib:Hor.Lor.Pel.}
}



@article{bib:Hor.Lew.Pak.,
  author    = {P. Hornung and M. Lewicka and R. Pakzad},
  title     = {Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells},
  journal   = {Journal of Elasticity},
  volume    = {111},
  number    = {1},
  year      = {2013},
  label     = {bib:Hor.Lew.Pak.}
}
\bibitem{} P. Hornung, M. Lewicka and R. Pakzad. Infinitesimal isometries on developable surfaces and asymptotic theories for thin developable shells. \textit{Journal of Elasticity,} Volume 111, Number 1 (2013).
\label{bib:Hor.Lew.Pak.}


@article{bib:Horgan,
  author    = {C. O. Horgan},
  title     = {Korn's inequalities and their applications in continuum mechanics},
  journal   = {SIAM Review},
  volume    = {37},
  number    = {4},
  pages     = {491--511},
  year      = {1995},
  label     = {bib:Horgan}
}
\bibitem{horg95}
C.~O. Horgan.
\newblock Korn's inequalities and their applications in continuum mechanics.
\newblock {\em SIAM Rev.}, 37(4):491--511, 1995.
\label{bib:Horgan}


@article{bib:Hun.Lor.Pel.,
  author    = {G. Hunt and G. J. Lord and M. A. Peletier},
  title     = {Cylindrical shell buckling: a characterization of localization and periodicity},
  journal   = {Discrete and Continuous Dynamical Systems Series B},
  volume    = {3},
  number    = {4},
  pages     = {505--518},
  year      = {2003},
  label     = {bib:Hun.Lor.Pel.}
}

@article{bib:Hun.Net.1,
  author    = {G. W. Hunt and E. L. Neto},
  title     = {Localized buckling in long axially-loaded cylindrical shells},
  journal   = {Journal of the Mechanics and Physics of Solids},
  volume    = {39},
  number    = {7},
  pages     = {881--894},
  year      = {1991},
  label     = {bib:Hun.Net.1}
}

@article{bib:Hun.Net.2,
  author    = {G. W. Hunt and E. L. Neto},
  title     = {Maxwell critical loads for axially loaded cylindrical shells},
  journal   = {Trans. ASME},
  volume    = {60},
  pages     = {702--706},
  year      = {1993},
  label     = {bib:Hun.Net.2}
}
\bibitem{} G. W. Hunt and E. L. Neto. Maxwell critical loads for axially loaded cylindrical shells. \textit{Trans. ASME,} 60:702--706, 1993.
\label{bib:Hun.Net.2}

@phdthesis{bib:Koiter,
  author    = {W. T. Koiter},
  title     = {On the stability of elastic equilibrium},
  school    = {Technische Hogeschool (Technological University of Delft)},
  address   = {Delft, Holland},
  year      = {1945},
  label     = {bib:Koiter}
}
\bibitem{koit45}
W.~T. Koiter.
\newblock {On the stability of elastic equilibrium}.
\newblock PhD thesis, Technische Hogeschool (Technological University of
  Delft), Delft, Holland, 1945.
\label{bib:Koiter}
  
@article{bib:Lan.Cal.Pal.,
  author    = {E. Lancaster and C. Calladine and S. Palmer},
  title     = {Paradoxical buckling behaviour of a thin cylindrical shell under axial compression},
  journal   = {International Journal of Mechanical Sciences},
  volume    = {42},
  number    = {5},
  pages     = {843--865},
  year      = {2000},
  label     = {bib:Lan.Cal.Pal.}
}

@article{bib:Lew.Mor.Pak.,
  author    = {M. Lewicka and M. G. Mora and M. Pakzad},
  title     = {The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells},
  journal   = {Arch. Ration. Mech. Anal.},
  volume    = {200},
  number    = {3},
  pages     = {1023--1050},
  year      = {2011},
  label     = {bib:Lew.Mor.Pak.}
}
\bibitem{} M. Lewicka, M.G. Mora and M. Pakzad. The matching property of infinitesimal isometries on elliptic surfaces and elasticity of thin shells, \textit{Arch. Ration. Mech. Anal.} (3) Vol. 200 (2011), 1023--1050.
\label{bib:Lew.Mor.Pak.}

@article{bib:Lor.Cha.Hun.,
  author    = {G. Lord and A. Champneys and G. J. Hunt},
  title     = {Computation of localized post buckling in long axially compressed cylindrical shells},
  journal   = {Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences},
  volume    = {355},
  number    = {1732},
  pages     = {2137--2150},
  year      = {1997},
  label     = {bib:Lor.Cha.Hun.}
}  

  
@article{bib:Lorenz,
  author    = {R. Lorenz},
  title     = {Die nicht achsensymmetrische knickung d{\"u}nnwandiger hohlzylinder},
  journal   = {Physikalische Zeitschrift},
  volume    = {12},
  number    = {7},
  pages     = {241--260},
  year      = {1911},
  label     = {bib:Lorenz}
}
\bibitem{lorenz11}
R.~Lorenz.
\newblock Die nicht achsensymmetrische knickung d{\"u}nnwandiger hohlzylinder.
\textit{Physikalische Zeitschrift}, 12(7):241--260, 1911.
\label{bib:Lorenz}


@techreport{bib:Tennyson,
  author    = {R. C. Tennyson},
  title     = {An experimental investigation of the buckling of circular cylindrical shells in axial compression using the photoelastic technique},
  institution = {University of Toronto},
  address   = {Toronto, ON, Canada},
  number    = {102},
  year      = {1964},
  label     = {bib:Tennyson}
}


@article{bib:Timoshenko,
  author    = {S. Timoshenko},
  title     = {Towards the question of deformation and stability of cylindrical shell},
  journal   = {Vesti Obshestva Tekhnologii},
  volume    = {21},
  pages     = {785--792},
  year      = {1914},
  label     = {bib:Timoshenko}
}
\bibitem{timosh14}
S.~Timoshenko.
\newblock Towards the question of deformation and stability of cylindrical shell.
\textit{Vesti Obshestva Tekhnologii}, 21:785--792, 1914.
\label{bib:Timoshenko}

@book{bib:Tim.Woi.,
  author    = {S. Timoshenko and S. Woinowsky-Krieger},
  title     = {Theory of plates and shells},
  volume    = {2},
  publisher = {McGraw-Hill},
  address   = {New York},
  edition   = {2nd},
  year      = {1959},
  label     = {bib:Tim.Woi.}
}
\bibitem{} S. Timoshenko and S. Woinowsky-Krieger. \textit{Theory of plates and shells,} volume 2. McGraw-hill New York, second edition, 1959.
\label{bib:Tim.Woi.}

@article{bib:Wei.Mor.Sei.,
  author    = {V. I. Weingarten and E. J. Morgan and P. Seide},
  title     = {Elastic stability of thin-walled cylindrical and conical shells under axial compression},
  journal   = {AIAA Journal},
  volume    = {3},
  pages     = {500--505},
  year      = {1965},
  label     = {bib:Wei.Mor.Sei.}
}
\bibitem{wms65}
V.~I. Weingarten, E.~J. Morgan, and P.~Seide.
\newblock Elastic stability of thin-walled cylindrical and conical shells under axial compression.
\textit{AIAA J}, 3:500--505, 1965.
\label{bib:Wei.Mor.Sei.}


@article{bib:Xu.Mic.,
  author    = {F. Xu and M. Potier-Ferry},
  title     = {On axisymmetric/diamond-like mode transitions in axially compressed core-shell cylinders},
  journal   = {Journal of the Mechanics and Physics of Solids},
  volume    = {94},
  pages     = {68--87},
  year      = {2016},
  label     = {bib:Xu.Mic.}
}
\bibitem{Xu-Potier-Ferry2016} F. Xu and M. Potier-Ferry. On axisymmetric/diamond-like mode transitions in axially compressed core-shell cylinders, \textit{Journal of the Mechanics and Physics of Solids,} Volume 94, September 2016, Pages 68--87.
\label{bib:Xu.Mic.}

@book{bib:Yamaki,
  author    = {N. Yamaki},
  title     = {Elastic Stability of Circular Cylindrical Shells},
  series    = {Appl. Math. Mech.},
  publisher = {North Holland},
  year      = {1984},
  volume    = {27},
  label     = {bib:Yamaki}
}
\bibitem{} N. Yamaki. \textit{Elastic Stability of Circular Cylindrical Shells,} volume 27 of Appl. Math. Mech. North Holland, 1984.
\label{bib:Yamaki}

@article{bib:Yoshimura,
  author    = {Y. Yoshimura},
  title     = {On the mechanism of buckling of a circular shell under axial compression},
  journal   = {Technical Report 1390, National advisory committee for aeronautics},
  year      = {1955},
  label     = {bib:Yoshimura}
}

 
  
@article{bib:Zhu.Man.Cal.,
  author    = {E. Zhu and P. Mandal and C. Calladine},
  title     = {Buckling of thin cylindrical shells: an attempt to resolve a paradox},
  journal   = {International Journal of Mechanical Sciences},
  volume    = {44},
  number    = {4},
  pages     = {661--682},
  year      = {2002},
  label     = {bib:Zhu.Man.Cal.}
}


%%%%%%%% Papers From BVBD chapter %%%%%%%%%%%%%%%

@book{evansGa,
  title={Measure Theory and Fine Properties of Functions, Revised Edition},
  author={Evans, L.C. and Gariepy, R.F.},
  %isbn={9781482242386},
  lccn={2015297289},
  series={Textbooks in Mathematics},
  year={2015},
  publisher={Taylor \& Francis}
}
@book{RogerBook,
  title={Mathematical Problems in Plasticity},
  author={Temam, R.},
  isbn={9780471918592},
  lccn={86164738},
  series={Modern Applied Mathematics Series},
  year={1985},
  publisher={Gauthier-Villars}
}
@book{RogerPaper,
  title={Functions of Bounded Deformation},
  author={Temam, R. and Strang, G.},
  series={Prepublications mathematiques d'Orsay},
  year={1978},
  publisher={Departement de mathematique}
}
@article{Ambrosio1997,
  author = { Ambrosio, Luigi and Coscia, Alessandra and Dal Maso, Gianni },
  title = { Fine Properties of Functions with Bounded Deformation },
  journal = { Arch. Rational Mech. Anal. },
  year = { 1997 },
  volume = { 139 },
  pages = { 201-238 },
  %URL = { http://cvgmt.sns.it/paper/858/ },
  note = { cvgmt preprint}
}
@article{SBD1,
  author    = {Bellettini, G. and Coscia, A. and Dal Maso, G.},
  title     = {Compactness and lower semicontinuity properties in $SBD(\Omega)$},
  journal   = {Mathematische Zeitschrift},
  year      = {1998},
  volume    = {228},
  number    = {2},
  pages     = {337-351},
  issn      = {1432-1823},
  %doi       = {10.1007/PL00004617},
  %url       = {https://doi.org/10.1007/PL00004617}
}
@article{SBD2,
title = {An approximation result for special functions with bounded deformation},
journal = {Journal de Mathématiques Pures et Appliquées},
volume = {83},
%number = {7},
pages = {929-954},
year = {2004},
issn = {0021-7824},
%doi = {https://doi.org/10.1016/j.matpur.2004.02.004},
%url = {https://www.sciencedirect.com/science/article/pii/S0021782404000285},
author = {Antonin Chambolle},
keywords = {Functions with bounded deformation, Free discontinuity problems, Brittle fracture},
}
@article{SBD3, title={Which special functions of bounded deformation have bounded variation?}, volume={148}, DOI={10.1017/S030821051700004X}, number={1}, journal={Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, publisher={Royal Society of Edinburgh Scotland Foundation}, author={Conti, Sergio and Focardi, Matteo and Iurlano, Flaviana}, year={2018}, pages={33–50}}
@book{SBV1,
  author    = {Giusti, E.},
  title     = {Minimal Surfaces and Functions of Bounded Variation},
  publisher = {Birkh\"auser},
  year      = {1984}
}
% pass H. Federer, Geometric Measure Theory, Springer, 1969. to bib
@book{SBV2,
  author    = {Federer, H.},
  title     = {Geometric Measure Theory},
  publisher = {Springer},
  year      = {1969}
}

% L. Ambrosio and E. De Giorgi, “New functionals in the calculus of variations” (Italian), Atti Accad. Naz. Lincei Rend. Cl.
Sci. Fis. Mat. Natur. 82 (1988), 199–210.
@article{SBV3,
author = {Ambrosio, Luigi and De Giorgi, Ennio},
year = {1988},
issn = {0001-4346},
journal = {Atti Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali},
pages = {199--210},
title = {New functionals in the calculus of variations},
}



%%%%%%%%%%%%%%%%%%% Presentation
@article{Bucklingdavit,
title = {Scaling Instability in Buckling of Axially Compressed Cylindrical Shells},
journal = {Journal of Elasticity},
volume = {26},
number = {1},
pages = {83-119},
year = {2016},
issn = {1432-1467},
author = {Yury Grabovsky and Davit Harutyunyan},
keywords = {Korn's inequality, Shells, Nonlinear elasticity, Cones, Cylinders},
}
@article{flipside,
author = {Grabovsky, Yury and Truskinovsky, Lev},
year = {2007},
month = {08},
pages = {211-243},
title = {The flip side of buckling},
volume = {19},
journal = {Continuum Mechanics and Thermodynamics},
}
@article{exactexp,
title = {Exact Scaling Exponents in Korn and Korn-Type Inequalities for Cylindrical Shells},
journal = {SIAM J. Math. Anal},
volume = {46(5)},
number = {1},
pages = {3277-3295},
year = {2014},
issn = {1095-7154},
author = {Yury Grabovsky and Davit Harutyunyan},
keywords = {Korn's inequality, Shells, Nonlinear elasticity, Cones, Cylinders},
}

 @article{Bucklingparadox,
  author = {E. Zhua, P. Mandalb, and C.R. Calladine},
  title = {Buckling of thin cylindrical shells: an attempt to resolve a paradox},
  journal = {International Journal of Mechanical Sciences},
  month = March,
  year = 2002,
  volume = 44,
  pages = {1583--1601}}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Chapter Weighted Korn %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

@article{conti0,
title = {The Tapering Length of Needles in Martensite/Martensite Macrotwins},
journal = { Archive for Rational Mechanics and Analysis},
author = {Conti, Sergio and Zwicknagl, Barbara},
volume = {247},
number = {1},
pages = {63},
year = {2023},

doi = {10.1007/s00205-023-01882-9},
}

@book{stein,
 ISBN = {9780691080796},
 %URL = {http://www.jstor.org/stable/j.ctt1bpmb07},
 %abstract = {Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself.Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance.Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, includingReal Analysisin 2005.},
 author = {ELIAS M. STEIN},
 publisher = {Princeton University Press},
 title = {Singular Integrals and Differentiability Properties of Functions (PMS-30)},
 %urldate = {2023-09-15},
 year = {1970}
}
@article{wKorn1,
    author    = {Harutyunyan, Davit},
    title     = {Sharp Weighted Korn and Korn-Like Inequalities and an Application to Washers},
    journal   = {Journal of Elasticity},
    year      = {2017},
    volume    = {127},
    number    = {1},
    pages     = {59--77},
    issn      = {1573-2681},
    %url       = {https://doi.org/10.1007/s10659-016-9596-z},
    %doi       = {10.1007/s10659-016-9596-z}
}
@article{wKorn2,
author = {Harutyunyan, Davit and Mikayelyan, Hayk},
year = {2017},
month = {09},
pages = {},
title = {Weighted asymptotic Korn and interpolation Korn inequalities with singular weights},
volume = {147},
journal = {Proceedings of the American Mathematical Society},
doi = {10.1090/proc/14533}
}

@article{wKorn3,
  title={Weighted anisotropic Korn's inequality for a junction of a plate and a rod},
  author={Serguei A. Nazarov},
  journal={Sbornik Mathematics},
  year={2004},
  volume={195},
  pages={553-583},
  %url={https://api.semanticscholar.org/CorpusID:122504340}
}
@article{wKorn4,
    author    = {Nazarov, Sergey A. and Slutskij, Andrey S. and Sweers, Guido H.},
    title     = {Korn Inequalities for a Reinforced Plate},
    journal   = {Journal of Elasticity},
    year      = {2012},
    volume    = {106},
    number    = {1},
    pages     = {43--69},
    abstract  = {Asymptotically optimal Korn inequalities are derived for a composite material that consists of two families of stiff rods surrounded by a homogeneous soft material. The composite plate is fixed through the protruding stiff rods only. The asymptotic behaviour is shown to be crucially different for families of connected rods and for those where the rods are isolated.},
    issn      = {1573-2681},
    %url       = {https://doi.org/10.1007/s10659-010-9289-y},
    %doi       = {10.1007/s10659-010-9289-y}
}

@article{wKorn5,
author = {Akimova, E.},
year = {2005},
month = {08},
pages = {3519-3522},
title = {Weighted Korn’s Inequality for an Arbitrary Plate},
volume = {129},
journal = {Journal of Mathematical Sciences},
doi = {10.1007/s10958-005-0289-6}
}

@article{surveyQuasilinearSystems,
    author    = {Narbut, M. A.},
    title     = {On A.I. Koshelev’s work on regularity theory of generalized solutions of quasilinear systems of elliptic and parabolic type},
    journal   = {Differential Equations},
    year      = {2011},
    volume    = {47},
    number    = {13},
    pages     = {1916--1928},
    abstract  = {This is a survey of A.I. Koshelev’s studies in the theory of regular solutions of boundary value problems, based on iterative processes converging in both the energy norm and the strong norm as well as on a priori estimates in weighted function spaces. In numerous cases, Koshelev’s estimates contain explicitly computable and sometimes sharp (unimprovable) constants. The results obtained for a broad class of problems were adapted by Koshelev to the study of boundary value problems of nonlinear elasticity and problems of hydrodynamics of viscous fluids.},
    issn      = {1608-3083},
    %url       = {https://doi.org/10.1134/S0012266111130039},
    %doi       = {10.1134/S0012266111130039}
}

%%%% Chapter 2 %%%%%%%%%%
@book{MS1, 
place={Cambridge}, 
title={The Mechanics and Thermodynamics of Continua}, 
publisher={Cambridge University Press}, 
author={Gurtin, Morton E. and Fried, Eliot and Anand, Lallit}, 
year={2010}
}




@book{MSP,
  author    = {Ciarlet, Philippe G.},
  title     = {Mathematical Elasticity: Three-Dimensional Elasticity},
  publisher = {Society for Industrial and Applied Mathematics},
  year      = {2021},
  address   = {Philadelphia, PA}
}

@book{MSP3,
author = {Ciarlet, Philippe G.},
title = {Mathematical Elasticity: Theory of Shells},
publisher = {Society for Industrial and Applied Mathematics},
year = {2021},
%doi = {10.1137/1.9781611976823},
address = {Philadelphia, PA},
edition   = {},
%URL = {https://epubs.siam.org/doi/abs/10.1137/1.9781611976823},
%eprint = {https://epubs.siam.org/doi/pdf/10.1137/1.9781611976823}
}

@book{MSP2,
author = {Ciarlet, Philippe G.},
title = {Mathematical Elasticity: Theory of Plates},
publisher = {Society for Industrial and Applied Mathematics},
year = {2021},
%doi = {10.1137/1.9781611976809},
address = {Philadelphia, PA},
edition   = {},
%URL = {https://epubs.siam.org/doi/abs/10.1137/1.9781611976809},
%eprint = {https://epubs.siam.org/doi/pdf/10.1137/1.9781611976809}
}

@book{MS2,
  added-at = {2017-06-29T07:13:07.000+0200},
  address = {New York},
  author = {Timoshenko, Stephen and Goodier, James N.},
  %biburl = {https://www.bibsonomy.org/bibtex/298322dfd9f036bafd6634f2934d3713e/gdmcbain},
  comment = {(private-note)Holdings: pers.},
  edition = {Third},
  interhash = {c5eb4d02ad0062de18981f5ff95decc3},
  intrahash = {98322dfd9f036bafd6634f2934d3713e},
  keywords = {74f05-solid-mechanics-thermal-effects 74b05-classical-linear-elasticity},
  posted-at = {2009-08-18 05:05:30},
  priority = {2},
  publisher = {McGraw-Hill},
  timestamp = {2019-04-16T07:26:31.000+0200},
  title = {{Theory of Elasticity}},
  year = 1970
}

@book{MS3,
  added-at = {2017-06-29T07:13:07.000+0200},
  address = {New York},
  author = {Love, A. E. H.},
  %biburl = {https://www.bibsonomy.org/bibtex/24da8b766bc5ed3849bd9dd27273f90fd/gdmcbain},
  comment = {(private-note)Holdings: pers. (order 24823 2010-09-08 from seashellbooks.com 21.14 AUD)},
  edition = {Fourth},
  interhash = {aab950d79c6ee73c8c2121441a7fa8e8},
  intrahash = {4da8b766bc5ed3849bd9dd27273f90fd},
  isbn = {0486601749},
  keywords = {74k20-plates 74-01-mechanics-of-deformable-solids-instructional-exposition 74-02-mechanics-of-deformable-solids-research-exposition 74b05-classical-linear-elasticity 74k10-rods 74g60-bifurcation-and-buckling},
  posted-at = {2010-09-08 01:27:18},
  priority = {2},
  publisher = {Dover Publications},
  timestamp = {2019-04-16T07:28:59.000+0200},
  title = {{A Treatise on the Mathematical Theory of Elasticity}},
  %url = {http://www.worldcat.org/isbn/0486601749},
  year = 1944
}
@book{MS4,
  title={An Introduction to the Theory of Elasticity},
  author={Atkin, R.J. and Fox, N.},
  isbn={9780486442419},
  lccn={2005041433},
  series={Dover Books on Physics},
  year={2005},
  publisher={Dover Publications}
}

@book{MSVariational,
  title={The Variational Principles of Mechanics},
  author={Lanczos, C.},
  isbn={9780486650678},
  lccn={lc85029168},
  series={Dover Books On Physics},
  year={1986},
  publisher={Dover Publications}
}


@phdthesis{KohnThesis,
  author  = "Kohn, Robert V.",
  title   = "New Estimates for Deformations in Terms of their Strains",
  school  = "Princeton University",
  year    = "1979"
}

@book{Korn1,
  title={Die Eigenschwingungen eines elastichen Korpers mit ruhender Oberflache},
  author={Korn, Arthur},
  year={1906}
}
@inproceedings{Korn2,
  title={Solution g{\'e}n{\'e}rale du probl{\`e}me d'{\'e}quilibre dans la th{\'e}orie de l'{\'e}lasticit{\'e}, dans le cas ou les efforts sont donn{\'e}s {\`a} la surface},
  author={Korn, Arthur},
  booktitle={Annales de la Facult{\'e} des sciences de Toulouse: Math{\'e}matiques},
  volume={10},
  pages={165--269},
  year={1908}
}

@inproceedings{john,
  title={Bounds for Deformations in Terms of Average Strains},
  author={Jan Fritz},
  year={1985},
  %url={https://api.semanticscholar.org/CorpusID:185708475}
}

@article{andre,
    author    = {Harutyunyan, Davit and Rodrigues, Andre Martins},
    title     = {The Buckling Load of Cylindrical Shells Under Axial Compression Depends on the Cross-Sectional Curvature},
    journal   = {Journal of Nonlinear Science},
    year      = {2023},
    volume    = {33},
    number    = {2},
    pages     = {27},
    date      = {2023-01-10}
}

@article{KornProof2,
    author    = {Conti, Sergio and Dolzmann, Georg and Müller, Stefan},
    title     = {Korn’s second inequality and geometric rigidity with mixed growth conditions},
    journal   = {Calculus of Variations and Partial Differential Equations},
    year      = {2014},
    volume    = {50},
    number    = {1},
    pages     = {437--454},
    issn      = {1432-0835},
    %url       = {https://doi.org/10.1007/s00526-013-0641-5},
    %doi       = {10.1007/s00526-013-0641-5}
}

@article{KornProof1,
%doi = {10.1070/RM1988v043n05ABEH001945},
%url = {https://dx.doi.org/10.1070/RM1988v043n05ABEH001945},
year = {1988},
publisher = {},
volume = {43},
number = {5},
pages = {65},
author = {V A Kondrat'ev and  O A Oleinik},
title = {Boundary-value problems for the system of elasticity theory in unbounded domains. Korn's inequalities},
journal = {Russian Mathematical Surveys},
}

@article{KornProof3,
    author    = {Mikhailova, T. V.},
    title     = {The mixed boundary value problem for the elasticity theory system in unbounded domains},
    journal   = {Journal of Mathematical Sciences},
    year      = {1996},
    volume    = {80},
    number    = {5},
    pages     = {2052--2083},
    issn      = {1573-8795},
  }

@misc{kornBD1,
  author    = {Antonin Chambolle and Sergio Conti and Gilles Francfort},
  title     = {Korn-Poincaré inequalities for functions with a small jump set},
  year      = {2014},
  note      = {ffhal-01091710f}
}

@article{kornBD2,
title = {Approximation of functions with small jump sets and existence of strong minimizers of Griffith's energy},
journal = {Journal de Mathématiques Pures et Appliquées},
volume = {128},
pages = {119-139},
year = {2019},
issn = {0021-7824},
author = {Antonin Chambolle and Sergio Conti and Flaviana Iurlano},
keywords = {Special functions of bounded deformation, Brittle fracture, Free discontinuity problems, Strong solutions},
abstract = {We prove that special functions of bounded deformation with small jump set are close in energy to functions which are smooth in a slightly smaller domain. This permits to generalize the decay estimate by De Giorgi, Carriero, and Leaci to the linearized context in dimension n and to establish the closedness (up to negligible sets for the (n−1)-dimensional Hausdorff measure) of the jump set for local minimizers of the Griffith energy.
Résumé
Nous montrons que les déplacements à déformation bornée dont l'ensemble de discontinuité est petit sont proches (ainsi que leur énergie élastique) de déplacements qui sont réguliers dans un domaine un peu plus petit. Ceci permet d'étendre une estimation de décroissance due à De Giorgi, Carriero, Leaci au contexte de l'élasticité linéarisée, en dimension n quelconque, et d'établir que l'ensemble de discontinuité de minimiseurs (locaux) de l'énergie de Griffith est essentiellement fermé pour la mesure de Hausdorff (n−1)-dimensionnelle.}
}

@article{kornBD3,
  author    = {Filippo Cagnetti and Antonin Chambolle and Lucia Scardia},
  title     = {Korn and Poincaré-Korn inequalities for functions with a small jump set},
  journal   = {Mathematische Annalen},
  year      = {2022},
  volume    = {383},
  number    = {3},
  pages     = {1179--1216},
  abstract  = {In this paper we prove a regularity and rigidity result for displacements in $GSBD^p$, for every $p>1$ and any dimension $n\ge 2$. We show that a displacement in $GSBD^p$ with a small jump set coincides with a $W^{1,p}$ function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in $GSBD^p$.},
  issn      = {1432-1807},
}

@article{kornBD5,
author = {Friedrich, Manuel},
title = {A Piecewise Korn Inequality in SBD and Applications to Embedding and Density Results},
journal = {SIAM Journal on Mathematical Analysis},
volume = {50},
number = {4},
pages = {3842-3918},
year = {2018},
doi = {10.1137/17M1129982},
}

Here's the provided citation in BibTeX format:

```bibtex
@article{kornBD4,
  author    = {Friedrich, M.},
  title     = {A Korn-type inequality in SBD for functions with small jump sets},
  journal   = {Mathematical Models and Methods in Applied Sciences},
  year      = {2017},
  volume    = {27},
  pages     = {2461--2484}
}
```

You can add this to your `.bib` file for use in your LaTeX document.

%%%%%%%%%%%%%%%Fracture Mechanics
@inproceedings{Frac2,
author = {Goyat, Vikas and Nain, Jagdeep and Verma, Suresh},
year = {2012},
month = {12},
pages = {},
title = {A Review on Fracture Mechanics and J-Integral}
}
@article{FracGrif,
 ISSN = {02643952},
 %URL = {http://www.jstor.org/stable/91192},
 author = {A. A. Griffith},
 journal = {Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character},
 number = {},
 pages = {163--198},
 publisher = {The Royal Society},
 title = {The Phenomena of Rupture and Flow in Solids},
 urldate = {2023-10-16},
 volume = {221},
 year = {1921}
}
@article{Frac3,
author = {Almi, Stefano and Belz, Sandro and Micheletti, Stefano and Perotto, Simona},
year = {2020},
month = {11},
pages = {},
title = {A Dimension-Reduction Model for Brittle Fractures on Thin Shells with Mesh Adaptivity},
volume = {31},
journal = {Mathematical Models and Methods in Applied Sciences},
doi = {10.1142/S0218202521500020}
}
@article{FracExistence,
author = {Chambolle, Antonin and Crismale, Vito},
year = {2019},
month = {07},
pages = {},
title = {Existence of strong solutions to the Dirichlet problem for the Griffith energy},
volume = {58},
journal = {Calculus of Variations and Partial Differential Equations},
doi = {10.1007/s00526-019-1571-7}
}
@article{FracBook,
author = {Bourdin, Blaise and Francfort, Gilles and Marigo, J.-J},
year = {2008},
month = {01},
pages = {},
title = {The Variational Approach to Fracture},
volume = {91},
isbn = {978-1-4020-6394-7},
journal = {Reprinted from J. Elasticity},
doi = {10.1007/978-1-4020-6395-4}
}

@article{Frac1,
title = {Revisiting brittle fracture as an energy minimization problem},
journal = {Journal of the Mechanics and Physics of Solids},
volume = {46},
number = {8},
pages = {1319-1342},
year = {1998},
issn = {0022-5096},
author = {G.A. Francfort and J.-J. Marigo},
abstract = {A variational model of quasistatic crack evolution is proposed. Although close in spirit to Griffith’s theory of brittle fracture, the proposed model however frees itself of the usual constraints of that theory : a preexisting crack and a well-defined crack path. In contrast, crack initiation as well as crack path can be quantified, as demonstrated on explicitly computable examples. Furthermore the model lends itself to numerical implementation in more complex settings.}
}