IntroductionIdeas.tex

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\input{tex/preamble.tex}
\begin{document}

\section{To do List:}
\begin{enumerate}
   \item Read Fracture Mechanics paper
   \item Fix Introduction
   \item Fix Abstract
   \item Fix Acknowledgements
   \item why the fracture part will be important?
   \item Why is the weighted korn inequality important?
\end{enumerate}

\section{thesis organization}

\section{Introduction Material science and korn inequalities}
why are korn inequalites important?  Stability result? priori estimates for deformation
shows coercivity of bilinear form (Introduction Nitsche.Korn-Inequality  )

%korn gives uniqueness( chapter 4- V_A_Kondrat'ev_1988_Russ._Math._Surv._43_65.pdf)



Originally, Korn inequalities were used to prove existence, uniqueness and well-posedness of boundary value problems of linear elasticity (see e.g., [11,1]). Nowadays, often, as in our particular case, it is the best Korn constant
K(,V ) in the first Korn inequality that is of central importance (e.g., [2,12,13,15,16,10]). Specifically, we are interested in the asymptotic behavior of the Korn constant K(h,Vh) for shells with zero Gaussian curvature as a function
of their thickness h for subspaces Vh of W1,2 functions satisfying various boundary conditions at the thin edges of
the shell. In [6,5] we have shown that K(h,Vh) represents an absolute lower bound on safe loads for any slender
structure. For a classical circular cylindrical shell we have proved in [3] that K(h,Vh)  h3/2 for a broad class of
boundary conditions at the thin edges of the shell.

The
 Korn inequalities are essential in establishing coerciveness of the differential operators of lin-
 ear elastostatics and thus form the basis for existence results in that theory [9], [15]. Among
 other areas in elasticity where a consideration of Korn's inequalities arise, we cite, for ex-
 ample, fundamental studies on the mathematical foundations of finite elements [4], [8], [13],
 [49], stability theory [12], [33], a priori estimates for solutions in terms of boundary data [58],
 estimation of vibration frequencies [25], [41], [42], error estimates for plate theories [43], and
 qualitative analyses of solution behavior, such as those involved in the investigation of Saint-
 Venant's principle [5], [24]-[26], [43], [46], [51], [57], [58]. In these areas of application,
 information on the dimensionless optimal constants appearing in the inequalities, the Korn
 constants, is of importance.
\section{Hyperelasticity and Korn inequalities}

Solid bodies, in general are not absolutely rigid, so when suitable forces are applied both size and shape of the body change. When  the induced changes are considerable, the body migh not return to its original shape. However, depending on the material, the body might return to its original shape when the forces are removed. This property is called elasticity. 

Understanding the deformation of elastic materials has long been a fundamental pursuit in science and engineering. Traditional linear elastic models, have proven successful in describing the behavior of many materials under small deformations. However, these models fail to capture the intricate responses observed in substances that undergo large deformations. Hyperelasticity offers a more robust framework by incorporating nonlinear strain-energy functions that better represent the complex behavior of these materials. 

Any body, or any portion of a body, can possess energy in various way. If it is moving, it has kinetic energy. If it is in a gravitational field, it has gravitational potential energy. If it is stretched, compressed, or deformed in any way it has strain energy. 

\section{Shells/ buckling}
Look into our paper on buckling of shells.
look into "TOPICS IN THE MATHEMATICAL THEORY OF NONLINEAR
ELASTICITY" thesis

"Both shells and plates are an important element in
structural design, and the extensive usage of rubbery
structures makes it necessary to comprehend the
nonlinear dynamics of the hyperelastic plate and shell
structures. For this reason, this section focuses on the
investigations undertaken to comprehend the nonlinear dynamics of such structures



\section{Fracture Mechanics}

\begin{enumerate}
   \item What does the trivial branch represent?
   \item This means that second variations at two different variations $\BGf_1$ and $\BGf_2$ differing by an infinitesimal value compared to  
   $\Gl_{\text {crit }}(h) \frac{\partial\left(\delta^{2} E\right)}{\partial \lambda}\left(\BGf_{h}^* ; h, \lambda_{\text {crit }}(h)\right)$ should not be distinguished. This observation leads to the following new definition of buckling loads and buckling modes in the broader sense. 
   \item flip instability through infinitesimal rotations
\end{enumerate}
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