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\title{Here goes the title of your abstract}
\author{Author1, \underline{Presenting author2} \\Affiliation of author1 \\Affiliation of author2}
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This contribution presents ideas, how crack propagation in three-dimensional
solids composed of anisotropic materials can be predicted using the Griffith
energy principle. Since the work of Irwin the change of potential energy
$\mathbf{\triangle U}$ caused by a straight elongation of a crack in an
isotropic two-dimensional homogeneous structure can be expressed in quadratic
terms of the stress intensities at the crack tip. This result was generalized
in the last decades to anisotropic and also inhomogeneous materials using
methods of asymptotic analysis by many authors \cite{key-1}. With the energy
release rate at hand, quasi-static scenarios of crack propagation can be
simulated for plane problems \cite{key-3}.
While crack propagation for plane scenarios is widely discussed in the
literature \cite{key-1}, this is much more complicated in three dimensions
\cite{key-2}. Mathematical models for crack prediction are based on the
asymptotic behavior of the displacements at the crack front. If the crack front
is a smooth curve completely contained inside a solid, displacements are of
well-known square-root type also in three dimensions. We show, how the
asymptotic structure can be calculated, now depending on the geometry of the
crack surface. Using methods from asymptotic analysis, we generalize the
results from {[}2{]} and derive a representation of the change of potential
energy caused by a small elongation of a three-dimensional smooth crack surface
with arbitrary curvature and torsion:
\[
\triangle
U=-\frac{1}{2}t\left(\int_{\Gamma}h\left(s\right)\left(\sum_{i,j=1}^{3}K_{i}\left(s\right)M_{i,j}\left(\vartheta\left(s\right),s\right)K_{j}\left(s\right)\right)ds\right)+ \cdots
\]
Here, $K_{i}\left(s\right)$ are the stress intensity factors at arc length s on
the crack front $\Gamma$ and $M_{i,j}$ are so-called local characteristics,
depending on the material properties and the geometry of the elongated
crack. The quantity $th\left(s\right)$ is the length of the crack elongation at
the crack front at arc length $s$ to direction $\vartheta\left(s\right)$. The
number $t$ can be interpreted as a time-like parameter which is always small.
\begin{thebibliography}{1}
\bibitem[1]{key-1}I.I Argatov, S.A. Nazarov. Energy release caused by the
kinking of a crack in a plane anisotropic solid. J. Appl. Maths. Mechs. 66
(2002), 491\textendash 503.
\bibitem[2]{key-2}M. Bach, S.A. Nazarov, W.L. Wendland. Stable propagation of a
mode-1 crack in an isotropic elastic space. Comparison of the Irwin and the
Griffith approaches. Problemi attuali dell\textquoteright analisi e della
fisica matematica. (2000), 167-189.
\bibitem[3]{key-3}A.A. Griffith. The phenomena of rupture and flow in
solids. Philos. Trans. Roy. Soc. London 221 (1921), 163\textendash 198.
\end{thebibliography}
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