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{\it Lower semicontinuity results in SBD and Barenblatt's theory}

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{\bf Elvira Zappale} 
Universit\'a di Salerno
{\tt ezappale@unisa.it} 
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\noindent {\bf Abstract}:  We consider surface energies framed in the
Barenblatt's theory of Fracture Mechanics, and which take into account the
fractures appearing under the regime of linearized elasticity.  These surface
integrals  the ''solid'' region, the second term is a surface term, which
represent the energy dissipated in the fracture process.  They may have
explicit dependence on the normal component of the opening of the fracture and
include a constraint which prevents infinitesimal interpenetration. 
              
In details, the problems concern energy minimization for brittle solids where
there is
 explicit dependence on the detachment or dependence both on the jump of the
 deformation and on the normal unit vector to the unknown crack site.  From
 the mathematical view point they consist of looking for sufficient conditions
 for lower semicontinuity in the space of special functions of bounded
 deformation, of surface integrands depending on the real amplitude of the
 jump or more generally on the pair (jump of the deformation, normal to the
 jump set), i.e. 
\begin{equation}\label{uno} 
\int_{J_{u}} \phi([u] \cdot \nu_{u})d {\cal H}^{N-1}
\enspace, \enspace [u]\cdot \nu_u  \geq 0
\enspace {\cal H}^{N-1}-\hbox{ a. e. on }J_u. 
\end{equation} 
and  
\begin{equation}\label{due}  
\int_{J_{u}} \Psi([u], \nu_{u})d {\cal H}^{N-1}
\enspace, \enspace [u]\cdot \nu_u  \geq 0
\enspace {\cal H}^{N-1}-\hbox{ a. e. on }J_u. 
\end{equation} 

Characterizations of the considered integrands, in terms of suitable
'convexity' conditions, are also given and a comparison between the two
classes  in (\ref{uno}) and (\ref{due}) has been performed.

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