| Abstract: |
| We consider a plane elasticity problem in a
domain $\Omega\subset \mathbb{R}^2$ with an
internal straight crack. Dealing with small strains is an essential assumption while
using the linear elasticity theory, however, the asymptotic behavior of the solution
$u$ near the crack tips leads to unbounded
strains (and stresses) which needs further physical interpretation. In the model of Dugdale it is assumed that the
stress cannot overcome a certain yield stress $\sigma_c$. Therefore
ahead of the crack $\Lambda$ a (one dimensional)
plastic zone appears, where $\sigma_{22}=-\sigma_c$ and $\sigma_c$
is large in comparison with the external loading. Leonov and Panasyuk used the
argument that in the crack mouth there exist strong adhesive forces which must
be overcome to initiate the growth of the crack. Both models lead to the same
nonlinear mathematical problem
and the same fracture criterion
for crack growth under Mode I loading. It can be generalized to the case of orthotropic media, like
in the case of isotropic media it coincides with the classical energy criterion (and Irwin`s criterion, resp.)
asymptotically. With asymptotic analysis it can be shown how the geometry of the problem influences the solution, this
leads to a refined Dugdale-criterion. By introducing a generalized energy functional this refined criterion can be interpreted
as an energy criterion as well. |
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