Explicit energy-minimizers of incompressible elastic brittle bars under uniaxial extension

A Derivation of Linearized Griffith Energies from Nonlinear Models

“…Analogous results for the case of expansive boundary values have been obtained in [37] and [31]. We thus do not repeat all the steps of these proofs but rather concentrate on the additional arguments necessary in our general setting (see (2.11)) in which we particularly can extend the aforementioned results to the case of compression.…”

“…Let us emphasize that the cleavage law is derived for a special geometry of by solving a static, global minimization problem similarly as in [10,30,37]. An accurate prediction of crack propagation under tensile loading is beyond the scope of the present contribution.…”

“…We recall that the proof of the cleavage laws in [29,30,37] fundamentally relied on the application of certain slicing techniques which were not suitable to treat the case of compression. Having general compactness and -convergence results we can now complete the picture about cleavage laws by extending the results to the case of uniaxial compression.…”

“…Finally, in Section 6.2 we prove a cleavage law and extend the results obtained in [29,30,37] to the case of uniaxial compression, where we essentially follow the proof in [31,37], in particular using a piecewise rigidity result in SBD (see [15]) and a structure theorem on the boundary of sets of finite perimeter (see [25]). It turns out that in the linearized limit the behavior for compression and extension is virtually identical.…”

“…As discussed before, the identification of critical loads and the investigation of crack paths is a challenging problem particularly for nonlinear models. The arguments in [29,30,37], where boundary value problems of uniaxial extension for brittle materials were investigated, fundamentally relied on the application of certain slicing techniques and due to the lack of convexity were not adapted to treat the case of compression. Our general -limit result can now be applied to also solve boundary value problems of uniaxial compression which is, like the uniaxial tension test, a natural and interesting problem.…”