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Analysis of stability and bifurcation in nonlinear mechanics with dissipation. (English) Zbl 1229.74068

Summary: The analysis of stability and bifurcation is studied in nonlinear mechanics with dissipative mechanisms: plasticity, damage, fracture. The description is based on introduction of a set of internal variables. This framework allows a systematic description of the material behaviour via two potentials: the free energy and the potential of dissipation. In the framework of standard generalized materials the internal state evolution is governed by a variational inequality which depends on the mechanism of dissipation. This inequality is obtained through energetic considerations in an unified description based upon energy and driving forces associated to the dissipative process. This formulation provides criterion for existence and uniqueness of the system evolution. Examples are presented for plasticity, fracture and for damaged materials.

MSC:

74H60 Dynamical bifurcation of solutions to dynamical problems in solid mechanics
37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
35Q74 PDEs in connection with mechanics of deformable solids
74C15 Large-strain, rate-independent theories of plasticity (including nonlinear plasticity)
74R10 Brittle fracture
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