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Diffusion in poro-plastic media. (English) Zbl 1095.74011
Summary: A model is developed for the flow of a slightly compressible fluid through a saturated inelastic porous medium. The initial-boundary value problem is a system that consists of the diffusion equation for the fluid coupled to the momentum equation for the porous solid together with a constitutive law which includes a possibly hysteretic relation of elasto-viscoplastic type. The variational form of this problem in Hilbert space is a nonlinear evolution equation for which the existence and uniqueness of a global strong solution is proved by means of monotonicity methods. Various degenerate situations are permitted, such as incompressible fluid, negligible porosity, or a quasi-static momentum equation. The essential sufficient conditions for the well-posedness of the system consist of an ellipticity condition on the term for diffusion of fluid and either a viscous or a hardening assumption in the constitutive relation for the porous solid.

MSC:
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
74S05 Finite element methods applied to problems in solid mechanics
74H20 Existence of solutions of dynamical problems in solid mechanics
74H25 Uniqueness of solutions of dynamical problems in solid mechanics
35Q72 Other PDE from mechanics (MSC2000)
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
76R50 Diffusion
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