On a Viscoelastoplastic Porous Medium Problem with Nonlinear Interaction

Phase transitions in porous media

“…5.5 , we do not keep the exponent k bounded, but we let in a controlled way. As in ( 5.28 ), we define auxiliary functions and rewrite ( 5.25 ) for as with a constant independent of k , with given by ( 5.26 ), and with It follows from ( 5.32 ) or ( 5.33 ), ( 5.35 ), and ( 5.36 ) that and Repeating exactly the argument of the proof of [ 8 , Proposition 6.2] we obtain the following result.…”

“…This represents an enormous increase of mathematical complexity. While the momentum balance in the shear-stress-free case in [ 15 ] can be reduced to an ODE, here, we need to exploit deeper results from the theory of PDEs to control the interactions between individual components of the system, as well as a generalized Moser iteration scheme from [ 8 ]. Additional difficulties are due to the effects of the three heat sources produced by mechanical hysteresis dissipation (plasticity, capillarity, phase transitions).…”

Forward and inverse problems for creep models in viscoelasticity

“…We refer to the studies by Bulíček et al [22] and de Hoop et al [23] for the well-posedness of viscoelastic models; when accounting for contact in Han et al [24]; the study by Itou & Tani [25] for cracks; the studies by Khludnev & Kovtunenko [26] and Khludnev & Sokolowski [27] for cracks with contact. Furthermore, the studies by Anderssen et al [28] and Gavioli & Krejčí [29] are referred to address the hysteresis issues in viscoelasticity.…”

“…[24]; the study by Itou & Tani [25] for cracks; the studies by Khludnev & Kovtunenko [26] and Khludnev & Sokolowski [27] for cracks with contact. Furthermore, the studies by Anderssen et al [28] and Gavioli & Krejčí [29] are referred to address the hysteresis issues in viscoelasticity.…”

Phase transitions in porous media

“…This represents an enormous increase of mathematical complexity. While the momentum balance in the shear-stress-free case in [13] can be reduced to an ODE, here, we need to exploit deeper results from the theory of PDEs to control the interactions between individual components of the system, as well as a generalized Moser iteration scheme from [6]. Additional difficulties are due to the effects of the three heat sources produced by mechanical hysteresis dissipation (plasticity, capillarity, phase transitions).…”