Derivation of a poroelastic flexural shell model

In this paper we investigate the limit behavior of the solution to quasi-static Biot's equations in thin poroelastic flexural shells as the thickness of the shell tends to zero and extend the results obtained for the poroelastic plate by Marciniak-Czochra and Mikeli\'c. We choose Terzaghi's time corresponding to the shell thickness and obtain the strong convergence of the three-dimensional solid displacement, fluid pressure and total poroelastic stress to the solution of the new class of shell equations. The derived bending equation is coupled with the pressure equation and it contains the bending moment due to the variation in pore pressure across the shell thickness. The effective pressure equation is parabolic only in the normal direction. As additional terms it contains the time derivative of the middle-surface flexural strain. Derivation of the model presents an extension of the results on the derivation of classical linear elastic shells by Ciarlet and collaborators to the poroelastic shells case. The new technical points include determination of the 2 X 2 strain matrix, independent of the vertical direction, in the limit of the rescaled strains and identification of the pressure equation. This term is not necessary to be determined in order to derive the classical flexural shell model.Comment: 32 page

Numerička aproksimacija modela poroelastične ljuske

U ovom radu uveli smo model poroelastične ljuske Naghdijevog tipa, koji je najsličniji modelu elastične ljuske opisanom u [9, 8]. Za razliku od modela poroelastične ljuske opisanom u [5] model opisan u radu uz fleksijske dopušta i membranske efekte. Glavni rezultat u radu su dokaz egzistencije i jedinstvenosti sustava. Dokaz jedinstvenosti rješenja temelji se na metodi energija, dok se egzistencija temelji na iterativnom postupku sličnom numeričkoj metodi opisanoj u [4]. Na kraju smo i pokazali neke numeričke aproksimacije rješenja modela u dva slučaja zanimljiva u primjeni, poput filtera u automobilima. Metoda dokaza egzistencije rješenja i numerička metoda kojom smo dobili rezultate u zadnjem poglavlju nisu ista metoda. Stoga je cilj daljnje istraživanje usmjeriti ka implementaciji prve numeričke metode i formalnom opravdanju druge.In this thesis we introduced the poroelastic shell model of Naghdi’s type, which is the most similar to the elastic shell model described in [9, 8]. While the poroelastic shell model described in [5] captures only bending effects, the model introduced in this thesis captures membrane effects, as well. The main result in this thesis is the proof of existence and uniqueness of the system of equations. The proof of uniqueness is based on the energy method, while the proof of existence is based on iterative procedure similar to the numerical method described in [4]. At the end, we showed numerical approximation of solutions of the model in two cases interesting in application, such as car filters. The method which we used in the proof of existence of solutions and numerical method which we used to obtain numerical results are not the same method. Thus, our goal is to extend our research to implementation of the first method and justification of the second one

Numerička aproksimacija modela poroelastične ljuske

U ovom radu uveli smo model poroelastične ljuske Naghdijevog tipa, koji je najsličniji modelu elastične ljuske opisanom u [9, 8]. Za razliku od modela poroelastične ljuske opisanom u [5] model opisan u radu uz fleksijske dopušta i membranske efekte. Glavni rezultat u radu su dokaz egzistencije i jedinstvenosti sustava. Dokaz jedinstvenosti rješenja temelji se na metodi energija, dok se egzistencija temelji na iterativnom postupku sličnom numeričkoj metodi opisanoj u [4]. Na kraju smo i pokazali neke numeričke aproksimacije rješenja modela u dva slučaja zanimljiva u primjeni, poput filtera u automobilima. Metoda dokaza egzistencije rješenja i numerička metoda kojom smo dobili rezultate u zadnjem poglavlju nisu ista metoda. Stoga je cilj daljnje istraživanje usmjeriti ka implementaciji prve numeričke metode i formalnom opravdanju druge.In this thesis we introduced the poroelastic shell model of Naghdi’s type, which is the most similar to the elastic shell model described in [9, 8]. While the poroelastic shell model described in [5] captures only bending effects, the model introduced in this thesis captures membrane effects, as well. The main result in this thesis is the proof of existence and uniqueness of the system of equations. The proof of uniqueness is based on the energy method, while the proof of existence is based on iterative procedure similar to the numerical method described in [4]. At the end, we showed numerical approximation of solutions of the model in two cases interesting in application, such as car filters. The method which we used in the proof of existence of solutions and numerical method which we used to obtain numerical results are not the same method. Thus, our goal is to extend our research to implementation of the first method and justification of the second one

Numerička aproksimacija modela poroelastične ljuske

U ovom radu uveli smo model poroelastične ljuske Naghdijevog tipa, koji je najsličniji modelu elastične ljuske opisanom u [9, 8]. Za razliku od modela poroelastične ljuske opisanom u [5] model opisan u radu uz fleksijske dopušta i membranske efekte. Glavni rezultat u radu su dokaz egzistencije i jedinstvenosti sustava. Dokaz jedinstvenosti rješenja temelji se na metodi energija, dok se egzistencija temelji na iterativnom postupku sličnom numeričkoj metodi opisanoj u [4]. Na kraju smo i pokazali neke numeričke aproksimacije rješenja modela u dva slučaja zanimljiva u primjeni, poput filtera u automobilima. Metoda dokaza egzistencije rješenja i numerička metoda kojom smo dobili rezultate u zadnjem poglavlju nisu ista metoda. Stoga je cilj daljnje istraživanje usmjeriti ka implementaciji prve numeričke metode i formalnom opravdanju druge.In this thesis we introduced the poroelastic shell model of Naghdi’s type, which is the most similar to the elastic shell model described in [9, 8]. While the poroelastic shell model described in [5] captures only bending effects, the model introduced in this thesis captures membrane effects, as well. The main result in this thesis is the proof of existence and uniqueness of the system of equations. The proof of uniqueness is based on the energy method, while the proof of existence is based on iterative procedure similar to the numerical method described in [4]. At the end, we showed numerical approximation of solutions of the model in two cases interesting in application, such as car filters. The method which we used in the proof of existence of solutions and numerical method which we used to obtain numerical results are not the same method. Thus, our goal is to extend our research to implementation of the first method and justification of the second one