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Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models

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Abstract

We consider the initial and boundary value problem for a system of partial differential equations describing the motion of a fluid–solid mixture under the assumption of full saturation. The ability of the fluid phase to flow within the solid skeleton is described by the permeability tensor, which is assumed here to be a multiple of the identity and to depend nonlinearly on the volumetric solid strain. In particular, we study the problem of the existence of weak solutions in bounded domains, accounting for non-zero volumetric and boundary forcing terms. We investigate the influence of viscoelasticity on the solution functional setting and on the regularity requirements for the forcing terms. The theoretical analysis shows that different time regularity requirements are needed for the volumetric source of linear momentum and the boundary source of traction depending on whether or not viscoelasticity is present. The theoretical results are further investigated via numerical simulations based on a novel dual mixed hybridized finite element discretization. When the data are sufficiently regular, the simulations show that the solutions satisfy the energy estimates predicted by the theoretical analysis. Interestingly, the simulations also show that, in the purely elastic case, the Darcy velocity and the related fluid energy might become unbounded if indeed the data do not enjoy the time regularity required by the theory.

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, SAS Hall 3236, Raleigh, NC, 27695, USA

    Lorena Bociu

  2. Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, 402 N. Blackford St, Indianapolis, IN, 46202, USA

    Giovanna Guidoboni

  3. Dipartimento di Matematica, Politecnico di Milano, Via Bonardi 9, 20133, Milano, Italy

    Riccardo Sacco

  4. Department of Mathematics, College of Charleston, 66 George St, Charleston, SC, 29424, USA

    Justin T. Webster

Authors
  1. Lorena Bociu
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  2. Giovanna Guidoboni
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  3. Riccardo Sacco
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  4. Justin T. Webster
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Correspondence to Giovanna Guidoboni.

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Communicated by I. Fonseca

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Bociu, L., Guidoboni, G., Sacco, R. et al. Analysis of Nonlinear Poro-Elastic and Poro-Visco-Elastic Models. Arch Rational Mech Anal 222, 1445–1519 (2016). https://doi.org/10.1007/s00205-016-1024-9

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