On the role of compressibility in poroviscoelastic models

Lorena Bociu; Giovanna Guidoboni; Riccardo Sacco; Maurizio Verri; Lorena Bociu; Giovanna Guidoboni; Riccardo Sacco; Maurizio Verri

doi:10.3934/mbe.2019308

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 5: 6167-6208. doi: 10.3934/mbe.2019308

Previous Article Next Article

Research article

On the role of compressibility in poroviscoelastic models

1.
Department of Mathematics, NC State University, SAS Hall 3236, Raleigh, NC 27695, USA
2.
Department of Electrical Engineering and Computer Science, University of Missouri, 201 Naka Hall, Columbia, MO 65211, USA
3.
Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received: 15 March 2019 Accepted: 19 June 2019 Published: 03 July 2019

In this article we conduct an analytical study of a poroviscoelastic mixture model stemming from the classical Biot's consolidation model for poroelastic media, comprising a fluid component and a solid component, coupled with a viscoelastic stress-strain relationship for the total stress tensor. The poroviscoelastic mixture is studied in the one-dimensional case, corresponding to the experimental conditions of confined compression. Upon assuming (i) negligible inertial effects in the balance of linear momentum for the mixture, (ii) a Kelvin-Voigt model for the effective stress tensor and (iii) a constant hydraulic permeability, we obtain an initial value/boundary value problem of pseudo-parabolic type for the spatial displacement of the solid component of the mixture. The dimensionless form of the differential equation is characterized by the presence of two positive parameters $\gamma$ and $\eta$, representing the contributions of compressibility and structural viscoelasticity, respectively. Explicit solutions are obtained for different functional forms characterizing the boundary traction. The main result of our analysis is that the compressibility of the components of a poroviscoelastic mixture does not give rise to unbounded responses to non-smooth traction data. Interestingly, compressibility allows the system to store potential energy as its components are elastically compressed, thereby providing an additional mechanism that limits the maximum of the discharge velocity when the imposed boundary traction is irregular in time.
- porous media flow,
- viscoelasticity,
- explicit solutions,
- well-posedness,
- compressible components
Citation: Lorena Bociu, Giovanna Guidoboni, Riccardo Sacco, Maurizio Verri. On the role of compressibility in poroviscoelastic models[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 6167-6208. doi: 10.3934/mbe.2019308

Related Papers:

Abstract

In this article we conduct an analytical study of a poroviscoelastic mixture model stemming from the classical Biot's consolidation model for poroelastic media, comprising a fluid component and a solid component, coupled with a viscoelastic stress-strain relationship for the total stress tensor. The poroviscoelastic mixture is studied in the one-dimensional case, corresponding to the experimental conditions of confined compression. Upon assuming (i) negligible inertial effects in the balance of linear momentum for the mixture, (ii) a Kelvin-Voigt model for the effective stress tensor and (iii) a constant hydraulic permeability, we obtain an initial value/boundary value problem of pseudo-parabolic type for the spatial displacement of the solid component of the mixture. The dimensionless form of the differential equation is characterized by the presence of two positive parameters $\gamma$ and $\eta$, representing the contributions of compressibility and structural viscoelasticity, respectively. Explicit solutions are obtained for different functional forms characterizing the boundary traction. The main result of our analysis is that the compressibility of the components of a poroviscoelastic mixture does not give rise to unbounded responses to non-smooth traction data. Interestingly, compressibility allows the system to store potential energy as its components are elastically compressed, thereby providing an additional mechanism that limits the maximum of the discharge velocity when the imposed boundary traction is irregular in time.

References

[1]	L. Bociu, G. Guidoboni, R. Sacco, et al., Analysis of nonlinear poro-elastic and poro-viscoelastic models. Arch. Rational Mech. Anal., 222 (2016), 1445–1519.
[2]	C.-Y. Huang, V. C. Mow and G. A. Ateshian, The role of flow-independent viscoelasticity in the biphasic tensile and compressive responses of articular cartilage. J. Biomech. Eng., 123 (2001), 410–417.
[3]	C.-Y. Huang, M. A. Soltz, M. Kopacz, et al., Experimental verification of the roles of intrinsic matrix viscoelasticity and tension-compression nonlinearity in the biphasic response of cartilage. J. Biomech. Eng., 125 (2003), 84–93.
[4]	M. Verri, G. Guidoboni, L. Bociu, et al., The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics. Math. Biosci. Eng., 15 (2018), 933.
[5]	D. Prada, A. Harris, G. Guidoboni, et al., Autoregulation and neurovascular coupling in the optic nerve head. Surv. Ophthalmol., 61 (2016), 164–186.
[6]	P. Causin, G. Guidoboni, A. Harris, et al., A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head. Math. Biosci., 257 (2014), 33–41.
[7]	J. C. Gross, A. Harris, B. A. Siesky, et al., Mathematical modeling for novel treatment approaches to open-angle glaucoma. Expert Rev. Ophthalmol., 12 (2017), 443–455.
[8]	A. Harris, G. Guidoboni, J. C. Arciero, et al., Ocular hemodynamics and glaucoma: the role of mathematical modeling. Eur. J. Ophthalmol., 23 (2013), 139–146.
[9]	J. H. Kim and J. Caprioli,. Intraocular pressure fluctuation: Is it important? J. Ophthalmic Vis. Res., 13 (2018), 170.
[10]	G. Guidoboni, F. Salerni, A. Harris, et al., Ocular and cerebral hemo-fluid dynamics in microgravity: a mathematical model. Invest. Ophth. Vis. Sci., 58 (2017), 3036–3036.
[11]	A. G. Lee, T. H. Mader, C. R. Gibson, et al., Space flight-associated neuro-ocular syndrome (sans). Eye, 32 (2018), 1164–1167.
[12]	M. Schanz and A.-D. Cheng. Dynamic analysis of a one-dimensional poroviscoelastic column. J. Appl. Mech., 68 (2001), 192–198.
[13]	R. E. Showalter. Diffusion in poro-elastic media. J. Math. Anal. Appl., 251 (2000), 310–340.
[14]	M. Biot. General theory of three-dimensional consolidation. J. Appl. Phys., 12 (1941), 155–164.
[15]	E. Detournay and A.-D. Cheng. Fundamentals of poroelasticity. In J. A. Hudson, editor, Comprehensive Rock Eng., 2 (1993), 113–171. Pergamon.
[16]	M. A. Soltz and G. A. Ateshian. Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an impermeable contact interface in confined compression. J. Biomech., 31 (1998), 927–934.
[17]	F. Treves. Basic Linear Partial Differential Equations. Academic Press, 1975.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)