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August 2018, 15(4): 933-959. doi: 10.3934/mbe.2018042

The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics

1. 

Politecnico di Milano, Dipartimento di Matematica, Italy

2. 

University of Missouri, Department of Electrical Engineering and Computer Science, USA

3. 

NC State University, Department of Mathematics, USA

4. 

Politecnico di Milano, Dipartimento di Matematica, Italy

* Corresponding author: lvbociu@ncsu.edu.

Received  July 24, 2017 Accepted  December 27, 2017 Published  March 2018

The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

Citation: Maurizio Verri, Giovanna Guidoboni, Lorena Bociu, Riccardo Sacco. The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics. Mathematical Biosciences & Engineering, 2018, 15 (4) : 933-959. doi: 10.3934/mbe.2018042
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P. Augat and S. Schorlemmer, The role of cortical bone and its microstructure in bone strength, Age and Ageing, 35 (2006), ii27-ii31. doi: 10.1093/ageing/afl081.

[2]

H. T. BanksK. Bekele-MaxwellL. BociuM. Noorman and G. Guidoboni, Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data, Quart. Appl. Math. In press, 75 (2017), 697-735. doi: 10.1090/qam/1475.

[3]

H. BarucqM. Madaune-Tort and P. Saint-Macary, On nonlinear Biot's consolidation models, Nonlinear Anal Theory Methods Appl., 63 (2005), e985-e995, Invited Talks from the Fourth World Congress of Nonlinear Analysts (WCNA 2004). doi: 10.1016/j.na.2004.12.010.

[4]

H. BarucqM. Madaune-Tort and P. Saint-Macary, Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models, Nonlinear Anal Theory Methods Appl., 61 (2005), 591-612. doi: 10.1016/j.na.2004.10.023.

[5]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164. doi: 10.1063/1.1712886.

[6]

L. BociuG. GuidoboniR. Sacco and J. Webster, Analysis of nonlinear poro-elastic and poro-viscoelastic models, Arch. Rational Mech. Anal., 222 (2016), 1445-1519. doi: 10.1007/s00205-016-1024-9.

[7]

S. CanicJ. TambacaG. GuidoboniA. MikelicC. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006), 164-193. doi: 10.1137/060651562.

[8]

Y. CaoS. Chen and A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, Discrete Continuous Dyn Syst Ser B, 18 (2013), 1253-1273. doi: 10.3934/dcdsb.2013.18.1253.

[9]

P. CausinG. GuidoboniA. HarrisD. PradaR. Sacco and S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Math. Biosci., 257 (2014), 33-41. doi: 10.1016/j.mbs.2014.08.002.

[10]

P. CausinR. Sacco and M. Verri, A multiscale approach in the computational modeling of the biophysical environment in artificial cartilage tissue regeneration, Biomech. Model. Mechanobiol, 12 (2013), 763-780. doi: 10.1007/s10237-012-0440-5.

[11]

D. ChapelleJ. Sainte-MarieJ.-F. Gerbeau and I. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 46 (2010), 91-101. doi: 10.1007/s00466-009-0452-x.

[12]

O. Coussy, Poromechanics, John Wiley & Sons Ltd, 2004. doi: 10.1002/0470092718.

[13]

S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999), 217-238.

[14]

R. de Boer, Theory of Porous Media. Highlights in the Historical Development and Current State, Springer, Berlin/New York, 2000.

[15]

E. Detournay and A. H.-D. Cheng, Poroelastic response of a borehole in a non-hydrostatic stress field, Int J Rock Mech Min Sci Geomech Abstr, 25 (1988), 171-182. doi: 10.1016/0148-9062(88)92299-1.

[16]

E. Detournay and A. H.-D. Cheng, Fundamentals of poroelasticity, Comprehensive rock engineering, 2 (1993), 113-171. doi: 10.1016/B978-0-08-040615-2.50011-3.

[17]

J. C. DownsJ. K. SuhK. A. ThomasA. J. BellezzaR. T. Hart and C. F. Burgoyne, Viscoelastic material properties of the peripapillary sclera in normal and early-glaucoma monkey eyes, Invest. Ophthalmol. Vis. Sci., 46 (2005), 540-546. doi: 10.1167/iovs.04-0114.

[18]

J. W. FreemanM. D. WoodsD. A. CromerL. D. Wright and C. T. Laurencin, Tissue engineering of the anterior cruciate ligament: The viscoelastic behavior and cell viability of a novel braid-twist scaffold, J Biomater Sci Polym Ed, 20 (2009), 1709-1728. doi: 10.1163/156856208X386282.

[19]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier/Academic Press, Amsterdam, 2015.

[20]

A. P. GuérinB. PannierS. J. Marchais and G. M. London, Arterial structure and function in end-stage renal disease, Curr. Hypertens. Rep., 10 (2008), 107-111.

[21]

W. M. LaiJ. S. Hou and V. C. Mow, A triphasic theory for the swelling and deformation behaviors of articular cartilage, ASME J. Biomech. Eng., 113 (1991), 245-258. doi: 10.1115/1.2894880.

[22]

V. C. MowS. C. KueiW. M. Lai and C. G. Armstrong, Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments, ASME J. Biomech. Eng., 102 (1980), 73-84. doi: 10.1115/1.3138202.

[23]

H. T. NiaL. HanY. LiC. Ortiz and A. Grodzinsky, Poroelasticity of cartilage at the nanoscale, Biophys. J., 101 (2011), 2304-2313. doi: 10.1016/j.bpj.2011.09.011.

[24]

M. S. OsidakE. O. OsidakM. A. AkhmanovaS. P. Domogatsky and A. S. Domogatskaya, Fibrillar, fibril-associated and basement membrane collagens of the arterial wall: Architecture, elasticity and remodeling under stress, Curr. Pharm. Des., 21 (2015), 1124-1133. doi: 10.2174/1381612820666141013122906.

[25]

S. Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids, 15 (2010), 42-56. doi: 10.1177/1081286508090966.

[26]

N. Özkaya, M. Nordin, D. Goldsheyder and D. Leger, Fundamentals of Biomechanics. Equilibrium, Motion, and Deformation, Springer, New York, 1999.

[27]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅰ: The continuous in time case, Comput Geosci, 11 (2007), 131-144. doi: 10.1007/s10596-007-9045-y.

[28]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅱ: The continuous in time case, Comput Geosci, 11 (2007), 145-158. doi: 10.1007/s10596-007-9044-z.

[29]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity, Comput Geosci, 12 (2008), 417-435. doi: 10.1007/s10596-008-9082-1.

[30]

L. Recha-Sancho, F. T. Moutos, J. Abell, F. Guilak and C. E. Semino, Dedifferentiated human articular chondrocytes redifferentiate to a cartilage-like tissue phenotype in a poly($\varepsilon$-caprolactone)/self-assembling peptide composite scaffold, Materials, 9 (2016), p472.

[31]

R. SaccoP. CausinP. Zunino and M. T. Raimondi, A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor, Biomech. Model. Mechanobiol., 10 (2011), 577-589. doi: 10.1007/s10237-010-0257-z.

[32]

I. SackB. BeierbachJ. WuerfelD. KlattU. HamhaberS. PapazoglouP. Martus and J. Braun, The impact of aging and gender on brain viscoelasticity, NeuroImage, 46 (2009), 652-657. doi: 10.1016/j.neuroimage.2009.02.040.

[33]

A. P. S. Selvadurai, On the mechanics of damage-susceptible poroelastic media, Key Engineering Materials, 251/252 (2003), 363-374. doi: 10.4028/www.scientific.net/KEM.251-252.363.

[34]

A. Settari and D. A. Walters, Advances in Coupled Geomechanical and Reservoir Modeling with Applications to Reservoir Compaction, Technical report, SPE Reservoir Simulation Symposium, 1999. doi: 10.2118/51927-MS.

[35]

R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000), 310-340. doi: 10.1006/jmaa.2000.7048.

[36]

R. E. Showalter and N. Su, Partially saturated flow in a poroelastic medium, Discrete Continuous Dyn Syst Ser B, 1 (2001), 403-420. doi: 10.3934/dcdsb.2001.1.403.

[37]

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.

[38]

K. Terzaghi, Erdbaumechanik auf Bodenphysikalischer Grundlage, Deuticke, Wien, 1925.

[39]

A. A. Tofangchi Mahyari, Computational Modelling of Fracture and Damage in Poroelastic Media, PhD thesis, Department of Civil Engineering and Applied Mechanics, McGill University, 1997.

[40]

A. Zenisek, The existence and uniqueness theorem in Biot's consolidation theory, Apl Mat, 29 (1984), 194-211.

show all references

References:
[1]

P. Augat and S. Schorlemmer, The role of cortical bone and its microstructure in bone strength, Age and Ageing, 35 (2006), ii27-ii31. doi: 10.1093/ageing/afl081.

[2]

H. T. BanksK. Bekele-MaxwellL. BociuM. Noorman and G. Guidoboni, Sensitivity analysis in poro-elastic and poro-visco-elastic models with respect to boundary data, Quart. Appl. Math. In press, 75 (2017), 697-735. doi: 10.1090/qam/1475.

[3]

H. BarucqM. Madaune-Tort and P. Saint-Macary, On nonlinear Biot's consolidation models, Nonlinear Anal Theory Methods Appl., 63 (2005), e985-e995, Invited Talks from the Fourth World Congress of Nonlinear Analysts (WCNA 2004). doi: 10.1016/j.na.2004.12.010.

[4]

H. BarucqM. Madaune-Tort and P. Saint-Macary, Some existence-uniqueness results for a class of one-dimensional nonlinear Biot models, Nonlinear Anal Theory Methods Appl., 61 (2005), 591-612. doi: 10.1016/j.na.2004.10.023.

[5]

M. A. Biot, General theory of three-dimensional consolidation, J. Appl. Phys., 12 (1941), 155-164. doi: 10.1063/1.1712886.

[6]

L. BociuG. GuidoboniR. Sacco and J. Webster, Analysis of nonlinear poro-elastic and poro-viscoelastic models, Arch. Rational Mech. Anal., 222 (2016), 1445-1519. doi: 10.1007/s00205-016-1024-9.

[7]

S. CanicJ. TambacaG. GuidoboniA. MikelicC. J. Hartley and D. Rosenstrauch, Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow, SIAM J. Appl. Math., 67 (2006), 164-193. doi: 10.1137/060651562.

[8]

Y. CaoS. Chen and A. J. Meir, Analysis and numerical approximations of equations of nonlinear poroelasticity, Discrete Continuous Dyn Syst Ser B, 18 (2013), 1253-1273. doi: 10.3934/dcdsb.2013.18.1253.

[9]

P. CausinG. GuidoboniA. HarrisD. PradaR. Sacco and S. Terragni, A poroelastic model for the perfusion of the lamina cribrosa in the optic nerve head, Math. Biosci., 257 (2014), 33-41. doi: 10.1016/j.mbs.2014.08.002.

[10]

P. CausinR. Sacco and M. Verri, A multiscale approach in the computational modeling of the biophysical environment in artificial cartilage tissue regeneration, Biomech. Model. Mechanobiol, 12 (2013), 763-780. doi: 10.1007/s10237-012-0440-5.

[11]

D. ChapelleJ. Sainte-MarieJ.-F. Gerbeau and I. Vignon-Clementel, A poroelastic model valid in large strains with applications to perfusion in cardiac modeling, Comput. Mech., 46 (2010), 91-101. doi: 10.1007/s00466-009-0452-x.

[12]

O. Coussy, Poromechanics, John Wiley & Sons Ltd, 2004. doi: 10.1002/0470092718.

[13]

S. C. Cowin, Bone poroelasticity, J. Biomech., 32 (1999), 217-238.

[14]

R. de Boer, Theory of Porous Media. Highlights in the Historical Development and Current State, Springer, Berlin/New York, 2000.

[15]

E. Detournay and A. H.-D. Cheng, Poroelastic response of a borehole in a non-hydrostatic stress field, Int J Rock Mech Min Sci Geomech Abstr, 25 (1988), 171-182. doi: 10.1016/0148-9062(88)92299-1.

[16]

E. Detournay and A. H.-D. Cheng, Fundamentals of poroelasticity, Comprehensive rock engineering, 2 (1993), 113-171. doi: 10.1016/B978-0-08-040615-2.50011-3.

[17]

J. C. DownsJ. K. SuhK. A. ThomasA. J. BellezzaR. T. Hart and C. F. Burgoyne, Viscoelastic material properties of the peripapillary sclera in normal and early-glaucoma monkey eyes, Invest. Ophthalmol. Vis. Sci., 46 (2005), 540-546. doi: 10.1167/iovs.04-0114.

[18]

J. W. FreemanM. D. WoodsD. A. CromerL. D. Wright and C. T. Laurencin, Tissue engineering of the anterior cruciate ligament: The viscoelastic behavior and cell viability of a novel braid-twist scaffold, J Biomater Sci Polym Ed, 20 (2009), 1709-1728. doi: 10.1163/156856208X386282.

[19]

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Elsevier/Academic Press, Amsterdam, 2015.

[20]

A. P. GuérinB. PannierS. J. Marchais and G. M. London, Arterial structure and function in end-stage renal disease, Curr. Hypertens. Rep., 10 (2008), 107-111.

[21]

W. M. LaiJ. S. Hou and V. C. Mow, A triphasic theory for the swelling and deformation behaviors of articular cartilage, ASME J. Biomech. Eng., 113 (1991), 245-258. doi: 10.1115/1.2894880.

[22]

V. C. MowS. C. KueiW. M. Lai and C. G. Armstrong, Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments, ASME J. Biomech. Eng., 102 (1980), 73-84. doi: 10.1115/1.3138202.

[23]

H. T. NiaL. HanY. LiC. Ortiz and A. Grodzinsky, Poroelasticity of cartilage at the nanoscale, Biophys. J., 101 (2011), 2304-2313. doi: 10.1016/j.bpj.2011.09.011.

[24]

M. S. OsidakE. O. OsidakM. A. AkhmanovaS. P. Domogatsky and A. S. Domogatskaya, Fibrillar, fibril-associated and basement membrane collagens of the arterial wall: Architecture, elasticity and remodeling under stress, Curr. Pharm. Des., 21 (2015), 1124-1133. doi: 10.2174/1381612820666141013122906.

[25]

S. Owczarek, A Galerkin method for Biot consolidation model, Math. Mech. Solids, 15 (2010), 42-56. doi: 10.1177/1081286508090966.

[26]

N. Özkaya, M. Nordin, D. Goldsheyder and D. Leger, Fundamentals of Biomechanics. Equilibrium, Motion, and Deformation, Springer, New York, 1999.

[27]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅰ: The continuous in time case, Comput Geosci, 11 (2007), 131-144. doi: 10.1007/s10596-007-9045-y.

[28]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity Ⅱ: The continuous in time case, Comput Geosci, 11 (2007), 145-158. doi: 10.1007/s10596-007-9044-z.

[29]

P. J. Phillips and M. F. Wheeler, A coupling of mixed and continuous Galerkin finite-element methods for poroelasticity, Comput Geosci, 12 (2008), 417-435. doi: 10.1007/s10596-008-9082-1.

[30]

L. Recha-Sancho, F. T. Moutos, J. Abell, F. Guilak and C. E. Semino, Dedifferentiated human articular chondrocytes redifferentiate to a cartilage-like tissue phenotype in a poly($\varepsilon$-caprolactone)/self-assembling peptide composite scaffold, Materials, 9 (2016), p472.

[31]

R. SaccoP. CausinP. Zunino and M. T. Raimondi, A multiphysics/multiscale 2D numerical simulation of scaffold-based cartilage regeneration under interstitial perfusion in a bioreactor, Biomech. Model. Mechanobiol., 10 (2011), 577-589. doi: 10.1007/s10237-010-0257-z.

[32]

I. SackB. BeierbachJ. WuerfelD. KlattU. HamhaberS. PapazoglouP. Martus and J. Braun, The impact of aging and gender on brain viscoelasticity, NeuroImage, 46 (2009), 652-657. doi: 10.1016/j.neuroimage.2009.02.040.

[33]

A. P. S. Selvadurai, On the mechanics of damage-susceptible poroelastic media, Key Engineering Materials, 251/252 (2003), 363-374. doi: 10.4028/www.scientific.net/KEM.251-252.363.

[34]

A. Settari and D. A. Walters, Advances in Coupled Geomechanical and Reservoir Modeling with Applications to Reservoir Compaction, Technical report, SPE Reservoir Simulation Symposium, 1999. doi: 10.2118/51927-MS.

[35]

R. E. Showalter, Diffusion in poro-elastic media, J. Math. Anal. Appl., 251 (2000), 310-340. doi: 10.1006/jmaa.2000.7048.

[36]

R. E. Showalter and N. Su, Partially saturated flow in a poroelastic medium, Discrete Continuous Dyn Syst Ser B, 1 (2001), 403-420. doi: 10.3934/dcdsb.2001.1.403.

[37]

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.

[38]

K. Terzaghi, Erdbaumechanik auf Bodenphysikalischer Grundlage, Deuticke, Wien, 1925.

[39]

A. A. Tofangchi Mahyari, Computational Modelling of Fracture and Damage in Poroelastic Media, PhD thesis, Department of Civil Engineering and Applied Mechanics, McGill University, 1997.

[40]

A. Zenisek, The existence and uniqueness theorem in Biot's consolidation theory, Apl Mat, 29 (1984), 194-211.

Figure 1.  Illustration of a novel hypothesis on potential causes of microstructure damage in deformable porous media. In the case that (ⅰ) the medium is subjected to a time-discontinuous mechanical load and (ⅱ) structural viscoelasticity is reduced or absent, then the fluid velocity within the porous medium will experience a blow-up, possibly leading to microstructural damage.
Figure 2.  Schematic representation of the one-dimensional problem considered in this article. The deformable porous medium may be either poro-elastic or poro-visco-elastic. The forcing term $P(t)$ may have discontinuities in time.
Figure 3.  Schematic representation of the dimensionless step pulse $\hat{P}(\hat{t}~)$ defined in (36). Here, the signal is discontinuous at the switch on time.
Figure 4.  Dimensionless displacement $\hat{u}_{\hat{\eta}}$ as a function of $\hat{x}$ and $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$.
Figure 5.  Dimensionless discharge velocity $\hat{v}_{\hat{\eta}}$ as a function of $\hat{x}$ and $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$. In order to highlight the velocity blow-up at $\hat{x}=1$, $\hat{t}=0$, the $\log_{10}$ plot of $|\hat{v}_{\hat{\eta}}|$ is plotted in both panels.
Figure 6.  Dimensionless power density $\hat{\mathcal{P}}_{\hat{\eta}}$ as a function of $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$. In order to highlight the power density blow-up at $\hat{t}=0$, the $\log_{10}$ plot of $\hat{\mathcal{P}}_{\hat{\eta}}$ is plotted in both panels.
Figure 7.  Dimensionless discharge velocity $\hat{v}_{\hat{\eta}}(\hat{x},0)$ as a function of $\hat{x}$. In order to highlight the velocity blow-up at $\hat{x}=1$, $\hat{t}=0$, in the case $\hat{\eta}=0$, the $\log_{10}$ plot of $|\hat{v}_{\eta}|$ is plotted.
Figure 8.  Left panel: maximal discharge velocity as a function of $\hat{\eta}$. Right panel: power density as a function of $\hat{\eta}$. Log$_{10}$-scale is used on the $y$-axis to better highlight blow-up of both quantities as $\hat{\eta} \rightarrow 0$.
Figure 9.  Left panel: dimensional maximal discharge velocity as a function of $\hat{\eta}$. Right panel: dimensional power density as a function of $\hat{\eta}$. Log$_{10}$-scale is used on the $y$-axis to better highlight blow-up of both quantities as $\hat{\eta} \rightarrow 0$. We set $K_0/L = 1 ~ \mathrm{m^2 s Kg^{-1}}$. The black arrows indicate increasing values of $P_{\text{ref}}$ in the range $[10^{-3}, 10^3] ~ \mathrm{N m^{-2}}$.
Figure 10.  Schematic representation of the dimensionless trapezoidal pulse $\hat{P}(\hat{t})$ defined in (41). Here, the signal switch on and switch off are characterized by linear ramps.
Figure 11.  Dimensionless discharge velocity $\hat{V}_{\hat{\eta}}\left( \hat{x},\hat{t} \right) $ for $\hat{\eta}=0.1$, $\hat{\varepsilon}=0.2$, $\hat{\tau}=1$.
Figure 12.  Dimensionless maximal discharge velocity $\hat{V}_{\max }\left( \hat{\eta},\hat{\varepsilon}\right)$ as a function of $\hat{\eta}$ and $\hat{\varepsilon}$.
Figure 13.  Schematic representation of a confined compression chamber.
Figure 14.  Comparison between the dimensionless maximum velocity $\hat{V}_{\max}(\hat{\eta},\hat{\varepsilon})$ obtained in the case of trapezoidal pulse using expression (44) and the threshold velocity $\hat{V}_{\text{th}} = 16.3$ typical of confined compression experiments [37].
Figure 15.  Colormap of the difference $ \hat{V}_{\text{th}}-\hat{V}_{\max }$ as a function of $\hat{\eta}$ and $\hat{\varepsilon}$. As in Fig. 14, $\hat{V}_{\max}(\hat{\eta},\hat{\varepsilon})$ is obtained using expression (44) and $\hat{V}_{\text{th}} = 16.3$ is the typical velocity of confined compression experiments [37]. The curve in the parameter space for which $ \hat{V}_{\text{th}}=\hat{V}_{\max }$ is reported in a thick black mark.
Table 1.  Numerical values of model parameters in the confined compression experiment for articular cartilage reported in [37].
symbol value units
$L$ $0.81 \cdot 10^{-3}$ $\mathrm{m}$
$\mu$ $0.97 \cdot 10^6$ $\mathrm{N m^{-2}}$
$\eta$ $0$ $\mathrm{N ~s ~m^{-2}}$
$K_0$ $2.9 \cdot 10^{-16}$ $\mathrm{m^4 N^{-1} s^{-1}}$
$P_{\text{ref}}$ $6 \cdot 10^{4}$ $\mathrm{N m^{-2}}$
symbol value units
$L$ $0.81 \cdot 10^{-3}$ $\mathrm{m}$
$\mu$ $0.97 \cdot 10^6$ $\mathrm{N m^{-2}}$
$\eta$ $0$ $\mathrm{N ~s ~m^{-2}}$
$K_0$ $2.9 \cdot 10^{-16}$ $\mathrm{m^4 N^{-1} s^{-1}}$
$P_{\text{ref}}$ $6 \cdot 10^{4}$ $\mathrm{N m^{-2}}$
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