On an isotropic porous solid cylinder: the analytical solution and sensitivity analysis of the pressure

doi:10.1007/s10483-024-3144-7

摘要/Abstract

Abstract:

Within this work, we perform a sensitivity analysis to determine the influence of the material input parameters on the pressure in an isotropic porous solid cylinder. We provide a step-by-step guide to obtain the analytical solution for a porous isotropic elastic cylinder in terms of the pressure, stresses, and elastic displacement. We obtain the solution by performing a Laplace transform on the governing equations, which are those of Biot's poroelasticity in cylindrical polar coordinates. We enforce radial boundary conditions and obtain the solution in the Laplace transformed domain before reverting back to the time domain. The sensitivity analysis is then carried out, considering only the derived pressure solution. This analysis finds that the time t, Biot's modulus M, and Poisson's ratio ν have the highest influence on the pressure whereas the initial value of pressure P₀ plays a very little role.

Key words: sensitivity analysis, Laplace transform, cylindrical polar coordinate, Biot's modulus, cylinder

中图分类号:

. [J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(9): 1499-1522.

H. ASGHARI, L. MILLER, R. PENTA, J. MERODIO. On an isotropic porous solid cylinder: the analytical solution and sensitivity analysis of the pressure[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(9): 1499-1522.

图/表 8

参考文献 67

1	BIOT, M. A. Theory of elasticity and consolidation for a porous anisotropic solid. Journal of Applied Physics, 26 (2), 182- 185 (1955)
2	BIOT, M. A. General solutions of the equations of elasticity and consolidation for a porous material. Journal of Applied Physics, 23 (1), 91- 96 (1956)
3	BIOT, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid, Ⅱ: higher frequency range. The Journal of the Acoustical Society of America, 28 (2), 179- 191 (1956)
4	BIOT, M. A. Mechanics of deformation and acoustic propagation in porous media. Journal of Applied Physics, 33 (4), 1482- 1498 (1962)
5	COWIN, S. C. Bone poroelasticity. Journal of Biomechanics, 32 (3), 217- 238 (1999)
6	WEINER, S., and WAGNER, H. D. The material bone: structure-mechanical function relations. Annual Review of Materials Science, 28 (1), 271- 298 (1998)
7	COOKŠÒN, A., LEE, J., MICHLER, C., CHABINIOK, R., HYDE, E., NORDSLETTEN, D., SINCLAIR, M., SIEBES, M., and SMITH, N. A novel porous mechanical framework for modelling the interaction between coronary perfusion and myocardial mechanics. Journal of Biomechanics, 45 (5), 850- 855 (2012)
8	MAY-NEWMAN, K., and MCCULLOCH, A. D. Homogenization modeling for the mechanics of perfused myocardium. Progress in Biophysics and Molecular Biology, 69 (2), 463- 481 (1998)
9	BUKAC, M., YOTOV, I., ZAKERZADEH, R., and ZUNINO, P. Effects of Poroelasticity on Fluid-Structure Interaction in Arteries: a Computational Sensitivity Study, Springer International Publishing, Berlin, 197–220 (2015)
10	CHAPELLE, D., GERBEAU, J. F., SAINTE-MARIE, J., and VIGNON-CLEMENTEL, I. E. A poroelastic model valid in large strains with applications to perfusion in cardiac modeling. Computational Mechanics, 46 (1), 91- 101 (2010)
11	BOTTARO, A., and ANSALDI, T. On the infusion of a therapeutic agent into a solid tumor modeled as a poroelastic medium. Journal of Biomechanical Engineering, 134 (8), 084501 (2012)
12	FLESSNER, M. F. The role of extracellular matrix in transperitoneal transport of water and solutes. Peritoneal Dialysis International, 21 (3), S24- S29 (2001)
13	CHALASANI, R., POOLE-WARREN, L., CONWAY, R. M., and BEN-NISSAN, B. Porous orbital implants in enucleation: a systematic review. Survey of Ophthalmology, 52 (2), 145- 155 (2007)
14	KARAGEORGIOU, V., and KAPLAN, D. Porosity of 3D biomaterial scaffolds and osteogenesis. Biomaterials, 26 (27), 5474- 5491 (2005)
15	KÜMPEL, H. J. Poroelasticity: parameters reviewed. Geophysical Journal International, 105 (3), 783- 799 (1991)
16	WANG, H. F. Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology, Princeton University Press, Princeton (2017)
17	CHENG, A. H. D. Analytical Solution, Springer International Publishing, Berlin 229- 396 (2016)
18	LEEUW, D. The theory of three-dimensional consolidation applied to cylindrical bodies. Proceeding of the 6th ICSMFE, Shanghai (1965)
19	CARCIONE, J. M., MORENCY, C., and SANTOS, J. E. Computational poroelasticity: a review. Geophysics, 75 (5), 75A229- 75A243 (2010)
20	SALTELLI, A., RATTO, M., ANDRES, T., CAMPOLONGO, F., CARIBONI, J., GATELLI, D., SAISANA, M., and TARANTOLA, S. Global Sensitivity Analysis: the Primer, John Wiley and Sons, New York (2008)
21	MEI, C., and VERNESCU, B. Homogenization Methods for Multiscale Mechanics, World Scientific, Singapore (2010)
22	SALTELLI, A., TARANTOLA, S., and CHAN, K. S. A quantitative model-independent method for global sensitivity analysis of model output. Technometrics, 41 (1), 39- 56 (1999)
23	ANSTETT-COLLIN, F., MARA, T., and BASSET, M. Application of global sensitivity analysis to a tire model with correlated inputs. Simulation Modelling Practice and Theory, 44, 54- 62 (2014)
24	SAISANA, M., SALTELLI, A., and TARANTOLA, S. Uncertainty and sensitivity analysis techniques as tools for the quality assessment of composite indicators. Journal of the Royal Statistical Society: Series A, 168 (2), 307- 323 (2005)
25	ASGHARI, H., TOPOL, H., MARKERT, B., and MERODIO, J. Application of sensitivity analysis in extension, inflation, and torsion of residually stressed circular cylindrical tubes. Probabilistic Engineering Mechanics, 73, 103469 (2023)
26	BECKER, W., OAKLEY, J. E., SURACE, C., GILI, P., ROWSON, J., and WORDEN, K. Bayesian sensitivity analysis of a nonlinear finite element model. Mechanical Systems and Signal Processing, 32, 18- 31 (2012)
27	NARASIMHA, K. V., KUMAR, R. K., and BOHARA, P. C. A sensitivity analysis of design attributes and operating conditions on tyre operating temperatures and rolling resistance using finite element analysis. Proceedings of the Institution of Mechanical Engineers Part D: Journal of Automob, 220 (5), 501- 517 (2006)
28	SCHAFER, F., STURDY, J., MESEK, M., SINEK, A., BIALECKI, R., OSTROWSKI, Z., MELKA, B., NOWAK, M., and HELLEVIK, L. R. Uncertainty quantification and sensitivity analysis during the development and validation of numerical artery models. ACM Transactions on Modeling and Computer Simulation, 32, 256- 262 (2022)
29	FREY, H. C., and PATIL, S. R. Identification and review of sensitivity analysis methods. Risk Analysis, 22 (3), 553- 578 (2002)
30	ASGHARI, H., TOPOL, H., MARKERT, B., and MERODIO, J. Application of the extended Fourier amplitude sensitivity testing (FAST) method to inflated, axial stretched, and residually stressed cylinders. Applied Mathematics and Mechanics (English Edition), 44 (12), 2139- 2162 (2023) doi: 10.1007/s10483-023-3060-6
31	COWIN, S. C., GAILANI, G., and BENALLA, M. Hierarchical poroelasticity: movement of interstitial fluid between porosity levels in bones. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (1902), 3401- 3444 (2009)
32	PERRIN, E., BOU-SAID, B., and MASSI, F. Numerical modeling of bone as a multiscale poroelastic material by the homogenization technique. Journal of the Mechanical Behavior of Biomedical Materials, 91, 373- 382 (2019)
33	COWIN, S. C., and CARDOSO, L. Blood and interstitial flow in the hierarchical pore space architecture of bone tissue. Journal of Biomechanics, 48 (5), 842- 854 (2015)
34	SÁNCHEZ, M. T., PÉREZ, M. Á., and GARCÍA-AZNAR, J. M. The role of fluid flow on bone mechanobiology: mathematical modeling and simulation. Computational Geosciences, 25, 823- 830 (2021)
35	SOLEIMANI, K., GHASEMLOONIA, A., and SUDAK, L. J. Interstitial fluid transport in cortical bone porosities: effects of blood pressure and mass exchange using porous media theory. Mechanics of Materials, 193, 104981 (2024)
36	GHIASI, M. S., CHEN, J., VAZIRI, A., RODRIGUEZ, E. K., and NAZARIAN, A. Bone fracture healing in mechanobiological modeling: a review of principles and methods. Bone Reports, 6, 87- 100 (2017)
37	LIU, C., CARRERA, R., FLAMINI, V., KENNY, L., CABAHUG-ZUCKERMAN, P., GEORGE, B. M., HUNTER, D., LIU, B., SINGH, G., and LEUCHT, P. Effects of mechanical loading on cortical defect repair using a novel mechanobiological model of bone healing. Bone, 108, 145- 155 (2018)
38	SALTELLI, A., TARANTOLA, S., and CAMPOLONGO, F. Sensitivity analysis as an ingredient of modeling. Statistical Science, 15, 377- 395 (2000)
39	EFRON, B., and STEIN, C. The jackknife estimate of variance. Annals of Statistics, 9, 586- 596 (1981)
40	SALTELLI, A. Making best use of model evaluations to compute sensitivity indices. Computer Physics Communications, 145 (2), 280- 297 (2002)
41	ARCHER, G. E. B., SALTELLI, A., and SOBOL, I. M. Sensitivity measures, anova-like techniques, and the use of bootstrap. Journal of Statistical Computation and Simulation, 58 (2), 99- 120 (1997)
42	SOBOL, I. M. Sensitivity analysis for non-linear mathematical models. Mathematical Modeling and Computational Experiment, 1, 407- 414 (1993)
43	JANON, A., KLEIN, T., LAGNOUX, A., NODET, M., and PRIEUR, C. Asymptotic normality and efficiency of two Sobol index estimators. ESAIM-Probability and Statistics, 18, 342- 364 (2014)
44	IOOSS, B., VAN, D. F., and DEVICTOR, N. Response surfaces and sensitivity analyses for an environmental model of dose calculations. Reliability Engineering and System Safety, 91 (10-11), 1241- 1251 (2006)
45	SALTELLI, A., ANNONI, P., AZZINI, I., CAMPOLONGO, F., RATTO, M., and TARANTOLA, S. Variance-based sensitivity analysis of model output: design and estimator for the total sensitivity index. Computer Physics Communications, 181 (2), 259- 270 (2010)
46	JANSEN, M. J. W. Analysis of variance designs for model output. Computer Physics Communications, 117 (1-2), 35- 43 (1999)
47	PUY, A., PIANO, S. L., SALTELLI, A., and LEVIN, S. A. Sensobol: an “R” package to compute variance-based sensitivity indices. Journal of Statistical Software, 102, 1- 37 (2021)
48	LIU, R., and OWEN, A. B. Estimating mean dimensionality of analysis of variance decompositions. Journal of the American Statistical Association, 101 (474), 712- 721 (2006)
49	JOHNSON, N. L., KOTZ, S. I., and BALAKRISHNAN, N. Beta Distributions, John Wiley and Sons, New York 221- 235 (1994)
50	DEHGHANI, H., PENTA, R., and MERODIO, J. The role of porosity and solid matrix compressibility on the mechanical behavior of poroelastic tissues. Materials Research Express, 6 (3), 035404 (2018)
51	BURRIDGE, R., and KELLER, J. B. Poroelasticity equations derived from microstructure. The Journal of the Acoustical Society of America, 70 (4), 1140- 1146 (1981)
52	LÉVY, T. Propagation of waves in a fluid-saturated porous elastic solid. International Journal of Engineering Science, 17 (9), 1005- 1014 (1979)
53	PENTA, R., MILLER, L., GRILLO, A., RAMÍREZ-TORRES, A., MASCHERONI, P., and RODRÍGUEZ-RAMOS, R. Porosity and diffusion in biological tissues: recent advances and further perspectives. Constitutive Modelling of Solid Continua, Springer, Berlin, 311–356 (2020)
54	MILLER, L., and PENTA, R. Effective balance equations for poroelastic composites. Continuum Mechanics and Thermodynamics, 32 (6), 1533- 1557 (2020)
55	MILLER, L., and PENTA, R. Double poroelasticity derived from the microstructure. Acta Mechanica, 232, 3801- 3823 (2021)
56	MILLER, L., and PENTA, R. Investigating the effects of microstructural changes induced by myocardial infarction on the elastic parameters of the heart. Biomechanics and Modelling in Mechanobiology, 22 (3), 1019- 1033 (2023)
57	MILLER, L., and PENTA, R. Micromechanical analysis of the effective stiffness of poroelastic composites. European Journal of Mechanics/A Solids, 98, 104875 (2023)
58	RODRIGUEZ, J., and MERODIO, J. A new derivation of the bifurcation conditions of inflated cylindrical membranes of elastic material under axial loading: application to aneurysm formation. Mechanics Research Communications, 38, 203- 210 (2010)
59	TOPOL, H., AL-CHLAIHAWI, M. J., DEMIRKOPARAN, H., and MERODIO, J. Bulging initiation and propagation in fiber-reinforced swellable mooney-rivlin membranes. Journal of Engineering Mathematics, 128, 1- 15 (2021)
60	TOPOL, H., AL-CHLAIHAWI, M. J., DEMIRKOPARAN, H., and MERODIO, J. Bifurcation of fiber-reinforced cylindrical membranes under extension, inflation, and swelling. Applied Computational Mechanics, 9, 113- 128 (2023)
61	SEDDIGHI, Y., and HAN, H. C. Buckling of arteries with noncircular cross sections: theory and finite element simulations. Frontiers of Physics, 12, 712636 (2021)
62	FU, Y. B., LIU, J. L., and FRANCISCO, G. S. Localized bulging in an inflated cylindrical tube of arbitrary thickness — the effect of bending stiffness. Journal of the Mechanics and Physics of Solids, 90, 45- 60 (2016)
63	JHA, N. K., MORADALIZADEH, S., REINOSO, J., TOPOL, H., and MERODIO, J. On the helical buckling of anisotropic tubes with application to arteries. Mechanics Research Communications, 128, 104067 (2023)
64	AMBROSI, D., BEN-AMAR, M., CYRON, C. J., DESIMONE, A., GORIELY, A., HUMPHREY, J. D., and KUHL, E. Growth and remodeling of living tissues: perspectives, challenges, and opportunities. Journal of the Royal Society Interface, 16, 20190233 (2019)
65	TOPOL, H., DEMIRKOPARAN, H., and PENCE, T. J. On collagen fiber morphoelasticity and homeostatic remodeling tone. Journal of the Mechanical Behavior of Biomedical Materials, 113, 104154 (2021)
66	TOPOL, H., DEMIRKOPARAN, H., and PENCE, T. J. Fibrillar collagen: a review of the mechanical modeling of strain-mediated enzymatic turnover. Applied Mechanics Reviews, 73, 050802 (2021)
67	SAINI, K., CHO, S., DOOLING, L. J., and DISCHER, D. E. Tension in fibrils suppresses their enzymatic degradation-a molecular mechanism for ‘use it or lose it’. Matrix Biology, 85-86, 34- 46 (2020)

[1]	. [J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(11): 2037-2054.
[2]	M. MADHU, N. S. SHASHIKUMAR, B. MAHANTHESH, B. J. GIREESHA, N. KISHAN. Heat transfer and entropy generation analysis of non-Newtonian fluid flow through vertical microchannel with convective boundary condition[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(9): 1285-1300.
[3]	K. RAMESH, O. OJJELA. Entropy generation analysis of natural convective radiative second grade nanofluid flow between parallel plates in a porous medium[J]. Applied Mathematics and Mechanics (English Edition), 2019, 40(4): 481-498.
[4]	K. R. RAGHUNATHA, I. S. SHIVAKUMARA. Stability of triple diffusive convection in a viscoelastic fluid-saturated porous layer[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(10): 1385-1410.
[5]	M. A. SHEREMET, R. TRÎMBIŢAŞ, T. GROŞAN, I. POP. Natural convection of an alumina-water nanofluid inside an inclined wavy-walled cavity with a non-uniform heating using Tiwari and Das' nanofluid model[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(10): 1425-1436.
[6]	P. K. YADAV, S. JAISWAL, B. D. SHARMA. Mathematical model of micropolar fluid in two-phase immiscible fluid flow through porous channel[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(7): 993-1006.
[7]	K. R. RAGHUNATHA, I. S. SHIVAKUMARA, SOWBHAGYA. Stability of buoyancy-driven convection in an Oldroyd-B fluid-saturated anisotropic porous layer[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(5): 653-666.
[8]	K. R. RAGHUNATHA, I. S. SHIVAKUMARA, B. M. SHANKAR. Weakly nonlinear stability analysis of triple diffusive convection in a Maxwell fluid saturated porous layer[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(2): 153-168.
[9]	Jiaxuan LIU, Xinchun SHANG, Weiyao ZHU. Investigation on nonlinear multi-scale effects of unsteady flow in hydraulic fractured horizontal shale gas wells[J]. Applied Mathematics and Mechanics (English Edition), 2018, 39(2): 181-192.
[10]	V. MISHRA, B. R. GUPTA. Motion of a permeable shell in a spherical container filled with non-Newtonian fluid[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(12): 1697-1708.
[11]	Lüwen ZHOU, Yuqian ZHANG, Xiaolong DENG, Moubin LIU. Dissipative particle dynamics simulation of flow through periodic arrays of circular micropillar[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(11): 1431-1440.
[12]	P. RANA, M. J. UDDIN, Y. GUPTA, A. I. M. ISMAIL. Two-component modeling for non-Newtonian nanofluid slip flow and heat transfer over sheet: Lie group approach[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(10): 1325-1340.
[13]	R. KANDASAMY, R. MOHAMMAD, I. MUHAIMIN. Carbon nanotubes on unsteady MHD non-Darcy flow over porous wedge in presence of thermal radiation energy[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(8): 1031-1040.
[14]	Yizhao WAN, Yuewu LIU, Weiping OUYANG, Guofeng HAN, Wenchao LIU. Numerical investigation of dual-porosity model with transient transfer function based on discrete-fracture model[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(5): 611-626.
[15]	Xi CHEN, Ying DAI. Differential transform method for solving Richards' equation[J]. Applied Mathematics and Mechanics (English Edition), 2016, 37(2): 169-180.