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A semi-analytical solution for the confined compression of hydrated soft tissue. (English) Zbl 1254.74084
Summary: Confined compression is a common experimental technique aimed at gaining information on the properties of biphasic mixtures comprised of a solid saturated by a fluid, a typical example of which are soft hydrated biological tissues. When the material properties (elastic modulus, permeability) are assumed to be homogeneous, the governing equation in the axial displacement reduces to a Fourier equation which can be solved analytically. For the more realistic case of inhomogeneous material properties, the governing equation does not admit, in general, a solution in closed form. In this work, we propose a semi-analytical alternative to finite element analysis for the study of the confined compression of linearly elastic biphasic mixtures. The partial differential equation is discretised in the space variable and kept continuous in the time variable, by use of the finite difference method, and the resulting system of ordinary differential equations is solved by means of the Laplace transform method.

##### MSC:
 74L15 Biomechanical solid mechanics 74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 76S05 Flows in porous media; filtration; seepage 74S20 Finite difference methods applied to problems in solid mechanics 92C10 Biomechanics
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##### References:
 [1] Almeida ES, Spliker RL (1998) Finite element formulations for hyperelastic transversely isotropic soft tissues. Comput Methods Appl Mech Eng 151:513–538 · Zbl 0920.73350 · doi:10.1016/S0045-7825(97)82246-3 [2] Ateshian GA, Warden WH, Kim JJ, Grelsamer RP, Mow VC (1997) Finite deformation biphasic material properties of bovine articular cartilage from confined compression experiments. J Biomech 30:1157–1164 · doi:10.1016/S0021-9290(97)85606-0 [3] Clark AL, Barclay LD, Matyas JR, Herzog W (2003) In situ chondrocyte deformation with physiological compression of the feline patellofemoral joint. J Biomech 36:553–568 · doi:10.1016/S0021-9290(02)00424-4 [4] Federico S, Grillo A, La Rosa G, Giaquinta G, Herzog W (2005) A transversely isotropic, transversely homogeneous microstructural-statistical model of articular cartilage. J Biomech 38(10):2008–2018 · doi:10.1016/j.jbiomech.2004.09.020 [5] Fung YC (1993) Biomechanics–mechanical properties of living tissues. Springer, New York [6] Holmes MH, Mow VC (1990) Finite deformation of a soft tissue: analysis of a mixture model in uni-axial compression. J Biomech 23:1145–1156 · doi:10.1016/0021-9290(90)90007-P [7] Holzapfel GA, Gasser TC, Ogden RW (2000) A new constitutive framework for arterial wall mechanics and a comparative study of material models. J Elast 61:1–48 · Zbl 1023.74033 · doi:10.1023/A:1010835316564 [8] Maroudas A, Bullough P (1968) Permeability of articular cartilage. Nature 219:1260–1261 · doi:10.1038/2191260a0 [9] Mow VC, Guo XE (2002) Mechano-electrochemical properties of articular cartilage: their inhomogeneities and anisotropies. Annu Rev Biomed Eng 4:175–209 · doi:10.1146/annurev.bioeng.4.110701.120309 [10] Mow VC, Kuei SC, Lai WM, Armstrong CG (1980) Biphasic creep and stress relaxation of articular cartilage, theory and experiment. J Biomech Eng 102:73–84 · doi:10.1115/1.3138202 [11] Nigg BM, Herzog W (2007) Biomechanics of the musculo-skeletal system, 3rd ed. Wiley, Chirchester [12] Preziosi L, Farina A (2002) On Darcy’s law for growing porous media. Int J Non-Linear Mech 37(3):485–491 · Zbl 1346.76194 · doi:10.1016/S0020-7462(01)00022-1 [13] Schinagl, RM, Gurskis D, Chen AC, Sah RL (1997) Depth-dependent confined compression modulus of full-thickness bovine articular cartilage. J Orthop Res 15:499–506 · doi:10.1002/jor.1100150404 [14] Wilson W, van Donkelaar CC, van Rietbergen B, Huiskes R (2005) A fibril-reinforced poroviscoelastic swelling model for articular cartilage. J Biomech 38:1195–1204 · doi:10.1016/j.jbiomech.2004.07.003 [15] Wu JZ, Herzog W (2000) Finite element simulation of location- and time-dependent mechanical behavior of chondrocytes in unconfined compression tests. Ann Biomed Eng 28:318–330 · doi:10.1114/1.271
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