Skip to main content
Log in

A semi-analytical solution for the confined compression of hydrated soft tissue

  • Published:
Meccanica Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Almeida ES, Spliker RL (1998) Finite element formulations for hyperelastic transversely isotropic soft tissues. Comput Methods Appl Mech Eng 151:513–538

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  5. Fung YC (1993) Biomechanics—mechanical properties of living tissues. Springer, New York

    Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  MATH  MathSciNet  Google Scholar 

  8. Maroudas A, Bullough P (1968) Permeability of articular cartilage. Nature 219:1260–1261

    Article  ADS  Google Scholar 

  9. Mow VC, Guo XE (2002) Mechano-electrochemical properties of articular cartilage: their inhomogeneities and anisotropies. Annu Rev Biomed Eng 4:175–209

    Article  Google Scholar 

  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

    Article  Google Scholar 

  11. Nigg BM, Herzog W (2007) Biomechanics of the musculo-skeletal system, 3rd ed. Wiley, Chirchester

    Google Scholar 

  12. Preziosi L, Farina A (2002) On Darcy’s law for growing porous media. Int J Non-Linear Mech 37(3):485–491

    Article  MATH  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

  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

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Walter Herzog.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Federico, S., Grillo, A., Giaquinta, G. et al. A semi-analytical solution for the confined compression of hydrated soft tissue. Meccanica 44, 197–205 (2009).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: