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An analytical solution for the radial and tangential displacements on a thin hemispherical layer of articular cartilage

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Abstract

A simplified analytical solution has been obtained for the radial and tangential displacements on the surface of a thin, hemispherical layer of porous-elastic articular cartilage firmly bonded to a rigid foundation. A static pressure distributed according to a paraboloid of revolution is applied simulating cartilage compression by a porous indenter. The solution method is in the form of an asymptotic series and uses Laplace transforms. The analytical predictions are in qualitative agreement with the behaviour of biphasic articular cartilage reported in the literature. A direct comparison with numerical simulations using commercially available Finite Element Modelling (FEM) software was also carried out for conditions relevant to natural hip joints and the results show a good quantitative agreement overall.

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Abbreviations

d :

cartilage layer thickness

E :

Young’s modulus

H :

aggregate modulus = λ s + 2μ s

k :

permeability \({ = {\varphi^f}^2/K}\)

K :

diffusive resistance coefficient

p :

fluid pressure

P t :

total applied pressure

r :

radius of the hemispherical layer

r a , r b :

radius of the interior and exterior layer surfaces

t :

time

u :

displacements vector

v :

velocities vector

x, y, z:

cartesian coordinates

α c :

radius of applied pressure

\({\epsilon}\) :

strain tensor \({ = \frac{1}{2}\left[(\nabla{\bf u}) + (\nabla{\bf u})^T \right]}\)

\({\theta, \phi}\) :

zenith and azimuth angles in the spherical coordinates

λ s , μ s :

Lamé’s constants

ν :

Poisson’s ratio

σ :

stress vector

\({\varphi}\) :

porosity

\({\left(\,\hat{}\,\right)}\) :

Dimensionless variables

\({\left(\,\bar{}\,\right)}\) :

Laplace transformed dimensionless quantities

s :

solid phase

f :

fluid phase

t :

total values

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Authors and Affiliations

  1. School of Mechanical Engineering, The University of Leeds, Leeds, LS2 9JT, UK

    A. Félix Quiñonez, J. L. Summers, J. Fisher & Z. M. Jin

Authors
  1. A. Félix Quiñonez
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  2. J. L. Summers
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  3. J. Fisher
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  4. Z. M. Jin
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Correspondence to J. L. Summers.

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Félix Quiñonez, A., Summers, J.L., Fisher, J. et al. An analytical solution for the radial and tangential displacements on a thin hemispherical layer of articular cartilage. Biomech Model Mechanobiol 10, 283–293 (2011). https://doi.org/10.1007/s10237-010-0234-6

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