Investigation of foot plantar pressure: experimental and numerical analysis

Abstract

The analysis of interaction phenomena occurring between the plantar region of the foot and insole was investigated using a combined experimental–numerical approach. Experimental data on the plantar pressure for treadmill walking of a subject were obtained using the Pedar^® system. The plantar pressure resultant was monitored during walking and adopted to define the loading conditions for a subsequent static numerical analysis. Geometrical configuration of the foot model is provided on the basis of biomedical images. Because the mechanical behaviour of adipose tissues and plantar fascia is the determinant factor in affecting the paths of the plantar pressure, specific attention was paid to define an appropriate constitutive model for these tissues. The numerical model included sole and insole, providing for friction contact conditions between foot–insole and insole–sole pairs as well. Two different numerical analyses were performed with regards to different loading conditions during the gait cycle. The plantar pressure peaks predicted by the numerical model for the two loading conditions are 0.16 and 0.12 MPa, and 0.09 and 0.12 MPa in the posterior and anterior regions of the foot, respectively. These values are in agreement with experimental evidence, showing the suitability of the model proposed.

Appendix

The constitutive model of the soft tissues, apart from the adipose tissue of the foot plant, was defined by the Ogden isotropic almost-incompressible hyperelastic model, with a strain energy function W of the type:

$$ W = U\left( J \right) + \tilde{W}\left( {\tilde{I}_{1} } \right) = \sum\limits_{i = 1}^{N} {{\frac{1}{{D_{i} }}}\left( {J - 1} \right)^{2i} + } \sum\limits_{i = 1}^{N} {{\frac{{2\mu_{i} }}{{\alpha_{i}^{2} }}}\left( {\tilde{\lambda }_{1}^{{\alpha_{i} }} + \tilde{\lambda }_{2}^{{\alpha_{i} }} + \tilde{\lambda }_{3}^{{\alpha_{i} }} - 3} \right)} $$

where the term U refers to the volumetric deformation and $\tilde{W}$ refers to the iso-volumetric deformation of the tissue according to standard procedures. The Jacobian J of the deformation is the root square of the determinant of the right Cauchy–Green strain tensor C, while $\tilde{\lambda }_{i}$ indicates the principal stretches of the iso-volumetric part of the right Cauchy–Green tensor J ^−2/3 C. The stress–strain behaviour is deduced through the relationship S, where as S = 2∂W/∂C the second Piola–Kirchhoff stress tensor.

According to the highly non-linear response, the adipose soft tissues of the foot plant were modelled with a specific isotropic hyperelastic constitutive model defined by the strain energy function:

$$ W\left( {\tilde{I}_{1} ,J} \right) = U\left( J \right) + \tilde{W}\left( {\tilde{I}_{1} } \right). $$

The non-linear response of the tissue is well fitted by assuming the following specific forms for the volumetric and the iso-volumetric terms:

$$ U\left( J \right) = {\frac{{K_{v} }}{{2 + r\left( {r + 1} \right)}}}\left[ {\left( {J - 1} \right)^{2} + J^{ - r} + rJ - \left( {r + 1} \right)} \right] $$

$$ \tilde{W}\left( {\tilde{I}_{1} } \right) = {\frac{{C_{1} }}{{\alpha_{1} }}}\left\{ {\exp \left[ {\alpha_{1} \left( {\tilde{I}_{1} - 3} \right)} \right] - 1} \right\}. $$

The elastic constants of the model K _ν , r, C ₁ and α ₁ were set considering the average loading rate for the configurations analysed, deduced from the experimental tests, and the mechanical response of the tissue considered as visco-elastic material [23].

The general procedure of the parameter identification for all the constitutive models adopted consists in fitting predicted model results to specific experimental data. The approach provides an inverse analysis that uses the stress–strain history given by experimental data and attempts to estimate the parameter values that would yield the best fit for the constitutive model. This action is performed using an optimization procedure based on a specific algorithm [30] that couples stochastic and deterministic techniques.