A Fourier-series-based virtual fields method for the identification of 3-D stiffness distributions and its application to incompressible materials

We present an inverse method to identify the spatially-varying stiffness distributions in three-dimensions (3-D). The method is an extension of the classical Virtual Fields Method (VFM) – a numerical technique which exploits information from full-field deformation measurements to deduce unknown material properties – in the spatial frequency domain, which we name the Fourier-series-based Virtual Fields Method (F-VFM). Three-dimensional stiffness distributions, parameterised by a Fourier series expansion, are recovered after a single matrix inversion. A numerically efficient version of the technique is developed, based on the Fast Fourier Transform. The proposed F-VFM is also adapted to deal with the challenging situation of limited or even non-existent knowledge of boundary conditions. The 3-D F-VFM is validated with both numerical and experimental data. The latter came from a phase contrast MRI experiment containing material with Poisson’s ratio close to 0.5; such a case requires a slightly different interpretation of the F-VFM equations, to enable the application of the technique to incompressible materials.