A Fourier‐series‐based virtual fields method for the identification of three‐dimensional stiffness distributions and its application to incompressible materials

Abstract We present an inverse method to identify the spatially varying stiffness distributions in 3 dimensions. The method is an extension of the classical Virtual Fields Method—a numerical technique that 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 three‐dimensional F‐VFM is validated with both numerical and experimental data. The latter came from a phase contrast magnetic resonance imaging 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.