A Fourier‐series‐based virtual fields method for the identification of three‐dimensional 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

Fast Fourier virtual fields method for determination of modulus distributions from full-field optical strain data

Inspection of parts for manufacturing defects or in-service damage is often carried out by full-field optical techniques (e.g., digital speckle pattern interferometry, digital holography) where the high sensitivity allows small anomalies in a load-induced deformation field to be measured. Standard phase shifting and phase unwrapping algorithms provide full-field displacement and hence strain data over the surface of the sample. The problem remains however of how to quantify the spatial variations in modulus due, for example, to porosity or damage-induced micro-cracking. Finite element model updating (FEMU) is one method to solve problems of this type, by adjusting an approximate finite element model until the responses it produces are as close to those acquired from experiments as possible