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

This is the peer reviewed version of the following article: Nguyen, TT and Huntley, JM and Ashcroft, IA and Ruiz, PD and Pierron, F (2017) A Fourier-series-based virtual fields method for the identification of three-dimensional stiffness distributions and its application to incompressible materials. Strain, 53 (5). e12229-e12229 which has been published in final form at 10.1111/str.12229 This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving." 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.