A Direct Approach to In-Plane Stress Separation using Photoelastic Ptychography.

Determining stresses in solids is crucial for evaluating and predicting mechanical behaviour. Experimentally measuring the individual spatially varying components of the stress tensor however is often extremely challenging. The most common method of stress determination is currently Finite Element Modelling (FEM) where the solution is highly dependent on the loading and boundary conditions, both of which can be difficult to determine experimentally. The drive for new methods to measure stress in optically transparent materials, e.g. thin films, has led to the development of a variety of approaches including photoelasticity and thermoelasticity. Such full-field techniques are routinely used to obtain images of fringe patterns that qualitatively relate to the samples’ stress distribution. However, the data obtained using these methods typically provide either the difference or sum of the principal stress components obtained from the isochromatics (polariscope) or isopachics (holographic photoelasticity) respectively. Since the isochromatics and isopachics are directly coupled to one another, separation of the individual stress components is extremely challenging. To address the problem of stress separation in full-field photoelasticity measurements, several approaches have been proposed, all of which combine theoretical or numerical analysis and inverse approaches1,2,3,4. More recently the combination of digital photoelasticity combined with interferometric approaches has been experimentally realised which allows a complete experimental solution to the problem of stress separation5,6,7,8. For example, both Lei et al.9 and Yoneyama et al.10 have demonstrated stress separation in two-dimensions by combining circular polarimetry and interferometric photoelasticity. In their approach the phase values of the isochromatics and isoclinics are determined using a circular polariscope, whilst the phase of the isopachics is obtained using interferometry. The use of multiple interferometry measurements is required to account for phase shift errors caused by linear and quadratic reference phase deviations. Ptychographic coherent diffractive imaging (CDI) is an iterative technique for quantitatively reconstructing the amplitude and phase of the complex wavefield exiting a sample11. In ptychographic CDI, instead of an in-focus image, the intensity distribution in the far-field associated with coherently scattered photons from the sample is recorded. Direct inversion of this diffraction pattern is impossible since the unique phase information accompanying the scattered wave is lost during its detection. As a result, an iterative phase retrieval process must be applied to recover the phase map and hence recover an image of the object under investigation. Unlike conventional visible light microscopy or other types of phase contrast optical microscopy such as Zernike, CDI recovers the quantitative amplitude and phase information of the wave exiting the sample (similar to Fourier Transform holography). Crucially for the present application, ptychography is able to simultaneously reconstruct the exit surface wave for both probe and object separately, entirely eliminating the need for multiple interferometry measurements12. In the X-ray regime ptychography has already been applied to a range of applications within materials science13,14. Recently Ferrand et al.15 showed in simulation the potential for anisotropic ptychography to recover birefringent sample information, however the combination of ptychography and optical photoelasticity has not been experimentally reported in the literature. In this paper we demonstrate that a straightforward approach to quantitative in-plane stress determination can be achieved by combining the circular polariscope with ptychographic coherent diffractive imaging (CDI). Unlike previous approaches to stress separation, only one experimental setup is required for analysis of the phases associated with the isochromatics, isoclinics and isopachics. In addition to the ease of the optical set up, ptychography offers a number of other advantages when compared to previous interferometric based approaches: (1) There are no errors due to non-linearity or misalignment of the set up leading to a higher quality result for the isopachic parameter. (2) Ptychography can cope with complex, highly scattering (though not multiply scattering) samples which can cause problems for interferometry. (3) For very small (sub-mm) specimens the total time to collect the data (including sample translations) can be less than for interferometric based set ups. To confirm the validity of our proposed method, in common with Lei et al. and Yoneyama et al.9,10, we apply our method to the model problem of a diametrically compressed elastic disc. Unwrapping of the phases associated with the isochromatics and isopachics is readily achieved using a ‘quality guided’ phase unwrapping algorithm based on the approach of Ghiglia et al.16. The validity of our approach is verified by direct comparison of the experimental results to the analytic solution for a stressed disc under point loading first derived by Hertz17.