Multiple-length-scale deformation analysis in a thermoplastic polyurethane

Abstract

Thermoplastic polyurethane elastomers enjoy an exceptionally wide range of applications due to their remarkable versatility. These block co-polymers are used here as an example of a structurally inhomogeneous composite containing nano-scale gradients, whose internal strain differs depending on the length scale of consideration. Here we present a combined experimental and modelling approach to the hierarchical characterization of block co-polymer deformation. Synchrotron-based small- and wide-angle X-ray scattering and radiography are used for strain evaluation across the scales. Transmission electron microscopy image-based finite element modelling and fast Fourier transform analysis are used to develop a multi-phase numerical model that achieves agreement with the combined experimental data using a minimal number of adjustable structural parameters. The results highlight the importance of fuzzy interfaces, that is, regions of nanometre-scale structure and property gradients, in determining the mechanical properties of hierarchical composites across the scales.

Figure 1. Schematic diagram of the experimental set-up.

The sample was subjected to repeated uniaxial loading within the tensile stage. Radiographic images and SAXS and WAXS patterns were recorded at each loading step. After collecting each radiographic image, the SAXS detector was placed in the beam to collect SAXS patterns, and then WAXS detector was translated into the beam to collect XRD data.

Figure 2. Hierarchical structural evolution of 2D X-ray images and diffraction patterns.

(a) Selected radiographic images acquired by the ‘X-ray Eye’ detector, and (b) SAXS and (c) WAXS patterns of TPUs during repeated loading, captured during the first loading stage from 0 to 6 N, then unloading stage from 6 to 0.8 N, and then the re-loading stage from 0.8 to 8 N. Each loading value is indicated in the label. The black line observed in c for each WAXS pattern is the beam stop. (d) The family of equivalent 1D profiles of scattered SAXS intensity (arbitrary units) versus q during monotonic loading between 0.8 and 8 N shows an appreciable peak shift towards lower q values, corresponding to the increase of linear dimension due to sample extension.

Figure 3. Multiple-length-scale strains visualization.

(a) Evolution of the macroscopic longitudinal strain measured by radiography X-ray eye. (b) Evolution of the nano-scale and sub-nano-scale strains interpreted from SAXS/WAXS patterns along both longitudinal x (solid) and transverse y axes (dash). The negative transverse strains reveal the Poisson effect that was observed in the experiment. The comparison of the macroscopic strain, sub-nano-scale strain and nano-scale strains along the loading direction (x axis) was shown in c and d, where c shows the original results and d shows the modified results.

Figure 4. Image-based FE modelling.

(a) The original TEM image. (b) A representative volume element (RVE) with 332 × 311 pixels was selected from a. (c) Representative 5 × 5 pixels (elements) were chosen to demonstrate the modelling procedure. Each pixel in c corresponds to an element in (d) the finite element model generated and meshed in ABAQUS.

Figure 5. Results of image-based finite element modelling and FFT analysis of TPUs.

(a) The same greyed values from the RVE image were assigned on the finite element model. Both the un-deformed and deformed shapes were shown. (b) Simulated 2D SAXS patterns of the selected region marked by a red square in a using FFT. (c) The 1D FFT analysis of electron density evolution along x axis in the selected region marked by a yellow rectangular in a.

Figure 6. Fuzzy interfaces.

(a) Schematic view of the TPU nanostructure and (b) the average electron density image obtained by ‘blurring’.

Figure 7. Property gradient representation.

Profiles prescribed by the function given in equation (4), with the values of parameter c set equal to 70, 50, 20 and 10.