A quantitative study on using digital photoelasticity for characterising the effect of the stretching speed on the necking phenomenon

3.1 Comparison between birefringence measurements using two-beam interferometry and digital photoelasticity

In this work, we measured the photoelastic order which is a good indicator of birefringence values at different points inside the fibre. The two-beam Pluta polarising interference microscope is one of the major techniques for measuring the variations of birefringence inside thermally and mechanically treated fibres [20]. On the other hand, RGB photoelastic measurements of birefringence have the advantage over two-beam interferometry in many aspects. In order to clarify this point, the Pluta polarising interference microscope in its uniform-field mode is adjusted to produce photoelastic patterns of the test sample by replacing the Wollaston prisms with a microscope objective lens. After capturing the photoelastic patterns, with the sample in its place, the Pluta polarising interference microscope is switched back to its nonduplicating two-beam interference mode using the Wollaston prisms. Hence, a two-beam interference pattern is produced at the same region captured previously in the photoelastic mode.

Figures 2a, b depicts the captured patterns for the case of two-beam interference and a photoelastic pattern showing the same necking region in order to compare the calculated phase map at this region. Photoelastic lookup tables are used to map the pixel colour values to the photoelastic order, N, at any point, see Figure 2c. It consists of pairs; each pair consists of a colour (RGB value) and a photoelastic order, N, value. Each colour pixel is compared against the lookup table pairs to find the nearest colour match by finding the minimum colour distance (error function) [19]:

where the subscript (t) denotes a lookup table value. The colour of the pixel is replaced with the corresponding photoelastic order value, N. The final result is a map of the photoelastic orders based on the colours of the photoelastic pattern. Accordingly, the captured photoelastic pattern is compared pointwise to a preevaluated colour-order pairs known as a lookup table. Each (RGB) value of a pixel will be replaced by a value (N) denoting the order value at this pixel based on its colour value.

Figure 2:

The captured patterns for the case of (a) two-beam interference and (b) a photoelastic pattern of the same necking region. (c) The lookup table used to calculate the photoelastic order, N, represented as a relation between the photoelastic order, N, represented by the x-axis and the RGB colour values of each pixel along the x-axis.

In Figure 3, we illustrate the main advantage of digital photoelasticity which is the pointwise calculation of birefringence values. On the other hand, online two-beam interferometry usually uses Fourier transform for obtaining the phase distribution across the fibre. Usually, Fourier transform produces acceptable phase maps for low-gradient phase values, e.g. uniform fibre samples. But for the case of increased phase gradients, as in necking regions, the measured phase values show deviations from the true phase values due to the nature of filtering in the frequency domain.

Figure 3:

The birefringence values calculated using (a) Fourier analysis of two-beam interferogram, (b) photoelastic analysis, (c) the birefringence profile calculated from the two-beam interference pattern (solid line measured at a cross-sectional line at a distance = 680 μm measured from the left of the field of view, and the birefringence profile calculated from the photoelastic pattern [dashed line]) measured at the same cross-sectional line.

A careful inspection of the birefringence values produced by the digital RGB photoelasticity showed a smooth transition of the phase values at the neck region (which are considered as high phase gradient regions). Accordingly, the digital RGB photoelastic determination of the birefringence values is more reliable than two-beam interferometric techniques for online characterisation of birefringence profile of fibres, in addition to being a more direct and rapid technique. Also, the fibre geometry can be determined using digital photoelastic patterns better than two-beam interference patterns since digital photoelasticity does not depend on the presence of carrier fringes, while two-beam interference patterns contains reference fringes in addition to the deformed regions which makes fibre-boundary recognition process more difficult. Thus, digital photoelasticity is recommended for online characterisation of a necking deformation. On the other hand, for smooth regions along the fibre, e.g. before and after the neck region, there is a negligible deviation between the birefringence profile obtained using digital photoelasticity and the birefringence values obtained using digital interferometry, as depicted in Figure 3c. Accordingly, digital photoelasticity provides a reliable determination of the birefringence profiles of high accuracy nearly equal to the obtained profiles using the two-beam Pluta polarising interference microscope.

3.2 The effect of stretching speeds on necking initiation

A monofilament of the iPP fibre is fixed to the clamps of the stretching motors. The heater of the thermomechanical assembly is used to fix the ambient temperature to 19 °C. The motors are rotated to stretch the iPP fibre until a neck is formed. The stretching continues until the neck propagates and is no longer visible in the observed field of the microscope. The process is repeated for identical monofilaments at different stretching speeds: v = 0.16, 0.79, 1.1, 1.42, 1.73, 2.05 (mm/s). Figure 4 depicts the photoelastic patterns of iPP fibres mechanically stretched at different speeds. For each speed, photoelastic patterns are captured for three different fibre states. The initial undrawn fibre state, a pattern showing a necking deformation nearly in the middle of the field view and a pattern for the region just after the necking region leaves the field of view.

Figure 4:

The photoelastic patterns of isotactic polypropylene (iPP) fibres mechanically stretched at different speeds: v = 0.16 mm/s; v = 0.79 mm/s; v = 1.1 mm/s; v = 1.42 mm/s; v = 1.73 mm/s and v = 2.05 mm/s.

In order to enhance the accuracy of calculations, each photoelastic pattern is processed for reducing noise and enhancing colour features along the fibre using suitable image processing techniques [4]. Then, the captured photoelastic patterns are analysed using lookup tables to find the photoelastic order, N, which is the main parameter for calculating the intrinsic stress anisotropy, ΔS. Figure 5 shows the calculated photoelastic order, N, for iPP samples stretched at different speeds. The black pixels are points having photoelastic orders, N = 0, while the white pixels denote photoelastic orders, N = 3: the intermediate grey pixels refer to orders between 0 and 3. Accordingly, it is clear that undrawn patterns showed relatively small photoelastic orders (N < 1) indicating low intrinsic stress anisotropy, ΔS.

Figure 5:

The photoelastic order of iPP fibres mechanically stretched at different speeds: v = 0.16 mm/s; v = 0.79 mm/s; v = 1.1 mm/s; v = 1.42 mm/s; v = 1.73 mm/s and v = 2.05 mm/s. The x-axis represents the distance along the fibre in microns, and the y-axis represents the distance across the fibre axis in microns. iPP, isotactic polypropylene.

Upon further stretching, the intrinsic stress anisotropy, ΔS, increases and becomes localised at the neck, reaching photoelastic orders between 1.5 and 2.5. The photoelastic patterns captured after the necking regions showed higher photoelastic orders (N ≈ 3) with radial symmetry across the fibre indicating that the intrinsic stresses which are higher at the central layers gradually decrease toward the outer layers. Also, a noticeable decrease of the intrinsic stress anisotropy is observed by increasing the stretching speeds for photoelastic patterns containing necking and after the passage of the necking. This implies that the increased speed of stretching results in lower localised intrinsic stresses during the formation of necking. Although after the observed necking region there is an increasing trend of the intrinsic stresses, the overall intrinsic stresses have lower values for higher speeds of stretching.

In Figure 6, a gradual increase of the fibre stress anisotropy is detected, which extends from the undrawn part of the fibre to the neck region. The highest value of the intrinsic stress anisotropy is localised at the centre of the neck. Figure 7a shows the calculated birefringence values from the measured intrinsic stress anisotropy at the middle of the necking region at different stretching speeds. This figure clearly shows that, the necking is initiated at relatively high intrinsic stress anisotropy, ΔS, for lower stretching speeds, which decreases by increasing the stretching speed. Another advantage of photoelastic patterns is the direct detection of necking initiation times. Figure 7b shows the necking initiation times as a function of the stretching speed. It is noticed that increasing the stretching speed decreases the time required to initiate a neck region inside the fibre. Accordingly, increasing the stretching speeds causes faster alignment of the molecular chains inside the fibre, and the result is that less time required to initiate the necking region.

Figure 6:

The intrinsic stress anisotropy in terms of birefringence of a typical necking observed at low stretching speed (ν = 0.157 mm/s). (a) The photoelastic pattern of the formed necking phenomenon; (b) a 2D map of the distribution of the birefringence values along the fibre axis; (c) the birefringence values at the central layer of the fibre along its axis during the formation of the necking phenomenon.

Figure 7:

(a) The calculated birefringence values from the measured intrinsic stress anisotropy at the middle of the necking region at different stretching speeds: v = 0.16 mm/s; v = 0.79 mm/s; v = 1.1 mm/s; v = 1.42 mm/s; v = 1.73 mm/s and v = 2.05 mm/s. (b) The necking initiation times as a function of the stretching speeds: v = 0.16 mm/s; v = 0.79 mm/s; v = 1.1 mm/s; v = 1.42 mm/s; v = 1.73 mm/s and v = 2.05 mm/s.

3.3 Early stages of necking growth

Necking is a strong precursor of material failure. Understanding the necking behaviour of fibres is essential to control the conditions leading to material failure. In this work, we were able to observe the birth of necking photoelastically. This observation provides information on the state of stress at regions where necking is expected to initiate. Consequently, we were able to predict the initiation of necking based on intrinsic stress measurements.

Careful stretching of the iPP fibres allowed us to detect the birth and growth of necking. The fibre samples are fixed to the stretching motors, and the fibre is moved so that its centre point is observed at the objective lens field of view. Slow stretching (speed = 0.16 mm/s) of the fibre at this point facilitates detection of the early stages of necking growth. Figure 8 shows the captured photoelastic patterns of the fibre at the early stages of necking formation. These photoelastic patterns are enhanced and analysed to find the photoelastic order at the region of the growing necking. This figure also depicts the calculated photoelastic orders of the fibre during the formation of the necking region, which is a measure of the intrinsic stress anisotropy, ΔS. The necking phenomenon initiates as a localised rise of the intrinsic stress at a certain region. Continuous stretching concentrates the intrinsic stress at a small area of the selected region. The concentrated localised stresses are propagated to the adjacent areas, and the intrinsic stress increases rapidly at the selected region compared to other parts of the stretched fibre. Finally, the region of high localised intrinsic stress anisotropy expands to form a neck-shoulder morphology, and a necking is visually observed. These stages are described in terms of the calculated birefringence values at different stages of necking initiation which is depicted in Figure 9.

Figure 8:

(a) The captured photoelastic patterns and (b) the calculated photoelastic orders of the fibre at early stages of necking formation.

Figure 9:

The birefringence values of the fibre at the early stages of necking formation.

3.4 Formation of multiple necking regions

In our study, the necking phenomenon may be initiated at different regions synchronously during stretching. In other words, in some cases, the localised increase in the intrinsic stress anisotropy responsible for the formation of necking may occur at different positions rather than at one region [21]. We are able to detect the formation of multiple necking by stretching an iPP fibre sample at a stretching speed of 0.471 mm/s and at temperature T = 25 °C. The stretched sample is investigated using a polariscope to capture the photoelastic patterns of the fibre during stretching.

The continuous capturing of the photoelastic patterns during stretching allowed us to reconstruct the necking regions along the fibre during stretching. Figure 10a shows the reconstructed photoelastic pattern of the multiple necking phenomenon. Our designed software is able to stitch, enhance and analyse the captured photoelastic frames. The photoelastic order, N, is calculated along the fibre during the formation of multiple necking, then the birefringence is calculated at different points along the multiple necking region, which is given in Figures 10b, c. According to this figure, although these synchronous necking regions were observed at different positions, they nearly initiated at the same birefringence values, Δn, approximately at the value of the photoelastic order, N ≈ 4. The same photoelastic order is observed for all the necking regions observed in a multiple necking deformation, which implies that the same stress state should be met in order to form a necking. This condition is also suggested in case of single necking formation which confirms that the current stress state can be used as a reliable indication of necking formation.

Figure 10:

(a) The photoelastic pattern of the multiple-necking phenomenon constructed by digitally stitching several photoelastic patterns captured along the multiple necking region. (b) The 3D birefringence profile of the multiple necking region and (c) the values of the birefringence for the central layers along the fibre.