Influence of Contact Stresses on Crack-Tip Stress Field: A Multiparameter Approach Using Digital Photoelasticity

Abstract

Background

The interaction of stress fields between cracks or cracks with discontinuities like holes, etc., has been widely studied. Another crucial class of problems include cracks interacting with contact stresses but there has been no work to study them systematically.

Objective

This study aims to understand the role of contact stress in influencing the crack-tip stress field which is essential for reliable estimation of stress intensity factors (SIFs) experimentally.

Method

The contact stress influence on crack-tip isochromatic features is initially discussed using an experimental result for a moderately-deep beam with a small crack. SIFs are evaluated using the over-deterministic nonlinear least squares method. The crack-contact stress interaction is then studied by a superposed crack-contact analytical solution. Photoelastic experiments are conducted for a cracked moderately-deep beam subjected to three-point bending. The SIFs evaluated using the multiparameter solution compare well with finite element predictions. Subsequently, multiple interaction configurations are experimentally examined in a cracked moderately-slender beam by varying the magnitude and position of the contact load relative to the crack.

Results

Even a small crack shows a noticeable change in isochromatics due to influence of contact stress and a two-parameter solution is inadequate here. A multiparameter crack-tip solution is observed to capture the isochromatic fringe field very effectively towards SIF evaluation.

Conclusion

The changes in isochromatics at a crack-tip due to contact stresses are significant. A systematic analysis shows that with appropriate data collection, the multiparameter solution provides SIFs with very little uncertainty in the presence of contact stresses with varying complexities.

Appendices

Appendix A

This appendix shows the implementation of the steps discussed in "Methods of Analysis" for SIF evaluation from an experimental isochromatic image. As an example, a single load configuration (S = 2 mm; P₂ = 83 N) is considered. This example is chosen as it is representative of the complex geometric fringe features seen near the crack-tip in the presence of nearby contact load.

Obtaining whole field fringe order (N) data using twelve fringe photoelasticity (TFP)

From the dark field isochromatics captured in the photoelastic experiment (Fig. 16(a)), the region of interest is decided and the remaining portion to be excluded from the analysis is masked out. Using the colour difference formula and error minimisation (Equation (1)), the fringe order N at every pixel in the region of interest is initially evaluated. Generally, there would be jumps observed in this initial evaluation as seen in Fig. 16(b) due to repetition of colours which requires further refinement. The advanced FRSTFP scheme is used to refine the fringe order variation. The respective values for refinement parameters, namely, window span and kernel size are taken as 0.4 and 11 as per the general recommendations which works for all the cases in this study [44]. The correct fringe order variation obtained after refinement is shown in Fig. 16(c). The N-data is subsequently smoothened with the parameters values for span and iterations taken as 10 and 5, respectively using the NLR smoothing scheme [32]. The smoothened whole field fringe order data at every pixel in the region of interest is shown in Fig. 16(d). These steps for obtaining N-data are sequentially carried out using an in-house software, DigiTFP^® [34]_.

Fig. 16

(a) Dark field isochromatic image, (b) whole field N-data generated using CDF, (c) refined N-data and (d) smoothed N-data

Data collection and non-linear least squares analysis

As discussed in "Data Collection for SIF Evaluation", the availability of whole field N-data is advantaegeous for flexible data collection. With the whole field data, one should be able to pick random data for regression and obtain results. However, research carried out over the years indicates that random inputs to the nonlinear algorithm do not always guarantee correct results and the algorithm requires to be guided with proper data, preferably collected along fringes. Different fringe fields, namely, dark field, mixed-field and composite field fringes as shown in Fig. 17 can be used to extract datapoints based on the photoelastic response. In this paper, mixed-field fringes are used for data extraction for all cases.

Fig. 17

Experimental fringes in grayscale (a) dark field, (b) mixed-field and (c) composite field

The positional coordinates and corresponding fringe order (r, θ, N) of the datapoints are passed as inputs for the iterative regression. Modules dedicated to data collection and regression in an in-house software, PSIF [37] are used for this purpose. The software also facilitates the plotting of the fringe field using the parameters obtained at any particular level of convergence during the analysis. The collected datapoints can be echoed back on this plot for readily assessing the quality of the solution.

The crack-tip, which is the origin for the coordinates, is user specified in this process and is error prone. The convergence of the solution and the accuracy of evaluated SIFs are found to be sensitive to variation in the crack-tip, even by a few pixels. Considering the crack-tip as an additional unknown in regression introduces unnecessary computational difficulties. To circumvent this issue and to identify the correct crack-tip location, a crack-tip refinement (CTR) [32, 45] procedure is deployed after the number of parameters for the problem are frozen based on the analysis. A 5 $\times$ 5 pixel mask surrounding the initially specified crack-tip is considered and the convergence error is recalculated by shifting the origin to each of these pixels. The pixel location giving the least error now serves as the centre of a new 5 $\times$ 5 pixel mask and the procedure is repeated until the estimated origin with least error becomes the centre of the mask. The procedure helps to identify the crack-tip coordinates accurately.

Stress intensity factors evaluated can be deemed reliable only if these are independent of the choice of data. Uncertainty for any quantity is defined as the ratio of standard deviation to the square root of dataset count. Hence, accuracy of SIFs can be gauged based on the measure of uncertainty with different input datasets. Towards this, each case is checked by processing six independent datasets for SIF evaluation. These datasets are created by systematic elimination of data from a master dataset at regular intervals. This elimination process introduces variability while preserving the geometry of fringe features necessary to guide the algorithm. Initially, SIFs are evaluated using all the six datasets and the one giving the least convergence error is considered. CTR is performed on this dataset and the correct crack-tip coordinates are identified. For the remaining five datasets, least squares analysis is repeated using the corrected crack-tip coordinates. The SIF values closest to the mean of all the six trials after CTR are deemed as final and results are reported along with uncertainty. More details about the procedures for crack-tip refinement and uncertainty analysis is available in Ref. [32].

The parameter-wise reconstruction of the complete solution for the example case is shown in Fig. 18 along with the convergence error (CE) with 167 datapoints. It can be observed the fringe field gets better captured as the number of parameters is increased. The converged solution requires 10 parameters with a convergence of 0.026 and a good reconstruction. The evaluated values in MPa√m for K_I and K_II are 0.465 and 0.105 with an uncertainty of 0.003 and 0.0013, respectively.

Fig. 18

Parameter-wise variation in the theoretical reconstruction of the fringe field with the experimental datapoints echoed back in red for the case of P₂ = 83 N and S = 2 mm

Appendix B

Within linear elasticity, a set of combined crack-contact stress field equations are obtained by linear superposition. The contact stress field equations relating normal and tangential loads by the friction law [19] and the singular crack-tip equations [40] for a planar condition are superposed. With suitable independent placements of the respective origins, namely, the contact load application point and the crack-tip, and appropriate transformations, the combined field equations in accordance with Fig. 

Fig. 19

(a) General schematic showing equation variables to be superposed to obtain combined crack-contact stress field equations and (b) radii of contacting bodies made of two materials

19 are presented in Equations. (4) to (6).

$${\sigma }_{x}=-\dfrac{{a}_{c}}{\pi \zeta }\left[\left({{a}_{c}}^{2}+{2x}^{2}+{2y}^{2}\right)\dfrac{{y\psi }_{1}}{{a}_{c}}-\dfrac{2\pi y}{{a}_{c}}-{3xy\psi }_{2}+\mu \left\{\left({2x}^{2}-{{2a}_{c}}^{2}-{3y}^{2}\right){\psi }_{2}+\dfrac{2\pi x}{{a}_{c}}+2\left({{a}_{c}}^{2}-{x}^{2}-{y}^{2}\right)\dfrac{{x\psi }_{1}}{{a}_{c}}\right\}\right]+\dfrac{{K}_{I}}{\sqrt{2\pi r}}{\text{cos}}\dfrac{\theta}{2}\left\{1-{\text{sin}}\dfrac{\theta }{2}{\text{sin}}\dfrac{3\theta }{2}\right\}$$

(4)

$${\sigma }_{y}=-\dfrac{{a}_{c}y}{\pi \zeta }\left[{a}_{c}{\psi }_{1}-{x\psi }_{2}+{\mu y\psi }_{2}\right]+\dfrac{{K}_{I}}{\sqrt{2\pi r}}{\text{cos}}\dfrac{\theta }{2}\left\{1+{\text{sin}}\dfrac{\theta }{2}{\text{sin}}\dfrac{3\theta }{2}\right\}$$

(5)

$$\begin{aligned}{\tau }_{xy}=-&\dfrac{{a}_{c}}{\pi \zeta }\left[{y}^{2}{\psi }_{2}+\mu \left\{\left({{a}_{c}}^{2}+{2x}^{2}+{2y}^{2}\right)\dfrac{{y\psi }_{1}}{{a}_{c}}-\dfrac{2\pi y}{{a}_{c}}-{3xy\psi }_{2}\right\}\right]\\&+\dfrac{{K}_{I}}{\sqrt{2\pi r}}{\text{cos}}\dfrac{\theta }{2}\left\{{\text{sin}}\dfrac{\theta }{2}{\text{cos}}\dfrac{3\theta }{2}\right\}\end{aligned}$$

(6)

where,

$$\begin{aligned}\zeta &\quad=\frac{1}{\dfrac{1}{2}\left(\dfrac{1}{{R}_{1}}+\dfrac{1}{{R}_{2}}\right)}\left[\dfrac{1-{\nu}_{1}^{2}}{{E}_{1}}+\dfrac{1-{\nu}_{2}^{2}}{{E}_{2}}\right];\\&{r}_{\mathrm{1,2}}={\sqrt{({a}_{c}\pm x)^{2}+{y}^{2}}};\\&{\psi }_{\mathrm{1,2}}=\dfrac{\pi ({r}_{1}\pm {r}_{2})}{{r}_{1}{r}_{2}\sqrt{2{r}_{1}{r}_{2}+{2x}^{2}+{2y}^{2}-{{2a}_{c}}^{2}}}\end{aligned}$$

Using the stress-optic law [32], the principal stress difference is expressed as

$${\sigma }_{1}-{\sigma }_{2}=\dfrac{{NF}_{\sigma}}{h}$$

(7)

where, N, F_σ and h represent the fringe order, material stress fringe value and specimen thickness, respectively. Hence, using a colour spectrum, the combined stress field can be plotted in the form of isochromatics by employing Equation. (7) as shown in Fig. 2.