High-throughput experiments for rare-event rupture of materials

Abstract

•Develop a high-throughput experiment to study rare-event rupture•Fabricate 1,000 samples and stretch them simultaneously•Identify rupture of individual samples by image processing•Analyze the data from the high-throughput experiment by extreme value statistics Engineers place a high premium on unexpected rupture of a material subjected to a small stretch or a small number of cycles. Under such conditions, rupture is a rare event. Predicting such a rare event requires tests of many samples and is extremely time-consuming. This paper describes a high-throughput experiment to obtain large datasets for the conditions of rupture. We load 1,000 samples of a material simultaneously, either monotonically or cyclically, and identify the rupture of individual samples by processing images automatically to detect ruptures. We analyze the data using extreme-value statistics. It is hoped that this work will motivate further the development of high-throughput experiments to predict rare events. The conditions for rupture of a material commonly vary from sample to sample. Of great importance to applications are the conditions for rare-event rupture, but their measurements require many samples and consume much time. Here, the conditions for rare-event rupture are measured by developing a high-throughput experiment. For each run of the experiment, 1,000 samples are printed under the same nominal conditions and pulled simultaneously to the same stretch. Identifying the rupture of individual samples is automated by processing the video of the experiment. Under monotonic load, the rupture stretch for each sample is recorded. Under cyclic load, the number of cycles to rupture for each sample is also recorded. Rare-event rupture is studied by using the Weibull distribution and the peak-over-threshold method. This work reaffirms that predicting rare events requires large datasets. The high-throughput experiments enable the prediction of rare events with high accuracy and confidence. The conditions for rupture of a material commonly vary from sample to sample. Of great importance to applications are the conditions for rare-event rupture, but their measurements require many samples and consume much time. Here, the conditions for rare-event rupture are measured by developing a high-throughput experiment. For each run of the experiment, 1,000 samples are printed under the same nominal conditions and pulled simultaneously to the same stretch. Identifying the rupture of individual samples is automated by processing the video of the experiment. Under monotonic load, the rupture stretch for each sample is recorded. Under cyclic load, the number of cycles to rupture for each sample is also recorded. Rare-event rupture is studied by using the Weibull distribution and the peak-over-threshold method. This work reaffirms that predicting rare events requires large datasets. The high-throughput experiments enable the prediction of rare events with high accuracy and confidence. Predicting rupture of materials is of paramount importance. Rupture is a complex process taking place over many time and length scales. Despite decades of intense efforts, theory and computation alone can seldom predict the conditions for rupture.1Creton C. Ciccotti M. Fracture and adhesion of soft materials: a review.Rep. Prog. Phys. 2016; 79: 046601https://doi.org/10.1088/0034-4885/79/4/046601Crossref PubMed Scopus (328) Google Scholar, 2Li W.D. Liaw P.K. Gao Y.F. Fracture resistance of high entropy alloys: a review.Intermetallics. 2018; 99: 69-83https://doi.org/10.1016/j.intermet.2018.05.013Crossref Scopus (84) Google Scholar, 3Rajabi A. Ghazali M.J. Daud A.R. Chemical composition, microstructure and sintering temperature modifications on mechanical properties of TiC-based cermet – a review.Mater. Des. 2015; 67: 95-106https://doi.org/10.1016/j.matdes.2014.10.081Crossref Scopus (110) Google Scholar, 4Domun N. Hadavinia H. Zhang T. Sainsbury T. Liaghat G.H. Vahid S. 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Observing rare events is extremely time consuming. A potential solution is to conduct high-throughput experiments.13Liu Y.H. Hu Z.H. Suo Z.G. Hu L.Z. Feng L.Y. Gong X.Q. Liu Y. Zhang J.C. High-throughput experiments facilitate materials innovation: a review.Sci. China-Technol. Sci. 2019; 62: 521-545https://doi.org/10.1007/s11431-018-9369-9Crossref Scopus (25) Google Scholar This solution poses challenges in developing methods to fabricate and test a large number of samples. Here, we develop a high-throughput experiment to study rare-event rupture of materials (Figure 1). We print 1,000 samples under the same nominal conditions. We program the printer to print the samples in five layers, along with six connective bars. We design a kinematic mechanism of one degree of freedom to deform all the samples by the same amount simultaneously. The length of a deformed sample divided by that of the undeformed sample defines the stretch λ. For such a large number of samples, it is impractical to identify the rupture of individual samples by the human eye. We record the video of each run of the experiment and write software that processes the video to identify rupture of individual samples. We conduct four runs of the experiment under monotonic load to a stretch of λ = 2.2, observe that 3,596 out of 4,000 samples rupture, and we record the rupture stretch of each sample. We also run the experiment under cyclic load to four amplitudes of stretch λ = 1.6, 1.7, 1.8, and 1.9, observe that 3,996 samples rupture, and record the number of cycles to rupture of each individual sample. The large datasets enable us to analyze rare-event rupture by using the Weibull distribution and peak-over-threshold method from extreme-value statistics. High-throughput experiments have long been developed. Such an experiment must resolve two main challenges: fabricate a large number of samples, and test them. Massive sample preparation is often achieved using methods such as printing and photolithography.14Burger B. Maffettone P.M. Gusev V.V. Aitchison C.M. Bai Y. Wang X. Li X. Alston B.M. Li B. Clowes R. et al.A mobile robotic chemist.Nature. 2020; 583: 237-241https://doi.org/10.1038/s41586-020-2442-2Crossref PubMed Scopus (210) Google Scholar, 15Libonati F. Gu G.X. Qin Z. Vergani L. Buehler M.J. Bone-inspired materials by design: toughness amplification observed using 3D printing and testing.Adv. Eng. Mater. 2016; 18: 1354-1363https://doi.org/10.1002/adem.201600143Crossref Scopus (112) Google Scholar, 16Ngo T.D. Kashani A. Imbalzano G. Nguyen K.T.Q. Hui D. Additive manufacturing (3D printing): a review of materials, methods, applications and challenges.Compos. B Eng. 2018; 143: 172-196https://doi.org/10.1016/j.compositesb.2018.02.012Crossref Scopus (2569) Google Scholar, 17Xiang Gu G. Su I. Sharma S. Voros J.L. Qin Z. Buehler M.J. Three-dimensional-printing of bio-inspired composites.J. 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Rodelas J.M. et al.Automated high-throughput tensile testing reveals stochastic process parameter sensitivity.Mater. Sci. Eng. a-Struct. Mater. Prop. Microst. Process. 2020; 772: 138632https://doi.org/10.1016/j.msea.2019.138632Crossref Scopus (18) Google Scholar Such sequential methods are unsuitable for processes that take a long time. Examples of prolonged processes include rupture under cyclic load, creep, and slow crack growth. The time of tests can be reduced by loading many samples simultaneously. For example, in 1930, Cooper studied fatigue rupture of a rubber by cyclically loading eight samples simultaneously.27Cooper L.V. Laboratory evaluation of flex-cracking resistance.Ind. Eng. Chem. 1930; 2: 0391-0394https://doi.org/10.1021/ac50072a018Crossref Scopus (1) Google Scholar In a previous paper, we studied fatigue rupture of a hydrogel by cyclically loading six samples simultaneously.28Zhou Y. Hu J. Zhao P. Zhang W. Suo Z. Lu T. Flaw-sensitivity of a tough hydrogel under monotonic and cyclic loads.J. Mech. Phys. Sol. 2021; 153: 104483https://doi.org/10.1016/j.jmps.2021.104483Crossref Scopus (3) Google Scholar Sun et al. studied delamination and cracking of many microfabricated inorganic islands on a plastic substrate.18Sun J.Y. Lu N.S. Oh K.H. Suo Z.G. Vlassak J.J. Islands stretch test for measuring the interfacial fracture energy between a hard film and a soft substrate.J. Appl. Phys. 2013; 113: 223702https://doi.org/10.1063/1.4810763Crossref Scopus (10) Google Scholar We first stretch 1,000 samples monotonically (Video S1), setting the stretch rate as 0.15/min. In each run of the experiment, we stretch 1,000 samples from λ = 1 to λ = 2.2 and record the number of ruptured samples as a function of λ. The experiment is repeated four times. A representative experiment is illustrated with three snapshots. In the unstretched state, λ = 1, the 1,000 samples are intact (Figure 2A ). At λ = 1.8, 34 samples have ruptured (Figure 2B). At λ = 2.2, 947 samples have ruptured (Figure 2C). https://www.cell.com/cms/asset/a02c8fb7-58aa-42ab-8bdf-7b79155d0c2d/mmc4.mp4Loading ... Download .mp4 (30.53 MB) Help with .mp4 files Video S1. Video of high-throughput rupture experiments At a given stretch λ, let F be the number of ruptured samples divided by the total number of samples (1,000). We plot the cumulative distribution function (cdf), F(λ), for each run of the experiment (Figure S1). The curves of four runs of the experiment nearly coincide, indicating that the experiment is statistically reproducible. We then aggregate the data of the four runs of the experiment. Of the 4,000 samples tested, a total of 3,596 samples rupture. Each ruptured sample corresponds to a data point in the F−λ plane (Figure 3A ). Following a common practice of statistics for rupture,29Doremus R.H. Fracture statistics - a comparison of the normal, Weibull, and Type-I extreme value distributions.J. Appl. Phys. 1983; 54: 193-198https://doi.org/10.1063/1.331731Crossref Scopus (80) Google Scholar we fit the measured cdf to the three-parameter Weibull distribution:30Weibull W. Fatigue testing and analysis of results. Elsevier, 2013Google Scholar,31Coles S. An introduction to statistical modeling of extreme values. Springer, 2001Crossref Google ScholarF(λ)=1−exp{−[(λ−α)/β]γ}(Equation 1) where α, β, and γ characterize the location, scale, and shape of the distribution. We determine the fitting parameters as α = 1.6709, β = 0.4175, and γ = 3.4404 using the maximum-likelihood Weibull fitting method.32Millar R.B. Maximum Likelihood Estimation and Inference: With Examples in R, SAS and ADMB. John Wiley & Sons, 2011Crossref Scopus (145) Google Scholar The measured cdf approximately follows the Weibull distribution (solid curve) in the full range of data (Figure 3A). For a given value of the cumulative probability F, we further calculate the 95% confidence interval (CI) of the rupture stretch λ and plot in the F-λ plane as two dashed curves. The 95% CI appears to be narrow in the full range of data. However, the experimental data mostly lie outside the 95% CI. This indicates that the Weibull model is not appropriate to approximate the whole range of the cdf. To discuss rare-event rupture, we magnify the plots in Figure 3A in the range 0 ≤ F(λ) ≤ 1% (Figure 3B). Similar to the overall fitting, in the range of rare events, a large portion of the experimental data, especially in the tail of interest, lies outside the 95% CI. That is, the Weibull fit using the data of all 4,000 tested samples is unable to predict the experimental data, including the rare events, with high confidence. The same conclusion is reached in an alternative way of evaluating the fitting results (Figure S2). To achieve a prediction for rare-event rupture with accuracy and confidence, we adopt a procedure in the extreme-value statistics, called the peak-over-threshold method.31Coles S. An introduction to statistical modeling of extreme values. Springer, 2001Crossref Google Scholar This method focuses on the statistics of the tail by imposing a threshold stretch. We apply the method to our data, and find the threshold stretch to be 1.87 (Figure S3). Of the 4,000 tested samples, 255 samples rupture at stretches below the threshold and are used to fit the Weibull distribution. With this method, all the experimental data in the range 0 ≤ F(λ) ≤ 1% fall within the 95% CI (Figure 3C). For example, we specify a rare event by the cumulative probability F(λ) = 0.1%, corresponding to the first four ruptured samples among the 4,000 tested samples. For the rare event of “0.1% rupture,” the measured rupture stretch is λ = 1.7111, the Weibull fit is λ = 1.7166, and the 95% CI is 1.7056 < λ < 1.7288. This high level of confidence as well as the narrow range of stretch are likely to satisfy most applications. By use of the peak-over-threshold method, the Weibull fit is reliable for prediction of rare-event rupture. Normally, such a large dataset of rupture of testing 4,000 samples is unavailable, and the engineer needs to predict the rare-event rupture using whatever number of tested samples. To represent a smaller dataset, from the 4,000 tested samples we randomly select 200 samples. The peak-over-threshold method determines a threshold stretch λ = 2. Of the 200 samples, 82 samples rupture at stretches below the threshold and are used to fit the Weibull distribution and calculate the 95% CI (Figure 3D). Of the 200 samples, only one sample belongs to the range of 0 ≤ F(λ) ≤ 1%, and this sample falls within the 95% CI. However, the 95% interval is perhaps too wide to satisfy some engineers. We next test the prediction of rare-event rupture by using the 200 samples. We plot the experimental data of all the 4,000 tested samples in the range of 0 ≤ F(λ) ≤ 1%. All the first 40 ruptured samples of the 4,000 samples fall within the 95% CI, even though only one of the 200 samples belongs to the range 0 ≤ F(λ) ≤ 1%. Thus, the dataset of the 200 samples can predict rare events with high confidence, but with a wide interval. To narrow the 95% CI, datasets with more samples are needed. From the 4,000 tested samples we randomly select 500 and 1,000 samples, and repeat the statistical procedure (Figures 3E and 3F). As the number of selected samples increases, the 95% CI becomes narrower. The large dataset can predict rare events with both high confidence and narrow interval. We next conduct high-throughput experiments of rupture under cyclic load (Video S2). The stretch rate is 0.2/s. In each run of the experiment, 1,000 samples are cycled between the undeformed state and a prescribed amplitude of stretch, λ. Such a test is conducted with four prescribed amplitudes of stretch, λ = 1.6, 1.7, 1.8, and 1.9. Each test is terminated when the number of cycles reaches 30,000. For each run of the experiment, snapshots are presented at four numbers of cycles (Figure 4). For example, for the test with the prescribed amplitude of stretch λ = 1.6, no sample ruptures after the first cycle, seven samples rupture after 500 cycles, 49 samples rupture after 1,000 cycles, and only four samples survive after 30,000 cycles. The number of cycles to rupture for individual samples, N, is obtained by processing the images from the recorded video. https://www.cell.com/cms/asset/5c1cbdd8-7ff5-4990-b078-9afd1920a2e3/mmc5.mp4Loading ... Download .mp4 (28.34 MB) Help with .mp4 files Video S2. Video of high-throughput fatigue experiments We characterize the probability of fatigue rupture by the cdf, Fλ(N), defined as the number of ruptured samples divided by the total number of samples at a given amplitude of stretch λ (Figure 5A ). For each amplitude of stretch λ, we fit the cdf, Fλ(N), to the three-parameter Weibull distribution function:Fλ(N)=1−exp{−[(ln(N)−α)/β]γ}(Equation 2) where α, β, γ are the location, scale, and shape parameters. The scatter in the number of cycles to rupture N is large. Following a common practice, we use ln(N) instead of N in the Weibull distribution (Figure 5A). Using the maximum likelihood Weibull fitting method, we find that the parameters are α = 5.4835, β = 2.7763, γ = 3.7668 for λ = 1.6; α = 3.4563, β = 3.6866, γ = 4.0735 for λ = 1.7; α = 2.9149, β = 3.5968, γ = 4.6706 for λ = 1.8; and α = 1.6939, β = 3.5867, γ = 6.4619 for λ = 1.9. The measured cdf under cyclic loads approximately follows the Weibull distribution in the full range of the data, but a large portion of the data lies outside the 95% CI (Figure 5A). Figure 5B magnifies the plots in Figure 5A to show the cdf for λ = 1.6 up to Fλ(N) = 1%. Similar to the fitting results under monotonic loads, using all the experimental data under cyclic loads, the Weibull fit is unable to predict fatigue rupture with high confidence. We next use the data of the first 200 ruptured samples to fit the Weibull distribution and find that all the ruptured samples of the 1,000 tested samples fall within the 95% CI (Figure 5C). The large dataset gives a remarkably narrow interval of high confidence (Figure 5C). For example, for the rare event of “1% fatigue rupture,” that is, the first 10 ruptured samples among the 1,000 tested samples, the measured number of cycles to rupture is N = 517, the Weibull fit is N = 566, and the 95% CI is 480 < N < 701. Obtaining a large dataset of fatigue rupture is much more time consuming than monotonic rupture. It is very often seen in the literature that only a few dozen samples are tested.10Syzrantseva K. Syzrantsev V. Determination of parameters of endurance limit distribution law of material by the methods of nonparametric statistics and kinetic theory of high-cycle fatigue.Key Eng. Mater. 2017; 736: 52-57https://doi.org/10.4028/www.scientific.net/KEM.736.52Crossref Scopus (8) Google Scholar To mimic this common practice, we randomly select 50 samples from the 1,000 tested samples, and use the first 10 ruptured samples of the 50 samples to fit the Weibull distribution and calculate the 95% CI. To test the prediction of rare events, we plot the ruptured samples among all the 1,000 tested samples in the range of 0 ≤ F(λ) ≤ 1%. Selecting 50 samples from the 1,000 samples is random, so we repeat this procedure by selecting 50 samples from the 1,000 samples three times (Figures 5D–5F). The three fitting results show clear inconsistency: sometimes the experimental data fall outside the 95% CI (Figure 5E), and sometimes the 95% CI is very wide (Figure 5F). As far as the Weibull statistics is concerned, a small dataset is unable to predict rare events with high confidence and narrow interval. Our experiments invalidate this common practice in the literature. We also plot the prescribed amplitude of stretch, λ, against the number of cycles to rupture of each individual sample, N, in the λ−N plane (Figure S4). This practice has been widely used to report fatigue data for small numbers of samples. For example, if each run of the test consists of only six samples, only six data points appear in the λ−N plane.28Zhou Y. Hu J. Zhao P. Zhang W. Suo Z. Lu T. Flaw-sensitivity of a tough hydrogel under monotonic and cyclic loads.J. Mech. Phys. Sol. 2021; 153: 104483https://doi.org/10.1016/j.jmps.2021.104483Crossref Scopus (3) Google Scholar These few data points visually display the mean and the scatter of fatigue life. However, when each run of the test has 1,000 samples, such a λ−N plot becomes less meaningful (Figure S4). The large number of data points appear as a continuous line, which no longer visually displays the mean and the scatter of fatigue life. In one run of the above experiments, either monotonic or cyclic, the distribution of ruptured samples may not be uniform. For example, more samples fail in the top and left parts in Figure 2C. Such nonuniformity makes testing 1,000 samples together in a high-throughput experiment different from testing 1,000 samples one by one in a conventional experiment. Both types of experiments introduce variations from sample to sample, but the causes for the variations can be different. The causes for variations in the high-throughput experiment deserve a careful study. The video of an experiment can be inspected by human eyes and analyzed by a combination of image processing and statistics to identify nonuniform distribution in ruptured samples. The identified nonuniformity can be used to pinpoint its cause. For example, the observation that more samples fail in the top and left parts suggests insufficient rigidity of the polymer brackets and aluminum plates. Other possible causes for nonuniformity include friction and vibration. Once identified, a cause for nonuniformity may be minimized by improving the experimental design. This iterative approach is fundamental to the design of the high-throughput experiment and will be reported in a subsequent work. The current experiment uses 3D printed samples, but the high-throughput experiment can be modified to accommodate samples prepared by other methods, such as cutting and molding. Consider the case where a large piece of a material exists, such as a polyethylene film. The piece can be cut into a pattern of many dogbone-shaped samples connected at their ends and then mounted simultaneously to the test frame. The high-throughput experiment can generate large datasets for fatigue of polyethylene. Next, consider the case where a large piece of a material does not exist, such as a type of biological tissue. A high-throughput experiment will require mounting individual samples to the test frame. The experiment will still save time in characterizing fatigue of the tissue. As noted before, multiple samples have long been tested simultaneously in fatigue experiments, although the number of samples has been small.27Cooper L.V. Laboratory evaluation of flex-cracking resistance.Ind. Eng. Chem. 1930; 2: 0391-0394https://doi.org/10.1021/ac50072a018Crossref Scopus (1) Google Scholar The method of image processing described here will allow such an experiment to be conducted for a large number of samples. The current high-throughput experiment is displacement controlled, whereas conventional mechanical tests sometimes are force controlled. We will report on force-controlled high-throughput experiments in a subsequent work. In summary, we have developed a high-throughput experiment to study rare-event rupture of materials. We print a large number of samples under the same nominal conditions, design a kinematic mechanism of one degree of freedom to pull the samples simultaneously to the same stretch, and write software that analyzes the videos of the experiment to identify the rupture of individual samples. The large datasets are used to study the rare events of rupture at small stretches and small numbers of cycles. Our work reaffirms a truism: predicting the statistics of rare events with a narrow interval of high confidence requires large datasets. It is hoped that further studies will soon take place to advance high-throughput experiments and statistical methods to predict rare events.