Assessment of statistical responses of multi-scale damage events in an acrylic polymeric composite to the applied stress Gang Qi a,b,n , Jianyu Li a , Ming Fan b , Jihui Li c , Steven F. Wayne b a School of Mechanical Engineering, Tianjin University of Science and Technology, Tianjin, China b Department of Mechanical Engineering, University of Memphis, Memphis, TN 38152, USA c Department of Orthopaedic Surgery, Inova Hospital, Falls Church, VA 22042, USA article info Article history: Received 26 July 2012 Received in revised form 10 February 2013 Accepted 19 March 2013 Available online 6 April 2013 Keywords: Acoustic emission Correlations Damage Multivariate analysis Statistical ensemble Stochastic process abstract Assessments of the statistics of damage ensemble are essential steps to develop accurate modeling and predictions of material failures. Events of random damage constitute a damage system that resides in the microstructures of the materials. Characterization and evaluation of such a system involve assessing the evolving the cascading damage events from hierarchical microstructures of the solids, and there currently lacks an experimental means to do so. To address this need, we established an approach to acquire the events of random damage (ERD) by employing a measureable multi-variate D A defined in our previous work based on acoustic emission. It was found that the responsive events of random damage created by pure tension and three-point bending correlated strongly across all multiscale column vectors of D A in spacetime. The correlation strength is much stronger under tension than that under bending, and much stronger in early loading stages across the column scale vectors of the D A variate. ERD were found to be in clear distinct statistical populations by Andrews' exploratory data analysis plots under tension and bending, and in different stages of loading, which suggests that damage mechanisms are not only “physical”, but also “statistical”. Furthermore, our data showed that the strongly coupled multiscale column vectors of D A can be transformed orthogonally to becoming decoupled principal components, PCs, which may facilitate the constitutive modeling. However, a PC indexes nearly evenly all scale vectors of D A , which implicates, in conjunction with the findings of correlation and Andrews' plot, can be unidirectional, bi-directional, and or interwoven, but is a complicated index variable to describe the cascading multiscale damage events in evolving hierarchical microstructures of semicrystalline polymers. & 2013 Elsevier Ltd. All rights reserved. 1. Introduction A consequence of applying load to a solid material is the occurrence of permanent damage events which ultimately results in failure of the material. In semicrystalline polymers, these events occur in mechanisms such as void nucleation, fibril breakage, craze, cracks, and debonding [1–6], they respond instantaneously to the applied stress, and are dependent on stages of loading and types of the materials [4,7,8]. To analyze material failure, engineers and scientists are concerned with the presence of a crack/flaw and the subsequent failure mechanisms in terms of formation, propa- gation, propagation rate, energy release, and stress singularity as the function of the applied stress [9–11]. But, assessment has been largely overlooked of the statistical responses of these damage events to the applied stress, especially experimental, which may have significant impacts on analyzing the performance and failure of solid materials. In semicrystalline polymers, damage events present as fine cracks, which are the fundamental element in the sequence of the irreversible damage processes. The occurrence of these events is highly random in variables such as time, space, physical character- istics of the material, instrumentation and physical conditions (i.e. temperature, applied stress) [12–18]. Therefore, each event is in essence a function of aforementioned variables. Fig. 1 shows schematically the types, the scale ranges, and the random nature in polymers. The ensemble studies of such damage events would differ fundamentally from the deterministic approaches, i.e. the studies of a single crack scenario that were specified in micro-, meso-, and macroscopic scales [9,10,19–27], and even in the atomic scale [28,29], particularly, when the occurrence of these events correlates and interacts with each other. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/probengmech Probabilistic Engineering Mechanics 0266-8920/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.probengmech.2013.03.006 Abbreviations: AE, acoustic emission; EDA, exploratory data analysis; ERD, events of random damage; KS, Kolmogorov–Smirnov goodness-of-fit; PC, principal com- ponent; PCA, principal component analysis. n Corresponding author at: Department of Mechanical Engineering, University of Memphis, Memphis, TN 38152, USA. Tel.: +1 901 678 2978; fax: +1 901 678 5459. E-mail address: gangqi@memphis.edu (G. Qi). Probabilistic Engineering Mechanics 33 (2013) 103–115