A statistical model for predicting size effects on the yield strength in dislocation-mediated crystal plasticity
Introduction
Since the first report on the now-commonly-known mantra of “smaller is stronger” (Brenner, 1956), numerous experiments have been carried out to investigate the yield strength dependence on both extrinsic (i.e. sample dimensions) and intrinsic (i.e. microstructural dimensions) length scales (Arzt, 1998, Uchic et al., 2004, Meyers et al., 2006, Uchic et al., 2009, Greer and De Hosson, 2011, El-Awady, 2015, Ovid’ko et al., 2018). Meanwhile, many computational studies have also been conducted to shed light on the underlying mechanisms that control these size effects, including, atomistic simulations (cf. Weinberger and Cai, 2008, Xiong et al., 2012, Zhu et al., 2013b), discrete dislocation dynamics (DDD) simulations (cf. Espinosa et al., 2005, Rao et al., 2008, El-Awady et al., 2009, Zhou et al., 2011, El-Awady et al., 2016) and strain gradient simulations (cf. Smyshlyaev and Fleck, 1996, Evers et al., 2002, Keller et al., 2012). Such studies provided a wealth of the understanding for the different mechanisms that control the mechanical properties of both single and polycrystalline materials (El-Awady et al., 2016).
Several attempts have been made in literature to incorporate the stochastic variation of the extrinsic and intrinsic dimensions in materials to predict the size effects on the yield strengths in single crystals. Parthasarathy et al. used a single-arm dislocation model to rationalize size effects by considering the randomness of dislocation source lengths in single crystal samples (Parthasarathy et al., 2007). Ng and Ngan performed Monte Carlo simulations in single crystal Al micropillars to study burst probability (Ng and Ngan, 2008b). They also showed the correlation between the operative slip system and the Schmid factor for different sample sizes (Ng and Ngan, 2008a). Ngan and coworkers also proposed a general probability model to rationalize the power-law size effect on the yield strength of single crystal micropillars (Ngan et al., 2006, Ngan, 2011, Gu and Ngan, 2013). In this model, the survival probability of a micro-pillar describes the fraction of samples that has not yielded at any given applied stress level. They showed that the power-law scaling of the size effect results from a power-law dislocation source length distribution. They have also investigated the controlling factors for the power exponents of the strength-size relation. Phani et al. proposed a two-dimensional (2D) statistical model studying the role of randomness in both dislocation spacing (in the presence of dislocations but also in their absence) and Schmid factor on strength (Phani et al., 2013). El-Awady derived a generalized size-dependent Taylor-strengthening law to predict average strengths as a function of crystal/grain size and the dislocation density based on three-dimensional (3D) DDD simulations with randomly-generated initial dislocation microstructures (El-Awady, 2015). Gao and Bei developed a model considering both the thermally activation and spatial stochasticity with random distributions of dislocation nucleation sources and dislocations (Gao and Bei, 2016).
On the other hand, for polycrystalline samples, it is well established that the strength of polycrystals depends on grain size (Miyazaki et al., 1979, Meyers et al., 2006, Di Leo and Rimoli, 2019). As described by the Hall–Petch relation, the strength of polycrystalline metals increases with decreasing grain size (Hall, 1951, Petch, 1953). This is primarily due to the spatial inhomogeneous plastic deformation in each grain. Strain gradient models were utilized in the past to interpret this size effect (Fleck et al., 1994, Smyshlyaev and Fleck, 1996, Gao et al., 1999). It was also shown that the polycrystal size effect can be correlated to the single crystal size effect, where the Hall–Petch relation was directly derived from the single crystal generalized Taylor law by coupling with an appropriate description of the evolution of the dislocation density in polycrystals (El-Awady, 2015). Nevertheless, the role of the stochastic nature of the microstructures on material strengths is not reflected in the Hall–Petch relationship. To address this, the effect of grain size distribution on the yield stress in heterogeneous materials has been numerically investigated by Berbenni et al. (2007). Keller et al. also studied the influence of the ratio between the sample thickness and grain size on the strengths in Ni using mechanical tests and TEM observations (Keller et al., 2012). Shao et al. developed a probabilistic model for the onset of plasticity to account for the statistical variation of the initial dislocation content (Shao et al., 2014). Askari et al. combined Monte Carlo methods with polycrystal continuum dislocation dynamics models to study the stochastic mechanical properties of polycrystals (Askari et al., 2015) and identified that dislocation scarcity, grain size distribution and crystal orientation cause the uncertainty in strengths. Eastman et al. carried out experiments on René 88 and attributed the stochastic response in strengths to both constraints in surface grains and biased distributions of Schmid factor values (Eastman et al., 2016).
One significant limitation of the existing models and calculations on size effects is that all implementations are based on specific distributions or assumptions of microstructures (e.g. Parthasarathy et al. adopted uniform dislocation length distributions and a single-ended dislocation model (Parthasarathy et al., 2007); Rinaldi et al. adopted the Weibull distribution, which has been used for many years to understand the size-dependent fracture strength of brittle materials, to describe the statistics of the “weak-link” concept and examine the plastic behavior of nanocrystalline Ni (Rinaldi et al., 2008); Phani et al. assumed uniform Schmid factor distributions and focused on infinitely long straight edge dislocations Phani et al., 2013; numerically-generated maximum Schmid Factor distributions and the identical grain size assumption are employed in Ref. Eastman et al. (2016)). With regard to the randomness of dislocation source length, Weibull distributions (El-Awady et al., 2009), log-normal distributions (Shishvan and Van der Giessen, 2010), and uniform distributions of the FR source lengths (Rao et al., 2008) are all commonly used. Regarding grain sizes in polycrystalline materials, rather than log-normal grain size density probability functions that are often used to fit real polycrystalline materials (Valiev et al., 2000, Berbenni et al., 2007), there are a considerable number of studies that report other types of grain size distributions (Sevillano and Aldazabal, 2004, Raeisinia et al., 2008, Humphreys and Hatherly, 2012).
Given that there are different possible statistical distributions of microstructural quantities, which inevitably affect the macroscopic properties of crystalline materials, there is a need to develop a comprehensive model that can account for all statistical distributions of the different microstructrual variables. Here, we present a generic weakest-link-based statistical model for dislocation-mediated crystal plasticity that predicts the size effects on the yield strength of single crystals, columnar-grain structured thin films (i.e. polycrystalline thin films with an average grain size equal to or larger than the thin film thickness), and polycrystals by accounting for the statistical distributions of different microstructural variables (including grain size, grain orientation, and dislocation length distributions). The paper is organized as follows. In Section 2, relevant order statistics formulations are summarized and used to derive the statistical model of the yield strength based on the extrinsic and intrinsic dimensions of the material. In Section 3, the developed model is used to predict the size effects in Ni single crystal, thin films, and polycrystals and the results are discussed in view of published experimental and simulation results. Finally, a summary and conclusions are given in Section 4.
Section snippets
Theoretical model
In this section, order statistics are used to develop a weakest-link statistical model to predict the effect of extrinsic and intrinsic dimensions on the yield strength of materials in which plasticity is governed by dislocation nucleation/slip. The focus here is on three particular cases: single crystals, columnar-grain structured thin films, and polycrystals, as shown schematically in Fig. 1. In the following analysis, the crystal is assumed to have an idealized rectangular prism geometry
Numerical results
In this section, the probability yield functions defined earlier are utilized to predict the size effects on the yield strength of FCC Ni single crystals, columnar-grain structured thin films, and polycrystals. The parameters used for the numerical calculations are listed in Table 1. Different CDFs of dislocation source length and PDFs of grain size distributions are also considered as listed in Table 2. Finally, two different grain orientation distribution are also considered as shown in Fig. 2
Summary and conclusions
In this work, a statistical dislocation-mediated crystal plasticity model was developed to study the size-affected yield strength in single crystals, columnar-grain structured thin films, and polycrystals. The developed model systematically takes into account the randomness of extrinsic sizes (sample sizes) and intrinsic sizes (dislocation densities and grain sizes) as well as grain orientations to study the size effects on the yield strength. The model accurately predicts that the strength of
CRediT authorship contribution statement
Yejun Gu: Conceptualization, Methodology, Formal analysis, Investigation, Software, Visualization, Writing - original draft, Writing - review & editing. David W. Eastman: Conceptualization, Investigation, Writing - review & editing. Kevin J. Hemker: Conceptualization, Writing - review & editing, Funding acquisition. Jaafar A. El-Awady: Conceptualization, Methodology, Visualization, Writing - original draft, Writing - review & editing, Funding acquisition.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgments
This work was supported by the National Science Foundation CAREER Award #CMMI-1454072 and the Center of Excellence on Integrated Materials Modeling (CEIMM) at Johns Hopkins University awarded by the Air Force Research Laboratory: the Air Force Office of Scientific Research and the Materials and Manufacturing Directorate . The authors gratefully acknowledge Dr. Luoning Ma (JHU), Dr. Yaxing Yang (XMU) and Dr. Wenxin Zhou (UCSD) for fruitful discussions.
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