Applied machine learning to predict stress hotspots II: Hexagonal close packed materials
Graphical abstract
Introduction
In polycrystalline materials, an applied stress is distributed inhomogeneously, resulting in stress concentrations, termed stress hot spots. An important mechanism for ductile fracture in metals and their alloys is by the growth and coalescence of microscopic voids, which nucleate near stress hotspots (Hull and Rimmer (1959)). In face centered cubic (fcc) materials under uniaxial tensile deformation, stress hotspots tend to form near microstructural features and usually occur in textures corresponding to maxima in the Taylor factor (Rollett et al. (2010); Mangal and Holm (2018a)). Crystalline anisotropy, which determines the ”hard” and ”soft” directions; also plays an important role. In FCC materials where these directions change between elastic and plastic regimes, the elastic hotspots can become plastic coldspots (Lebensohn et al. (2012)).
The elastic/plastic behavior of hexagonal close packed (HCP) materials is more complex due to the inherent anisotropy of a non-cubic crystal structure. As shown in Fig. 1, HCP materials deform plastically by slip on 4 slip systems: basal , prismatic , pyramidal and pyramidal , each with different critical resolved shear stress (CRSS) values (Thornburg and Piehler (1975)). (Deformation twinning also adds to the complexity but has been ignored in this work.) Deformation textures developed in HCP materials vary due to the unique slip and twinning systems that are activated based on the c/a ratio and the critically resolved shear stress (CRSS) of basal and non basal slip modes.
To understand polycrystal plasticity and texture development in terms of single crystals, the concept of the single crystal yield surface (SCYS) was developed. The SCYS determines the shears that are activated in a grain and depends on the CRSS ratios between deformation modes, as well as the stress state. The SCYS has been analyzed and derived in detail for BCC materials in Orlans-Joliet et al. (1988), for FCC materials in Kocks et al. (1983) and HCP materials in Tome and Kocks (1985). Chin and Mammel (1970) showed that the SCYS is topologically invariant in certain domains of CRSS ratios, and leads to a simplified analysis of deformation when slip modes harden at different rates.
The CRSS ratio is defined with respect to the basal slip resolved shear strength () as:where and are the CRSS of prismatic and pyramidal slip systems respectively. Even if the CRSS of a mode is very high, it might be activated to complete the yield surface to achieve the 5 independent slip modes required by the Taylor and Von-Mises criteria, resulting in a highly anisotropic macroscopic response (Taylor (1938); Piehler (2009)). The situation is worsened by the need to satisfy compatibility and equilibrium conditions between neighboring grains, and results in the material selecting a spatially inhomogeneous solution to accommodate the macroscopic boundary conditions.
Changing the texture of the material will have the same effect of making some slip systems more favorable than others. Hence in order to understand the evolution of stress hotspots, it is necessary to look into a combination of all these variables: texture, grain shape, c/a ratio, CRSS ratios, slip hardening, twinning, temperature and stress state. In this work, we keep the temperature constant, and uniaxial tensile deformation is constrained to occur only by 3 slip modes: prismatic, basal and pyramidal without any twinning or anisotropic slip hardening. The microstructure consists of equiaxed grains and the c/a ratio is fixed. Thus, we can vary the CRSS ratio and crystallographic texture to analyze their impact on stress hotspot formation.
Machine learning (ML) techniques are gaining popularity and have been applied successfully to various fields (LeCun et al. (2015); Bose and Mahapatra (2001); Lavecchia (2015); McMahan et al. (2013); Mangal and Kumar (2016)) to gain insights and relationships between features or attributes of different kinds. These techniques are finding their way into the materials science domain (Rajan (2015); Fedorov and Shamanaev (2017); Gómez-Bombarelli et al. (2016)), in areas such as molecular informatics (Yao et al. (2017)), predicting deformation twinning based on the local structure (Orme et al. (2016)) and predicting phase diagrams (Meredig et al. (2014)). In a companion paper, we have used ML methods to analyze stress hotspots in FCC materials (Mangal and Holm (2018a)). Our model was based on local microstructural features that describe the crystallography (Euler angles, Schmid factor, misorientations) and geometry (grain shape, grain boundary types). The target was predict whether a grain becomes a stress hotspot based on a feature vector whose components are the local microstructural descriptors. In this work, we extend this approach to study stress hotspots in HCP materials as a function of texture and compare them among two different HCP materials: an Equal CRSS ratio case where the CRSS ratio is and an Unequal CRSS ratio case of . The Equal CRSS case is hypothetical and is analyzed purely for model development and analysis. We then compare the performance of machine learning models and delineate the microstructural features that contribute the most in predicting stress hotspots.
Section snippets
Dataset generation
We use the Dream.3D package (Groeber and Jackson (2014)) to generate a dataset of synthetic polycrystalline microstructures with a mean grain size of 2.7 microns consisting of grains each. We study 8 representative textures shown in Fig. 2. For each representative texture, between 6 and 9 stochastic microstructure instantiations were created, resulting in between to grains per texture. The texture intensity for each microstructure instantiation varied from weak (˂ 5 MRD) to
Results and discussion
For the equal CRSS material, the ratio of basal : prismatic : pyramidal CRSS is 1:1:1. It is worth noting that this CRSS ratio is not observed in α-Ti, and represents an ideal HCP material with isotropic slip systems. Fig. 3a shows the representative grain averaged stress distribution in each texture class for the Equal CRSS ratio case: the stress distributions are all right tailed.
For the Unequal CRSS ratio case, uniaxial tensile deformation is simulated with the same
Conclusions
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Stress hotspots can be predicted with 82.5% AUC in HCP materials with Equal CRSS ratio, and 81.18% AUC in HCP materials with Unequal CRSS ratio using random forest models. We observe that the performance of Mixed-models is comparable to or better than Partition-models. This could mean the existence of common factors independent of the macro-texture which cause stress hotspots in a material.
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A change in material composition will result in altered constitutive parameters, and consequently, the
Acknowledgements
This work was performed at Carnegie Mellon University and has been supported by the United States National Science Foundation award number DMR-1307138 and DMR-1507830. Ricardo Lebensohn of the Los Alamos National Laboratory is acknowledged for the use of the MASSIF (EVPFFT) code.
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