Large rotations of the grain-scale stress tensor during yielding set the stage for failure

https://doi.org/10.1016/j.ijplas.2021.103087Get rights and content

Highlights

  • Rotations of the stress tensor in individual grains are experimentally quantified.

  • These stress rotations are an order of magnitude larger than lattice reorientation.

  • The stress state rotates mainly about rather than towards the loading axis.

  • The stress tends to rotate towards the yield surface vertices.

  • These stress rotations reduce triaxiality in certain grains and determine failure.

Abstract

The stress-state within individual grains in a polycrystal determine the fate of the aggregate including mechanical failure. By tracking the evolution of the stress tensor throughout the elastoplastic transition, large rotations of the stressstate, which have long been theorized to occur, are observed experimentally for the first time. These stress rotations (∼15°) are more than an order of magnitude larger than the concomitant crystallographic lattice reorientations (∼0.9°) well-known to occur during metal plasticity. Furthermore, these rotations are accompanied by a decrease in stress triaxiality within certain grains, promote strain softening, and set the stage for failure at an early stage of deformation. These results provide a completely new perspective through which to contemplate the question of “hot-spots” responsible for failure of high-performance structural materials.

Introduction

Highly engineered polycrystalline, metallic alloys are an integral part of the modern society and are at the heart of infrastructure, manufacturing, energy production, and transportation sectors. Sustainable performance relies on the ability of these materials to withstand design loads and resist failure. Scientists and engineers have increasingly employed physics-based, multiscale modeling approaches to obtain a detailed description and understanding of polycrystalline deformation behavior. This strategy necessitates well-designed experiments for model validation (Sangid, 2020). At low homologous temperatures, under high stresses, the motion of dislocations is the primary mechanism for plastic deformation in most crystalline materials. Thus, quantifying the stress-state within the individual grains of a polycrystalline material is key for understanding plastic deformation. Furthermore, these stresses play a critical role in determining damage via initiation of shear bands, cracks and voids, in a loaded polycrystal. Notably, the accumulation of damage due to localized plasticity during cyclic loading leads to fatigue, which is one of the main failure mechanisms in structural components (e.g. Gagg and Lewis, 2009). The elastoplastic transition (also known as yielding) is particularly relevant, since fatigue can occur at stresses well below the macroscopic yield strength, where only a fraction of grains may undergo dislocation motion during cyclic loading. Due to the inherent elastic and plastic anisotropy of single crystals, the stress (and strain) levels within the individual grains of a polycrystal can strongly vary, e.g. stiffer grains share a larger stress. As a result, significant grain-scale heterogeneity develops where, e.g., a distinct dislocation substructure evolution may occur in grains of different orientations (Jakobsen et al., 2006, 2007).

A Single Crystal Yield (hyper) Surface (SCYS) is a map in six-dimensional stress space, which delineates the elastic conditions from those which give rise to plasticity. A slip system induces a (hyper) facet of the SCYS in stress space, and the direction of plastic straining on that system is perpendicular to that facet. Facets from all slip systems of highly symmetric crystals form a closed surface, the inner envelope of which is the SCYS. For the rate independent case, slip occurs if the resolved shear stress (RSS) on a slip system is equal to the critical resolved shear stress (CRSS) on that system. For the rate dependent case, the CRSS is a function of temperature and strain-rate and the strain rate sensitivity of the material is an important parameter that determines the “roundness” of the single crystal yield potential (SCYP). The seminal theoretical works of Kocks (Kocks, 1970; Kocks et al., 1983) followed by finite element-based full-field polycrystal plasticity simulations of Dawson's research group (e.g., Ritz et al., 2010) highlighted the fact that the crystal stress-state may strongly reorient during the elastoplastic transition. These rotations are posited to occur along ‘facets’ and ‘edges’ of the SCYS towards stress ‘vertices’ where multiple slip systems are simultaneously activated, in order to accommodate arbitrary straining directions and maintain compatibility among the grains. Since then, several other finite element-based simulation studies have also shown that the crystal stressstate tends to align with a vertex of the SCYS (Han et al., 2012, 2013; Han and Chung, 2012). However, the experimental evidence for such predictions have been lacking.

Fig. 1 shows a schematic two-dimensional SCYS defined by three slip systems (one per pair of parallel facets). A portion of the SCYS is magnified to illustrate the concept of stress rotation. For simplicity, isotropic elasticity and perfect plasticity are assumed, since extension to the completely general case (e.g. Lee et al., 2017, 2018) does not alter the concept. Prior to yielding (at 1), the material is elastic, and as soon as the material yields (at 2) the total straining direction is now (within an infinitesimal strain framework) the algebraic sum of two components; the plastic strain increment, which is normal to the facet of the yield surface, and the elastic strain increment. With deformation, the stress and strain tensor evolve in a manner such that stress equilibrium, strain compatibility, and the grain's boundary conditions are satisfied simultaneously. Due to interactions with neighboring grains, the total straining direction of a grain may change with deformation and, consequently, the stress will rotate in order to produce the correct total strain increment and maintain compatibility among the grains (e.g., at 3). If the stressstate rotates to a vertex where multiple slip systems intersect, then a range of plastic straining directions can be accommodated by a linear combination of the involved slip systems (at 4). This implies that the crystal stress-state will tend toward the vertices (or at least edges) of the SCYS, rather than only rotating toward the macroscopic applied stress (Kocks, 1970; Kocks et al., 1983; Ritz et al., 2010).

The specific vertex towards which the grain stress-state rotates will depend on both the boundary conditions imposed on a specific grain and its crystallographic orientation. Kocks (1970) emphasized that, in order to maintain compatibility among the grains in a polycrystal, the resulting conditions of “polyslip” are the norm beyond the elastoplastic transition.

An important point to note is that as the stress-state rotates, the magnitudes of the stresses may also change. Thus, the rotation of the stress tensor is potentially accompanied by an alteration of the nature of the stressstate, e.g. the stress triaxiality, which is known to have a significant effect on failure of materials (Lou et al., 2014; Papasidero et al., 2015).

Section snippets

Sample and ex-situ characterization

A metastable β-Ti alloy, Timetal-18 with composition, 5.5Al-5V-5Mo-2.4Cr-0.75Fe-0.15O (in wt.%) (Fanning, 2011; Lebrun et al., 2014), was used for this study because a relatively low modulus and high yield strength provide the best combination for minimizing the experimental uncertainty in the diffraction-based assessments of lattice strain (and therefore stress). Rectangular sections were machined out from the as-received rolled plate with bimodal microstructure, which were then subjected to β

Results

The constitutive response obtained from the in-situ HEDM experiment is shown in Fig 2a. The numbered points denote different stages throughout the elastoplastic transition. The 3D nearfield reconstructed microstructure and relationship to the macroscopic sample are presented in Fig. 2b.

To visualize the orientation of the stress tensor of a grain with respect to the polycrystalline specimen, it is convenient to use the stereographic projection and to map the eigenvectors (directions) of the

Discussion

The experimental results highlight that the rotations of the stress-state predicted by theory are observable. Furthermore, they are large, relative to the rigid body and lattice rotations which crystals undergo during the elastoplastic transition. Furthermore, although the stresses tend to rotate toward a vertex, the rotations saturate by the end of the elastoplastic transition, and the stress-states are still far from the vertices (∼20°). This is expected because of the following reasons:

  • 1

    Most

Concluding remarks

The results from MASSIF full-field crystal plasticity simulations clearly show that the stress rotations presented in this paper is not a result of experimental errors or uncertainties. It is thus possible that combining knowledge of such stress rotations with statistical descriptions of microstructure could reveal previously unrealized vulnerabilities in materials. New techniques of nondestructive evaluation could be developed to help avoid premature failures. Alternatively, microstructures

Funding

The research at University of Virginia was sponsored by the US Department of Energy, Basic Energy Sciences, Mechanical Behavior and Radiation Effects Program led by Dr. John Vetrano, Grant # DE-SC0018923. The data for this paper was collected at the Center for High Energy X-ray Sciences (CHEXS) which is supported by the National Science Foundation under award DMR-1829070.

Data and materials availability

The raw data is available as diffraction patterns which are stored at the CHESS repository.

CRediT authorship contribution statement

Jishnu J. Bhattacharyya: Conceptualization, Investigation, Methodology, Formal analysis, Writing – original draft, Visualization. Darren C. Pagan: Conceptualization, Investigation, Methodology, Formal analysis, Software, Writing – review & editing. Sean R. Agnew: Conceptualization, Investigation, Methodology, Resources, Writing – review & editing, Supervision, Funding acquisition, Project administration.

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

We thank Prof. Haitham El-Kadiri and Dr. Andrew Oppedal (Mississippi State University) and the Army Research Laboratory (ARL) for providing us with the Timetal-18 samples.

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