Modeling slip system strength evolution in Ti-7Al informed by in-situ grain stress measurements

Abstract

Far-field high-energy X-ray diffraction microscopy is used to asses the evolution of slip system strengths in hexagonal close-packed (HCP) Ti-7Al during tensile deformation in-situ. The following HCP slip system families are considered: basal $〈a〉$, prismatic $〈a〉$, pyramidal $〈a〉$, and first-order pyramidal $〈c+a〉$. A 1 mm length of the specimen's gauge section, marked with fiducials and comprised of an aggregate of over 500 grains, is tracked during continuous deformation. The response of each slip system family is quantified using ‘slip system strength curves’ that are calculated from the average stress tensors of each grain over the applied deformation history. These curves, which plot the average resolved shear stress for each slip system family versus macroscopic strain, represent a mesoscopic characterization of the aggregate response. A short time-scale transient softening is observed in the basal $〈a〉$, prismatic $〈a〉$, and pyramidal $〈a〉$ slip systems, while a long time-scale transient hardening is observed in the pyramidal $〈c+a〉$ slip systems. These results are used to develop a slip system strength model as part of an elasto-viscoplastic constitutive model for the single crystal behavior. A suite of finite element simulations is performed on a virtual polycrystal to demonstrate the relative effects of the different parameters in the slip system strength model. The model is shown to accurately capture the macroscopic stress-strain response using parameters that are chosen to capture the mesoscopic slip system responses.

Introduction

The adoption of an integrated computational materials engineering (ICME) approach for the design of critical polycrystalline structural components requires the development of rigorously validated micromechanical models. These models must accurately predict elasto-plastic response in-operando across multiple length scales [1], [2]. An important length scale for understanding and characterizing critical phenomena, such as incipient damage nucleation and growth, is that which comprises a local neighborhood of grains. Heterogeneities in the local intergranular stresses and strains arise even under simple macroscopic loading conditions due to both the anisotropic elastic and plastic properties of the constituent grains (particularly for hexagonal metals) as well as complex interactions among them. These heterogeneities can vary significantly from the mean fields and give rise to ‘hot spots’ that have a higher probability of nucleating and propagating voids or cracks [3], [4]. Therefore, a mesoscopic characterization of both the individual crystal responses as well as the aggregate interactions is critical to the formulation and validation of accurate micromechanical models. The measured volume at a given macroscopic load must therefore span an aggregate large enough to sample the full range of grain loading states. Far-field high-energy X-ray diffraction microscopy (ff-HEDM) provides such a mesoscopic characterization of evolving microstructure and micromechanical state for polycrystalline specimens subject to thermo-mechanical processing in-situ [5]. We demonstrate this by using lattice strain data measured by ff-HEDM to directly develop and calibrate a single crystal constitutive model capable of also capturing the aggregate response; specifically a per-mechanism slip system strength model for Ti-7Al.

Using the ff-HEDM technique at third generation synchrotron sources, the stress states in aggregates of up to ≈1000 grains have been studied during in-situ deformation. These data, taken as an ensemble, are a boon to the advancement of predictive models for the mechanical response of polycrystalline materials [6], [7]. Relevant to this work, the technique has been applied to the measurement of stresses in individual grains of titanium alloys during in-situ deformation [8], [9], [10], [11]. In practice however, extracting relevant quantities from these large, high-dimensional data sets present an open challenge; new methods must be developed to distill down these massive quantities of data so that they can be used to enhance our physical understanding of deformation processes, and in turn inform micromechanical models. Towards this goal, we focus on a method to extract the per-family evolution of slip system strengths in Ti-7Al using ff-HEDM and use these data to formulate and calibrate a micromechanical model that accurately captures the meso- and macroscale flow responses.

Crystal plasticity formulations are based on the principle of restricted slip. A family of slip systems is comprised of all symmetrically equivalent combinations of slip plane and direction that can accommodate shear deformation. Strength, in rate-dependent viscoplastic formulations, is a state based quantity relating the resolved shear stress to shear rate. Hardening is represented with the evolution equations for the strength quantities. Accurate values for initial slip system strength are necessary for proper representation of the development of incipient plastic flow. Notably, the onset of yield is a critical portion of the mechanical response during both fatigue and fracture processes in ductile crystalline materials. Modeling the evolution of slip system strengths also requires accurate modeling parameters. However, while slip system strength and its evolution are important components of nearly all microscale plasticity models, most modeling efforts settle for calibrations that are based on indirect quantities such as flow strength and texture evolution.

The Ti-7Al tested herein exists in the hexagonal close-packed (HCP) α phase at room temperature and is similar in composition to the α phase of several commercial alloys, including Ti-6Al-4V. Ti-7Al serves as a model for materials that exhibit both significant non-uniformities among slip system strengths and evolution of the slip system strengths with increasing strain. Plastic deformation in Ti-7Al is accommodated primarily by slip on the basal $〈a〉$ ($〈11\overline{2}0〉$(0001)) and prismatic $〈a〉$ $\left(〈11\overline{2}0〉\left\{1\overline{1}00\right\}\right)$ slip systems and less frequently, slip is also observed on the pyramidal $〈a〉$ $\left(〈11\overline{2}0〉\left\{1\overline{1}01\right\}\right)$ slip systems [12]. The final deformation mode necessary to close the single crystal yield surface is provided by the glide of $〈c+a〉$ dislocations on $\left\{10\overline{1}1\right\}$ and $\left\{11\overline{2}2\right\}$ families of planes (first order and second order pyramidal slip respectively) [13], [14], [15], rather than by twinning as in pure titanium. With the inclusion of these slip systems, the von Mises criterion is satisfied and all modes of plastic deformation can be accommodated. Previous work has shown that various families of slip systems in titanium exhibit significant variation in initial slip system strengths, producing significant anisotropy of the shape of the single crystal flow surface [12].

In addition to variation of initial slip system strengths, aging of Ti-7Al leads to short-range ordering (SRO) and the formation of nanoscale coherent ${\alpha }_2$ precipitates (Ti₃Al) [16]. The existence of SRO and these ${\alpha }_2$ precipitates is known to influence the mechanical response of titanium alloys with aluminum content greater than 5% [12]. First, the presence of the ${\alpha }_2$ precipitates has been reported to suppress twinning [17]. Also, slip is enhanced on the basal plane [18], promoting intense slip localization and banding [19], [20], [15]. Lastly, the SRO and ${\alpha }_2$ precipitates can serve as dislocation obstacles that raise the strength of the material [12], but as slip occurs these obstacles are sheared by dislocation motion and lose their effectiveness as slip barriers [19], effectively softening the material with increasing macroscopic strain [21].

Measuring the initial slip system strengths of a material such as Ti-7Al is a difficult process, and quantifying the hardening or softening behavior of a slip system even more so. If possible, a single crystal is grown and oriented such that uniaxial deformation will be accommodated primarily by a single slip system. The strength is found by calculating the resolved shear stress applied to the assumed active slip system when elastic behavior ends [15]. In these single crystal tests, the evolution of the strength becomes more difficult to quantify because other slip systems activate and begin to interact with one another as the applied load increases, particularly in HCP metals. Furthermore, the single crystal experiments are typically conducted under uniaxial stress loading conditions, whereas grains inside of a deforming polycrystal are subject to a wide array of local deformation conditions that may influence the hardening response. In lieu of calibration to macroscopic quantities, we propose a method that uses the individual stress responses of an ensemble of grains (single crystals) in a deforming aggregate to constrain the model. Each grain probes different portions of the single crystal flow surface; as an ensemble, they can be used to determine the flow surface's shape and evolution.

In this paper, we isolate the effects of individual slip mechanisms by extracting the evolution of slip system strengths from the ensemble data gathered from an aggregate of individually resolved grains via ff-HEDM. Virtual samples are generated using a simple Voronoi tessellation of measured grain centroids and lattice orientations, facilitating direct comparisons between resolved shear stress distributions from both the simulated and experimental results. The slip system strength data are then used to test a single crystal strength evolution model for Ti-7Al as part of an elasto-viscoplastic constitutive model. The effects of different components of the slip system strength model are explored by modulating individual parameters in polycrystal plasticity simulations.