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Crystal Plasticity Framework for Single Crystals

Bridging the gap between continuum and crystal scale is done primarily through the auspices of the modern crystal plasticity mathematical framework. Crystal plasticity for polycrystal modelling and simulation is built on the foundation of single crystal plasticity, where a monocrystal representative volume element that deforms solely by slip is consid- ered. Beginning with this foundation and considering deformation accomodated by the shear glide of a single slip system, the Schmid orientation tensor for single crystal lattice is calculated as:
m = bn (2.1a).
mij = binj (2.1b).
Here, the vectors b and n are the Burger’s vector and the vector normal to the plane of slip. Using the Schmid tensor, the velocity gradiant may be composed from the shear strain rate of of the considered slip system as Lij = mij (2.2).
For the case of the deformation of a single crystal with multiple slip systems s, the total strain rate for the crystal can be described as the summation of shear deformation on each system as: ij = N 1 ms + ms s (2.3).

Approaches to the Modeling of Polycrystals

Extrapolating the behavior of polycrystalline materials from the behavior of their single crystal counterparts is no mean feat, and a variety of approaches have been developed in order to address this issue. These approaches primarily differ in the number and type of simplifying assumptions they make in order to capture specific intergranular phenomena, and how these assumptions dictate the type of numerical methods that might be applied, thereby relating the behavior of grains with a single crystal orientation to the behavior of the entire polycrystal aggregate.
Finite Element Crystal Plasticity (FECP) directly models the stress and strain fields of polycrystals by recreating the polycrystal geometry directly. This method uses finite element meshes of sufficient resolution to allow for greater precision in capturing the ge- ometry of the polycrystal contained in a representative volume element. Computational costs, however, limit the number of grains and the complexity of their geometry. This rep- resents the primary limitation of this method. More recently, a method of using fast Fourier transforms (FFT) has been used to solve for the averaged stress and strain fields within a representative volume element (RVE).
The Self-Consistent (SC) methods utilize micromechanics to approximate the behavior of each grain relative to an averaged polycrystal by treating as an inclusion problem. The  limitations of this method are that grain geometry must be approximated in the course of micromechanical formulation. Therefore, great care must be taken in order to insure physical admissibility of obtained results.
Alone, the crystal plasticity framework is insufficient to describe the behavior of poly- crystalline materials. This was the primary motivation for the development of Self Consis- tent Methods. These models were later adapted to include new loading paths in the work of [15]. The treatment of polycrystals changed with the development of micromechanics, in-troduced by [30] in 1957. Micromechanics utilizes the concept of eigenstrains to construct schemes for the homogenization of mechanical behavior of materials with geometrically complex microstructures. The work of [30] led to the development of the homogenization of elastic polycrystal behavior.
These homogenization techniques were adapted for the simulation of nonlinear me-chanical behavior in the works of [21] and [58], and later, the affine linearization approach was developed by [46] as a series of arbitrarily small linear deformation steps meant to approximate nonlinear behavior. Secant and tangent linearizations were later developed in the works of [10] and [76], respectively.
In 1993, the work of [63] adapted the tangent formulation from [76] for the simulation of fully anisotropic materials, including HCP materials. This formed the basis for their Viscoplastic Self-Consistent (VPSC) method and its corresponding code. In this method, the homogeneous polycrystal medium was not assumed to be isotropic, contrasting with other contemporaneous self consistent codes [77]. This approach was expanded by [67] and again in 2000, in the work of [73], to include a generalized affine linearization scheme. The most contemporary form of the VPSC method and code was finally developed in the works of [66].

Kinematic Overview: Correspondence Method

Originally presented in the 1965 work of Bilby [14], the kinematics of twinning are fre- quently described in terms of a plane of shear, S, a twin or composition plane K1 containing the vector 1, and the conjugate planes and vectors K2 containing 2 for the untwinned lat- tice that moves to K02 containing 20 in the new twin lattice.

Twin Grain Boundary Interactions

The complexity of deformation processes in polycrystal materials is predicated on a number of phenomena. Even when considering only slip, the number of ways in which slip dislocations behave in polycrystals is substantial. Dislocations may initiate and annihilate in response to mechanical and thermal loading, or as a response to interactions between other dislocations. Grain boundaries may migrate with annealing, or facilite sliding of grains past each other. In both cases, dislocations may pile up at grain boundaries resulting in increased hardening refered to as the Hall-Petch mechanism, or may initiate cross-slip, thereby bypassing the grain boundaries. When twinning is considered, the complexity of dislocation interactions and their effects on the mechanical behavior of polycrystal materi- als can be staggering.
Slip dislocations encountering twin grain boundaries have been observed in experi- ments and simulations to behave in a number of ways. Molecular dynamics simulations of FCC materials have shown that dislocations may transmute across twin boundaries or they may dissociate or be incorporated into the boundary itself. In other cases they may pile up in a manner consistent with Hall-Petch behavior, and in other cases they may even be emitted on the opposing side of the twin boundary as multiple slip dislocations [31]. Ex- perimentally, dislocations in FFC copper-aluminum alloys have been observed to crossover into advancing twin volume fractions. In this case, dislocations encountering twin grain boundaries were posited to pile-up at said boundaries. As the twin grain boundary ad- vanced in order to facilitate twin volume growth, these dislocations were overtaken andbecame sessile dislocations inside the twin volume fraction [9] These finding contrast with the observed behavior of TWIP steels, in which dislocations are seen simple to pile up at twin grain boundaries, thus demonstrating the wide spectrum of behavior even in materials with the same crystal structure [38].
Currently, debate exists regarding the nature of the dislocation-twin grain boundary interactions taking place in deforming magnesium. While nano-indentation studies of twinned magnesium showed no evidence of an increased dislocation density inside of twin volume fraction that would be indicative of dislocation transmutation or Basinski mechanism at work, the observed“gulf” between saturation stress levels of magnesium un- der different load paths suggests that more hardening mechanisms than simple mechanical anisotropy are at work [48] This has supported the assumption that the evolution of harden- ing in heavily twinning magnesium is motivated primarily by Hall-Petch effects, with slip dislocations building up at twin grain boundaries [11]. Counter to this assumption, VPSC simulations of rolled magnesium using a twinned storage factor to empirically account for the effects of increased dislocation density inside twin volume fractions were able to suc- cessfully recreate the hardening evolution of the material across multiple load paths [55].
The suggestion that dislocation density inside the twinned volume fraction was further sup- ported by molecular dynamics simulations, in which basal dislocations were observed to both transmute across and dissociate to aid the migration of tensile twin boundaries [8, 53].

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TWinning Induced Plasticity Steels

Discovered in 1888 by Sir Robert Hadfield, TWIP steels are a type of austenetic FFC steel whose general mechanical behavior combines high yield strength with high elonga- tion (ductility) properties relative to other steels. These traits make TWIP steels highly desirable for implementation in vehicle design and extensive efforts to characterize the material behavior of TWIP steels across a broad range of temperatures, strain rates, and across a spectrum of compositions have been made in the past two decades. Ultimate strength at failure has been observed in the range of 650-1000 MPa with an elongation at failure ranging from 60%-92% [32, 5].

Table of contents :

1.1 Role of Computational Methods in the Materials Genome Initiative for Global Competitiveness
1.2 Twinning Polycrystals: TWIP Steel and Magnesium
1.2.1 Twinning Induced Plasticity Steel
1.2.2 Magnesium
1.3 Simulation Methods for Polycrystals: An Overview
1.4 Research Goals and Project Overview
1.4.1 Research Design
2.1 Approaches to Modeling Polycrystals
2.1.1 Crystal Plasticity Framework for Single Crystals
2.1.2 Approaches to the Modeling of Polycrystals
2.2 Twinning Polycrystals
2.2.1 Kinematic Overview: Correspondence Method
2.2.2 Twin Grain Boundary Interactions
2.2.3 TWinning Induced Plasticity Steels
2.2.4 Magnesium
2.3 Modeling Twin Volume Growth
2.4 Constitutive Approaches to Hardening Evolution
3.1 Introduction
3.2 Methods
3.2.1 Experimental Overview
3.2.2 Twin Volume Transfer Constitutive Model
3.2.3 Models for Hardening Evolution
3.2.4 Implementation
3.2.5 Simulation
3.2.6 Calibration
3.3 Results
3.3.1 1st ECAP Pass
3.4 Anaylsis
3.5 Conclusion
4.1 Introduction
4.2 Methods
4.2.1 Model
4.3 Implementation and Calibration
4.3.1 Correspondence Method for Transmutation for the Construction of
4.3.2 Parameters for Dissociation Parameters for Dislocation Generation and Twin Nucleation and Propagation
4.3.3 Simulation
4.4 Simulation Results
4.4.1 TTC Load Path
4.4.2 IPC Load Path
4.4.3 Comparison of Approaches
4.4.4 Sensitivity
4.5 Analysis and Conclusions
5.1 Contributions
5.2 For Further Research


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