•An extended implementation of the Inductive Design Exploration Method for multilevel robust design of hierarchical materials.•The IDEM framework is scripted via Python (pyDEM) to allow for widespread adaption and further alterations/modifications.•Extended for disjoint and nonconvex feasible design spaces that satisfy ranged sets of performance requirements.•Two test cases demonstrate the capability of pyDEM: (i) four-level UHPC panel, and (ii) WEDM process of titanium.The emergence of multiscale design of materials and products has necessitated development of inductive robust design methods to rapidly develop and deploy new material systems. In addition, practical applications require robust designs which ensure performance goals are satisfied while accounting for model, noise, and control factor uncertainties. Recognizing the utility of a robust platform for design exploration, the Python Design Exploration Module (pyDEM) has been developed. The purpose of this work is to present this improved, generalized implementation of the Inductive Design Exploration Method (IDEM) to support integrated multiscale materials, process, and product design. The capabilities of pyDEM are highlighted and demonstrated via two test cases: (i) four-level Ultra High Performance Concrete (UHPC) panel and (ii) wire electric discharge machining (WEDM) process of titanium.Download high-res image (197KB)Download full-size image

•In HCF, plastic strains are orders of magnitude smaller than elastic strains.•Compute elastic strains using an efficient localization linkage.•Compute cyclic plastic strains using a forward Euler scheme.•Compute extreme value distributions (EVDs) of fatigue indicator parameters (FIPs).•Rank-order HCF resistance of microstructures by their FIP EVDs.Traditionally, crystal plasticity finite element method (CPFEM) simulations have been used to capture the variability in the microstructure-scale response of polycrystalline metals. However, these types of simulations are computationally expensive and require significant resources. To explore the large space of microstructures (reflecting a variety of grain shape, size, and orientation distributions) within the practical constraints of computational resources, a more efficient strategy is required. The purpose of this work is to explore the viability of leveraging the recently established, high-throughput Materials Knowledge System (MKS) for fast evaluation of high cycle fatigue (HCF) performance of candidate microstructures. More specifically, we explore the feasibility of estimating the mesoscale strain fields in hexagonal close packed (HCP) α-titanium polycrystals during HCF loading conditions using the computationally low-cost MKS approach, and subsequently estimating the slip system activities via decoupled numerical integration of the relevant crystal plasticity (CP) constitutive relations. The computed slip activities are then used to arrive at extreme value distributions (EVDs) of fatigue indicator parameters (FIPs). As critical validation of this reduced-cost computational strategy, it is shown that the FIP distributions in the HCF regime estimated using this novel strategy are in reasonable agreement with those computed directly using the conventional CPFEM approach. Additionally, the computational advantages of the MKS and decoupled numerical integration approach over the traditional, computationally-expensive, CPFEM approach are presented and discussed.Download high-res image (211KB)Download full-size image

Computationally efficient structure-property (S-P) linkages (i.e., reduced order models) are a necessary key ingredient in accelerating the rate of development and deployment of structural materials. This need represents a major challenge for polycrystalline materials, which exhibit rich heterogeneous microstructure at multiple structure/length scales, and exhibit a wide range of properties. In this study, a novel framework is described for extracting S-P linkages in polycrystalline microstructures that are obtained using 2-point spatial correlations (also called 2-point statistics) to quantify the material's microstructure, and principal component analysis (PCA) to represent this information in a reduced dimensional space. Additionally, it is demonstrated that the use of generalized spherical harmonics (GSH) as a Fourier basis for functions defined on the orientation space leads to a compact and computationally efficient representation of the desired S-P linkages. In this study, these novel protocols are developed and demonstrated for elastic stiffness and yield strength predictions for α− Ti microstructures using a dataset produced through microscale finite element simulations.Download high-res image (206KB)Download full-size image

Slip transfer via sequential pile-up dislocations across grain boundaries (GBs) plays an important role in plastic deformation in polycrystalline face-centered cubic (FCC) metals. In this work, large scale concurrent atomistic-continuum (CAC) method simulations are performed to address the slip transfer of mixed character dislocations across GBs in FCC Ni. Two symmetric tilt GBs, a Σ3{111} coherent twin boundary (CTB) and a Σ11{113} symmetric tilt GB (STGB), are investigated using five different fits to the embedded-atom method (EAM) interatomic potential to assess the variability of predicted dislocation-interface reaction. It is shown that for the Σ3 CTB, two of these potentials predict dislocation transmission while the other three predict dislocation absorption. In contrast, all five fits to the EAM potential predict that dislocations are absorbed by the Σ11 STGB. Simulation results are examined in terms of several slip transfer criteria in the literature, highlighting the complexity of dislocation/GB interactions and the significance of multiscale modeling of the slip transfer process.

In this work we present a molecular dynamics study of a symmetrically strained SiGe superlattice subjected to compression. We measure the velocity of a dislocation nucleated at a step in the superlattice on a slip system that cuts across the interfaces, and compare it with that in single crystal Si and Ge for the same geometry. Simulation results of this work show that the phase interfaces in the short-period SiGe superlattice are free of misfit dislocations and that the average dislocation velocity in the SiGe superlattice sample is about two orders of magnitude lower than that in the Si and Ge single crystals for the same geometry and size. The effect of the coherent phase interfaces on the average dislocation velocity is quantified, and the underlying mechanism is identified. This work demonstrates that the coherency stress plays a dominant role on dislocation migration in coherent SiGe superlattices. It shows that a phase interface can impede or assist the migration of a dislocation depending on whether the sign of the stress field along the interface opposes or reinforces that of the dislocation core.

•Finite element framework developed to model deformation of Mo–Si–B alloys.•Cohesive elements used within intermetallics and at Moss/intermetallic interfaces.•Three-phase microstructures were instantiated using weighted Voronoi tessellation.•Alloys have optimal strength and ductility with Moss vol. fraction in the range 0.5–0.65.A computational framework is developed to study the role of microstructure on the deformation behavior of Mo–Si–B alloys. A parametric range of idealized multi-phase microstructures of Mo–Si–B alloys are instantiated in 2D using Voronoi tessellation schemes and their deformation behavior modeled with the use of the finite element method. Continuum elements are used to model the constituent phases, while cohesive elements are used to model debonding at the interfaces of the intermetallic (A15 and T2) phases with the solid solution-strengthened Moss matrix and cleavage fracture within the intermetallic phases. The deformation behavior of Mo–Si–B alloys is studied in terms of the simulated stress-strain response and microstructure evolution characteristics. Effects of various microstructure parameters, such as composition and clustering of intermetallic phases, on the tensile strength and ductility are also studied.

Formation of slip bands plays an important role in deformation and fatigue processes of duplex Ti–6Al–4V. In this study, shear-enhanced crystal plasticity constitutive relations are proposed to account for the slip softening due to breakdown of the short-range order between titanium and aluminum atoms. A hybrid strategy is developed which allows the softening to occur in slip bands only within the primary α phase, with the degree of localization depending on the specific polycrystalline initial-boundary-value problem and the requirements for compatibility of each grain or phase with its neighbors. The proposed model is calibrated by performing finite-element (FE) simulations on an experimentally studied Ti–6Al–4V alloy. The slip behavior of a Ti–6Al–4V sample subjected to an in situ (scanning electron microscopy (SEM)) tensile test is investigated. A two-dimensional (2-D) FE with 3-D crystal plasticity relations is constructed to represent the microstructure of the Ti–6Al–4V sample. Due to the lack of access to fully 3-D microstructure, a generalized plane-strain condition is used in the FE model which assumes columnar grains that are free of net traction in the direction normal to the surface. The assumption of columnar grains significantly reduces the computational cost. The contours of effective plastic strain are compared with the surface SEM micrographs from experiments at various strain levels. It is shown that the proposed approach for modeling slip bands qualitatively captures experimentally observed slip band behavior.

Atomistic simulations of dislocation nucleation from grain boundaries provide an insight into dislocation sources in nanocrystalline copper. Simulations show that dislocation sources emit single partial dislocation loops, with half absorbed into the boundary and half emitted into the lattice. The specific boundary dislocation content determines whether the absorbed half-loop annihilates pre-existing boundary dislocations or increases boundary dislocations. Atomistic studies of this type provide details of the emission sequence that enhance our understanding of dislocation sources in high angle boundaries.

Atomistic simulations are used to investigate how grain boundary structure influences dislocation nucleation under uniaxial tension and compression for a specific class of symmetric tilt grain boundaries that contain the E structural unit. After obtaining the minimum energy grain boundary structure, molecular dynamics was employed based on an embedded-atom method potential for copper at 10 K. Results show several differences in dislocation nucleation with respect to uniaxial tension and compression. First, the average nucleation stress for all 〈1 1 0〉 symmetric tilt grain boundaries is over three times greater in compression than in tension for both the high strain rate and quasistatic simulations. Second, partial dislocations nucleate from the boundary on the {1 1 1} slip plane under uniaxial tension. However, partial and full dislocations nucleate from the boundary on the {1 0 0} and {1 1 1} slip planes under uniaxial compression. The full dislocation nucleation on the {1 0 0} plane for boundaries with misorientations near the coherent twin boundary is explained through the higher resolved shear stress on the {1 0 0} plane compared to the {1 1 1} plane. Last, individual dislocation nucleation mechanisms under uniaxial tension and compression are analyzed. For the vicinal twin boundary under tension, the grain boundary partial dislocation is emitted into the lattice on the same {1 1 1} plane that it dissociated onto. For compression of the vicinal twin, the 1/3〈1 1 1 〉 disconnection is removed through full dislocation emission on the {1 0 0} plane and partial dislocation emission parallel to the coherent twin boundary plane, restoring the boundary to the coherent twin. For the Σ19Σ19 boundary, the nearly simultaneous emission of numerous partial dislocations from the boundary result in the formation of the hexagonal close-packed (HCP) phase.

Engineering design has historically been taught using the paradigm of selecting materials on the basis of tabulated databases of properties (mechanical, physical, chemical, etc.). Recent trends have moved toward concurrent design of material composition and microstructure together with the component/system level. The goal is to tailor materials to meet specifi ed ranges of performance requirements of the overall system. Often these multiple performance requirements are in confl ict in terms of their demands on composition and microstructure. This paper explores the elements of a decision-based robust design framework for concurrent design of materials and products, focusing on enhancing the fraction of decisions supported by modeling and simulation.

•We create and apply a micromechanics based model to shock loading of aluminum.•We compare simulated wave profiles with experimental data from 2 to 110 GPa.•We quantify effects of polycrystallinity on the observed response.•We propose a coarse-grained model to capture the polycrystal response.•We emphasize need for microstructural characterization in experiments.A thermoelastic–viscoplastic constitutive model has been developed to model high strain rate deformation in single crystal metals. The thermoelastic formulation employs a material Eulerian strain measure, which has recently been shown to converge more rapidly than traditional Lagrangian strain measures for material undergoing large compression. The viscoplastic formulation is based on the physics of dislocation glide and generation as well as their interaction. This model has been implemented into a one-dimensional, extended finite-difference formulation for anisotropic materials and is used to model shock wave propagation in aluminum single crystals, polycrystals, and pre-textured polycrystals for peak shock pressures ranging from 2 to 110 GPa. The model was able to reproduce experimentally measured particle velocity profiles from both plate impact and laser shock experiments performed on single crystals and polycrystals. Simulations showed that the orientation distribution in vapor-deposited polycrystalline samples can affect the observed elastic precursor decay by a factor of two, as well as change the observed response in laser shock experiments from a dual to single-wave shock structure. Simulations performed on cold rolled aluminum samples showed that an increase in the cold rolling reduction caused a decrease in the number of active slip systems, as well as a decrease in the heterogeneity of total accumulated slip. Finally, a coarse-grained analytical model was developed from results of plane wave simulations and was shown to effectively reproduce plastic heterogeneity induced by single crystal orientations. Simulations in this work showed that single crystal effects play a key role in dictating the macroscopically observed response, which suggests high strain rate experiments that omit detailed initial microstructural characterization do not provide sufficient information for complete mechanistic understanding or model validation.

Despite the large amount of research that has been performed to quantify the high strain rate response of Aluminum, few studies have addressed effects of crystal orientation and subsequent crystal-level microstructure evolution on its high strain rate response. To study orientation effects in single crystal Al, both a constitutive model and novel numerical method have been developed. A plane wave formulation is developed so that materials undergoing anisotropic viscoplastic deformation can be modeled in a thermodynamically consistent framework. Then, a recently developed high strain rate viscoplastic model is extended to include single crystal effects by incorporating higher order crystal-based thermoelasticity, anisotropic plasticity kinetics, and distinguishing influences of forest and parallel dislocation densities. Steady propagating shock waves are simulated for [100], [110], and [111] oriented single crystals and compared to existing experimental wave profile and strength measurements. Finally, influences of initial orientation and peak pressure ranging from 0 to 30 GPa are quantified. Results indicate that orientation plays a significant role in dictating the high rate response of both the wave profile and the resultant microstructure evolution of Al. The plane wave formulation can be used to evaluate microstructure-sensitive constitutive relations in a computationally efficient framework.

Research trends in metal plasticity over the past 25 years are briefly reviewed. The myriad of length scales at which phenomena involving microstructure rearrangement during plastic flow is discussed, along with key challenges. Contributions of the author’s group over the past 30 years are summarized in this context, focusing on the statistical nature of microstructure evolution and emergent multiscale behavior associated with metal plasticity, current trends and models for length scale effects, multiscale kinematics, the role of grain boundaries, and the distinction of the roles of concurrent and hierarchical multiscale modeling in the context of materials design.

A rate dependent crystal plasticity model for the α/β Ti–Al alloy Ti–6Al–4V with duplex microstructure is developed and presented herein. Duplex Ti–6Al–4V is a dual-phase alloy consisting of an hcp structured matrix primary α-phase and secondary lamellar α + β domains that are composed of alternating layers of secondary α laths and bcc structured residual β laths. The model accounts for distinct three-dimensional slip geometry for each phase, anisotropic and length scale dependent slip system strengths, the non-planar dislocation core structure of prismatic screw dislocations in the primary α-phase, and crystallographic texture. The model is implemented in the general purpose finite element code (ABAQUS, 2005. Ver 6.5, Hibbitt, Karlsson, and Sorensen, Inc., Pawtucket, RI) via a UMAT subroutine.

A rate dependent, microstructure-sensitive crystal plasticity model is formulated for correlating the mechanical behavior of a polycrystalline Ni-base superalloy IN 100 at 650 °C. This model has the capability to capture first order effects on the stress–strain response due to (a) grain size, (b) γ′ precipitate size distribution, and (c) γ′ precipitate volume fraction. Experimental fatigue data with variable strain rates are used to calibrate the model for several distinct IN 100 microstructures (grain size, precipitate size distributions and volume fractions) obtained from thermomechanical processing. Physically based hardening laws are employed to evolve the dislocation densities for each slip system, taking into consideration the dislocation interaction mechanisms.The calibrated crystal plasticity model is used to inform microstructure dependent parameters of a macroscopic internal state variable (ISV) model, which is computationally feasible for use in component scale notch root analyses. A hierarchical methodology is outlined to embed this microstructure-dependence in the macroscale model.

Atomistic simulations are used to investigate how the stress required for homogeneous nucleation of partial dislocations in single crystal copper under uniaxial loading changes as a function of crystallographic orientation. Molecular dynamics is employed based on an embedded-atom method potential for Cu at 10 and 300 K. Results indicate that non-Schmid parameters are important for describing the calculated dislocation nucleation behavior for single crystal orientations under tension and compression. A continuum relationship is presented that incorporates Schmid and non-Schmid terms to correlate the nucleation stress over all tensile axis orientations within the stereographic triangle. Simulations investigating the temperature dependence of homogeneous dislocation nucleation yield activation volumes of ≈0.5–2b3 and activation energies of ≈0.30eV. For uniaxial compression, full dislocation loop nucleation is observed, in contrast to uniaxial tension. One of the main differences between uniaxial tension and compression is how the applied stress is resolved normal to the slip plane on which dislocations nucleate—in tension, this normal stress is tensile, and in compression, it is compressive. Last, the tension–compression asymmetry is examined as a function of loading axis orientation. Orientations with a high resolved stress normal to the slip plane on which dislocations nucleate have a larger tension–compression asymmetry with respect to dislocation nucleation than those orientations with a low resolved normal stress. The significance of this research is that the resolved stress normal to the slip plane on which dislocations nucleate plays an important role in partial (and full) dislocation loop nucleation in FCC Cu single crystals.

Grain boundary influence on material properties becomes increasingly significant as grain size is reduced to the nanoscale. Nanostructured materials produced by severe plastic deformation techniques often contain a higher percentage of high-angle grain boundaries in a non-equilibrium or energetically metastable state. Differences in the mechanical behavior and observed deformation mechanisms are common due to deviations in grain boundary structure. Fundamental interfacial attributes such as atomic mobility and energy are affected due to a higher non-equilibrium state, which in turn affects deformation response. In this research, atomistic simulations employing a biased Monte Carlo method are used to approximate representative non-equilibrium bicrystalline grain boundaries based on an embedded atom method potential, leveraging the concept of excess free volume. An advantage of this approach is that non-equilibrium boundaries can be instantiated without the need of simulating numerous defect/grain boundary interactions. Differences in grain boundary structure and deformation response are investigated as a function of non-equilibrium state using Molecular Dynamics. A detailed comparison between copper and aluminum bicrystals is provided with regard to boundary strength, observed deformation mechanisms, and stress-assisted free volume evolution during both tensile and shear simulations.

A hierarchical multiscale approach is presented for modeling microstructure evolution in heterogeneous materials. Preservation of momentum across each scale transition is incorporated through the application of the principle of virtual velocities at the fine scale giving rise to the appropriate continuum momentum balance equations at the coarse scale. In addition to satisfying momentum balance and invariance of momentum among scales, invariance of elastic free energy, stored free energy, and dissipation between two scales of observation is regarded as crucial to the physics of each scale transition. The preservation of this energy partitioning scheme is obtained through construction of constitutive relations within the framework of internal state variable theory. Internal state variables that are directly computed from the fine scale response are introduced to augment the state equations and describe the inelastic energy storage and dissipation within the fine scale. By virtue of a second gradient kinematic decomposition, the framework naturally gives rise to couple stresses.

•A continuum damage framework is developed to model failure initiation in irradiated bcc polycrystalline ensembles.•Constitutive equations for vacancy condensation, void nucleation and growth are developed.•The framework is used to model damage at dislocation channel interfaces ahead a sharp notch.•The fracture toughness of irradiated specimens is calculated for various loading histories and notch geometries.•Crack growth resistance of the irradiated specimens is computed and compared to that of the virgin specimens.A continuum damage framework is developed and coupled with an existing crystal plasticity framework, to model failure initiation in irradiated bcc polycrystalline materials at intermediate temperatures. Constitutive equations for vacancy generation due to inelastic deformation, void nucleation due to vacancy condensation, and diffusion-assisted void growth are developed. The framework is used to simulate failure initiation at dislocation channel interfaces and grain boundaries ahead of a sharp notch. Evolution of the microstructure is considered in terms of the evolution of inelastic deformation, vacancy concentration, and void number density and radius. Evolution of the damage, i.e., volume fraction of the voids, is studied as a function of applied deformation. Effects of strain rate and temperature on failure initiation are also studied. The framework is used to compute the fracture toughness of irradiated specimens for various loading histories and notch geometries. Crack growth resistance of the irradiated specimens are computed and compared to that of virgin specimens. Results are compared to available experimental data.

Highlights•Third term is added to the multiplicative decomposition of the deformation gradient.•The third term is consistent with lattice accommodation of dislocations.•Elasticity solutions illustrate importance of this term for single dislocations.•The multiscale nature of the three term decomposition is explored.•Implications for crystal plasticity are considered.In finite inelasticity, the gradient of total deformation is typically split into a product of two terms: a reversible (elastic) term whose strain vanishes upon load removal at some scale, and an irreversible (plastic) component that remains. In this work it is argued that this decomposition represents a limiting case for which defects are absent from the volume of interest, with compatible elastic distortion associated with externally applied stress and plastic deformation associated with history of dislocation glide through the element. An additional (third) term should be incorporated in the multiplicative decomposition when applied to an element of material of any realistic volume, accounting for local lattice distortion due to defects within. In the limiting case that this volume approaches a few lattice spacings, the probability of interior defects tends towards zero, but a very small volume element containing a few defects, or a larger element containing a large density of defects, requires a third term in the multiplicative decomposition to represent contributions of defects to residual lattice distortion. Physical experiments and reported atomic and continuum calculations support these theoretical arguments. The magnitude of distortion from the “missing” third term is estimated analytically using elasticity solutions for straight dislocations. Advances to crystal plasticity theory involving a three-term decomposition are suggested.

Two new formulations of micropolar single crystal plasticity are presented within a geometrically linear setting. The construction of yield criteria and flow rules for generalized continuum theories with higher-order stresses can be done in one of two ways: (i) a single criterion can be introduced in terms of a combined equivalent stress and inelastic rate or (ii) or individual criteria can be specified for each conjugate stress/inelastic kinematic rate pair, a so-called multi-criterion theory. Both single and multi-criterion theories are developed and discussed within the context of dislocation-based constitutive models. Parallels and distinctions are made between the proposed theories and some of the alternative generalized crystal plasticity models that can be found in the literature. Parametric numerical simulations of a constrained thin film subjected to simple shear are conducted via finite element analysis using a simplified 2-D version of the fully 3-D theory to highlight the influence of specific model components on the resulting deformation under both loading and unloading conditions. The deformation behavior is quantified in terms of the average stress–strain response and the local shear strain and geometrically necessary dislocation density distributions. It is demonstrated that micropolar single crystal plasticity can qualitatively capture the same range of behaviors as slip gradient-based models, while offering a simpler numerical implementation and without introducing plastic slip rates as generalized traction-conjugate velocities subject to an additional microforce balance.

Atomistic simulations were used to investigate dislocation nucleation from Σ3 asymmetric (inclined) tilt grain boundaries under uniaxial tension applied perpendicular to the boundary. Molecular dynamics was employed based on embedded atom method potentials for Cu and Al at 10 K and 300 K. Results include the grain boundary structure and energy, along with mechanical properties and mechanisms associated with dislocation nucleation from these Σ3 boundaries. The stress and work required for dislocation nucleation were calculated along with elastic stiffness of the bicrystal configurations, exploring the change in response as a function of inclination angle. Analyses of dislocation nucleation mechanisms for asymmetric Σ3 boundaries in Cu show that dislocation nucleation is preceded by dislocation dissociation from the boundary. Then, dislocations preferentially nucleate in only one crystal on the maximum Schmid factor slip plane(s) for that crystal. However, this crystal is not simply predicted based on either the Schmid or non-Schmid factors. The synthesis of these results provides a better understanding of the dislocation nucleation process in these faceted, dissociated grain boundaries.

A theoretical framework for the hierarchical multiscale modeling of inelastic response of heterogeneous materials is presented. Within this multiscale framework, the second gradient is used as a nonlocal kinematic link between the response of a material point at the coarse scale and the response of a neighborhood of material points at the fine scale. Kinematic consistency between these scales results in specific requirements for constraints on the fluctuation field. The wryness tensor serves as a second-order measure of strain. The nature of the second-order strain induces anti-symmetry in the first-order stress at the coarse scale. The multiscale internal state variable (ISV) constitutive theory is couched in the coarse scale intermediate configuration, from which an important new concept in scale transitions emerges, namely scale invariance of dissipation. Finally, a strategy for developing meaningful kinematic ISVs and the proper free energy functions and evolution kinetics is presented.

Viscoplastic behavior of metallic materials is manifested by collective behavior of lattice defects at multiple length scales. The full gamut of length scales, from atomic to macroscopic, is considered in terms of both concurrent and hierarchical multiscale modeling of plasticity of metals. Particular challenges are identified in the realm between atomistic simulations and the scale of dislocation substructure formation in establishing next generation constitutive models. Bottom–up modeling is useful in applications such as fracture or dislocation mediation by interfaces. This is differentiated from top–down modeling that supports design of a material as a hierarchy of sub-systems. Examples of hierarchical microstructure-sensitive models are presented for both Ni-base superalloys and α–β Ti alloys, with emphasis on cyclic deformation behavior.Future directions to advance both discrete dislocation and crystal plasticity theories are discussed, emphasizing the need for additional focus on dislocation sources and dislocation line curvature in modeling scale dependent behavior. Difficulties with low order approximations of the dislocation density distribution, such as second-order gradient theories, are discussed in terms of predictive capability and transferability among geometric configurations. The multiscale nature of the decomposition of the deformation gradient into elastic and plastic parts is discussed in light of sources of incompatibility at each scale considered, with interpretation offered at different scales of Burgers circuits that highlight dislocation substructures and polycrystals, respectively. Recent atomistic studies of dislocation nucleation at grain boundaries are outlined and some thoughts are offered towards potentially fruitful directions to incorporate this understanding into statistical continuum models that account for the role of grain boundary structure as an element of the evolving incompatibility field.