Strain Gradient Crystal Plasticity Framework Research Articles

Study of grain boundary orientation gradients through combined experiments and strain gradient crystal plasticity modeling

A combined experimental and strain gradient crystal plasticity framework is presented for studying the development of orientation and misorientation gradients at the grain boundaries during plastic deformation. The regions in the vicinity of prior-deformation grain boundaries experience localized plastic deformation and are referred to as Near Boundary Gradient Zones (NBGZs). Deformed microstructures of an aluminum–magnesium (Al–6wt.% Mg) alloy are studied using Electron Back Scattered Diffraction (EBSD) during progressive tensile deformation. Orientation and misorientation gradients at the prior-deformation grain boundaries are quantified on grain pairs with different combinations of crystallographic orientations. Realistic virtual microstructures are then simulated using a strain gradient crystal plasticity finite element framework to study the evolution of local misorientations, stresses and strains and their correlation with the underlying substructure. The simulated microstructures indicate that the average profiles of the Geometrically Necessary Dislocation (GND) density correlates with the local misorientation, while the average profiles of the Statistically Stored Dislocation (SSD) density correlates with the effective plastic strain in these NBGZs. The combined experimental and modeling analysis reveals that the width (mean with deviation) of the distribution of the length of NBGZs, normalized by the grain size, scales with the initial grain-average Schmid factor. Further, our results also suggest a possibly strong correlation between the length of the NBGZ and the grain size.

Misorientation and grain boundary orientation dependent grain boundary response in polycrystalline plasticity

This paper studies the evolution of intergranular localization and stress concentration in three dimensional micron sized specimens through the Gurtin grain boundary model (J Mech Phys Solids 56:640–662, 2008) incorporated into a three dimensional higher-order strain gradient crystal plasticity framework (Yalcinkaya et al. in Int J Solids Struct 49:2625–2636, 2012). The study addresses continuum scale dislocation-grain boundary interactions where the effect of crystal orientation mismatch and grain boundary orientation are taken into account through the grain boundary model in polycrystalline metallic specimens. Due to the higher-order nature of the model, a mixed finite element formulation is used to discretize the problem in which both displacements and plastic slips are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated and implemented together with the bulk plasticity model. The capabilities of the framework is demonstrated through 3D polycrystalline examples considering grain boundary conditions, grain boundary strength, the orientation distribution and the specimen size. A detailed grain boundary condition and stress concentration analysis is presented. The advantages and the disadvantages of the model is discussed in detail through numerical examples.

Inter-granular cracking through strain gradient crystal plasticity and cohesive zone modeling approaches

Even though intergranular fracture is generally regarded as a macroscopically brittle mechanism, there are various cases where the fracture occurs at the grain boundaries with considerable plastic deformation at the macroscopic scale. There exists several microstructural reasons for grain boundaries to host crack initiation. They can interact with impurities and defects, can provide preferential location for precipitation, can behave as a source of dislocations and can impede the movement of dislocations as well. The understanding of the crack initiation and propagation at the grain boundaries requires the analysis of the grain boundary orientation and the orientation mismatch between the neighboring grains and the related the stress concentration, which is only possible through the combination of micro-mechanical plasticity and fracture mechanics. For this reason the current work studies the evolution of plasticity in three dimensional Voronoi based microstructures through a strain gradient crystal plasticity framework (see e.g. Yalcinkaya et al., 2011; Yalçinkaya et al., 2012; Yalçinkaya, 2016) and incorporates a potential based cohesive zone model (see Park et al., 2009; Cerrone et al., 2014) at the grain boundaries for the crack initiation and propagation. The numerical examples considers the effect of the orientation distribution, the grain boundary conditions, the specimen size and the fracture energy parameter on the intergranular fracture behavior of micron-sized specimens. The study presents important conclusions for the modeling of fracture at this length scale.

An investigation of back stress formulations under cyclic loading

A single material length parameter governs both the material size dependence and the predicted micro-structural behavior in most existing micro-structurally based continuum theories. As a consequence, smoothly varying field quantities are predicted which contrast recent years' experimental observations. In a previous work by the authors, addressing this matter, two new back stress formulations were proposed, and demonstrated to offer novel modeling capabilities in the localization behavior of geometrically necessary dislocation pile-up under monotonic loading. However, the cyclic behavior of these formulations remains to be investigated. The present work studies the new back stress formulations, within a non-work conjugate type higher order strain gradient crystal plasticity framework, and demonstrates their performance through the idealized single slip simple shear case. At high values of the material length scale parameter, a seemingly anomalous cyclic response is observed when deviating from the conventional type back stress formulation. Similar observations have recently been reported in other numerical studies, and the present work offers a discussion of the physical justification of such material behavior. It is found that the properties of the adopted formulations, in fact, open the possibility for modeling complex material behavior, tied to the presence of long range internal stresses due to dislocation pile-up. Moreover, the present study extends the discussion on micro-structure predictions as a consequence of the adopted back stress models.

Micromechanical Modeling of Inter-Granular Localization, Damage and Fracture

The recent developments in the production of miniaturized devices increases the demand on micro-components where the thickness ranges from tens to hundreds of microns. Various challenges, such as size effect and stress concentrations at the grain boundaries, arise due to the deformation heterogeneity observed at grain scale. Various metallic alloys, e.g. aluminum, exhibit substantial localization and stress concentration at the grain boundaries. In this regard, inter-granular damage evolution, crack initiation and propagation becomes an important failure mechanism at this length scale. Crystal plasticity approach captures intrinsically the heterogeneity developing due to grain orientation mismatch. However, the commonly used local versions do not possess a specific GB model and leads to jumps at the boundaries. Therefore, a more physical treatment of grain boundaries is needed. For this purpose, in this work, the Gurtin GB model (Gurtin (2008)) is incorporated into a strain gradient crystal plasticity framework (Yalcinkaya et al. (2011), Yalçinkaya et al. (2012), Yalçinkaya (2017)), where the intensity of the localization and stress concentration could be modelled considering the effect of grain boundary orientation, the mismatch and the strength of the GB. A zero thickness 12-node interface element for the integration of the grain boundary contribution and a 10-node coupled finite element for the bulk response are developed and implemented in Abaqus software as user element subroutines. 3D grain microstructure is created through Voronoi tessellation and the interface elements are automatically inserted between grains. After obtaining the localization, the mechanical behavior of the GB is modelled through incorporation of a potential based cohesive zone model (see Park et al. (2009), Cerrone et al. (2014)). The numerical examples present the performance of the developed tool for the intrinsic localization, crack initiation and propagation in micron-sized specimens.

Micromechanical modeling of intrinsic and specimen size effects in microforming

Size effect is a crucial phenomenon in the microforming processes of metallic alloys involving only limited amount of grains. At this scale intrinsic size effect arises due to the size of the grains and the specimen/statistical size effect occurs due to the number of grains where the properties of individual grains become decisive on the mechanical behavior of the material. This paper deals with the micromechanical modeling of the size dependent plastic response of polycrystalline metallic materials at micron scale through a strain gradient crystal plasticity framework. The model is implemented into a Finite Element software as a coupled implicit user element subroutine where the plastic slip and displacement fields are taken as global variables. Uniaxial tensile tests are conducted for microstructures having different number of grains with random orientations in plane strain setting. The influence of the grain size and number on both local and macroscopic behavior of the material is investigated. The attention is focussed on the effect of the grain boundary conditions, deformation rate and the grain size on the mechanical behavior of micron sized specimens. The model is intrinsically capable of capturing both experimentally observed phenomena thanks to the incorporated internal length scale and the crystallographic orientation definition of each grain.

Micromechanical Modelling of Size Effects in Microforming

Abstract This paper deals with the micromechanical modelling of the size dependent mechanical response of polycrystalline metallic materials at micron scale through a strain gradient crystal plasticity framework. The model is implemented into a Finite Element software as a coupled implicit user element subroutine where the plastic slip and displacement fields are taken as global variables. Uniaxial tensile tests are conducted for microstructures having different number of grains with random orientations in plane strain setting. The influence of the grain size and number on both local and macroscopic behavior of the material is investigated. The model is capable of capturing both size effect due to statistical distribution of the grains and their size taking into account the grain boundary conditions.

Modeling of dislocation–grain boundary interactions in a strain gradient crystal plasticity framework

This paper focuses on the continuum scale modeling of dislocation---grain boundary interactions and enriches a particular strain gradient crystal plasticity formulation (convex counter-part of Yalcinkaya et al., J Mech Phys Solids 59:1---17, 2011; Int J Solids Struct 49:2625---2636, 2012) by incorporating explicitly the effect of grain boundaries on the plastic slip evolution. Within the framework of continuum thermodynamics, a consistent extension of the model is presented and a potential type non-dissipative grain boundary description in terms of grain boundary Burgers tensor (see e.g. Gurtin, J Mech Phys Solids 56:640---662, 2008) is proposed. A fully coupled finite element solution algorithm is built-up in which both the displacement $${{\varvec{u}}}$$ u and plastic slips $$\gamma ^{\alpha }$$ ? ? are considered as primary variables. For the treatment of grain boundaries within the solution algorithm, an interface element is formulated. The proposed formulation is capable of capturing the effect of misorientation of neighboring grains and the orientation of the grain boundaries on slip evolution in a natural way, as demonstrated by bicrystal specimen examples.

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).

Multi-Scale Modeling of Microstructure Evolution Induced Anisotropy in Metals

This paper presents two crystal plasticity based computational constitutive models for the intrinsic formation of plastic microstructure during monotonic loading and its altered evolution under strain path changes in metal forming operations. The formation step is modeled via a non-convex strain gradient crystal plasticity framework which could simulate the intrinsic development of the plastic microstructures. The evolution under strain path changes is modeled via phenomenologically based constitutive equations incorporated into crystal plasticity framework. The latter is capable of simulating the transient anisotropy effects (e.g. cross hardening, Bauschinger effect) depending on the change in the strain path. The paper discusses the unification of such models for the continuous modeling of microstructure formation and evolution processes.

Intragranular deformation patterning through non‐convex rate dependent strain gradient crystal plasticity

AbstractA rate dependent strain gradient crystal plasticity framework is presented where the displacement and the plastic slip fields are considered as primary variables. These coupled fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equation, which is derived in a thermodynamically consistent manner. The formulation is based on the 1D theory presented in Yalcinkaya et al. (2011), where the patterning of plastic slip is obtained in a system with non‐convex energetic hardening through a phenomenological double‐well plastic potential. In the current multi‐dimensional multi‐slip analysis the non‐convexity enters the framework through a latent hardening potential presented in Ortiz and Repettto (1999) where the microstructure evolution is obtained explicitly via a lamination procedure. The current study aims the implicit evolution of deformation patterns due to the incorporated physically based non‐convex potential. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Non-convex rate dependent strain gradient crystal plasticity and deformation patterning

A rate dependent strain gradient crystal plasticity framework is presented where the displacement and the plastic slip fields are considered as primary variables. These coupled fields are determined on a global level by solving simultaneously the linear momentum balance and the slip evolution equation, which is derived in a thermodynamically consistent manner. The formulation is based on the 1D theory presented in Yalcinkaya et al. (2011), where the patterning of plastic slip is obtained in a system with non-convex energetic hardening through a phenomenological double-well plastic potential. In the current multi-dimensional multi-slip analysis the non-convexity enters the framework through a latent hardening potential presented in Ortiz and Repettto (1999) where the microstructure evolution is obtained explicitly via a lamination procedure. The current study aims the implicit evolution of deformation patterns due to the incorporated physically based non-convex potential.

Probing mesoscopic lattice misorientation by strain gradient crystal plasticity modelling and micro-beam Laue diffraction experiments

A lengthscale and rate-dependent strain gradient crystal plasticity framework was employed to simulate the inter- and intra-granular deformation at the meso- to microscopic level in a large-grained polycrystalline Ni foil. The sample was characterised by micro-beam Laue diffraction experiments. Post-processing of the model was carried out both to forward predict the Laue diffraction patterns on the detector and to perform inverse analysis of the local lattice misorientation in the crystal. The anisotropic broadening of the Laue spots ('streaking') was correctly captured by the simulation and further analysed to reveal the local lattice arrangement.

Dislocation-based plasticity model and micro-beam Laue diffraction analysis of polycrystalline Ni foil: A forward prediction

A physically-based, rate and length-scale dependent strain gradient crystal plasticity framework was employed to simulate the polycrystalline plastic deformation at the microscopic level in a large-grained, commercially pure Ni sample. The latter was characterised in terms of the grain morphology and orientation (in the bulk) by micro-beam Laue diffraction experiments carried out on beamline B16 at Diamond Light Source. The corresponding finite element model was developed using a grain-based mesh with the specific grain orientation assignment appropriate for the sample considered. Sample stretching to 2% plastic strain was simulated, and a post-processor was developed to extract the information about the local lattice misorientation (curvature), enabling forward-prediction of the Laue diffraction patterns. The ‘streaking’ phenomenon of the Laue spots (anisotropic broadening of two-dimensional (2D) diffraction peaks observed on the 2D detector) was correctly captured by the simulation, as constructed by direct superposition of reflections from different integration points within the diffraction gauge volume. Good agreement was found between the images collected from experiments and simulation patterns at various positions in the sample.