Crystal Plasticity-based Finite Element Research Articles

OXFORD-UMAT: An efficient and versatile crystal plasticity framework

The crystal plasticity-based finite element method is widely used, as it allows complex microstructures to be simulated and allows direct comparison with experiments. This paper presents the OXFORD-UMAT for Abaqus®, a novel crystal plasticity code that is publicly available online for researchers interested in using crystal plasticity. The model is able to simulate a wide range of materials and incorporates two different solvers based on the solution of slip increments and Cauchy stress, with variants of state update procedures including explicit, semi-implicit, and fully-implicit for computational efficiency that can be set by the user. Constitutive laws are available for a range of materials with single or multiple phases for slip, creep, strain hardening, and back stress. The model includes geometrically necessary dislocations that can be computed using finite element interpolation functions by four alternative methods, including the total form with and without a correction for the dislocation flux, a widely used rate form, and a slip-gradient formulation. In addition, the initial strengthening and subsequent softening seen in irradiated materials can also be simulated with the model. The analysis is available in 2D (plane stress and plane strain) and 3D, including linear and quadratic elements. Here we include full derivations of the key equations used in the code and then demonstrate the capability of the code by modeling single-crystal and large-scale polycrystal cases. Comparison of OXFORD-UMAT with other available crystal plasticity codes for Abaqus® reveals the efficiency of the proposed approach, with the backup solver offering greater versatility for handling convergence issues commonly found in practical applications.

Crystal plasticity-based finite element simulations of load reversals and hat-shaped draw-bending for predicting the springback behavior of dual-phase steel sheets

This work is concerned with predicting geometrical shape changes in sheet metal forming using a multi-level simulation framework that considers the directionality of deformation mechanisms acting at the single-crystal level and microstructural evolution. The multi-level model is an elasto-plastic self-consistent (EPSC) homogenization of single-crystal behavior giving the constitutive response at each finite element (FE) material point. Numerical solution of a boundary value problem over geometry is then obtained using continuum finite elements at the macro-level. First, a set of model parameters for the evolution of slip resistance of ferrite and martensite and backstress are established by fitting a comprehensive set of mechanical data for dual-phase (DP) steels 590, 780, and 1180 using one-element model. Next, the potential of the FE-EPSC modeling framework is illustrated by carrying out a set of hat-shaped draw-bending simulations of the steel sheets. The evolution of geometry after hat-shaped draw-bending and springback is predicted and verified with experimental measurements for as-received DP 780. In doing so, the role of accounting for backstress is revealed as critical for the accurate prediction of the part geometry. The same process simulation involving a pre-strained sheet of DP 780 is compared with a corresponding experiment to reveal the role of strain hardening and residual stress on the subsequent part shape changes after the hat-shaped draw-bending test and springback. Finally, the same process simulations involving DP 590 and DP 1180 are performed to confirm the effect of strength on the geometrical shape changes of the sheets after springback.

Restraining geometrically-necessary dislocations to the active slip systems in a crystal plasticity-based finite element framework

Strain gradients have been cast in the form of geometrically-necessary dislocations (GND) to relate the length-scale dependence of strength and to determine potential sites for failure initiation. The literature contains various different incompatibility measures, the main ones being: the total form (∇×Fp), the rate form for large displacements (∇×γ̇anaFp), and the slip gradient form (∇γ̇a). Here, these different approaches are compared rigorously for the first time. Obtaining GND densities when using the total form is a rank-deficit linear problem, solved by singular value decomposition (SVD) known as the Least Squares Minimization (L2 method). Alternative methods for finding GND densities such as Karush–Kuhn–Tucker (KKT) optimization are also investigated. Both L2 and KKT methods predict unrealistic GND densities on inactive slip systems leading to excessive strain hardening; even for a single crystal single slip case. Therefore, the restriction of GNDs to the active slip systems by using a threshold based on the total slip is proposed. This restriction reveals relatively consistent results for various single crystal single slip cases including: simple shear, uniaxial tension, and four-point bending. In addition, the small numerical differences in the slip leads to large discrepancies in the flow stress due to error accumulation, even for strain-gradient-free uniaxial tension, hence a threshold for the GND density increment (2×102 m−2) is used in all models to avoid formation of erroneous GND densities. Finally, the proposed method is applied to the evolution of the GND density for a grain inside a polycrystal aggregate that posses a complex stress state. The lowest incompatibility error is obtained by both of the total forms that use the curl of the plastic deformation gradient with the active slip system restriction suggesting them to be the most reliable GND measures.

Evolution of in-plane strain fields onset of a plastically deforming crack in zirconium

Understanding the fracture behaviour of a material is essential for the sustainable design of structural components. The complex nature of microscale mechanical testing techniques makes identifying the fracture behaviour experimentally challenging at a microscopic scale. Validating microscale simulations with experimental studies can be difficult using traditional experimental techniques. High-resolution digital image correlation combined with electron backscatter diffraction describes the total in-plane deformation field at a microstructural scale, which can be compared against finite element models for verification. The present study determines the in-plane strain distribution ahead of a plastically deforming crack during an in-situ experiment. A crystal plasticity-based finite element model is used to simulate the experimental crack tip behaviour, after which quantitative and qualitative comparisons are made. A concentration of normal strains ahead of the crack tip is observed in both experimental and simulated results, defining the fracture process zone. Slip localization behaviour around the crack is also evaluated using Schmid factor analysis. A good correlation between the experimental slip bands and theoretical predictions has been observed. The strain path dependence on slip activation is also identified. Strain localization in deforming zirconium is initially limited with dense and low-intensity slip occurring up to a maximum in-plane shear strain of 0.2. Formation of well-defined equidistant slip bands occurs with further deformation. Equidistant slip bands are observed for prismatic <a> and pyramidal <c+a> slip systems, having an interplane distance of ∼0.60±0.05 µm and ∼0.46±0.04 µm, respectively.

In-situ assessment of microscale crack tip fields in zirconium

High-resolution electron backscatter diffraction (HR-EBSD) can provide an estimate of elastic strains at a microscopic scale. Due to the need for high-quality Kikuchi patterns, HR-EBSD-based elastic studies have primarily been limited to the examination of small or residual strains. Due to this constraint, the scope for the use of HR-EBSD in the presence of large plastic deformation has not been extensively explored. In the present study, we investigate the elastic strain field in front of a plastically deforming crack by conducting an in-situ experiment inside a scanning electron microscope (SEM). Further, the experimental results are compared with crystal plasticity-based finite element (CPFE) simulations. A reasonable degree of qualitative similarity has been observed. Furthermore, the effect of a large strain floor on the relative elastic measurements is examined via simulations. A qualitative agreement has been observed in measuring relative elastic strains with or without a non-trivial strain floor, which rationalizes the effectiveness of relative elastic measurements in the presence of high applied strains.

Origins of improved elongation to fracture in cyclic bending under tension of AA6022-T4 sheets as revealed using crystal plasticity modeling

Plastic instabilities can be suppressed by imposing cyclic bending during tension to increase elongation-to-fracture (ETF) of metallic sheets. This work is a combined experimental and modeling study into the origins of the improved ETF in cyclic bending during tension facilitated by a continuous-bending-under-tension (CBT) test. Load-displacement is measured during the CBT test under an optimized combination of band depth and pull speed process parameters for aluminum alloy (AA) 6022-T4. Monotonic and load reversal response of the alloy is also characterized under strain histories that resemble the history in CBT. Electron-backscattered diffraction and neutron diffraction are employed to characterize the initial microstructure and texture evolution of the alloy after CBT and simple tension (ST). Monotonic, load reversal, and texture data are used to adjust and verify the parameters of an elasto-plastic self-consistent crystal plasticity-based finite element model for simulating CBT. Additionally, a flow stress curve based on a bulge test is used to calibrate the model to capture the large strain hardening behavior of the alloy. The model is then used to simulate the CBT test while predicting the load-displacement and texture evolution data. The model successfully reproduced the succession of spikes and plateaus during CBT validating the prediction of the extrapolated hardening. After the validation, the development of stress and strain is probed through the sheet thickness to determine the underlying profiles as well as along the sheet to elucidate the location and onset of failure axially. The model reveals the development of more strain and dislocation density at the surface then at the center of the sheet and correctly predicts the strain localizations. Comparisons of texture evolution and mechanical fields developed in ST and CBT reveal very similar mechanical states in the sheet. These results in combination with experimental results suggest that the origins of improved ETF during CBT pertain primarily to the incremental deformation structural effects and secondary to the microstructural evolution effects.

A novel multiscale computational methodology for numerical material testing based on finite element polycrystal model

In this paper, a novel computational procedure for numerical material testing method based on a strain rate-independent crystal plasticity model and an optimization scheme is presented. In order to simulate deformation of materials that exhibit mechanical anisotropy with high accuracy, it is necessary to apply material models that include many material parameters. Therefore, if difficult-to-perform material tests such as biaxial tests can be replaced by numerical material tests, the cost and time required for experiments can be significantly reduced. The proposed method was developed using a crystal plasticity-based finite element model, known as finite element polycrystal model (FEPM), and an optimization scheme based on a genetic algorithm (GA) for microscopic parameters. FEPM is a model that performs analysis while determining the activity of all slip systems without using the strain rate-dependent constitutive law that is commonly used and is suitable for numerical material testing because it has less physical ambiguity. In the optimization process, deformation anisotropy is related to the grain orientation expressed by the Euler angle, and yield stress anisotropy is related to the work-hardening coefficients of the slip system. In addition, by generating pseudo-anisotropic crystal orientation by numerical rolling, the need for crystal orientation observation is eliminated. In this paper, after giving a description of the proposed method, experimental verifications on aluminum alloys to prove the effectiveness of the proposed method are presented. Data will be available on reasonable request.

A Comparison of Statistically Equivalent and Realistic Microstructural Representative Volume Elements for Crystal Plasticity Models

Two methods used to construct a microstructural representative volume element (RVE) were evaluated for their accuracy when used in a crystal plasticity-based finite element (CP-FE) model. The RVE-based CP-FE model has been shown to accurately predict the complete tensile stress–strain response of a Ti–6Al–4V alloy manufactured by laser powder bed fusion. Each method utilized a different image-based technique to create a three-dimensional (3D) RVE from electron backscatter diffraction (EBSD) images. The first method, referred to as the realistic RVE (R-RVE), reconstructed a physical 3D microstructure of the alloy from a series of parallel EBSD images obtained using serial-sectioning (or slicing). The second method captures key information from three orthogonal EBSD images to create a statistically equivalent microstructural RVE (SERVE). Based on the R-RVEs and SERVEs, the CP-FE model was then used to predict the complete tensile stress–strain response of the alloy, including the post-necking damage progression. The accuracy of the predicted stress–strain responses using the R-RVEs and SERVEs was assessed, including the effects of each microstructure descriptor. The results show that the R-RVE and the SERVE offer comparable accuracy for the CP-FE purposes of this study.

A deep learning method for predicting microvoid growth in heterogeneous polycrystals

In heterogeneous polycrystals, microvoid growth presents inherent randomness and dispersion, which generally follows a statistical law. This statistical characteristic intrinsically arises from randomly distributed grain-orientations around microvoids. It remains a huge challenge to explicitly depict the inherent correlation between microvoid growth and grain-orientation distribution by conventional deterministic damage models. In recent years, deep learning has gained in popularity in materials science and has been demonstrated to exhibit excellent data-mining abilities. To our best knowledge, deep learning has not been applied to investigate the statistical damage-evolution issues hitherto. In this work, a novel microvoid growth model based on deep neural network is creatively designed, incorporating both convolutional and long short-term memory components. The former extracts the spatial grain-orientation information, and the latter captures the causal effect of strain history on the microvoid growth. Moreover, to train and test the deep learning-based model, a microvoid-growth database is generated through a large number of crystal plasticity-based finite element simulations, incorporating randomly-oriented grains and different void locations. All the sample data (i.e., the grain-orientation distributions, microvoid locations and microvoid-growth curves) are processed by specific methods (e.g., the pixel-based method) to be amenable for the training process. Our results show that this novel model well captures the statistical characteristic of the microvoid growth in heterogeneous polycrystals. It is expected that the deep learning-based method can provide a new way to predict the microvoid growth at the grain-level.

Tuning lamellar structure distribution around elliptical notch in Ti-6Al-4V for fatigue performance

The paper is aimed to investigate the effect of lamellar structure distribution around elliptical notch in bimodal titanium alloys of Ti-6Al-4V. In this regard, within the framework of crystal plasticity based finite element (CPFE), Representative volume element (RVE) model with elliptical notch for bimodal Ti–6Al–4V is developed, to address the effect of lamellar structure on fatigue resistance performance under external fatigue loading. Lamellar structure distribution around elliptical notches with different aspect ratios are investigated. Strain along radial and circular paths are extracted to identify the possible crack initiation points. The numerical results show that, Strain distribution around notches can be adjusted through lamellar structure distribution such that strain uniformity is achieved. Increase of strain oscillation caused by lamellar structure leads to the improvement of fatigue resistance. With the increase of notch aspect ratio, elliptical notch eventually evolves into line crack and tuning performance of lamellar structure appreciably declines. This numerical study provides a new insight into lamellar structure tuning in dual-phase titanium alloys.

Numerical simulations of mixed-mode (I and II) notch tip fields in basal-textured magnesium alloys

Crystal plasticity based finite element (CPFE) simulations employing Taylor homogenization for a basal-textured Mg alloy are performed under plane strain, small scale yielding (SSY) conditions to study mixed-mode (I and II) notch tip fields. Two notch orientations are considered. In one case, the notch line and normal to the flat surfaces of the notch are taken parallel to the transverse direction (TD) and rolling direction (RD), whereas in the other, they are aligned with normal direction (ND) and TD. The results are compared against those obtained from an isotropic (von Mises) plasticity model with identical stress–strain behavior under RD tension. The maximum plastic zone size at a given level of effective stress intensity factor |K|, enhances with mode II component (i.e., with decrease in mode mixity parameter Me), and is larger for ND–TD orientation. Further, {101̄2} or tensile twinning (TT) occurs near the stretched part of the notch surface, and is more profuse as Me reduces especially for the ND–TD case. It is found that for a given Me, the notch orientation can profoundly affect the distribution of equivalent plastic strain and hydrostatic stress around the notch tip, which underscores the importance of plastic anisotropy of Mg alloys. Finally, by employing appropriate fracture criteria and the computed near-tip fields, the variations of fracture toughness and operative fracture mechanism with Me are predicted for the TD–RD case, which corroborate well with a recent experimental study.

Crystal plasticity-based finite element modeling of twin transmission across grain boundaries in magnesium

For polycrystalline materials with the deformation mechanism of twinning, the interaction between the twins and grain boundaries are substantial to their deformation behavior. In the current work, the twin transmission across the grain boundaries in a magnesium bicrystal is investigated with the aid of crystal plasticity finite element model. The influencing factors, such as the crystallographic orientation of grains, GB misorientation, critical resolved shear stress (CRSS), dislocation slip, and twin-induced stress relaxation are considered. The commonly used geometric factors of macroscopic Schmid factor (MSF), geometric compatibility factor (GCF), and the lately proposed composite Schmid factor (CSF) are assessed by comparing the simulations and available experiments. The CSF can interpret the twin transmission better than either the MSF or the GCF. Moreover, the available twin transmission ability characterized in terms of the fraction of twin transmitted GBs is well captured by the CSF.

A crystal plasticity based finite element framework for RVE calculations of two-phase materials: Void nucleation in dual-phase steels

A crystal plasticity based finite element (CPFE) framework is developed for performing representative volume element (RVE) calculations on two-phase materials. The present paper investigates the mechanical response and the evolution of microstructure of dual-phase (DP) steels under uniaxial tensile loading, with a special focus on void nucleation. The spatial distribution and morphology of the ferrite and martensite grains in DP steels are explicitly accounted for by generating three-dimensional RVEs with Voronoi tessellations. The effects of microstructural parameters—the volume fraction, morphology and spatial distribution of the martensite phase, and the grain size and orientation distribution of the ferrite phase—on the initiation and spread of localized plastic deformation (leading to void nucleation) are investigated in detail. The soft ferrite phase is modelled by a local crystal plasticity theory with anisotropic elasticity, and the hard martensite phase by the J2 flow theory with isotropic elasticity. The developed CPFE framework successfully predicts both the overall and the local mechanical response, and is perfectly capable of distinguishing between different void nucleation mechanisms experimentally observed for DP steels.

Modeling Ti–6Al–4V using crystal plasticity, calibrated with multi-scale experiments, to understand the effect of the orientation and morphology of the α and β phases on time dependent cyclic loading

Classically, crystal plasticity modeling has used a range of constitutive equations, in which the incorporation of additional physics-based relationships typically results in additional model parameters. These additional parameters need to be reliably calibrated, which often necessitates the use of a range of experimental data acquired at multiple length scales. In this work, a crystal plasticity based finite element (CPFE) model for a dual-phase Titanium alloy, Ti–6Al–4V, is developed. The α and β phases of the microstructure are explicitly modeled. The model is calibrated using a systematic optimization routine and experimental data that consist of macroscopic stress-strain curves coupled with lattice strains on different crystallographic planes for the two phases. These experimental data were obtained from in situ high energy X-ray diffraction experiments for multiple material pedigrees, with varying crystallographic orientation distribution and β volume fractions. Depending on the thermomechanical-processing route and the heat treatment used to manufacture the alloy, Ti–6Al–4V can exist in a wide number of microstructural forms, which often results in the α and β phases either having well aligned slip systems (following the Burgers orientation relationship (BOR)) or possessing no alignment of the slip systems across the interphase boundary (not following the BOR). In this study, the fully-calibrated CPFE model is used to gain a comprehensive understanding of the deformation behavior of Ti–6Al–4V, specifically, the effect of microstructures that follow the BOR (or not) on time-dependent cyclic loading (including the effects of dwell hold times).

Crystal plasticity based study to understand the interaction of hydrogen, defects and loading in austenitic stainless-steel single crystals

A crystal plasticity-based finite element study is performed to understand hydrogen effects on void growth in single crystals of austenitic stainless steel. The model assumes plastic deformation is driven primarily by dislocation motion and captures the influence of hydrogen. Hydrogen effects are incorporated by assuming agreement with the hydrogen enhanced localised plasticity (HELP) mechanism. Despite experimental evidence, hydrogen effect on face centred cubic (FCC) crystals has hitherto not been considered in a numerical void growth model for a wide range of stress states. For the first time, the influence of hydrogen on void growth for different Lode parameters at single crystalline levels is investigated for a range of stress triaxialities in FCC crystals. Hydrogen was found to increase equivalent stresses and hardening responses for various stress triaxialities and Lode parameters. Hydrogen also induces higher void growth response at different stress states, and this was more pronounced at high stress triaxialities.

Estimation of Errors in Stress Distributions Computed in Finite Element Simulations of Polycrystals

The accuracy of the stresses predicted from crystal plasticity-based finite element formulation depends on estimation and control of the errors associated with the discretization. In the current work, the errors in the stress distribution are estimated in virtual polycrystalline samples of α-phase titanium (hexagonal close-packed phase of Ti–6Al–4V). To estimate the error, the stress field, which does not possess inter-element continuity, is smoothed over a grain using an $$L_2$$ projection, thereby providing continuous stress distributions with inter-element continuity. The differences between the continuous (smooth) and discontinuous (raw) stress fields are calculated at individual Gauss quadrature points and used to estimate errors for corresponding elements and grains. Error estimations are performed for a Voronoi-tessellated microstructure, an equiaxed microstructure, and two microstructures with varying grain sizes for tensile loading extending into the fully plastic regime ($$\approx $$ 5% extension). Magnitudes of the errors are found to depend on microstructural characteristics, particularly the shape and size of grains. Samples having variations in grain size or having less spherical grains exhibited higher errors than samples with uniformly sized, equiaxed grains, with the size variations having a more pronounced effect. Errors correlate with proximity to grain boundaries at small (elastic) strains and with deformation-induced features (deformation bands) at large strains.

Analysis of size dependent earing evolution in micro deep drawing of TWIP steel by using crystal plasticity modeling

As a promising microforming process, micro deep drawing has attracted great attention for manufacturing of microparts with unique cup-features. However, the occurrence of size effect (SE) and the intrinsic characteristics of crystallographic orientation can lead to an inevitable and uncertain plastic anisotropy during the micro deep drawing and further result in the formation of earing. In this research, a size dependent polycrystalline hardening model using crystal plasticity (CP) theory considering the interactions between slip and twinning was developed via incorporating the size scaling factor into the surface layer model to represent and model the SE and further analyze the entire micro deep drawing process of the TWIP steel. Then the micro deep drawing of polycrystalline TWIP steel with different grain sizes and thicknesses was conducted using the elaborately designed tooling sets. The size dependent crystal plasticity-based finite element modeling in conjunction with a virtual polycrystalline microstructure was employed to study the earing evolution induced by grain size, orientation and geometrical size. It is revealed that an obvious earing profile occurred in the workpiece with the larger size scaling factor. In addition, it is noted that the earing at micro scale is obviously higher than that at macro scale. The asymmetric distribution of earing profile is induced by the Goss and Brass orientations, while the Cubic type orientation leads to the symmetric distribution of earing profile. This issue is mainly attributed to the symmetric distribution of slip activities than that of twinning activities, since the activated quantity of twinning volume fraction is much smaller. Therefore, the proposed size dependent polycrystalline hardening modeling is efficient in predicting the earing formation at micro scale. The study thus provides an in-depth understanding of micro deep drawing of sheet metals and accurate prediction of shape microforming of microparts.

Parametric Study of Cyclic Plasticity Behavior in a Directionally Solidified Superalloy with Partial Recrystallization by Crystal Plasticity Finite Element Simulation

To understand the influence of recrystallization on the low-cycle fatigue behavior of directionally solidified nickel-based DZ4 superalloy, crystal plasticity-based finite element (CPFE) analysis was carried out to study the cyclic plastic behaviors. Using the design of experiments (DOE) technique, a series of CPFE simulations were designed and performed for DZ4 superalloy with recrystallized grains to examine the cyclic plastic shear deformation at the carbide/matrix interface, at the recrystallized grain boundary/matrix, and in the matrix. Seven influence factors, i.e., applied strain, strain ratio, carbide modulus, dwell time, misorientation angle, carbide aspect ratio, and carbide size, were considered, and the effects of which were quantitatively evaluated by the maximum accumulated plastic shear strains via CPFE-based DOE analysis. The results showed that carbide and recrystallized grain boundary had competitively significant effects on fatigue crack initiation. Applied strain was the most influential parameter among all the studied parameters.

Crystal plasticity analysis of deformation anisotropy of lamellar TiAl alloy: 3D microstructure-based modelling and in-situ micro-compression

Detailed microstructure characterisation and in-situ micropillar compression were coupled with crystal plasticity-based finite element modelling (CP-FEM) to study the micro-mechanisms of plastic anisotropy in lamellar TiAl alloys. The consideration of microstructure in both simulation and in-situ experiments enables in-depth understanding of micro-mechanisms responsible for the highly anisotropic deformation response of TiAl on the intra-lamella and inter-lamella scales. This study focuses on two specific configurations of γ/α2 lamellar microstructure with the γ/α2 interfaces being aligned 25o and 55o to the loading direction. Microstructure-based CP-FEM shows that longituginal slip of super and ordinary dislocations are most responsible for the plastic anisotropy in the 25o micropillar while the anisotropy of the 55o micropillar is due to longitudinal superdislocations and longitudinal twins. In addition, transversal superdislocations were more active, making the deformation in the 25o micropillar less localised than that in the 55o micropillar. Moreover, the CP-FEM model successfully predicted substantial build-up of internal stresses at γ/α2 interfaces, which is believed to be detrimental to the ductility in TiAl. However, as evidenced by the model, the detrimental internal stresses can be significantly relieved by the activation of transverse deformation twinning, suggesting that the ductility of TiAl can be improved by promoting transverse twins.

Numerical Investigation of the Origin of Anomalous Tensile Twinning in Magnesium Alloys

It has been observed that tension twins (TTs) are triggered in rolled polycrystalline magnesium alloys under tensile loading applied along the rolling direction (RD) or the transverse direction. This is surprising because these alloys have a near-basal texture, and TTs would therefore cause extension (instead of contraction) along the normal direction. In this work, the origin of these anomalous TTs is first examined by performing crystal plasticity-based finite element simulations using model textures, wherein the c-axis in one grain is systematically tilted toward the loading direction (RD), with the other grains maintained in ideal basal orientation. It is shown that strong basal slip is triggered in the former, which through its effect on the local stress distribution plays a catalytic role in activating TTs. The above behavior is also observed in a simulation performed with an actual texture pertaining to a rolled AZ31 Mg alloy. Most importantly, when basal slip is suppressed, evolution of TTs is found to be very much retarded. The present results corroborate well with experimental observations.