Strain Gradient Crystal Plasticity Research Articles

Revealing the fatigue strengthening and damage mechanisms of surface-nanolaminated gradient structure

Extending the fatigue life of metals is a critical concern for maintaining material and component integrity in engineering systems. The integration of gradient structures within materials represents a highly promising approach to enhance the fatigue properties in metallic materials, while a detailed mechanistic understanding of the fatigue damage evolution of such structures is yet to be developed. Here, we report that the surface-nanolaminated gradient structure comprised of nanolaminates and hierarchical twins imparts remarkable resistance to both low-cycle and high-cycle fatigue. A dislocation-based strain gradient crystal plasticity model is developed to investigate the strengthening and damage mechanisms of our gradient structure. The size dependence of the initial dislocation density, its evolution and back stress hardening are taken into account and verified by the experimental data. The simulation results reveal that the strain delocalization and back stress hardening induced by the structure gradient significantly mitigate the fatigue damage accumulation. Additionally, in contrast to conventional gradient structures, the mechanical stability of the present structure enables these strengthening mechanisms to persist until crack initiation. These effects, combined with the sequential toughening mechanisms activated in the surface-nanolaminated gradient structure, ensure a marked life extension under low-cycle fatigue (by a factor of four), outperforming conventional gradient and other microstructural design strategies. Finally, a multiscale anti-fatigue design principal for damage homogenization is given based on the prior quantitative analysis.

Advanced modeling of higher-order kinematic hardening in strain gradient crystal plasticity based on discrete dislocation dynamics

An extensive study of size effects on the small-scale behavior of crystalline materials is carried out through discrete dislocation dynamics (DDD) simulations, intended to enrich strain gradient crystal plasticity (SGCP) theories. These simulations include cyclic shearing and tension-compression tests on two-dimensional (2D) constrained crystalline plates, with single- and double-slip systems. The results show significant material strengthening and pronounced kinematic hardening effects. DDD modeling allows for a detailed examination of the physical origin of the strengthening. The stress–strain responses show a two-stage behavior, starting with a micro-plasticity regime with a steep hardening slope leading to strengthening, and followed by a well-established hardening stage. The scaling exponent between the apparent (higher-order) yield stress and the geometrical size h varies depending on the test type. Scaling relationships of h−0.2 and h−0.3 are obtained for respectively constrained shearing and constrained tension-compression, aligning with some experimental observations. Notably, the DDD simulations reveal the occurrence of the uncommon type III (KIII) kinematic hardening of Asaro in both single- and double-slip cases, emphasizing the relevance of this hardening type in the realm of small-scale plasticity. Inspired by insights from DDD, two advanced SGCP models incorporating alternative descriptions of higher-order kinematic hardening mechanisms are proposed. The first model uses a Prager-type higher-order kinematic hardening formulation, and the second employs a Chaboche-type (multi-kinematic) formulation. Comparison of these models with DDD simulation results underscores their ability to effectively capture the observed strengthening and hardening effects. The multi-kinematic model, through the use of quadratic and non-quadratic higher-order potentials, shows a notably better qualitative congruence with DDD findings. This represents a significant step towards accurate modeling of small-scale material behaviors. However, it is noted that the proposed models still have limitations, especially in matching the DDD scaling exponents, with both models producing h−1 scaling relationships (i.e., Orowan relationship for precipitate size effects). This indicates the need for further improvements in gradient-enhanced theories in order to guarantee their suitability for practical engineering applications.

Slip-discreteness-corrected strain gradient crystal plasticity (SDC-SGCP) theory

Strain gradient plasticity theory addresses the plastic strain gradient induced hardening by considering the internal stress and Taylor hardening associated with the geometrically necessary dislocations (GNDs). However, the continuum description of internal stress associated with GNDs is inaccurate due to the coarsening of discrete dislocations. Corrections are thus derived as the difference between the stresses produced by the continuous configuration and the discrete configuration. We further demonstrate the capability of this correction in effectively capturing the internal stress induced strengthening effect associated with GNDs, and elucidate that its role in strengthening is to homogenize the deformation and extend the influence of grain boundaries into the interior of grains within polycrystals. This capability to capture intragranular slip distribution is validated through the simulation of a polycrystalline tensile experiment. This work explains the limitations of classical crystal plasticity theory under high strain gradients and offers a straightforward yet robust slip discreteness correction to crystal plasticity with transparent input from dislocation theory, opening a new perspective for the connections between continuum crystal plasticity theory and dislocation theory.

Crystal plasticity quantification of anisotropic tensile and fatigue properties in laser powder bed fused Inconel 718 superalloy

The high thermal gradients during the laser powder bed fusion (LPBF) process induce an inhomogeneous microstructure with anisotropic mechanical properties. This study developed a dislocation-based strain-gradient crystal plasticity finite element (CPFE) model to reveal the microstructure-based deformation mechanisms and quantify their strengthening contributions to the mechanical property anisotropy of LPBF-ed Inconel 718 (IN718) superalloy. The analyzed microstructural attributes included 1- crystallographic orientations/misorientations, 2- solid solutions (SS), 3- grain size and morphologies, and 4- dislocation dynamics, such as statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs). The investigated mechanical properties contained tensile strength, fatigue resistance, fracture mechanisms, and their anisotropy under loading towards building (BD) and transverse (TD) directions. The experimentally obtained microstructural features of the as-LPBF-ed IN718 were synthesized into the crystal plasticity representative volume elements (CPRVE) to model the deformation behavior using a user-defined material subroutine (UMAT) in the ABAQUS solver. The energy-based fatigue indicator parameter (FIP) captured the fatigue properties and failure mechanisms. The main slip-strengthening contributions stemmed from SSDs, grain size, SS, and GNDs. in TD. The corresponding order in BD was SSDs, SS, GNDs, and grain size because the grain dimensions were larger in BD than in TD. The grain-size effect had the highest contribution to the tensile property anisotropy, followed by SSDs, GNDs, and SS. Aside from their strengthening effects, crystal texture and grain size induced mechanical property heterogeneity and fatigue sensitivity at grain boundaries, reducing their contributions to fatigue strength relative to tensile. In contrast to the previous findings, local accumulations of SSDs and GNDs at softer sides of grain boundaries during cyclic loading reduced the strength heterogeneity in these areas, improving fatigue properties. The highest slip-strengthening effect for fatigue resistance and anisotropy was under SSDs, followed by GNDs, SS, and grain size.

A robust nano-indentation modeling method for ion-irradiated FCC single crystals using strain-gradient crystal plasticity theory and particle swarm optimization algorithm

Addressing the challenge of identifying an appropriate set of material and irradiation parameters for accurate simulation models using crystal plasticity finite element method (CPFEM), this study proposes a novel two-stage method for nano-indentation modeling of ion-irradiated face-centered cubic (FCC) materials. It includes implementing the strain-gradient crystal plasticity (SGCP) theory with irradiation effects and the calibration of simulation parameters using the particle swarm optimization (PSO) algorithm with experimental data. The proposed method consists of two stages: establishing CPFEM without irradiation effects in stage 1 and modeling irradiation effects based on CPFEM in stage 2. Modeling the nano-indentation test of ion-irradiated stainless steel 304 (SS304) using real experimental data is conducted to evaluate the efficiency of the proposed method. The accuracy of the calibration method using PSO is verified through comparisons between simulation and experimental results for force-indentation depth and hardness-indentation depth relationships under both unirradiated and irradiated conditions. Moreover, effect of ion-irradiation on the mechanical behavior during the nano-indentation of single crystal SS304 is also examined to demonstrate that the proposed method is a powerful approach for nano-indentation modeling of ion-irradiated FCC single crystals using SGCP theory and the PSO algorithm.

Effect of grain boundary on scratch behavior of polycrystalline copper

The anisotropy in the elastoplastic response of crystalline solids influences substantially their scratch resistance at the microscale. While grain boundaries typically act as obstacles to the motion of dislocations at the grain level, the scratch behavior in the vicinity of the grain boundary is influenced by other factors such as crystallographic orientation and pile-up topography. We performed a systematic set of nano-scratch simulations using the mechanism-based strain gradient crystal plasticity to study the evolution of scratch force and depth near grain boundaries. To validate our model, we conducted a nano-scratch test on a polycrystalline copper sample and compared the scratch depth profile for a specific grain boundary with the simulated result, where a good qualitative agreement was achieved. Our simulations showed that the scratch depth, force, and hardness vary between the corresponding values for the individual constitutive grains, and the variation decreases by reducing the grain size. Additionally, we analyzed the scratch response in terms of the pile-up topography and subsurface stresses and distinguished different regimes when the indenter approaches and scratches across the boundary. Our findings offer a new direction to optimize the scratch resistance of polycrystalline metals by tailoring the size and orientation of grains near the surface. Additionally, it introduces scratching as a robust material characterization method.

Nanoscratching of polycrystalline copper examined through strain gradient crystal plasticity

Nanoscratch testing is a method pivotal for evaluating the mechanical and tribological characteristics of materials which involves the controlled scratching of specimens with a nano-scale indenter. This research delves into the analysis of effect of grain size on the deformation mechanisms and material responses during nanoscratch tests on polycrystalline copper. By utilizing finite element method and adopting a lower-order strain gradient crystal plasticity framework, this study investigates results such as reaction forces on the indenter, apparent friction coefficients, and alterations in pile-up topography. These factors are examined with the aforementioned strain gradient theory, employing calculations of the density of geometrically necessary dislocations to obtain size-dependent material response. The crystal plasticity framework is implemented into ABAQUS as a user material subroutine (UMAT), and the model's accuracy is affirmed through comparisons with experimental data from single crystal copper studies available in the literature. 3D geometries are generated to model a single crystal and three polycrystal materials with average grain diameters of 5 µm, 15 µm, and 50 µm. These specimens are subjected to deformation by a rigid Berkovich indenter to simulate nanoscratching tests in the numerical examples, where the key point is to examine the grain size effects while keeping fixed any other variables that may influence the results.

A numerical study on the visco-plastic regularization of a rate-independent strain gradient crystal plasticity formulation

A common practice in computational investigations of rate-independent plasticity is to approximate the rate-independent behavior by a visco-plastic regularization. As the fidelity of the approximation increases, the numerical solution of the non-linear problem becomes challenging and it can eventually lead to divergence; thus, there is a numerical limit to the regularization parameter. This limit may be exacerbated by complex material models and critical values of material parameters. Due to these constraints, the regularization may be rendered insufficient and the artificially introduced rate-dependency may lead to a behavior that can be mistakenly attributed to the material model and that we thus identify as spurious in a rate-independent context. To study these spurious effects and their onset, here the problem of an infinite strip under shear loading is numerically solved using a visco-plastic regularized gradient crystal plasticity formulation. The required accuracy of the rate-dependent approximation is found to vary with respect to the type of loading. Furthermore, a set of sigmoid functions used for the regularization is investigated and a subset is shown to deliver improved approximation of the rate-independent case.

Managing the inevitable microstructural and property heterogeneity in powder bed fusion Ti-6Al-4V parts via heat treatment

This study investigates the effects of the commonly-observed microstructure heterogeneity from the metal electron beam powder bed fusion (PBF-EB) processes and how it evolves following the associated heat treatment (HT), on the tensile behaviour of Ti-6Al-4V alloy. A strain gradient crystal plasticity finite element (CPFE) method is employed using physically-based dislocation mechanics to capture the microstructure-sensitive size effect. Microstructural characterization is performed using electron backscatter diffraction (EBSD) of the as-built and post-HT specimens, with measurements of grain morphology, phase, and texture extracted from the bottom and top regions of specimens, used to construct the CPFE models. The dual-phase CPFE model represents α (hcp) and β (bcc) basket-weave morphology, typical in Ti-6Al-4V, with accurate PBF-induced texture definitions defined directly from the measured EBSD data. Key microstructural features exhibiting a gradient include lath width and phase fraction. The effects of as-built microstructural gradient in a single component include an increase of lath width by 43 %, and the beta phase fraction, from bottom to top. The CPFE models successfully predict the effect of HT and microstructural gradient on tensile response, specifically yield stress and initial hardening rate. The role of dislocation density and the associated influence of lath width on yield stress are empirically correlated for the structure-property design of PBF-EB metals. This study also demonstrates the homogenising effect of the heat treatment; effectively eradicating the microstructural gradient observed in as-built specimens.

Size-dependent yield criterion for single crystals containing spherical voids

A micromechanics-based yield criterion is derived for a porous single crystal containing a random distribution of spherical voids, using homogenization and limit analysis of a hollow spherical representative volume element obeying a strain gradient crystal plasticity model. The criterion captures the effect of plastic anisotropy due to crystallographic slip on a set of discrete slip systems, as well as the void size dependence of yielding. The yield criterion is formally similar to the well known Gurson model for isotropic porous materials, and reduces to existing results in the literature in the limiting case of large void sizes relative to the length scale in the gradient plasticity model. The yield criterion is validated by comparison with rigorous upper bound yield loci obtained using a numerical limit analysis procedure. Predictions for the size dependence of void growth are obtained by integrating the porous plasticity model, and shown to be consistent with the observations from lower scale discrete dislocation dynamics simulations. The loading path dependence of void growth in the size-independent limit are compared with finite element simulations of void growth in a single crystal using the unit cell model.

Effect of stress-states on non-classical twinning in three-point bending of magnesium alloys

The influence of stress state on the drivers of non-classical twin nucleation in three-point bending of Magnesium alloy AZ31 is presented in this study. Deformed microstructures and textures at the tensile and, for the first time, shear regions are investigated in detail. The experimental observations coupled with the strain-gradient crystal plasticity analyses show that the shear nucleates more twins compared to the tensile and compressive stress-states at near yield-point. Stored energy density provides mechanistic insight into the local drivers of non-classical twin nucleation observed in the shear stress-state region. Further, the twin variant selection follows the locally stored shear energy density criterion, as opposed to the twin resolved shear stress, indicating that their nucleation is driven by local energy and defect sources.

Deciphering phase stress partition and its correlation to mechanical anisotropy of laser powder bed fusion AlSi10Mg

Laser powder bed fusion AlSi10Mg exhibits severe mechanical anisotropy, which hinders its applications in industrial field. In the present work, we investigated the mechanisms of anisotropy in the light of phase stress partition, using traditional crystal plasticity and mechanism-based strain gradient crystal plasticity models. Two representative volume elements of the Al-Si cellular structures, corresponding to 35 ℃ and 200 ℃ build platform temperatures, were built to assess the influences of Al matrix strength and strain gradient on phase stress partition and anisotropy. In addition, the maximum principal stress distribution of the Si-rich network was probed, based on which the potential damage sites and densities were evaluated. The results show that the anisotropic strength and damage both originate from the elongated Si-rich network. According to the comparison between the 35 ℃ and 200 ℃ materials, an increase in Al phase strength can change the phase stress partition and effectively mitigate the anisotropy degree. Moreover, it is revealed that the strain gradient tends to alleviate anisotropy and delay damage evolution. The findings of this work can pave the way to developing high-performance LPBF Al alloys with lower degrees of anisotropy.

An investigation of slip transmission during uniaxial tensile deformation of a coarse-grained copper via EBSD characterization and crystal plasticity modelling

This study explores the phenomenon of slip transmission across grain boundaries in coarse-grained copper. The associated effects on the deformation gradient are also investigated, using both experiment and simulation. An in-situ tensile test of a fully annealed copper specimen was conducted first using a scanning electron microscope (SEM). The deformation gradient and geometrically necessary dislocation (GND) maps at selected tensile strain values were obtained and characterized by electron backscatter diffraction (EBSD). In addition, a physically based strain gradient crystal plasticity finite element method model has been developed to simulate the early stage of tensile deformation behaviour of copper. In this model, slip transmission across the grain boundary is either allowed or blocked, depending on the alignment of adjacent grains. Slip transmission is allowed at grain boundaries between suitably aligned grains according to the Luster-Morris factor, residual Burgers vector is applied to consider the slip misalignment at GB region, and dislocation accumulation at grain boundaries is included following the Kocks-Mecking law. It was found that dislocation pile-up, lattice rotation and stress concentration were much higher near the impenetrable grain boundaries than at the penetrable ones. Due to the slip transmissions across grain boundaries, the stress and strain concentrations can be partially relaxed, and the slip transmission induced relaxation is associated with the number of active slip systems and shear strain on the best aligned slip systems. This study provides an effective strategy to investigate the early stage of plastic deformation, and a good agreement between experimental observations and simulation results has been achieved.

On the size effect in scratch and wear response of single crystalline copper

Indentation hardness is a well-known material parameter for determining the resistance of a material to deformation. It is however known that indentation hardness alone cannot fully characterize anisotropic material strength in particular for single crystalline materials. Recently, it is shown that scratch hardness is a more suitable material property to characterize the wear response of crystalline materials. In this study, we use the mechanism-based strain gradient crystal plasticity to explore the size effect in the scratch behavior of single crystalline copper. Our simulations picture strong size effects in the scratch hardness value and pile-up magnitude. Interestingly, it shows that the degree and nature of size-dependency for these two parameters varies significantly and oppositely by the scratch direction. For instance, while [001] direction shows the highest degree of size effect in scratch hardness, it presents the lowest pile-up size effect. This opposite effect is explained in terms of the contribution of different slip systems to dislocation resistance and work hardening. We also show that scratch hardness, which takes into account both the size effect and crystallographic direction dependence, is an appropriate material property for wear characterization.

Coherency loss marking the onset of degradation in high temperature creep of superalloys: Phase-field simulation coupled to strain gradient crystal plasticity

A dislocation density based crystal plasticity — phase-field model is applied to investigate directional coarsening during creep in CMSX-4 Ni-based superalloys in the high temperature and low stress regime. Coherency between the ordered γ′ precipitates and the disordered γ channels prevents the generation of geometrically necessary dislocations, since the precipitate can be considered undeformable in the low stress regime. After coherency loss between the γ matrix phase and the γ′ precipitates the constraint against generation of geometrically necessary dislocations is relaxed, causing rotation of the crystal lattice under uniaxial load, known as “Schmid rotation”. As a consequence, the creep rate in the matrix increases, whereby degradation can be measured by the number density of geometrically necessary dislocations. The state of coherency loss is associated with the minimum creep rate in a creep experiment under constant load. The presented simulations start from a coherent γ′ precipitates distribution with random size and position generated during a precipitation heat treatment process. Simulations of N-type and P-type rafting under tensile and compressive load respectively are presented. The effect of coherency loss, coalescence of precipitates and lattice rotation due to generation of geometrically necessary dislocations is discussed in correlation with experimental findings.

Role of particles and lattice rotation in tension–compression asymmetry of aluminium alloys

Plastic deformation of aluminium alloy AA6061 under tension and compression is investigated using a strain gradient crystal plasticity model implemented with the finite element method. This approach takes into account the microstructure as determined experimentally by electron backscatter diffraction (EBSD). The asymmetry of stress strain curves in loading of tension and compression (TC) is revealed for the FCC crystal structure. Plastic slip on activated slip systems leads to lattice rotation of crystal matrix and particles towards specific orientations, resulting in variation of moduli. The evolution of two different types of dislocations, statistically stored dislocations (SSD) and geometrically necessary dislocations (GND), is investigated to reveal the TC asymmetry due to slip resistance. It is found that the TC asymmetry at the yield strain is caused by lattice rotation of both crystal matrix and particles. When strain increases, the TC asymmetry is due to a combination of lattice rotation of crystal matrix and particle and variation of the GND density. As particle content increases, the TC asymmetry is mainly attributed to lattice rotation of particles at large strain.

An alternative way to describe thermodynamically-consistent higher-order dissipation within strain gradient plasticity

In the context of strain gradient plasticity (SGP), description of higher-order dissipation is the subject of extensive on-going discussions. In most existing SGP theories including thermodynamically-consistent higher-order dissipation, higher-order dissipative processes are described based on the decomposition of the higher-order stresses into recoverable and unrecoverable parts. This higher-order stress decomposition represents the basis of the so-called non-incremental (Gurtin-type) SGP theories, which are the most commonly used in the literature. As formulated, these theories satisfy the thermodynamic requirement of non-negative dissipation. However, they generally lead to unusual effects for some boundary value problems, such as the occurrence of elastic gaps under non-proportional loading conditions. The present work proposes an alternative way to describe higher-order dissipative effects, with an illustration within strain gradient crystal plasticity (SGCP) framework. Inspired by rheological models in series like Maxwell model, the higher-order stress decomposition is replaced by a decomposition of the plastic slip gradients into recoverable and unrecoverable parts. Effects of this decomposition technique are studied and compared with those obtained using higher-order stress decomposition. Capabilities of such a technique to deal with elastic gaps are also investigated.

Size-dependent microvoid growth in heterogeneous polycrystals

Microvoid growth involves a strong size effect, i.e., smaller microvoid presents a lower growth rate. In polycrystalline materials, the size ratio between microvoids and grains may also affect microvoid growth behavior. However, most previous studies treated material matrix surrounding microvoids as homogeneous. It turned out that such treatment cannot effectively depict the influence of the abovementioned void-grain size ratio on damage evolution. In the present study, both classical local and non-local strain-gradient crystal plasticity finite element simulations are performed to study size-dependent microvoid growth in heterogeneous polycrystals. The results indicate that both void-grain size ratio and absolute microvoid size influence microvoid growth significantly, referred to as first (induced by grain-scale heterogeneous deformation) and second kinds of (induced by plastic strain gradient) size effects, respectively. Besides, macroscopic stress triaxiality T has a significant influence on the size effect of microvoid growth, while Lode parameter L exhibits a negligible effect. Due to random grain-orientation distribution and grain-geometric characteristic, a smaller microvoid within polycrystalline environments may even grow faster than a larger one, implying that the size effect of microvoid growth should be understood from a statistical point of view in polycrystalline environments. The present study provides a fundamental understanding on the intrinsic mechanism of the size-dependent microvoid growth in heterogeneous polycrystals.

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.

Modeling size effects in microwire torsion: A comparison between a Lagrange multiplier-based and a [formula omitted] gradient crystal plasticity model

The size-dependent response of metallic microwires under monotonic and cyclic torsion is modeled taking a reduced-order strain gradient crystal plasticity approach involving a single scalar-valued micromorphic variable. It is compared with the response predicted by the CurlFp model proposed in Kaiser and Menzel (2019a), which is based on the complete dislocation density tensor. It is shown that, in cyclic non-uniform plastic deformation processes, the gradient of the scalar-valued internal variable in the reduced-order model predicts isotropic hardening in contrast to kinematic-type hardening produced by the CurlFp model due to a dislocation-induced back-stress component. The arising size effect in the monotonic torsion tests is described by the normalized torque T/R3 as a function of the ratio of the microwire radius R and the characteristic length scale ℓ. In the size-dependent domain, characterized by an inflection point on the corresponding curve, the scaling law T/R3∼(R/ℓ)n can be identified, and explicit relations are found for the power n. The relative evolution of Statistically Stored Dislocation (SSD) and Geometrically Necessary Dislocation (GND) densities during torsion is described in detail.