Strain Gradient Plasticity Research Articles

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations

A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations

Strain-gradient and damage failure behavior in particle reinforced heterogeneous matrix composites

A strain gradient plasticity model with strengthening effect and damage effect (SGSED) is developed under the continuum medium framework for particle-reinforced heterogeneous matrix composites (PRHMCs). Boron carbide (B4C) particle-reinforced ultrafine-grained (UFG)/fine-grained (FG) heterogeneous matrix composites (B4Cp/(UFG/FG)) are fabricated, in which the UFG region consisted of carbon nanotubes (CNTs)/6061 aluminum (Al) flakes with grains in the ultrafine range and the FG region is processed via 6061Al. The SGSED model is written into the user subroutines using commercial finite element (FE) calculation software, and the three-dimensional (3D) FE representative volume element (RVE) for B4Cp/(UFG/FG) composites is established, from which the distribution of the interface-affected-zone (IAZ) formed of the strain gradient caused by the uncoordinated deformation of the UFG-FG heterogeneous matrix and reinforced phase-matrix is calculated. The evolution of the strain gradient in the deformation process of composites and the influence of the strain gradient on the progressive damage and crack evolution of composites are analyzed, and the strain gradient strengthening-toughening mechanism of composites is revealed. It is found that the IAZ has a considerable strengthening-toughening effect on the composites, which can reduce stress concentration at the interface between the reinforced phase and the matrix, and slow down the crack propagation of the matrix.

Mesoscopic coupled plasticity-damage model of rough surface considering size-dependent plasticity

Metallic materials exhibit significant size effect at mesoscopic scale have been revealed by many experiments. Thus, it is essential to consider size effect when conducting mesoscopic scale studies in tribology. However, there is no constitutive model considering both ductile fracture and size effects of rough surface in the literature. At the same time, the simulation of friction and wear at the mesoscopic scale suffer from computational inefficiency using the conventional theory of mechanism-based strain gradient (CMSG) plasticity theory. To address the aforementioned challenges, we propose a mesoscopic coupled plasticity-damage model by combining the coupled plasticity-damage model and the simplified CMSG plasticity theory. This model can take into account not only the effect of stress state, temperature, strain rate on plasticity and fracture, but also the effect of the effective plastic strain gradient. Meanwhile, the model is able to solve the effective plastic strain gradient by a simplified method, thereby enhancing the computational efficiency. Then, model parameters for bearing bushing material of an engine are calibrated. Finally, scratch simulation and scratch tests of bearing bushing material at the mesoscopic scale are conducted to validate the effectiveness of the mesoscopic constitutive model. The residual scratch depth and width, coefficient of friction at different normal loads are investigated by experiment and simulation, respectively. Results proved the effectiveness of mesoscopic coupled plasticity-damage model, which is applicable for investigating issues related to wear, friction, and contact.

Mechanical Properties of Silicon Nitride in Different Morphologies: In Situ Experimental Analysis of Bulk and Whisker Structures.

Silicon nitride (Si3N4) is widely used in structural ceramics and advanced manufacturing due to its excellent mechanical properties and high-temperature stability. These applications always involve deformation under mechanical loads, necessitating a thorough understanding of their mechanical behavior and performance under load. However, the mechanical properties of Si3N4, particularly at the micro- and nanoscale, are not well understood. This study systematically investigated the mechanical properties of bulk Si3N4 and Si3N4 whiskers using in situ SEM indentation and uniaxial tensile strategies. First, nanoindentation tests on bulk Si3N4 at different contact depths ranging from 125 to 450 nm showed significant indentation size effect on modulus and hardness, presumably attributed to the strain gradient plasticity theory. Subsequently, in situ uniaxial tensile tests were performed on Si3N4 whiskers synthesized with two different sintering aids, MgSiN2 and Y2O3. The results indicated that whiskers sintered with Y2O3 exhibited higher modulus and strength compared to those sintered with MgSiN2. This work provides a deeper understanding of the mechanical behavior of Si3N4 at the micro- and nanoscale and offers guidance for the design of high-performance Si3N4 ceramic whiskers.

Generalized grain boundary constitutive description implemented in a strain-gradient large-strain FFT-based formulation: Application to nano-metallic laminates

This paper presents a general treatment of grain boundary constitutive behavior in the context of strain-gradient (SG) plasticity, and its numerical implementation in a large-strain (LS) elasto-viscoplastic (EVP) fast Fourier transform (FFT)-based micromechanical model. Two novel grain boundary constitutive equations are proposed, allowing for more accurate description of the Burgers vector flow at the grain boundary. The capabilities of the generalized SG-LS-EVPFFT formulation are illustrated for the case of kink-band formation during layer-parallel compression of nano-metallic laminates (NMLs), requiring consideration of the interaction between dislocations and interfaces.

Achieving exceptional strain-hardening ability in nanocrystalline Ni–W coatings through compositionally modulated multilayer approach

Nanocrystalline Ni–W coatings show exceptional strength; however, they exhibit poor toughness due to limited ductility and high residual stresses which impede their use in practical applications as a potential hard chrome replacement. In the present study, compositionally modulated multilayer (CMM) coatings with alternate layers of Ni-1 at% W (soft) and Ni-15 at% W (hard) of three different layer thicknesses of 1, 0.5, and 0.1 μm are developed by pulse reverse electrodeposition to achieve the strength-ductility synergy, thereby enhanced wear resistance. Microcompression test results reveal that all the CMM coatings exhibit remarkably higher strain-hardening ability than the homogenous coatings. Significant fluctuations in the strain-hardening rate and a positive shift in the onset stress for the work-hardening stages were observed with the layer thickness. This suggests a strong interplay between the hard and soft layers on the deformation characteristics of multilayer coatings. Detailed electron microscopy of the cross-section of the deformed micropillars shows substantial evidence of co-deformation of both layers. A mechanism is proposed to explain the enhanced strain-hardening in multilayers involving the strength incompatibility arising in the layers leading to lateral constraints, resulting in strain gradient plasticity, stabilizing of local shear by the soft layers, and the barrier effect of the interfaces.

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.

Developing evolutionary models for deformation band formation in high-porosity suprasalt sandstones: An example from Paradox Basin, Utah

Pervasive deformation adjacent to salt diapirs can drive significant porosity and permeability reduction in potential reservoir units, yet predicting the intensity and spatial distribution of these subseismic-scale features has remained as a persistent challenge. Previously, Jurassic sandstones adjacent to salt-cored anticlines in the Paradox Basin, eastern Utah, have been used to characterize deformation banding formation. However, these host rocks have undergone significant structural and diagenetic modification following band formation, which adds significant complexity to analyses of this type. Here, we integrate detailed structural transect analysis, petrologic analysis of bands and host rocks, depth-compaction relationships, critical state deformation models, and regional burial evolution models to constrain the timing and conditions of band formation. This analysis demonstrates that bands in this region formed prior to pervasive cementation and are generally expressed as two plastic strain gradients; one characterized by distributed deformation bands across the crest of salt-cored anticlines and a second wherein deformation band density increases near mesoscale normal faults. Comparison of these results with numerical structural evolution models of similar structures indicates that band gradients may have formed as a result of outer-arc extensional stress induced by the rising diapir and are linked with normal faults. These results highlight that the integration of material behaviors with kinematic evolution can constrain models for deformation band formation and may provide a useful workflow for understanding the development of these features in geologically young petroleum systems like those of the Atlantic passive margin.

Modelling of size-dependent plasticity in polymer-based composites based on nano- and macroscale experimental results

A number of experimental evidences indicate that the local response of polymer matrices near fibres is inadequately represented by classical continuum models relying on the bulk polymer behaviour. This results from size-dependency associated to large plastic strain gradients, complex interphase behaviour and/or changes of polymer structure. Classical multiscale models require artificial tuning of the properties to provide realistic macroscale predictions. We demonstrate that an unprecedented modelling approach based on a micromorphic theory is able to capture such size effects in long-fibre composites. The model is identified via nano digital image correlation strain fields and validated by predicting the strengthening found in transverse compression of UD composites, not captured by classical models. Micro-shear bands are properly regularised by the model, thus correctly handling the size-dependent plasticity and softening effects. The improved prediction of the strain localisation pattern in the matrix opens avenues to more accurately model interfacial failure and damage processes.

Nanoindentation study on early-stage radiation damage in single-phase concentrated solid solution alloys

Concentrated solid solution alloys (CSAs) comprising multiple components have unlocked novel pathways for materials design, particularly in enhancing radiation tolerance. It is imperative to detect early-stage radiation damage in CSAs to gain insights into damage initiation and accumulation mechanisms. In this study, nanoindentation is employed to assess the impact of irradiation on deformation mechanisms in single crystal CSAs, specifically NiCo, NiFe, and NiCoFeCr. It is discovered that pile-up behavior in CSAs significantly affected by irradiation: pile-up induces 10–20 % increase in the contact area before irradiation that is independent of the penetration depth but 20–30 % after irradiation, which substantially affects hardness analyses. Within the context of strain gradient plasticity theory, distinct radiation-induced hardening in three CSAs is interpreted by two components: one is from the radiation-induced defects, and the other is the increase in density of geometrically necessary dislocations (GNDs) associated with indentation size effect (ISE). Quantitative analysis shows that the radiation-induced defects produce obvious hardening in NiFe and NiCoFeCr sample, but not in the NiCo sample. Meanwhile, the irradiation induces a higher GND density in NiCo and NiFe, but not in NiCoFeCr, which is interpreted by the volume change of the plastic zone.

Mechanics of gradient nanostructured metals

The emergence of heterogeneous nanostructured materials (HNMs) offers exciting opportunities to achieve outstanding mechanical properties. Among these materials, gradient nanotwinned (GNT) Cu is a prominent class of HNMs, demonstrating superior strengths by gaining extra strengths compared to non-gradient counterparts. Its layered gradient structure provides a simplified quasi-one-dimensional model system for understanding the extra strengthening effects of structural gradients and resulting plastic strain gradients. This paper presents a comprehensive report for recent experimental and modeling studies on the mechanics of GNT Cu, covering advances in controlled material processing, back stress measurement, deformation field characterization, dislocation microstructure analysis, and strain gradient plasticity modeling. These studies unveil the spatiotemporal evolution of both plastic strain gradients and extra back stresses originating from structural gradients. Direct connections are established between the sample-level extra strength of GNT Cu and the synergistic strengthening effects induced by local nanotwin structures and their gradients. We emphasize the critical role of the size of the representative volume element in assessing the effects of plastic strain gradient and extra back stress. Moreover, lower-order strain gradient plasticity models are validated through experimental characterizations of GNT Cu, paving the way for future investigation into the mechanics of gradient nanostructured metals. Finally, we provide an outlook on research needs for understanding the mechanics of gradient nanostructured metals and, more broadly, HNMs, towards achieving exceptional mechanical properties.

Microstructural and mechanistic insights into the Tension–Compression asymmetry of rapidly solidified Fe–Cr alloys: A phase field and strain gradient plasticity study

Rapid solidification in Additively Manufactured (AM) metallic materials results in the development of significant microscale internal stresses, which are attributed to the printing induced dislocation substructures. The resulting backstress due to the Geometrically Necessary Dislocations (GNDs) is responsible for the observed Tension–Compression (TC) asymmetry. We propose a combined Phase Field (PF)-Strain Gradient J2 Plasticity (SGP) framework to investigate the TC asymmetry in such microstructures. The proposed PF model is an extension of Kobayashi’s dendritic growth framework, modified to account for the orientation-based anisotropy and multi-grain interaction effects. The SGP model has consideration for anisotropic temperature-dependent elasticity, dislocation strengthening, solid solution strengthening, along with GND-induced directional backstress. This model is employed to predict the solute segregation, dislocation substructure and backstress development during solidification and the post-solidification anisotropic mechanical properties in terms of the TC asymmetry of rapidly solidified Fe–Cr alloys. It is observed that higher thermal gradients (and hence, cooling rates) lead to higher magnitudes of solute segregation, GND density, and backstress. This also correlates with a corresponding increase in the predicted TC asymmetry. The results presented in this study point to the microstructural factors, such as dislocation substructure and solute segregation, and mechanistic factors, such as backstress, which may contribute to the development of TC asymmetry in rapidly solidified microstructures.

Micromechanism of strength and damage trade-off in second-phase reinforced alloy by strain gradient plasticity theory

The strength and damage are mutually exclusive in most alloys. It is still a challenge to understand the role of microstructural features in affecting strength and damage tolerance simultaneously. In this study, we provide a general approach to unveil the micromechanism of strength and damage trade-off in second-phase reinforced alloy. Concretely, a microstructure-based constitutive model incorporating the strain gradient plasticity theory is developed to evaluate the overall mechanical behavior and local deformation behavior of particle-reinforced alloys by finite element simulations. The strain gradient effect exacerbates the non-uniform stress distribution, particularly enhancing the local stress level around the particle-matrix interface, thereby improving the total strain hardening of the alloys. Importantly, the strain gradient effect can significantly expand regions with local stress levels surpassing the critical stress for microcrack nucleation, ultimately leading to damage initiation and failure. This elucidates the micromechanism underpinning the trade-off between strength and damage. By revealing the competitive relationship between local stress concentration and microcrack nucleation stress, the comprehensive effects of particle size and volume fraction on the strength, strain hardening and damage tolerance are quantitatively predicted. It is recommended to adjust the particle volume fraction based on reducing the particle size to obtain the desired properties. The present work discerns the inherent origin of the strength-damage trade-off in particle-reinforced alloy and sheds light on the role of the particle size and volume fraction on the strength and damage tolerance, which can offer valuable guidance for developing high-performance particle-reinforced materials.

On the role of higher-order condition of strain gradient plasticity in the cyclic torsion of thin metallic wires: Experiments and modeling

Cyclic torsion tests are performed on micron-scale copper wires with and without surface passivation to study the role of the higher-order condition in the plastic behavior of thin wires under non-proportional loading. A typical strengthening size effect is observed in the symmetric cycles. More obvious strength enhancement exists in the torsional response of passivated copper wires. An unusual Bauschinger effect is found during the loading-unloading cycles, which is more pronounced in passivated wires. The finite element implementation based on Gudmundson's strain gradient plasticity theory is developed for wire torsion to characterize the observed size-dependent phenomena. The higher-order boundary conditions are introduced to simulate the passivated surface. The predicted radial distributions of plastic strain, stress components, and geometrically necessary dislocation density for the passivated and unpassivated wires are given and compared. This work provides a reasonable basis for understanding the role of higher-order conditions of strain gradient plasticity.

Cosserat-phase-field modeling of grain nucleation in plastically deformed single crystals

Thermomechanical processing of crystalline materials induces microstructural evolution such as grain nucleation and growth. In the numerical simulation of these processes, grain nucleation is generally treated as an additional ad hoc step in which circular or spherical grains are added in regions where a critical dislocation density, stress or strain are reached. In this paper, systematic finite element simulations are performed showing that the Kobayashi–Warren–Carter (KWC) phase field model and its coupling with Cosserat crystal plasticity predict spontaneous nucleation of new grains in single crystals in the presence of lattice orientation/rotation gradients. The numerical analysis of the stability of gradients of lattice rotation and dislocation-based stored energy indicates that a gradient of stored energy alone is not sufficient to trigger grain formation. As an application, the KWC–Cosserat model is used to simulate the torsion and annealing of a copper single crystal bar with a circular cross section. This mechanical loading produces a large, fairly uniform axial rotation gradient which induces nucleation in the form of a stack of cylindrical grains. Plastic strain gradients in cross-sections predicted by the 3D finite element simulation, are not strong enough to compete with the longitudinal nucleation process, as confirmed by experimental observations from the literature.

Elastic-Gap Free Formulation in Strain Gradient Plasticity Theory

Abstract This article presents an elastic-gap free isotropic higher-order strain gradient plasticity theory that effectively captures dissipation associated to plastic strain gradients. Unlike conventional methods that divide the higher-order stress, this theory focuses on dividing the plastic strain gradient into energetic and dissipative components. The moment stress that arises from minimizing a dissipating potential demonstrates a nonlinear evolution over time, resembling the Armstrong–Frederick nonlinear kinematic hardening rule in classical plasticity. The thermodynamically consistent framework establishes additional dissipation in the dissipation inequality. The energetic moment stress saturates as the effective plastic strain increases during plastic flow. In contrast to the Gurtin-type nonincremental model, the proposed model smoothly captures the apparent strengthening at saturation without causing a stress jump. A passivated shear layer is analytically assessed to demonstrate that the proposed theory exhibits the same amount of dissipation as the existing Gurtin-type model when they show similar shear responses at saturation. It is also shown that the plastic flow remains continuous under nonproportional loading conditions using an intermediately passivated shear layer problem. Finally, the proposed theory is validated against a recent experiment involving combined bending torsion of an L-shaped beam using a 3D finite element solution. Overall, the proposed model provides an alternative approach to evaluating the size effect within the nonincremental isotropic strain gradient plasticity theory without introducing any stress jump.

Potential regulation of strength-ductility by nucleation and growth mechanisms of shear bands in heterogeneous laminates

Shear bands (SBs) expand rapidly in hard domains and are suppressed in soft domains, effectively improving the tensile plasticity of the material. Significant evolution of SBs during elastoplastic deformation exists in heterogeneous laminates (HLs). Molecular dynamics (MD) simulation results indicate that the activities of grain boundary (GB) and dislocation jointly mediate the nucleation and growth of SBs. SBs tend to nucleate near the interface, and then grow rapidly under the promotion of GB activities and dislocation slips, resulting in a rapid increase in plastic strain gradient. Ultimately, the density and morphology of SBs determine the magnitude of the plastic strain gradient. In structures with lower layer thickness, the band-like SBs are densely distributed, which achieves higher strain gradient strengthening capability. However, the interface serves as an accumulation site for geometrically necessary dislocations (GNDs), too low a layer thickness will induce overlapping of the interface-affected-zone (IAZ), thereby weakening the strain hardening efficiency. The simulations track the nucleation and growth of SBs at the atomic scale, and discover that the structure can better balance HL's strength and ductility when the layer thickness is three times the grain size of the soft domain.

Unravelling the extra-hardening in chemically architectured high entropy alloys

Chemically architectured high entropy alloys are a new concept of multi-scale microstructure originally including a 3D network of composition fluctuations, named interphase. To unravel the strengthening contribution of each entity of the microstructure, chemically architectured alloys were processed, their microstructure and mechanical properties were characterized and then they were modelled by the finite element method. Nanoindentation measurements reveal a local extra-hardening at the interphase. Conventional modelling, even when considering three phases in a full-field approach, could not reproduce the compression properties, indicating again the existence of an extra-hardening. Then, the chemical and plastic strain gradients effects were included in the model and an agreement was reached with experimental data. Both experimental and modelling results prove that chemical gradients are at the origin of a new strengthening mechanism. This chemical gradient strengthening depends on the many microstructural parameters of chemically architectured alloys and opens the way for tuning and optimization of mechanical properties.

Effect of stress/strain partition on the mechanical behavior of heterostructured laminates: A strain gradient plasticity modeling

Based on the structure of natural nacres, laminated metal materials have been successfully developed to possess desirable mechanical properties for diverse engineering applications. These heterostructured (HS) laminates consist of layers with different mechanical performances, resulting in asynchronous deformation and intricate stress/strain partitioning behavior. These distinctive phenomena are crucial in achieving the strength-ductility synergy of HS laminates. Nonetheless, further investigations are imperative to gain a comprehensive understanding of local stress and strain evolutions during deformation, and to thoroughly explore the influence of stress/strain partition on the mechanical behavior of the HS laminates. This study utilized the conventional mechanism-based strain gradient (CMSG) plasticity theory to conduct finite element simulations. The tensile deformation of HS laminate comprising nanostructured (NS) bronze and coarse-grained (CG) copper layers was simulated as a benchmark. The simulation results revealed the following: (1) Stress partition and stress transfer between the layers occur during deformation and are influenced by the layer thickness; (2) Both strain partition and strain banding exist in the HS laminates with relatively thinner layers, and the formation of dispersed strain bands is related to the plastic strain gradient; (3) Heterogeneous deformation promotes strain delocalization and strain hardening, consequently leading to improved uniform elongation. This study provides valuable insights into the significant effect of stress/strain partition on the strength-ductility synergy of HS laminates from the perspective of constitutive modeling.

Understanding extra strengthening in gradient nanotwinned Cu using crystal plasticity model considering dislocation types and strain gradient effect

Gradient nanotwinned (GNT) Cu, composed of various homogeneous nanotwinned (HNT) components, successfully achieves the synergistic enhancement of strength and ductility due to the combined advantages of gradient and nanotwinned structures. However, the complicated dislocation-twin boundary (TB) interactions in each HNT component and extra strengthening mechanism in GNT Cu remain elusive. In this study, a crystal plasticity model, which takes into account the reaction characteristics of various dislocation types at the TB and the extra size dependent geometrically necessary dislocations (GNDs) induced by plastic strain gradients, is developed for understanding both HNT and GNT Cu. The developed model successfully captures the mechanical behavior and microstructure evolution of HNT Cu with different TB orientations and grain sizes (and twin thicknesses). The simulation results emphasize the importance of incorporating dislocation types when describing dislocation-TB interactions. Furthermore, the intrinsic mechanism for the extra strengthening in the GNT Cu is revealed through the analysis of deformation contours and microstructural evolutions. The results show that the HNT components with the lower original strength have the higher extra back stress increment with the increase of structural gradient. This study provides valuable insights into predicting and further optimizing the strength of the GNT Cu through manipulating the gradient microstructure.