Strain Gradient Plasticity Theory Research Articles

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.

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.

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.

Toward the development of plasticity theories for application to small-scale metal structures.

Experiments are performed on micron-scale single-crystal prototypical structural elements experiencing combined torsion and bending to gather data on their load-carrying capacity in the range of size and strain relevant to micron-scale structures for which little data are available. The observed strengthening dependence on size for the structural elements is in general accord with trends inferred from prior tests such as indentation and pure torsion. In addition, the experiments systematically reveal the strengthening size-dependence of structural elements whose surface has been passivated by a very thin Cr coating, an effect shown to have substantial strengthening potential. A state-of-the-art strain gradient plasticity theory is used to analyze the structural elements over the entire range of size and loading. While the computed trends replicate the experimental trends with reasonable fidelity, the predictive exercise, which is representative of those that will be required in micron-scale structural analysis, brings to light constitutive and computational issues which will have to be addressed before micron-scale plasticity theory can serve as effectively at the micron scale as conventional plasticity does at larger scales.

Non-Quadratic Strain Gradient Plasticity Theory and Size Effects in Constrained Shear

Abstract A previously proposed strain gradient plasticity theory is extended to incorporate a non-quadratic power law function of the plastic strain gradient in the free energy expression with an exponent of N + 1. The values of N are taken to vary from N = 1 to N = 0. A simple shear problem of a metal layer between rigid boundaries is analyzed. Two stages of plastic deformation are considered. In stage I, the plastic strain is taken to be zero at the boundaries. Stage I ends when a specified magnitude of the plastic strain gradient is attained at the boundaries. In stage II, the magnitude of the plastic strain gradient at the boundaries is fixed at the specified value. With N = 0, a critical plastic strain gradient cannot be specified at the boundaries because the plastic strain gradient is infinite at the boundaries. The theory with N = 0 predicts a constant plateau stress immediately after initial yield, and the dependence of the plateau stress on the layer thickness can fit experimentally observed plateau stress values. However, with N = 0, a stress gap occurs between the initial yield stress and the plateau stress. The theory with 0 < N ≤ 1 and with stage II also can reproduce the experimentally observed dependence of the plateau stress on the layer thickness for any value of N in that range, with an appropriate value of critical plastic strain gradient at the boundaries. The solution for 0 < N ≤ 1 includes that for N = 0 as a limiting case.

The coupling effects of strain gradient and damage on Mode I crack tip stress fields

The cleavage fracture of metallic materials is determined by the crack tip stress field depending on plastic strain gradient and microstructural damage. The synergistic effects of strain gradient and damage on the stress fields near a Mode I crack tip in metallic materials are investigated using a modified conventional theory of mechanism-based strain gradient plasticity (CMSG). Besides preserving the essence of classical CMSG, a stress-dependent damage variable is introduced to characterize the effect of microstructural damage on the material's intrinsic length, elastic modulus and plastic yielding criterion. Based on the present theoretical model, it is found that the strain gradient and microstructural damage effects both are prominent in a small region near the Mode I crack tip, at which the damage evolution is determined by the strain gradient. Although the crack tip stress field with strain gradient and damage effects is lower than the existing ones influenced only by strain gradient, it is still significantly higher compared with the yielding strength of the material so that the cleavage fracture at the crack tip can be well explained. The damage-induced stress level reduction indicates that the stress concentration at the crack tip can be alleviated to inhibit the crack propagation. As a result, when the damage effect is considered, not only a larger external load is required to induce crack propagation but also the plastic region in front of the crack tip is enlarged, consequently leading to increases of fracture strength and energy dissipation in the material. The present research provides a more precise prediction of the crack tip stress distribution in metallic materials, which is helpful for better understanding the microscopic mechanism of the cleavage fracture phenomenon.

On elastic gaps in strain gradient plasticity: 3D discrete dislocation dynamics investigation

Although presenting attractive features in dealing with small-scale size effects, strain gradient plasticity (SGP) theories can lead to uncommon phenomena for some boundary value problems. Almost all non-incremental (Gurtin-type) SGP theories including thermodynamically-consistent higher-order dissipation predict elastic gaps under certain non-proportional loading conditions. An elastic gap is defined as a finite change in the equivalent yield stress after an infinitesimal change in the strain conditions, at the occurrence of the non-proportional loading source. The existence of such gaps in reality is largely questioned and represents a major source of uncertainty preventing the development of robust SGP theories for real small-scale applications. Using 3D discrete dislocation dynamics (3D-DDD), the present paper aims at investigating size effects within micron-scale single crystal structures under various non-proportional loading conditions, including tension–compression–passivation, bending–passivation and tension–bending. An in-depth investigation of the occurrence of elastic gaps under these conditions, which are known to entail such gaps when using classical non-incremental SGP theories, is conducted. The obtained 3D-DDD results reproduce well known experimentally confirmed size effects like Hall–Petch effect, Asaro’s type III kinematic hardening and reversible plasticity. However, no evidence of the phenomenon of elastic gaps is found, which constitutes a first indication that this phenomenon may not exist in reality. The simulations are performed on face-centered cubic (FCC) Nickel single grains with cuboid shapes ranging from 2μm to 15μm and different orientations.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

The particle/matrix interface of metal matrix composites (MMCs) can give rise to extra strength and then affect the local deformation behavior. This strengthening effect originates from the plastic strain gradients due to the incompatibility of plastic deformation between the particle and matrix. However, only limited researches utilized the strain gradient plasticity to study the damage evolution and fracture behavior, in which only one or two of the damage mechanisms (i.e., matrix damage, interface debonding, and particle fracture) was considered. In this work, all of these damage mechanisms were coupled into the finite element model under conventional theory of mechanism-based strain-gradient (CMSG) plasticity. Besides, a new numerical algorithm of geometrically necessary dislocations (GNDs) was proposed for a multi-scale model, and then this model was used to analyze the damage evolution and failure behavior of SiCp/AZ91 composite. The results show that the strengthening effects of plastic strain gradients can describe interface debonding, break the monotonicity of the effective plastic strain with the effective stress at the local area close to the particle/matrix interface, and give a more reasonable distribution of stress and plastic strain compared to the classical J2 flow theory. If the probability distribution of the interfacial strength is considered, the CMSG model has the potential capacity to capture the crack initiation in the matrix. When using the continuum damage mechanics approach to describe the fracture process based on the multi-scale model, the weakening exponent value should be considered.

Incremental strain gradient plasticity model and torsion simulation of copper micro-wires

Experimental observations show that the torsional deformation of copper micro-wires exhibits obvious sample and grain size effects. In this work, based on the framework of the cyclic plastic J2 flow rule, an incremental higher-order strain gradient constitutive model is established to describe the size effect observed in the torsional deformation of copper micro-wires. A novel kinematic hardening evolution rule is constructed in which the coupling effect of sample and grain sizes on the plastic hardening has been considered. In terms of numerical implementation, a finite element iterative algorithm which can independently solve the gradient field and plastic strain increment is proposed. Finally, the proposed model is implemented into the finite element software ABAQUS using a three-dimensional user-defined element and user-defined material subroutine. Simulated results show that the proposed model can capture the size-dependent torsional deformation of copper micro-wires since both the effects of sample and grain sizes are considered reasonably. This provides a good basis for the combination of strain gradient plasticity theory and cyclic plasticity constitutive model.

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.

Dynamic size effects across the scales: Localization in round notched bars

The strengthening and hardening of metals increase with diminishing size due to plastic strain gradient accompanied by Geometrically Necessary Dislocations (GNDs). In contrast, the inertia forces under severe dynamic loading conditions decrease. To shed the light on the interaction between the two mechanisms, the present work focuses on the localization in notched bars subject to high strain-rate tensile loading and first lays out a new computational model where the Fleck and Willis strain gradient plasticity theory is implemented into a dynamic finite element framework. The new framework allows accounting for strain gradient strengthening through the non-local plasticity material model and models the “smaller is stronger” mechanism (controlled by a built-in length parameter), while material inertia governs the “larger is more dynamic” effect. The work by Needleman (2018) [Effect of size on necking of dynamically loaded notched bars, Mech. Materials, 116:180-188] is taken as a starting point, and the present study first conveys the message that the scale-dependent dynamic effect observed in visco-plastic tensile bars also exists for rate-independent materials across scales (questioned by Needleman, 2018). Subsequently, the notched bar’s size and corresponding loading are scaled to the micrometer size range where plastic strain gradients influence the material response. Focusing attention on the shift between plastic localization in the notch and away from the notch (in the region subject to uni-axial tension) allows cementing this shift at the micron scale. The study shows that gradient strengthening owing the GNDs stabilizes the early localization in the notch and delays the shift between the localization mechanisms when the size of the notched bar diminishes while keeping the material length parameter constant. Moreover, severe gradient strengthening gives rise to a slow-growing neck-like (micron scale) localization mechanism where a neck-type deformation evolves near the notch — but not in the notch as oppose to larger bar sizes.

Elastic and nonlinear crack tip solutions comparison with respect to failure probability

This study represents a methodology to assess the probability of failure based on three the driving force formulations defined by the corresponding brittle and ductile fracture criteria for compact and bending specimens made of 34XH3MA and S55C steels. The elastic stress intensity factor (SIF) and two types of the non-linear plastic SIFs were considered as the driving force or generalized parameter (GP) to determine the probability of failure assuming a three-parameter Weibull distribution. The elastic SIF were experimentally obtained for studied materials and specimen geometries whereas the plastic SIFs were numerically calculated for the same material properties, specimen configurations and loading conditions according to classical J2 and strain gradient plasticity theories. Different specimen types with varying relative crack lengths and thicknesses were investigated. Proposed the normalized generalized parameter accounting for brittle or ductile fracture can be used as a suitable failure variable that is confirmed by comparison of the obtained failure cumulative distribution functions based on the three studied GPs.

Strain gradient plasticity modelling of cyclic loading in dispersion hardened materials

An analytical model, based on an isotropic strain gradient plasticity theory, describing work hardening during cyclic straining in a metal reinforced by a dispersion of non shearable particles is presented. The yield criterion is expressed in terms of isotropic and kinematic hardening contributions and the model is validated against full field finite element (FE) solutions on a 2D axi-symmetric unit cell model. Excellent agreement between analytical and FE results is obtained. The theory presented includes mixed energetic/dissipative contributions from higher order stresses in both bulk and at particle/matrix interfaces. In particular, the influence of a quadratic interface free energy that transitions into a linear form at some threshold value of plastic strain is investigated. It is shown that such an energy is capable of capturing the experimentally observed phenomenon of inflections in the reverse stress–strain curve. It is argued, based on the well known phenomenon where particles are shielded by Orowan dislocation loops during reverse strain, that an energetic interface contribution could be physically relevant for low plastic strains.

Investigation of the mechanical response of a micro-textured surface using a higher order strain gradient plasticity theory

Investigation of the mechanical response of a micro-textured surface using a higher order strain gradient plasticity theory

Insights into hardening, plastically deformed zone and geometrically necessary dislocations of two ion-irradiated FeCrAl(Zr)-ODS ferritic steels: A combined experimental and simulation study

As one of promising candidate materials for fuel claddings and structural components in the Gen-IV fission reactors, FeCrAl(Zr)-ODS ferritic steels were studied to well understand the radiation hardening behavior. Nanoindentation (NI) hardness and plastically deformed zone (PDZ), geometrically necessary dislocations (GNDs) as well as microstructures were investigated for two FeCrAl(Zr)-ODS ferritic steels irradiated with 6.4 MeV Fe3+ at room temperature (RT) up to the nominal damages of 2, 10 and 50 dpa. Irradiation-induced hardening, which was estimated by using the Nix-Gao model regardless of the damage gradient effect (DGE), increased continuously with increasing the nominal displacement damage. When taking the DGE into account, the dependence of the irradiation hardening (in MPa) on a local damage level (dpa) obtained by finite element method (FEM) simulations was 153.65 × (dpa)0.26 and 158.35 × (dpa)0.25 for the non-Zr steel and the Zr-added one, respectively. The hardening caused by ion-irradiation was discussed in terms of the loss of oxide particles, the formation of dislocation loops and the solid solution hardening mainly by dissolved oxygen. The addition of Zr reduced the ion-irradiation hardening of steel to some extent mainly by suppressing the formation of dislocation loops. Meantime, as the FEM simulations revealed, the PDZ size underneath a conical indenter at the equivalent plastic strain εeq > 1.9% can reasonably estimate the density of GNDs. According to the strain gradient plasticity (SGP) theory, the densities of GNDs at an indentation depth of 200 – 300 nm were slightly higher in the case of the 50 dpa/nominal than the unirradiated, in spite of similarity for the two FeCrAl(Zr)-ODS steels.

Inversion of dislocation densities under mixed mode fracture

Mixed-mode crack growth is here investigated through experiments and computations for 34X and P2M steels, 7050 aluminium, and Ti-6Al-4V alloys in a compact tension shear (CTS) specimen. In this study, we use the mechanism-based strain gradient (MSG) plasticity theory to evaluate both crack tip dislocation density behaviour and the coupled effect of the material plastic properties and the intrinsic material length on local stress distributions. The constitutive relations are based on Taylor’s dislocation model, which allows to gain insights into the role of the increased dislocation density associated with large gradients in plastic strain near cracks. The material model is implemented in a commercial finite element (FE) software package using a user subroutine, and the nonlinear stress intensity factors (SIFs) are evaluated as a function of the intrinsic material length. As a result of the FE calculations of dislocation density distributions, the effects of both the fracture mode and the stress–strain state are determined. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. Dislocation densities, stress fields and nonlinear SIF solutions are determined for experimental curvilinear crack paths by taking into account the transition from the initial Mode II crack to the mixed-mode fracture.

Plastic buckling of columns at the micron scale

Clamped-clamped compressed wide metal columns with length to thickness ratios such that they undergo plastic buckling are considered. Further, the thickness of the columns is assumed to be in the range of microns to tens of microns, typical of elements comprising some of the small-scale lattice materials currently being produced. The strengthening effects associated with plastic strain gradients are expected to influence buckling behavior, and the columns are analyzed using several versions of the available strain gradient plasticity theories. The columns are assumed to be infinitely wide and subject to plane strain deformations. Approximate one-dimensional and exact two-dimensional analyses are presented. The primary focus is the onset of plastic buckling as predicted by bifurcation from the state of uniform compression. However, a numerical post-buckling study is carried out for one class of strain gradient theories for columns with initial imperfections to ascertain if the buckling predictions stemming from this class of theory are realistic. The paper highlights the fact that one class of strain gradient theories does not appear to be adequate for analyzing plastic buckling.

Continuum plasticity modelling of work hardening for precipitation-hardened martensitic steel guided by atom probe tomography

An analytical flow stress model, based on isotropic strain gradient plasticity theory, for precipitation hardened materials, is proposed and evaluated against tensile data on a 15 wt% Cr - 5 wt% Ni (15-5) PH stainless steel. The 15-5 PH material was aged at 500 °C for 1 h, 2 h, 5 h and 50 h to obtain a wide range of precipitate sizes. Detailed characterisation of precipitates was obtained using atom probe tomography (APT). A second material, a 15-5 stainless steel without added Cu was heat treated to obtain a similar matrix microstructure as in the 15-5 PH, but without Cu precipitates. Tensile testing revealed that the heat treated 15-5 PH material covered the full range from under- to overaged conditions. The analytical model, which accounts for stress reducing effects of plastic relaxation around particles, manages to capture the experimental data in a very satisfying manner using only a total of three tunable parameters. It is believed that the proposed model can offer an alternative to the much more commonly used work hardening models based on the internal variable approach.

Loading rate and holding load dependent room temperature nanoindentation creep behavior of 60NiTi alloy: Individual and coupling effects

60NiTi alloy is highly promising for aerospace bearings and mechanisms. Shock load that is inevitably generated during launching process brings about an quick loading process and transient load duration, and may further results in an unnoticed but detrimental creep deformation. On this account, the nanoindentation creep behavior of solution-treated 60NiTi alloy at room temperature and the corresponding influencing mechanism were studied. The results show that creep displacement was dominated by strain rate at the very beginning of holding stage. Strain rate and flow stress at the very beginning of holding stage were increased with an increased loading rate but declined with increasing holding load or with simultaneously increasing loading rate and holding load. The underlying mechanisms were provided as nucleation and absorption of geometrically necessary dislocations (GNDs) associated with strain gradient plasticity theory rather than the model based on the consumption of creep deformation at loading stage.