Strain Gradient Plasticity Theory Research Articles

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.

Micro-cutting force model of micro-jet induced by cavitation collapse in the ultrasonic field at micro-nano scale

The micro-jet is the main mechanism of cavitation on materials. More and more ultrasonic cavitation assisted machining uses the micro-jet effect in the industrial field. In this paper, the micro-jet impact is considered the micro-cutting process. To deeply analyze the micro-cutting mechanism of micro-jet, the size effect of materials was considered, and the micro-cutting force model of micro-jet was established and solved based on the spherical indentation test theory and the strain gradient plasticity theory. The results showed that the micro-cutting force and impact pressure of micro-jet with size effect are larger than those without size effect and the micro-cutting force of single micro-jet is within 2.35 N. The micro-cutting force of micro-jet increases exponentially with the increase of pit radius, but only slightly increases with the increase of pit depth. When the size effect is considered or not, the impact pressure of micro-jet is 1616–2922 MPa or 1615–1980 MPa, respectively. The increased ratio of micro-cutting force with size effect is \(\left(\sqrt{1+\frac{2{\overline{m} }^{2}{a}^{2}{G}^{2}bh}{\left({a}^{2}+{h}^{2}\right){\sigma }_{\mathrm{JC}}^{2}}-1}\right)\times 100\%\), which can directly reflect the strength of size effect. The size effect is more obvious when the pit size is smaller, and the maximum increased ratio is 47.54%. The increased ratio increases nonlinearly with the decrease of pit radius and the increase of pit depth. It has a strong correlation with pit radius and pit depth. This paper can provide a theoretical viewpoint and support for the quantitative description of cavitation effect in ultrasonic-assisted machining.

Mixed mode crack paths in terms of plastic stress intensity factors based on conventional and strain gradient plasticity

The effect of an initial mode II loading on subsequent mixed-mode crack path was investigated through experiments and computations for 34X and P2M steels, 7050 aluminum, and Ti-6Al-4V alloys in a compact tension shear (CTS) specimen. An optical microscope and the drop potential method were used to determine the actual position of the crack tip on the curvilinear crack path monitoring and continuous measurements of the crack size were made along the path in the CTS specimen. In accordance with these experimental observations, the main feature of the initial mode II fracture is the crack growth on a curvilinear path. For subsequent mixed-mode crack propagation, the crack front continuously changes the shape and direction with each loading cycle. For a mode II loaded initial crack, the scenario becomes more complex because of the change in the propagation direction when a kinked crack is formed. The experimental data on crack growth for all tested materials are represented in terms of the elastic and new plastic stress intensity factors (SIFs). To this end, both the classical and the conventional mechanism-based strain-gradient plasticity (CMSGP) theories were employed. The material constitutive equation is implemented in a finite element code and the elastic and plastic fracture resistance parameters are calculated as a function of the position along the curvilinear crack path in all tested CTS specimens produced from 34X and P2M steels, 7050 aluminum and Ti-6Al-4V alloys. As a result of the finite element calculations, nonlinear amplitude factor solutions are determined across a wide range of material work hardening conditions, and the role of the Taylor’s intrinsic material length and both the coupled effect of these parameters are established. The potential of fundamental and practical applications of introduced fracture resistance parameters are discussed.

ひずみ勾配塑性論に基づく弾粘塑性動的有限要素解析

One mechanical property of metallic materials that has yet to be clarified is the size dependence of material strength on the micrometer scale. Conventional elasto-viscoplasticity theories do not represent any size dependent behavior predicted by dislocation theory-based gradient crystal plasticity models. This study presents a formulation of dynamic finite element method that incorporates strain gradient plasticity theory, which can take into account the size dependence of mechanical properties, and shows several computational results. We analyze dynamic tensile deformation of a plane strain block with geometrical initial imperfections. It is confirmed that the size dependent property from the strain gradient effect manifests in plastic flow localization behavior, but does not affect the position where diffuse neck occurs.

Indentation size effects in aluminum and titanium alloys

This paper presents the results of study on indentation size effects in structural metals/alloys (6061 aluminum alloy, titanium and Ti–6Al–4V) at the micron scale. The size dependence of indentation hardness is explained using strain gradient plasticity theories introduced by Nix and Gao (J. Mech. Phys. Solids., 46 (1998), 411–425). Resulting material length scales are also compared with the underlying microstructures and previously reported values in the literature. The impacts of strain hardening models from stress-strain curves on the predicted length scales are explored. Implications of the results are discussed for the modeling of plasticity and hardness at the micron-scale.

Crack root radius effect on stress fields under nonlinear

The influence of crack root radius and plastic material properties on the crack tip fields was evaluated by strain gradient plasticity theory, and classical plasticity theory. A conventional mechanism-based strain gradient (CMSG) plasticity theory was used in the study. The stress fields in single edge tension specimen are investigated numerically for a wide range of the crack tip radius variations. Five values of the crack tip curvature radius ρ = 0 (mathematical notch); ρ = 25 nm, ρ = 30 nm, ρ = 60 nm and ρ = 100 nm were investigated. The strain hardening exponent varied from N = 0.075 to N = 0.4. A stress analysis by CMSG plasticity was performed for two values of the Taylor’s length parameter l = 1μm and l = 10μm. The boundaries of the local areas, where the influence of the crack tip radius is realized were determined for both HRR and CMSG plasticity. The sizes of the dominance area of the strain gradient plasticity were established and their approximation equations were presented. The influence of the plastic properties and the Taylor’s length scale parameter on the stress fields in the vicinity of rhe crack tip was estimated for the strain gradient plasticity.

Analysis of stress fields at the crack tip and fracture resistance parameters under conditions of gradient plasticity

This paper presents a numerical analysis based on previously obtained experimental data on crack growth for P2M and 34X steels, aluminum 7050 and titanium alloy Ti-6Al-4V compact samples (CTS) with a one-sided lateral incision, within the framework of the conventional theory of strain gradient plasticity (CMSG), based on the Taylor dislocation model. In this study, the initial points of the curved crack trajectory were considered for two classical states: plane strain and plane stress with normal separation and pure shear. The constitutional equations of material behavior for CMSGP theories were introduced into the finite element computational complex and new fields of stress-strain state parameters for the conditions of strain gradient plasticity were obtained. The results of FE calculations show a significant increase in the magnitude of the true stress fields at the crack tip, taking into account the plastic deformation gradients and the internal characteristic length of the material. The numerical results also show that the singularity in the crack tip region is different for the model of gradient plasticity of deformations and depends on the mode I/II. An important conclusion regarding the numerical results regarding the parameters of the material's fracture resistance is that the new plastic stress intensity coefficients for gradient plasticity differ for plane strain and plane stress, and also show significant sensitivity to the plastic properties of the material and to the scale parameter of the intrinsic material length, which is attractive from the point of view of practical application and further fundamental research.

Strain gradient elasto-plasticity model: 3D isogeometric implementation and applications to cellular structures

In the present work, we combine Mindlin’s strain gradient elasticity theory and Gudmundson–Gurtin–Anand strain gradient plasticity theory to form a unified framework. The gradient plasticity model is enriched by including the gradient of elastic strains into the expression of the internal virtual work and free energy. This augments the modelling capabilities by incorporating elasticity-related length scales along with plasticity-related energetic and dissipative ones. The strong form governing equations are derived via the principle of virtual work addressing a complete set of boundary conditions. The fourth-order boundary value problem of the gradient elasto-plasticity model is then formulated in a variational form within an H2 Sobolev space setting. Conforming Galerkin discretizations for numerical results are obtained utilizing an isogeometric approach with NURBS basis functions of degree p≥2 providing Cp−1-continuity. The implementation follows a viscoplastic constitutive framework and adopts the backward Euler time integration scheme. A set of benchmark examples is considered to illustrate convergence properties and to accomplish parameter studies. It is shown that the elastic length scale parameter controls the slope of the elastic part and causes an additional hardening in the plastic part of the material response curves. Finally, an illustrative example is considered in order to demonstrate the applicability of both the continuum model and the numerical method in capturing the size-dependent torsion response of cellular structures.

Rate-dependent indentation size effect on hardness and creep behavior of a titanium metallization film on alumina substrate

Nanoindentation tests were performed at room temperature to investigate the indentation size effect (ISE) and creep behavior of a titanium metallization film on alumina substrate. The ISE of hardness was examined at different loading strain rate (LSR). The effects of LSR and penetrating depth on nanoindentation creep were evaluated at the steady creep state. Inverse ISE in titanium film was observed at indentation depth below 200 nm while normal ISE was observed at penetrating depth ranging from 200 nm to 1000 nm. Elastic contact theory and strain gradient plasticity theory were applied to interpret the variation trends of hardness respectively. In addition, the intrinsic hardness and characteristic length were found to be strongly dependent on LSR. Except for creep strain rate, creep displacement and creep stress exponent showed relatively conspicuous correlations to LSR and/or penetrating depths.

Mode I and mode II stress intensity factors and dislocation density behaviour in strain gradient plasticity

In this study, we use the mechanism-based strain gradient 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 non-linear amplitude factors. The two planar classical stress–strain states are examined, namely, plane strain and plane stress, both under pure mode I and pure mode II loading conditions. The constitutive relations are based on Taylor’s dislocation model, which enables gaining 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 (SIF) are evaluated as a function of the intrinsic material length, characterising the scale at which gradient effects become significant. 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. In pure mode I, the geometrically necessary dislocation (GND) density is located symmetrically with respect to the blunted crack tip. On the contrary, under pure mode II, the GND density becomes concentrated in the blunted and sharp parts of the crack tip. In this case, fracture initiation is shown to be likely to occur near the blunted region of the crack tip, where both the stress triaxiality and the GND density are at their maximum. The relation between the equilibrium state of dislocation densities and the intrinsic material length as well as the plastic SIF as a function of the work hardening exponent is discussed.

Aggravated stress fluctuation and mechanical size effects of nanoscale lamellar bone pillars

The size effects of mechanical properties influence the microdeformation behaviors and failure mechanisms of hierarchical lamellar bones. Investigations of the continuous deformation behaviors and structure–behavior–property relationships of nanoscale lamellar bones provide essential data for reducing the risk of fracture. Here, five pillars with diameters ranging from 640 to 4971 nm inside a single lamella were fabricated. In situ pillar compressive tests inside a scanning electron microscope directly revealed the diameter-dependent enhanced strength, ductility, and stress fluctuation amplitude. Real-time observations also revealed the segmented deformation and morphological anisotropy of pillars with smaller diameters and the slight elastic recovery of pillars with larger diameters. The critical diameter leading to the brittle-to-ductile transition was confirmed. The “analogous to serrated flow” stress fluctuation behaviors at the nanoscale exhibited a significant size effect, with coincident fluctuation cycles independent of diameter, and each cycle of the fluctuation manifested as a slow stress increase and a rapid stress release. The discontinuous fracture of collagen fibrils, embedded enhancement of hydroxyapatite crystals, and layered dislocation movement on the basis of strain gradient plasticity theory were expected to induce cyclical stress fluctuations with different amplitudes.

Analytical predictions of yield stress of a strain gradient plasticity material reinforced by small elastic particles

Theories describing the important role of small particles for strengthening of metals have evolved since the pioneering work of Orowan. Here, this problem is analysed by a strain gradient plasticity (SGP) theory. The structure of the governing equations on non-dimensional form reveals that the plastic strain in the matrix material is to zeroth order approximation constant for a sufficiently small particle size a in comparison to material length scale ℓ. Based on this observation, a perturbation solution has been developed by expansions of all field variables in terms of a/ℓ and the volume fraction of particles f. The simple structure of the plastic strain field is also exploited to derive an upper bound solution from the principles of virtual work and maximum plastic dissipation. These analytical solutions are then used to derive expressions for the yield stress taking into account a random distribution of particles of various size and shape with elastic constants that differ from the matrix. The accuracy and range of validity of these solutions are demonstrated by comprehensive 2D and 3D finite element analyses of material volumes containing realistic distributions of particles of spherical and spheroidal shape of various elastic modulus. The results show that significant strengthening will arise provided that the representative particle size is smaller than the material length scale ℓ of the SGP material.

Fracture predictions based on a coupled chemo-mechanical model with strain gradient plasticity theory for film electrodes of Li-ion batteries

High-capacity electrodes in Li-ion batteries inevitably undergo a large volume deformation originating from high diffusion-induced stresses during charging and discharging processes. In this paper, we firstly develop a new elastoplastic model for describing diffusion-induced deformation in the framework of high-density dislocation defects generated due to the migration of Li atoms. Then, we analyze the film size effect, diffusion-induced stress, plastic yielding, and hardening of electrode materials based on the evolutions of Li concentration by a strategy combining the strain gradient plasticity theory and finite element simulations. Finally, according to the traction-separation law, interface damage and debonding are characterized in the active film materials (with a thickness of 150, 200, and 250 nm, respectively) on a rigid substrate.

Application of stress gradient plasticity model in different passivated problems

In this study, the theory of low-order stress-gradient plasticity deformation is considered to model elastic and plastic size-dependent deformation. In classical plasticity models, the physical length scale is not considered to control the size effects. Strain gradient plasticity models include one or more length scales that control size effects. Stress gradient plasticity model is introduced with a specific physical length scale and does not include any additional parameters. Torsion of a hollow tube, pure shear of a slab and torsion of a rod with and without passivation is modeled by considering elasto-plastic hardening model. The effects of passivation on materials deformation are studied. By increasing the material length scale and keeping the other parameters constant, the average equivalent plastic strain is decreased and the average stress is increased, as well as the hardening rate of the stress-strain curves. The obtained results are compared with the experimental results of Liu et al. (2013) and the results obtained from the theory of strain gradient plasticity for different problems (Niordson and Hutchinson, 2003; Lubarda, 2016; Hua and Liu, 2020). The results of the Stress gradient plasticity model is also compared with the strain gradient plasticity models.

Research on Deformation Mechanism of Cutting Nickel-Based Superalloy Inconel718 Based on Strain Gradient Theory

Abstract The traditional material constitutive model can effectively simulate the mechanical properties during the cutting process. However, the scale characteristics contained in materials are not considered in the traditional cutting model, and the inherent scale effect of materials is also ignored. Therefore, the traditional cutting constitutive model cannot effectively reflect the size effect in the cutting process, and then cannot obtain the accurate stress, strain, and temperature. In this present paper, a material constitutive model which can reflect the scale effect is established based on the strain gradient plasticity theory. Through the established model and secondary development of abaqus, the two-dimensional dynamic finite element simulation model of cutting Inconel 718 is established. By comparing the cutting experiment results with the simulation results, the established simulation model can more accurately reflect the effects of temperature, strain gradient effect, equivalent stress, and its scale effect on cutting deformation during the machining process.

Damage and fracture with strain gradient plasticity for high-capacity electrodes of Li-ion batteries

In consideration of the high-density dislocations from the lithiation process of high-capacity electrodes in Li-ion batteries, in this paper, a new elastoplastic model is established to describe the diffusion-induced deformation and damage fracture. With the help of the relative physical quantities and state of charge, the surface damage and fracture behaviors of electrode materials are discussed based on the elastic-perfectly plastic (PP) and the strain gradient plasticity (SGP) theories, respectively. The results show that the lithiation deformation could be alleviated by reducing the electrode scale, and the plastic flow can play an essential role in the extrusion ratcheting effect relating to the upper surface fracture. Furthermore, the interface damage is more likely to appear by increasing the initial bond stiffness at the upper surface, which has little effect on the later fracture. A strong size effect is also found in the damage and fracture critical curves for the PP and SGP models.