Mechanism-based Strain Gradient Plasticity Theory Research Articles

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.

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.

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.

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.

Three-Dimensional Elastoplastic Contact Analysis of Rough Surface Considering a Micro-Scale Effect

Abstract The micro-surface asperity scale of grinding metal parts is within several microns. When two grinding surfaces are in contact, the unevenness of the plastic deformation of the asperities at the micro-scale leads to greater plastic hardening strength of the material. The results of the nano-indentation experiment conducted in this paper confirmed this phenomenon. Based on conventional mechanism-based strain gradient (CMSG) plasticity theory, the micro-scale plastic constitutive equation of materials is given and then is verified by the nano-indentation experiment. Finite element software abaqus and the user-defined element (UEL) subroutine are used to build three-dimensional rough surface elastoplastic contact models. By calculating the grinding rough surface contact in the macro-scale constitutive model based on J2 theory and in the CMSG plasticity constitutive model, the influence law of plastic micro-scale effect on contact performance is obtained.

Effect of crack curvature radius on the stress fields under nonlinear deformation

The influence of crack curvature radius and plastic properties of material on the crack tip fields is evaluated by conventional mechanism-based strain gradient (CMSG) plasticity theory, and classical plasticity theory. Stress fields in single edge tension specimen are investigated numerically for a wide range of the crack curvature radius variations. The strain hardening exponent varied from N=0.075 to N=0.4. The local areas of the influence of the crack tip curvature radius are established for the strain gradient plasticity and classical plasticity. The couple effects of material length scale parameter and plastic properties of material on crack tip fields are established for strain gradient plasticity. The boundaries of CMSG plasticity dominated zone are determined.

Crack tip fields and fracture resistance parameters based on strain gradient plasticity

The crack tip mechanics of strain gradient plasticity solids is investigated analytically and numerically. A first-order mechanism-based strain gradient (MSG) plasticity theory based on Taylor’s dislocation model is adopted and implemented in the commercial finite element package ANSYS by means of a user subroutine. Two boundary value problems are considered, a single edge tension specimen and a biaxially loaded plate. First, crack tip fields are characterized. Strain gradient effects associated with dislocation hardening mechanisms elevate crack tip stresses relative to conventional plasticity. A parametric study is conducted and differences with conventional plasticity predictions are quantified. Moreover, the asymptotic nature of the crack tip solution is investigated. The numerical results reveal that the singularity order predicted by the first-order MSG theory is equal or higher to that of linear elastic solids. Also, the crack tip field appears not to have a separable solution. Moreover, contrarily to what has been shown in the higher order version of MSG plasticity, the singularity order exhibits sensitivity to the plastic material properties. Secondly, analytical and numerical approaches are employed to formulate novel amplitude factors for strain gradient plasticity. A generalized J-integral is derived and used to characterize a nonlinear amplitude factor. A closed-form equation for the analytical stress intensity factor is obtained. Amplitude factors are also derived by decomposing the numerical solution for the crack tip stress field. Nonlinear amplitude factor solutions are determined across a wide range of values for the material length scale l and the strain hardening exponent N. The domains of strain gradient relevance are identified, setting the basis for the application of first-order MSG plasticity for fracture and damage assessment.

Establishment of damage estimation rules for brittle fracture after cyclic plastic prestrain in steel

The microprocess of brittle fracture in steel is suggested to occur because microcracks appearing in the brittle layer at the grain boundary propagate driven by increased piled-up dislocation. Thus, loading plastic deformation, equivalent to the increase in dislocation, is directly connected to the increase in brittle fracture risk. Because of these mechanisms, prestraining means approaching the limits of the material, which could lead to deterioration of fracture toughness. For a lifetime evaluation of steel structure, it is important to generalize the effect of various cyclic prestrains.In this paper, material toughness changes when tensile and compression prestrain occur were investigated. A general three-point bending test was employed to evaluate fracture. Synthetic analysis using Conventional Mechanism-based Strain Gradient Plasticity (CMSGP) theory, in which dislocation density is used to determine the amount of material damage, was carried out as well as employing conventional macroscopic material damage consideration rules. The consistency of the simulation and the experimental results were supported by the Electron Back Scattered Diffraction Pattern (EBSD) observation. Additionally, critical stress was calculated, and the change in critical stress for prestrain conditions was investigated. Then, the mechanism of critical stress changes could be expressed by the yield point increase and dislocation density.

Gradient-enhanced statistical analysis of cleavage fracture

We present a probabilistic framework for brittle fracture that builds upon Weibull statistics and strain gradient plasticity. The constitutive response is given by the mechanism-based strain gradient plasticity theory, aiming to accurately characterize crack tip stresses by accounting for the role of plastic strain gradients in elevating local strengthening ahead of cracks. It is shown that gradients of plastic strain elevate the Weibull stress and the probability of failure for a given choice of the threshold stress and the Weibull parameters. The statistical framework presented is used to estimate failure probabilities across temperatures in ferritic steels. The framework has the capability to estimate the three statistical parameters present in the Weibull-type model without any prior assumptions. The calibration against experimental data shows important differences in the values obtained for strain gradient plasticity and conventional J2 plasticity. Moreover, local probability maps show that potential damage initiation sites are much closer to the crack tip in the case of gradient-enhanced plasticity. Finally, the fracture response across the ductile-to-brittle regime is investigated by computing the cleavage resistance curves with increasing temperature. Gradient plasticity predictions appear to show a better agreement with the experiments.

A new model for permanent flexural deflection of cantilever MEMS actuator by conventional mechanism-based strain gradient plasticity framework

In two past decades, there are so many works representing to elastic behaviors of cantilever MEMS actuators in the micro and nanoscales for various loading conditions and geometries. However, there are little efforts available for the permanent plastic behavior of the MEMS structures and actuators. In addition, the most size—dependent plasticity considerations based on gradient models have been limited to primary concepts on material modeling or microstructural evolution as compared with structural analyses. In this paper, a conventional mechanism-based strain gradient (CMSG) plasticity theory is applied to investigate permanent behavior of the cantilever micro-beam actuators, by determining the effect of length scale on flexural displacement. If the beam is scaled to micron size, mechanical behavior follows from material dimension or length scale. While the model consisted of a multiple plastic work hardening, its kinematics is established on the Euler–Bernoulli hypothesis. The deflection of cantilever MEMS actuator is determined for different cases of the loading and length scales, compared with other relevant theoretical and experimental observations. Also, the effect of the elastic medium of the environmental foundation is considered for the actuator, dedicatedly.

Size-dependent plastic buckling behavior of micro-beam structures by using conventional mechanism-based strain gradient plasticity

Since the actuators with small- scale structures may be exposed to external reciprocal actions lead to create undesirable loads causing instability, the buckling behaviors of them are interested to make reliable or accurate actions. Therefore, the purpose of this paper is to analyze plastic buckling behavior of the micro beam structures by adopting a Conventional Mechanism-based Strain Gradient plasticity (CMSG) theory. The effect of length scale on critical force is considered for three types of boundary conditions, i.e. the simply supported, cantilever and clamped - simply supported micro beams. For each case, the stability equations of the buckling are calculated to obtain related critical forces. The constitutive equation involves work hardening phenomenon through defining an index of multiple plastic hardening exponent. In addition, the Euler-Bernoulli hypothesis is used for kinematic of deflection. Corresponding to each length scale and index of the plastic work hardening, the critical forces are determined to compare them together.

Size effects in elastic-plastic functionally graded materials

We develop a strain gradient plasticity formulation for composite materials with spatially varying volume fractions to characterize size effects in functionally graded materials (FGMs). The model is grounded on the mechanism-based strain gradient plasticity theory and effective properties are determined by means of a linear homogenization scheme. Several paradigmatic boundary value problems are numerically investigated to gain insight into the strengthening effects associated with plastic strain gradients and geometrically necessary dislocations (GNDs). The analysis of bending in micro-size functionally graded foils shows a notably stiffer response with diminishing thickness. Micro-hardness measurements from indentation reveal a significant increase with decreasing indenter size. And large dislocation densities in the vicinity of a crack substantially elevate stresses in cracked FGM components. We comprehensively assess the influence of the length scale parameter and material gradation profile to accurately characterize the micro-scale response and identify regimes of GNDs relevance in FGMs.

Non-local plasticity effects on notch fracture mechanics

We investigate the influence of gradient-enhanced dislocation hardening on the mechanics of notch-induced failure. The role of geometrically necessary dislocations (GNDs) in enhancing cracking is assessed by means of a mechanism-based strain gradient plasticity theory. Both stationary and propagating cracks from notch-like defects are investigated through the finite element method. A cohesive zone formulation incorporating monotonic and cyclic damage contributions is employed to address both loading conditions. Computations are performed for a very wide range of length scale parameters and numerous geometries are addressed, covering the main types of notches. Results reveal a strong influence of the plastic strain gradients in all the scenarios considered. Transitional combinations of notch angle, radius and length scale parameter are identified that establish the regimes of GNDs-relevance, laying the foundations for the rational application of gradient plasticity models in damage assessment of notched components.

Application of strain gradient plasticity in coining simulations of commemorative coins

Size effect plays an important role in coining processes of commemorative coins. To deal with the size effect and improve the simulation accuracy, a constitutive model incorporating the conventional mechanism-based strain gradient (CMSG) plasticity theory was established in this study, with the nonlinear coupling between the statistically stored dislocation (SSD) density and the geometrically necessary dislocation (GND) density determined experimentally. Via the least square method, a bilinear equivalent plastic strain field was constructed for the evaluation of equivalent plastic strain gradient. In this way, the proposed constitutive model was implemented within a home-developed explicit finite element code for coining analyses. Uniaxial tensile tests and a nanoindentation test were conducted to obtain the mechanical properties of the Ag999 material, which was selected to verify the performance of the proposed constitutive model. By simulating the nanoindentation test and the coining process of a typical commemorative coin, it was demonstrated that the proposed constitutive model could provide accurate results in applications involving size effect, outperforming the classical plasticity constitutive model. Therefore, the proposed constitutive model based on strain gradient plasticity theory is suitable for coining simulations of commemorative coins.

Simulation of ballistic performance of a two-layered structure of nanostructured metal and ceramic

Two-layered structures, especially the metal/ceramic structure, have proved to have far better ballistic performance than a single-layered one. In this paper, a theoretical framework and numerical scheme, based on the combination of a mechanism-based strain gradient plasticity theory, micromechanics of composite model, the Johnson–Cook failure model, and the extended Drucker–Prager plasticity model, are set up to investigate a two-layered structure of nanostructured (NS) metal and ceramic. This study focuses on the different roles the two layers play during the protection process and clarifies the intrinsic significance of the ceramic layer and the metal layer according to different criteria of structural design. It is found that, in cases of targets with the same total thickness, the function of wearing and slowing down the projectile is mainly realized by the NS metal layer, while in cases of targets with a constant areal density, the function of wearing the projectile is mainly realized by the ceramic layer. Our simulations also show that, under the same impact velocity, the resistance of the target composed of the NS metal and ceramic laminates is remarkably stronger than that composed of coarse-grained metal and ceramic laminates.

Strain gradients in Cu–Fe thin films and multilayers during micropillar compression

Plastic strain gradients can influence the work-hardening behaviour of metals due to the accumulation of geometrically necessary discolations at the micron/submicron scale. A finite element model based on the conventional theory of mechanism-based strain-gradient plasticity has been developed to simulate the micropillar compression of Cu–Fe thin films and multilayers. The modelling results show that the geometric constraints lead to inhomogeneous deformation in the Cu layers, which agrees well with the bulging of Cu layers observed experimentally. Plastic strain gradients develop inside the individual layers, leading to extra work-hardening due to the accumulation of geometrically necessary dislocations. In the multilayer specimens, the Cu layers deform more severely than the Fe layers, resulting in the development of tensile stresses in the Fe layers. It is proposed that these tensile stresses are responsible for the development of micro-cracks in the Fe layers.

Numerical simulation of ballistic performance of bimodal nanostructured metals

Nanostructured (NS) metals with bimodal grain size distribution, composed of coarse grained (CG) and nanograined (NG)/ultrafine-grained (UFG) regions, not only have high strength and good ductility, but they can also provide excellent ballistic performance. In this work we use numerical simulations to study the influence of microstructural attributes such as the distribution and shape of the CG inclusions on the ballistic performance of bimodal NS Cu. Our formulation is based on the mechanism-based strain gradient plasticity theory and the Johnson–Cook failure model. We also consider the effects of constitutive and failure parameters of the NG phase as well as boundary constraint of the specimen. Our results on the effect of inclusion attributes indicate that microstructures can significantly affect the velocity history of the bullet as well as limit velocity and maximum displacement of the specimen. The analysis also suggests that, to improve the ballistic performance of the bimodal NS metal, the CG inclusions need to have regular distribution and, under the condition of same distribution, they also need to have a longer projection perpendicular to the direction of impact. The simulations show that when the abrasion effect is not considered, the ballistic performance of the microstructures depend heavily on their ductility rather than their strength. The results on the boundary constraint show that when this constraint is released, the ballistic performance is reduced tremendously. It is believed that these results could provide useful insights into the development of advanced NS materials for ballistic protection.

Local Strain Energy Density Applied to Bainitic Functionally Graded Steels Plates Under Mixed-Mode (I + II) Loading

In this paper, the averaged value of the strain energy density (SED) over a control volume is used to predict the critical load of V-notched specimens made of functionally graded steels (FGSs) under mixed-mode loading. The studied FGSs contain ferritic and austenite phases in addition to bainitic layer produced by electroslag remelting. The mechanism-based strain gradient plasticity theory is used to determine the flow stress (yield stress or ultimate stress) of each layer. The Young’s modulus and the Poisson’s ratio have been assumed to be constant, while other mechanical properties vary exponentially along the specimen width. The control volume is centered in relation to the maximum principal stress present on the notch edge and assumes a crescent shape. The points belonging to the volume perimeter are obtained numerically. In the present contribution, the effects of notch radius and notch depth on the SED and the critical load are studied. The notch radius varies from 0.2 to 2.0 mm, and the notch depth varies from 5 to 7 mm. By using the SED approach and finite element simulations, the critical load is determined, and the obtained results show a sound agreement with the experimental results.

Computational micromechanics analysis of toughening mechanisms of particle-reinforced bulk metallic glass composites

Based on the interaction between the dislocation-sliding and shear-banding, numerical simulations were performed to elucidate the toughening mechanisms of bulk metallic glass (BMG) composites under tension. During the numerical modeling, the free volume theory was applied for the BMG matrix, and free volume density acts as an internal state variable to describe the evolution of shear banding. While the mechanism-based strain gradient plasticity theory was adopted to describe the crystalline second phase, and dislocation density was employed to track the dislocation sliding. With the aid of the user material subroutines (UMAT) in the ABAQUS code, these theories were implemented for predicting the tensile deformation of BMG composites. Numerical results showed that the proposed model could successfully describe the enhanced ductility of BMGs by introducing varied sized of crystalline particles. The predictions were in good agreement with the experimental results. Additionally, the toughing micro-mechanism was clearly clarified from the strong interaction between the dislocation propagation and shear bands. It is further noted that the tensile ductility of a BMG composite is very sensitive to the parameters of the back stress, Taylor factor and particle size. The present results will shed some light on optimizing the microstructures in effectively improving the tensile ductility of BMG composites.

Micromechanical simulation of fracture behavior of bimodal nanostructured metals

Nanostructured (NS) metals with bimodal grain size distribution that consist of coarse grained (CG) and nano-grained (NG) regions have proved to have both high strength and good ductility. In this paper a numerical investigation, using the combination of a mechanism-based strain gradient plasticity theory, a micromechanics composite model, and the Johnson–Cook failure model, is conducted to investigate the effects of the distribution of the CG inclusions and their shape on fracture behavior of a bimodal NS copper. Load–response relations are employed to evaluate the loading history stability, while apparent crack length and strain energy history are used to analyze the fracture resistance. This study shows that both crack bridging in the CG inclusions and crack deflection in the NG matrix can significantly toughen the bimodal NS Cu. Our simulations also show that there exists a critical volume fraction of CG inclusions for some microstructures at which the fracture resistance of the bimodal NS Cu is at its minimal state and thus it should be avoided in material design.