Gradient Plasticity Model Research Articles

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.

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.

Size-dependent yield criterion for single crystals containing spherical voids

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

“Implicit” vs “Explicit” gradient plasticity models: Do they always remove mesh dependence in softening materials?

Finite element quasi-static solutions of rate-independent softening plasticity models are known to depend on the mesh size (i.e., are unreliable) due to loss of ellipticity of the governing partial differential equations, which allows for the development of non-smooth solutions, such as shear bands of zero thickness. A “regularization” scheme that has been proposed is the use of “non-local” (Bažant et al., 1984) or “strain-gradient” theories (Aifantis, 1984). We examine in detail the families of “explicit” and “implicit” gradient plasticity models and show that use of such theories does not always eliminate mesh dependence of finite element solutions in rate-independent softening materials. Depending on the value of the non-local hardening modulus, the mathematical problem may still lose ellipticity and allow for the development of solutions with discontinuous spatial velocity gradients, which are responsible for the mesh dependence of numerical solutions. As an example, a rate-independent “implicit” gradient version of the von Mises yield condition with the associated flow rule is implemented in ABAQUS, and the formation of shear bands in plane strain tension is analyzed numerically; it is shown that, unless the material parameters in the non-local model are such that ellipticity is retained, the finite element solutions depend on the mesh size. The example of isotropic porous metal plasticity with a hardening matrix material is considered (e.g., the Gurson (1977) model); it is shown that regularization is achieved if an implicit non-local model is used, in which the local porosity f is replaced by its non-local (neighborhood average) counterpart f̄ in the yield condition. In all non-local plasticity models, dimensional consistency requires the introduction of one or more “material lengths” ℓi, which multiply the additional terms with the higher-order spatial derivatives introduced by the non-local models. The boundary layers associated with the singular perturbation problem when the material lengths ℓi→0 are examined. A linear stability analysis is carried out to study the stability of homogeneous problems under small harmonic perturbations; it is shown that the stability range of the non-local models is always larger than that of the corresponding local models.

Strain gradient plasticity based on saturating variables

Standard scalar-based strain gradient plasticity models generally incorporate the gradient of the cumulative plastic strain into the constitutive modeling. The following limitations of this approach have been pointed out in the literature: Unlimited enhanced hardening at large strains, unlimited cyclic hardening in bending, strain localization band broadening at large applied strains for softening materials with saturation, and possible vanishing of the yield stress for too high values of the Laplace term in the enhanced hardening function. The present work proposes an alternative scalar-based micromorphic approach that solves most of the listed difficulties. The theory incorporates the effect of the gradient of saturating variables instead of the gradient of ever-increasing cumulative plastic strain. A phenomenological theory is presented that includes an exponentially saturating internal variable and its gradient. The approach is also applied to dislocation density based plasticity where the dislocation density is known to saturate at large strains. The benefits of the new theory are demonstrated by finite element simulations of various boundary value problems: Monotonic and cyclic tension and simple glide of an infinite strip, strain localization in a plate, bending and torsion of bars. Both hardening and softening plasticity laws are addressed illustrating the size-dependent hardening in the former case and the evolution of the finite width of localization bands in the latter. Weak points of the approach, namely the possibility of negative dissipation and of negative yield stress, are also identified.

One-Dimensional Failure Modes for Bodies with Non-convex Plastic Energies

In this paper, a complete picture of the different plastic failure modes that can be predicted by the strain gradient plasticity model proposed in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) is drawn. The evolution problem of the elasto-plastic strain is formulated in Del Piero et al. (J. Mech. Mater. Struct. 8:109–151, 2013) as an incremental minimization problem acting on an energy functional which includes a local plastic term and a non-local gradient contribution. Here, an approximate analytical solution of the evolution problem is determined in the one-dimensional case of a tensile bar. Different solutions are found describing specific plastic strain processes, and correlations between the different evolution modes and the convexity/concavity properties of the plastic energy density are established.The variety of solutions demonstrates the large versatility of the model in describing many failure mechanisms, ranging from brittle to ductile. Indeed, for a convex plastic energy, the plastic strain diffuses in the body, while, for a concave plastic energy, it localizes in regions whose amplitude depends on the internal length parameter included into the non-local energy term, and, depending on the convexity properties of the first derivative of the plastic energy, the localization band expands or contracts. Complex failure processes combining different modes can be reproduced by assuming plastic energy functionals with specific convex and concave branches.The quasi-brittle failure of geomaterials in simple tension tests was reproduced by assuming a convex-concave plastic energy, and the accuracy of the analytical predictions was checked by comparing them with the numerical results of finite element simulations.

Dispersed strain bands promote the ductility of gradient nano-grained material: A strain gradient constitutive modeling considering damage effect

Gradient nano-grained (GNG) metals have achieved superior strength-ductility synergy than their homogeneous counterparts. The high strength is usually attributed to the grain size effect and hetero-deformation-induced strengthening. However, incommensurate work on ductility leads to an incomplete understanding of the strength-ductility combination. In this work, a dislocation density-based strain gradient plasticity model coupling with a damage model is developed to describe the strain hardening and softening behavior of GNG material. A grain size-dependent back stress model derived from the generation of dislocation pileups is invoked to describe the widely-concerned back stress hardening. Finite element implementation of the model quantitatively predicts the tensile response of GNG nickel with various degrees of grain size gradient. The results reveal that dispersed strain bands propagate stably in the nano-grained surface layer of GNG material, which is totally different from those occurring in a freestanding nano-grained material. The stabilization of dispersed strain bands enables the nano-grained layer to deform uniformly, thus premature failure of the whole GNG material is suppressed and improved ductility is achieved. Furthermore, increasing the grain size gradient renders the strain bands more stable, leading to enhanced ductility. The method developed in this work is helpful for understanding the strength-ductility synergy of GNG materials and for optimizing the microstructure gradient in GNG materials.

Characterization of gradient plastic deformation in gradient nanotwinned Cu

Gradient nanostructured (GNS) metals exhibit high overall extra strengths relative to their non-gradient counterparts. However, the spatial distribution of local extra strengths stemming from plastic strain gradients remains elusive. This work is focused on characterizing the gradient distribution of plastic strains in a representative GNS metal of gradient nanotwinned (GNT) Cu. Full-field strain mapping reveals the gradient distributions of lateral strains in the transverse cross section of GNT Cu samples undergoing uniaxial tensile deformation. We find that the lateral strain gradient increases but the maximum lateral strain difference decreases in GNT samples with increasing structural gradient. The latter arises because the softest layer with the lowest initial yield strength gains the largest local extra strength during tensile deformation, and vice versa. Such a gradient distribution of local extra strengths results from the combined strengthening effects of plastic strain gradient and grain size. These experimental results are used to inform a strain gradient plasticity model for revealing the gradient distributions of local extra back stresses and local extra strengths with increasing load. The coupled experimental and modeling characterization of gradient plastic deformation provides an in-depth mechanistic understanding of the spatial-temporal evolution of gradient strengthening effects in gradient nanostructures.

3D printed Inconel mechanical response related to volumetric energy density

This work aims to provide a methodology for producing Inconel 718 specimens ISO 527–2-5B with a tailored mechanical response, by Selective Laser Melting (SLM). Since the macroscopic mechanical behavior of 3D printed metals is dictated by their microstructure, the effect of the printing parameters such as the laser power, the scan speed, the hatching distance and the layer thickness was considered. To evaluate the maximum dimensional accuracy and optimum Volumetric Energy Density (VED) value, the specimens were printed with a constant layer height, hatching distance and spot size and with a varying combination of laser power with scan speed. The 3D printed specimens were subjected to uniaxial tension. The experiments have shown an increased reproducibility of the specimens’ microstructure that led to a more consistent mechanical behavior, when lower scan speed and power were used. Specimens printed with higher speed and power seem to have a more heterogeneous microstructure leading to a more varying mechanical response. While the differences in the mechanical response between specimens of different batches are attributed to different laser power and scan speed, different response between specimens of the same batch is attributed to the inherent inhomogeneity of their microstructure. A simple elasticity model was used to describe their elastic behavior, while a 1D gradient plasticity model was utilized to describe their post-yield tensile response, in which the effect of microstructural heterogeneity on macroscopic behavior was taken into account through a respective internal length. The combined elasto-plastic model was used to capture mechanical response until failure, in good agreement with the experiments. The predicted mechanical characteristics were related to the specimens’ VED to come up with simple relations providing optimum 3D printing parameters for producing specimens with a certain mechanical response. For a complete characterization, the specimens’ roughness was studied and its VED dependence was performed.

Toward robust scalar-based gradient plasticity modeling and simulation at finite deformations

Strain gradient plasticity has been the subject of extensive research in the past 40 years in order to model size effects in metal plasticity, on the one hand, and provide finite width shear bands in the simulation of localization phenomena, on the other hand. However, the use of the emerging models is still limited to academic applications and has not yet been adopted by industry practitioners. The present paper systematically reviews the pros and the cons of gradient plasticity at finite strains based on the gradient of scalar plastic variables, in particular the gradient of the cumulative plastic strain. It proposes benchmark tests addressing both size effect modeling and plastic strain localization simulation. It includes new analytical solutions for validation of FE implementation. It focuses on the micromorphic approach to gradient plasticity, as a convenient method for implementation in FE codes. New features of the analysis include the comparison of three distinct formulations of rate-independent gradient plasticity at finite deformations, based on the multiplicative decomposition of the deformation gradient and on quadratic potentials with respect to gradient terms. The performance of micromorphic and Lagrange-multiplier-based strain gradient plasticity models is evaluated for various monotonic and cyclic loading conditions including confined plasticity in simple glide and tension, bending and torsion at large deformations. Limitations are pointed out in the case of bending and torsion, which can be overcome for instance by the use of the gradient of equivalent plastic strain model.

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.

Toward selecting optimal predictive multiscale models

This work presents a systematic strategy for selecting an “optimal” predictive computational model among a set of possible mechanistic models of a physical system. To this end, the Occam-Plausibility Algorithm (Farrell-Maupin et al., 2015) is extended by introducing a method for the design of model-specific validation experiments to provide data that portray the features of the prediction quantities of interest. Leveraging Bayesian inference and model plausibility, the framework adaptively balances the trade-off between complexity and validity of the models while taking into account the uncertainty in data, model parameters, and the choice of the model itself for making reliable computational predictions. An application of this framework is demonstrated in discrete-continuum multiscale modeling of size-dependent plasticity in polycrystalline materials, involving dislocation dynamics simulations and strain gradient plasticity models.This study suggests that the effectiveness of validation experiments relies upon the choice of the model; thus, a validation data set that adequately informs a predictive computational model may not be effective for validating other models. Additionally, the multiscale modeling results show that, given the discrete dislocation dynamics data, the optimal strain gradient plasticity model for predicting the deformation of a microelectromechanical system is obtained by excluding the isotropic hardening mechanism.

Stress gradient plasticity theory development for cyclic and monotonic loading of thin-walled structures: Back-stress, size effect, passivation effect

Small scale plasticity behaviour is modelled by strain or stress gradient plasticity models. The stress gradient plasticity theory, proposed by Chakravarthy and Curtin, is a constitutive consideration of the stress gradient that affects the initial flow as well as strain hardening rate. We apply the lower-order stress gradient plasticity model to different monotonic problems and cyclic torsion, under small strain and rotation assumption. The stress gradient can be due to the problem geometry and loading and/or due to passivation. Passivation is considered in different monotonic loadings of classical problems. To use the stress gradient plasticity model in cyclic loading problems, a back-stress model is used and the kinematic hardening is considered. The developed model in microscale plasticity considers dislocation pile-ups, grain size, and dislocation density. The Bauschinger effect is also considered during reverse loading by the consideration of the kinematic hardening. The cyclic torsion of thin copper wires of different diameters is modelled by the developed model. Comparisons with the experimental results and distortion strain gradient plasticity model are also performed. It is shown that the simple developed model can capture the essential microscale plasticity behaviour of metals, with only one length scale parameter.

Mechanics of micropillar confined thin film plasticity

Micropillar compression experiments probing size effects in confined plasticity of metal thin films, including the indirect imposition of ‘canonical’ simple shearing boundary conditions, show dramatically different responses in compression and shear of the film. The Mesoscale Field Dislocation Mechanics (MFDM) model is confronted with this set of experimental observations and shown to be capable of modeling such behavior, without any ad-hoc modification to the basic structure of the theory (including boundary conditions), or the use of extra fitting parameters. This is a required theoretical advance in the current state-of-the art of strain gradient plasticity models. It is also shown that significantly different inhomogeneous fields can display qualitatively similar size effect trends in overall agreement with the experimental results. The (plastic) Swift and (elastic) Poynting finite deformation effects are also demonstrated.

An Improved Mechanism-Based Strain Gradient Plasticity Model and Its Application to Size Effect Under Complex Loading

The ingredient devices tend to be designed and fabricated with microscale, high accuracy, and complex geometry and are subjected to complex loading. Many experiments have proved that the size effect plays a significant role in designing and manufacturing an engineering component. This size effect is largely attributed to the grain size and the strain gradient. While the grain size effect was omitted in conventional strain gradient theories, an extended model that considers both effects of grain size and strain gradient has been proposed in the current work (denoted as GMSG). The proposed GMSG model has been implemented into the finite element method (denoted as GMSG-FEM) to investigate the size effect of copper wires under complex working conditions. The simulation results show that the hollow structure can improve the bearing capacity of the micro-wires under torsion. Among micro-wires with different cross-sections, the bearing capacity of the micro-wire with a circular section is the largest, followed by those with square and triangle sections. Complex loading also has an important influence on stress distribution. Based on the current study, it can be envisaged that the GMSG-FEM could be a useful tool in engineering applications where the size effect has to be considered.

Generalized Aifantis strain gradient plasticity model with internal length scale dependence on grain size, sample size and strain

Generalized Aifantis strain gradient plasticity model with internal length scale dependence on grain size, sample size and strain

Particle size effects in ductile composites: An FFT homogenization study

We present a computational homogenization study on the particle size effect in ductile composites. The micromechanical formulation is based on non-local models through (i) the incorporation of a lower-order strain gradient plasticity model and (ii) the application of an implicit gradient regularization technique to the Gurson–Tvergaard–Needleman ductile damage model for metals. In this way, the extended model is equipped with two length-scale parameters, one for each non-local extension, which modulate the size dependent character of the formulation. The problem consists of a system of partial differential equations in which two Helmholtz-type equations for the damage regularization are coupled with the balance of linear momentum through the stress, which depends on the non-local variables and on the plastic strain gradient.A series of numerical simulations are conducted to investigate the behavior of three-dimensional microstructures representative of particle reinforced metal matrix composites. The change in strengthening and ductility, as a function of the particle size, is first analyzed by means of a parametric study in which the considered non-local extensions act both independently and together. Finally a comparative study with experimental results demonstrates that the particle size induced strengthening in metal matrix composites can be quantitatively captured by the considered model.

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.

A localizing gradient plasticity model for ductile fracture

The conventional gradient enhanced plasticity model uses a nonlocal gradient regularization to avoid mesh sensitivity issues present in the continuum-based local plasticity models. These conventional gradient plasticity models are found to manifest spurious spreading of localization zone during the post-peak regime, leading to non-physical structural and fracture response. Therefore, in the present work, a localizing gradient plasticity (LGP) model is proposed to avoid these non-physical responses. The LGP model avoids spurious responses by including the effects of the fracture processes (at micro and macro level) in the model description through the micromorphic framework. The fracture processes are typically found to initiate as a diffused region of plastic deformation (called necking region), which localizes to form cracks upon further loading. An interaction function is adopted in the LGP model to incorporate the localization of the fracture processes. The enhanced capability of the LGP model is demonstrated through one-dimensional and planar problems with tensile, compressive and mixed-mode loading. The numerical results show that the LGP model can arrest the spurious widening of the localization region resulting in a narrow region of high deformation representing failure.