Nonlocal Plasticity Research Articles

Solution for the general integral‐type nonlocal plasticity model with Tikhonov regularization

SummaryA new solution approach, based on Tikhonov regularization on the Fredholm integral equations of the first kind, is proposed to find the approximate solutions of the strain softening problems. In this approach, the consistency condition is regularized with the Tikhonov stabilizers along with a regularization parameter, and the internal variable increments are solved from the resulting Euler's equations. It is shown that, as the regularization parameter is increased, the solutions converge to a unique one. A nonlocal yield condition and a nonlocal return mapping algorithm are proposed to carry out the integration of constitutive equations in the time and spatial domains. A global plastic dissipation principle is proposed to relax the classical local plastic dissipation postulate. Numerical examples show that the proposed approach leads to objective, mesh‐independent solutions of the softening‐induced localization problems. A comparison of the results from the proposed approach with those from the gradient‐dependent plasticity model shows that the two models give close solutions.

Dynamic Model of Elastoplastic Normal Collision of Spherical Particles under Nonlocal Plasticity

The problem of normal collision of a spherical particle with a half-space is considered with allowance for nonlocal plastic deformation in the case where the strength limit depends on the contact radius, as well as for the strengthening effect in the deformed material. The dimensionless coefficient of normal velocity restitution has been calculated numerically as a function of the initial velocity of the spherical particle. The obtained data coincide well with experimental results available in the literature.

Динамическая модель упруго-пластического нормального столкновения сферических частиц при нелокальной пластичности

AbstractThe problem of normal collision of a spherical particle with a half-space is considered with allowance for nonlocal plastic deformation in the case where the strength limit depends on the contact radius, as well as for the strengthening effect in the deformed material. The dimensionless coefficient of normal velocity restitution has been calculated numerically as a function of the initial velocity of the spherical particle. The obtained data coincide well with experimental results available in the literature.

Plastic deformation and failure mechanisms in nano-scale notched metallic glass specimens under tensile loading

In this work, numerical simulations using molecular dynamics and non-local plasticity based finite element analysis are carried out on tensile loading of nano-scale double edge notched metallic glass specimens. The effect of acuteness of notches as well as the metallic glass chemical composition or internal material length scale on the plastic deformation response of the specimens are studied. Both MD and FE simulations, in spite of the fundamental differences in their nature, indicate near-identical deformation features. Results show two distinct transitions in the notch tip deformation behavior as the acuity is increased, first from single shear band dominant plastic flow localization to ligament necking, and then to double shear banding in notches that are very sharp. Specimens with moderately blunt notches and composition showing wider shear bands or higher material length scale characterizing the interaction stress associated with flow defects display profuse plastic deformation and failure by ligament necking. These results are rationalized from the role of the interaction stress and development of the notch root plastic zones.

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.

Analysis of Tikhonov Regularization on the Integral-Type Nonlocal Plasticity Model

AbstractTikhonov regularization on the integral-type nonlocal plasticity model is proposed to obtain a mesh-independent solution of strain localization. A detailed parametric study on the Tikhonov-...

Ultimate load prediction of MMNCs structures

Abstract The paper proposes a nonlocal limit analysis procedure to evaluate the load bearing capacity of metal matrix nanocomposites (MMNCs) structural elements. The promoted procedure rephrases and extends, in a nonlocal context, a static limit analysis numerical method, known in literature as elastic compensation method. Due to the ductility and the intrinsic nonlocal behaviour of MMNCs materials, a nonlocal elastic-perfectly plastic constitutive behaviour is hypothesized with the further hypothesis that nonlocality belongs only to the elastic phase. An associative von Mises type yield condition together with a nonlocal elasticity model of integral type are adopted. A nonlocal formulation of the finite element method is used throughout and a numerical application is presented and critically discussed with the aim to test the capability of the method in capturing the main features of the novel nonlocal limit analysis approach.

A non-local plasticity theory for slow granular flows

Motivated by experimental observations of apparent non-locality in the mechanical response of granular materials, a few recent studies have proposed non-local rheological models. Though their predictions agree with some aspects of experimental observations, the tenets of the models are not experimentally verified, nor do they have an experimentally verifiable or plausible mechanical basis. Here we present a non-local model that is based on the simple idea of volume averaging, and show that it overcomes many of the shortcomings of the previously proposed models, and is superior in its predictive capabilities.

Recycled aggregate concrete: Localized failure assessment in thermodynamically consistent non-local plasticity framework

A gradient plasticity theory is extended to predict the mechanical failure behavior of concrete with recycled aggregates. A relevant feature of the novel constitutive formulation is its full consistency with the thermodynamic laws regarding hardening and softening response behaviors. The proposed model includes a formulation of a maximum strength surface which is dependent on the recycled aggregate content. While the hardening law is formulated in the framework of classical-local elastoplasticity, the post-peak behavior is described by means of a non-local gradient and fracture-energy-based law. Thereby, the gradient and fracture-energy characteristic lengths are described in terms of the acting confining pressure and the recycled aggregate content. The formulation of the proposed constitutive theory is complemented by the description of the associated localization indicator, based on the acoustic or localization tensor. Finally, numerical results are presented to demonstrate the predictive capabilities of the proposed model and the performance of the localization indicator for different critical stresses and recycled aggregate contents.

Notch sensitivity in nanoscale metallic glass specimens: Insights from continuum simulations

Recent experiments have shown that nano-sized metallic glass (MG) specimens subjected to tensile loading exhibit increased ductility and work hardening. Failure occurs by necking as opposed to shear banding which is seen in bulk samples. Also, the necking is generally observed at shallow notches present on the specimen surface. In this work, continuum finite element analysis of tensile loading of nano-sized notched MG specimens is conducted using a thermodynamically consistent non-local plasticity model to clearly understand the deformation behavior from a mechanics perspective. It is found that plastic zone size in front of the notch attains a saturation level at the stage when a dominant shear band forms extending across the specimen. This size scales with an intrinsic material length associated with the interaction stress between flow defects. A transition in deformation behavior from quasi-brittle to ductile becomes possible when this critical plastic zone size is larger than the uncracked ligament length. These observations corroborate with atomistic simulations and experimental results.

Energy-based non-local plasticity models for deformation patterning, localization and fracture

This paper analyses the effect of the form of the plastic energy potential on the (heterogeneous) distribution of the deformation field in a simple setting where the key physical aspects of the phenomenon could easily be extracted. This phenomenon is addressed through two different (rate-dependent and rate-independent) non-local plasticity models, by numerically solving two distinct one-dimensional problems, where the plastic energy potential has different non-convex contributions leading to patterning of the deformation field in a shear problem, and localization, resulting ultimately in fracture, in a tensile problem. Analytical and numerical solutions provided by the two models are analysed, and they are compared with experimental observations for certain cases.

Microstructural Modeling of Dual Phase Steel Using a Higher-Order Gradient Plasticity-Damage Model

This work focuses on the application of a higher-order gradient-dependent plasticity-damage model for microstructural modeling of dual-phase (DP) steels. Damage evolution is governed by the evolution of a nonlocal plasticity measure which is a function of the local equivalent plastic strain rate and its corresponding first-order gradient. Different RVEs of DP microstructures are virtually generated and simulated in order to predict the macroscopic mechanical response. Size effects and additional hardening due to evolution of geometrically necessary dislocations are predicted.

Measuring Plasticity in Confined Cu Thin Films at Different Geometries

We report quantitative measurement of plasticity in confined Cu thin films with a new micro-pillar testing protocol. Polycrystalline Cu and CrN thin films were sequentially sputter deposited onto Si (100) substrates, forming thin film assemblies in which polycrystalline Cu thin films of various thicknesses were confined between non-deforming Si and CrN. Cylindrical micro-pillars of CrN/Cu/Si were fabricated through scripted focused Ga+ ion beam milling, with the interfaces either normal to the axial direction or at a 45° inclination. The CrN/Cu/Si micro-pillars were compression loaded in the axial direction with a flat diamond punch on an instrumented nanoindenter. The axial compression loading caused extensive plasticity within the thin Cu interlayers with the interfaces in both the normal and inclined orientations, but with distinctly different responses. We show that significant plastic flow occurs within the confined Cu thin films in both normal compression and shear loading. The flow stress of the confined Cu films is dependent on the Cu layer thickness and the deformation geometry. The presently described micro-pillar testing protocol offers quantitative evaluation of the plastic response of thin metal films under different deformation geometries. The present results offer new experimental examples of scale-dependent plasticity in thin metal films, and new experimental test cases for non-local plasticity theories.

Microstructural modeling of dual phase steel using a higher-order gradient plasticity–damage model

This paper focuses on the application of a higher-order gradient-dependent plasticity–damage model for microstructural modeling of dual-phase (DP) steels. Damage evolution is governed by the evolution of a nonlocal plasticity measure which is a function of the local equivalent plastic strain rate and its corresponding first-order gradient. Different two-dimensional representative volume elements of the DP microstructure are virtually generated by varying the martensite phase volume fraction, distribution, and size, and volume fraction and size of dispersed hard inclusions. It is shown that the employed modeling framework is capable of addressing three main issues that are not considered by the current studies on microstructural modeling of advanced high strength steels (AHSS); finite element mesh-dependency, size effects, and additional hardening due to plastic strain gradients. It is concluded that based on the employed mechanical properties of each phase in the DP steel, strength and ductility is governed by damage evolution and not necessarily by plastic strain localization alone. Therefore, it is shown that including nonlocal damage evolution is critical for accurate prediction of the strength and ductility of DP steels. It is also shown that dispersing 5% volume fraction of hard inclusions in the DP steel optimizes both strength and ductility such that a new generation of AHSS might be attained.

Micro-pillar measurements of plasticity in confined Cu thin films

Sputtering was utilized to deposit polycrystalline Cu and CrN films sequentially onto Si(100) substrates, forming specimen assemblies in which predominantly 〈111〉 oriented Cu thin films of varying thicknesses were confined between Si and CrN. Cylindrical micro-pillars of CrN/Cu/Si(100) were fabricated through focused ion beam milling, with the interfaces either normal to the axial direction or at an inclination of 45°. Axial compression loading of the micro-pillars produced extensive plasticity within the thin Cu interlayers in both cases, but with distinctly different responses involving combined compression and shear when the interfaces are normal to the compression axis and constrained shear when the interfaces are inclined. Significant size effects were observed, offering new experimental evidence of scale-dependent plasticity for thin film layers, and new experimental test cases for non-local plasticity theories.

Ductility enhancement in nanoglass: role of interaction stress between flow defects

Using a thermodynamically consistent non-local plasticity model, the mechanistic origin of enhancement in ductility and suppression of dominant shear banding in nanoglasses (NGs) is analysed. It is revealed that the interaction stress between flow defects plays a central role in promoting global plasticity of NGs. Specifically, we find that the intrinsic length associated with this stress provides a scaling for the shear band width and its coupling with grain size governs the level of enhancement in the deformation behaviour of NGs. The present work may provide useful insights in developing highly ductile NGs for practical engineering applications.

비국부 이론을 이용한 입자 강화 복합재 이중후방응력 소성 구성방정식 모델 및 전단밴드 분석

2개의 상으로 구성된 입자 강화 복합재에 대한 균질화와 내부 상태 변수에 대해 2차 미분항이 포함된 비구역적 이론을 적용하여 탄소성 구성 방정식을 제안하였다. 열역학과 소성 포텐셜을 통해 내부 상태 변수에 대한 전개식 또한 본 논문에 포함되었다. 연속체 결함 모델을 이용, 결함 인자에 따른 물성 저하 현상도 감안되었으며 이중 후방응력이 조합된 전개식 또한 제시하였다. 일부 예에 대한 수치해석 결과, 비구역적 변수의 영향이 증가할수록 전단밴드는 감소하나 반면 특정 후방응력 전개가 지배적일수록 소성변형 집중이 증가함이 관찰되었다. 더욱이 두 개의 강소성 상으로 이루어진 복합재의 경우 강성이 높은 게재물의 비중이 증가함에 따라 전단밴드 형성이 용이한 것으로 나타났다. 그 밖에 제어변수들의 변화에 따른 전단밴드 형성에 대한 분석 결과는 Rice 소성 불안정성 분석결과와 잘 일치함 또한 밝혀졌다.

Thickness dependence of flow stress of Cu thin films in confined shear plastic flow

Abstract

Buckling and post-buckling of gradient and nonlocal plasticity columns experiencing softening

The buckling and the post-buckling behaviors of a perfect axially loaded column are analytically investigated through a global bilinear moment–curvature elastoplastic constitutive law. Three plasticity cases are studied, namely the linear hardening plasticity law, the perfect elastoplastic case and the softening case. The applications of such a study can be found in various structural engineering problems, including reinforced concrete, steel, timber or composite structures. It is analytically shown that for all kinds of elastoplastic behaviors, the plasticity phenomena lead to a global softening branch in the load–deflection diagram. The propagation of the plasticity zone during the post-buckling process is analytically characterized in case of linear hardening or softening plasticity laws. However, it is shown that the unphysical elastic unloading solution necessarily occurs in presence of local softening moment–curvature constitutive law. A nonlocal plasticity moment–curvature softening law is then used to control the localization branch in the post-buckling stage. This nonlocal plasticity law includes the explicit and the implicit gradient plasticity law. Higher-order plasticity boundary conditions are derived from an extended variational principle. Some parametric studies finally illustrate the main findings of this paper, including the plasticity modulus effect on the post-buckling behavior of these plasticity structural systems.

Plastic Slip Patterns through Rate-Independent and Rate-Dependent Plasticity

Plastic deformation induces various types of dislocation microstructures at different length scales, which eventually results in a heterogeneous deformation field in metallic materials. Development of such structures manifests themselves as macroscopic hardening/softening response and plastic anisotropy during strain path changes, which is often observed during forming processes. In this paper we present two different non-local plasticity models based on non-convex potentials to simulate the intrinsic rate-dependent and rate-independent development of plastic slip patterns, which is the simplified mechanism for the intrinsic microstructure development. For the sake of mechanistic understanding, the formulation and the simulations will be conducted in one-dimension which does not exclude its extension to multi-dimensions resulting in a crystal plasticity framework.