Strain Gradient Plasticity Research Articles

A micromechanically motivated lower order strain gradient model for plastic behavior of functionally graded crystalline micro beam structures

A size-dependent micromechanically motivated formulation is introduced for titanium-boride/titanium functionally graded crystalline material at the microscale based on the Chen–Wang strain gradient plasticity (CWSGP). A lower order flow stress is proposed to define the brittle to ductile phases reflecting a combined effective length scale associated to stress and strain fields. Since the characteristics of two-phase ceramic–metal structure vary continuously along its thickness; the TTO composite model is applied for intermediate mixture law to indicate brittle-to-ductile transition in the titanium-boride/titanium FG structures. The mentioned TTO model addresses this phase transition through an eventually equilibrium mixed definition of ductile–like phase. Then, the governing differential equation for the energy functional is derived to solve for different boundary conditions. In addition, the influence of the elastic foundation and thermal treatment on the plastic deformations is investigated. Moreover, the accuracy of the model was demonstrated through 3 case studies. This study offers an efficient size – dependent gradient model for two-phase microscale structures without complexity on the crystal plasticity.

Geometrically Nonlinear Constitutive Equations of the Plastic Flow Theory in Terms of Asymmetric Stress and Strain Measures

A formulation of the geometrically nonlinear plastic flow theory (PFT) based on asymmetric measures of stress and strain states is proposed. A main emphasis is placed on the physically reasonable decomposition of the deformation gradient into three components: elastic distortions, which determine stresses, an orthogonal tensor characterizing the quasi-rigid motion of a material and the plastic strain gradient. The quasi-rigid motion of the material is defined by introducing for a representative volume element a generalized lattice, which represents its symmetry elements. The hypoelastic anisotropic law is introduced in terms of the movable coordinate system associated with the material. The rate of plastic deformations is determined by the associated law of plastic flow. As a result, the closed system of constitutive equations of the geometrically nonlinear PFT of is obtained.

Evolution of nano-laminated structure formed by the thermally-assisted plastic deformation in dry sliding wear

Abstract The evolution of nano-laminated structure generated in dry sliding wear of a high strength martensite steel was investigated by using Scanning Electronic Microscopy (SEM) and Transmission Electron Microscopy (TEM). Gradient microstructure along the depth was observed with the top layer consisting of nano-laminated structure. During the sliding wear-induced plastic deformation, grains were refined and lamellar structures were formed as a result of generation of dislocation cell blocks bounded by geometrically necessary boundaries (GNBs) and incidental dislocation boundaries (IDBs). With increasing plastic shear strain and strain gradient, there is a continuous reduction in the spacing and length in dislocation cell blocks, leading to nano-laminated structures. The ultrafine nano-laminated structure with laminate spacing of 20 nm was observed at the sliding time of 120 min. However, the grain refinement is not a merely mechanical deformation process. The formation of Fe3O4 and FeO gave direct evidence that the highest temperature due to frictional heat could reach up to 600 °C. The result suggested that the formation of nano-laminated structure was governed by thermally-assisted plastic deformation.

Strain-gradient elasticity and gradient-dependent plasticity with hierarchical refinement of NURBS

Higher-order strain-gradient models are relevant for engineering materials which exhibit size-dependent behaviour as observed from experiments. Typically, this class of models incorporate a length scale - related to micro-mechanical material properties - to capture size effects, remove stress singularities, or regularise an ill-posed boundary value problem resulting from localisation of deformation. The higher-order continuity requirement on shape functions can be met using NURBS discretisation, as is considered herein. However, NURBS have a tensor-product nature which makes selective refinement cumbersome. To maintain accuracy and efficiency in analysis, a finer mesh may be required, to capture a localisation band, certain geometrical features, or in regions with high gradients. This work presents strain-gradient elasticity and strain-gradient plasticity, both of second-order, with hierarchically refined NURBS. Refinement is performed based on a multi-level mesh with element-wise hierarchical basis functions interacting through an inter-level subdivision operator. This ensures a standard finite-element data structure. Suitable marking strategies have been used to select elements for refinement. The capability of the numerical schemes is demonstrated with two-dimensional examples.

Some issues concerning the use of a single, material specific length scale parameter in theories of higher order strain gradient plasticity

Higher order strain gradient plasticity theories extend conventional ones with a view to explain the observed elevation of plastic flow stress in micron-sized structures. The essential elements of this modifications are the definition of a measure of plastic strain gradients, introduction of a conjugate higher order stress and introduction of one or more length scale parameters. We adopt a particular elasto-viscoplastic version of the higher order strain gradient theory based on a single scalar length scale parameter, formulate a large deformation based Finite Element model capable of handling strain gradients and higher order boundary conditions and simulate for three different materials, two sets of experiments where size effects manifest. The experiments differ in the level of plasticity that is induced during deformation — ranging from the order of the yield strain to approximately ten times its value. The materials, namely Cu, Al and Ni, have grain sizes that differ widely. We determine the scalar length scale parameter by demanding that the simulations match the experimental load deformation responses closely. The results indicate that single, scalar length scale parameter based higher order strain gradient theories cannot describe experimental observations across a wide range of geometries, boundary conditions, and plastic strain levels. The length scale is not intrinsic to the material but depends on the level of plastic strain induced by the deformation and also possibly on the number of grains involved in the deformation.

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.

Study on one-dimensional softening with localization via integrated MPM and SPH

Nonlocal elastoplasticity and/or damage models in terms of gradient or integral of strain or internal state variables have been proposed to regularize softening with localization in model-based simulation of failure evolution. For transient problems, the numerical oscillations due to semi-discretization in space make the evaluation of higher order terms a difficult task in avoiding numerical failure such as premature cracking. In this preliminary study, an integrated MPM (material point method) and SPH (smoothed particle hydrodynamics) numerical scheme is proposed to implement a nonlocal model via the gradient of plastic strain, with an application to a one-dimensional bar under impact. The MPM is used to discretize the continuum region without the need for master/slave nodes at the contact surface, while the smoothing operator and particle approximation based on the SPH are adopted on the material points to reduce the unphysical oscillations and facilitate the evaluation of strain gradient. A parametric investigation is performed to illustrate the effects of controlling parameters on the evolution of localized softening. It is demonstrated that integrating nonlocal constitutive modeling with spatial discretization might yield an effective procedure for predicting and evaluating failure evolution.

Isotropic strain gradient plasticity model based on self-energies of dislocations and the Taylor model for plastic dissipation

A dislocation mechanics based isotropic strain gradient plasticity model is developed. The model is derived from self-energies of dislocations and the Taylor model for plastic dissipation. It is shown that the same microstructural length scale emerges for both the energetic and the dissipative parts of the model. Apart from a non-dimensional factor of the order of unity, the length scale is defined as the Burgers vector divided by the strain for initiation of plastic deformation. When the structural length scale approaches this microstructural length scale, strengthening effects result. The present model predicts an increased initial yield stress that is controlled by the energetic contribution. For larger plastic strains, the hardening is governed by the dissipative part of the model. The theory is specialized to the simple load cases of tension with a passivation layer that prohibits plastic deformation on the surfaces as well as pure bending with free and fixed boundary conditions for plastic strain. Simulations of initial yield stress for varying thicknesses are compared to experimental observations reported in the literature. It is shown that the model in a good way can capture the length scale dependencies. Also upper bound solutions are presented for a spherical void in an infinite volume as well as torsion of a cylindrical rod. The model is as well applied to derive a prediction for the Hall–Petch effect.

Analysis of local grain boundary strengthening utilizing the extrinsic indentation size effect

Abstract

An analytical model for the flat punch indentation size effect

An analytical approximation for the indentation size effect (ISE) due to plane strain flat punch nanoindentation is derived. The flat punch ISE differs from that observed for self-similar (pointed) and spherical indenters in a number of ways: (1) the contact area does not change; (2) the contact pressure depends on two length scales not just one (the punch width and the indentation depth); (3) the profile of the punch is not differentially continuous, resulting in singular plastic strain gradients at the sharp edges, such that (4) the shape and connectivity of the plastic zones change with indentation depth and punch width, resulting in (5) changes in the proportion of the deformation accommodated by elasticity and plasticity are important, meaning that a fully elastoplastic model is required. Complete loading-unloading curves are modelled, with the calibration of geometrical parameters from finite element strain gradient plasticity simulations. As the punch width decreases, it is observed that there are increases in the indentation pressure, the relative size of the plastic zone(s) and the elastic component of the deformation. These predictions are found to compare favourably with experimental measurements in the literature. The model is extended to incorporate the consequence of imposing natural limitations on the maximum dislocation density at the edges. It is suggested that observable changes in the plastic zone morphology with the ISE make this an experimentally interesting area for the validation of size effects in plasticity.

A Study on the Prediction of Strength of Trimodal Composites

Particle-reinforced trimodal composites, which typically consist of three constituent phases where at least one phase forms itself a composite with ceramic particles, feature a length-scale-dependent strengthening. Joshi and Ramesh [1] presented a secant Mori–Tanaka method to describe the strength of trimodal composites. They, however, seem to have used excessively high strength by mismatch in coefficients of thermal expansion between ultrafine-grained 5083 aluminum and B4C particles. Since the ultrafine-grained 5083 aluminum features high strength, an additional enhancement of strength by this kind of thermoelastic mismatch between matrix and particle may hardly occur. Using finite-element methods with strain-gradient plasticity, this problem is examined computationally and an enhanced procedure of strength prediction for trimodal composites is suggested.

Numerical investigations of strain-gradient plasticity with reference to non-homogeneous deformations

In this work, a higher-order irrotational strain gradient plasticity theory is studied in the small strain regime. A detailed numerical study is based on the problem of simple shear of a non-homogeneous block comprising an elastic-plastic material with a stiff elastic inclusion. Combinations of micro-hard and micro-free boundary conditions are used. The strengthening and hardening behaviour is explored in relation to the dissipative and energetic length scales. There is a strong dependence on length scale with the imposition of micro-hard boundary conditions. For micro-free conditions there is marked dependence on dissipative length scale of initial yield, though the differences are small in the post-yield regime. In the case of hardening behaviour, the variation with respect to energetic length scale is negligible. A further phenomenon studied numerically relates to the global nature of the yield function for the dissipative problem; this function is given as the least upper bound of a function of plastic strain increment, and cannot be determined analytically. The accuracy of an upper-bound approximation to the yield function is explored, and found to be reasonably sharp in its prediction of initial yield.

Nonlinear plastic buckling analysis of Micro–Scale thin plates established on higher order mechanism-based strain gradient plasticity framework

The present study investigates a nonlinear plastic buckling analysis of microplate by using the Mechanism-based Strain Gradient (MSG) plasticity. The MSG as a higher order theory is based on the Taylor model of dislocation interaction, consisting a multiple plastic work hardening. It extends a virtual work postulating the existence of higher-order stress work-conjugate at the in-plane traction of the microplate to the plastic strain gradient. The energy minimizing method is applied to obtain plastic buckling critical forces per each length scales and plastic hardening exponents. An iterative quadratic extrapolation (IQE) is implemented for linearization of nonlinear terms generated due to spatial variations of constitutive equations. The results indicate significant dependence of the critical force on the length scale as well as plastic work hardening exponent. Verifications are carefully done among two limit states of the elastic case and particular plastic analysis of a macro plate.

Structure and Deformation Behavior of Ti-SiC Composites Made by Mechanical Alloying and Spark Plasma Sintering.

Combining high energy ball milling and spark plasma sintering is one of the most promising technologies in materials science. The mechanical alloying process enables the production of nanostructured composite powders that can be successfully spark plasma sintered in a very short time, while preserving the nanostructure and enhancing the mechanical properties of the composite. Composites with MAX phases are among the most promising materials. In this study, Ti/SiC composite powder was produced by high energy ball milling and then consolidated by spark plasma sintering. During both processes, Ti3SiC2, TiC and Ti5Si3 phases were formed. Scanning electron microscopy, energy-dispersive X-ray spectroscopy and X-ray diffraction study showed that the phase composition of the spark plasma sintered composites consists mainly of Ti3SiC2 and a mixture of TiC and Ti5Si3 phases which have a different indentation size effect. The influence of the sintering temperature on the Ti-SiC composite structure and properties is defined. The effect of the Ti3SiC2 MAX phase grain growth was found at a sintering temperature of 1400–1450 °C. The indentation size effect at the nanoscale for Ti3SiC2, TiC+Ti5Si3 and SiC-Ti phases is analyzed on the basis of the strain gradient plasticity theory and the equation constants were defined.

Plastic microstress for a defect energy dependent on Burgers tensor

1. Aifantis E.C., 1984, On the microstructural origin of certain inelastic models, Journal of Engineering Materials and Technology, Transactions of the ASME, 106, 326-330. Google Scholar

Plasticity as the $${\Gamma}$$ Γ -Limit of a Two-dimensional Dislocation Energy: The Critical Regime Without the Assumption of Well-Separateness

In this paper, a strain-gradient plasticity model is derived from a mesoscopic model for straight parallel edge dislocations in an infinite cylindrical crystal. The main difference to existing work is that in this work the well-separateness of disloactions is not assumed. In order to prove meaningful lower bounds the ball construction technique, which was developed in the context of Ginzburg-Landau by Jerrard and Sandier, is adapted and modified. To overcome the difficulty of a loss of rigidity on thin annuli during the ball construction a combination of combinatorial arguments and local modifications of the occurring elastic strains is presented.

Investigation of a gradient enriched Gurson-Tvergaard model for porous strain hardening materials

Size effects in a strain hardening porous solid are investigated using the Gurson-Tvergaard (GT) model enriched by a constitutive length parameter, as proposed by Niordson and Tvergaard [C.F. Niordson, V. Tvergaard, A homogenised model for size effects in porous metals, J. Mech. Phys. Solids (2019)]. The results are compared with unit cell calculations of regularly distributed voids embedded in a strain gradient enhanced matrix material. The strain gradient plasticity theory proposed by Fleck and Willis [N.A. Fleck, J.R. Willis, A mathematical basis for strain gradient plasticity theory. Part II: tensorial plastic multiplier, J. Mech. Phys. Solids 57 (2009) 1045–1057], extended to finite strains, is adopted for the cell model, consistent with the gradient enriched Gurson model. The gradient model allows for a material length parameter to enter the constitutive framework for dimensional consistency, while the enriched GT model has the same length parameter introduced through prefactors of the usual q1 and q2 factors. The continuum model featuring size-dependent Tvergaard-constants is used to investigate a strain hardening material with the strain gradient plasticity enriched cell model as reference. The two models are compared for three triaxialities, three initial void volume fractions, and three hardening exponents. The enriched GT model captures the effect of elevated yield point and suppressed void growth with increasing length parameter for all the cases investigated. The agreement between the models is good until severe void distortion or plastic flow localisation between neighbouring voids. The response curves and void growth curves for the enriched GT model deviate from those of the cell model at high axial strains. Void shape plots, which are only available for the cell model, show that the length parameter influences the shape of the void which in turn has impact on the material response curves and the void evolution. This is not captured by the enriched GT model as the voids are accounted for solely through a volume fraction parameter.

A finite strain FE-Implementation of the Fleck-Willis gradient theory: Rate-independent versus visco-plastic formulation

This paper presents a numerical basis for finite strain analysis using the higher order flow theory by Fleck and Willis [2009, A mathematical basis for strain-gradient plasticity theory - part II: tensorial plastic multiplier, J. Mech. Phys. Solids, 57, 1045–1057]. The rate-dependent (visco-plastic) formulation shares similarities with its rate-independent (generalized J2-flow) counterpart, and a combined presentation is laid out. The visco-plastic formulation [2018, A homogenized model for size-effects in porous metals, J. Mech. Phys. Solids, DOI: 10.1016/j.jmps.2018.09.004] is revisited to set the scene for the development of the rate-independent model, but the two formulations differ substantially in the numerical implementation as the rate-independent framework possesses a challenge in determining the yield of individual plastic zones. The framework developed readily accounts for repeated loading/unloading and handles the interaction of multiple plastic zones. The method presented exploits the image analysis techniques that was first combined with the Fleck-Willis theory in Nielsen and Niordson [2014, A numerical basis for strain-gradient plasticity theory: rate-independent and rate-dependent formulations, J. Mech. Phys. Solids 63, 113–127], but extends it to finite strains by adopting an updated Lagrangian framework. Besides somewhat standard extensions to deal with finite deformations, the framework largely resembles that of small strains. In fact, the adopted image analysis can be conducted in the undeformed configuration without any approximations as it relates only to the connectivity of plastic zones on the finite element mesh. The new framework is focused on the deformation and localization in both uniform and notched round bars by formulating the numerical model in an axi-symmetric setup. Comparisons between the visco-plastic and the rate-independent formulations are presented.

Fractional strain-gradient plasticity

We develop a strain-gradient plasticity theory based on fractional derivatives of plastic strain and assess its ability to reproduce the scaling laws and size effects uncovered by the recent experiments of Mu et al. (2014, 2016, 2017) on copper thin layers undergoing plastically constrained simple shear. We show that the size-scaling discrepancy between conventional strain-gradient plasticity and the experimental data is resolved if the inhomogeneity of the plastic strain distribution is quantified by means of fractional derivatives of plastic strain. In particular, the theory predicts that the size scaling exponent is equal to the fractional order of the plastic-strain derivatives, which establishes a direct connection between the size scaling of the yield stress and fractionality.

Mode I crack tip fields: Strain gradient plasticity theory versus J2 flow theory

The mode I crack tip asymptotic response of a solid characterised by strain gradient plasticity is investigated. It is found that elastic strains dominate plastic strains near the crack tip, and thus the Cauchy stress and the strain state are given asymptotically by the elastic K-field. This crack tip elastic zone is embedded within an annular elasto-plastic zone. This feature is predicted by both a crack tip asymptotic analysis and a finite element computation. When small scale yielding applies, three distinct regimes exist: an outer elastic K field, an intermediate elasto-plastic field, and an inner elastic K field. The inner elastic core significantly influences the crack opening profile. Crack tip plasticity is suppressed when the material length scale ℓ of the gradient theory is on the order of the plastic zone size estimation, as dictated by the remote stress intensity factor. A generalized J-integral for strain gradient plasticity is stated and used to characterise the asymptotic response ahead of a short crack. Finite element analysis of a cracked three point bend specimen reveals that the crack tip elastic zone persists in the presence of bulk plasticity and an outer J-field.