As an extension to strain-gradient models of size-dependent plastic behaviour, this work proposes a model for a stress-gradient theory. The model is distinguished from earlier works on the topic by its being embedded in a thermodynamically consistent framework. The development is carried out in the context of single-crystal plasticity, and draws on thermodynamically consistent models for single-crystal conventional and strain-gradient plasticity. The model is explored numerically using the example of torsion of a thin wire comprising a face centred cubic crystal, and its behaviour compared with that based on a recent disequilibrium density model of size-dependent plasticity.
There is a long standing technological problem in which a stress dwell during cyclic loading at room temperature in Ti causes a significant fatigue life reduction. It is thought that localised time dependent plasticity in soft grains oriented for easy plastic slip leads to load shedding and an increase in stress within a neighbouring hard grain poorly oriented for easy slip. Quantifying this time dependent plasticity process is key to successfully predicting the complex cold dwell fatigue problem. This work uses a novel approach of in situ synchrotron X-ray diffraction during stress relaxation…
The statistics and origin of the first discrete plastic event in a one dislocation dynamics simulation are studied. This is done via a linear stability analysis of the evolving dislocation configuration up to the onset of irreversible plasticity. It is found, via a fold catastrophe, the dislocation configuration prior to loading directly determines the stress at which the plastic event occurs and that between one and two trigger dislocations are involved. The resulting irreversible plastic strain arising from the instability is found to be highly correlated with these triggering dislocations.
Selective laser melting is receiving increasing interest as an additive manufacturing technique. Residual stresses induced by the large temperature gradients and inhomogeneous cooling process can favour the generation of cracks. In this work, a crystal plasticity finite element model is developed to simulate the formation of residual stresses and to understand the correlation between plastic deformation, grain orientation and residual stresses in the additive manufacturing process. The temperature profile and grain structure from thermal-fluid flow and grain growth simulations are implemented …
The first plastic event occurring in discrete intermittent plasticity, as for example a pop-in seen in nano-indentation, is evaluated with extreme value statistics. It is found that when the same deformation is repeated many times, the average of the stress at this first event is related to the deforming volume via an exponentially truncated power-law. The present work demonstrates this trend and the expected Weibull fluctuation around it. The statistical formalism is shown to apply to the nano-indentation data of Morris {\em et al}, Phys. Rev. Lett. 106, 165502 (2011), as well as to dislocati…
Localization of plastic strain induced by softening can be objectively described by a regularized plasticity model that postulates a dependence of the current yield stress on a nonlocal softening variable defined by a differential (gradient) expression. This paper presents analytical solutions of the one-dimensional localization problem under certain special nonuniform stress distributions. The one-dimensional problem can be interpreted as describing either a tensile bar with variable cross section, or a beam subjected to a nonuniform bending moment. Explicit as well as implicit gradient formu…
Voids can limit the life of engineering components. This motivates us to understand local plasticity around voids in a nickel base superalloy combining experiments and simulations. Single crystal samples were deformed in tension with in-situ high angular resolution electron back scatter diffraction to probe the heterogeneous local stress field under load; the reference stress is informed by crystal plasticity finite element simulations. This information is used to understand the activation of plastic deformation around the void. Our investigation indicates that while the resolved shear stress …
In an attempt to bridge the gap between atomistic and continuum plasticity simulations of hydrogen in iron, we present three dimensional discrete dislocation plasticity simulations incorporating the hydrogen elastic stress and a hydrogen dependent dislocation mobility law. The hydrogen induced stress is incorporated following the formulation derived by Gu and El-Awady (2018) which here we extend to a finite boundary value problem, a microcantilever beam, via the superposition principle. The hydrogen dependent mobility law is based on first principle calculations by Katzarov et al. (2017) and w…
We study the effect of Stacking Fault Energy (SFE) on the deformation behaviour of copper and copper-aluminium alloys using Molecular Dynamics (MD) simulation. We find that both yield stress and the magnitude of stress drop at yield decrease with increasing Al content. This "anomalous" softening behaviour is explained on the basis of nucleation controlled yielding behaviour. Further, the decrease in stress drop is rationalised in terms of the stored energy available at yielding - we show that this decreases with increasing Al. As a result, the maximum dislocation density is found to decrease w…
We extend our earlier shear-transformation-zone (STZ) theory of amorphous plasticity to include the effects of thermally assisted molecular rearrangements. This version of our theory is a substantial revision and generalization of conventional theories of flow in noncrystalline solids. As in our earlier work, it predicts a dynamic transition between jammed and flowing states at a yield stress. Below that yield stress, it now describes thermally assisted creep. We show that this theory accounts for the experimentally observed strain-rate dependence of the viscosity of metallic glasses, and that…
The influence on macroscopic work hardening of small, spherical, elastic particles dispersed within a matrix is studied using an isotropic strain gradient plasticity framework. An analytical solution, based on a recently developed yield strength model is proposed. The model accounts for random variations in particle size and elastic properties, and is numerically validated against FE solutions in 2D/3D material cell models. Excellent agreement is found as long as the typical particle radius is much smaller than the material length scale, given that the particle volume fraction is not too large…
Electroplasticity is defined as the reduction in flow stress of a material undergoing deformation on passing an electrical pulse through it. The lowering of flow stress during electrical pulsing has been attributed to a combination of three mechanisms: softening due to Joule-heating of the material, de-pinning of dislocations from paramagnetic obstacles, and the electron-wind force acting on dislocations. However, there is no consensus in literature regarding the relative magnitudes of the reductions in flow stress resulting from each of these mechanisms. In this paper, we extend a dislocation…
Because of the enormous range of time and space scales involved in dislocation dynamics, plastic modeling at macroscale requires a continuous formulation. In this paper, we present a rigorous formulation of the transition between the discrete, where plastic flow is resolved at the scale of individual dislocations, and the continuum, where dislocations are represented by densities. First, we focus on the underlying coarse-graining procedure and show that the emerging correlation-induced stresses are scale-dependent. Each of these stresses can be expanded into the sum of two components. The firs…
Finite element analysis of stress about a blunt crack tip, emphasizing finite strain and phenomenological and mechanism-based strain gradient plasticity (SGP) formulations, is integrated with electrochemical assessment of occluded-crack tip hydrogen (H) solubility and two H-decohesion models to predict hydrogen environment assisted crack growth properties. SGP elevates crack tip geometrically necessary dislocation density and flow stress, with enhancement declining with increasing alloy strength. Elevated hydrostatic stress promotes high-trapped H concentration for crack tip damage; it is impe…
Here we present a model to study the micro-plastic regime of a stress-strain curve. In this model an explicit dislocation population represents the mobile dislocation content and an internal shear-stress field represents a mean-field description of the immobile dislocation content. The mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm, whilst the shear-stress field is chosen to be a sinusoidal function of distance along the mat direction. The latter, defined by a periodic length and a shear-stress amplitude, represents a pre-…
A long standing postulate in crystal plasticity of metals is that yielding commences once the resolved shear stress on a slip plane reaches a critical value. This assumption, known as Schmid law, implies that the onset of plasticity is independent of the normal stress acting on the slip plane. We examine the validity of this assumption in single crystal perfect lattices at zero temperature by subjecting them to a wide range of combined normal and shear stresses and identifying the onset of plasticity. We employ phonon stability analysis on four distinct single crystal metals and identify the o…
High-energy synchrotron x-rays are used to track grain rotations and the micromechanical evolution of a hexagonal Ti-7Al microstructure as it is cyclically loaded below its macroscopic yield stress. The evolution of the grains through 200 cycles reveals a continual change in von Mises stress and orientation across the entire specimen indicating a slow accumulation of plasticity even though the sample was cycled below its macroscopic yield. Grain reorientation is consistent with the development of a tension texture despite the negligible magnitude of (macroscopic) plastic strain. It is observed…
Dislocation based modeling of plasticity is one of the central challenges at the crossover of materials science and continuum mechanics. Developing a continuum theory of dislocations requires the solution of two long standing problems: (i) to represent dislocation kinematics in terms of a reasonable number of variables and (ii) to derive averaged descriptions of the dislocation dynamics (i.e. material laws) in terms of these variables. The kinematic problem (i) was recently solved through the introduction of continuum dislocation dynamics (CDD), which provides kinematically consistent evolutio…
In this work, the well known power-law relation between strength and sample size, $d^{-n}$, is derived from the knowledge that a dislocation network exhibits scale-free behaviour and the extreme value statistical properties of an arbitrary distribution of critical stresses. This approach yields $n=(τ+1)/(α+1)$, where $α$ reflects the leading order algebraic exponent of the low stress regime of the critical stress distribution and $τ$ is the scaling exponent for intermittent plastic strain activity. This quite general derivation supports the experimental observation that the size effect paradig…
Digital image correlation (DIC) and crystal plasticity simulation were utilised to study cold dwell behaviour in a coarse grain Ti-6Al alloy at 3 different temperatures up to 230 C. Strains extracted from large volume grains were measured during creep by DIC and were used to calibrate the crystal plasticity model. The values of critical resolved shear stresses (CRSS) of the two main slip systems (basal and prismatic) were determined as a function of temperature. Stress along paths across the boundaries of two grain pairs, (1) a `rogue' grain pair and (2) a `non-rogue' grain pair, were determin…
A 3D unit cell model containing eight different spherical particles embedded in a homogeneous strain gradient plasticity (SGP) matrix material is presented. The interaction between particles and matrix is controlled by an interface model formulated within the higher order SGP theory used. Strengthening of the particle reinforced material is investigated in terms of the increase in macroscopic yield stress. The results are used to validate a closed form strengthening relation proposed by the authors, which suggests that the increase in macroscopic yield stress is governed by the interface stren…
In a recent publication, we derived the mesoscale continuum theory of plasticity for multiple-slip systems of parallel edge dislocations, motivated by the statistical-based nonlocal continuum crystal plasticity theory for single-glide due to Yefimov et al. (2004b). In this dislocation field theory (DiFT) the transport equations for both the total dislocation densities and geometrically necessary dislocation densities on each slip system were obtained from the Peach-Koehler interactions through both single and pair dislocation correlations. The effect of pair correlation interactions manifested…
In this paper we present a thermodynamically consistent material model which is capable of modelling ductile-to brittle failure mode transition in ductile material undergoing deformations at high strain rates, and demonstrate the performance of the model in a numerical study using a fully coupled thermal-structural finite element analysis of a notched aluminium alloy specimen loaded in cyclic tension. The model is based on an objective representation of the deformation and stress measures and on a rate type constitutive equations. It does not only complies with the principles of material model…
The present report describes a big data numerical study of crystal plasticity finite element (CPFE) modelling using static and grain-based meshing to investigate the dependence of the results on the discretization approach. Static mesh refers to the integration point-based representation of the microstructure in which the integration points (IPs) within a finite element may belong to different grains, while in the grain-based meshing the minimum discretization unit is an element that may only belong to one grain. The crystal plasticity constitutive law was coded using UMAT subroutine within co…
This work presents a three-dimensional constitutive model for shape memory alloys considering the TRansformation-Induced Plasticity (TRIP) as well as the Two-Way Shape Memory Effect (TWSME) through a large deformation framework. The presented logarithmic strain based model is able to capture the large strains and rotations exhibited by SMAs under general thermomechanical cycling. By using the martensitic volume fraction, transformation strain, internal stress, and TRIP strain tensors as internal state variables, the model is capable to capture the stress-dependent TRIP generation when SMAs are…
The nonequilibrium dynamics of diffusion-mediated plasticity and creep in materials subjected to constant load at high homologous temperatures is studied atomistically using Phase Field Crystal (PFC) methods. Creep stress and grain size exponents obtained for nanopolycrystalline systems, $m \simeq 1.02$ and $p \simeq 1.98$, respectively, closely match those expected for idealized diffusional Nabarro-Herring creep. These exponents are observed in the presence of significant stress-assisted diffusive grain boundary migration, indicating that Nabarro-Herring creep and stress-assisted boundary mig…
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 es…
A toy model of two dimensional nanoindentation in finite crystals is proposed. The crystal is described by periodized discrete elasticity whereas the indenter is a rigid strain field of triangular shape representing a hard knife-like indenter. Analysis of the model shows that there are a number of discontinuities in the load vs penetration depth plot which correspond to the creation of dislocation loops. The stress vs depth bifurcation diagram of the model reveals multistable stationary solutions that appear as the dislocation-free branch of solutions develops turning points for increasing str…
In this work, the influence of the plastic size effect on the fracture process of metallic materials is numerically analyzed using the strain-gradient plasticity (SGP) theory established from the Taylor dislocation model. Since large deformations generally occur in the vicinity of a crack, the numerical framework of the chosen SGP theory is developed for allowing large strains and rotations. The material model is implemented in a commercial finite element (FE) code by a user subroutine, and crack-tip fields are evaluated thoroughly for both infinitesimal and finite deformation theories by a bo…
In this work a general framework for damage and fracture assessment including the effect of strain gradients is provided. Both mechanism-based and phenomenological strain gradient plasticity (SGP) theories are implemented numerically using finite deformation theory and crack tip fields are investigated. Differences and similarities between the two approaches within continuum SGP modeling are highlighted and discussed. Local strain hardening promoted by geometrically necessary dislocations (GNDs) in the vicinity of the crack leads to much higher stresses, relative to classical plasticity predic…
The microstructure evolution due to thermomechanical treatment of metals can largely be described by viscoplastic deformation, nucleation and grain growth. These processes take place over different length and time scales which present significant challenges when formulating simulation models. In particular, no overall unified field framework exists to model concurrent viscoplastic deformation and recrystallization and grain growth in metal polycrystals. In this work a thermodynamically consistent diffuse interface framework incorporating crystal viscoplasticity and grain boundary migration is …
There is a long-standing technological problem in which a stress dwell during cyclic loading at room temperature in Ti causes a significant fatigue life reduction. It is thought that localised time dependent plasticity in soft grains favourably oriented for easy plastic slip leads to load shedding and an increase in stress within a neighbouring hard grain poorly oriented for easy slip. Quantifying this time dependent plasticity process is key to understand the complex cold dwell fatigue problem. Knowing the effect of operating temperature and oxygen content on cold dwell fatigue will be benefi…
In this paper, we present a thorough stress analysis of the Cu-Zr metallic-glass composite with embedded B2 particles subject to a martensitic transformation. Within the framework of the Eshelby theory, we are able to explain, in a quantitative manner, (1) the formation of three types of shear bands with distinct morphologies as observed experimentally in the severely deformed Cu-Zr metallic-glass composite and (2) the work hardening ability of the Cu-Zr metallic-glass composite as related to the coupled effects of elastic back stress and elastic mismatch caused by the martensitic transformati…
A crystal plasticity theory was developed for use in simulations of dynamic loading at high pressures and strain rates. At pressures of the order of the bulk modulus, compressions o(100%) may be induced. At strain rates o(10^9)/s or higher, elastic strains may reach o(10%), which may change the orientation of the slip systems significantly with respect to the stress field. Elastic strain rather than stress was used in defining the local state, providing a more direct connection with electronic structure predictions and consistency with the treatment of compression in initial value problems in …
A Crystal Plasticity Finite Element (CPFE) framework is proposed for modeling the non-Schmid yield behavior of L12 type Ni3Al crystals and Ni-based superalloys. This framework relies on the estimation of the non-Schmid model parameters directly from the orientation- and temperature-dependent experimental yield stress data. The inelastic deformation model for Ni3Al crystals is extended to the precipitate phase of Ni-based superalloys in a homogenized dislocation density based crystal plasticity framework. The framework is used to simulate the orientation- and temperature-dependent yield of Ni3A…
The existence of a well defined yield stress, where a macroscopic piece of crystal begins to plastically flow, has been one of the basic observations of materials science. In contrast to macroscopic samples, in micro- and nanocrystals the strain accumulates in distinct, unpredictable bursts, which makes controlled plastic forming rather difficult. Here we study by simulation, in two and three dimensions, plastic deformation of submicron objects under increasing stress. We show that, while the stress-strain relation of individual samples exhibits jumps, its average and mean deviation still spec…
We present a thermodynamic description of crystal plasticity. Our formulation is based on the Langer-Bouchbinder-Lookman thermodynamic dislocation theory (TDT), which asserts the fundamental importance of an effective temperature that describes the state of configurational disorder and therefore the dislocation density of the crystalline material. We extend the TDT description from isotropic plasticity to crystal plasticity with many slip systems. Finite-element simulations show favourable comparison with experiments on polycrystal fcc copper under uniaxial compression, tension, and simple she…
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 …
Three-dimensional discrete dislocation dynamics simulations are used to study strain-controlled plastic deformation of face-centered cubic aluminium single crystals. After describing the rate and size dependence of the average stress-strain curves, we study the power-law distributed strain bursts and the average avalanche shapes, and find a universal power-law exponent $τ\approx 1.0$ for all imposed strain rates and system sizes, characterizing both the event sizes and their durations. We discuss the dependence of our results on loading rate and compare these with previous studies of strain-co…
In this work, distortion gradient plasticity is used to gain insight into material deformation ahead of a crack tip. This also constitutes the first fracture mechanics analysis of gradient plasticity theories adopting Nye's tensor as primal kinematic variable. First, the asymptotic nature of crack tip fields is analytically investigated. We show that an inner elastic region exists, adjacent to the crack tip, where elastic strains dominate plastic strains and Cauchy stresses follow the linear elastic stress singularity. This finding is verified by detailed finite element analyses using a new nu…
Crystal plasticity occurs by deformation bursts due to the avalanche-like motion of dislocations. Here we perform extensive numerical simulations of a three-dimensional dislocation dynamics model under quasistatic stress-controlled loading. Our results show that avalanches are power-law distributed, and display peculiar stress and sample size dependence: The average avalanche size grows exponentially with the applied stress, and the amount of slip increases with the system size. These results suggest that intermittent deformation processes in crystalline materials exhibit an extended critical-…
The complex incremental behavior of granular materials is explored with multi-directional loading probes. An advanced discrete element model (DEM) was used to examine the reversible and irreversible strains for small loading probes, which follow an initial monotonic axisymmetric triaxial loading. The model used non-convex non-spherical particles and an exact implementation of the Hertz-like Cattaneo-Mindlin model for the contact interactions. Rectilinear true-triaxial probes were used in the study (i.e., no direct shear strain or principal stress rotation), with small strains increments of $2\…
Uncertainty-Aware Machine-Learning Framework for Predicting Dislocation Plasticity and Stress-Strain Response in FCC Alloys
Johns Hopkins University（约翰斯·霍普金斯大学）；Institute of High Performance Computing, Agency for Science, Technology and Research（高性能计算研究所，科技研究局）；Xi’an Jiaotong University（西安交通大学）；City University of Hong Kong（香港城市大学）
We propose a theoretical framework for dealing with a transient polymer network undergoing small deformations, based on the rate of breaking and re-forming of network crosslinks and the evolving elastic reference state. In this framework, the characteristics of the deformed transient network at microscopic and macroscopic scales are naturally unified. Microscopically, the breakage rate of the crosslinks is affected by the local force acting on the chain. Macroscopically, we use the classical continuum model for rubber elasticity to describe the structure of the deformation energy, whose refere…
The sequence of deformation bursts during plastic deformation exhibits scale-free features. In addition to the burst or avalanche sizes and the rate of avalanches the process is characterized by correlations in the series which become manifest in the resulting shape of the stress-strain curve. We analyze such features of plastic deformation with 2D and 3D simulations of discrete dislocation dynamics models and we show, that only with severe plastic deformation the ensuing memory effects become negligible. The role of past deformation history and dislocation pinning by disorder are studied. In …
Micron-scale single crystalline materials deform plastically via large intermittent strain bursts that make the deformation process unpredictable. Here we investigate this stochastic phenomenon by analysing the plastic response of an ensemble of specimens differing only in the initial arrangement of dislocations. We apply discrete dislocation dynamics simulations and microcompression tests on identically fabricated Cu single crystalline micropillars. We find that a characteristic yield stress can be defined in the average sense for a given specimen ensemble, where the average and the variance …
Built on the tenets of rational thermodynamics, this article proposes a theory of strain gradient thermo-visco-plasticity for isotropic polycrystalline materials under high strain rates. The effect of micro-inertia, which arises due to dynamically evolving microstructural defects, is brought to bear on the macro-continuum through a micro-force balance. Constitutive modelling of dissipative micro-stresses incorporates relaxation time parameters to account for the time lags of the dissipative fluxes in attaining a steady state. Augmentation of the micro-force balance with constitutive relations …
This work explains the occurrence of transformation-induced plasticity via stress-assisted martensite, when designing ultrafine-grained duplex steels. It is found that, when the austenite is reduced to a fine scale of about 300 nm, the initial deformation-induced microstructure can be dominated by parallel lamellae of epsilon martensite or mechanical twinning, which cannot efficiently provide nucleation sites for strain-induced martensite. Hence, alpha martensite nucleation occurs independently by a stress-assisted process that enhances transformation-induced plasticity in ultrafine-grained au…
Plastic deformation in microscale differs from the macroscopic plasticity in two respects: (i) the flow stress of small samples depends on their size (ii) the scatter of plasticity increases significantly. In this work we focus on the scatter of plasticity. We statistically characterize the deformation process of micropillars under tension using results from discrete dislocation dynamics (DDD) simulations. We then propose a stochastic microplasticity model which uses the extracted information from the above statistical characterization to make statistical predictions regarding the micropillar …
Structural transitions are invariably affected by lattice distortions. If the body is to remain crack-free, the strain field cannot be arbitrary but has to satisfy the Saint-Venant compatibility constraint. Equivalently, an incompatibility constraint consistent with the actual dislocation network has to be satisfied in media with dislocations. This constraint can be incorporated into strain-based free energy functionals to study the influence of dislocations on phase stability. We provide a systematic analysis of this constraint in three dimensions and show how three incompatibility equations …
Simulations are used to examine the microscopic origins of strain hardening in polymer glasses. While stress-strain curves for a wide range of temperature can be fit to the functional form predicted by entropic network models, many other results are fundamentally inconsistent with the physical picture underlying these models. Stresses are too large to be entropic and have the wrong trend with temperature. The most dramatic hardening at large strains reflects increases in energy as chains are pulled taut between entanglements rather than a change in entropy. A weak entropic stress is only obser…
Dislocations are the carriers of plasticity in crystalline materials. Their collective interaction behavior is dependent on the strain rate and sample size. In small specimens, details of the nucleation process are of particular importance. In the present work, discrete dislocation dynamics (DDD) simulations are performed to investigate the dominant yielding mechanisms in single crystalline copper pillars with diameters ranging from 100 to 800 nm. Based on our simulations with different strain rates and sample size, we observe a transition of the relevant nucleation mechanism from "dislocation…
Crystal plasticity theory is often employed to predict the mesoscopic states of polycrystalline metals, and is well-known to be costly to simulate. Using a neural network with convolutional layers encoding correlations in time and space, we were able to predict the evolution of the stress field given only the initial microstructure and external loading. In comparison to our recent work we were able to predict not only the spatial average of the stress response but the field itself. We show that the stress fields and their rates are in high fidelity with the crystal plasticity data and have no …
We deform representative volume elements of amorphous carbon obtained from melt-quenches in molecular dynamics calculations using bond-order and machine learning interatomic potentials. A Drucker-Prager law with a zero-pressure flow stress of $41.2$~GPa and an internal friction coefficient of $0.39$ describes the deviatoric stress during flow as a function of pressure. We identify the mean coordination number as the order parameter describing this flow surface. However, a description of the dynamical relaxation of the quenched samples towards steady-state flow requires an additional order para…
We present a dislocation density-based strain hardening model for single crystal copper through a systematic coarse-graining analysis of more than 200 discrete dislocation dynamics (DDD) simulations of plastic deformation under uniaxial tension. The proposed constitutive model has two components: a generalized Taylor relation connecting resolved shear stresses to dislocation densities on individual slip systems, and a generalized Kocks-Mecking model for dislocation multiplication. The DDD data strongly suggests a logarithmic dependence of flow stress on the plastic shear strain rate on each sl…
