Forest Dislocation Density Research Articles

Exploring grain size influence on tensile behavior of 316 H austenitic stainless steel at high temperature: A phenomenological dislocation model

The high temperature mechanical behavior of 316 H austenitic stainless steel with different grain size was investigated in this study through the use of a novel phenomenological dislocation model finite element method. The study was conducted by producing steel specimens with varying grain sizes through adjustments in annealing temperature and time, followed by high temperature tensile tests at 550°C. The finite element method, when integrated with a phenomenological dislocation model that considers grain size parameters, effectively captured the mechanical response under different grain size conditions. Additionally, the two-variable Kocks-Mecking(KM) model accurately captured the grain size effect in 316 H austenitic steel and effectively depicted its tensile flow behavior. The model’s predictions were based on the evolution of forest dislocation density and mobile dislocation density, proving to be a reliable tool for analyzing the microstructural evolution of 316 H austenitic steel specimens with varying grain sizes. This study provides insight into the effect of grain size on the high temperature strength of austenitic stainless steels and demonstrates the utility of a novel phenomenological model finite element method for predicting the mechanical behavior of polycrystalline materials.

Nonlinear unloading-reloading behavior and macro/micro models during multi-pass cyclic hot compression of aluminum alloys

Unloading-reloading behavior is of crucial importance in multi-pass forming of metal materials, the nonlinear of which creates challenges for the analysis of springback and residual stress of the precision manufacture components. Especially for the hot bulk-formed material at elevated temperature, the thermal-mechanical coupling effect makes the unloading-reloading process more complicated and thus more difficult to model and control, which remains to be solved. In this paper, the unloading-reloading behavior dependent on strain rate and deformation temperature and its model of hot compressive deformed aluminum alloys at elevated temperature were studied from the macroscopic and microscopic perspectives. The effect of deformation temperature, strain rate and strain on the unloading-reloading behavior of hot compressive deformed aluminum alloys were revealed, and corresponding micro-mechanism was elucidated and then discussed. Macroscopically, a strain-, strain rate- and temperature-dependent nonlinear macroscopic model describing unloading–reloading process was established, which can be used to describe the unloading stress relaxation and predict springback during multi-pass hot bulk forming. Microscopically, it is suggested that the relaxations of back stress and forest dislocation hardening stress collectively affect the unloading stress relaxation, due to the decrease of mobile and immobile (forest) dislocation density in unloading, while the dislocation-related behavior before yield, i.e. the dislocations’ bowing-out under pinning, affects the reloading behavior. Based on the micro-mechanism, a physically-based micro-model for describing unloading–reloading was constructed which further is used to clarify the physical mechanism of internal stress relaxation and inelastic strain recovery under multi-pass hot compression deformation. The comparison of the calculation results of the models with those of experiments indicated that the established unloading–reloading macroscopic and microscopic models are able to predict the stress–strain response upon the unloading–reloading loop of multi-pass cyclic hot compression deformation.

Yield strength modeling of an Al-Cu-Li alloy through circle rolling and flow stress superposition approach

In the present work, the shear stress required to move a dislocation through an array of randomly distributed T1 precipitates (Al2CuLi), forest dislocations and solutes was computed using the circle rolling method. The evolution of shear stress as a function of swept area was determined to calculate the shear resistance of T1 precipitates, forest dislocations and solutes for different ageing conditions. The inputs for the circle rolling simulations, such as the critical breaking angle of the obstacles, were obtained by performing detailed microstructural analysis at different length scales to estimate forest dislocation density, diameter, thickness, and number density of T1 precipitates. The individual shear resistance of precipitates, dislocations and solutes were suitably superimposed using a flow stress addition law, where the superposition exponent is dependent on the critical breaking angles of the T1 precipitates, forest dislocations and solutes. The present approach outperforms previous models based on the linear superposition of strengthening components in predicting the yield strength evolution of AA 2195 alloy as a function of pre-strain.

Re-examination and comparison of activation volume, strain rate sensitivity, and dislocation density for coarse grain and CGP-processed UFG materials

The frequently used metallurgical parameters such as activation volume, internal stress, strain rate sensitivity, and dislocation density are identified in ultrafine-grained (UFG) materials through erudite analytical relations. It is difficult to analyze such parameters optically since the material consists of highly distorted and inhomogeneous microstructure due to the large strain imposed during the constrained groove pressing (CGP) process. Hence, transient tests like stress relaxation tests are advantageous over the optical to use. Apart from the identification of parameters in low-carbon steel materials using uniaxial loading and stress relaxation tests, the relationship between the parameters is nobly investigated in as-received and UFG structure samples produced through the CGP process. In the relationship, the exponential-power law constitutive to describe the accurate stress–strain curves, identification of mobile and forest dislocation density during strain hardening, and exponential variation between grain size and activation volume are proposed. The replenishment or exhaustion of mobile and forest dislocation density significantly affects the metallurgical parameters during uniaxial tensile loading followed by stress relaxation testing. In this experimental and analytical work, the CGP process produces a UFG material that has lower activation volume and higher stress drop, mobile dislocation density, forest dislocations density, strain rate sensitivity, and internal stress.

The effect of size, orientation and temperature on the deformation of microcast silver crystals

Monocrystalline microwires of silver are produced by means of a novel casting process that combines precise mold production by silicon microfabrication with molten metal pressure infiltration. The wires have a shape amenable to tensile testing and a diameter from a few to several tens of micrometer. Displacement-controlled in-situ tensile tests are conducted at room-temperature, 200 °C and 400 °C. Data are compared with bulk counterparts of the same silver cast to have a diameter near 1 mm. Results show clear evidence of small-scale plasticity. The influence of size, crystal orientation, and temperature on the deformation, yield, strain burst statistics and strain hardening of these micrometric dense metal samples is analyzed. Yield stress values agree with the single arm source model provided that the density of forest dislocations comprises a scale-dependent component likely resulting from thermal mismatch stresses. Statistics of strain burst amplitudes agree with an exponentially truncated power-law complementary cumulative distribution. The power-law exponent agrees with predictions of mean-field theory, while the cutoff intensity exhibits no visible dependence on sample diameter or test temperature and agrees with predictions of current theory.

An advanced mean field dislocation density reliant physical model to predict the creep deformation of 304HCu austenitic stainless steel

Physical based creep modelling enables to understand the life-limiting factors that are required for a safe and economic operation of power plant components. Thus, herein an improved physical approach to address the creep behavior of 304HCu austenitic stainless steel is presented. This approach combines a dislocation density reliant physical model with a continuum damage mechanics (CDM) model. Two different dislocation densities: mobile and forest, and dislocation mean free path are used to describe the substructure in order to model the creep strain. The original Orowan’s equation for estimating creep strain rate is modified employing CDM based softening parameters to take account of damage causing tertiary creep. The model is advantageous in the sense that with the ongoing creep, the evolution of different variables that are dislocation densities, dislocation mobility, dislocation velocity, internal stress, effective stress and damage evolution is tracked and discussed thoroughly. Furthermore, the model output is corroborated with experimental creep data of 304HCu steel. The predicted values of forest dislocation density, mobile dislocation density, mean free path, internal stress, effective stress, dislocation mobility and dislocation velocity are in the range of 7.91 × 1011 – 1.01 × 1013 m−2, 8.16 × 1010 – 4.56 × 1011 m−2, 9.35 – 9.80 µm, 11.70 – 35.50 MPa, 86.0 – 165.0 MPa, 1.68 × 10−9 – 2.11 × 10−7 Pa−1s−1 and 6.48 × 10−11 – 7.45 × 10−9 m/s, respectively, at the end of simulation.

Grain size effect of FCC polycrystal: A new CPFEM approach based on surface geometrically necessary dislocations

A multiscale modeling methodology involving discrete dislocation dynamics (DDD) and crystal plasticity finite element method (CPFEM) was used to study the grain size effect in FCC polycrystalline plasticity. The developed model is based on the dislocation density storage-recovery framework and is expanded to the scale of slip systems. DDD simulations were used to establish a constitutive law incorporating the main dislocation mechanisms that are involved in the strain hardening process observed in monotonically deformed FCC polycrystals. This was achieved by calculating the key features controlling the accumulation of the forest dislocation density within the grains and the polarized dislocation density at the grain boundaries during plastic deformation. The model was then integrated with a CPFEM model at the polycrystalline aggregate scale to compute short- and long-range internal stresses within the grains. These simulations quantitatively reproduced the deformation curves of the FCC polycrystals as a function of grain size. Because of its predictive ability to reproduce the Hall–Petch effect in a physically justified approach, the proposed framework has significant potential for further applications.

On the origin of the Hall–Petch law: A 3D-dislocation dynamics simulation investigation

3D-DD simulations are performed with cubic grains ranging from 1 to 10 µm to investigate the physical mechanisms at the origin of the Hall–Petch law. In particular, the long-range stress (back stress) induced by the density of polarized dislocations (GNDs) accumulated at GBs is quantified separately from the short-range stress associated with the forest dislocation (SSDs) density. We show that the back stress and the associated strain hardening is independent of grain size at low strain. Hence, the grain size effect reproduced by 3D-DD simulations is controlled by an increase of the CRSS when decreasing grain size. Such evolution of the CRSS amplitude is controlled by two competing strengthening mechanisms justifying the generic 1/d dependent form of the Hall–Petch law observed in simulations and experiments.

Tensile mechanical properties, deformation mechanisms, fatigue behaviour and fatigue life of 316H austenitic stainless steel: Effects of grain size

AbstractTo explore the effect of grain size on the tensile behaviour, symmetrical strain control fatigue behaviour and ratcheting fatigue behaviour of the new austenitic steel 316H, a series of tensile tests, symmetrical strain control fatigue tests and ratcheting fatigue tests were conducted at 25°C within an average grain size range from 14 to 122 μm. The average dislocation free path, forest dislocation density and mobile dislocation density of 316H steel with different grain sizes under tension were obtained. Moreover, the decrease of grain size led to saturation of mobile dislocations at lower strains. The decrease of grain size led to the increase of the cyclic stress and affected softening and secondary hardening process. In addition, under the same stress control conditions, the reduction in grain size caused a reduction in ratcheting strain under the same cycle. The grain refinement of 316H steel contributed to the improvement of ratcheting fatigue life.

Thermoviscoplasticity in Body-Centered Cubic Metals: A Two-Temperature Model With Grain Boundary Evolution

Abstract In this work, we develop a thermo-viscoplasticity model for body-centered cubic (BCC) metals based on a two-temperature theory of nonequilibrium thermodynamics. Modeling the plastic deformation here involves two subsystems, viz., a configurational subsystem related to grain growth, dislocation motion, and a kinetic vibrational subsystem describing the vibration of atoms. Due to a separation of the time scales, the two subsystems are described by two different temperatures. In this study, we introduce a grain boundary density, in addition to the mobile and forest dislocation densities, as an internal variable. The focus in this paper is on how large plastic deformation is affected by the evolving grain boundaries. In order to check the predictive quality of the model, numerical simulations are conducted and validated against available experimental evidence wherever possible.

A nonequilibrium thermodynamic model for viscoplasticity coupled with damage for BCC metals

We present a physically enhanced ductile damage model applicable for body centered cubic (BCC) metals. The current proposition extends the authors’ recent work on thermo-viscoplasticity based on two-temperature thermodynamics and physics of disparate types of dislocation densities. The description of the thermodynamic system involves primarily two types of variables (or degrees of freedom, DOFs) representing several micro/meso-scopic processes occurring in two separable time-scales during ductile damage. Processes of rearrangement and movement of defects, namely dislocations, voids, micro-cracks, take place in a time scale much slower than that of the vibration of atoms about their equilibrium positions in the lattice. Consequently, they appear in the thermodynamic theory in terms of slow configurational DOFs and the fast kinetic vibrational DOFs respectively. While we consider physics based internal variables, e.g., mobile and forest dislocation densities, for modeling viscoplasticity alone, material degradation due to ductile damage is treated in a phenomenological fashion taking recourse to the framework of continuum damage mechanics. In order to assess the performance of our proposal, numerical experiments on boundary value problems of viscoplasticity with or without damage are carried out and validated against available experimental evidence.

On the Origin of the Hall-Petch Law: A 3D-Dislocation Dynamics Simulation Investigation

3D -DD simulations are performed with cubic grains ranging from 1 to 10 μm to investigate the physical mechanisms at the origin of the Hall- Petch law. In particular, the long-range stress (back stress) induced by the density of polarized dislocations ( GNDs ) accumulated at GBs is quantified separately from the short-range stress associated with the forest dislocation ( SSDs ) density. We show that the back stress and the associated strain hardening is independent of grain size at low strain. Hence, the grain size effect reproduced by 3D-DD simulations is controlled by an increase of the CRSS when decreasing grain size. Such evolution of the CRSS amplitude is controlled by two competing strengthening mechanisms justifying the generic 1/√d dependent form of the Hall- Petch law observed in simulations and experiments.

Dislocation density based modeling of three-stage work hardening behaviour of type 316LN SS with varying nitrogen content and its finite element implementation for different notch radii

Dislocation density based modified Kocks-Mecking approach has been employed to understand the influence of nitrogen on kinetics of dislocation storage and annihilation processes during tensile deformation of type 316LN stainless steel at 300 K. As compared to original Kocks-Mecking approach, the modified version based on the evolution of forest dislocation density and mean free path of dislocations provides better description of three-stage hardening behaviour of the steel at all the nitrogen content. The effect of increase in nitrogen on work hardening behaviour is clearly reflected in systematic increase in flow stress, mobile and forest dislocation densities and decrease in mean free path at a given plastic strain. The predicted decrease in mean free path with increasing nitrogen has not only influenced the enhancement of dislocation-dislocation interactions leading to higher storage of dislocation but also provides a large driving force for higher dynamic recovery in the steel. The material model is implemented into finite element code via user-defined material subroutine using implicit algorithm to study the influence of notch radius on tensile properties of type 316LN stainless steel for different nitrogen content. Detailed analysis showed that notch-strengthening effect increases with increase in nitrogen and decrease in notch radius in the steel.

On the tensile flow stress response of 304 HCu stainless steel employing a dislocation density based model and electron backscatter diffraction measurements

ABSTRACTTensile flow stress response of 304 HCu stainless steel is investigated using a dislocation density based model and electron backscatter diffraction (EBSD) studies. The model considers two types of dislocations explicitly, i.e. mobile and forest, to model the flow behaviour in the temperature range 300–973 K, at strain rates 3×10 −3 and 3×10 −5 s-1. The flow behaviour indicates the predominance of thermal recovery at higher temperatures, except at 923 K/3 × 10−5 s−1. The dislocation model predicts a rapid increase in both mobile and forest dislocation densities at the early stages of deformation and thereafter reach saturation/steady state. Higher strain rate leads to an increase in dislocation densities with concomitant increase in peak flow stress, indicating significant dislocation multiplication. The dislocation densities decreased with an increasing temperature and is attributed to thermally activated recovery by glide (slip and cross-slip) and climb of dislocations. However, an offset to the thermal recovery is observed at 873–923 K at 3 × 10−5 s−1 wherein the magnitudes of peak flow stress, boundary and forest dislocation densities are of similar magnitude at both the temperatures, thereby signifying the occurrence of matrix hardening by DSA and precipitation. Boundary dislocation densities estimated from EBSD data have shown affinity to that of predicted forest dislocation densities.

Implementation of Internal-State-Variable Based Model into Finite Element Code and its Applicability to Study the Tensile Behaviour of Type 316LN SS With Different Notch Radii

An attempt has been made towards the implementation of two-internal-variable model based on the evolution of forest dislocation density and mean free path with the plastic strain into finite element code via user-defined material subroutine using implicit Euler backward algorithm with radial-return scheme. The rate independent two-internal-variable model considers isotropic hardening and the parameters associated with the model are optimised by fitting uniaxial true stress-true plastic strain data. Applicability of the implemented scheme has been examined by simulating the tensile behaviour of type 316LN austenitic stainless steel with different U-notch radii as 1.25, 2.5 and 5 mm at 300 K. Good agreement between the predicted and experimental data has been observed. Detailed analysis showed notch-strengthening effect increases with decrease in notch radius at 300 K.

Large-scale dislocation dynamics simulations of strain hardening of Ni microcrystals under tensile loading

The strain hardening in FCC Ni was studied along low index directions using 3-dimensional discrete dislocation dynamics. Large (20 × 20 × 50 μm) Ni microcrystals were simulated using rectangular parallelepiped-cells loaded in tension along four low-index directions ([111], [001], [110] and [112]) to shear strains of ∼0.01–0.02. Loading was at a constant strain rate of 10/sec, and all surfaces of the cell are treated as free surfaces. The FCC dislocation mobility routines were modified to include thermally activated cross-slip processes, as a function of three different stress components, using the results of previous atomistic simulations. These include bulk cross slip (cross-slip at atomic jogs), intersection cross slip (attractive and repulsive) as well as surface cross slip. One of these, repulsive intersection cross-slip, has zero activation energy and is present at all simulated deformation temperatures. The simulations were performed for three different temperatures, 5, 150 and 300 K. The strain-hardening rate is independent of temperature and of the order of μ/200 – μ/400, in agreement with experimental data for the <111> and <001> orientations of deformation. For the [001] orientation, at 5 and 150 K, the strain hardening rate decreases considerably when repulsive intersection cross-slip is removed from the simulations. The [110] and [112] orientations exhibit single-slip glide and a very low strain-hardening rate (∼μ/3000). Heterogeneity of dislocation microstructure develops spontaneously at the higher temperatures as a result of increased cross slip. Even though the strain-hardening rate is independent of temperature, the increase in dislocation density with shear strain is larger at higher temperatures. It is proposed that higher temperatures deformation produces larger dislocation-microstructure heterogeneities, providing for higher average dislocation densities and regions of low density where deformation can proceed. Also, the strain hardening rate at 300 K is controlled by the rate of increase of forest dislocation density in these lean regions.

Combined Effects of Texture and Grain Size Distribution on the Tensile Behavior of α-Titanium.

This work analyzes the role of both the grain size distribution and the crystallographic texture on the tensile behavior of commercially pure titanium. Specimens with different microstructures, especially with several mean grain sizes, were specifically prepared for that purpose. It is observed that the yield stress depends on the grain size following a Hall–Petch relationship, that the stress–strain curves have a tendency to form a plateau that becomes more and more pronounced with decreasing mean grain size and that the hardening capacity increases with the grain size. All these observations are well reproduced by an elasto-visco-plastic self-consistent model that incorporates grain size effects within a crystal plasticity framework where dislocations’ densities are the state variables. First, the critical resolved shear stresses are made dependent on the individual grain size through the addition of a Hall–Petch type term. Then, the main originality of the model comes from the fact that the multiplication of mobile dislocation densities is also made grain size dependent. The underlying assumption is that grain boundaries act mainly as barriers or sinks for dislocations. Hence, the smaller the grain size, the smaller the expansion of dislocation loops and thus the smaller the increase rate of mobile dislocation density is. As a consequence of this hypothesis, both mobile and forest dislocation densities increase with the grain size and provide an explanation for the grain size dependence of the transient low work hardening rate and hardening capacity.

A Novel Approach to Modelling Nanoindentation Instabilities

We review the recently developed models for load fluctuations in the displacement controlled mode and displacement jumps in the load controlled mode of indentation. To do this, we devise a method for calculating plastic contribution to load drops and displacement jumps by setting-up a system of coupled nonlinear time evolution equations for the mobile and forest dislocation densities by including relevant dislocation mechanisms. These equations are then coupled to the equation defining constant displacement rate or load rate. The model for the displacement controlled mode using a spherical indenter predicts all the generic features of nanoindentation such as the elastic branch followed by several force drops of decreasing magnitudes and residual indentation depth after unloading. The stress corresponding to the elastic force maximum is close to the yield stress of an ideal solid. The predicted numbers for all the quantities match experiments on single crystals of Au using a spherical indenter. We extend the approach to model the load controlled nanoindentation experiments that employ a Berkovich indenter. We first identify the dislocation mechanisms contributing to different regions of the F − z curve as a first step for obtaining a good fit to a given experimental F − z curve. This is done by studying the influence of the parameters associated with various dislocation mechanisms on the model F − z curves. The study also demonstrates that the model predicts all the generic features of nanoindentation such as the existence of an initial elastic branch followed by several displacement jumps of decreasing magnitudes and residual plasticity after unloading for a range of model parameter values. Furthermore, an optimized set of parameter values can be easily determined that give a good fit to the experimental load–displacement curves for Al single crystals of ( 110 ) and ( 133 ) orientations. Our model also predicts the indentation size effect in a region where the displacement jumps disappear. The good agreement of the results of the models with experiments supports our view that the present approach can be used as an alternate method to simulations. The approach also provides insights into several open questions.

Effect of Processing on Nix-Gao Bilinear Indentation Results Obtained for High Purity Iron

Instrumented indentation of a high purity Fe surface with unresolved surface deformation due to mechanical polishing is compared to the same grain surface annealed at increasing time and temperature. The differences in indentation size effect behavior with annealing are correlated with hardness and electron backscatter diffraction measurements as independent measures of surface layer deformation. It is found that the Nix Gao plot evolves from non-linear (bilinear) towards the predicted linear relationship as the surface deformation is removed. The experimental observations are rationalized by inclusion of a depth dependent, polishing induced forest dislocation density within the Nix-Gao model.

Metal viscoplasticity with two-temperature thermodynamics and two dislocation densities

Posed within the two-temperature theory of non-equilibrium thermodynamics, we propose a model for thermoviscoplastic deformation in metals. We incorporate the dynamics of dislocation densities–mobile and forest—that play the role of internal state variables in the formulation. The description based on two temperatures appears naturally when one recognizes that the thermodynamic system undergoing viscoplastic deformation is composed of two weakly interacting subsystems, viz. a kinetic-vibrational subsystem of the vibrating atomic lattices and a configurational subsystem of the slower degrees of freedom relating to defect motion, each with its own temperature. Starting with a basic model that involves only homogeneous deformation, a three-dimensional model for inhomogeneous viscoplasticity applicable to finite deformation is charted out in an overstress driven viscoplastic deformation framework. The model shows how the coupled evolutions of mobile and forest dislocation densities, which are critically influenced by the dynamics of configurational temperature, govern the strength and ductility of the metal. Unlike most contemporary models, the current proposal also affords a prediction of certain finer details as observed in the experimental data on stress–strain behaviour of metals and this in turn enhances the understanding of the evolving and interacting dislocation densities.