Multiscale Modeling Of Plasticity Research Articles

Dislocation descriptors of low and high angle grain boundaries with convolutional neural networks

Multiscale modeling of plasticity in polycrystalline metals is a long-standing challenge in part because of the lack of an accepted grain boundary descriptor, which has hindered the bridging of scales between atomistic simulations and meso-scale discrete dislocation dynamics (DDD) models. While grain boundary dislocations (GBDs) for low angle grain boundaries can be ascertained by Burgers circuit analyses, the dislocation structures of high angle grain boundaries have remained elusive because of overlapping dislocation core fields. Here, we use convolutional neural networks (CNNs) to establish the locations of GBDs responsible for the misorientations of <001> symmetrical-tilt Cu grain boundaries, from the local atomistic stress fields modeled with molecular dynamics (MD) simulations. We achieve accurate CNN predictions of GBDs with sub-angstrom resolution across an extensive set of low- to high-tilt-angle grain boundaries through a training dataset, generated from superposing continuum-representations of the MD stress fields of single dislocations that consider the contributions of both the Volterra and dislocation core fields. The approach paves the way for dislocation representations of the atomistic grain boundary structures modeled by MD simulations or density functional theory (DFT) calculations in mesoscale DDD models to elucidate multiscale plasticity effects.

A predictive discrete-continuum multiscale model of plasticity with quantified uncertainty

Multiscale models of materials, consisting of upscaling discrete simulations to continuum models, are unique in their capability to simulate complex materials behavior. The fundamental limitation in multiscale models is the presence of uncertainty in the computational predictions delivered by them. In this work, a sequential multiscale model has been developed, incorporating discrete dislocation dynamics (DDD) simulations and a strain gradient plasticity (SGP) model to predict the size effect in plastic deformations of metallic micro-pillars. The DDD simulations include uniaxial compression of micro-pillars with different sizes and over a wide range of initial dislocation densities and spatial distributions of dislocations. An SGP model is employed at the continuum level that accounts for the size-dependency of flow stress and hardening rate. Sequences of uncertainty analyses have been performed to assess the predictive capability of the multiscale model. The variance-based global sensitivity analysis determines the effect of parameter uncertainty on the SGP model prediction. The multiscale model is then constructed by calibrating the continuum model using the data furnished by the DDD simulations. A Bayesian calibration method is implemented to quantify the uncertainty due to microstructural randomness in discrete dislocation simulations (density and spatial distribution of dislocations) on the macroscopic continuum model prediction (size effect in plastic deformation). The outcomes of this study indicate that the discrete-continuum multiscale model can accurately simulate the plastic deformation of micro-pillars, despite the significant uncertainty in the DDD results. Additionally, depending on the macroscopic features represented by the DDD simulations, the SGP model can reliably predict the size effect in plasticity responses of the micropillars with below 10% of error.

A multi-scale model of plasticity and damage for rock-like materials with pores and inclusions

A multi-scale model is developed for modeling of plastic deformation and induced damage in rock-like materials containing pores and mineral inclusions. An analytical macroscopic plastic yield criterion is first determined from three steps of homogenization. This criterion explicitly takes into account respective effects of two populations of pores and mineral inclusions at three different scales. The accuracy of criterion is evaluated by comparisons with numerical results issued from direct simulations on representative volume elements. Then, an induced damage evolution law is defined to describe the onset and growth of micro-cracks inside the solid matrix of materials. The induced damage affects both elastic and strength properties of the solid matrix and its evolution is coupled with the plastic deformation. The performance of the proposed model is first assessed through a series of numerical examples, clearly showing influences of pores and inclusions on macroscopic responses of heterogeneous materials. Finally, the proposed model is applied to describe mechanical behaviors of clayey rocks. Comparisons between numerical results and experimental data are presented.

Hierarchical top-down bottom-up calibration with consideration for uncertainty and inter-scale discrepancy of Peierls stress of bcc Fe

A hierarchical multiscale model of plasticity in single crystal bcc Fe, framed at the mesoscale pertaining to a statistically representative set of dislocations for each slip system, relies on specification of a set of parameters which are informed using both bottom-up (BU) (atomistic) and top-down (TD) (experimental) information. We term this specification process a ‘connection’ between BU and TD pathways to inform the mesoscale model. The connection is considered in the presence of error, uncertainty, and discrepancy between the models. We expand upon a previously developed reconciled TD and BU calibration method to account for anticipated discrepancy between information from different length scales. The results of a previously formulated likelihood-based connection test of the multiscale model suggest a ‘missing link’ may be responsible for part of the inter-scale discrepancy. In this case, this link is assumed to be a relation between the unit-process (single dislocation line) of coordinated kink-pair nucleation on a screw dislocation segment and the many-body dislocation process which manifests in the onset of experimentally observed plastic deformation in a single crystal. A physics-informed discrepancy layer is formulated to improve the connection in the presence of additional persistent uncertainties. This physics-informed hypothesis testing is demonstrated as an alternative to fully data-driven search methods, particularly applicable to datasets of limited size that are often encountered in such multiscale material modeling applications. Discrete dislocation dynamics simulations are used to investigate the effect of the Peierls stress on straight screw dislocations and in the operation of Frank–Read sources. Discussion concerns the critical role of uncertainty quantification in providing a basis for this form of incremental hypothesis testing.

On the synergy between physical and virtual sheet metal testing: calibration of anisotropic yield functions using a microstructure-based plasticity model

This paper scrutinizes in detail the synergy between physical and virtual testing of the mechanical response of sheet metal. In this context, physical testing refers to the usage of physical samples onto which mechanical tests are conducted, while virtual testing refers to multi-scale plasticity simulations onto a model representation of the metallic microstructure derived from microstructural measurement. An extensive experimental campaign was conducted to capture the plastic material response of mild steel sheet in the first quadrant of the biaxial stress space, i.e. for tensile stresses along the Rolling Direction (RD) and Transverse Direction (TD). The experimental data was acquired using state-of-the-art stress-controlled material tests enabling to probe the material up to large plastic strains in the first quadrant of stress space. Identical stress paths were simulated using the Virtual Experimentation Framework (VEF) software suite adopting the ALAMEL multi-scale plasticity model. The ALAMEL model was calibrated solely on the basis of the initial crystallographic texture of the material and a reference hardening curve obtained through a uniaxial tensile test in the rolling direction. The predictive accuracy of the ALAMEL model to reproduce the experimentally acquired material response has been thoroughly assessed. To avoid any bias in the assessment due to extrapolation of the material behavior, predictions were limited to the pre-necking regime of the material. The predictions of the VEF show good agreement with the physical test results. Subsequently, the experimental and virtual test data were used to calibrate Hill’s quadratic yield criterion and the Yld2000-2d yield function. The calibration accuracy of these yield criteria is thoroughly assessed by comparing theoretical predictions and experimental data. In addition, the calibrated yield functions are used to simulate a hydraulic bulge test and the results are compared with experimental observations. It is shown that the adopted virtual material testing procedure has reached a sufficient level of maturity to potentially serve as a viable alternative for physical material testing of steel sheet.

A Three-Dimensional Multi-scale Plasticity Model for Metal-Intermetallic Laminate Composites Containing Phases of the L12 Structure

A three-dimensional plasticity model was developed and applied to metal-intermetallic laminate composites containing phases of the L12 structure. A multi-scale approach that combined the methods of continuum mechanics and dislocation kinetics was used. This model takes account of the different mechanisms of self-locking superdislocations, the dislocations and the dislocation walls’ density storage for each type of layer at the micro-scale. At the meso-scale, the solutions to the dislocation kinetics equations, in the form of stress–strain curves, were used to create the properties of a three-dimensional representative element. The numerical simulation study of the macroscopic deformation was carried out with the finite element method using the dynamic model of continuum mechanics, which included the classical conservation laws, constitutive equations and the equation of state. It was shown that the simulation results generated using this model were in good agreement with the mechanical tests conducted on the single crystals of the L12 structure. The model provides an excellent description of the high-temperature plastic strain superlocalization effect of single crystal intermetallics of the L12 structure. This paper describes the numerical results of the study of the tension and compression tests of metal-intermetallic laminate composites containing phases of the L12 structure. The model allows the description of the distribution of the accumulated plastic strain inhomogeneities and is capable of predicting the strengthening properties and plastic behaviour of the metal-intermetallic laminate composites containing phases of the L12 structure.

Hierarchical multiscale modeling of plasticity in copper: From single crystals to polycrystalline aggregates

Modeling the deformation behavior of polycrystalline materials using the information embedded at grain level is a recent research area of high interest. In view of this, an attempt has been made to imbue the predictions of polycrystalline deformation with the single crystal behavior within the framework of hierarchical multiscale modeling scheme. Face centered cubic Cu has been selected as the model material for demonstration. At the nanoscale, behavior of a single dislocation is quantified in terms of dislocation drag coefficient using atomistic simulations, which is then transferred to dislocation dynamics simulations at the microscale. The collective behavior of a huge dislocation population is then simulated to quantify the necessary hardening parameters to be passed on to crystal plasticity simulations at the mesoscale. Crystal plasticity simulations are performed to simulate the uniaxial tensile behavior of single crystal Cu in multiple crystallographic orientations. The calibrated hardening parameter set for various orientations of single crystal Cu is then refined statistically to introduce a single parameter set, that could adequately capture the uniaxial tensile behavior of polycrystalline Cu. The scheme, therefore, couples atomistics to discrete dislocations to single crystal plasticity to polycrystalline plasticity within the framework of hierarchical multiscale modeling scheme. Experimental data of Takeuchi (1975) and Bronkhorst et al. (1992) has been employed to access the predictive capabilities of our approach. A good agreement has been obtained between these experiments and our simulation results, thereby validating our methodology.

Experiment-based validation and uncertainty quantification of coupled multi-scale plasticity models

Purpose– Highly anisotropic zirconium is a material used in the cladding of nuclear fuel rods, ensuring containment of the radioactive material within. The complex material structure of anisotropic zirconium requires model developers to replicate not only the macro-scale stresses but also the meso-scale material behavior as the crystal structure evolves; leading to strongly coupled multi-scale plasticity models. Such strongly coupled models can be achieved through partitioned analysis techniques, which couple independently developed constituent models through an iterative exchange of inputs and outputs. Throughout this iterative process, biases, and uncertainties inherent within constituent model predictions are inevitably transferred between constituents either compensating for each other or accumulating during iterations. The paper aims to discuss these issues.Design/methodology/approach– A finite element model at the macro-scale is coupled in an iterative manner with a meso-scale viscoplastic self-consistent model, where the former supplies the stress input and latter represents the changing material properties. The authors present a systematic framework for experiment-based validation taking advantage of both separate-effect experiments conducted within each constituent’s domain to calibrate the constituents in their respective scales and integral-effect experiments executed within the coupled domain to test the validity of the coupled system.Findings– This framework developed is shown to improve predictive capability of a multi-scale plasticity model of highly anisotropic zirconium.Originality/value– For multi-scale models to be implemented to support high-consequence decisions, such as the containment of radioactive material, this transfer of biases and uncertainties must be evaluated to ensure accuracy of the predictions of the coupled model. This framework takes advantage of the transparency of partitioned analysis to reduce the accumulation of errors and uncertainties.

XFEM–dislocation dynamics multi-scale modeling of plasticity and fracture

An extended three-dimensional multi-scale framework is presented to model plasticity in precipitate containing materials in the presence of cracks. The present framework is constructed in two scales. At the micro scale, plasticity is computed via the line dislocation dynamics (DD) methodology in which the penetrable and impenetrable precipitates are modeled. At the macro scale, application of the extended finite element method (XFEM) allows for accurate analysis of mixed-mode cohesive crack propagation without the need for any remeshing procedure. This is a significant improvement especially when two different scales are involved. While the former simulations have so far been limited to the study of only mode-I crack propagation to avoid the expensive remeshing procedure, in the present work, the effects of loading rates and precipitate density on general crack propagations are studied by utilizing the XFEM–DD multi-scale framework to illustrate the efficiency and capability of the proposed approach.

Multiscale modeling of plasticity in a copper single crystal deformed at high strain rates

AbstractA hierarchical multiscale modeling approach is presented to predict the mechanical response of dynamically deformed (1100 s−1−4500 s−1) copper single crystal in two different crystallographic orientations.Anattempt has been made to bridge the gap between nano-, micro- and meso- scales. In view of this, Molecular Dynamics (MD) simulations at nanoscale are performed to quantify the drag coefficient for dislocations which has been exploited in Dislocation Dynamics (DD) regime at the microscale. Discrete dislocation dynamics simulations are then performed to calculate the hardening parameters required by the physics based Crystal Plasticity (CP) model at the mesoscale. The crystal plasticity model employed is based on thermally activated theory for plastic flow. Crystal plasticity simulations are performed to quantify the mechanical response of the copper single crystal in terms of stressstrain curves and shape changes under dynamic loading. The deformation response obtained from CP simulations is in good agreement with the experimental data.

Prediction of Transient Hardening after Strain Path Change by a Multi-scale Crystal Plasticity Model with Anisotropic Grain Substructure

Multi-scale modelling offers physical insights in the relationship between microstructure and properties of a material. The macroscopic anisotropic plastic flow may be accounted for by consideration of (a) the polycrystalline nature and (b) the anisotropic grain substructure. The latter contribution to anisotropy manifests itself most clearly in the event of a change in the strain path, as occurs frequently in multi-step forming processes. Under monotonic loading, both the crystallographic texture and the loading-dependent strength contribution from substructure influence the macroscopically observed strength.The presented multi-scale plasticity model for BCC polycrystals combines a crystal plasticity model featuring grain interaction with a substructure model for anisotropic hardening of the individual slip systems. Special attention is given to how plastic deformation is accommodated: either by slip of edge dislocation segments, or alternatively by dislocation loop expansion. Resultsof this multi-scale modelling approach are shown for a batch-annealed IF steel. Whereas both model variants are seen to capture the transient hardening after different types of strain path changes, the dislocation loop model offers more realistic predictions under a variety of monotonic loading conditions.

Multiscale modeling of plasticity based on embedding the viscoplastic self-consistent formulation in implicit finite elements

This paper is concerned with the multiscale simulation of plastic deformation of metallic specimens using physically-based models that take into account their polycrystalline microstructure and the directionality of deformation mechanisms acting at single-crystal level. A polycrystal model based on self-consistent homogenization of single-crystal viscoplastic behavior is used to provide a texture-sensitive constitutive response of each material point, within a boundary problem solved with finite elements (FE) at the macroscale. The resulting constitutive behavior is that of an elasto-viscoplastic material, implemented in the implicit FE code ABAQUS. The widely-used viscoplastic selfconsistent (VPSC) formulation for polycrystal deformation has been implemented inside a user-defined material (UMAT) subroutine, providing the relationship between stress and plastic strain-rate response. Each integration point of the FE model is considered as a polycrystal with a given initial texture that evolves with deformation. The viscoplastic compliance tensor computed internally in the polycrystal model is in turn used for the minimization of a suitable-designed residual, as well as in the construction of the elasto-viscoplastic tangent stiffness matrix required by the implicit FE scheme. Uniaxial tension and simple shear of an FCC polycrystal have been used to benchmark the accuracy of the proposed implicit scheme and the correct treatment of rotations for prediction of texture evolution. In addition, two applications are presented to illustrate the potential of the multiscale strategy: a simulation of rolling of an FCC plate, in which the model predicts the development of different textures through the thickness of the plate; and the deformation under 4-point bending of textured HCP bars, in which the model captures the dimensional changes associated with different orientations of the dominant texture component with respect to the bending plane.

Multiscale modelling of MgO plasticity

We propose a multiscale model of plasticity of pure MgO single crystals. The core structure of the rate controlling ½〈1 1 0〉 screw dislocations has been modelled by the Peierls–Nabarro–Galerkin method. This model relies on γ-surfaces calculated ab initio for the {1 1 0}, {1 0 0} and {1 1 1} planes. The ½ 〈1 1 0〉 screw dislocations spread mostly in the {1 1 0} planes. The Peierls friction values are 150 MPa and 1.6 GPa for glide on the {1 1 0} and {1 0 0} planes, respectively. The kink pair theory is applied to model thermal activation of dislocation glide over the Peierls barrier below the athermal temperature T a and to build a velocity law for this regime. The critical resolved shear stresses are deduced below T a from the Orowan law. Above T a the athermal stress τ μ is obtained from discrete dislocation dynamics simulations to account for dislocation–dislocation interactions. This model is found to satisfactorily reproduce the critical resolved shear stresses observed experimentally, provided the contribution of impurities (unavoidable in experiments) is subtracted.

Stable splitting scheme for general form of associated plasticity including different scales of space and time

An efficient implementation of the operator split procedure for boundary value solution with plastic flow computation is presented for a general form of associated plasticity model. We start with the general form of phenomenological model of plasticity where the yield criterion is not restricted to a simple (quadratic) form, and the elasticity tensor does not have constant entries. We then turn to the multi-scale model of plasticity which employs the fine scale representation of the plastic deformation along with the homogenization procedure for stress computation. We also visit the plasticity model with rate sensitive plastic response where plastic flow computation is carried out at fine scale in time. We proved herein the sufficient and necessary conditions for the proposed operator split procedure to converge, for any such general form of associated plasticity. Moreover, we presented a systematic manner for constructing the main ingredients for the plastic flow computation and the global Newton’s iteration, such as the consistent elastoplastic tangent.

Multi-scale plasticity modeling: Coupled discrete dislocation and continuum crystal plasticity

A hierarchical multi-scale model that couples a region of material described by discrete dislocation plasticity to a surrounding region described by conventional crystal plasticity is presented. The coupled model is aimed at capturing non-classical plasticity effects such as the long-range stresses associated with a density of geometrically necessary dislocations and source limited plasticity, while also accounting for plastic flow and the associated energy dissipation at much larger scales where such non-classical effects are absent. The key to the model is the treatment of the interface between the discrete and continuum regions, where continuity of tractions and displacements is maintained in an average sense and the flow of net Burgers vector is managed via “passing” of discrete dislocations. The formulation is used to analyze two plane strain problems: (i) tension of a block and (ii) crack growth under mode I loading with various sizes of the discrete dislocation plasticity region surrounding the crack tip. The computed crack growth resistance curves are nearly independent of the size of the discrete dislocation plasticity region for region sizes ranging from 30 μ m × 30 μ m to 10 μ m × 5 μ m . The multi-scale model can reduce the computational time for the mode I crack analysis by a factor of 14 with little or no loss of fidelity in the crack growth predictions.

Viscoplasticity of heterogeneous metallic materials

Viscoplastic behavior of metallic materials is manifested by collective behavior of lattice defects at multiple length scales. The full gamut of length scales, from atomic to macroscopic, is considered in terms of both concurrent and hierarchical multiscale modeling of plasticity of metals. Particular challenges are identified in the realm between atomistic simulations and the scale of dislocation substructure formation in establishing next generation constitutive models. Bottom–up modeling is useful in applications such as fracture or dislocation mediation by interfaces. This is differentiated from top–down modeling that supports design of a material as a hierarchy of sub-systems. Examples of hierarchical microstructure-sensitive models are presented for both Ni-base superalloys and α–β Ti alloys, with emphasis on cyclic deformation behavior. Future directions to advance both discrete dislocation and crystal plasticity theories are discussed, emphasizing the need for additional focus on dislocation sources and dislocation line curvature in modeling scale dependent behavior. Difficulties with low order approximations of the dislocation density distribution, such as second-order gradient theories, are discussed in terms of predictive capability and transferability among geometric configurations. The multiscale nature of the decomposition of the deformation gradient into elastic and plastic parts is discussed in light of sources of incompatibility at each scale considered, with interpretation offered at different scales of Burgers circuits that highlight dislocation substructures and polycrystals, respectively. Recent atomistic studies of dislocation nucleation at grain boundaries are outlined and some thoughts are offered towards potentially fruitful directions to incorporate this understanding into statistical continuum models that account for the role of grain boundary structure as an element of the evolving incompatibility field.

Stochastic flow rule for granular materials

There have been many attempts to derive continuum models for dense granular flow, but a general theory is still lacking. Here, we start with Mohr-Coulomb plasticity for quasi-two-dimensional granular materials to calculate (average) stresses and slip planes, but we propose a "stochastic flow rule" (SFR) to replace the principle of coaxiality in classical plasticity. The SFR takes into account two crucial features of granular materials-discreteness and randomness-via diffusing "spots" of local fluidization, which act as carriers of plasticity. We postulate that spots perform random walks biased along slip lines with a drift direction determined by the stress imbalance upon a local switch from static to dynamic friction. In the continuum limit (based on a Fokker-Planck equation for the spot concentration), this simple model is able to predict a variety of granular flow profiles in flat-bottom silos, annular Couette cells, flowing heaps, and plate-dragging experiments--with essentially no fitting parameters--although it is only expected to function where material is at incipient failure and slip lines are inadmissible. For special cases of admissible slip lines, such as plate dragging under a heavy load or flow down an inclined plane, we postulate a transition to rate-dependent Bagnold rheology, where flow occurs by sliding shear planes. With different yield criteria, the SFR provides a general framework for multiscale modeling of plasticity in amorphous materials, cycling between continuum limit-state stress calculations, mesoscale spot random walks, and microscopic particle relaxation.

Modelling the dynamic deformation and patterning in fcc single crystals at high strain rates: dislocation dynamics plasticity analysis

The deformation process in copper and aluminium single crystals under shock loading is investigated using a multiscale model of plasticity that couples discrete dislocation dynamics and finite element analyses. Computer simulations are carried out to mimic loading condition of high strain rates ranging from 105 to 107 s−1, and short pulse durations of few nanoseconds involved in recent laser based experiments. The effects of strain rate, shock pulse duration and the nonlinear elastic properties are investigated. Relaxed configurations using dislocation dynamics show formation of dislocation micro bands and weak dislocation cells. Statistical analyses of the dislocation microstructures are preformed to study the characteristics of the local dislocation densities and the distribution of the instantaneous dislocations velocities.

Coupled Atomistic/Discrete Dislocation Simulations of Nanoindentation at Finite Temperature

Simulations of nanoindentation in single crystals are performed using a finite temperature coupled atomistic/continuum discrete dislocation (CADD) method. This computational method for multiscale modeling of plasticity has the ability of treating dislocations as either atomistic or continuum entities within a single computational framework. The finite-temperature approach here inserts a Nose-Hoover thermostat to control the instantaneous fluctuations of temperature inside the atomistic region during the indentation process. The method of thermostatting the atomistic region has a significant role on mitigating the reflected waves from the atomistic/continuum boundary and preventing the region beneath the indenter from overheating. The method captures, at the same time, the atomistic mechanisms and the long-range dislocation effects without the computational cost of full atomistic simulations. The effects of several process variables are investigated, including system temperature and rate of indentation. Results and the deformation mechanisms that occur during a series of indentation simulations are discussed.

Multiscale plasticity modeling: coupled atomistics and discrete dislocation mechanics

A computational method (CADD) is presented whereby a continuum region containing dislocation defects is coupled to a fully atomistic region. The model is related to previous hybrid models in which continuum finite elements are coupled to a fully atomistic region, with two key advantages: the ability to accomodate discrete dislocations in the continuum region and an algorithm for automatically detecting dislocations as they move from the atomistic region to the continuum region and then correctly “converting” the atomistic dislocations into discrete dislocations, or vice-versa. The resulting CADD model allows for the study of 2d problems involving large numbers of defects where the system size is too big for fully atomistic simulation, and improves upon existing discrete dislocation techniques by preserving accurate atomistic details of dislocation nucleation and other atomic scale phenomena. Applications to nanoindentation, atomic scale void growth under tensile stress, and fracture are used to validate and demonstrate the capabilities of the model.