Continuum Crystal Plasticity Model Research Articles

Deformation, dislocation evolution and the non-Schmid effect in body-centered-cubic single- and polycrystal tantalum

A physically-informed continuum crystal plasticity model is presented to elucidate deformation mechanisms, dislocation evolution and the non-Schmid effect in body-centered-cubic (bcc) tantalum widely used as a key structural material for mechanical and thermal extremes. We show the unified structural modeling framework informed by mesoscopic dislocation dynamics simulations is capable of capturing salient features of the large inelastic behavior of tantalum at quasi-static (10−3 s−1) to extreme strain rates (5000 s−1) and at low (77 K) to high temperatures (873 K) at both single- and polycrystal levels. We also present predictive capabilities of the model for microstructural evolution in the material. To this end, we investigate the effects of dislocation interactions on slip activities, instability and the non-Schmid behavior at the single crystal level. Furthermore, ex situ measurements on crystallographic texture evolution and dislocation density growth are carried out for polycrystal tantalum specimens at increasing strains. Numerical simulation results also support that the modeling framework is capable of capturing the main features of the polycrystal behavior over a wide range of strains, strain rates and temperatures. The theoretical, experimental and numerical results at both single- and polycrystal levels provide critical insight into the underlying physical pictures for micro- and macroscopic responses and their relations in this important class of refractory bcc materials undergoing large inelastic deformations.

Strain hardening in molecular crystal cyclotetramethylene-tetranitramine (β-HMX): a theoretical evaluation

The molecular crystal cyclotetramethylene-tetranitramine (β-HMX) is a broadly used energetic material. Its plastic deformation is important when describing the detonation behavior. This work aims to clarify the importance of strain hardening for the plastic deformation of this crystal. To this end, we use a line tension model to evaluate the strength of junctions formed by dislocations moving in different slip systems. We evaluate analytically the contribution to the flow stress of repulsive interactions between dislocations. Further, we test using atomistic models and confirm the conjecture that neutral core–core interactions of crossing dislocations do not contribute to the flow stress. This information is used to define the hardening matrix which can be further used in continuum crystal plasticity models. We conclude that strain hardening is weak at all realistic dislocation densities, and leads to a modest increase of the flow stress above the critical resolved shear stress corresponding to the vanishing dislocation density limit. A procedure is provided which allows extrapolating these results from ambient conditions to pressures and temperatures relevant for shock loading.

A comparison of different crystal plasticity finite-element models on the simulation of nickel alloys

ABSTRACT Continuum crystal plasticity models based on the kinematics of plastic slip are widely developed and used in recent years to describe the anisotropic elasto-viscoplastic behaviour of metallic material. Various microstructural physical-based models have been proposed to reproduce experimental evolutions. This work aims to analyse and compare different crystal plasticity finite element models, especially those based on dislocation densities, from the simulations of stress–strain curves of a nickel base super alloy CMSX-4. Special attention is paid on hardening matrix. The adequacy between the simulated results and experimental data is commented, and it is found that a strain-rate- dependent double hardening matrix model can give a better prediction of the CMSX-4 behaviours.

Strain Avalanches in Microsized Single Crystals: Avalanche Size Predicted by a Continuum Crystal Plasticity Model**Supported by the China Postdoctoral Science Foundation under Grant No 2015M572118, the Outstanding Young Talent Research Fund of Zhengzhou University under Grant No 1521327001, and the National Natural Science Foundation of China under Grant No 51405452.

Plastic deformation of small crystals occurs by power-law distributed strain avalanches whose universality is still debated. In this work we introduce a continuum crystal plasticity model for the deformation of microsized single crystals, which is able to reproduce the main experimental observations such as flow intermittency and statistics of strain avalanches. We report exact predictions for scaling exponents and scaling functions associated with random distribution of avalanche sizes. In this way, the developed model provides a routine for a quantitative characterization of the statistical aspects of strain avalanches in microsized single crystals.

Extended modelling of dislocation transport-formulation and finite element implementation

In the past decade, several higher-order crystal plasticity models have been developed to properly capture size effects, dislocation pile-up and patterning. Here we consider a formulation which accounts for the presence and behavior of both positive and negative dislocations in terms of densities. We derive an implicit finite element implementation for the continuum crystal plasticity model including dislocation transport, using a generalised continuum expression for the short-range dislocation interactions, by discretizing the two governing non-linear transport equations in time and space. The resulting non-linear algebraic equations are solved by an incremental-iterative solution scheme. We compare the resulting numerical solutions with discrete dislocation simulations. This analysis shows the capabilities of the implicit FEM framework to solve continuum dislocation transport in crystal plasticity with the added energetic dislocation interactions.

Review on slip transmission criteria in experiments and crystal plasticity models

A comprehensive overview is given of the literature on slip transmission criteria for grain boundaries in metals, with a focus on slip system and grain boundary orientation. Much of this extensive literature has been informed by experimental investigations. The use of geometric criteria in continuum crystal plasticity models is discussed. The theoretical framework of Gurtin (2008, J. Mech. Phys. Solids 56, p. 640) is reviewed for the single slip case. This highlights the connections to slip transmission criteria from the literature that are not discussed in the work itself. Different geometric criteria are compared for the single slip case with regard to their prediction of slip transmission. Perspectives on additional criteria, investigated in experiments and used in computational simulations, are given.

Micromorphic modelling of grain size effects in metal polycrystals

AbstractA long‐standing problem in the modelling of polycrystal behavior from the knowledge of the constitutive equations of single crystals is the incorporation of grain size effects. Standard ho‐mogenization procedures efficiently take into account grain to grain interaction primarily induced by strain incompatibilities at grain boundary and grain morphology effects. Continuum crystal plasticity models are available that satisfactorily describe dislocation forest hardening associated with multiplication and dynamic recovery of dislocation populations. One strong limitation of these techniques is that the local crystal plasticity constitutive parameters at the grain scale are assumed to be known. Conversely, these material parameters can be identified from the macroscopic polycrystal response by an inverse approach involving the homogenization model. But then the found crystal plasticity model will be valid only for the considered grain size and the model can hardly be used for prediction of polycrystal behavior for another grain state of the material. Generalized continuum approaches arise as the suitable mechanical framework for formulating enhanced crystal plasticity models that intrinsically include grain size dependent local material responses. They incorporate the dislocation density tensor as a consitutive variable. Recent developments have shown the limitations of existing strain gradient models in the case of a two–phase laminate microstructure under single or double slip [1]. They plead for a theory even more general than the strain gradient or Cosserat approach, namely the micromorphic model presented here. The effect of the dislocation density tensor is introduced into the classical crystal plasticity framework by means of a micromor‐phic theory of single crystals. A computational homogenization strategy is presented in order to simulate the global and local responses of two–dimensional polycrystalline aggregates for grain sizes ranging from 1 to 200 microns. The model is shown to induce a size–dependent kinematic hardening component which is responsible for the observed strong size effects. The yield stress at a given averaged plastic deformation is shown to follow a power law scaling relation for grain sizes larger than a critical grain size. The field of plastic deformation is also strongly affected by grain size, micron–size grains leading to the formation of intense slip bands crossing several grains. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Crystal plasticity model of shear and kink bands–energetic approach

We present a continuum crystal plasticity model of a lamellar deformation substructure of shear and kink bands. An evolutionary problem for the development of a spontaneous structural inhomogeneity is formulated in the framework of energetic solutions. Conti and Theil proved that in the case of an isothermal single-slip crystal, rigid plasticity with no hardening lamellaea form an optimal microstructure. Moreover, their model predicts the existence of a boundary layer which accommodates the lamellar substructure to displacement boundary conditions. It is suggested that the width of the shear and kink bands is a compromise: the minimization of bulk energy tends to decrease their size, while the energy of the band interfaces or the inner structure of the bands opposes this tendency.

A continuum plasticity model that accounts for hardening and size effects in thin films

We conducted three-dimensional finite element simulations of the mechanical response of passivated single crystal copper thin films with a continuum crystal plasticity model. The model introduces the formation of high density dislocation layers close to the substrate and passivation interfaces obtained from dislocation dynamics simulations. These dislocation structures are responsible for an increase in strain hardening as the film thickness decreases. The model predicts an increase in strain hardening as the film thickness decreases in agreement with experimental observation in films with thickness in the range 0.2 to 2 µm.

A geometric approach to modeling microstructurally small fatigue crack formation: II. Physically based modeling of microstructure-dependent slip localization and actuation of the crack nucleation mechanism in AA 7075-T651

The objective of this paper is to develop further a framework for computationally modeling microstructurally small fatigue crack growth in AA 7075-T651 (Bozek et al 2008 Modelling Simul. Mater. Sci. 16 065007). The focus is on the nucleation event, when a crack extends from within a second-phase particle into a surrounding grain, since this has been observed to be an initiating mechanism for fatigue crack growth in this alloy. It is hypothesized that nucleation can be predicted by computing a non-local nucleation metric near the crack front. The hypothesis is tested by employing a combination of experimentation and finite element modeling in which various slip-based and energy-based nucleation metrics are tested for validity, where each metric is derived from a continuum crystal plasticity formulation. To investigate each metric, a non-local procedure is developed for the calculation of nucleation metrics in the neighborhood of a crack front. Initially, an idealized baseline model consisting of a single grain containing a semi-ellipsoidal surface particle is studied to investigate the dependence of each nucleation metric on lattice orientation, number of load cycles and non-local regularization method. This is followed by a comparison of experimental observations and computational results for microstructural models constructed by replicating the observed microstructural geometry near second-phase particles in fatigue specimens. It is found that orientation strongly influences the direction of slip localization and, as a result, influences the nucleation mechanism. Also, the baseline models, replication models and past experimental observation consistently suggest that a set of particular grain orientations is most likely to nucleate fatigue cracks. It is found that a continuum crystal plasticity model and a non-local nucleation metric can be used to predict the nucleation event in AA 7075-T651. However, nucleation metric threshold values that correspond to various nucleation governing mechanisms must be calibrated.

Influence of Crystallographic Orientation on Energy Dissipation During Sliding

The aim of this study is to evaluate a methodology for modeling the influence of crystallographic grain orientation in sliding contacts. The simulations of translating interfering cylindrical asperities, using finite element analysis, were conducted using two different plasticity models for copper: a conventional isotropic, homogeneous J2 plasticity model and a continuum crystal plasticity model. Using crystal plasticity, the dependence of crystallographic orientation on plastic deformation and energy dissipation can be determined. The relative trends predicted using crystal plasticity are consistent with experiments that show friction depends on crystallographic orientation when plastic deformation is one of the primary energy dissipation mechanisms.

A semi-discrete and non-local crystal plasticity model for nanocrystalline metals

The application of traditional continuum crystal plasticity models to nanocrystalline metals is discussed and compared to a novel semi-discrete crystal plasticity model. The new model exhibits larger stress heterogeneities and greater amounts of intragranular plasticity. (c) 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Thermomechanical Fatigue Behavior of a Directionally Solidified Ni-Base Superalloy

A continuum crystal plasticity model is used to simulate the material behavior of a directionally solidified Ni-base superalloy, DS GTD-111, in the longitudinal and transverse orientations. Isothermal uniaxial fatigue tests with hold times and creep tests are conducted at temperatures ranging from room temperature (RT) to 1038°C to characterize the deformation response. The constitutive model is implemented as a User MATerial subroutine (UMAT) in ABAQUS (2003, Hibbitt, Karlsson, and Sorensen, Inc., Providence, RI, v6.3) and a parameter estimation scheme is developed to obtain the material constants. Both in-phase and out-of-phase thermo-mechanical fatigue tests are conducted. A physically based model is developed for correlating crack initiation life based on the experimental life data and predictions are made using the crack initiation model.

On the evolution of crystallographic dislocation density in non-homogeneously deforming crystals

A set of evolution equations for dislocation density is developed incorporating the combined evolution of statistically stored and geometrically necessary densities. The statistical density evolves through Burgers vector-conserving reactions based in dislocation mechanics. The geometric density evolves due to the divergence of dislocation fluxes associated with the inhomogeneous nature of plasticity in crystals. Integration of the density-based model requires additional dislocation density/density-flux boundary conditions to complement the standard traction/displacement boundary conditions. The dislocation density evolution equations and the coupling of the dislocation density flux to the slip deformation in a continuum crystal plasticity model are incorporated into a finite element model. Simulations of an idealized crystal with a simplified slip geometry are conducted to demonstrate the length scale-dependence of the mechanical behavior of the constitutive model. The model formulation and simulation results have direct implications on the ability to explicitly model the interaction of dislocation densities with grain boundaries and on the net effect of grain boundaries on the macroscopic mechanical response of polycrystals.

Characteristics of plastic deformation field in polycrystalline fretting contacts

The plastic deformation field produced by fretting occurs over subsurface distances comparable to the grain size. Therefore, a more physically realistic constitutive model should be used to take into account the discrete grains and their crystallographic orientation distribution in this region to better capture the heterogeneity of the dislocation slip in the fretting damage process volume. This is accomplished by using a continuum crystal plasticity material model developed to represent these effects in Ti–6Al–4V. A two-dimensional finite element analysis is used to simulate an ideal fretting condition. This study examines how the characteristics of the plasticity produced by fretting changes with key fretting parameters including the normal contact force, the tangential force amplitude, and the coefficient of friction at the contact. The results produced using a continuum crystal plasticity model show that the distribution of plastic strain is highly heterogeneous and it contains both cyclic and ratchetting components.

Measurement of deformation fields in polycrystalline OFHC copper

Validation of continuum crystal plasticity models for polycrystalline materials requires experiments with sub-grain scale resolution, performed on materials with realistically small grain sizes. To date only limited experiments have been performed to investigate displacement fields with resolution well below the grain scale over a wide field for polycrystals with realistic grain sizes. This paper explores high-resolution in-plane material stretch and rotation fields measured at the sub-grain scale for ensembles of 10–20 grains in 70 μm grain-size OFHC Cu. Using a novel photolithographic technique, a series of micron-scale grids were deposited onto the surface specimens that were subsequently subjected to compression or torsion to effective strains up to unity. The resulting grid patterns for compression exhibit a tendency toward localization of stretch and rotation fields, with larger grains less highly deformed. For the case of shear, there is a reduced tendency towards microstructural scale localization of the in-plane stretch and rotation fields.

A comparative discrete-dislocation/crystal-plasticity analysis of bending of a single-crystalline microbeam

Bending of a micron-size single-crystalline beam is analyzed using both discrete-dislocation plasticity and crystal-plasticity formulations. Within the discrete-dislocation plasticity formulation, dislocations are treated as infinitely long straight-line defects residing within a linear elastic continuum. Evolution of the dislocation structure during bending is simulated by allowing the dislocations to glide in response to long-range interactions between different dislocations, and between dislocations and the applied stresses, and by incorporating various short-range reactions which can result in dislocation nucleation, annihilation or pinning. At each stage of bending, the stress and deformation fields are obtained by superposing the dislocation fields and the complementary fields obtained as a solution of the corresponding linear-elastic boundary value problem. The results obtained show that there is a continuing accumulation of “geometrically necessary” dislocations during bending which is expected due to the gradient in the strain throughout the beam height. In addition, it is found that localization of plastic flow into slip bands is a salient feature of materials deformation at the micron-length scale. Within the crystal-plasticity analysis, of beam bending, a small displacement gradient formulation is used and the material parameters selected in such a way that plastic flow localizes into deformation bands at low strains. It is found that, while the global response of the beam predicted by the two approaches can be quite comparable, fine details of the dislocation-based stress and deformation fields cannot be reproduced by the continuum crystal-plasticity model.

Molecular dynamics study of crack processes associated with dislocation nucleated at the tip

Atomic scale changes of structure around a crack-tip in an f.c.c. crystal under in-plane shear (mode II) loading are analyzed by molecular dynamics simulation (MD simulation), and its counterpart in the continuum crystal plasticity model is discussed. A ductile fracture process involving dislocation nucleation from the stressed tip is always observed. The nucleated dislocation is driven away from the crack and a dislocation-free zone develops in the near-tip region. Time averages of local stress in the near-tip region, both before and after dislocation nucleation, coincide well with the linear elastic prediction. The critical stress intensity factor estimated by MD simulation agrees well with Rice's theoretical prediction derived from the unstable stacking energy concept. The temperature dependence of the critical factor can be explained primarily as a thermally activated process.