Scale transition of a higher order plasticity model – A consistent homogenization theory from meso to macro

Standard plasticity models cannot capture the microstructural size effect associated with grain sizes, as well as structural size effects induced by external boundaries and overall gradients. Many higher-order plasticity models introduce a length scale parameter to resolve the latter limitation – microstructural influences are not explicitly account for. This paper adopts two distinct length scales in the formulation, i.e. an intrinsic length scale (l) governing micro-processes such as dislocation pile-up at internal boundaries, as well as the characteristic grain size (L), and aims to unravel the interaction between these two length scales and the characteristic specimen size (H) at the macro level. At the meso-scale, we adopt the strain gradient plasticity model developed in Gurtin (2004) [Gurtin, M.E., 2004. A gradient theory of small-deformation isotropic plasticity that accounts for the Burgers vector and for dissipation due to plastic spin. J. Mech. Phys. Solids 52, 2545–2568] which accounts for the direct influence of grain boundaries. Through a novel homogenization theory, the plasticity model is translated consistently from meso to macro. The two length scale parameters (l and L) manifest themselves naturally at the macro scale, hence capturing both types of size effects in an average sense. The resulting (macro) higher-order model is thermodynamically consistent to the meso model, and has the same structure as a micromorphic continuum. Finally, we consider a bending example for the two limiting cases – microhard and microfree conditions at grain boundaries – and illustrate the excellent match between the meso and homogenized solutions.     

Fracture of bicrystal metal/ceramic interfaces: A study via the mechanism-based strain gradient crystal plasticity theory

Two continuum mechanical models of crystal plasticity theory namely, conventional crystal plasticity theory and mechanism-based crystal plasticity theory, are used to perform a comparative study of stresses that are reached at and ahead of the crack tip of a bicrystal niobium/alumina specimen. Finite element analyses are done for a stationary crack tip and growing cracks using a cohesive modelling approach. Using mechanism-based strain gradient crystal plasticity theory the stresses reached ahead of the crack tip are found to be two times larger than the stresses obtained from conventional crystal plasticity theory. Results also show that strain gradient effects strongly depend on the intrinsic material length to the size of plastic zone ratio (l/R₀). It is found that the larger the (l/R₀) ratio, the higher the stresses reached using mechanism-based strain gradient crystal plasticity theory. An insight into the role of cohesive strength and work of adhesion in macroscopic fracture is also presented which can be used by experimentalists to design better bimaterials by varying cohesive strength and work of adhesion.     

Modeling the plastic response of thin metal membranes

Using a dislocations-based model of slip and crystal plasticity, we show by illustrative examples that the experimentally observed increase in the yield stress of very thin metallic membranes most likely is due to the variation of grain orientations through the thickness of the membrane, as well as the surface hardness due to oxidation or contamination, both of which generally are insignificant when there is a sufficient number of interior crystals through the membrane thickness; the overall effect may well be produced by a combination of these two causes. We show that crystal plasticity models can account for such size effects without a need for resorting to phenomenological strain-gradient models. We illustrate this using Nemat-Nasser's dislocations-based slip-induced crystal plasticity model that inherently includes length scales, although other rate-dependent slip models, e.g., the classical power-law slip model, most likely would qualitatively produce similar results. Our numerical results, based on the experimentally supported dislocation-induced slip model and the values of the model parameters given in Nemat-Nasser and Li [1998. Flow stress of F.C.C. polycrystals with application to OFHC Cu. Acta Mater. 46, 565-577], correlate well, both qualitatively and quantitatively, with the experimental results reported by Hommel and Kraft [2001. Deformation behavior of thin copper films on deformable substrates. Acta Mater. 49, 3935-3947] and Espinosa et al. [2004. Plasticity size effect in free-standing submicron polycrystalline FCC films subjected to pure tension. J. Mech. Phys. Solids 52, 667-689] for thin copper membranes, suggesting that, for submicron-sized samples, the classical crystal plasticity with slip models, does qualitatively account well for the small-size effects, and that quantitative predictions are obtained when, in addition, a physics-based dislocation model that includes length scales, is used. It is thus concluded that the length-scale effect and the size effect are two separate issues in metal plasticity, both of which are nicely accounted for by physics-based dislocation models of crystal plasticity without a need to include the plastic strain gradient.     

Cosserat modeling of cellular solidsModélisation des solides cellulaires selon Cosserat

Cellular solids inherit their macroscopic mechanical properties directly from the cellular microstructure. However, the characteristic material length scale is often not small compared to macroscopic dimensions, which limits the applicability of classical continuum-type constitutive models. Cosserat theory, however, offers a continuum framework that naturally features a length scale related to rotation gradients. In this paper a homogenization procedure is proposed that enables the derivation of macroscopic Cosserat constitutive equations based on the underlying microstructural morphology and material behavior. To cite this article: P.R. Onck, C. R. Mecanique 330 (2002) 717–722.     

A stochastic approach to capture crystal plasticity

A stochastic crystal plasticity model is proposed and applied within the rate-independent regime. As opposed to conventional deterministic algorithms wherein multiple slip systems are activated and redundant constraints may exist, the new Monte Carlo plasticity (MCP) paradigm is based on a stochastic chain of singly activated slip systems and thus avoids the possible ill-condition associated with multi-slip algorithms. The choice of the activated slip system is made at each Monte Carlo (MC) step based on the Metropolis algorithm. The MCP model is implemented within a Material Point Method (MPM) as a constitutive model to capture the elasto-plastic behavior of polycrystalline materials. A comparison with a commonly used singular value decomposition (SVD) algorithm indicates that MCP offers superior computational efficiency while maintaining comparable accuracy.     

Strain gradient crystal plasticity effects on flow localization

In metal grains one of the most important failure mechanisms involves shear band localization. As the band width is small, the deformations are affected by material length scales. To study localization in single grains a rate-dependent crystal plasticity formulation for finite strains is presented for metals described by the reformulated Fleck–Hutchinson strain gradient plasticity theory. The theory is implemented numerically within a finite element framework using slip rate increments and displacement increments as state variables. The formulation reduces to the classical crystal plasticity theory in the absence of strain gradients. The model is used to study the effect of an internal material length scale on the localization of plastic flow in shear bands in a single crystal under plane strain tension. It is shown that the mesh sensitivity is removed when using the nonlocal material model considered. Furthermore, it is illustrated how different hardening functions affect the formation of shear bands.     

Direct calculations of material parameters for gradient plasticity

Length scale parameters introduced in gradient theories of plasticity are calculated in closed form with a continuum dislocation based theory. The similarity of the governing equations in both models for the evolution of plastic deformation of a constrained thin film makes it possible to identify parameters of the gradient plasticity theory with the dislocation based model. A one-to-one identification is not possible given that gradient plasticity does not account for individual dislocations. However, by comparing the mean plastic deformation across the film thickness we find that the length scale parameter, l, introduced in the gradient plasticity theory depends on the geometry as well as material constants.     

A robust integration algorithm for implementing rate dependent crystal plasticity into explicit finite element method

Severe numerical instability in the integration of rate dependent crystal plasticity (RDCP) model is one of the main problems for implementing RDCP into finite element method (FEM), especially for simulating dynamic/transient forming process containing complicated contact conditions under large step length, large strain and high strain rate. In order to overcome the problem, an implicit model is deduced with the primary unknowns of shear strain increments of slip systems under the corotational coordinate system in the paper. The homotopy auto-changing continuation method combined with the Newton–Raphson (N–R) iteration is adopted. The subroutine VUMAT is developed for implementing RDCP model in ABAQUS/Explicit. Simulation results show that the algorithm is stable and accurate in 3D FE simulations on both dynamic simple loading and complicated loading process containing nonlinear contacts under the conditions of the maximal step length of 3.5 × 10⁻⁶ s, the maximal strain of 1.05, the maximal loading speed of 120 mm s⁻¹, and the minimal material rate sensitivity coefficient of 0.01. The predictions of the model on crystal behaviors of anisotropy, rate sensitivity and elasticity, as well as ear profiles in deep cup drawing are in agreement with experiments.     

Numerical simulation of welding induced damage and residual stress of martensitic steel 15-5PH

This paper is about the modeling of damage and residual stresses induced by a complex history of thermo-elasto-plastic multiphase in welding heat affected zone (HAZ). A two-scale model for elasto-plastic damaged multiphase is developed. The constitutive equations of the model are coupling ductile damage, elasto-plastic strains, phase transformation and transformation plasticity. In this study, an equation of damage evolution is proposed based on the continuum damage mechanics. In order to validate the proposed damage evolution, a set of flat notched bars with various notches are to simulated and then compared to experimental results. An example of laser heated disk is simulated to elaborate the two-scale model and the simulated results of the two-scale model in CAST3M^® is compared the results calculated by a homogenized macroscopic model in SYSWELD^®. The study shows the two-scale model provides valuable freedom to choose the material or mechanical law for each phase. The proposed damage equation extends the application of existing ductile damage models. In the calculation of residual stresses, damage decreases the residual stresses in HAZ by a small amount.     

A unified theory for cavity expansion in cohesive-frictional micromorphic media

This paper presents a unified theory for both cylindrical and spherical cavity expansion problems in cohesive-frictional micromorphic media. A phenomenological strain-gradient plasticity model in conjunction with a generalized Mohr–Coulomb criterion is employed to characterize the elasto-plastic behavior of the material. To solve the resultant two-point boundary-value problem (BVP) of fourth-order homogeneous ordinary differential equation (ODE) for the governing equations which is not well-conditioned in certain cases, several numerical methods are developed and are compared in terms of robustness, efficiency and accuracy. Using one of the finite difference methods that shows overall better performance, both cylindrical and spherical cavity expansion problems in micromorphic media are solved. The influences of microstructural properties on the expansion response are clearly demonstrated. Size effect during the cavity expansion is captured. The proposed theory is also applied to a revisit of the classic problem of stress concentration around a cavity in a micromorphic medium subjected to isotropic tension at infinity, for which some conclusions made in early studies are revised. The proposed theory can be useful for the interpretation of indentation tests at small scales.     

Multiscale modeling of the plasticity in an aluminum single crystal

This paper describes a numerical, hierarchical multiscale modeling methodology involving two distinct bridges over three different length scales that predicts the work hardening of face centered cubic crystals in the absence of physical experiments. This methodology builds a clear bridging approach connecting nano-, micro- and meso-scales. In this methodology, molecular dynamics simulations (nanoscale) are performed to generate mobilities for dislocations. A discrete dislocations numerical tool (microscale) then uses the mobility data obtained from the molecular dynamics simulations to determine the work hardening. The second bridge occurs as the material parameters in a slip system hardening law employed in crystal plasticity models (mesoscale) are determined by the dislocation dynamics simulation results. The material parameters are computed using a correlation procedure based on both the functional form of the hardening law and the internal elastic stress/plastic shear strain fields computed from discrete dislocations. This multiscale bridging methodology was validated by using a crystal plasticity model to predict the mechanical response of an aluminum single crystal deformed under uniaxial compressive loading along the [4 2 1] direction. The computed strain-stress response agrees well with the experimental data.     

Numerical simulations of necking during tensile deformation of aluminum single crystals

Finite element simulations are used to study strain localization during uniaxial tensile straining of a single crystal with properties representative of pure Al. The crystal is modeled using a constitutive equation incorporating self- and latent-hardening. The simulations are used to investigate the influence of the initial orientation of the loading axis relative to the crystal, as well as the hardening and strain rate sensitivity of the crystal on the strain to localization. We find that (i) the specimen fails by diffuse necking for strain rate exponents m < 100, and a sharp neck for m > 100. (ii) The strain to localization is a decreasing function of m for m < 100, and is relatively insensitive to m for m > 100. (iii) The strain to localization is a minimum when the tensile axis is close to (but not exactly parallel to) a high symmetry direction such as [1 0 0] or [1 1 1] and the variation of the strain to localization with orientation is highly sensitive to the strain rate exponent and latent-hardening behavior of the crystal. This behavior can be explained in terms of changes in the active slip systems as the initial orientation of the crystal is varied.     

Plasticity in polycrystalline fretting fatigue contacts

Plastic deformation at the scale of microstructure plays an important role in fretting fatigue failure of metals under cyclic loading. In this study, crystal viscoplasticity theory with a planar triple slip idealization is employed to represent crystallographic plasticity in two-dimensional fretting analyses of Ti-6Al-4V. Subsurface deformation maps, fretting maps, and shakedown maps are constructed based on application of J₂ plasticity theory for the polycrystalline substrate. Comparisons are then made with polycrystal viscoplasticity simulations, the latter suggesting that plastic ratchetting plays a significant role in the fretting fatigue process.     

Two stochastic mean-field polycrystal plasticity methods

In this work, we develop two mean-field polycrystal plasticity models in which the crystal velocity gradients L^c are approximated stochastically. Through comprehensive CPFEM analyses of an idealized tantalum polycrystal, we verify that the L^c tend to follow a normal distribution and surmise that this is due to the crystal interactions. We draw on these results to develop the stochastic Taylor model (STM) and the stochastic no-constraints model (SNCM), which differ in the manner in which the crystal strain rates are prescribed. Calibration and validation of the models are performed using data from tantalum compression experiments. Both models predict the compression textures more accurately than the fully constrained model (FCM), and the SNCM predicts them more accurately than the STM. The STM is extremely computationally efficient, only slightly more expensive than the FCM, while the SNCM is three times more computationally expensive than the STM.