Characterization of macroscopic tensile strength of polycrystalline metals with two-scale finite element analysis

The objective of this contribution is to develop an elastic-plastic-damage constitutive model for crystal grain and to incorporate it with two-scale finite element analyses based on mathematical homogenization method, in order to characterize the macroscopic tensile strength of polycrystalline metals. More specifically, the constitutive model for single crystal is obtained by combining hyperelasticity, a rate-independent single crystal plasticity and a continuum damage model. The evolution equations, stress update algorithm and consistent tangent are derived within the framework of standard elastoplasticity at finite strain. By employing two-scale finite element analysis, the ductile behaviour of polycrystalline metals and corresponding tensile strength are evaluated. The importance of finite element formulation is examined by comparing performance of several finite elements and their convergence behaviour is assessed with mesh refinement. Finally, the grain size effect on yield and tensile strength is analysed in order to illustrate the versatility of the proposed two-scale model.     

A multiscale approach for modeling scale-dependent yield stress in polycrystalline metals

Modeling of scale-dependent characteristics of mechanical properties of metal polycrystals is studied using both discrete dislocation dynamics and continuum crystal plasticity. The initial movements of dislocation arc emitted from a Frank-Read type dislocation source and bounded by surrounding grain boundaries are examined by dislocation dynamics analyses system and we find the minimum resolved shear stress for the FR source to emit at least one closed loop. When the grain size is large enough compared to the size of FR source, the minimum resolved shear stress levels off to a certain value, but when the grain size is close to the size of the FR source, the minimum resolved shear stress shows a sharp increase. These results are modeled into the expression of the critical resolved shear stress of slip systems and continuum mechanics based crystal plasticity analyses of six-grained polycrystal models are made. Results of the crystal plasticity analyses show a distinct increase of macro- and microscopic yield stress for specimens with smaller mean grain diameter. Scale-dependent characteristics of the yield stress and its relation to some control parameters are discussed.     

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.     

Multiscale strain measurements of plastically deforming polycrystalline titanium: Role of deformation heterogeneities

The purpose of this work was to characterize the spatial distribution of residual deformation at the mesoscale (a few grains) and at the macroscale (hundreds of grains) in titanium subjected to cyclic tensile loading. Using ex situ digital image correlation, we compared the axial residual strain fields obtained at optical magnifications ranging from 3.2× to 50×. To compare the results obtained at different optical magnifications, numerous images at higher magnification had to be assembled to encompass the same field-of-view observed at lower magnifications. The strain fields at the highest optical magnification revealed deformation patterns that were not detectable at lower magnifications. These deformation patterns appeared as inclined slip bands near grain boundaries and grain boundary triple points, with the bands sometimes crossing into neighboring grain interiors. Measurements made at optical magnifications greater than 10× captured an underlying deformation pattern, however, considerably more detail within grains was obtained at 50× magnification. The strain fields obtained at 10× and 50× magnifications were subsequently used to estimate the length scale of a representative volume element (RVE) based on the standard deviation of the average residual strain. The estimated RVE length scale was nearly three times the average grain diameter if extracted from the 50× results. The estimate of the RVE length scale was smaller at lower magnification which was due to a homogenizing effect caused by the low measurement resolution. Thus, care must be taken when experimentally obtaining RVE length scale estimates.     

Multiscale modeling of the dynamics of solids at finite temperature

We develop a general multiscale method for coupling atomistic and continuum simulations using the framework of the heterogeneous multiscale method (HMM). Both the atomistic and the continuum models are formulated in the form of conservation laws of mass, momentum and energy. A macroscale solver, here the finite volume scheme, is used everywhere on a macrogrid; whenever necessary the macroscale fluxes are computed using the microscale model, which is in turn constrained by the local macrostate of the system, e.g. the deformation gradient tensor, the mean velocity and the local temperature. We discuss how these constraints can be imposed in the form of boundary conditions. When isolated defects are present, we develop an additional strategy for defect tracking. This method naturally decouples the atomistic time scales from the continuum time scale. Applications to shock propagation, thermal expansion, phase boundary and twin boundary dynamics are presented.     

Multiscale modeling of strain-hardening cementitious composites

This research involves the multiscale characterization of strain-hardening cementitious composites under tensile loading. The sensitivity of cracking behavior to fiber dispersion is studied using a special form of lattice model, in which each fiber is explicitly represented. It is shown that the nonlocal modeling of fiber bridging forces is essential for obtaining realistic patterns of crack development and strain-hardening behavior. Crack count and crack size are simulated for progressively larger levels of tensile strain. The influence of fiber dispersion is clearly evident: regions with significantly fewer fibers act as defects, reducing strength and strain capacity of the material.     

Effects of lattice misorientations on strain heterogeneities in FCC polycrystals

It is well documented that the highly heterogeneous deformation behaviour and lattice rotation typically observed within grains in a polycrystal are attributed to microstructural features such as grain structure, topology, size, etc. In this work, the effects of low- and high-angle grain boundaries on the mechanical behaviour of FCC polycrystals are investigated using a micro-mechanical model based on crystal plasticity theory. The constitutive framework relies on dislocation mechanics concepts to describe the plastic deformation behaviour of FCC metallic crystals and is validated by comparing the measured and predicted local and macroscopic deformation behaviour in a thin Al-0.5% Mg polycrystal tensile specimen containing a relatively small number of surface grains. Comparisons at the microscopic (e.g. local slip distribution) and macroscopic (e.g. average stress-strain response) levels elucidate the role of low-angle grain boundaries, which are found to have a profound effect on both the local and average deformation behaviour of FCC polycrystals with a small number of grains. However, this effect diminishes when the number of grains increases and becomes negligible in bulk polycrystals. In light of the widely accepted view that high-angle grain boundaries strongly influence the mechanical behaviour of very fine-grained metals, this work has shown that low-angle grain boundaries can also play an equally important role in the deformation behaviour of polycrystals with a relatively small number of grains.     

Multiscale cohesive failure modeling of heterogeneous adhesives

A novel multiscale cohesive approach that enables prediction of the macroscopic properties of heterogeneous thin layers is presented. The proposed multiscale model relies on the Hill's energy equivalence lemma, implemented in the computational homogenization scheme, to couple the micro- and macro-scales and allows to relate the homogenized cohesive law used to model the failure of the adhesive layer at the macro-scale to the complex damage evolution taking place at the micro-scale. A simple isotropic damage model is used to describe the failure processes at the micro-scale. We establish the upper and lower bounds on the multiscale model and solve several examples to demonstrate the ability of the method to extract physically based macroscopic properties.     

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 the induced plastic anisotropy in bcc metals

This paper presents a new framework to predict the qualitative and quantitative variation in local plastic anisotropy due to crystallographic texture in body-centered cubic polycrystals. A multiscale model was developed to examine the contribution of mesoscopic and local microscopic behaviour to the macroscopic constitutive response of bcc metals during deformation. The model integrated a dislocation-based hardening scheme and a Taylor-based crystal plasticity formulation into the subroutine of an explicit dynamic FEM code (LS-DYNA). Numerical analyses using this model were able to predict not only correct grain rotation during deformation, but variations in plastic anisotropy due to initial crystallographic orientation. Optimal results were obtained when {1 1 0}〈1 1 1〉, {1 1 2}〈1 1 1〉, and {1 2 3}〈1 1 1〉 slip systems were considered to be potentially active. The predicted material heterogeneity can be utilised for research involving any texture-dependent work hardening behaviour, such as surface roughening.     

A self-consistent plasticity theory for modeling the thermo-mechanical properties of irradiated FCC metallic polycrystals

A self-consistent theoretical framework is developed to model the thermo-mechanical behaviors of irradiated face-centered cubic (FCC) polycrystalline metals at low to intermediate homologous temperatures. In this model, both irradiation and temperature effects are considered at the grain level with the assist of a tensorial plasticity crystal model, and the elastic-visocoplastic self-consistent method is applied for the scale transition from individual grains to macroscopic polycrystals. The proposed theory is applied to analyze the mechanical behaviors of irradiated FCC copper. It is found that: (1) the numerical results match well with experimental data, which includes the comparison of results for single crystals under the load in different directions, and for polycrystals with the influences of irradiation and temperature. Therefore, the feasibility and accuracy of the present model are well demonstrated. (2) The main irradiation effects including irradiation hardening, post-yield softening, strain-hardening coefficient (SHC) dropping and the non-zero stress offset are all captured by the proposed model. (3) The increase of temperature results in the decrease of yield strength and SHC. The former is attributed to the weakened dislocation–defect interaction, while the latter is due to the temperature-strengthened dynamic recovery of dislocations through the thermally activated mechanism. The present model may provide a theoretical guide to predict the thermo-mechanical behaviors of irradiated FCC metals for the selection of structural materials in nuclear equipment.     

Multiscale modeling of fracture of MgO: Sensitivity of interatomic potentials

Three different interatomic potentials, namely, B-G I Model, B-G II Model and L-C Model, are used in multiscale modeling and simulation of a center-cracked specimen made of magnesia subjected to monotonically increasing loading. The specimen is decomposed into a far field, a near field and a crack-tip region. The analytical solution in the far field from linear elastic fracture mechanics (LEFM) is utilized. The solution of the near field is based on a multiscale field theory. In the crack-tip region, molecular dynamics (MD) simulation is employed. These methodologies are integrated to simulate mixed mode fracture of magnesia (MgO). Three different interatomic potentials are examined and the interatomic potential and interatomic force between Mg-Mg, Mg-O and O-O are shown. The numerical results of crack propagation demonstrate that (1) crack closure is witnessed in B-G I Model but not in B-G II Model and L-C Model, (2) B-G II Model and L-C Model diverge in the early stage. The cause of instability and the remedy are also discussed.     

Micromechanical modeling of the elasto-viscoplastic behavior of semi-crystalline polymers

A micromechanically based constitutive model for the elasto-viscoplastic deformation and texture evolution of semi-crystalline polymers is developed. The model idealizes the microstructure to consist of an aggregate of two-phase layered composite inclusions. A new framework for the composite inclusion model is formulated to facilitate the use of finite deformation elasto-viscoplastic constitutive models for each constituent phase. The crystalline lamellae are modeled as anisotropic elastic with plastic flow occurring via crystallographic slip. The amorphous phase is modeled as isotropic elastic with plastic flow being a rate-dependent process with strain hardening resulting from molecular orientation. The volume-averaged deformation and stress within the inclusions are related to the macroscopic fields by a hybrid interaction model. The uniaxial compression of initially isotropic high density polyethylene (HDPE) is taken as a case study. The ability of the model to capture the elasto-plastic stress-strain behavior of HDPE during monotonic and cyclic loading, the evolution of anisotropy, and the effect of crystallinity on initial modulus, yield stress, post-yield behavior and unloading-reloading cycles are presented.     

Multiscale micromechanical modeling of the constitutive response of carbon nanotube-reinforced structural adhesives

Appropriate formulations are developed to allow for the atomistic-based continuum modeling of nano-reinforced structural adhesives on the basis of a nanoscale representative volume element that accounts for the nonlinear behavior of its constituents; namely, the reinforcing carbon nanotube, the surrounding adhesive and their interface. The newly developed representative volume element is then used with analytical and computational micromechanical modeling techniques to investigate the homogeneous dispersion of the reinforcing element into the adhesive upon both the linear and nonlinear properties. Unlike our earlier work where the focus was on developing linear micromechanical models for the effective elastic properties of nanocomposites, the present approach extends these models by describing the development of a nonlinear hybrid Monte Carlo Finite Element model that allows for the prediction of the full constitutive response of the bulk composite under large deformations. The results indicate a substantial improvement in both the Young’s modulus and tensile strength of the nano-reinforced adhesives for the range of CNT concentrations considered.     

Constitutive modeling of cyclic plasticity deformation of a pure polycrystalline copper

Cyclic plasticity experiments were conducted on a pure polycrystalline copper and the material was found to display significant cyclic hardening and nonproportional hardening. An effort was made to describe the cyclic plasticity behavior of the material. The model is based on the framework using a yield surface together with the Armstrong–Frederick type kinematic hardening rule. No isotropic hardening is considered and the yield stress is assumed to be a constant. The backstress is decomposed into additive parts with each part following the Armstrong–Frederick type hardening rule. A memory surface in the plastic strain space is used to account for the strain range effect. The Tanaka fourth order tensor is used to characterize nonproportional loading. A set of material parameters in the hardening rules are related to the strain memory surface size and they are used to capture the strain range effect and the dependence of cyclic hardening and nonproportional hardening on the loading magnitude. The constitutive model can describe well the transient behavior during cyclic hardening and nonproportional hardening of the polycrystalline copper. Modeling of long-term ratcheting deformation is a difficult task and further investigations are required.     

An explicit formulation for multiscale modeling of bcc metals

Many materials for specialized applications exhibit a body-centered cubic structure; e.g., tantalum, vanadium, barium and chromium. In addition, the successful modeling of body-centered cubic (bcc) metals is a necessary step toward modeling of common structural materials such as iron. Implicit formulations for this class of materials exist [e.g., Stainier, L., Cuitiño, A., Ortiz, M., 2002. A micromechanical model of hardening, rate sensitivity, and thermal softening in bcc crystals. Journal of the Mechanics and Physics of Solids 50 (7), 1511–1545; Kuchnicki, S., Radovitzky, R., Cuitiño, A., Strachan, A., Ortiz, M., 2007. A pressure-dependent multiscale model for bcc metals], but are impractical to resolve large-scale dynamic deformation processes. In this article, we describe a procedure analogous to Kuchnicki et al. [Kuchnicki, S., Cuitiño, A., Radovitzky, R., 2006. Efficient and robust constitutive integrators for single-crystal plasticity modeling. International Journal of Plasticity 22 (10), 1988–2011]. wherein we construct an explicit formulation for the multiscale physics models. This update is based on the model of Kuchnicki et al. (in preparation) using a power law representation for the plastic slip rates. The existing implicit form of the model provides qualitative matching with experiments at quasi-static strain rates. The model is recast in an explicit form and applied first to a high quasi-static strain rate to verify that the two forms of the model return similar predictions for similar input parameters. The explicit model is also applied to several high strain rates, showing that it captures characteristic features observed in experimental tests of high-rate deformations, such as the drop in stress immediately after yield that is present in split Hopkinson pressure bar (SHPB) experiments. This test provides qualitative evidence that the model is suitable for high-strain-rate applications. The utility of the model is further demonstrated by a one-dimensional simulation of a SHPB test. Finally, a test case modeling pressure impact of a Tantalum plate using 600,000 elements is shown. The simulations show that the explicit model is capable of recovering the salient features of the experiments while integrating the constitutive update in a robust manner.     

Multiscale dislocation dynamics simulations of shock compression in copper single crystal

Ultra short pulse shock wave propagation, plastic deformation and evolution of dislocations in copper single crystals with (0 0 1), (0 1 1) and (1 1 1) orientations are investigated using multiscale dislocation dynamics plasticity analyses. The effects of peak pressure, pulse duration, crystal anisotropy and the nonlinear elastic properties on the interaction between shock wave and dislocations are investigated. The results of our calculations show that the dislocation density has a power law dependence on pressure with a power of 1.70 and that the dislocation density is proportional to pulse duration and sensitive to crystal orientation. These results are in very good agreement with the analytical predications of Meyers et al. [Meyers, M.A., Gregori, F., Kad, B.K., Schneider, M.S., Kalantar, D.H., Remington, B.A., Ravichandran G., Boehly, T., Wark, J., 2003. Laser-induced shock compression of monocrystalline copper: characterization and analysis. Acta Materialia 51, 1211–1228] and the experimental results of Murr [Murr, L.E., 1981. Residual microstructure-mechanical property relationships in shock loaded metals and alloys. In: Meyers, M.A., Murr, L.E. (Eds.), Shock Waves and High Strain Rate Phenomena in Metals. Plenum, New York, pp. 607–673]. It is shown also that incorporating the effect of crystal anisotropy in the elastic properties results in orientation dependent wave speed and peak pressure. The relaxed configurations of dislocation microstructures show the formation of microbands coincident with the slip planes.     

Mechanics modeling for deformation of nano-grained metals

The electro-deposition technique is capable of producing nano-grained bulk copper specimens that exhibit superplastic extensibility at room temperature. Metals of such small grain sizes deform by grains squeezing past each other, with little distortion occurring in the grain cores. Accommodation mechanisms such as grain boundary diffusion and grain rotation control the kinetics of the process. A model of a 9-grain cluster is proposed that incorporates both the Ashby-Verrall mechanism and the 30° rotation of closely linked grain pairs. A constitutive relation is derived that relates the creep strain rate linearly to the difference between the applied stress and a threshold stress. The creep rate and the threshold stress predicted by the model are in quantitative agreement with the experimental data.     

Cohesive modeling of low-velocity impact damage in layered functionally graded beams

Numerical analysis of the low-velocity impact damage of a layered composite beam with a functionally graded core is performed using the multiple-isoparametric cohesive volume finite element (MCVFE) scheme. A mixed-mode intrinsic cohesive zone model is used to simulate the spontaneous damage initiation and growth in this work. The inhomogeneous Young’s modulus variation is assumed to be symmetric about the neutral plane. Our parametric simulations showed that the energetics of damage is altered by the presence of a functionally graded core. The effect of including a functionally graded core is to advance the time of fracture initiation compared to a cross-ply (90°) core. The assumed symmetry and linear inhomogeneity leads to the energetics for the graded core to be similar to those observed for a 45° core ply-orientation.