Grain size effects in multiphase steels assisted by transformation-induced plasticity

The influence of the austenitic grain size on the overall stress–strain behavior in a multiphase carbon steel is analyzed through three-dimensional finite element simulations. A recently developed multiscale martensitic transformation model is combined with a plasticity model to simulate the transformation-induced plasticity effects of a grain of retained austenite embedded in a ferrite-based matrix. Grain size effects are included via a surface energy term in the Helmholtz energy. Tensile simulations for representative orientations of the grain of retained austenite show that the initial stability of the austenite increases as the grain size decreases. Consequently, the effective strength is initially higher for smaller grains. The influence of the grain size on the evolution of the transformation process strongly depends on the grain orientation. For “hard” orientations, the transformation rate is higher for larger grains. In addition, the phase transformation is partially suppressed as the grain size decreases. In contrast, for “soft” orientations, the transformation rate is lower for larger grains. The phase transformation is more homogeneous for smaller grains and, consequently, the effective transformation strain is larger. Nevertheless, in multiphase carbon steels with a relatively low percentage of retained austenite, the influence of the austenitic grain size on the overall constitutive response is smaller than the influence of the austenitic grain orientation.     

Multiscale boundary conditions in crystalline solids: Theory and application to nanoindentation

This paper presents a systematic approach to treating the interfaces between the localized (fine grain) and peripheral (coarse grain) domains in atomic scale simulations of crystalline solids. Based on Fourier analysis of regular lattices structures, this approach allows elimination of unnecessary atomic degrees of freedom over the coarse grain, without involving an explicit continuum model for the latter. The mathematical formulation involves compact convolution operators that relate displacements of the interface atoms and the adjacent atoms on the coarse grain. These operators are defined by geometry of the lattice structure, and interatomic potentials. Application and performance are illustrated on quasistatic nanoindentation simulations with a crystalline gold substrate. Complete atomistic resolution on the coarse grain is alternatively employed to give the benchmark solutions. The results are found to match well for the multiscale and the full atomistic simulations.     

Theoretical properties of cubic crystals at arbitrary pressure—III. Stability

A complete account is presented of the application of the principles of bifurcation analysis for general materials to the particular case of cubic crystals subjected to hydrostatic loading. The treatment of crystal stability is classical in that (i) the loading environment is fully specified, to sufficient order and in both its active and passive modes, and (ii) the potential energy of the system as a whole is examined in all the nearby, possibly inhomogeneous, configurations allowed by the kinematic constraints. Computations are made of the pressures and the bulk and shear moduli of the entire Morse-model family of fcc, bcc, and sc monatornic crystals under pure hydrostatic compression and tension. The stable range of each lattice as well as the potential bifurcations at the range limits are presented and discussed in terms of the role of the particular lattice structure and the effective range of the interatomic potential function (as specified by the parameter log β). The fee lattices are stable in compression and in tension up to an all-round stretch λ = Λ_k, at which point the bulk modulus vanishes; Λ_k, is a monotonically decreasing function of log β. The bcc lattices are stable for λ < Λ_CR, where the bulk modulus or the shear modulus μ vanishes (depending upon the value of log β) at λ = Λ_CR. For very large values of log β a second range of bcc stability is located in a region of hydrostatic expansion. The sc crystals are stable only in a range of hydrostatic tension and only for relatively short-range interatomic interactions (large log β); the present work appears to be the first in which a theoretical range of stability of sc crystals has been revealed. The question of the possibility of assessing lattice stability under load with the aid of higher order moduli at zero load is given consideration quantitatively for the fee lattices and the bcc lattices that are stable at zero load. Finally, the present approach to crystal stability is distinguished from some simplistic notional criteria based upon local convexity of strain energy and, for the Morse-model cubic crystals, quantitative comparisons are made with the present classical treatment of stability in a hydrostatic environment.     

On the continuum modeling of carbon nanotubes

We have recently proposed a nanoscale continuum theory for carbon nanotubes. The theory links continuum analysis with atomistic modeling by incorporating interatomic potentials and atomic structures of carbon nanotubes directly into the constitutive law. Here we address two main issues involved in setting up the nanoscale continuum theory for carbon nanotubes, namely the multi-body interatomic potentials and the lack of centrosymmetry in the nanotube structure. We explain the key ideas behind these issues in establishing a nanoscale continuum theory in terms of interatomic potentials and atomic structures.     

晶体非比例双滑移的一般运动学

本文对非比例双滑移的变形规律和运动学进行了一般的数学分析。提出了双线性双滑移、逐段线性双滑移的变形梯度张量的解析公式。导出了线元切向量的变形方程以及拉伸轴在极射赤面投影图上位置变化计算公式。 对于任意形式的双滑移,本文提出了运动学分析的精确的数值积分方法。 最后结合面心立方和体心立方晶体,计算了几种典型情况下的拉伸轴旋转。     

Size effects in martensitic microstructures: Finite-strain phase field model versus sharp-interface approach

A finite-strain phase field model for martensitic phase transformation and twinning in shape memory alloys is developed and confronted with the corresponding sharp-interface approach extended to interfacial energy effects. The model is set in the energy framework so that the kinetic equations and conditions of mechanical equilibrium are fully defined by specifying the free energy and dissipation potentials. The free energy density involves the bulk and interfacial energy contributions, the latter describing the energy of diffuse interfaces in a manner typical for phase-field approaches. To ensure volume preservation during martensite reorientation at finite deformation within a diffuse interface, it is proposed to apply linear mixing of the logarithmic transformation strains. The physically different nature of phase interfaces and twin boundaries in the martensitic phase is reflected by introducing two order-parameters in a hierarchical manner, one as the reference volume fraction of austenite, and thus of the whole martensite, and the second as the volume fraction of one variant of martensite in the martensitic phase only. The microstructure evolution problem is given a variational formulation in terms of incremental fields of displacement and order parameters, with unilateral constraints on volume fractions explicitly enforced by applying the augmented Lagrangian method. As an application, size-dependent microstructures with diffuse interfaces are calculated for the cubic-to-orthorhombic transformation in a CuAlNi shape memory alloy and compared with the sharp-interface microstructures with interfacial energy effects.     

A general kinematical analysis of double slip

General kinematic solutions for double slip in fcc and bcc crystals are presented which are free from constitutive assumptions: that is, the analysis does not presuppose equal amounts of slipping on equallystressed slip systems, in contrast to the standard solutions (wherein Taylor hardening is implicitly assumed). The axis rotations and limiting positions on a stereographic projection are illustrated for several different slip-system combinations, initial axis positions (none on a symmetry line), and proportional slip ratios in both fcc and bcc crystals. It is suggested that the solutions have particular application to the experimental study of double slip in the tensile test of prestrained crystals.     

A variational multiscale constitutive model for nanocrystalline materials

This paper presents a variational multi-scale constitutive model in the finite deformation regime capable of capturing the mechanical behavior of nanocrystalline (nc) fcc metals. The nc-material is modeled as a two-phase material consisting of a grain interior phase and a grain boundary effected zone (GBAZ). A rate-independent isotropic porous plasticity model is employed to describe the GBAZ, whereas a crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The constitutive models of both phases are formulated in a small strain framework and extended to finite deformation by use of logarithmic and exponential mappings. Assuming the rule of mixtures, the overall behavior of a given grain is obtained via volume averaging. The scale transition from a single grain to a polycrystal is achieved by Taylor-type homogenization where a log-normal grain size distribution is assumed. It is shown that the proposed model is able to capture the inverse Hall-Petch effect, i.e., loss of strength with grain size refinement. Finally, the predictive capability of the model is validated against experimental results on nanocrystalline copper and nickel.     

Microstructure-performance relations of ultra-high-performance concrete accounting for effect of alpha-quartz-to-coesite silica phase transformation

The effect of the α-quartz-to-coesite silica phase transformation on the load-carrying and energy-dissipation capacities of ultra-high-performance concrete (UHPC) under dynamic loading with hydrostatic pressures of up to 10 GPa is evaluated. The model resolves essential deformation and failure mechanisms and provides a phenomenological account of the transformation. Four modes of energy dissipated are tracked, including inelastic deformation, distributed cracking, interfacial friction, and the energy dissipated through transformation of the quartz aggregate. Simulations are carried out over a range of volume fractions of the constituent phases. Results show that the phase transformation has a significant effect on the energy-dissipation capacity of UHPC for the conditions studied. Although transformation accounts for less than 2% of the total energy dissipation, the transformation leads to a twofold increase in the crack density and yields almost an 18% increase in the overall energy dissipation. Structure-response relations that can be used for materials design are established.     

Band gaps of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress

In this paper, the stop band properties of elastic waves in three-dimensional piezoelectric phononic crystals with initial stress are studied taking the mechanical and electrical coupling into account. The band gap characteristics for three kinds of lattice arrangements (i.e. sc, bcc and fcc) are investigated by the plane wave expansion (PWE) method. Regarding the variables of mechanical and electrical fields as the elements of the generalized state vector, the expression of the generalized eigenvalue equation for three-dimensional piezoelectric periodic structures is derived. Numerical calculations are performed for the PZT-2/polymer and ZnO/polymer phononic crystals. It can be observed from the results that the fcc lattice is more favorable to create the stop band than the sc and bcc lattices for the piezoelectric phononic crystals, which has also been proved for the pure elastic periodic structures. Compared with the PZT-2/polymer systems, the band gap of the sc lattice for the ZnO/polymer structures is narrower. However, the widths of the bcc and fcc lattices for the ZnO/polymer phononic crystals are much larger than those for the PZT-2/polymer structures. The lattice arrangements and the piezoelectricity have remarkable influences on the stop band behaviors.     

Modeling dynamic plasticity and spall fracture in high density polycrystalline alloys

The dynamic thermomechanical response of a tungsten heavy alloy is investigated via modeling and numerical simulation. The material of study consists of relatively stiff pure tungsten grains embedded within a more ductile matrix alloy comprised of tungsten, nickel, and iron. Constitutive models implemented for each phase account for finite deformation, heat conduction, plastic anisotropy, strain-rate dependence of flow stress, thermal softening, and thermoelastic coupling. The potentially nonlinear volumetric response in tungsten at large pressures is addressed by a pressure-dependent effective bulk modulus. Our framework also provides a quantitative prediction of the total dislocation density, associated with cumulative strain hardening in each phase, and enables calculation of the fraction of plastic dissipation converted into heat energy. Cohesive failure models are employed to represent intergranular fracture at grain and phase boundaries. Dynamic finite element simulations illustrate the response of realistic volume elements of the polycrystalline microstructure subjected to compressive impact loadings, ultimately resulting in spallation of the material. The relative effects of mixed-mode interfacial failure criteria, thermally-dependent fracture strengths, and grain shapes and orientations upon spall behavior are weighed, with interfacial properties exerting a somewhat larger influence on the average pressure supported by the volume element than grain shapes and initial lattice orientations within the bulk material. Spatially resolved profiles of particle velocities at the free surfaces of the volume elements indicate the degree to which the incident and reflected stress waves are altered by the heterogeneous microstructure.     

Non-equilibrium grain boundary structure and inelastic deformation using atomistic simulations

Grain boundary influence on material properties becomes increasingly significant as grain size is reduced to the nanoscale. Nanostructured materials produced by severe plastic deformation techniques often contain a higher percentage of high-angle grain boundaries in a non-equilibrium or energetically metastable state. Differences in the mechanical behavior and observed deformation mechanisms are common due to deviations in grain boundary structure. Fundamental interfacial attributes such as atomic mobility and energy are affected due to a higher non-equilibrium state, which in turn affects deformation response. In this research, atomistic simulations employing a biased Monte Carlo method are used to approximate representative non-equilibrium bicrystalline grain boundaries based on an embedded atom method potential, leveraging the concept of excess free volume. An advantage of this approach is that non-equilibrium boundaries can be instantiated without the need of simulating numerous defect/grain boundary interactions. Differences in grain boundary structure and deformation response are investigated as a function of non-equilibrium state using Molecular Dynamics. A detailed comparison between copper and aluminum bicrystals is provided with regard to boundary strength, observed deformation mechanisms, and stress-assisted free volume evolution during both tensile and shear simulations.     

Atomistically determined phase-field modeling of dislocation dissociation,stacking fault formation,dislocation slip,and reactions in fcc systems

The purpose of the current work is the development of a phase field model for dislocation dissociation, slip and stacking fault formation in single crystals amenable to determination via atomistic or ab initio methods in the spirit of computational material design. The current approach is based in particular on periodic microelasticity (Wang and Jin, 2001, Bulatov and Cai, 2006, Wang and Li, 2010) to model the strongly non-local elastic interaction of dislocation lines via their (residual) strain fields. These strain fields depend in turn on phase fields which are used to parameterize the energy stored in dislocation lines and stacking faults. This energy storage is modeled here with the help of the ”interface” energy concept and model of Cahn and Hilliard (1958) (see also Allen and Cahn, 1979, Wang and Li, 2010). In particular, the “homogeneous” part of this energy is related to the “rigid” (i.e., purely translational) part of the displacement of atoms across the slip plane, while the “gradient” part accounts for energy storage in those regions near the slip plane where atomic displacements deviate from being rigid, e.g., in the dislocation core. Via the attendant global energy scaling, the interface energy model facilitates an atomistic determination of the entire phase field energy as an optimal approximation of the (exact) atomistic energy; no adjustable parameters remain. For simplicity, an interatomic potential and molecular statics are employed for this purpose here; alternatively, ab initio (i.e., DFT-based) methods can be used. To illustrate the current approach, it is applied to determine the phase field free energy for fcc aluminum and copper. The identified models are then applied to modeling of dislocation dissociation, stacking fault formation, glide and dislocation reactions in these materials. As well, the tensile loading of a dislocation loop is considered. In the process, the current thermodynamic picture is compared with the classical mechanical one as based on the Peach-Köhler force.     

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.     

On the energy conservation during the active deformation in molecular dynamics simulations

In this paper, we examined the energy conservation for the current schemes of applying active deformation in molecular dynamics (MD) simulations. Specifically, two methods are examined. One is scaling the dimension of the simulation box and the atom positions via an affine transformation, suitable for the periodic system. The other is moving the rigid walls that interact with the atoms in the system, suitable for the non-periodic system. Based on the calculation of the external work and the internal energy change, we present that the atom velocities also need to be updated in the first deformation method; otherwise the energy conservation cannot be satisfied. The classic updating scheme is examined, in which any atom crossing the periodic boundary experiences a velocity delta that is equal to the velocity difference between the opposite boundaries. In addition, a new scheme which scales the velocities of all the atoms according to the strain increment is proposed, which is more efficient and realistic than the classic scheme. It is also demonstrated that the Virial stress instead of its interaction part is the correct stress definition that corresponds to Cauchy stress in the continuum mechanics.     

The elastic modulus of single-wall carbon nanotubes: a continuum analysis incorporating interatomic potentials

A nanoscale continuum theory is established to directly incorporate interatomic potentials into a continuum analysis without any parameter fitting. The theory links interatomic potentials and atomic structure of a material to a constitutive model on the continuum level. The theory is applied to study the linear elastic modulus of a single-wall carbon nanotube. The Young's modulus predicted by this nanoscale continuum theory agrees well with prior experimental results and atomistic studies.     

Constitutive modeling of strain rate effects in nanocrystalline and ultrafine grained polycrystals

We present a variational two-phase constitutive model capable of capturing the enhanced rate sensitivity in nanocrystalline (nc) and ultrafine-grained (ufg) fcc metals. The nc/ufg-material consists of a grain interior phase and a grain boundary affected zone (GBAZ). The behavior of the GBAZ is described by a rate-dependent isotropic porous plasticity model, whereas a rate-independent crystal-plasticity model which accounts for the transition from partial dislocation to full dislocation mediated plasticity is employed for the grain interior. The scale bridging from a single grain to a polycrystal is done by a Taylor-type homogenization. It is shown that the enhanced rate sensitivity caused by the grain size refinement is successfully captured by the proposed model.     

Size dependent mechanical behaviour of tantalum

The size dependence of deformation of Ta was studied using compression tests of focused ion beam (FIB) machined microcolumns. Columns with diameters between 0.5 and 8 μm with 〈1 1 1〉 and 〈1 0 0〉 orientations along the column axis were tested. By comparing results of bcc Ta columns with results from previous experiments on fcc metals it was found that Ta shows significantly higher normalized yield stresses in combination with a weaker sample size dependence. The differences between bcc and fcc metals can be attributed to the different dislocation behaviour of bcc metals, especially to the lower mobility of screw dislocations.     

Coarse-graining and renormalization of atomistic binding relations and universal macroscopic cohesive behavior

We present two approaches for coarse-graining interplanar potentials and determining the corresponding macroscopic cohesive laws based on energy relaxation and the renormalization group. We analyze the cohesive behavior of a large—but finite—number of interatomic planes and find that the macroscopic cohesive law adopts a universal asymptotic form. The universal form of the macroscopic cohesive law is an attractive fixed point of a suitably-defined renormalization-group transformation.