Geometrically necessary dislocations in viscoplastic single crystals and bicrystals undergoing small deformations

In this study we develop a gradient theory of small-deformation single-crystal plasticity that accounts for geometrically necessary dislocations (GNDs). The resulting framework is used to discuss grain boundaries. The grains are allowed to slip along the interface, but growth phenomenona and phase transitions are neglected. The bulk theory is based on the introduction of a microforce balance for each slip system and includes a defect energy depending on a suitable measure of GNDs. The microforce balances are shown to be equivalent to nonlocal yield conditions for the individual slip systems, yield conditions that feature backstresses resulting from energy stored in dislocations. When applied to a grain boundary the theory leads to concomitant yield conditions: relative slip of the grains is activated when the shear stress reaches a suitable threshold; plastic slip in bulk at the grain boundary is activated only when the local density of GNDs reaches an assigned threshold. Consequently, in the initial stages of plastic deformation the grain boundary acts as a barrier to plastic slip, while in later stages the interface acts as a source or sink for dislocations. We obtain an exact solution for a simple problem in plane strain involving a semi-infinite compressed specimen that abuts a rigid material. We view this problem as an approximation to a situation involving a grain boundary between a grain with slip systems aligned for easy flow and a grain whose slip system alignment severely inhibits flow. The solution exhibits large slip gradients within a thin layer at the grain boundary.     

Self-consistent modeling of large plastic deformation, texture and morphology evolution in semi-crystalline polymers

A self-consistent model for semi-crystalline polymers is proposed to study their constitutive behavior, texture and morphology evolution during large plastic deformation. The material is considered as an aggregate of composite inclusions, each representing a stack of crystalline lamellae with their adjacent amorphous layers. The deformation within the inclusions is volume-averaged over the phases. The interlamellar shear is modeled as an additional slip system with a slip direction depending on the inclusion's stress. Hardening of the amorphous phase due to molecular orientation and, eventually, coarse slip, is introduced via Arruda-Boyce hardening law for the corresponding plastic resistance. The morphology evolution is accounted for through the change of shape of the inclusions under the applied deformation gradient. The overall behavior is obtained via a viscoplastic tangent self-consistent scheme. The model is applied to high density polyethylene (HDPE). The stress-strain response, texture and morphology changes are simulated under different modes of straining and compared to experimental data as well as to the predictions of other models.     

Transmission electron microscopy investigation of dislocation slip during superelastic cycling of Ni-Ti wires

Superelastic deformation of thin Ni-Ti wires containing various nanograined microstructures was investigated by tensile cyclic loading with in situ evaluation of electric resistivity. Defects created by the superelastic cycling in these wires were analyzed by transmission electron microscopy. The role of dislocation slip in superelastic deformation is discussed. Ni-Ti wires having finest microstructures (grain diameter <100 nm) are highly resistant against dislocation slip, while those with fully recrystallized microstructure and grain size exceeding 200 nm are prone to dislocation slip. The density of the observed dislocation defects increases significantly with increasing grain size. The upper plateau stress of the superelastic stress-strain curves is largely grain size independent from 10 up to 1000 nm. It is hence claimed that the Hall-Petch relationship fails for the stress-induced martensitic transformation in this grain size range. It is proposed that dislocation slip taking place during superelastic cycling is responsible for the accumulated irreversible strains, cyclic instability and degradation of functional properties. No residual martensite phase was found in the microstructures of superelastically cycled wires by TEM and results of the in situ electric resistance measurements during straining also indirectly suggest that none or very little martensite phase remains in the studied cycled superelastic wires after unloading. The accumulation of dislocation defects, however, does not prevent the superelasticity. It only affects the shape of the stress-strain response, makes it unstable upon cycling and changes the deformation mode from localized to homogeneous. The activity of dislocation slip during superelastic deformation of Ni-Ti increases with increasing test temperature and ultimately destroys the superelasticity as the plateau stress approaches the yield stress for slip. Deformation twins in the austenite phase ({1 1 4} compound twins) were frequently found in cycled wires having largest grain size. It is proposed that they formed in the highly deformed B19′ martensite phase during forward loading and are retained in austenite after unloading. Such twinning would represent an additional deformation mechanism of Ni-Ti yielding residual irrecoverable strains.     

3-D simulation of spatial stress distribution in an AZ31 Mg alloy sheet under in-plane compression

A complete 3-D crystal plasticity finite element method (CPFEM) that considered both crystallographic slip and deformation twinning was applied to simulate the spatial distribution of the relative amount of slip and twin activities in a polycrystalline AZ31 Mg alloy during in-plane compression. A microstructure mapping technique that considered the grain size distribution and microtexture measured by electron backscatter diffraction (EBSD) technique was used to create a statistically representative 3-D microstructure for the initial configuration. Using a 3-D Monte Carlo method, a 3-D digital microstructure that matched the experimentally measured grain size distribution was constructed. Crystallographic orientations obtained from the EBSD data were assigned on the 3-D digital microstructure to match the experimentally measured misorientation distribution. CPFEM captured the heterogeneity of the stress concentration as well as the slip and twin activities of a polycrystalline AZ31 Mg alloy during in-plane compression.     

Yield stress strengthening in ultrafine-grained metals: A two-dimensional simulation of dislocation dynamics

The effect of grain size on the tensile plastic deformation of ultrafine-grained copper polycrystals is investigated using a two-dimensional simulation of dislocation dynamics. Emphasis is put on the elementary mechanisms governing the yield stress in multislip conditions. Whatever the grain size, the yield stress is found to follow a Hall-Petch law. However, the elementary mechanism controlling slip transmission through the grain boundaries at yield is observed to change with the grain size. For the larger grain sizes, the stress concentrations due to dislocations piled-up at grain boundaries are responsible for the activation of plastic activity in the poorly stressed grains. For the smaller grain sizes, the pile-ups contain less dislocations and are less numerous, but the strain incompatibilities between grains become significant. They induce high internal stresses and favor multislip conditions in all grains. Based on these results, simple interpretations are proposed for the strengthening of the yield stress in ultrafine grained metals.     

A crystal plasticity theory for latent hardening by glide twinning through dislocation transmutation and twin accommodation effects

Imagine a residual glide twin interface advancing in a grain under the action of a monotonic stress. Close to the grain boundary, the shape change caused by the twin is partly accommodated by kinks and partly by slip emissions in the parent; the process is known as accommodation effects. When reached by the twin interface, slip dislocations in the parent undergo twinning shear. The twinning shear extracts from the parent dislocation a twinning disconnection, and thereby releases a transmuted dislocation in the twin. Transmutation populates the twin with dislocations of diverse modes. If the twin deforms by double twinning, double-transmutation occurs even if the twin retwins by the same mode or detwins by a stress reversal. If the twin deforms only by slip, transmutation is single. Whether single or double, dislocation transmutation is irreversible. The multiplicity of dislocation modes increases upon strain, since the twin finds more dislocations to transmute upon further slip of the parent and further growth of the twin. Thus, the process induces an increasing latent hardening rate in the twin. Under profuse twinning conditions, typical of double-lattice structures, this rate-increasing latent hardening combined with crystal rotation to hard orientations by twinning is consistent with a regime of increasing hardening rate, known as Regime II or Regime B. In this paper, we formulate governing equation of the above transmutation and accommodation effects in a crystal plasticity framework. We use the dislocation density based model originally proposed by Beyerlein and Tomé (2008) to derive the effect of latent hardening in a transmuting twin. The theory is expected to contribute to surmounting the difficulty that current models have to simultaneously predict under profuse twinning, the stress-strain curves, intermediate deformation textures, and intermediate twin volume fractions.     

A micromechanical theory of grain-size dependence in metal plasticity

The effect of grain-size on the elastoplastic behavior of metals is investigated from the micromechanics standpoint. First, based on the observations that dislocation pile-ups, formation of cell structures, and other inelastic activities influenced by the presence of grain boundary actually take place transcrystallinely, a grain-size dependent constitutive equation is proposed for the slip deformation of slip systems. By means of a modified Hill's self-consistent relation the local stress of a grain is calculated, and used in conjunction with this constitutive equation to evaluate the plastic strain of each constituent grain. The grain-size effect on the plastic flow of polycrystals then can be determined by an averaging process. To check the validity of the proposed theory it was finally applied to predict the stress-strain curves and flow stresses of a copper at various grain-sizes. The obtained results were found to be in good agreement with experimental data.     

Numerical investigation of grain boundary effects on elevated-temperature deformation and fracture

A three-dimensional (3D) polycrystal intergranular model that accounts for grain boundary deformation and intergranular weakening at elevated temperatures is presented. The effects of grain boundaries on the accumulated slip deformation of grain interiors and lattice rotation have been investigated through a comparison between results from a model including grain boundary region (GBM) and a model representing only the grain interiors not the grain boundary region directly (NGBM). It is found that the presence of grain boundaries seems to suppress the grain interior slip deformation, and this suppressive role is reduced with increased relative thickness of the grain boundaries. In addition, grain boundaries promote the lattice rotation of individual grains in shear bands but suppress that of individual grains within non-shear bands. Mutual rotation of grains in both shear and non-shear bands is caused by the introduction of grain boundary regions. Rate-dependence of high-temperature plasticity could be more accurately captured by the GBM than by the NGBM. By considering creep damage of grain boundary, when the damage variable reaches a critical value, the corresponding grain boundary element is eliminated to describe dynamic intergranular fracture processes. The volume-averaged stress–strain curve by a model considering grain boundary damage (DGBM) showed better agreement with experimental results than that by a model not considering grain boundary damage (GBM).     

Cylindrical void in a rigid-ideally plastic single crystal. Part I: Anisotropic slip line theory solution for face-centered cubic crystals

The fracture toughness of ductile materials depends upon the ability of the material to resist the growth of microscale voids near a crack tip. Mechanics analyses of the elastic–plastic deformation state around such voids typically assume the surrounding material to be isotropic. However, the voids exist predominantly within a single grain of a polycrystalline material, so it is necessary to account for the anisotropic nature of the surrounding material. In the present work, anisotropic slip line theory is employed to derive the stress and deformation state around a cylindrical void in a single crystal oriented so that plane strain conditions are admitted from three effective in-plane slip systems. The deformation state takes the form of angular sectors around the circumference of the void. Only one of the three effective slip systems is active within each sector. Each slip sector is further subdivided into smaller sectors inside of which it is possible to derive the stress state. Thus the theory predicts a highly heterogeneous stress and deformation state. In addition, it is shown that the in-plane pressure necessary to activate plastic deformation around a cylindrical void in an anisotropic material is significantly higher than that necessary for an isotropic material. Experiments and single crystal plasticity finite element simulations of cylindrical voids in single crystals, both of which exhibit a close correspondence to the analytical theory, are discussed in a companion paper.     

Grain boundary interface mechanics in strain gradient crystal plasticity

Interactions between dislocations and grain boundaries play an important role in the plastic deformation of polycrystalline metals. Capturing accurately the behaviour of these internal interfaces is particularly important for applications where the relative grain boundary fraction is significant, such as ultra fine-grained metals, thin films and micro-devices. Incorporating these micro-scale interactions (which are sensitive to a number of dislocation, interface and crystallographic parameters) within a macro-scale crystal plasticity model poses a challenge. The innovative features in the present paper include (i) the formulation of a thermodynamically consistent grain boundary interface model within a microstructurally motivated strain gradient crystal plasticity framework, (ii) the presence of intra-grain slip system coupling through a microstructurally derived internal stress, (iii) the incorporation of inter-grain slip system coupling via an interface energy accounting for both the magnitude and direction of contributions to the residual defect from all slip systems in the two neighbouring grains, and (iv) the numerical implementation of the grain boundary model to directly investigate the influence of the interface constitutive parameters on plastic deformation. The model problem of a bicrystal deforming in plane strain is analysed. The influence of dissipative and energetic interface hardening, grain misorientation, asymmetry in the grain orientations and the grain size are systematically investigated. In each case, the crystal response is compared with reference calculations with grain boundaries that are either ‘microhard’ (impenetrable to dislocations) or ‘microfree’ (an infinite dislocation sink).     

用晶体弹塑性有限单元法研究双晶金属拉伸变形

本文从单晶体应力-应变关系的精确实验结果和多晶体滑移特性出发,建立相应的计算模型,并采用微观力学和晶体弹塑性有限单元法,研究双晶金属试样的拉伸变形,得到其应力-应变曲线,晶体内滑移变形和应力分布规律,以及晶界影响区对它们的影响。     

Higher-order stress and grain size effects due to self-energy of geometrically necessary dislocations

The higher-order stress work-conjugate to slip gradient in single crystals at small strains is derived based on the self-energy of geometrically necessary dislocations (GNDs). It is shown that this higher-order stress changes stepwise as a function of in-plane slip gradient and therefore significantly influences the onset of initial yielding in polycrystals. The higher-order stress based on the self-energy of GNDs is then incorporated into the strain gradient plasticity theory of Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5-32] and applied to single-slip-oriented 2D and 3D model crystal grains of size D. It is thus found that the self-energy of GNDs gives a D^-1-dependent term for the averaged resolved shear stress in such a model grain under yielding. Using published experimental data for several polycrystalline metals, it is demonstrated that the D^-1-dependent term successfully explains the grain size dependence of initial yield stress and the dislocation cell size dependence of flow stress in the submicron to several-micron range of grain and cell sizes.     

A crack on the grain boundary — An analysis based on crystal plasticity theory

The influence of the mismatch of the lattice orientation on the deformation and stress fields of a crack located on the grain boundary is studied by means of the finite-element analysis taking account of finite deformatio and finite lattice rotation. The plane strain calculations for an fcc crystal subjected to mode I loading are performed on the basis of the crystalline plasticity described by a planar three-slip model. For the crack-tip shapes and the dominant deformation modes on slip systems, results of all the cases analysed here are in qualitative agreement with the earlier analytical and numerical solutions. Our results indicate that the lattice orientation difference may greatly influence the shear stress along the grain boundary which is related to grain-boundary sliding, while the normal stress along the grain boundary, which may induce cleavage fracture, is virtually insensitive to it. The influence of the lattice orientations on the crack-tip fields is also investigated under small-scale-yielding conditions and the comparison with the results of finite deformation is made.     

Finite element analysis of grain-by-grain deformation by crystal plasticity with couple stress

Rigid–plastic crystal plasticity with the rate-sensitive constitutive behavior of a slip system has been formulated within the framework of a two-dimensional finite element method to predict the grain-by-grain deformation of single- and polycrystalline FCC metals. For that purpose, individual grains are represented by several numbers of finite elements to describe the sub-grain deformation behavior, and couple stress has been introduced into the equilibrium equation to be able to describe the size effect as well as to prevent mesh-dependent predictions. A modified virtual work-rate principle with an approximate interface constraint has been suggested to use a C ⁰-continuous element in the finite element implementation, and the couple stress work-rate has been formulated on the basis of an assumed constitutive behavior. Simulated plane-strain compressions of a single crystal cube show that the shearing and the deformation load are closely related to the imbedded lattice orientation of the crystal grain, and that the sub-grain deformation and the load magnitude can be controlled by the couple stress hardening. It is also confirmed that almost the same predictions are obtained for different mesh systems by considering the couple stress hardening. Simulated plane-strain compressions of a bi-crystal show considerably curved grain-by-grain surface profiles after large reduction for several combinations of the imbedded lattice orientation. The high couple stress hardening predicted around grain boundaries is supposed to be related to the grain size effect. It is also supposed that consideration of couple stress is necessary to predict the sub-grain or the grain-by-grain deformation, and the couple stress hardening may be used to describe the state of microstructures in grain.     

Deformation behaviour in advanced heat resistant materials during slow strain rate testing at elevated temperature

In this study, slow strain rate tensile testing at elevated temperature is used to evaluate the influence of temperature and strain rate on deformation behaviour in two different austenitic alloys. One austenitic stainless steel (AISI 316L) and one nickel-base alloy (Alloy 617) have been investigated. Scanning electron microscopy related techniques as electron channelling contrast imaging and electron backscattering diffraction have been used to study the damage and fracture micromechanisms. For both alloys the dominante damage micromechanisms are slip bands and planar slip interacting with grain bounderies or precipitates causing strain concentrations. The dominante fracture micromechanism when using a slow strain rate at elevated temperature, is microcracks at grain bounderies due to grain boundery embrittlement caused by precipitates. The decrease in strain rate seems to have a small influence on dynamic strain ageing at 650°C.     

Determination of Ti-6242 α and β slip properties using micro-pillar test and computational crystal plasticity

The properties and behaviour of an α−β colony Ti-6242 alloy have been investigated at 20 °C utilising coupled micro-pillar stress relaxation tests and computational crystal plasticity. The β-phase slip strength and intrinsic slip system strain rate sensitivity have been determined, and the β-phase shown to have stronger rate sensitivity than that for the α phase. Close agreement of experimental observations and crystal plasticity predictions of micro-pillar elastic-plastic response, stress relaxation, slip activation in both α and β-phases, and strain localisation within the α−β pillars with differing test strain rate, β morphology, and crystal orientations is achieved, supporting the validity of the properties extracted. The β-lath thickness is found to affect slip transfer across the α−β−α colony, but not to significantly change the nature of the slip localisation when compared to pure α-phase pillars with the same crystallographic orientation. These results are considered in relation to rate-dependent deformation, such as dwell fatigue, in complex multiphase titanium alloys.     

Superplastic characteristics of high purity yttria-stabilized zirconia polycrystals

The mechanical characteristics of superplastic yttria-stabilized zirconia polycrystals have been analyzed as a function of stress, temperature and grain size. The evolution of the stress exponent n with stress found in high purity materials is similar to that observed in superplastic metals. True creep parameters can be ascribed to the deformation mechanism at high stresses. By contrast, the creep parameters exhibit a continuous evolution with stress, temperature and grain size at low stresses. The threshold stress formalism used in conventional and high strain rate superplastic metals accounts for the mechanical characteristics observed in fine-grained zirconia polycrystals.     

On the partition of rate of deformation in crystal plasticity

Two different partitions of the rate of deformation tensor into its elastic and plastic parts are derived for elastic–plastic crystals in which crystallographic slip is the only cause of plastic deformation. One partition is associated with the Jaumann, and the other with convected rate of the Kirchhoff stress. Different expressions for the plastic part of the rate of deformation are obtained, and corresponding constitutive inequalities discussed. Relationship with the plastic part of the rate of the Lagrangian strain is also given.     

大应变率范围内AM80镁合金的变形行为及组织演变

采用INSTRON准静态压缩试验机和分离式霍普金森压杆装置，研究固溶态AM80镁合金在室温准静态和冲击载荷下的变形行为及组织演变。准静态载荷下，流变应力随应变率（3×10-5~4×10-1 s-1）的升高逐渐降低，表现为负应变率敏感性；冲击载荷下，流变应力随应变率（7.00×102~5.20×103 s-1）的升高而升高，呈现出明显的正应变率敏感性。冲击载荷下AM80镁合金的变形机制以基面滑移和孪生为主，大量细小致密的形变孪生以及适量非基面滑移的启动是AM80镁合金在冲击载荷下流变应力明显高于准静态载荷的重要原因。此外，随应变率的升高，AM80镁合金变形的均匀性明显增强，当应变速率升至3.65×103 s-1时，冲击变形所引起的局部绝热温升软化大于应变硬化与应变速率硬化的总和，部分晶粒产生了明显的动态回复，使得孪晶密度和变形均匀性反而降低。