The role of texture in tension–compression asymmetry in polycrystalline NiTi

The purpose of the present study is to thoroughly understand the stress–strain behavior of polycrystalline NiTi deformed under tension versus compression. To do this, a micro-mechanical model is used which incorporates single crystal constitutive relationships and experimentally measured polycrystalline texture into the self-consistent formulation. For the first time it is quantitatively demonstrated that texture measurements coupled with a micro-mechanical model can accurately predict tension/compression asymmetry in NiTi shape memory alloys. The predicted critical transformation stress levels and transformation stress–strain slopes under both tensile and compressive loading are consistent with experimental results. For textured polycrystalline NiTi deformed under tension it is demonstrated that the martensite evolution is very abrupt, consistent with the Luders type deformation experimentally observed. The abrupt transformation under tension is attributed to the fact that the majority of the grains are oriented along the [111] crystallographic direction, which is soft under tensile loading. Since single crystals of the [111] orientation are hard under compression it is also demonstrated that under compression the martensite in textured polycrystalline NiTi evolves relatively slower.     

形状记忆合金相变塑性的统一本构理论

本文根据单品相变时其宏观(长程)切应变(等效应变)与外加切应力(等效应力)问的幂函数关系假定,建立了形状记忆合金多晶体相变塑性的统一细观本构理论,它推广了现有细观本构理论的结果,消除了现有理论中存在的当外加偏应力趋于零时自由能函数的不连续性和宏观塑性变形的不连续性。这一本构理论将能统一地描述在不同温度和载荷下记忆合金所呈现的各种行为。     

Anisotropy of martensitic transformations in modeling of shape memory alloy polycrystals

Based on the knowledge of the anisotropy associated with the martensitic transformations obtained from tension/compression experiments with oriented CuAlNi single crystals, a simple constant stress averaging approach is employed to model the SMA polycrystal deformation behaviors. Only elastic and inelastic strains due to the martensitic transformation, variant reorientations in the martensite phase and martensite to martensite transformations in thermomechanical loads are considered. The model starts from theoretical calculation of the stress-temperature transformation conditions and their orientation dependence from basic crystallographic and material attributes of the martensitic transformations. Results of the simulations of the NiTi, NiAl, and Cu-based SMA polycrystals in stress–strain tests are shown. It follows that SMA polycrystals, even with randomly oriented grains, typically exhibit tension/compression asymmetry of the shape of the pseudoelastic σ−ε curves in transformation strain, transformation stress, hysteresis widths, character of the pseudoelastic flow and in the slope of temperature dependence of the transformation stresses. It is concluded that some macroscopic features of the SMA polycrystal behaviors originate directly from the crystallography of the undergoing MT's. The model shows clearly the crystallographic origin of these phenomena by providing a link from the crystallographic and material attributes of martensitic transformations towards the macroscopic σ−ε−T behaviors of SMA polycrystals.     

超弹性镍钛形状记忆合金单轴相变棘轮行为的宏观唯象本构模型

超弹性镍钛形状记忆合金因其良好的力学性能以及独特的超弹性和形状记忆效应已广泛应用于土木工程、航空航天和生物医疗等多个领域,在实际服役环境中超弹性镍钛合金元件不可避免地会承受不同应力水平的循环载荷作用,亟待建立描述相变棘轮行为(即峰值应变和谷值应变随着正相变和逆相变循环的进行不断累积)的循环本构模型.为此,基于已有的超弹性镍钛形状记忆合金在不同峰值应力下的单轴相变棘轮行为实验研究结果,在广义黏塑性框架下,对Graesser等提出的通过背应力非线性演化方程反映超弹性镍钛形状记忆合金超弹性行为的一维宏观唯像本构模型进行了拓展,考虑了正相变和逆相变过程中特征变量的差异及其随循环的演化,以非弹性应变的累积量为内变量引入了正相变开始应力、逆相变开始应力、相变应变和残余应变的演化方程,同时通过峰值应力与正相变完成应力的比值来确定演化方程中的相关系数,建立了描述超弹性镍钛合金单轴相变棘轮行为的本构模型.将模拟结果与对应的实验结果进行对比发现,建立的宏观唯像本构模型能够合理地描述超弹性镍钛形状记忆合金的单轴相变棘轮行为及其峰值应力依赖性,模型的预测结果和实验结果吻合得很好.     

Constitutive modeling of textured body-centered-cubic (bcc) polycrystals

A new latent hardening model for body-centered-cubic (bcc) single crystals motivated by the inapplicability of the Schmid law (Critical Resolved Shear Stress Criterion) is presented. This model is based on the asymmetry of shearing resistance of the {112} slip planes depending on the shearing direction in the sense of ‘twin’ and ‘anti-twin’. For the interpretation of deformation of polycrystalline aggregates depending upon initial texture, a constitutive law for bcc single crystals is developed. This law is based on a rigorous constitutive theory for crystallographic slip that accounts for the effects of strain hardening, rate-sensitivity and thermal softening. The deformation response of textured polycrystal is investigated by means of a Taylor type averaging scheme and an established numerical procedure. Results for textured tungsten polycrystals at low and high strain rates for two different textures [001] and [011] are presented and compared with experimental results. The predictions compare well with experimental observations for the [001] texture. In the [011] texture, due to the reduced symmetry of deformation, lateral tensile stresses develop even under uniaxial compression. These lateral tensile stresses are responsible for observed lack of ductility and transgranular failure in the [011] texture.     

A Simple Explicit Formula for the Voigt-Reuss-Hill Average of Elastic Polycrystals with Arbitrary Crystal and Texture Symmetries

The Voigt-Reuss-Hill (VRH) average provides a simple way to estimate the elastic constants of a textured polycrystal in terms of its crystallographic texture and the elastic constants of the constituting crystallites. Empirically the VRH estimates were found in most cases to have an accuracy comparable to those obtained by more sophisticated techniques such as self-consistent schemes. In this paper we determine, in the space of fourth-order tensors with major and minor symmetries, a special set of irreducible basis tensors, with which we obtain a simple explicit formula for the VRH average for elastic polycrystals with arbitrary crystal and texture symmetries. Our formula is correct to first order in the texture coefficients.     

A multivariant micromechanical model for SMAs Part 1. Crystallographic issues for single crystal model

A general 3-D multivariant model for shape memory alloy constitutive behavior is further developed in this paper. The model is based on the habit planes and transformation directions for variants of martensite and uses a thermodynamic and micromechanics approach to develop the governing equations for thermomechanical response. The model accounts for the self-accomodating group structure exhibited during martensitic plate formation and utilizes this concept in its calculation of the interaction energy between variants. In this paper, we expand the multivariant model to consider the impact of inclusion shape on model predictions. A direction selection scheme is proposed for penny shaped inclusions and is based on the fact that several habit plane variants tend to cluster about one of the {011} or {001} poles. We also explore in detail the crystallographic basis of material response and the impact of specific crystallographic changes on the macroscopic single crystal constitutive response. Differences between type I and type II twinning are examined and it is shown that choice of the proper twinning type is essential to capture experimental data. The grouping structure is examined and several different options published for a NiTi alloy are implemented and results compared. Several concepts, i.e. artificial merging, exclusive and non-exclusive grouping, are raised to assist exploration of NiTi grouping possibilities. The anisotropy of the single crystal material response is illustrated and implications on higher level modeling are discussed. It is noted that properly representing the details of the crystallographic microstructure is crucial to obtaining accurate macroscopic stress–strain predictions.     

Perturbation approach to elastic constitutive relations of polycrystals

Herein we obtain a formula for the effective elastic stiffness tensor C^eff of an orthorhombic aggregate of cubic crystallites by the perturbation method. The effective elastic stiffness tensor of the polycrystal gives the relationship between volume average stress and volume average strain. Under Voigt's model, Reuss’ model and Man's theory, the elastic constitutive relation accounts for the effect of the orientation distribution function (ODF) up to terms linear in the texture coefficients. However, the formula derived in this paper delineates the effect of crystallographic texture on elastic response and shows quadratic texture dependence. The formula is very simple. We also consider the influence of grain shape to elastic constitutive relations of polycrystals. Some examples are given to compare computational results of the formula with those given by Voigt's model, Reuss's model, the finite element method, and the self-consistent method. In Section 3, we also present an expression of the perturbation displacement field, in which Green's function for an orthorhombic aggregate of cubic crystallites is included.     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

Almost compatible X-microstructures in CuAlNi shape memory alloy

A systematic study of a specific martensitic microstructure, called the X-microstructure, is carried out with the focus on the CuAlNi shape memory alloy undergoing the cubic-to-orthorhombic transformation. The set of all crystallographically distinct candidate X-microstructures is determined, and it is shown that, according to the crystallographic theory of martensite, none of them is compatible. Almost compatible X-microstructures, which involve elastic strains, are thus examined. These microstructures are searched in the neighborhood of all candidate X-microstructures by minimizing the total elastic strain energy with respect to the microstructure parameters. Several low-energy X-microstructures are found, and it is shown that the total elastic strain energy correlates reasonably well with one of the indicators which characterize incompatibility of the corresponding candidate X-microstructure.     

Cyclic thermomechanical behavior of a polycrystalline pseudoelastic shape memory alloy

In this work, 3D finite element modeling is employed to examine the thermomechanical behavior of a polycrystalline Ni-Ti shape memory alloy in the pseudoelastic regime. It is shown that the tension-compression asymmetry during uniaxial cyclic loading is due to a preferred orientation of the crystallographic texture. In pure shear loading, the thermomechanical behavior exhibits symmetry in both senses of shear, due to the fiber texture of the specimen bar stock. It is also shown that the apparent strain rate-dependence is due to thermomechanical coupling with latent heat generation/absorption during phase transformation.     

Texture predictions using a polycrystal plasticity model incorporating neighbor interactions

A viscoplastic model is presented for distributing the deformation applied to a polycrystal in a non-uniform fashion among the constituent crystals. Interactions with surrounding crystals are incorporated in the calculation of the deformation rate of each crystal through an appropriately defined local neighborhood. A compliance tensor is computed for each crystal based on a viscoplastic constitutive relation for deformation by crystallographic slip. The compliance of the crystal relative to that of its neighborhood provides a means for partitioning the macroscopic deformation rate among the crystals. The deviation of the crystal deformation rate from the macroscopic value is suitably scaled to obtain the crystal spin. Polycrystal simulations of crystallographic texture development using this model are compared to the results of similar calculations using the Taylor model, to finite element simulations of a model polycrystal, and to experimental data. The model incorporating neighbor interactions is shown to result in improved texture predictions, in terms of both the intensity levels and the locations of certain texture components.     

Anisotropic yield criterion for polycrystalline metals using texture and crystal symmetries

An anisotropic yield criterion for polycrystalline metals which uses texture data and takes advantage of crystal symmetries is presented. A linear transformation is developed to map an anisotropic yield surface for a polycrystal to an appropriate isotropic yield surface. The transformation developed reflects the symmetry of the material being modeled. First, the transformation is determined. Then, information regarding the orientation distribution (texture) of the crystals in a polycrystalline aggregate is used to determine, via averaging, the transformation for the polycrystal. The transformation, along with appropriate isotropic yield surface, provides a phenomenological approach to modeling yield, yet accounts for microstructural texture. The approach reduces to the Hill (1950) anisotropic plasticity theory under certain conditions. The yield surfaces and R-values for various face-centered-cubic ( fcc) polycrystalline textures are computed by this method. Results compare favorably with those given by other theories, and with experiment. The method proves to have the computational efficiency of phenomenological approaches to modeling yield, while effectively incorporating the physics of more complex crystallographic approaches.     

Elastic/crystalline viscoplastic finite element analyses of single- and poly-crystal sheet deformations and their experimental verification

The elastic/crystalline viscoplastic constitutive equation, based on a newly proposed hardening-softening evolution equation, is introduced into the dynamic-explicit finite element code “Itas-Dynamic.” In the softening evolution equation, the effective distance and the angle between each slip system of a crystal are introduced to elucidate the interaction between the slip systems, which causes a decrease of dislocation density. The polycrystal sheet is modeled by Voronoi polygons, which correspond to the crystal grains; and by the selected orientations, which can relate to the texture, they are assigned to the integration points of the finite elements. We propose a direct crystal orientation assignment method, which means that each integration point of finite element has an assigned orientation, and its orientation can be rotated independently. Therefore, this inhomogeneous polycrystal model can consider the plastic induced texture development and subsequent anisotropy evolution. The parameters of the constitutive equation are identified by uni-axial tension tests carried out on single crystal sheets. Numerical results obtained for sheet tensions are compared with experimental ones to confirm the validity of our finite element code. Further, we investigate the following subjects: (1) how the initial orientation of single crystal affects slip band formation and strain localization; (2) how the grain size and particular orientations of the grain affect the strain localization in case of a polycrystal sheet. It is confirmed that the orientation of a single crystal can be related to the primary slip system and the deformation induced activation of that system, which in turn can be related to the slip band formation of the single crystal sheet. Further, in case of a polycrystal sheet, the larger the grain size, the more the strain localizes at a specific crystal, which has the particular orientation. It is confirmed through comparisons with experiments that our finite element code can predict the localization of strain in sheets and consequently can estimate the formability of sheet metals.     

一种描述形状记忆合金拟弹性变形行为的本构关系

本文给出一种描述形状记忆合金拟弹性变形现象的本构关系,可用于多晶材料在一般应力状态下及单晶材料在单轴应力下的变形情况。该本构关系采用弹性应变与相变应变迭加形式,物理意义明显,形式简洁。对 Cu-Zn-Sn 合金及 Ti-Ni 合金材料的变形行为进行了模拟计算,结果与实验值有较好的吻合。     

Application of a discrete polycrystal model to the analysis of cyclic straining in copper

A discrete polycrystal model, designed to simulate a metal aggregate macro-element, is applied to the study of cyclic straining in copper. The numerical method of solution (an adaptation of the “finite element method”) incorporates a convergent discrete Green's function within the constrained minimum principle which governs the (crystallographic) plastic shear increments at each load step. Isothermal elastic moduli of copper crystals and Taylor's hardening rule with constant hardening modulus are used in the calculations. Numerical results are obtained for macroscopic elastic properties, cyclic stress-strain curves (which indicate the contribution of aggregate heterogeneity to macroscopic hardening), macroscopic plastic work, and residual (latent) strain energy through four loading cycles between fixed macrostrain limits. Other estimates for elastic properties also are included, and all results are compared, both qualitatively and quantitatively, with published experiments. The predictions of the model are in general satisfactory.     

Nucleation of micro-cracks for polycrystalline metal with anisotropy: micro-stress evaluation

Presented is the local stresses on the crystallographic plane as they are influenced by the metal fracture with anisotropy. It is based on the nucleation of micro-cracks and its unstable equilibrium in a polycrystal with texture. Crystallographic texture causes non-uniform distribution of the crack nucleus orientations owing to their preference to expand or open on certain crystallographic planes. This is the main cause of anisotropy of cleavage fracture stress of textured metal. “Oriented” micro-stresses in textured metal contribute to the anisotropic effect. In view of what has been said, accumulated plastic strains at fracture is analysed.     

一种描述形状记忆合金拟弹性变形行为的本构关系

本文给出一种描述形状记忆合金拟弹性变形现象的本构关系,可用于多晶材料在一般应力状态下及单晶材料在单轴应力下的变形情况。该本构关系采用弹性应变与相变应变迭加形式,物理意义明显,形式简洁。对 Cu-Zn-Sn 合金及 Ti-Ni 合金材料的变形行为进行了模拟计算,结果与实验值有较好的吻合。