Crystallite Distribution Function Research Articles

Finite element simulation of metal forming operations with texture based material models

In this paper two different texture-dependent material models based on the Taylor assumption are discussed and applied to the simulation of deep drawing operations of aluminium. From the numerical point of view, large-scale FE computations based on the Taylor model are very time-intensive and storage-consuming if the crystallographic texture is approximated by several hundred discrete crystals. Furthermore, the Taylor model in its standard form, which is based on discrete crystal orientations, has the disadvantage that the anisotropy is significantly overestimated if only a small number of crystal orientations are used. We quantitatively analyse this overestimation of anisotropy and suggest two Taylor-type models which allow us to reduce the sharpness of the crystallite orientation distribution function related to single crystals or texture components. One model is an elastic-viscoplastic Taylor model based on discrete orientations. The sharpness is reduced by modelling the isotropic background texture by an isotropic material law. The other model is a rigid-viscoplastic material one, which is based on continuous model functions on the orientation space. This model allows for a direct incorporation of the scattering around an ideal texture component since the model contains the half-width as a microstructural parameter which can be biased. These models are used to compute yield stresses, R values and earing profiles. The predictions are compared with experimental data.

Crystallographic texture approximation by quadratic programming

This paper considers the problem of approximating a given crystallite orientation distribution function (codf) by a set of texture components. Problems of this type arise for example if the codf has to be reconstructed from discrete orientations or if one looks for a physical interpretation of the codf. The same problem is encountered if crystallographic texture based constitutive models have to be specified. The equivalence of these tasks to a mixed integer quadratic programming problem (MIQP) – a standard but challenging problem in optimization theory – is shown. Special emphasis is given to the generation of a class of approximations with an increasing number of texture components. Furthermore, the constraints resulting from the non-negativity, the normalization, and the symmetry of the codf are analyzed. Finally, a set of approximations of three different experimental textures determined with this solution scheme is presented and discussed. Based on these hierarchical solutions, the engineer can decide in what detail the microstructure is considered.

Segmental Orientation under Simultaneous Biaxially Stretching Using a Lattice Model and the Application of Oriented Crystallization of Polyethylene Films

A lattice model proposed before for uniaxial stretching of polyethylene films was applied to estimate the oriented crystallization of ultra-high molecular weight polyethylene (UHMWPE) dry gel films under simultaneous biaxially stretching. In this model system, the preferred axis associated with the preferred orientation of amorphous chain segments was chosen along the direction between two successive cross-linked points and the preferred axis was assumed to deform in an affine fashion with respect to the stretching direction. As the application of the proposed model, the orientation distribution function of crystallites was calculated on the basis of the lattice model and oriented crystallization model. The oriented crystallization model is based on the concept that a kinetically determined distribution of crystal chain axes as the normalized distribution of clusters found at the saddle point corresponding to a non-uniform orientation under conditions of a steady-state nucleation rate. Of course, the crystallites are oriented randomly with respect to the film normal direction. The parameter fitting for the formulated orientation distribution function of crystallites was done for the film which was prepared by the gelation/crystallization from solution with 0.9 g/100 mL concentration, the solvent being decalin, since the concentration assured the highest drawability under simultaneous biaxially stretching. The calculated orientation distribution functions 2πqj(cosθj) of the reciprocal lattice vector of the crystal planes were in good agreement with the observed ones. Thus the numerical calculations indicate that the orientation of the c-axes depends on that of amorphous chain segments and the orientation behavior of crystallites is strongly affected by their rotation around the c-axis.

A texture component model for anisotropic polycrystal plasticity

There are many crystallographic textures which can be approximated by a small number of texture components [see, e.g., Int. J. Mech. Sci. 31(7) (1989) 549]. In some cases, such texture components can be described by central distributions. Central distributions are characterized by a mean orientation and a half width. The classical Taylor model for viscoplastic polycrystals assumes that a discrete set of single crystals deforms homogeneously. If the viscoplastic version of the Taylor model is numerically implemented then the crystallite orientation distribution function (codf) is usually discretized by a set of Dirac distributions, where each of the Dirac distributions represents a single crystal. Due to the specific discretization of the codf this approach requires usually a large number of discrete crystal orientations even if the texture can be described by a small number of texture components. In the present work, we consider face-centered cubic (fcc) polycrystals and compare the classical upper bound model with an approach based on texture components. The texture components are modeled by Mises–Fischer distributions, which are central distributions. The stress of the polycrystal is obtained by a numerical integration of the single crystal stress state over the orientation space.

Application of the maximum entropy method in texture analysis

The problem of estimating the crystallite orientation distribution function (codf) based on the leading texture coefficients is considered. Problems of such a type are called moment problems, which are well known in statistical mechanics and other areas of science. It is shown how the maximum entropy method can be applied to estimate the codf. Special emphasis is given to a coordinate-free formulation of the problem. The codf is represented by a tensorial Fourier series. The equations, which have to be solved for the estimate of the distribution function, are derived for all tensor ranks of the Fourier coefficients. As a numerical example, a model codf is estimated based on a set of discrete crystal orientations given by a full-constrained Taylor type texture simulation.

Correlation between strength and microstructure of ball-milled Al–Mg alloys determined by X-ray diffraction

The microstructure of ball-milled Al base Al–Mg alloys is determined by the “Convolutional Multiple Whole Profile” fitting procedure proposed for the evaluation of X-ray diffraction peak profiles. The whole measured powder diffraction pattern is fitted by the sum of a polynomial background and physically well-established theoretical profile functions. The procedure provides the size distribution function of crystallites and the characteristic parameters of the dislocation structure. The mechanical strength of the specimens is correlated to the parameters of the microstructure by the Hall–Petch and the Taylor models. Both models show that the critical resolved shear stress saturates at a Mg-solute concentration of about 2 wt.% probably due to the clustering of the Mg-solute atoms.

Orientation Distribution Functions of the Three Principal Crystallographic Axes as well as Crystallites of Poly(ethylene terephthalate) Films under Biaxially Stretching

The orientation of the three principal crystallographic axes, the a-, b-, and c-axes of crystallites of poly(ethylene terephthalate) (PET) under simultaneous biaxial stretching was estimated in terms of the orientation distribution function. For most of crystalline polymers with a triclinic unit like PET, there are no crystal planes perpendicular to the a-, b-, and c-axes that can be detected directly by X-Ray diffraction techniques. Accordingly, the functions of the a-, b-, and c-axes must be calculated by the method proposed by Roe and Krigbaum. In doing so, the orientation functions of the reciprocal lattice vectors must be measured for a number of crystal planes. In this paper, as an example, the orientation of crystallites and the orientation of the a-, b-, and c-axes were estimated for a PET film under simultaneous biaxial stretching in terms of the orientation distribution function of crystallites because of considerable utilization rate of PET as commercial films. The estimated orientation functions of the b- and c-axes predicted the detailed information concerning uniplanner orientation of benzene rings parallel to the film surface.

Theoretical examination of ultrasonic pole figures via comparison with the results analyzed by finite element polycrystal model.

The ultrasonic wave velocities in a polycrystalline aggregate are sensitively influenced by texture development due to plastic deformation. According to Sayer's model, it is possible to construct ultrasonic pole figures via the crystallite orientation distribution function (CODF), which can be calculated by using ultrasonic wave velocities. In the previous papers, the theoretical modeling to simulate ultrasonic wave velocities propagating in solid materials under plastic deformation has been proposed by the authors and proved to be a good agreement with experimental results. Generally, wave velocities are dependent upon the propagating wave frequency; hence to evaluate texture development via ultrasonic pole figures it is necessary to examine an influence of frequency dependence on the ultrasonic wave velocities. In the present paper, the proposed theoretical modeling is applied to the texture characterization in polycrystalline aggregates of FCC metals under various plastic strain histories via ultrasonic pole figures, and also the frequency dependence is examined by using Granato-Lücke's dislocation strings model. Then the simulated ultrasonic pole figures are compared with the pole figures analyzed by the finite element polycrystal model (FEPM). The good qualitative agreement between both results suggests the sufficient accuracy of our proposed theoretical modeling.

Effect of c/a-Ratio on Crystallographic Texture and Mechanical Anisotropy of Hexagonal Close Packed Metals

Hexagonal close packed metals exhibit highly anisotropic mechanical properties, a thorough knowledge of which is a primary requisite for predictability of the dimensional changes in service as well as formability characteristics. We summarize our work on the effects of c/a-ratio on preferred orientations and the resulting anisotropic mechanical properties. Zr and Ti alloys are considered to represent low c/a-ratio while Mg alloy (AZ31B) was chosen as ideal; these materials were available in tubing form. Pure zinc sheet was used to represent high c/a-ratio. Distinct differences are noted in the basal pole distributions, and crystallite orientation distribution functions derived from the pole distributions were combined with crystal slip models to predict the mechanical anisotropy parameters. While in all cases the uniaxial yield stresses were evaluated along the RD and TD, the contractile strain ratios (CSR) were determined using grided tensile samples fabricated from sheets of Zircaloy and Zinc. The experimental results correlated with slip predictions based on predominant prism and basal slip modes respectively in Zicaloy and Zn. For Mg alloy, either pyramidal slip or a combination of basal and prism slip modes exhibited good agreement with the experimental results. In addition, the anisotropic creep behavior of thin-walled Zircaloy tubing is considered. Excellent agreement is noted, in recrystallized Zircaloy tubing, between the experimental creep locus and model predictions based on prism slip dominance. However, distinct deviations are noted in Ti3Al2.5V alloy albeit electron microscopy of deformed tubing revealed dislocations on prism ({10.0}) planes with Burgers vectors along .

Crystallographic texture induced anisotropy in copper: An approach based on a tensorial Fourier expansion of the CODF

We consider a tensorial Fourier expansion of the crystallite orientation distribution function for aggregates of cubic crystals. The coefficient of rank four is determined explicitly. It is shown that this coefficient governs the anisotropic bounds of the linear elastic behavior of polycrystals. Recently, a phenomenological model has been proposed [4, 6], that describes the evolving elastic anisotropy in copper. The model also reproduces the cyclic Swift effect, which is due to a texture induced plastic anisotropy [7]. In the model the anisotropic part of the effective elastic stiffness tensor is used to define a texture dependent quadratic yield function. The present paper gives a new interpretation of the aforementioned model in terms of an approximation of the crystallographic texture by means of a 4th-order coefficient of the tensorial Fourier expansion of the crystallite orientation distribution function.

3D Monte-Carlo simulation of texture-controlled grain growth

A three-dimensional (3D) Monte-Carlo (MC) routine was developed to quantify the interaction of grain growth and texture development during annealing. The program included special software to enable the input of the initial grain structure and texture and incorporated a description of the misorientation-dependence of the grain-boundary mobility. Outputs from the model quantified the evolving texture in terms of pole figures or crystallite orientation distribution functions and statistics on the grain structure such as the grain-size distribution and boundary-misorientation distribution function. The MC routine was applied to establish grain growth and texture development in materials with random or strongly textured starting conditions and isotropic or anisotropic grain-boundary mobility. Depending on the starting condition and material properties, normal grain growth or a behavior characterized by alternating cycles of fast and slow grain growth was predicted.

Comparison of Ultrasonic Pole Figures Based upon Ultrasonic Nondestructive Evaluation Method with Pole Figures Based upon Finite Element Polycrystal Model

The ultrasonic wave velocities in a polycrystalline aggregate are sensitively influenced by texture changes due to plastic deformation, and their relationship was systematically analyzed by Sayers [J. Phys. D: Appl. Phys. 15 (1982)]. According to Sayers's proposed model, it is possible to construct ultrasonic pole figures via the crystallite orientation distribution function (CODF), which can be derived by using ultrasonic wave velocity changes. In the previous paper, the theoretical modeling to simulate ultrasonic wave velocities propagating in solid materials under plastic deformation has been proposed by the authors and proved to be in good agreement with experimental results. In the present paper, the proposed theoretical modeling is applied to construct the ultrasonic pole figures based upon Sayers's model under various loading conditions of uniaxial tension, pure torsion, equi-biaxial tension-compression, biaxial compression and biaxial tension, respectively. To examine the accuracy and reliability of the ultrasonic pole figures simulated by the proposed theoretical modeling, the ultrasonic pole figures are compared with those analyzed by the finite element polycrystal model (FEPM). The results show a remarkable qualitative similarity among the two methods.

繊維配向を有するＴｉＮおよびＴｉＣＮ薄膜の熱による残留応力変化にともなう機械的強度への一影響

TiN and TiCN thin films are used as antiwear layer on parts such as metal cutting tools. It is reported that the thin films that are deposited ceramics and metals have preferred orientation, and that the residual stresses generate by the difference in the coefficients of thermal expansion between the thin films and the substrate. Tools coated with TiN and TiCN films are heated to high temperature during service due to the frictional heat. Under this condition, the texture and the residual stress should change due to the heat. In the material engineering, it is well known that the residual stress in the material affects the mechanical strength of the industrial products. The purpose of this study is to examine the effect of the residual stress on the mechanical strength of TiN and TiCN thin films having preferred orientation.In this study, specimens that are deposited TiN and TiCN respectively by Physical Vapor Deposition (PVD) were annealed in a furnace at temperature of 573K, 798K, 843K and 893K. Using X-ray diffraction technique, the crystallite orientation was evaluated by the pole figure and the crystallite orientation distribution function (ODF). The films exhibited {111} fiber texture. After that, the residual stress of thin films was measured using CoKα radiation with the two-exposure method at Ψ=39° and 75°. Since the relationship between the mechanical strength and the residual stress has not been revealed about the thin film materials, the effect of the residual stress on the mechanical strength of the TiN and TiCN thin films having preferred orientation was investigated by the dynamic hardness (DH) and the scratch test.As a result, although the ODF little changed due to the heat treatment, the residual compressive stress relaxed at higher temperatures than 573K. As for the hardness of thin films, the residual stress did not affect the hardness. However, the residual stress influenced the behavior of the scratch test. The abrasive wear and the micro cracks were decreased by highly residual compressive stress.

X-Ray Stress Measurements of TiCN Thin Films

TiCN thin films deposited by Physical Vapor Deposition (PVD) were heated to various temperatures. Using X-ray diffraction technique, the preferred orientation of the crystallites was evaluated from pole figures and the crystallite Orientation Distribution Function (ODF) was calculated from the pole figures. After that, the residual stress was analyzed using the X-ray elastic constant obtained from the ODF. As a result of the heat treatment, the residual compressive stress relaxed at temperatures higher than 573K.

Plastic Instability and Mechanical Anisotropy of Textured Zinc

Abstract The influence of the material parameters, strain hardening, and strain-rate hardening on the plastic instability of textured zinc sheets tested in uniaxial tension was investigated. The experimental instability strains are observed to be in accord with those evaluated from the analytical expressions involving the material parameters. The crystallite orientation distribution function obtained from the pole figure data is used in conjunction with a plasticity model to predict the anisotropy parameters (R and P) and subsequently the width strains in longitudinal and transverse zinc sheets. While the experimental R and P values matched well with those predicted for basal slip, the predicted width strains beyond uniform deformation are observed to be higher than the experimental values.

Neutron diffraction applied to geological texture and stress analysis

Texture is defined as preferred crystallographic orientation in a polycrystalline aggregate. The main mechanisms of geological texture formation are crystallization, sedimentation, plastic deformation, recrystallization and metamorphism. Textures of geomaterials are fingerprints of the earth's history. The complexity of geological texture analysis results mainly from the overprinting of different textures on several mineral components during different orogenic periods. Quantitative texture analysis, i.e., the calculation of a three-dimensional orientation distribution function of crystallites, is based on experimental pole figures which represent the orientation distributions of certain crystallographic directions and which can be obtained from X-ray or neutron-diffraction techniques. Due to the high penetration capability of neutrons, large specimens can be investigated resulting in global volume textures rather than local surface textures from X-rays. Neutron-diffraction pole figures are characterized by high grain statistics even on coarse-grained material. Individual pole figures can also be obtained from reflection-rich diffraction patterns of multiphase rocks and low-symmetry mineral constituents by using position-sensitive detectors and by pattern decomposition by means of profile-fitting methods. Neufron texture diffractometers are operated in the constant-wavelength mode at steady-state sources and using time-of-flight techniques at pulsed sources. Different experimental set-ups are discussed. Results of neutron-diffraction texture analyses on monomineralic and polymineralic rocks are presented, e.g., on calcite, quartzite, plagioclase, pyrrhotite ores, granite, and orthopyroxene-sillimanite-granulite. The wide application spectrum of neufron texture analysis includes objects of the outer solar system as in the low-temperature texture study on ice and the non-destructive investigations on rare pieces of meteorites. The application of neutron diffraction on strain measurements and residual stress analysis of geological material is discussed. Natural effects on rocks are orders of magnitude smaller than in technological material and drilling gives rise to stress relaxation. Experimental deformations can be observed during in situ measurements at various pressures and temperatures. Neutron strain diffractometers are described. Examples of recent results on the full strain/stress tensor in sandstone and on strain partitioning in polymineralic rocks are given. Future prospects of new high-intensity neutron sources are promising in performing combined structure, texture and stress analysis. New instruments are under construction and software packages on the basis of full-pattern Rietveld refinements are available.

Temperature dependence of the morphology and mechanical properties of poly(vinyl alcohol) drawn films prepared by gelation/crystallization from solutions by X-ray and solid state 13C NMR

The morphology and mechanical properties of PVA were studied using atactic poly(vinyl alcohol) (at-PVA) dry gel films prepared by crystallization from solutions in dimethyl sulfoxide (Me 2SO) and water (H 2O) mixtures, in which Me 2SO/H 2O composition was set to be 60:40. The hot homogenized solution was quenched by pouring it into an aluminum tray controlled at −80°C, thus generating a gel. The crystal lattice modulus along the chain axis was measured by X-ray diffraction. The values were 200–220 GPa, which is independent of temperature up to 170°C. However, the values became lower with further increase in temperature and the value at 200°C became 133 GPa. This tendency is different from the previous reports showing rapid decrease in the crystal modulus at higher temperature than 120°C. The difference means that crystallites within the specimens prepared by quenching the solution (Me 2SO/H 2O=60:40) at −80°C are in much more stable state than those of the previous specimens. In spite of the temperature independence of the crystal modulus along the chain axis and the crystallinity, the storage modulus similar to Young's modulus decreases gradually with increasing temperature. To study the mechanical properties and molecular orientation of the present specimens with the stable crystallites, the orientational behavior of crystallites was estimated in terms of the orientation distribution function of crystallites, and Young's modulus was calculated by using the generalized orientation factors of crystallites and amorphous chain segments estimated from the orientation functions of crystallites and amorphous chain segments. In doing so, the theoretical calculation was carried out by using a three-dimensional model, in which the oriented crystalline layers are surrounded by an anisotropic amorphous phase and the strains of the two phases at the boundary are identical. The theoretical values were in good agreement with the experimental ones. The 13C NMR measurements suggested that the inter-molecular hydrogen bonds may be broken by drawing and the intra-molecular hydrogen bonds are more preferably formed in the mm and mr sequences with draw ratio but no marked difference in the structure exists between the crystalline and non-crystalline regions, as judged from the hydrogen bonding in the triad sequences. Even so, the spin–lattice relaxation time, T 1C, decreased drastically with increasing temperature reflecting drastic activity of chain mobility. Accordingly, the drastic decrease in Young's modulus (the storage modulus) is thought to be due to the fact that chain mobility in the amorphous phase becomes more pronounced as temperature increases.

Comparison of Ultrasonic Pole Figures Based Upon Ultrasonic Nondestructive Evaluation Method With Pole Figures Based Upon FEPM.

The ultrasonic wave velocities in a polycrystalline aggregate are sensitively influenced by texture changes due to plastic deformation, and their relationship was systematically analyzed by Sayers. According to Sayers's proposed model, it is possible to construct ultrasonic pole figures via the crystallite orientation distribution function (CODF), which can be derived by using ultrasonic wave velocity changes. In the previous paper, the Sayers's model was examined by experimental work and the effect of plastic deformation on texture development of an aluminum alloy specimen was studied by using ultrasonic pole figures as one of its applications. In the present paper, the ultrasonic pole figures based upon Sayers's model are constructed under various loading conditions of uniaxial tension, pure torsion, biaxial tension-compression, biaxial compression and biaxial tension, respectively. To examine its accuracy and reliability, the ultrasonic pole figures are compared with ones analyzed by the finite element polycrystal (FEPM), which has already been considered as a well-developed technology for the analysis of microstructural behaviors due to plastic deformation. The results show a remarkable qualitative similarity among the two methods.

Complete Pole Figure Measurement Using Only Back-Reflection Method with Imaging Plate and Application to Three-Dimensional Analysis of Texture.

Complete pole figure measurement using only a back-reflection method with an imaging plate (IP) was attempted. In our method, complete pole figures were obtained using only the back-reflection method without the use of a transmission method. In particular, the complete pole figure theoretically is obtained when its diffraction angle 2θ is 90°. In the cases of other 2θ values, the complete pole figure is obtained by measuring two incomplete pole figures by the back-reflection method and combining them. Such incomplete pole figures could be simultaneously measured by means of the two-dimensional detection ability of IPs. In addition, the correction of X-ray diffraction intensity to compensate the deviation from the pseudofocusing condition was not needed in our method, unlike Schulz’s back-reflection method. Pole density data obtained from our complete pole figures were effectively applied to three-dimensional analysis of the texture of a dual-phase stainless steel sheet. Crystallite orientation distribution functions (ODFs) of each phase were obtained.

Variant selection and its effect on phase transformation textures in cold rolled titanium sheet

Abstract The recrystallization and phase transformation textures in cold rolled titanium sheets with different rolling reductions have been investigated by means of three dimensional crystallite orientation distribution function (CODF) analysis, the high temperature X-ray diffraction (HTXRD) and transmission electron microscopy (TEM) technique. The formation of phase transformation texture is discussed in terms of orientation variant selection between α and β phases. It is found that at a low rolling reduction, for example 30%, the transformation texture is composed of a more randomly distributed orientations without orientation variant selection occurs. After titanium sheet cold rolled 80% and subsequently α→β→α phase transformation, the main transformation textures is characterized by [ 2 110]‖ ND fibre texture due to orientation variant selection.