ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅰ) ──DIRECTIONAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS

In this two_part paper, a thorough investigation is made on Fourier expansions with irreducible tensorial coefficients for orientation distribution functions (ODFs) and crystal orientation distribution functions (CODFs), which are scalar functions defined on the unit sphere and the rotation group, respectively. Recently it has been becoming clearer and clearer that concepts of ODF and CODF play a dominant role in various micromechanically_based approaches to mechanical and physical properties of heterogeneous materials. The theory of group representations shows that a square integrable ODF can be expanded as an absolutely convergent Fourier series of spherical harmonics and these spherical harmonics can further be expressed in terms of irreducible tensors. The fundamental importance of such irreducible tensorial coefficients is that they characterize the macroscopic or overall effect of the orientation distribution of the size, shape, phase, position of the material constitutions and defects. In Part (Ⅰ), the investigation about the irreducible tensorial Fourier expansions of ODFs defined on the N_dimensional (N_D) unit sphere is carried out. Attention is particularly paid to constructing simple expressions for 2_ and 3_D irreducible tensors of any orders in accordance with the convenience of arriving at their restricted forms imposed by various point_group (the synonym of subgroup of the full orthogonal group) symmetries. In the continued work -Part (Ⅱ), the explicit expression for the irreducible tensorial expansions of CODFs is established. The restricted forms of irreducible tensors and irreducible tensorial Fourier expansions of ODFs and CODFs imposed by various point_group symmetries are derived.

ORIENTATION DISTRIBUTION FUNCTIONS FOR MICROSTRUCTURES OF HETEROGENEOUS MATERIALS (Ⅱ)──CRYSTAL DISTRIBUTION FUNCTIONS AND IRREDUCIBLE TENSORS RESTRICTED BY VARIOUS MATERIAL SYMMETRIES

The explicit representations for tensorial Fourier expansion of 3_D crystal orientation distribution functions (CODFs) are established. In comparison with that the coefficients in the mth term of the Fourier expansion of a 3_D ODF make up just a single irreducible mth_order tensor, the coefficients in the mth term of the Fourier expansion of a 3_D CODF constitute generally so many as 2m+1 irreducible mth_order tensors. Therefore, the restricted forms of tensorial Fourier expansions of 3_D CODFs imposed by various micro_ and macro_scopic symmetries are further established, and it is shown that in most cases of symmetry the restricted forms of tensorial Fourier expansions of 3_D CODFs contain remarkably reduced numbers of mth_order irreducible tensors than the number 2m+1 . These results are based on the restricted forms of irreducible tensors imposed by various point_group symmetries, which are also thoroughly investigated in the present part in both 2_ and 3_D spaces.

NUMERICAL SIMULATION OF ORIENTATION DISTRIBUTION FUNCTION OF CYLINDRICAL PARTICLE SUSPENSIONS

The orientation distribution function of cylindrical particle suspensions was deduced and numerically simulated, and an application was taken in a wedge-shaped flow field. The relationship between the orientation distribution function and particle orientation angles was obtained. The results show that comparing with the most probable angle distribution which comes to being in short time, the distribution of the steady state doesn't vary much. in range; the main difference is the anti-clockwise rotation in the right and upper field, that is, particles rotate more at the points where the velocity gradients are larger. The most probable orientations are close to the direction of local streamlines. In the direction of streamlines, with poleradius decreasing, the most probable angles increase, but the angles between their orientations and the local streamlines decrease.

Direction-dependent functions of physical properties for crystals and polycrystals

Some physical properties of crystals differ in direction n because crystal lattices are often anisotropic. A polycrystal is an aggregate of numerous tiny crystallites. Unless the polycrystal is an isotropic aggregate of crystallites, the physical properties of the polycrystal vary with n. The direction-dependent functions （DDF） for crystals and polycrystals are introduced to describe the variations of the physical properties in direction n. Until now there are few papers dealing systematically with relations between the DDF and the crystalline orientation distribution. Herein we give general expressions of the DDF for crystals and polycrystals. We discuss the applications of the DDF in describing the physical properties of crystals and polycrystals.

Influence of Rolling Reductions on Recrystallization Texture in Commercially Pure Al

The recrystallization texture in commercially pure Al is investigated after the samples are rolled differently and annealed.The samples with low rol- ling reductions could be generally recovered or recrystallized in situ and characterized by the re- tained G{110}<001>component.This recovery process decreases during annealing with the increase of reduction.The appearance of the B/R{359}. <132>and R{123}<634>components de- pending on the rolling reduction is investigated and the less developed Cube{100}<001>component in commercially pure ahninium is also discussed.

Rethinking about the Formulae of the Relationship between Euler Angles and Texture

The formulae of the relationships between Euler angles and texture are important for the orientation distribution functions of research materials.At present,it is found that the formulae for the orthogonal and hexagonal crystal structure proposed by Roe and Bunge are not in accord with the normal direction(ND)being perpendicular to the rolling direction(RD).In this paper,these formulae are deduced independently according to the notations proposed by Roe and Bunge.The results show that in addition to the coincidence with the formulae for the cubic crystal structure given by the Roe and Bunge,the other formulae are different from Roe’s and Bunge’s.The new formulae are in accord with ND,RD,and transverse direction(TD)at right angles.

Textural Characteristic of 70-30 Brass Rolled by Cross Shear Rolling in Single Direction

70-30 brass is rolled with 90% reduction by cross shear rolling in single direction with speed ra- tio 1.39.The sheet is divided into five layers along rolling plane normal to measure macroscopic statis- tical unsymmetric textures in every layer are des- cribed and analysed by means of three dimensional orientation distribution function.The results indi- cate that the main textures in every layer of brass rolled by cross shear rolling in single direction are the same as the main textures of brass rolled by conventional rolling.But the intensities,peak posi- tions and scatters of every texture component in {110}<112>are different,namely,there is a macroscopic statistical unsymmetry.It is found that the textures in every layer of brass rolled by cross shear rolling in single direction can be considered as the textures of brass rolled by common rolling in single direction at identical shear forces,the macroscopic statistical unsymmetry depends on the shear forces which are exerted on the layer.

STUDY OF TEXTURE EFFECT ON STRAIN LOCALIZATION OF BCC STEEL SHEETS

Using elastic crystalline viscoplastic finite element (FE) annlysis, the formability of BCC steel sheets was assessed. An orientation probability assignment method in the FE modeling procedure, which can be categorized as an inhomogenized material modeling, was newly proposed. In the study, the crystal orientations of three materials, mild steel, dual phase steel and the high strength steel, were obtained by Xray diffraction and orientation distribution function (ODF) analyses. The measured ODF results have revealed clearly different textures in the sheets, featured by orientation fibers, skeleton lines and selected orientations in Euler angle space, which are closely related to the plastic anisotropy. Then, the crystal orientations were assigned to FE integration points by using this ODF data, individually. The FE analyses of the standard limiting dome height (LDH) test show how the fiber textures affect the extent of strain localization in the forming processes. It was confirmed by comparison with experimental results that this FE code could predict the extreme strain localization and assess the sheet formability.

PREDICTION OF EARING IN TEXTURED ALUMINIUM SHEETS BASED ON CRYSTAL PLASTICITY

The phenomenon of earing is investigated in the present studybased on the theory of crystal plasticity with the dynamic explicityfinite element program developed. Firstly texture analysis is carriedout of rolled aluminum alloy A15052 by means of X-ray technique. Thenfrom the texture coefficients an analytical expression for theorientation distribution function (ODF) is derived making use of thecomputer algebraic language Mathematica4.0, which makes it easier todiscretize the ODF into a series of Eulerian angles repre- sentingthe distribution of lattices and further the preferred orientation(texture) of crystals of the original she- ets.

Rare earth texture analysis of rectangular extruded Mg alloys and a comparison of different alloying adding ways

The microstructures and textures of four rectangular extruded Mg alloys, Mg-3Gd, Mg-3Nd, Mg-1Gd-1Nd and Mg-1.5Gd-1.5Nd（all in wt%） were investigated by using optical microscopy（OM）, X-ray diffraction（XRD） and electron backscatter diffraction（EBSD）. The results demonstrate that the extruded alloys all reveal fully recrystallized microstructure and weak non-basal texture.For textures of the four extruded alloys, it is shown that the basal planes of majority grains tilt away from the sheet plane normal toward the extrusion direction. Compared with two binary alloys, the ternary alloys possess weaker texture strengths. Through orientation distribution function analysis, two types of rare earth textures, {0113} 〈0223〉and {1215} 〈1213〉, were found in the extruded alloys.Texture component {1215} 〈1213〉 is relatively weak, but it plays a critical role in the low texture intensities of two ternary alloys. A pairing effect of Gd and Nd elements has an influence on the texture modification. Additionally,there is a difference in the sensitivities of experimental parameters between the microstructure and texture.

Moving Mesh Kinetic Simulation for Sheared Rodlike Polymers with High Potential Intensities

The Doi-Hess equation that describes the evolution of an orientational dis-tribution function is capable of predicting several rheological features of nematic poly-mers.Since the orientational distribution function becomes sharply peaked as poten-tial intensity increases,powerful numerical methods become necessary in the relevant numerical simulations.In this paper,a numerical scheme based on the moving grid techniques will be designed to solve the orientational distribution functions with high potential intensities.Numerical experiments are carried out to demonstrate the effec-tiveness and robustness of the proposed scheme.