Terms Of Texture Coefficients Research Articles

Representation of Hashin–Shtrikman bounds of cubic crystal aggregates in terms of texture coefficients with application in materials design

The Hashin–Shtrikman bounds of aggregates of cubic crystals are explicitly represented in terms of tensorial texture coefficients. The formula is valid for arbitrary crystallographic textures and isotropic two-point statistics. The isotropy of the two-point statistics implies that the grain shape is isotropic on average. The new explicit representation has the advantage that the set of energetically admissible crystallographic textures and corresponding effective linear elastic properties can be directly determined and analyzed based on minimum principles of the elastic strain energy density. It is shown that all energetically admissible textures with maximum anisotropy have an effective elastic behavior with cubic sample symmetry. Furthermore, it is proven that there exist texture states without maximum anisotropy which have the extreme elastic properties peculiar to states with maximum anisotropy. This is an important result for the design of elastic material properties.

A Representation Theorem for Material Tensors of Weakly-Textured Polycrystals and Its Applications in Elasticity

Material tensors pertaining to polycrystalline aggregates should manifest also the influence of crystallographic texture on the material properties in question. In this paper we make use of tensors which form bases of irreducible representations of the rotation group and prove a representation theorem by which a given material tensor of a weakly-textured polycrystal is expressed as a linear combination of an orthonormal set of irreducible basis tensors, with the components given explicitly in terms of texture coefficients and a set of undetermined material parameters. Once the irreducible basis tensors that appear in the formula are determined, the representation formula, which is valid for all texture and crystal symmetries, will delineate quantitatively the effect of crystallographic texture on the material tensor in question. We present an integral formula and an orthonormalization process which serve as the basis for a procedure to determine explicitly the irreducible basis tensors required in the representation formula. For applications we determine a set of irreducible basis tensors for the elasticity tensor and a set for fourth-order tensors that define constitutive equations in incompressible elasticity and Hill’s quadratic yield functions in plasticity. We show that orientation averaging of a tensor can be done easily if we have in hand a set of irreducible basis tensors for the decomposition of the tensor in question. As illustration we derive a formula, which is valid for all texture and crystal symmetries, for the elasticity tensor under the Voigt model.

Elastic properties of polycrystalline microcomponents

The statistics of Young’s modulus of microspecimens subjected to a tensile load is examined experimentally, numerically and analytically. The material considered is Stabilor ®G, a dental alloy, mainly consisting of gold. Tensile tests with microspecimens produced by vacuum pressure casting have been performed using a self-designed universal micro-testing machine. The numerical approach is based on the finite element method with the microstructure being described by a periodic Voronoi tessellation with randomly oriented grains. The analytical approach uses an explicit formulation of the singular approximation in terms of texture coefficients and is valid for arbitrary sample symmetries. It is found that the finite element approach and the analytical approach reproduce the experimental findings. Furthermore, it is shown that the mean value and the standard deviation of Young’s modulus of microspecimens made of cubic crystals subjected to tensile loads can be described by a unified scaling law. The results imply that the statistics of apparent properties of cubic crystal aggregates can be determined using anisotropic effective medium approximations.

Representation of effective flow potentials for polycrystals based on texture data

In this paper, an effective micromechanically based flow potential for face-centered cubic crystals is pro- posed. In a first step, the single crystal flow potential is represented in terms of texture coefficients of the crystallite orientation distribution function. An optimization problem on SO(3) is formulated and solved in order to identify the best approximation of the single crystal flow potential. In a second step, assuming homogeneous stress or strain rate fields, the flow potential of the corresponding polycrystal is deduced explicitly from the tensorial representation of the single crystal potential. In contrast to phenomenological yield criteria, the advantage of such an approach lies in the fact that the macroscopic yield function can be estimated directly based on a single texture measurement and the critical resolved shear stress of the single crystal.

An analytical micro-macro model for textured polycrystals at large plastic strains

An analytical micro-macro model of evolving plastic anisotropy is presented that is suitable for numerical simulation of forming processes. The model is based on the combination of a polycrystal model and different analytical procedures for writing anisotropic plastic potentials, expressing their coefficients in terms of texture coefficients, and updating the texture coefficients as function of the (tensorial) strain increment. The use of a fourth-order dual plastic potential (“C4”) in the analytical micro-macro model is studied, and this use is compared with that of Hill's [1948] yield criterion and also with the usual run of the Taylor model. The coefficients of the C4 potential depend linearly on the texture coefficients, which are updated using a variational polycrystal model. The analytical operation of this updating lies on the method first proposed by Esling et al. [1984] and is described and checked in some detail. The predictions of the analytical micro-model compare well with measurements of the Lankford coefficient, provided the C4 potential is used. The predicted texture evolution is also in a good experimental agreement: a better one than with the Taylor model, which in some cases, gives a poor updating. The theoretical stress evolution during biaxial or plane-strain tension is experimentally consistent too, although in that case the C4 potential, closer to Taylor's model, makes no improvement as compared with Hill's quadratic criterion.

Effect of cold working and annealing on the texture of Zr-2·5Nb pressure tubes

Texture plays an important role in the commercial acceptability of Zr-2·5 wt% Nb pressure tubes used in nuclear reactors. A modified flow sheet for the fabrication of these pressure tubes involves a few additional steps viz. stress relieving, cold working and annealing, as compared to the conventional route. The evolution of texture during sequential fabrication steps of extrusion, stress relieving, cold working and annealing was studied in terms of texture coefficients and inverse pole figures. It was observed that crystallographic texture primarily developed during hot extrusion and the additional steps of stress relieving, cold working and annealing did not alter the texture significantly. The texture developed was one having a majority of the basal plane normals along the tangential direction of the tube.