Group theory and representation of microstructure and mechanical behavior of polycrystals

Computational Materials Science

2005

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.

Section: Tensorial Representation Of the Codfmentioning

confidence: 99%

“…The classical representation by generalized harmonic functions was introduced by Bunge [7] and Roe [13]. Later on, Adams et al [1] and Guidi et al [9] introduced a tensorial Fourier expansion of the codf (see also [21,22]). Both representations are generally equivalent.…”

Section: Introductionmentioning

confidence: 99%

Application of the maximum entropy method in texture analysis

Computational Materials Science

2005

“…This property implies the existence of a tensorial Fourier expansion. Adams et al [1] and Guidi et al [10] considered this expansion for the special case of a cubic crystal symmetry. Zheng and Fu [24,25] analyzed the expansion for arbitrary crystal and sample symmetries.…”

Section: Tensorial Fourier Expansion Of the Codfmentioning

confidence: 99%

“…Here an approach is presented that is based on a tensorial Fourier expansion of the codf [1,10]. The tensorial Fourier coefficients or texture coefficients can be considered as micro-mechanically based tensorial internal variables [5,6,22].…”

Section: Introductionmentioning

confidence: 99%

Texture simulation based on tensorial Fourier coefficients

2006

Computers & Structures

The main result of the paper is the derivation of the evolution equation of the tensorial texture coefficients of the crystallite orientation distribution function (codf). The evolution equation of each coefficient depends on the complete codf and the lattice spin, which is a constitutive quantity. For the solution of the differential equation based on a finite number of coefficients, the codf has to be estimated. This estimate is obtained here by the maximum entropy method. By this approach the texture evolution can be described by modeling some low-order Fourier coefficients. It will be shown that such a low-dimensional approach yields a reasonable description of the texture evolution and of mechanical properties like the Taylor factor.

“…The Fourier series is represented by scalar products of irreducible tensors of rank β, texture coefficients V β i and normalized reference tensors T β i [1,2]. The Rayleigh product ( ) denotes a rotation by an orthogonal tensor Q ∈ SO(3).…”

Section: Introductionmentioning

confidence: 99%

Effective Flow Potentials for Anisotropic Polycrystals

Tsotsova

Proc Appl Math and Mech

2009

In this paper a micromechanically based flow potential for anisotropic fcc polycrystals is derived that takes into account the crystallite orientation distribution function (codf) in terms of tensorial texture coefficients. The effective flow potential is based on a representation theorem for anisotropic scalar functions depending on a 2nd-order tensor. A priori unknown functions in the representation are determined by defining and solving explicitly a minimization problem over SO(3). Important analytical properties of the coefficient matrix of the minimization problem are discussed.