Orientation Density Function Research Articles

Effect of fiber orientation on the fracture toughness of brittle matrix composites

The effective fracture toughness of brittle matrix materials can be increased through the addition of short, poorly bonded fibers which bridge the growing crack. The orientation distribution of the fibers is likely to be biased, and not in an ideal random or aligned state. A micromechanical model is formulated for the post-cracking behavior using the force-displacement relation for an arbitrary fiber bridging a crack, the fiber orientation density function, and the fiber location density function. This model is then used to determine an effective traction law for the bridging fibers, as well as the steady state bridging toughness increment. In most cases, the results may be placed in the form of a product of the aligned fiber results times a modifying integrated orientation factor. The friction shear st ress on fiber pull-out is allowed to vary during pull-out, modeling the effects of matrix breakdown, fiber surface smoothing or wear debris accumulation. Results are presented for a variety of representative planar and three-dimensional fiber orientation states.

The toughness of brittle matrix composites reinforced with discontinuous fibers

The effective fracture toughness of brittle matrix materials can be increased through the addition of short, poorly bonded fibers which bridge the growing crack. Using a fundamental crack opening-restraining traction law for the resistance to crack propagation provided by an arbitrarily oriented fiber which bridges a crack, together with the fiber orientation and location density functions, a micromechanical model is developed. This model determines the effective crack opening-traction law of the bridging fibers. The steady-state toughness increment is obtained by integrating the contributions of the individual fiber pull out energies. This model includes the observed effect that fibers at large angles to the crack face normal may break rather than pull out. The shear stress at the fiber-matrix interface is allowed to vary during pull out, simulating fiber or matrix damage. Results are presented for a variety of representative fiber orientation states and interfacial shear cases.

Compressional Energy of the Random Fiber Assembly

In this paper, we describe the method used to evaluate the theory of the compressional energy of the random fiber assembly, which was developed mathematically in Part I. We also introduce a mathematical derivation of the method for updating the orientation density function of fibers in a general fiber assembly. In order to evaluate the energy, the distribution of the fiber segment lengths is characterized using the gamma distribution. In addition, the curvature of the fiber segments is characterized by an equation that relates the crimp and the effective diameter of the crimped fiber configuration. We use minimization technique to compute the compressional energy. Effects of the mechanical properties of fibers and the structural parameters of the assembly on the minimum compressional energy are computed for New Zealand wools. Except for fiber crimp, there is good agreement between the computed results and experimental results for the various fiber and structural parameters. The model shows that if only fiber crimp is increased for a given initial geometry, the tangent compression modulus actually decreases. However, this point cannot be tested because crimpier fibers cannot be brought to the same initial geometry without generating nonzero compressional strain energy. We plan further investigation of this point.

Some practical features for calculating the polycrystalline elastic properties from texture

The prediction of the elastic properties of a polycrystalline material requires the knowledge of its orientation density function (ODF), which represents mathematically the crystallographic texture. The real piece of texture information needed for such calculations depends on the modelling and is obtained from pole figures, the number of which is determined by crystal symmetry. According to these features different methods have been used to calculate (mean values, self-consistent approach) the elastic properties of low-symmetry materials with minimal texture information. Calculation of these properties for hexagonal materials becomes very simple and can be achieved directly from 00.1 pole figures.

Entropy optimization in quantitative texture analysis. II. Application to pole-to-orientation density inversion

Anisotropic behavior of single-phase polycrystalline material is controlled by its constituent crystal grains and their spatial orientation within the specimen. More specifically, a macroscopic physical property in a given direction is the mean value of the corresponding property of the individual crystallites with respect to the statistical distribution of their orientations. Thus, the statistical orientation distribution is a mathematical approach of describing and quantifying anisotropy. Unfortunately, the orientation density function (ODF) can generally not be measured directly. Therefore, it is common practice to measure pole density functions (PDFs) of several distinct reflections in x-ray- or neutron-diffraction experiments with a texture goniometer. Recovering an ODF from its corresponding PDFs is then the crucial prerequisite of quantitative texture analysis. This mathematical problem of texture goniometry is essentially a projection problem because the measured PDFs represent integral properties of the specimen along given lines; it may also be addressed as a tomographic problem specified by the crystal and statistical specimen symmetries and the properties of the diffraction experiment itself. Mathematically, it reads as a Fredholm integral equation of the first kind and was conventionally tackled by transform methods. Because of the specifics of the problem, these are unable to recover the part of the ODF represented by the odd terms of the (infinite) series expansion. In this situation, more sophisticated iterative methods, especially finite-series-expansion methods were developed within which the required non-negativity of the ODF to be recovered plays the prominent role. An efficient way to guarantee non-negativity is to employ entropy optimization.

Entropy Optimization in Texture Goniometry. II. Practical Applications

AbstractPursuing the approach of finite series expansion methods to the texture problem of recovering an orientation density function (ODF) from its pole density functions (PDF) measured in X‐ray or neutron diffraction experiments, the mathematical model of entropy maximization is applied to pole figure data and numerical results are presented. It is shown that this model as well as its results compare favorably with other methods of ODF restoration because it can yield a solution free of artifacts, also ghosts, if this is at all theoretically possible.

Practical application of entropy optimization in quantitative texture analysis

Abstract Finite series expansion methods are applied to the texture problem of recovering an orientation density function (odf) from its pole density functions (pdf) measured in X-ray or neutron diffraction experiments, the mathematical model of entropy maximization is applied to pole figure data and numerical results are presented as well as an interpretation of their geological significance.

繊維強化複合材料における繊維配向分布の表示法

This paper deals with a method for representing the distribution of fiber orientation in fiber reinforced composites. It is well known that the mechanical properties of fiber composites depend on the elastic properties and volume fraction of the constituent materials, and are affected by the fabrication processing. In particular, the fiber orientation and content can be strongly influenced by the material flow in molding. Therefore, a knowledge of the fiber orientation distribution is a requirement for predicting the mechanical behavior of the fiber composites. In this paper, the fiber orientation density function using an incomplete beta function is proposed in order to express the distribution of fiber orientation for the fiber composite, and the mold flow of fibers in SMC laminates is discussed by the proposed formulation. It is clear that various distributions of fiber orientation can be expresssd by selecting proper paramater values.

Does fabric tensor exist for a fabric?

It is shown that the mean intercept length distribution for planar fibre networks or for materials composed of a set of plates is not in general elliptic and cannot be expressed analytically in terms of a second-order tensor. However, our numerical computations indicate that the polar plot of the mean intercept length at the angle of measurement may become nearly ellipsoidal as the microstructure (fibres or plates) become less discretely organized, but yet remain orthotropic. The equations presented in this study may be used to obtain fibre (plate) orientation density functions from the experimental data on mean intercept length distribution.

On the orientation distribution function of nonspherical aerosol particles in a general shear flow: I. The laminar case

The problem of the orientation distribution of an important class of aerosol particles in a general laminar flow is solved. Various results are presented for significant cases characterized by an arbitrary particles' aspect ratio and an intermediate range of rotational Péclet numbers. This study constitutes a further step in the investigation of the orientation density function which deals with more realistic situations than have been considered until now.

Dynamic mechanical behavior of stress-graphitized and boron-doped pyrolytic carbon

The effects of stress-graphitizing and of co-deposition of boron on the dynamic mechanical properties of pyrolytic carbon have been investigated from 4 to 700°K. Stress-graphitizing was found to increase both the internal friction and the dynamic modulus in the strain-amplitude-independent region. The increase in modulus was consistent with an increase in the orientation density function. The order-of-magnitude increase in the internal friction may be explained by damping involving basal-plane dislocations associated with the increase in three-dimensional ordering. An internal friction peak, similar to the α peak previously reported in commercial graphites and pitch-bonded natural graphite crystals, was observed in the vicinity of room temperature for some of the stress-graphitized samples. It was found that low-dose reactor irradiation further increased the modulus, but that it caused the overall level of the internal friction to decrease. The internal friction of pyrolytic carbon with co-deposited boron (not stress-graphitized) somewhat resembled that of non-boronated, as-deposited pyrolytic carbon. However, values for the dynamic modulus of some of the boronated samples were significantly greater than those obtained on the non-boronated, as-deposited material, which may suggest that the codeposited boron resulted in some improvement in the crystalline structure.