Mean Intercept Research Articles

Role of Grain Growth in the Oxidation of Nickel

The isothermal growth of a scale upon nickel 270 at 800°–1000°C is accompanied by a coordinated growth of the grains of the oxide, wherein the mean intercept of the grains (measured parallel with the scale surface) is directly proportional to the thickness of the scale. The average grain volume of the columnar grains of the scale is directly proportional to the time of oxidation. Correspondingly, the scale thickness and the mean intercept of the grains increase in proportion to the cube root of the time. Markers maintain the same relative positions in the scale as the scale thickens, showing that the scale grows by swelling. These findings are explained by the formation of new oxide, mainly along grain boundaries, where oxygen diffusing inward upon grain boundaries meets nickel diffusing outward through the grains. The sweeping of the grain boundaries that accompanies grain growth serves to distribute the new oxide uniformly throughout the volume of the scale, resulting in growth by swelling.

Subject index to volume 8

Accounting for spatial variation of trabecular material anisotropy and orientation can improve the accuracy of quantitative computed tomography-based finite element (FE) modeling of bone. The objective of this study was to investigate the feasibility of quantifying trabecular material anisotropy and orientation using clinical computed tomography (CT). Forty four cubic volumes of interest were obtained from micro-CT images of the human radius. Micro-FE modeling was performed on the samples to obtain orthotropic stiffness entries as well as trabecular orientation. Simulated computed tomography images (0.32, 0.37, and 0.5 mm isotropic voxel sizes) were created by resampling micro-CT images with added image noise. The gray-level structure tensor was used to derive fabric eigenvalues and eigenvectors in simulated CT images. For ‘best case’ comparison purposes, Mean Intercept Length was used to define fabric from micro-CT images. Regression was used in combination with eigenvalues, imaged density and FE to inversely derive the constants used in Cowin and Zysset–Curnier fabric-elasticity equations, and for comparing image derived fabric-elasticity stiffness entries to those obtained using micro-FE. Image derived eigenvectors (which indicated trabecular orientation) were then compared to orientation derived using micro-FE. When using clinically available voxel sizes, gray-level structure tensor derived fabric combined with Cowin's equations was able to explain 94–97% of the variance in orthotropic stiffness entries while Zysset–Curnier equations explained 82–88% of the variance in stiffness. Image derived orientation deviated by 4.4–10.8° from micro-FE derived orientation. Our results indicate potential to account for spatial variation of trabecular material anisotropy and orientation in subject-specific finite element modeling of bone using clinically available CT.

Tensile strength and work hardening of ultrafine-grained high-purity copper

Sputter-deposited high-purity copper specimens with grain sizes from 8.4 to 0.056 μm were tested in tension to investigate the influence of grain size on yield strength for small grain sizes. The smallest grain size in the sputter-deposited copper was a factor of 40 smaller than the smallest grain size previously available for yield-strength–grain-size studies of high-purity copper. The 0.2% offset yield strengths varied from 73.4 MPa for the coarsest-grained copper to 481 MPa for the finest-grained copper. The hardening was related to grain- and twin-boundary spacings. The yield strength obeyed the Hall-Petch relation even for the finest grain size of 0.056 μm, giving σy=6.39+3.74(l̄)−1/2 MPa, where σy is the 0.2% offset yield strength and l̄ is the mean intercept length (mm) between boundaries. The data agreed with the Ashby model, σ∼ (ε/l̄)1/2, only at very low strains (ε<0.001) for the specimens with very fine grain sizes (l̄⩽0.077 μm).

The quantitative morphology of anisotropic trabecular bone.

SUMMARYDimensional measurements have been made on a small number of tracings of low magnification scanning electron micrographs of trabecular bone sections, using the Quantimet Image Analysing Computer, with a view to expressing the properties of the local bone pattern in a quantitative manner.Measurements of the orthogonally projected length of the boundary profile of the interface between bone and marrow were made as a function of the direction of projection, and these showed that the majority of the patterns were distinctly anisotropic.A formula is proposed for calculating the mean widths of the trabecular bands as they appear on the surfaces of the sections, from the mean area of bone and the boundary profile length. As the validity of this formula is not theoretically evident, the results have been compared with direct experimental measurements, and found to agree very well.The mean intercept lengths in the bone and in the marrow spaces have been calculated as a function of direction across the section. The polar diagrams of these lengths are found to have the shapes of ellipses, to an unexpected degree of accuracy. This makes it possible to express the departure from isotropy of the pattern by a single number, for instance by the ratio of the axes of the ellipse.The essential limitations of the methods used in these experiments are briefly discussed, in particular the impossibility of finding the exact symmetry properties of the surface patterns by experiments of this type.

Recent progress in automatic image analysis*

SUMMARYA general survey has been made of the present utility of computing equipment and automatic devices for common problems in stereology and morphometry. Three situations have been evaluated:(1) Commonly available computer facilities permit easy and rapid prediction of two‐dimensional measurement distributions characteristic of any mathematically defined spatial structure. Tabulated predictions can be computed by direct simulation of sampling processes and random perturbations may be introduced if desired. Resort to formal mathematical solutions is not normally required.(2) For purposes of controlling the quality of useful solid materials, the materials engineer can be satisfied by description of a Gestalt which characterizes the material as a whole and which can reasonably be presumed to control the behaviour of the material in service. Six stereologically valid parameters are sufficient to describe such a Gestalt. These are: volume fraction (Vv), mean free path in the matrix (l̄α), mean intercept width of particles (l̄β) and measures of material variability, general anistropy, and degree of patternness. Practical measurements of the latter parameters have been defined. Current scanner models are intrinsically suitable for these measurements but will benefit from standardized data outputs and improvements to increase measuring accuracy.(3) While Gestalt measurements may perhaps be applicable to some biological problems, many problems in this field are specifically concerned with the micro‐architecture of individual objects or cells. Micro‐architectural observations such as counting and measuring, or identifying, individual object sections are inefficiently performed in ordinary computers. Large programming effort is required for limited machine intelligence and the operating speeds are unsatisfactory. These machine operations are not yet competitive with human observers. A partnership combining human identification with machine measurement of identified objects is practical and equipment is presently available. There is reason to presume that computer methods of object identification which have been employed are not similar to the methods actually employed by humans. New hardware systems appear to be necessary either to obtain satisfactory operating speeds by known processes or, if possible, to economically simulate human processes of image recognition.

Examination of neutron-irradiated UO2 using the scanning electron microscope

Formulae are derived for the stereological determination of boundary length per unit area,BA, and surface density,Sv, in materials with elliptically shaped pores. These relationships are valid for two-phase solids with orthotropically distributed internal surfaces (surfaces that can be mapped onto an ellipsoid). Maximum and minimum mean intercept length measurements in a plane are needed to calculateBA. Mean intercept length measurements in the three principal directions of porous symmetry are necessary to calculateSv. If the principal directions are not known they can be found by calculating the eigenvectors of a second rank tensor called the mean intercept length tensor. In a material with a transversely isotropic distribution of internal surfaces (surfaces that can be mapped onto a spheroid), mean intercept length measurements along and transverse to the axis of symmetry are needed to calculateSv.

Stereology, or the quantitative evaluation of microstructures

SUMMARYA general survey is presented of the basic equations of stereology, that is, the quantitative relationships that exist between quantities measured on the two‐dimensional plane of polish or section and the magnitudes of microstructural features that occur in three‐dimensional space. A new and important branch of stereology concerns the quantitative relationships between structures in a thin foil and their projected images. Basic equations are given in the absence of overlap and truncation; then overlap and truncation corrections are indicated for simple cases. The methods of determining the size distribution of particles or grains are outlined. Finally, other important parameters are described, such as the mean free distance and the mean intercept length.The subject of stereology is not completely new to some, but there are still many experiments to be run and methods to be perfected. It is hoped that microscopists dealing with microstructures will be encouraged to apply quantitative methods in their work in order to advance our knowledge in this fascinating field.

Microstructural development during the liquid-phase sintering of two-phase alloys, with special reference to the NbC/Co system

An investigation was made of the grain growth and other microstructural changes occurring during the liquid-phase sintering of NbC alloys with ∼20 wt % cobalt. The effects of sintering time, sintering temperature, and small alloying additions were studied. It was found that the grain growth of NbC in liquid cobalt, at 1420° C, can be described by the equation: $$\bar d^3 - \bar d_0 ^3 = {\text{K}}t$$ where $$\bar d$$ is the mean linear intercept of the grains after time t, and $$\bar d_0$$ the initial mean intercept, K being a temperature-dependent constant with an “activation energy” of 95±15 kcal/mole. This equation suggests that grain growth occurs by a solution/ precipitation process controlled by diffusion in the liquid phase. Small alloying additions of WC, TiC or NbB2 inhibit the growth and/or alter the growth process, as well as affecting such properties as the shape and contiguity of the carbide grains. The relative significance of grain coalescence to grain growth in a liquid phase is discussed. By examining theoretically the effect of anisotropy of interface energy on the cube ⇌ sphere grain-shape change, it has been possible to explain the observed sensitivity of grain shape towards sintering conditions.