Biomedical Engineering Reference
In-Depth Information
Fig. 7.6 Test lines
superimposed on a cancellous
bone specimen. The test lines
are oriented at the angle y .
The mean intercept length
measured at this angle is
denoted L ( y ). From Cowin
and Mehrabadi ( 1989 )
the flow of liquid crystals a vector is often used to characterize the long axis of the
liquid crystal. In early liquid crystal theories the formation of the constitutive
equation for the liquid crystal follows the development outlined in Chap. 5 for
the constitutive equation for the Newtonian law of viscosity up to equation (5.4 N).
In the early liquid crystal theory the equation equivalent to (5.4 N) was assumed to
also depend upon the unit vector n , a vector coincident with the long axis of the
liquid crystal (de Gennes and Prost 1993 ).
The modeling of the microstructural architecture of a material with two distinct
constituents, one dispersed in the other, has been accomplished using a second rank
tensor called the fabric tensor. Fabric tensors may be defined in a number of ways; it
is required only that the fabric tensor be a positive definite tensor that is a
quantitative stereological measure of the microstructural architecture, a measure
whose principal axes are coincident with the principal microstructural directions
and whose eigenvalues are proportional to the distribution of the microstructure in
the associated principal direction. The fabric tensor is a continuum point property
(as usual its measurement requires a finite test volume or RVE) and is therefore
considered to be a continuous function of position in the material.
One type of fabric tensor is the mean intercept length (MIL) tensor. The MIL in a
material is the average distance, measured along a particular straight line, between
two interfaces of the two phases or constituents (Fig. 7.6 ). The value of the mean
intercept length is a function of the slope of the line,
, along which the measure-
ment is made in a specified plane. A grid of parallel lines is overlaid on the plane
through the specimen of the binary material and the distance between changes of
phase, first material to second material or second material to first material, are
counted. The average of these lengths is the mean intercept length at the angle
y
, the
angle characterizing the orientation of the set of parallel lines. Figure 7.6 illustrates
such measurements. It is frequently observed that when the mean intercept lengths
measured in the selected plane in the specimen are plotted in a polar diagram as a
function of
y
y
, producing a closed curve in the plane. If the test lines are rotated
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