By David Wells

A better half quantity to the author's "Dictionary of Curious and engaging Numbers", which makes a speciality of mathematics and quantity conception. The entries during this e-book hide curves, topology, tilings and all branches of aircraft and 3-dimensional geometry, from Euclid to fractals.

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Stress prediction still has to be related to type of composite as classiﬁed in Chap. 2. As a ﬁrst step to establish such relationship we observe that θk obviously is a function of not only nk , vP , vS , but also of the more detailed geometry of phase P and phase S. 6). 3) claiming that the ﬁrst expression (σP ) must equal the latter expression (σS ) when subscripts P and S are interchanged (including c ⇔ 1 − c). The geometry function just deﬁned is further restricted when speciﬁc composites are considered as they are classiﬁed in Chap.

3). It applies also for both isotropic composites and cubical composites. 6) can be written as follows where supscript 0 indicates stress at vanishing concentration c. 9) where κS , γS are Poisson parameters deﬁned in Appendix A and fk , fg are so-called stress functions. 8). 6) can be used as follows for stress prediction when stiﬀness is known. 12) Stresses in general (σij ) are determined from Appendix A combining these expressions. Stress prediction is exact if exact stiﬀness properties are known.

2). 9). In general, however, numerical methods have to be used to calculate shape parameters for arbitrary ﬁber aspect ratios A. 48 7 Quantiﬁcation of Geometry Fig. 7. 2 For this purpose such a method has been developed in Appendix B by which particles stress in isotropic dilute composites (with ellipsoidal inclusions) can be calculated for any stiﬀness ratio (n). 2) simulated by n = 10−30 and 1030 respectively. Examples of shape parameters determined in this way are shown in Figs. 8. Shape parameters do not depend very much on the phase P Poisson’s ratio vP .