The Penguin dictionary of curious and interesting geometry by David Wells

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 .

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