By Abel Flint
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Extra info for A system of geometry and trigonometry: Together with a treatise on surveying : teaching various ways of taking the survey of a field, also to protract ... using them : compiled from various authors
1) is exact. 13), however, is an approximation. No such simple, exact, and closed form relation is known from the literature. The approximation, however, is very good. This has been justiﬁed in a FEM stiﬀness analysis of particulate composites presented in . The approximation becomes exact at low phase P concentrations (c → 0) where the following solution has been presented [45, 46, 53]. 13) is considered exact in subsequent deductions with respect to the behavior of composite materials. Such procedure is strongly supported by some results obtained by Christoﬀersen  and Levin  in their analysis of materials reinforced with randomly distributed spherical particles.
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. 2 according to their critical concentrations cP and cS . The inﬂuence of composite type on the geometry function is subsequently determined by looking at a composite where phase P can be extremely stiﬀ (nk = ∞) or it can be extremely soft (nk = 0).
4 Composite Stress and Geometry It has been shown in Chap. 6), that any stiﬀness modulus of isotropic composite materials can be predicted if stress is known for one of the constituent phases. Therefore, the analysis of stiﬀness versus phase geometry is basically an analysis of internal stresses as a function of phase geometry. In this chapter a method is developed by which consistent estimates can be given on stress versus geometry as classiﬁed in Chap. 2 for various types of composites. The term, consistent, means that the stress estimates presented agree with exact quantities whenever such are known from the literature on composite materials – and that the stiﬀness predictions subsequently made in Chap.
A system of geometry and trigonometry: Together with a treatise on surveying : teaching various ways of taking the survey of a field, also to protract ... using them : compiled from various authors by Abel Flint