麻省理工学院：《polymer engineering》p 2

Prob. 6.2 unidirectionally reinforced with optimal fiber packing? Consider a triangular area inscribed on a close-packed section as shown. The enclosed fiber area includes half of the three circles located on the midsides and one-sixth of the three circles at the vertices The area of fibers in the triangle is they A:=(3*(1/2)+3*(1/6)°Pir^2 4=2兀 The area of the equilaterial triangle, with sides of 4r, is At:=4 grt(3) √3 cking density is ther Digits: =4; p: evalf(AfA[tD; =.9072

Prob. 6.2 What is the maximum fiber volume fraction Vf that could be obtained in a unidirectionally reinforced with optimal fiber packing? Consider a triangular area inscribed on a close-packed section as shown. The enclosed fiber area includes half of the three circles located on the midsides, and one-sixth of the three circles at the vertices. The area of fibers in the triangle is then A[f]:=(3*(1/2)+3*(1/6))*Pi*r^2; A := 2 π r2 f The area of the equilaterial triangle, with sides of 4r, is A[t]:=4*r^2*sqrt(3); A := 4 r2 3 t Packing density is then Digits:=4;p:=evalf(A[f]/A[t]); p := .9072