2 Extensive experimental and theoretical efforts have been made to estimate the required energy and force to fracture a brittle solid into fragments with desirable size distribution. In the experimental approach, one of the most popular tests is the compression test of spheres, either statically or dynamically, between two flat rigid platens. Under quasi-static loads, compression of spheres between two flat platens has been proposed for testing the deformability of elastic materials, hardness of ductile materials and crushing strength of brittle materials. For example, the crushing of spheres between the flat platens can also be used to estimate the tensile strength of brittle spheres. A comprehensive review is given by Chau et al. [2] and by Darvell [6]. Although there are numerous experimental studies, stress distribution within a sphere under compression between two rigid platens has not been studied comprehensively. The most popular theoretical model is that proposed by Hiramatsu and Oka [7], which has been applied by various authors. Chau et al. [2] also provided a extension of Hiramatsu-Oka solution to incorporating the Hertz contact stress under compression. For the dynamic impact of spheres between two rigid platens, although a informative crater analysis was also suggested by Chau et al. [2]. However, due to mathematical complexity, Chau et al. [2] did not consider the exact solution for the stress distribution within spheres under the double impact test. Therefore, the dynamic stress inside the sphere and its relationship to the final fragmentation is not well understood. Therefore, this paper outlines a new approach in which the Valanis [1] superposition principle for dynamic problems and the dynamic crack growth results considered by Freund [8] are combined to investigate the problem of dynamic fragmentation in spheres. The analysis is still on-going, only the essential idea and preliminary results will be reported. VALANIS (1966) SUPERPOSITION PRINCIPLE By applying the superposition principle put forward by Valanis [1], the problem of double impact test on spheres can be decomposed into two associated problems: the static problem and the free vibration problem of a sphere. Static Compression of Spheres The static solution for sphere can be generalized from that of Hiramatsu and Oka [7] (see Chau et al. [2]). In particular, by incorporating the Hertz contact stress, the static problem of compression of sphere between two rigid platens are: ( ) ( )( ) þ ý ü î í ì - + + - - + + - - = - ¥ = å 2 2 2 2 2 2 0 2 4 (2 1) 4 3 4 2 3 2 2 1 1 ) (cos n n n n n n rr A r n n C r n n n n n P m m l q s (1) ( ) ( ) ( ) ( ) ( ) þ ý ü î í ì + + + + + + - ¶ ¶ + þ ý ü î í ì + + + - - = - - ¥ = å 2 2 2 2 2 2 2 2 2 2 2 2 2 0 2 2(2 1)(4 3) 2 3 2 5 cos 2 4 4 3 2 3 2 1 ) (cos n n n n n n n n n n n A r C r n n n n P A r n C r n n n P m m l q q m m l q sqq (2) ( ) ( ) ( ) þ ý ü î í ì + - + + + + + - - ¶ ¶ = - ¥ = å 2 2 2 2 2 2 2 0 2(2 1) (2 1)(4 3) 4 1 4 4 1 cos n n n n n n r n C r A r n n n n n n P m m l q q sq (3) where the unknown constants can be obtained from the boundary condition as
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