We present two classes of exact solutions to a geometric model which describes the kinetics of fragmentation of d-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate a(${\mathit{x}}_1$,...,${\mathit{x}}_{\mathit{d}}$)=1 and daughter distribution function b(${\mathit{x}}_1$,...,${\mathit{x}}_{\mathit{d}}$‖${\mathit{x}}_1^{\prime }$,...,${\mathit{x}}_{\mathit{d}}^{\prime }$)=(${\mathrm{\alpha }}_1$+2). .(${\mathrm{\alpha }}_{\mathit{d}}$+2) ${\mathit{x}}_1^{{\mathrm{\alpha }}_1}$...${\mathit{x}}_{\mathit{d}}^{{\mathrm{\alpha }}_{\mathit{d}}}$/${\mathit{x}}_1^{\prime ({\mathrm{\alpha }}_1+1)}$... ${\mathit{x}}_{\mathit{d}}^{\prime ({\mathrm{\alpha }}_{\mathit{d}}+1)}$. The second class of exact solutions is described by a fragmentation rate a(${\mathit{x}}_1$,...,${\mathit{x}}_{\mathit{d}}$)= ${\mathit{x}}_1^{{\mathrm{\beta }}_1+1\mathrm{}}$...${\mathit{x}}_{\mathit{d}}^{{\mathrm{\beta }}_{\mathit{d}}+1\mathrm{}}$/$2^{\mathit{d}}$ and daughter distribution function b(${\mathit{x}}_1$,...,${\mathit{x}}_{\mathit{d}}$‖${\mathit{x}}_1^{\prime }$,...,${\mathit{x}}_{\mathit{d}}^{\prime }$)= $2^{\mathit{d}}$δ(x ${}_1{}^{}{}_{}{}^{}\mathit{x}_1^{\prime }{}_{}{}^{}$/2)...δ(${\mathit{x}}_{\mathit{d}}$-${\mathit{x}}_{\mathit{d}}^{\prime }$/2). Each class of exact solutions is analyzed in detail for the presence of scaling solutions and the occurrence of shattering transitions; the results of these analyses are also presented. © 1996 The American Physical Society.

• Received 25 October 1995

DOI:https://doi.org/10.1103/PhysRevE.53.3134