The Shape of Dented Rigid-plastic Ellipses and Ellipsoids

New Zealand Journal of Mathematics (Online)

Graham Weir

关键词: identical particles;    ellipse;    ellipsoid;    separating surface;    rigid-plastic impacts;    level surfaces;    deformation.;   

学科分类：社会科学、人文和艺术（综合）

来源: University of Auckland * Department of Mathematics

【 摘 要 】

Impacts between identical elliptical or ellipsoidal particles can produce plastic yield, permanently altering (denting) the shape of the particles in the region of impact. We assume that the surface separating the two particles is closely approximated by equality of the level surfaces for each particle, and we prove that these separating surfaces for ellipses are formed from sections of rectangular hyperbola, or straight lines; and for ellipsoids, the separating surfaces are formed from sections of either hyperboloids of one or two sheets,an elliptic cone, a hyperbolic paraboloid, a ruled surface with the line of striction being a rectangular hyperbola, or a plane. For ellipses, the separating surface is formed from straight line sections if and only if the second ellipse is moved relative to the first either by a pure translation, or for the fixed point of the combined rotation and translation to lie on a principal axis, or the two ellipses are both circles. When two ellipses contact, the separating surface comprises of straight line sections if and only if the two contact points are conjugate.If two ellipsoids just contact, then the separating surfaces are planar if and only if (at least) one of the principal directions is common to both ellipsoids, and the two contact points are conjugate. The separating surface can never have a positive definite metric, and are characterised by hyperbolic or planar geometries.

【 授权许可】

Unknown