Elastic displacements produced by cross-links in polyethylene crystals

An elastic model of a cross-link between adjacent molecules in a polyethylene crystal is formulated. The model consists of two point forces and two pairs of dipole forces acting on the axes of the cross-linked molecules. The elastic displacements arising from these forces are obtained by using the elastic Green displacement tensor, and the displacements near the cross-link are calculated by means of a simplified discrete atomic model. The actual values of the forces and dipole forces generating the elastic displacements are found by matching the discrete and elastic solutions, and the complete displacement field is thus obtained for a cross-link in a crystal of infinite dimensions. It is shown that the cross-link acts as a centre of elastic distortion with principal strains $\epsilon _{11}$ = -0.5, $\epsilon _{22}$ = 0.054 and $\epsilon _{33}$ = -0.055. The limitations of the calculation are discussed and it is concluded that the results are applicable with little error to cross-links lying beyond seven interatomic distances below the surface of thin lamellar crystals. Possible useful applications of the results are also indicated.