title: ÐÑÐ°ÐµÐ²Ð°Ñ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ñ Ð² ÑÐ¿ÑÑÐ³Ð¾Ð¼ ÑÐ°ÑÐµ
title: Ð²ÑÐ¿ÑÑÐºÐ½Ð°Ñ ÐºÐ²Ð°Ð»Ð¸ÑÐ¸ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð°Ñ ÑÐ°Ð±Ð¾ÑÐ° Ð¼Ð°Ð³Ð¸ÑÑÑÐ°
titleType: Subtitle
title: Edge dislocation in elastic sphere
Creators: name: ÐÐµÑÑÐ¾Ð², ÐÐ¼Ð¸ÑÑÐ¸Ð¹
Publisher: name: Ð¡Ð°Ð½ÐºÑ-ÐÐµÑÐµÑÐ±ÑÑÐ³ÑÐºÐ¸Ð¹ Ð¿Ð¾Ð»Ð¸ÑÐµÑÐ½Ð¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ½Ð¸Ð²ÐµÑÑÐ¸ÑÐµÑ ÐÐµÑÑÐ° ÐÐµÐ»Ð¸ÐºÐ¾Ð³Ð¾
Year: 2023
Subjects: Ð£Ð¿ÑÑÐ³Ð¾ÑÑÐ¸ ÑÐµÐ¾ÑÐ¸Ñ
Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð¸Ñ Ð½ÐµÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²Ð¸Ñ
Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸
ÑÐ¿ÑÑÐ³Ð°Ñ ÑÐ½ÐµÑÐ³Ð¸Ñ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸
misfit stress
dislocations
elastic energy of an edge dislocation

description: Ð Ð´Ð°Ð½Ð½Ð¾Ð¹ ÑÐ°Ð±Ð¾ÑÐµ Ð²Ð¿ÐµÑÐ²ÑÐµ Ð½Ð°Ð¹Ð´ÐµÐ½Ð¾ Ð°Ð½Ð°Ð»Ð¸ÑÐ¸ÑÐµÑÐºÐ¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ Ð·Ð°Ð´Ð°ÑÐ¸ Ð¾ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸, Ð¿ÑÐ¾ÑÐ¾Ð´ÑÑÐµÐ¹ ÑÐµÑÐµÐ· ÑÐµÐ½ÑÑ ÑÐ¿ÑÑÐ³Ð¾Ð³Ð¾ ÑÐ°ÑÐ°. ÐÐºÑÑÐ°Ð»ÑÐ½Ð¾ÑÑÑ Ð¿ÑÐ¾Ð±Ð»ÐµÐ¼Ñ ÑÐ²ÑÐ·Ð°Ð½Ð° ÑÐ¾ ÑÑÐ°Ð²Ð½Ð¸ÑÐµÐ»ÑÐ½Ð¾ Ð¼Ð°Ð»ÑÐ¼ ÐºÐ¾Ð»Ð¸ÑÐµÑÑÐ²Ð¾Ð¼ ÑÐµÑÐµÐ½Ð½ÑÑ ÑÑÐµÑÐ¼ÐµÑÐ½ÑÑ Ð·Ð°Ð´Ð°Ñ Ð´Ð»Ñ Ð»Ð¸Ð½ÐµÐ¹Ð½ÑÑ Ð´ÐµÑÐµÐºÑÐ¾Ð² Ð² ÐºÐ¾Ð½ÐµÑÐ½ÑÑ ÑÐµÐ»Ð°Ñ. ÐÐµÐ¾Ð±ÑÐ¾Ð´Ð¸Ð¼Ð¾ÑÑÑ ÑÐ°ÐºÐ¸Ñ ÑÐµÑÐµÐ½Ð¸Ð¹ Ð¿Ð¾ÑÐ²Ð»ÑÐµÑÑÑ Ð¿ÑÐ¸ ÑÐµÐ¾ÑÐµÑÐ¸ÑÐµÑÐºÐ¾Ð¼ Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ð¸ Ð½Ð¸Ð·ÐºÐ¾ÑÐ°Ð·Ð¼ÐµÑÐ½ÑÑ Ð³ÐµÑÐµÑÐ¾-ÑÑÑÑÐºÑÑÑ Ñ Ð¿Ð»Ð¾ÑÐºÐ¸Ð¼Ð¸ Ð²Ð½ÑÑÑÐµÐ½Ð½Ð¸Ð¼Ð¸ Ð³ÑÐ°Ð½Ð¸ÑÐ°Ð¼Ð¸. ÐÑÐ¾Ð±Ð¾Ðµ Ð²Ð½Ð¸Ð¼Ð°Ð½Ð¸Ðµ Ð² ÑÐ°Ð±Ð¾ÑÐµ ÑÐ´ÐµÐ»ÐµÐ½Ð¾ ÑÑÐ°Ð²Ð½ÐµÐ½Ð¸Ñ Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð½Ð¾Ð³Ð¾ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ñ Ð´Ð»Ñ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸ Ð² ÑÐ°ÑÐµ Ð¸ Ð² Ð±ÐµÑÐºÐ¾Ð½ÐµÑÐ½Ð¾Ð¼ ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐµ ÑÐ¾Ð³Ð¾ Ð¶Ðµ ÑÐ°Ð´Ð¸ÑÑÐ°. ÐÐ»Ñ ÑÑÐ¾Ð³Ð¾ Ð¿Ð¾ÑÑÑÐ¾ÐµÐ½Ñ ÐºÐ°ÑÑÑ Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð¸Ð¹ Ð² ÑÐ°Ð·Ð»Ð¸ÑÐ½ÑÑ ÑÐµÑÐµÐ½Ð¸ÑÑ ÑÐ°ÑÐ° Ð¸ Ð¿Ð¾Ð¿ÐµÑÐµÑÐ½Ð¾Ð¼ ÑÐµÑÐµÐ½Ð¸Ð¸ ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐ°, ÐºÐ¾ÑÐ¾ÑÑÐµ Ð¿Ð¾ÐºÐ°Ð·Ð°Ð»Ð¸, ÑÑÐ¾ Ð¾ÑÐ½Ð¾Ð²Ð½ÑÐ¼ Ð¾ÑÐ»Ð¸ÑÐ¸ÐµÐ¼ Ð² Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð½Ð¾Ð¼ ÑÐ¾ÑÑÐ¾ÑÐ½Ð¸Ð¸ ÑÑÐ¸Ñ ÑÐ¸ÑÑÐµÐ¼ ÑÐ²Ð»ÑÐµÑÑÑ Ð·Ð½Ð°ÑÐ¸ÑÐµÐ»ÑÐ½Ð°Ñ ÑÐµÐ»Ð°ÐºÑÐ°ÑÐ¸Ñ Ð°ÐºÑÐ¸Ð°Ð»ÑÐ½ÑÑ Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð¸Ð¹ Ð² ÑÐ°ÑÐµ. ÐÐ¿ÐµÑÐ²ÑÐµ Ð½Ð°Ð¹Ð´ÐµÐ½Ð¾ Ð²ÑÑÐ°Ð¶ÐµÐ½Ð¸Ðµ Ð´Ð»Ñ ÑÐ¿ÑÑÐ³Ð¾Ð¹ ÑÐ½ÐµÑÐ³Ð¸Ð¸ ÐºÑÐ°ÐµÐ²Ð¾Ð¹ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸ Ð² ÑÐ°ÑÐµ. ÐÐ¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð²ÐµÐ»Ð¸ÑÐ¸Ð½Ð° ÑÑÐµÐ´Ð½ÐµÐ¹ Ð¿Ð¾Ð³Ð¾Ð½Ð½Ð¾Ð¹ ÑÐ½ÐµÑÐ³Ð¸Ð¸ Ð¼ÐµÐ½ÑÑÐµ, ÑÐµÐ¼ Ð² ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐµ Ð½Ð° Ð²ÐµÐ»Ð¸ÑÐ¸Ð½Ñ Ð¿Ð¾ÑÑÐ´ÐºÐ° Ð¿Ð¾Ð»Ð¾Ð²Ð¸Ð½Ñ ÑÐ½ÐµÑÐ³Ð¸Ð¸ ÑÐ´ÑÐ° Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸. ÐÐ°Ð¹Ð´ÐµÐ½Ð½Ð¾Ðµ ÑÐµÑÐµÐ½Ð¸Ðµ ÑÐ°ÐºÐ¶Ðµ Ð±ÑÐ»Ð¾ Ð¸ÑÐ¿Ð¾Ð»ÑÐ·Ð¾Ð²Ð°Ð½Ð¾ Ð´Ð»Ñ Ð¾Ð¿Ð¸ÑÐ°Ð½Ð¸Ñ ÑÐµÐ»Ð°ÐºÑÐ°ÑÐ¸Ð¸ Ð½Ð°Ð¿ÑÑÐ¶ÐµÐ½Ð¸Ð¹ Ð½ÐµÑÐ¾Ð¾ÑÐ²ÐµÑÑÑÐ²Ð¸Ñ Ð¿ÑÑÐµÐ¼ Ð·Ð°ÑÐ¾Ð¶Ð´ÐµÐ½Ð¸Ñ Ð¿ÑÑÐ¼Ð¾Ð»Ð¸Ð½ÐµÐ¹Ð½Ð¾Ð¹ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¸ Ð½Ð° Ð¿Ð»Ð¾ÑÐºÐ¾Ð¼ ÑÑÐ°ÑÑÐºÐµ Ð¼ÐµÐ¶ÑÐ°Ð·Ð½Ð¾Ð¹ Ð³ÑÐ°Ð½Ð¸ÑÑ Ð² Ð´Ð²ÑÑ ÐºÐ¾Ð¼Ð¿Ð¾Ð·Ð¸ÑÐ½ÑÑ ÑÐ¸ÑÑÐµÐ¼Ð°Ñ: Ð² ÑÐ½ÑÑ-ÑÐ°ÑÑÐ¸ÑÐµ Ð¸ Ð² ÑÐ°ÑÑÐ¸ÑÐµ ÑÐ¸Ð¿Ð° Â«ÑÐ´ÑÐ¾-Ð¾Ð±Ð¾Ð»Ð¾ÑÐºÐ°Â» Ñ ÑÐ´ÑÐ¾Ð¼ Ð² Ð²Ð¸Ð´Ðµ ÑÐ¾Ð¾ÑÐ½Ð¾Ð³Ð¾ Ð¿Ð¾Ð»ÑÑÐ°ÑÐ¸Ñ. ÐÐ¾ÐºÐ°Ð·Ð°Ð½Ð¾, ÑÑÐ¾ Ð² Ð¿ÐµÑÐ²Ð¾Ð¼ ÑÐ»ÑÑÐ°Ðµ ÐºÑÐ¸ÑÐ¸ÑÐµÑÐºÐ¸Ð¹ ÑÐ°Ð´Ð¸ÑÑ ÑÐ°ÑÑÐ¸ÑÑ Ð¿ÑÐ¸Ð¼ÐµÑÐ½Ð¾ Ð½Ð° ÑÑÐµÑÑ Ð±Ð¾Ð»ÑÑÐµ Ð² ÑÐ½ÑÑ-ÑÐ°ÑÐµ, ÑÐµÐ¼ Ð² ÑÐ½ÑÑ-ÑÐ¸Ð»Ð¸Ð½Ð´ÑÐµ. ÐÐ¾ Ð²ÑÐ¾ÑÐ¾Ð¼ ÑÐ»ÑÑÐ°Ðµ Ð±Ð¾Ð»ÐµÐµ Ð²ÑÐ³Ð¾Ð´Ð½ÑÐ¼ Ð¼ÐµÑÐ°Ð½Ð¸Ð·Ð¼Ð¾Ð¼ ÑÐµÐ»Ð°ÐºÑÐ°ÑÐ¸Ð¸ Ð¾ÐºÐ°Ð·Ð°Ð»Ð¾ÑÑ Ð¸ÑÑÐ»ÐµÐ´Ð¾Ð²Ð°Ð½Ð½Ð¾Ðµ ÑÐ°Ð½ÐµÐµ Ð¾Ð±ÑÐ°Ð·Ð¾Ð²Ð°Ð½Ð¸Ðµ ÐºÑÑÐ³Ð¾Ð²Ð¾Ð¹ Ð¿ÑÐ¸Ð·Ð¼Ð°ÑÐ¸ÑÐµÑÐºÐ¾Ð¹ Ð´Ð¸ÑÐ»Ð¾ÐºÐ°ÑÐ¸Ð¾Ð½Ð½Ð¾Ð¹ Ð¿ÐµÑÐ»Ð¸ Ð²Ð¾ÐºÑÑÐ³ ÑÑÐµÑÐ¸ÑÐµÑÐºÐ¾Ð¹ ÑÐ°ÑÑÐ¸ ÑÐ´ÑÐ°.
description: For the first time, an analytical solution for a straight edge dislocation threading the center of an elastic sphere is derived. The work is relevant due to the lack of analytical solutions for linear defects in finite bodies. Such solutions are of high interest for the theoretical description of low-dimensional heterostructures with planar interfaces. The substantial part of the work is devoted to the comparison of the stress-states caused by a dislocation in a sphere and in an infinite cylinder with equal radii. For that purpose, the analysis of the stress maps in different cross-sections of the sphere and cylinder is provided, which shows that the main difference between the two cases is the strong relaxation of the axial stress in the sphere. For the first time, an expression for the strain energy of an edge dislocation in a sphere is derived. The value of the mean energy per unit length of the dislocation in the sphere is less than in the cylinder approximately by a half of the core energy value. Then, the relaxation mechanism with the formation of an edge misfit dislocation at the planar part of the interface is considered in two composite systems: Janus particle and core-shell particle with the core of the hemispherical form. In the first case, the critical radius of the Janus sphere is one third higher than that of the Janus cylinder. In the second case, it is shown that the formation of a straight misfit dislocation is less favorable than the earlier studied formation of a circular prismatic dislocation loop around the spherical part of the core-shell interface.