Elongation and rotation of a linear element under the continuum deformation

Abstract

Studying the deformations (see, e. g., [1], p. 72, or [2], p. 33), one uses relative elongations and angles of rotation of linear elements (material fibers). Solving many applied problems for technical objects, one measures the linear deformations, displacements, and angles of rotation of surface fibers and approximates the measuring linear (knife-edge) bearings by linear elements. In this paper we describe a method which enables one to determine the linear and angular deformations of a linear element in a homogeneous elastic medium under small deformations. We represent the angle of rotation of the linear element in a vector form and define its components in terms of those of the distortion tensor, the Levi-Civita pseudotensor, and the pairwise products of directional cosines of the linear element. Let the location of an arbitrary mass point Q of a solid (a continuum) in the three-dimensional Euclidean space be defined by the radius-vector x, whose components are the Lagrange coordinates xi (i = 1, 2, 3) in the Cartesian rectangular system. Assume that near the pointQ(xi) themedium is elastic and homogeneous. Let an external action move two infinitely close pointsQ(xi) andN(xi + dxi) of the solid (initially, the distance between them equals ds = |dx|) to the locations Q(xi) and N (xi + dxi). The linear element ds bounded by the pointsQ(xi) andN(xi + dxi) turns into the linear element ds′ = |dx′| bounded by the points Q(xi) andN (xi + dxi). Assume that the coordinates xi of the continuum points are continuous and one-valued functions of the coordinates x1, x2, x3 of the initial location: xi = x ′ i(x1, x2, x3). These functions have continuous derivatives with respect to all coordinates. We define the direction, along which the linear element ds goes out of the point Q(xi), with the