Plane elastic boundary value problem posed on orientation of principal stresses

Boundary Value Problems of Plane Elasticity Involving Orientations of Displacements and Tractions

“…The example of the latter is lack of data on stress, tractions or displacement magnitudes on the contour, while directions of these vectors can be given. In regard to elasticity, this type of problems was firstly introduced by Galybin and Mukhamediev [4]. This work addresses the case when principal directions (orientations of principal stresses) and its normal derivative are known on the smooth boundary of a plane isotropic domain.…”

“…This work addresses the case when principal directions (orientations of principal stresses) and its normal derivative are known on the smooth boundary of a plane isotropic domain. Determination of a finite number of linearly independent solutions from the analysis of principal directions on the boundary is the main result of [4]. Galybin [5] presented a somewhat similar analysis for an interior domain when displacement and traction orientations are given on its boundary (this case is generalized for an exterior domain in the present paper).…”

“…Data on orientations of principal stresses are found in the World Stress Map Project, [8], these data are widely used in indirect modeling (e.g., [2]) and also in the direct approach [14] developed on the basis of the results obtained in [4]. Displacement orientations are available, for instance, from GPS observations and geological studies (it should be noted that GPS data can only provide changes in displacements and therefore their full magnitudes remain unknown, while the data on directions are more reliable).…”

“…It should also be noted that introduction of fictitious boundary conditions with the aim to fit experimental data on stress/displacement orientations results in a highly undetermined problem (see the proof in [9] for the case when principal directions are known at discrete points within the domain). Although formulations in terms of orientations also lead to non-uniqueness, in such cases, as shown in [4,5], the solution depends on a finite number of arbitrary parameters. These parameters actually specify the number of linearly independent solutions, which is vital for the problem formulated by means of force, stress and displacement orientations.…”

Introduction of STEM for stress analysis in statically determined bodies

“…However the number of possible solutions is finite if this BC is complemented by an additional BC, say by continuity of the stress vector. One type of BC for plane elastic body has been investigated in [17], where the curvatures of stress trajectories have been used as the second condition. It has been shown that this BVP can have finite number of linearly independent solutions (stress states) or have no solutions, which depends on, socalled, index of the problem that is determined from the analysis of the principal directions on the boundary (curvatures of stress trajectories do not affect the index).…”