CONCENTRATION OF STRESSES NEAR V-SHAPED ELEMENTS

Abstract

The study of stress concentrators, especially those that cause the formation of singularities of σ→∞ form, is of great scientific and applied importance. The most famous representative of such stress concentrators is the crack, which is the subject of study of fracture mechanics. Cracks can be normal, located in a homogeneous material, interfacial, intersecting the phase plane between two different materials. Depending on this, stress fields are formed with different features. Another class of similar stress concentrators is V-shaped elements that form a dihedral angle with a disclosure angle greater than zero. V-shaped elements can be formed in both homogeneous and dissimilar materials. The singularity features depend on the mechanical properties of the materials and the angles occupied by these materials. If the opening angle is zero, then the corresponding formulas are transferred to the well-known formulas for cracks. The next class of stress concentrators forming the singularity is the internal elements with V-shaped material boundaries. Such elements are formed around non-metallic inclusions with sharp corners, in fibrous composite materials at the ends of the fibers. Unlike the previous class, in this case there are no empty sectors. However, the resulting formulas transfer into the well-known expressions, if the mechanical properties of one of the materials tend to zero. Thus, all classes of tasks related to the formation of a singularity of a σ→∞ form can be considered as one common task with an increased number of parameters. Such work is done in this article. This made it possible to obtain new patterns of stress state and to determine the directions for the further development of mechanics destruction. As an example, using the complex potentials method with the application of the Kolosov-Muskhelishvili equations, a theoretical analysis of the stress state at the top of the V-shaped boundary separating two homogeneous materials with different normal elastic moduli E1 and E2 was carried out. The calculation showed that in the local zone near the top of the angle, the straightened state acquires a singular character, the stress distribution is described by terms of the form σ ≈ К/rλ, and the parameters of the singularity λ1, λ2, λ3 depend on the ratio of the elastic characteristics of the materials. Relevant patterns are studied.