Comparison of the stress and strain intensity factors for the corner area of the structure boundary

The stress-strain state of structures in areas with corner cut-outs and cuts of boundaries features the occurrence of areas of stress concentration and requires assessment of strength and reliability of facilities, which is a relevant task in engineering practice. Theoretical analysis of stress-strain state (SSS) of corner cut-outs zones of the area boundary is reduced to the study of singular solutions of the elasticity theory problem with exponential features. At that, the concept of stress or strain concentration in an irregular point of the area boundary is meaningless. This paper considers the stress-strain state in the vicinity of the top of the corner cut-out of the flat area boundary, which is recorded using the intensity factors as limit values of stresses and strains. We give two approaches for obtaining the limit values for stress and strain in the vicinity of an irregular point of the plane area boundary using the stress intensity factors and the strain intensity factors. The stress-strain state in the corner cut-outs zone of structures and buildings boundary recorded in the form of limit values of stresses and strains may further be used to determine and record the influence of changing the factors of intensity of stresses and strains on SSS of structures, which is a separate task of solid mechanics. The difference in the expressions of stresses and displacements obtained for limit values of stresses and strains determines practical significance of the work when carrying out experiments and at determination of critical values of stresses and strains.


Introduction
The stress-strain state (SSS) of composite structures in zones of connection of elements under the action of forced strains being ruptural along the line (surface) of the elements contact is characterized by exponential feature. The stress-strain state in the irregular point zone of the area boundary under the action of forced strains, in particular temperature ones, is determined by solution of the homogeneous boundary value problem of elasticity theory. Solution of the homogeneous elasticity problem in the vicinity of an irregular point on a particular line is reduced [1][2][3][4] . Solution of plane problems in the polar coordinate system is searched by separation of variables [3][4][5][6].
The exponential feature of the SSS depends on the shape of the area boundary -a cutout, a cut, the order of which depends on the eigenvalues of the homogeneous elastic problem [4][5][6]. Eigenvalues of the homogeneous boundary value problem depend on the shape of boundary, type of boundary conditions, mechanical characteristics of the area material and have an array of values [5,[8][9][10]. In the fracture mechanics, for a perfect mathematical cut, singular stresses are characterized by the stress intensity factors [3,5,7,8,[11][12][13][14][15][16][17][18]. The purpose of this work is to determine the stress-strain state in the zone of corner cut-out of arbitrary opening of the area boundary using factors of intensity of stresses and strains as the limit values of stresses and strains.

Problem formulation. Plane strain case
We consider here a homogeneous boundary value problem of elasticity theory for a flat wedge-shaped area with a symmetrical opening 2 in the vicinity of the corner point of the boundary -a top of the corner cut-out of an arbitrary opening [3,5,[11][12][13]. The solution of the problem of plane strain when moving in polar coordinates system is searched as follows:  are unknown functions of angle  to be determined,  is the unknown parameter. If we put relations (1) to the Lame equations, we obtain expressions for displacements and stresses in the vicinity of an irregular point of the area boundary [3,5,13] containing four unknown factors A, B, C, D.
When meeting homogeneous boundary conditions: 0 r       at    , we get two homogeneous systems of linear equations relative to the unknown constants A, B, C, D. Determinants of the systems have the following form: Subject to the conditions (2a), we obtain relations for factors A and C: Subject to the conditions (2b), we obtain relations for factors B and D: SSS in the vicinity of the top of the corner cut-out of the area boundary taking into account the differences of the roots of the characteristic equations (2a), (2b) will be written as follows:   is Poisson factor and shear modulus, respectively, The unknown factors , , , A B C Darbitrary constants to be determined -satisfy relations (3), (4). Let's consider SSS in the form of (5) with the help of limit values of stresses and strains in the vicinity of the top of the corner cut-out of the boundary of arbitrary opening area called the stress and strain intensity factors, like the stress and strain intensity factors in fracture mechanics [3,8,12,13].

Area stress intensity factors
For stresses   in expression (5), let's proceed to the limit at 0   : Let's denote limit values of stresses (6) as follows: Factors A and C taking into account relations (3) and designations (7) will be written as follows: To find factors B and D in expression (5) for stresses r   , let's proceed to the limit at 0   : Let's denote limit values of stresses (10) as follows: Factors B and D taking into account relations (4) and designations (11) will be written as follows:           (k-)sin 1 K 2 1 sin 1 1 sin 1 Taking into account found constants A, B, C, D in the form of (8), (9), (12), (13) and designation of limit values of stresses I K  , I I K  (stress intensity factors) in the form of (7), (11), the initial stress -strain state (5) in the zone of corner cut-out of arbitrary opening of the area boundary will be written as a function of the stress intensity factors.

The area strain intensity factors
Let's write the expression for strain   in the zone of corner cut of the area border:

Conclusions
The stress-strain state in the corner cut-outs zone of structures and buildings boundary recorded in the form of limit values of stresses and strains further may be used to determine and record the influence of changing the factors of intensity of stresses and strains on SSS of structures, which is a separate task of solid mechanics.