Stress-Strain State in Elastic Body with Physical Cut

We consider the problem of distributing the stress-strain state (SSS) characteristics in the body arbitrarily loaded on the outer surface and weakened by a physical cut with a thickness of 0  . It is assumed that 0  parameter is the smallest possible size permitting the use of the hypothesis of continuity. The continuation of the physical cut divides the body into two parts interacting with one another by means of a contact with  -layer. Due to constant average stresses and strains over the layer thickness, the problem reduces to the system of variational equations for the displacement fields in the adjacent bodies. The geometry of the bodies under consideration has no singular points and, as a consequence, has no singularity of stresses. The use of average characteristics makes it possible to disregard a form of the physical cut end. The obtained solution can be used for processing of experimental data in order to establish the continuity scale 0  . The entered structure parameter for silicate glass is assessed using known mechanical characteristics.


Introduction
Strength calculations for structural parts and elements with various stress concentrators as part of the classical concepts of continuum mechanics (CM), as a rule, lead to unreal stress values in the neighbourhood of singular points in terms of strength characteristics.This is caused by the use of the hypothesis of continuity.While the cut curvature radius is large enough in comparison with the crystals of the matter, it has no effect on the stress distribution, but if the curvature is commensurate with the crystal sizes, the questions of whether it is reasonable to use the classical theory of elasticity arise.
Note that the crack in a solid body naturally generates a stress concentrator and, in this case, consideration of the medium structure permits eliminating some contradictions in the model representation related to the singularity of the stress field in the singular points.However, the question is how to determine the average characteristics on the entered structural elements.In this case, two approaches can be distinguished.The first one [1][2][3] uses singular solutions of the theory of elasticity for the crack model in the form of a mathematical cut, and averaging over the entered generic element is carried out on their basis.The second one [4][5][6][7] attributes a homogeneity property of the stress-strain state (SSS) to the structural element in a particular direction (e.g., orthogonally to the supposed direction of fracture) and the coupled problem [6,7] on SSS determination, both in the structural element and in the medium adjacent to it, is solved, where CM classical solutions are deemed to be feasible.Thus, in the paper [8], a layer with characteristic thickness 0  is singled out assuming the strain homogeneity through its thickness in the crack development trajectory.However, the paper [8] does not include 0  parameter esti- mations and statements of the relevant problems.
The model presenting the crack as a physical cut with a thickness of 0  is proposed in the papers [4][5][6][7][8] for SSS determination in the bodies with cracks.In addition, the model also includes a material layer on the extension of the cut.Material adjacent to the layer can be regarded within the framework of the classical CM concepts, using the layer boundary stresses as boundary conditions.The stress state of the layer is described by average and boundary stresses connected by equilibrium conditions [6,7].The use of the average characteristics allows not considering the geometry of the physical cut transition to the material layer.Defining relations within the layer are considered for average stresses and strains.The singularity of the physical cut model can be excluded by introducing a definite form of its end, i.e. a part of a circle or an ellipse.However, this case raises the question of the corresponding curvature radiuses.The book [9] deter-mines an experimental dependence of К IC on the sharpness (curvature radius of the notch base) of the stress concentrator.The obtained dependence shows that К IC quickly falls with a decrease in the notch base radius until it reaches some threshold value.The further decrease in the curvature radius has no effect on К IC characteristic.It demonstrates the presence of some typical size which would make the fracture beginning independent of the cut end geometry.In our situation, the physical cut thickness is associated with this typical size, so there's no point in discussing a form of its end.The introduction of average characteristics over the layer thickness makes it possible to dismiss the questions related both to infinite stresses on the physical cut extension in the continuous medium and to a form of the physical cut end, so the corresponding boundaries are shown in Figure 1 with a wavy line.The boundary stresses associated with average equilibrium conditions [6] are also considered for the layer.The layer/medium conjunctions are established by the layer boundary stresses on the surface which has no singular points and, as a consequence, has no singularity.The article [7] includes solutions for infinite linear-elastic medium with the physical cut.The paper [6] for perfectly elastoplastic behavior of the layer material solves the analogue of the Dugdale problem [10].This article provides general statement of the problem of damaged finite body straining.

Problem Statement
Let us consider loading of the finite body with the physical cut with a length of а and a thickness of 0  according to the diagram in Figure 1.The X-axis of the Cartesian system is to be associated with the direction of the cut, and the reference point is to be associated with its middle.Following the model presentation [4][5][6][7], let us consider the material layer overlying the extension of the physical cut.The corresponding areas are numbered 3  , , , P P P P , , , Q Q Let us use the following designations for the layer boundary stresses: We assume that the stress vectors on the layer adjoint boundaries are equal and opposite to the stress vectors of the body adjoint boundaries: We have a rigid coupling between the boundaries: And continuous displacements along the layer boundaries."b" index is related to the body areas adjacent to the layer.
Let us define average stresses, strains and displacements in the layer using their boundary values as follows:           We derive the expression of average shear strain along the layer from the expressions ( 8) and ( 9): Let us consider equilibrium condition for the body dispose out of the cut, using: where is work of external surface loads; are work of internal stresses in the relevant areas of the body.
where  P u u is vector of mean displacements on end; h is body thickness in the direction orthogonal to the plane of the formulae ( 10), (11), the expression (14) may be written as: Let us consider the work of internal stresses in area 3: where ,   are tensors of the layer average stresses an the average stress and strain tensor ( d strains; 3 S is layer area BD'DB' Using expressions (6), ( 7), (12) and the symmetry of The work of the internal stresses in area 4 may be obtained in a similar way: ), the work (16) may be presented as: Using the system (13) -(18), we obtain: The work of the distributed external load may be presented as: Using (10), (11), we can derive from the last equation (20) On the analogy with (20), the distributed external load may be presented as: and for body 2: Fo he relations between the aver- r the layer material, t age stresses and strains may be presented in the form of Hooke's law for the case of plane strain:  is Poisson's ratio.
Let us group the summands in the expression in relation to where Relation (23) may be written in a similar way: Specific virtual work of stresses in the bodies 1 and 2 located outside the layer is determined in accordance with Hooke's law (24)-(26) through the displacement field as follows: The system of variational Equations ( 27) and ( 28) with regard to the expression (29) allows us to determine the displacement field in the bodies 1 and 2 including displacement along the boundaries of the la e most ob

Determination of the Characteristic Size
The main problem of the proposed statement is the determination of .Alternatively, the entered parameter may be found ing the scheme of loading with concentrated load as shown in Figure 2 for brittle materials.
Due to the problem symmetry, it is sufficient to consider one half of the body 1.In this case, we have the following conditions for the layer boundaries: yer.The finite element method seems to be th vious way of solution this problem.
With regard to the conditions (30) and (31), the variational relation (27) for the right half of the body 1 may be written as: Under the following boundary conditions:  32) obtained by means of the finite element method.We used the ratic approximation of the displacement field on the element, the element size in the neighbourhood of po t O was equal 0  .
, follo Glass g strength wing the paper [12], varies from 1.5 × 10 7 Pa for severely damaged glass to 1500 × 10 7 Pa for undamaged glass.Furthermore, the latter value is close to the lower limit of the theoretical strength of glass, which varies within It is known that the glass tens less than the compression strengt cludes the bend loading of ac Further, we assum glass is 3 equal to about half a millimeter as shown in the paper [3].
The article [4] contains the following relationship for brittle materials:

Conclusion
Based on the results of the experiments for specific el tem of Equations ( 27) and (2 tion of SSS characteristics in in ach allows us to avoid the singularity of stresses and strains in the crack end in contrast to the classical representation of the crack in the form of the mathematical cut.interactive layer thickness astic materials, using the sys-8), we can find the distributhe finite body with a crack the form of the physical cut.The proposed model allows performing calculations for arbitrary external load.This appro

Figure 2 .
Figure 2. Scheme of loading with concentrated load.To define a value 0  , we can use the linear relationship in the elastic ween power P and SSS characteristics in the local area.We can adopt strain field bet

Figure 3 .
Figure 3. Dependence of breaking stress in the cut top on parameter δ 0 for critical force.