Brittle failure of incompressible material in plane strain

The stress–strain behavior of rock mass under plane strain is studied in the cases of incompressibility and perfect brittle failure. The characteristics of of differential equations of equilibrium and the relations at them are obtained using the condition of coaxiality of the stress and strain tensors. The boundary problem is formulated for determining the stress–strain behavior in the failure zone of rocks. By way of illustration, the equations of the perfect brittle post-limit deformation of rock mass in the form of a rectangular plate (pillar) under uniform compression are analyzed. It is shown that by monitoring displacement of the side boundary of the plate, it is possible to predict the plate failure.


Introduction
The stress-strain curve describes deformation resistance of a material [1][2][3][4]. An increase in the stress means the growing resistance; a decrease in the stress implies that the strength of the material drops, i.e. the material fails. The post-limit deformation curve may be smooth. The curve in the forma of a vertical straight line means perfect brittle failure [5][6][7][8][9][10][11]. Plane strain of a perfect brittle, incompressible material was studied in [8]. The stud used the incompressibility condition and two compatibility conditions of strains with derivatives of components of the rotation vector in the line of the axis z . Two characteristics of the system of differential equations for displacements and two relations at them, connecting the rotation vector component and the angle between the major axes of the strain tensor were obtained. However, the boundary problem was not formulated in [8], no equations were proposed for the determination of stresses. The present paper formulates the equations for stress estimation and the boundary problem to find both stresses and strains.

Determination of stresses and strain under plain strain deformation
Problem formulation. Let in medium in the Cartesian coordinates the equations below be fulfilled: x cos 2 , sin 2 , = p 2 where p Γ is the maximum tangent deformation: where Τ is the maximum shear stress (Figure 1).
It is assumed that during deformation and failure of the medium: where θ is the angle between the first major direction of the stress tensor σ Τ and axis x ; Ω is the angle between the first major direction of the strain tensor ε Τ and axis x . Equality (3) means that the major axes of the tensors σ Τ and ε Τ coincide during deformation of the medium.
In addition to (1)-(3), the compatibility of strains is conditioned to be true: where z ω is the rotation vector component: , , u v ω are the displacement vector components. Furthermore, it is assumed that the equilibrium equations are fulfilled: In [8] relations (1), (2) were inserted in (4), which produced:   (7) is hyperbolic and has the characteristics: , They have the same direction as the major axes of the strain tensor ε Τ . The relations at the characteristics are given by [5]: where , ξ η are the constants.
The problem solution needs knowing boundary values of the functions To this effect, it is assumed that the the boundary L the displacements are preset: where the coordinates x,y L ∈ . Calculation of the total differentials of u and v gives: Placement of (11) in (1), (2) where , u v s s ′ ′ are the derivatives of the displacements , v u along the tangent to L , and ϕ -\ is the angle between the normal to L and the axis x . From (13) we find: Formulas (14) Here, the right-hand sides are considered the known functions accurate to Τ . From the comparison of (16) and (7), the right-hand sides of (16) respectively equal the expressions below: When determined, the characteristics of (16) coincide with (8) The right-hand sides of (18) are the vector product: This means that when integrating the relations at the characteristics, the integrals will depend on the integrating path as (18) where ϕ is the angle between the normal to L and the axis x . From (19)  Thus, the stress state in the failure domain is found using (20) and (18).
By way of illustration, we present the equations and the simplest solution for the problem on postlimit deformation of a plate in the plain strain under uniform compression. Let during loading:

Comments
It is unclear from the previous analysis when Γ reaches the value p Γ . In order to solve this problem, the previous solutions should be considered, for instance, in elasticity ( Figure 1). We have for elasticity: ε ε ε ω σ σ τ . The system has a unique solution from which the value Γ is found.

Conclusion
The stress-strain behavior is determined in the domain of post-limit deformation in perfect brittle failure of incompressible medium. The solution to the problem on uniform compression of a plate under conditions of perfect brittle failure is analyzed. It is show that pillar control in rock mass needs monitoring the change in the lateral displacement of the pillar.