The problem solution on wedge penetration in an initially anisotropic medium within the rigid-plastic scheme

Two problems are solved in the paper: on ultimate loads in the initial stage of indentation of an absolutely rigid smooth wedge into a layer of an initially anisotropic plastic medium and in the final stage when the tool penetrates through the layer. The problems are solved with Chanyshev’s constitutive relations of plasticity of the initially anisotropic medium based on use of the eigen elasticity tensors.

The process of wedge tool penetration in rocks is a subject of many researches [1][2][3][4][5][6][7][8]. Sokolovsky [1] studied wedge penetration in a rigid-plastic half-plane earlier investigated by Hill, Lee and Tupper. Some foreign researchers modeled stress state of soil under wedge introduction [2] and solved the problem on penetration of a cylinder in multi-layered soil [3]. The model of interaction between a cone and soil with regard to penetration resistance is presented in [4]. Gareeva [5] offers experimental relations for static probing by standard procedures and determines resistances on front and side faces of indenters. Relations for fining optimal parameters of a cone tool penetrating in soil are given in [6], and it is illustrated that the cone is the most rational shape.
This study deals with the penetration of a rigid smooth wedge in a layer of an initially anisotropic material (plain strain deformation). Two limit loads are determined: initial state of the edge penetration and final state when the wedge penetrates the layer and comes out from the other side (an analogous problem for isotropic medium is solved in [9].
Let in the coordinate system xOy the initial medium is deformed elastically by the Hooke law: where ij a -elasticity constants characterizing ductility of the material. The choice of the Hooke law in the form of (1) with four different elasticity constant is only governed by the simplified formula of the solution; the authors will include the rest independent elastic constants ij a (six!) of the plain strain deformation in the further studies.
Aiming the construct plasticity equations for the medium (1), a basis of tensors is introduced [10]: (2) In the basis (the scalar product is determined as resultant of ij ij ε σ with summing over repeated indices), the coordinates of the tensors ε σ T T , are, respectively given by: For the symmetrical matrix A in (4), the eigenvalues [11] are real: The eigenvectors, pursuant to the symmetry of the matrix A, are mutually orthogonal: Then, thee stress tensor σ T is expanded by the basis (7). As a result, we have the coordinates of the tensor σ T in the new basis: For the coordinates i S , i Ω in the basisв (7), there exists proportional elastic relations: where i λ -the eigenvalues (5). Apparently, Poisson's ratio is absent, i.e. a force directed along a tensor i T causes deformation only along this direction.
The ductilities 1 λ , 2 λ , 3 λ for the initially isotropic medium are distributed so that 2 3 1 λ λ λ > = and plasticity develops in a plane intersecting the tensors 1 T , 3 T where there are the highest ductilities.
In our case of (1), considering (9) and the hypothesis that 1 we assume that plasticity in the anisotropic medium (1) also develops in the plane 1 T , 3 T where ductilities are higher than in the line of 2 T .
The plasticity criteria in the case of K -constants which are the elastic limits in the medium (1) in the lines of the tensors 1 T , 3 T (in the plane 1 S , 3 S the plasticity conditions is a rectangle with the sides (10)). The idea that the eigen tensors of elasticity are the eigen tensors of plasticity, creep and failure belongs to Anvar Chanyshev [12].
Suing the common scheme [13], we analyze the system of equilibrium equations: together with the plasticity conditions (10). Using (11) and the plasticity conditions (10) allows finding the axes in xOy as the straight lines: β ctg ± = dx dy (12) (the angle β is determined from (6)) and the equilibrium equations along them: Combined with (10), this yields the stress distribution along the lines (12): Using the second plasticity condition (10) with (11) produces the stress distributions below: where ϕ , f -arbitrary functions of the appropriate coordinates x, y.
In layered anisotropic medium, the state (14) conforms with the plastic shearing of the layers relative one another. The formulas (13) describe plastic deformation of the layers.
Finally, we use (13), (14) in solving a problem on penetration of a perfectly rigid smooth wedge in a layer composed of the material (1).
The first illustration is given in Figure 1. A half-plane xOy has a recess BCD where a wedge with a nose angle γ 2 is placed. It is required to find the limit load on the wedge faces BC and CD such that material starts yielding along the slide lines (12) subject to the conditions (13).
Denoting the angle between the characteristics (12) and the axis x as α brings the case of a sharp wedge when α γ π > − 2 / as in Figure 1 where t  -unit vector of a tangent. Considering (15), we find the stress vector p  on CD: The second equation (17) With the known load on CD, we find the force   The implications of (26) are quite evident: the penetration depth is higher at the higher initial velocity 0 v ; the penetration depth is lower with the higher rigidity of the medium in the line of the normal drawn to the wedge; the penetration depth is higher with the higher ratio 0 / h L (bulkier tail of the tool); the penetration depth is higher with the higher density of the wedge. The most interesting is the dependence of the medium resistance n p on the medium properties (angle β ) and the wedge nose angle γ . This study includes calculations of the resistance versus the listed parameters.

Conclusion
In the framework of a rigid-plastic body, the authors have constructed the mathematical model of penetration of a rigid tool in an initially anisotropic medium. The maximum possible penetration depth has been determined. The dependence of the penetration depth on geometry of the wedge and on the anisotropic parameters of the medium has been analyzed.