13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- where ( ( ,0, )) r z u u = u and σ are, respectively, the axisymmetric versions of the displacement vector and the stress tensor referred to the cylindrical polar coordinate system ( , , ) r z θ and Δ is the displacement of the test plate. In addition, the regularity conditions require that uand σreduce to zero as either r or z →∞. The mixed boundary value problem in elasticity defined by the set of equations (1) is a classical problem solved by Boussinesq [26] employing results of potential theory and by Harding and Sneddon [27] using the theory of dual integral equations. Details of the methods of solution are also given in [21-25] and [28, 29]. The result of interest to geomechanics is the relationship between the indentation displacement ( )Δ and the corresponding axial load( )P required to achieve the indentation. This can be obtained in exact closed form as (1 ) 4 P a ν μ − Δ= (2) where μ and νare, respectively, the linear elastic shear modulus and Poisson’s ratio of the geomaterial. As is evident from (2), the classical analysis of the plate load test provides only an estimate of /(1 ) μ ν− and additional information is needed to determine the parameters separately. When the plate adheres to the surface of the geomaterial, the resulting boundary value problem is described by the following boundary conditions: ( ,0) , (0, ) ( ,0) 0, [ , ); ( ,0) 0, (0, ) ( ,0) 0, ( , ) z zz r rz u r r a r r a u r r a r r a σ σ =Δ ∀ ∈ = ∀ ∈ ∞ = ∀ ∈ = ∀ ∈ ∞ ; ; (3) This mixed boundary value problem can be examined by appeal to the theory of integral equations where the problem can be reduced to the solution of the Hilbert problem involving singular integral equations. The elasticity problem of adhesive contact between a plate and an elastic halfspace region was examined by Mossakovskii [30] and Ufliand [31] and the exact closed form result is given by (1 2 ) 4 ln(3 4 ) P a ν μ ν − Δ= − (4) The Hilbert problem approach accounts for the oscillatory form of the stress singularity at the boundary of the rigid plate. Selvadurai [32] also examined the mixed boundary value problem defined by (3) but by replacing the oscillatory form of the stress singularity by a regular 2 2 1/2 ( ) a r − − type singularity, thus reducing the problem to the solution of a Fredholm integral equation of the second-kind. It was shown that the difference between the exact result based on the Hilbert problem formulation and the Fredholm integral equation formulation is less than 0.5% when 0 ν= and the results converge when 1/2 ν= . A further classical development is to consider that the entire surface of the halfspace is composed of an inextensible membrane, in which case the bonded boundary condition is automatically satisfied in the indentation zone and the shear tractions are non zero beyond the indented zone. The load-displacement relationship of the indenter can be obtained from the result for the problem of a rigid disc embedded in an elastic infinite space [33, 34]: i.e. (3 4 ) 16 (1 ) P a ν μ ν − Δ= − (5) It should be noted that in the limit of material incompressibility, (2) and (4) reduce to the same result. The analysis can be extended to include Coulomb friction at the contact zone [35] and the influence of depth of embedment of the test plate [36, 37].
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