Load Transfer Problem and Boundary Collocation

The acoustic and mechanical properties of cemented granular materials such as sedimentary rocks are directly related to the load transfer problem between two granules (Stoll). The theoretical description of granular materials has been based on the Hertzian contact problem between two elastic spherical inclusion in an elastic matrix, or its modifications; a review of the contact problem can be found in Johnson. In essence, the deformation problem resulting from a relative displacement between two nearby spherical elastic inclusions is studied, and the load transfer between the two is used to construct a constitutive theory for the particulate solid. In particular, Dvorkin et al. studied the deformation of an elastic layer between two spherical elastic grains, using a two-dimensional plane strain analysis similar to those of Tu and Gazis and Phan-Thien and Karihalo. They concluded that the elastic properties of the cemented system can depend strongly on the length of the cement layer and the stiffness of the cement. The main problem with the method is the assumption that the contribution to the load transfer between the granules comes from the region near contact. The assumption is well justified in the case where the Poisson’s ratio of the cement layer is 0.5 (incompressible), in which case the problem is equivalent to the corresponding Stokes flow problem where exact and asymptotic solutions are available (see, for example, Kim and Karrila). The Stokes asymptotic solution shows that the leading term in the load transfer is of O(є-1), where є is the dimensionless thickness of the cement layer. In the case where the Poisson’s ratio of the elastic layer is less than 0.5, it is not clear that the load is still strongly singular in є, and therefore a local stress analysis in the region of near contact may not necessarily yield an accurate answer, unless є is extremely small. The load transfer problem is pedagogic in that it allows us to demonstrate an effective technique often used in Stokes flow known as the reflection method, which has its basis in Faxén relations (discussed in the previous chapter).