Research On Crack Problems In Multilayered Elastic Media Based On Consecutive Stiffness Matrix Method

The problem of cracks in layered elastic media has a wide range of applications in engineering,especially in hydraulic fracturing engineering,which can increase the production of unconventional oil and gas.It is well accepted that one of the core components to model hydraulic fracturing is the ability to compute the elastic response of a pressurized crack intersecting a number of layers,each of which is assumed homogeneous and isotropic individually.Various numerical methods have been developed to compute the opening displacement of crack in the crack problems,such as displacement discontinuity method and classical boundary element method(direct method).The displacement discontinuity method(DDM)ensures an efficient computation,while it only applies to homogeneous medium or bi-materials(bounded two half planes),since the analytical fundamental solutions to the DDM only exist in the two cases.On the other hand,the direct method(DM)is versatile for general multilayered elastic media,but fails when the media contains an initial slit-like crack,of which one crack surface coincides with the other effectively.The detailed implementations of the DDM and DM for the crack problems under relatively complicated media conditions have not been found in current literatures.Thus,we have firstly derived the detailed procedures of the implementations of the two methods for the crack problems under relatively complicated media conditions.Then,we have proposed a DM-based method that divides the media along the slit-like crack surface,so that the DM that cannot be directly applied for crack problems turns out to be applicable.After that,we have derived a recursive formula that obtains a “stiffness matrix” for each layer by exploiting the chain-like structure of the system,enabling a sequential computation to solve the displacements on the crack surface in each layer “consecutively” in a descending order from the very top layer to the very bottom one.Thus,our method is named as consecutive stiffness method.In our method,the final system of equations only contains the unknown displacements on the crack surface and known boundary conditions,ensuring the efficiency of the method.The numerical examples demonstrate better accuracy and broader applicability of our method compared to the displacement discontinuity method and moreacceptable efficiency of our method compared to the conventional direct method.