Elastic Foundation Solution for the Energy Release Rate and Mode Partitioning of Sandwich Debonds by Use of the Timoshenko Beam Theory

Face/core debonding is a most common cause of failure in (tri-material) sandwich. These debonds can grow and completely delaminate the face sheet posing a threat to the structural integrity of the component. The aim of this paper is to present closed form expressions for the energy release rate and mode partitioning of face/core debonds in sandwich composites using elastic foundation analysis with the goal of enabling the design of damage tolerant structures as well as that of appropriate tests. The elastic foundation analysis used here is comprehensive and includes the deformation of the substrate part (core & bottom face) and is done for a general asymmetric sandwich construction. In tri-material sandwich beams particularly the ones with transversely flexible core, the shear deformation effects are significant. In order to capture this, the Timoshenko beam theory is used, which introduces an additional degree of freedom. A finite length sandwich is treated as having a “debonded” section where the debonded top face and the substrate are free and a “bonded” section where an elastic foundation is used between the face and the substrate. The interaction between the face and the substrate in the “bonded” section is modeled using both normal and shear spring distributions to account for vertical and rotational degrees of freedom. A Double-Cantilever Beam (DCB) specimen with external loads and moments is chosen to demonstrate the procedure and verify the results. Appropriate boundary and continuity conditions are used to solve the governing equations and obtain expressions for the displacement field. The energy release rate is obtained by use of the J-Integral, and the expression is derived in terms of the applied forces and moments. Another advantage of using the elastic foundation theory is that the energy released during crack growth can also be obtained from the energy stored in the springs due to deformations in the beam. The mode partitioning damage tolerant structures defined in the context of elastic foundation makes use of the ratio of axial and transverse displacements near the crack tip. For verification purposes, results from finite element analysis (ABAQUS) are produced by using isoparametric 8-node bi-quadratic plane stress elements (CPS8R) to model the sandwich beam. In ABAQUS, singular elements are used near the crack tip and the evaluation of the J-Integral and the stress intensity factors is done via a contour integral approach. The energy release rate values from both the J-integral method and the spring energy method show excellent agreement with the corresponding values from the finite element analysis. The mode partitioning measure used here approach also provides values close to the traditional mode mixity values obtained from the  stress intensity factor approach from ABAQUS. Both the energy release rate and mode partitioning results are also compared with the corresponding ones from an earlier study, in which the Euler-Bernoulli theory was used to model the face and substrate parts.

Acknowledgment

The financial support of the Office of Naval Research, Grant N00014-16-1-2831, and the interest and encouragement of the Grant Monitor, Dr. Y.D.S. Rajapakse, is greatefully acknowledged.