The finite element discretized symplectic method for composite mode III cracks

• Introducing a new finite discretized symplectic method to evaluate interface cracks.
• Anti-plane variables near the crack tip are written in the form of symplectic series.
• Eigenfunctions of displacements are used to reduce the computational cost.
• Exact solutions near the crack tip and SIFs are obtained without post processing.
• A reinforced structure with complex geometry in engineering practice is analyzed.

A finite element discretized symplectic method is proposed to compute the stress intensity factors (SIFs) of interface cracks in multi-material composites subjected to anti-plane loading. The whole body is divided into a near field containing a crack to be solved analytically and a far field to be solved by conventional finite elements. In the near field, analytical series solutions are found by the Hamiltonian formulation. The unknown displacements of the densely populated finite elements are transformed to a handful of unknown coefficients of the series. After combining with the far field, the SIFs are actually the first few coefficients.