Chapter 9 - Analytical methods for guided waves in aerospace composites

Abstract

This chapter discusses analytical methods for predicting guided waves in aerospace composites. The guided wave is modeled as an assembly of partial waves that share the same wavenumber parallel to the plate mid-surface but have different wavenumbers in the thickness direction. The problem setup is initiated by assuming six partial waves and imposing the coherence condition (generalized Snell’s law) wrt the guided-wave propagation. The resulting Christoffel equation yields a nonlinear wavespeed-dependent eigenvalue problem. For every wavespeed, the constrained Christoffel equation yields six eigenvalues and eigenvectors that are used to assemble the displacement and stress field matrices. These are multiplied by the six unknown partial-wave participation factors to assemble the physical displacements and stresses in each composite layer. The boundary conditions at the plate upper and lower faces and at the interfaces between the N layers yield a linear problem with 6N unknowns. If the 6N×6N problem is solved directly, then one has the global matrix method (GMM). However, if the GMM problem is too large to be solved directly, then one can apply a method of layer-by-layer transfer to reduce the problem size. If the transfer involves the displacement-stress state vector, then one has the transfer matrix method (TMM). The size of the TMM search determinant is 3×3 but it subject to numerical instability. If the layer transfer involves a recursive stiffness algorithm, then one has the stiffness transfer method [stiffness matrix method (SMM)] which has a 6×6 search determinant but a better numerical stability. In any case, the resulting search function is used to find the wavenumber–frequency–wavespeed combinations that give the guided-wave modes. The chapter contains a number of worked-out examples of solving the constrained Christoffel equation as well as the discussion of two software packages: (1) the GMM-based DISPERSE software from Imperial College, London, United Kingdom and (2) the GMM–TMM–SMM Dispersion Calculator software from ZLP-DLR in Augsburg, Germany.