A generalized finite element formulation for nonlinear frequency response analysis of viscoelastic sandwich beams using harmonic balance method

Harmonic balance method (HBM) is a popular computational tool for the nonlinear dynamic analysis of structural elements in the frequency domain. Its application in conjunction with the finite element (FE) procedure involves complexity in the formulation of the geometrically nonlinear equation of motion. Further complexity arises in the case of a viscoelastic structure as its constitutive model involves temporal derivative/integral of stress/strain. In this concern, the consideration of a few harmonic terms in HBM poses somewhat simplified formulation, but it may not provide a good theoretical estimation of nonlinear dynamics. Therefore, a large number of harmonic terms in HBM are to be considered despite the corresponding complexity, as well as a high computational cost. In this view, presently, two new formulation strategies are introduced toward a generalized FE formulation, especially for the consideration of an arbitrary number of harmonic terms in HBM. The first strategy lies in the formulation of the geometrically nonlinear stiffness matrix through a special factorization of the nonlinear strain–displacement matrix, while the second one lies in the analytical integration of system matrices/vectors over a time period by exploiting the orthogonality of Fourier basis functions. These formulation strategies provide not only the equation of motion with a reduced number of terms in the HBM-based expanded forms of system matrices/vectors but also a significantly reduced computational time. Additionally, various time–domain viscoelastic constitutive models are reduced into a generalized form for the periodic stress/strain to achieve a common HBM-based FE formulation for any of these viscoelastic material models.