Mixed‐mode delamination of layered structures modeled as Timoshenko beams with linked interpolation

Abstract

In this work, a finite-element formulation for modeling mixed-mode delamination in layered structures, consisting of two-node Timoshenko beam finite elements with quadratic linked interpolation and corresponding 4-node interface elements is presented and compared to a more common approach where linear Lagrange interpolation is used. The principal novelty of the proposed approach is that the vertical displacements of the beam elements, as well as the transversal relative displacements of the interface elements, are interpolated using a quadratic linked interpolation that also takes into account the nodal cross-sectional rotations of the beam elements. At the same time, the axial displacements and cross-sectional rotations are interpolated using linear Lagrange polynomials. A bilinear cohesive zone model is embedded in the interface finite elements for delamination modes I and II. Numerical analyses based on the examples from the literature with metal joints show that the formulation with quadratic linked interpolation improves the convergence and robustness of the solution with respect to the approach with linear interpolation. On the other hand, in case of composites with stiff adhesives this formulation exhibits a peculiar behavior with spurious oscillations of the normal interface tractions that leads to a poor performance in mode-I and mixed-mode tests. This problem can be easily solved by canceling the quadratic term in the interpolation function and using the standard Lagrange interpolation in such cases.

Hexahedral finite elements with enhanced fixed‐pole interpolation for linear static and vibration analysis of 3D micropolar continuum

“…In the so-called linked interpolation, the displacement field is interpolated using a one-degree higher polynomial than the polynomial that interpolates the rotational unknowns, and it has been widely used and thoroughly investigated in finite-element applications of the Timoshenko beams 33,36,38,39,66 and the Reissner-Mindlin plates. 42,54 The linked interpolation considered here is in its general, problem-independent, form for a beam with m nodes presented in Reference 32 as…”

“…In the so‐called linked interpolation, the displacement field is interpolated using a one‐degree higher polynomial than the polynomial that interpolates the rotational unknowns, and it has been widely used and thoroughly investigated in finite‐element applications of the Timoshenko beams 33,36,38,39,66 and the Reissner–Mindlin plates 42,54 . The linked interpolation considered here is in its general, problem‐independent, form for a beam with $$$ m $$$ nodes presented in Reference 32 as $$$$ {\mathbf{u}}^h\left({x}_1\right)\kern0.5em =\sum \limits_{i=1}^m{N}_i\left({x}_1\right)\left({\mathbf{u}}_i+\frac{1}{m}\hat{\boldsymbol{\phi} -{\boldsymbol{\phi}}_i}{\mathbf{r}}_{o,i}\right), $$$$ derived from the analytical solution of the differential equation of the spatial Timoshenko beam.…”