SUMMARYIn this work, a 2D finite element (FE) formulation for a multi-layer beam with arbitrary number of layers with interconnection that allows for mixed-mode delamination is presented. The layers are modelled as linear beams, while interface elements with embedded cohesive-zone model are used for the interconnection. Because the interface elements are sandwiched between beam FEs and attached to their nodes, the only basic unknown functions of the system are two components of the displacement vector and a cross-sectional rotation per layer. Damage in the interface is modelled via a bi-linear constitutive law for a single delamination mode and a mixed-mode damage evolution law. Because in a numerical integration procedure, the damage occurs only in discrete integration points (i.e. not continuously), the solution procedure experiences sharp snap backs in the force-displacements diagram. A modified arc-length method is used to solve this problem. The present model is verified against commonly used models, which use 2D plane-strain FEs for the bulk material. Various numerical examples show that the multi-layer beam model presented gives accurate results using significantly less degrees of freedom in comparison with standard models from the literature.

In this work we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linear-elastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the case when the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, . This formula, compared to the so-called ‘standard beam theory’ formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler–Bernoulli beam theory. In the numerical examples we compare results for Euler–Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters.

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