Issue 41

M.F. Funari et alii, Frattura ed Integrità Strutturale, 41 (2017) 524-535; DOI: 10.3221/IGF-ESIS.41.63 525 In order to simulate debonding phenomena in layered structures, several approaches have been proposed in the literature. However, among the most important ones, Fracture Mechanics (FM) and Cohesive Zone Model (CZM) are widely utilized in practice [4]. In FM, the total energy release rate and its individual mode components need to be evaluated to predict delamination growth. For general configurations, the energy release rates can be computed by using a very accurate mesh of solid finite elements and the Virtual Crack Closure Method (VCCM) [5]. Such models calculate the energy release rate as the work performed by the internal traction forces at the crack faces during a virtual crack advance of the tip. Moreover, in dynamic Fracture Mechanics, the VCCM is applied by using the modified form, in which the ERR, during the time evolution, is evaluated by the product between the reaction forces and the relative displacements at the crack tip and at the nodes closer to the crack tip front, respectively, [6,7]. The prediction of the energy release rate is strictly dependent from the mesh discretization of the crack tip. Alternatively, cohesive models propose an easy way to simulate debonding phenomena including also crack onset. Distributed or discrete interface elements are introduced between continuum elements based on traction separation damage laws. However, such modeling is strictly dependent from the mesh discretization since the direction of crack propagation is restricted by the element size and orientation adopted by the user [8]. Moreover, the presence of an initial finite stiffness may produce in brittle solids an excess of compliance and, in those cases in which a high stiffness is introduced, spurious traction oscillations [9]. Such problems may be partially circumvented by introducing a very fine discretization at the crack tip front, to obtain a high resolution of the characteristic fracture length of the interface [10]. However, the resulting model is affected by computational complexities, because of the large number of variables and nonlinearities involved along the interfaces. In order to avoid such problems, combined formulations based on fracture and moving mesh methodologies are proposed [11,12]. In particular, the former is able to evaluate the variables, which govern the conditions concerning the crack initiation and growth, whereas the latter is utilized to simulate the evolution of the crack growth by means of ALE formulation [13,14]. It is worth noting that the use of moving mesh method, combined with regularization and smoothing techniques, appears to be quite efficient to reproduce the evolution of moving discontinuities. However, existing models based on ALE and FM are based on a full coupling of the governing equations, arising in both structural and ALE domain. In this framework, material and mesh points in the structural domain produce convective contributions and thus nonstandard terms in both inertial and internal forces. In the proposed formulation, the use of a weak discontinuity approach avoids the modification of the governing equations arising from the structural model and thus a lower complexity in the governing equations and the numerical computation is expected. Despite exiting numerical methodologies based on pure CZM, the present approach reduces the nonlinearities involved in the governing equations to a small region containing the process zone, leading to a quite stable and efficient procedure to identify the actual solution in terms of both crack initiation and evolution. In order to verify the consistency of the proposed model, comparisons with existing formulations for several cases involving single and multiple delaminations are developed. The outline of the paper is as follows. At first, the formulation of the governing equations for the ALE and interface approach is presented and, subsequently, the numerical implementation of the finite element model is reported. Finally, comparisons and parametric results to investigate static and dynamic behavior of the debonding phenomena are proposed. T HEORETICAL FORMULATION OF THE MODEL he proposed model is presented in the framework layered structures, in which thin layers are connected through adhesive elements. In particular, a shear deformable beam model with a proper number of mathematical elements along the thickness direction is utilized to reproduce a high order zig-zag kinematic (Fig.1). However, each layer is connected to the adjoining ones by means of imperfect interfaces, in which debonding phenomena may affect the adhesion between layers introducing material discontinuities and traction forces along normal or sliding directions. In order to simulate debonding phenomena, a fundamental task to be achieved is to identify the position, in which the onset of interfacial mechanisms is produced and subsequently to simulate the evolution of the cracked length. The theoretical formulation is articulated into two steps, which are presented separately. At first, when the crack onset condition is not satisfied, the moving mesh elements are inactive and only the structural problem is considered. Subsequently, when the debonding process is triggered, the ALE elements are activated introducing moving traction forces which follows the process zone length. Governing equations The governing equations of the FE model are derived by means the principle of virtual works. In particular, for the general dynamic problem presented in Fig. 1 it is required that the total virtual work is stationary: T