Issue 41

M.F. Funari et alii, Frattura ed Integrità Strutturale, 41 (2017) 524-535; DOI: 10.3221/IGF-ESIS.41.63 529           1 1 1 1 1 1 1 1 1 1 0 0 i i i i i i i i i i i i X , at X ,L , X X X X X X X X                    (7) It is worth noting that   i i ,     can be considered as variable quantities for each crack path, that should be determined during the crack evolution. In particular, from the physical point of view, they represent the portion in which the traction separation laws are distributed. Since those regions are assumed to be moved rigidly, by means of the ALE strategy, the nonlinearities involved by the traction forces may be reduced to a small region close to the crack tip, avoiding as a result spurious and oscillatory effects typically documented in pure CZMs. From the numerical point of view, displacements of the debonding regions are determined on the basis of the values of the fracture function at their extremities by enforcing that during the crack growth a null value of the fracture function on the debonding region is achieved. Therefore, by using a linear approximation function along the debonding region, the current crack tip displacement can be expressed by means of the following relationship:       1 1 1 0 0 0 i i f i i i i i T T f f i i i i i f f g X X g , X g g X g X                    (8) N UMERICAL IMPLEMENTATION overning equations introduced in previous section are formulated by means of a numerical formulation based on the Finite Element (FE) approach. In particular, the derivation of the FE starts from the principle of virtual works in terms of displacements, integrating the equations on the volume of the elements. A Lagrange cubic approximation is adopted to describe both displacement and rotation fields, whereas linear interpolation functions are utilized for the axial displacements. Moreover, for the variables concerning moving mesh equations, quadratic interpolation functions are assumed to describe the mesh position of the computational nodes. The proposed approach takes the form of a set of nonlinear differential equations, whose solution is obtained by using a customized version of the finite element package Comsol Multiphysics combined with MATLAB script files [16]. The model can be solved in both static and dynamic framework, taking into account the time dependent effects produced by the inertial characteristics of the structure and the boundary motion involved by debonding phenomena. In both cases, since the governing equations are essentially nonlinear, an incremental-iterative procedure is needed to evaluate the solution [17]. In the case of static analysis, the resulting equations are solved by using a nonlinear methodology based on Newton-Raphson or Arc-length integration procedures. In the framework of a dynamic analysis, the algebraic equations are solved by using an implicit time integration scheme based on a variable step-size backward differentiation formula (BDF). A synoptic representation of the numerical procedure as well as the computational algorithm implemented in the FE environmental program are reported in Fig. 3. R ESULTS n this section, results are developed with the purpose to verify the consistency and the reliability of the proposed model. At first, a layered structured formed by four mathematical layers and three intact interfaces are investigated in the static framework. The main aim of the present analysis is to validate the proposed procedure to predict the onset conditions and crack growth for a case involving multiple debonding mechanisms. Subsequently, in order to validate the procedure to describe the crack front speed, dynamic debonding mechanisms concerning a FRP strengthened steel beam specimen have been investigated by means comparisons with numerical results arising from the literature. G I