Issue 31

J. Lopes et alii, Frattura ed Integrità Strutturale, 31 (2015) 67-79; DOI: 10.3221/IGF-ESIS.31.06 71 still unsatisfactory. Essentially, it has to be considered in the following investigations that these residual stresses increase with lower testing temperatures as the difference between manufacturing and testing temperature increases. N UMERICAL MODELS Numerical techniques in fracture modelling here are two main techniques used to model fracture onset and fracture propagation: VCCT and CZM also known as cohesive elements, or in some literature as interface elements. The VCCT technique is based on the principle that when a crack extends for a small amount the energy released in the crack propagation is equal to the work necessary to close the crack. This concept was first introduced by Rybicki and Kanninen [18] and developed further by Krueger [19]. In this technique the values of G I , G II , and G III are computed from the nodal forces and displacements obtained from the solution of the finite element model. VCCT enables the calculation of these parameters in a single simulation. It does require however complex meshing techniques and an initial delamination. Therefore VCCT can predict crack propagation but not crack initiation. Mendes [20, 21] examined the failure criteria for mixed mode delamination in glass/epoxy and CFRP/epoxy specimens. The purpose of an extensive program of tests was to determine the inter-laminar energy release rate of mode I, mode II, mixed-mode I+II, and mode III. The tests were DCB, ENF, MMB, and ECT. Mendes calculated the energy release rate analytically, through the beam theory, and numerically through the VCCT technique. By comparing the experimental results with numerical simulations using VCCT Mendes concluded that the Power Law [22] criteria showed reasonable results in modes I+II, B-K [23] criteria had better results when Mode III was present. B-K extended to mode III as proposed by Reader [24] seems more convincing. Cohesive elements are a more recent technique than VCCT. The concept of the cohesive elements is actual finite elements that are intended to model the resin layer between ply interfaces. Cohesive elements are able to predict the onset and propagation of delamination without requiring pre-cracks. However, cohesive elements must be placed along all possible interfaces where delamination may occur. In the proposed method [25, 26] a softening law for mixed-mode delamination can be applied to any interaction criterion. The constitutive equation of the cohesive elements uses a single variable, the maximum relative displacement, to track the damage at the interface under general loading conditions. The material properties required to define the element constitutive equation are: i) the inter-laminar fracture toughness; ii) the penalty stiffness, iii) and the strengths. The B–K interaction law requires additionally a material parameter  that is determined from standard delamination tests [25]. These elements have zero thickness and typically are rectangular elements with two nodes at each vertex as shown in Fig. 4. Figure 4 : Zero thickness cohesive element. Figure retrieved from [25]. Ankersen and Davies [27] discuss some advantages and limitations of the cohesive elements by comparing two different constitutive laws of the cohesive elements: The bi-linear law and the exponential law. According to this research both constitutive laws are identical in delamination prediction. Exponential constitutive law is more appropriate to use with dynamic implicit solvers whereas the bi-linear constitutive law is more suited with explicit solvers. Mesh size is also a critical parameter in the cohesive elements technique due to the high stress gradients ahead of the cohesive zone. However for a sufficiently refined mesh the results are mesh independent. The main characteristics of the two techniques are summarized in the Tab. 2. In this research the numerical modelling is based on previous researches where cohesive elements were proposed and developed [25, 26, 28]. This is the main reason why the cohesive elements were used. T