Issue 45

G. Gomes et alii, Frattura ed Integrità Strutturale, 45 (2018) 67-85; DOI: 10.3221/IGF-ESIS.45.06 68 an incremental analysis due to the repeated introduction of an artificial boundary leading to unwanted approximations along the interface between the sub-regions and the formation of larger systems of algebraic equations. The second authors proposed a technique based on the application of two boundary integral equations, for displacements and tractions, each applied to one crack surface. The latter technique, named DBEM, has the advantage of not requiring remeshing in the neighbourhood of the crack-tip, as required by the FEM and the sub-regions BEM. Other pioneering works in the field are those of Ingraffea et al. [3], Becker [4], Myiazaki et al. [5], Chen and Farris [6] and Bush [7], all of which are characterised by the application of the displacement boundary integral equation and the sub-regions approach. This paper deals with two-dimensional crack growth with the DBEM, using an in-house developed computer code called BemCracker2D for incremental analysis as well as for the calculation of stress intensity factors (SIFs) using the J-integral, the crack growth path and fatigue life. Two structural components under simple loading have been analysed, derived from experimental tests of fatigue crack propagation due to Miranda [8, 9], including their numerical results with the FEM codes Quebra2D [8, 9] and ViDa [10], for simulation of two-dimensional crack growth and the prediction of fatigue life, respectively, to support the BEM analysis. The functionality of these programs, as well as their advantages and disadvantages, will not be discussed here. Moreover, the whole treatment of the crack modelling and the visualisation of the crack path, the SIF diagrams, the f(a/w) curves and the fatigue life diagrams were conducted with the code BEMLAB2D GUI, a pre- and post-processing program that interacts with BemCracker2D . One main contribution of this work concerns the development of computer software for educational use that allows a complete numerical analysis of elastostatic engineering problems, ranging from CAD modelling ( BEMLAB2D GUI) to the visualisation of crack propagation results including the calculation of fatigue life through the BemCracker2D code, validated by experimental results and their FEM correlations This paper is organised as follows: initially, a brief review of the DBEM formulation for crack problems is presented, as well as the techniques used to obtain the SIFs, the crack-extension direction and fatigue life. This is followed by a discussion on the automation of the modelling process and calculations by the BemCracker2D code, together with visualisations by using the code BEMLAB2D GUI. The paper then presents the experimental methodology and the FEM models of Miranda [8], followed by numerical results with BemCracker2D and correlations between the DBEM and both experimental and FEM results. T HEORY he numerical technique adopted for the simulations is the dual boundary element method (DBEM) [11], which considers two independent boundary integral equations, for the displacements and tractions. Both integral equations use the same integration path for each pair of matching source points but are applied to distinct boundaries along the crack, giving rise to independent algebraic equations. These governing equations are described in more detail below. The displacement boundary integral equation, in the absence of body forces and assuming continuity of displacements at a boundary point x’ , is given by,     ' ' ( ', ) (x)d (x) ( ', ) (x)d (x) ij j ij j ij j c x u x CPV T x x u U x x t         (1) where i and j are Cartesian components;  ij (x’,x) and U ij (x’,x) represent Kelvin fundamental solutions for traction and displacement, respectively; CPV denotes Cauchy principal value, and c ij (x’) are geometric coefficients given by  ij /2 for a smooth boundary at point x’ , where  ij is the Kronecker delta. The integrals in eq. (1) are regular in case the distance r between the source point x’ and the field point x is non-zero. When r tends to zero, the fundamental solutions present strong singularities of order 1/r for  ij , and weak singularities of order ln(1/r) for U ij . In the absence of body forces and assuming continuity of displacements and tractions at a boundary point x’ on a smooth boundary, the stress components  ij are given by,               1 ' ', ', 2 ij ijk k ijk k x HPV S x x u x d x CPV D x x t x d x          (2) T