PSI - Issue 54

Francisco Q. de Melo et al. / Procedia Structural Integrity 54 (2024) 585–592 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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the BEM, Boundary Element Method to evaluate the flexibility factors of each Line-Spring Element bearing on the Reissner deformation model. 2. Equivalent method to model the deformation over a part-through crack existing in a flat plate 2.1. Original method by Rice and Levy revisited Previous research about the deformation field in a plate containing a part-through crack lent important data for the development of alternative solutions modelling the behaviour of the residual ligament in a part-through crack existing in a flat plate. Each side edge cracked blade has extensional and edge rotation flexibilities at the crack plane as depicted in Figure 1. In this step, the line-spring flexibility matrix is developed, which contains factors inherent to the structural deformation of each line spring element. The method refers only to a linear elastic material behaviour and aims to replace the structural behaviour of the remaining ligament over the part-through crack by a set of elastic elements as mentioned above. Such line-spring element, when subjected to extensional or bending remote loads, can be characterized in a matrix form as follows: { } = [ 11 12 21 22 ] { } { } = [ 11 12 21 22 ] { } (1) In this equation, and are the extension displacement and relative or mutual crack edge rotation over and x − of crack reference basis, respectively. These displacements are related with the specific load vector , the extension membrane stress and m, the bending stress by the compliance matrix [ ] defined as: [ ] = [ 11 12 21 22 ] [ ] = [ 11 12 21 22 ] (2) The flexibility factors in [ ] , have been worked out with former investigation about the deformation of an edge cracked plate by (Gross and Srawley 1965), using the method of Boundary Collocation to define the flexibility factors of a side edge cracked element. In the original form, an equation equivalent to form (1) was deduced by (Rice and Levy 1972) as follows: { } = 2(1− 2 )ℎ [ 6 ℎ 6 ℎ ] { } (3) where and are, respectively, the Young modulus, the Poisson’s factor of the plate material and is the plate thickness, or line-spring blade width. The previous compliance matrix is not symmetric, despite factors being symmetric, = . This is due to the specific loads acting on each line-spring element ends. In fact, such forces are respectively, the extension membrane stress and the bending stress . It is more coherent to represent those specific efforts by the effective values of tensile force and bending moment by the following equation: { } = { 6 /ℎ /ℎ 2 }=[ ℎ 1 0 06] { } (4) We recall that and , are respectively, the membrane stress and bending stress in a plate having a section × , where “ ” refers to a unitary thickness over − , as in Figure 1 and is the line-spring width, or plate thickness over − . Substituting the equivalent values for the RHS vector in (3) , a new expression is deployed, now with a symmetric flexibility matrix: { } = 2(1− 2 )ℎ [ ℎ 6 ℎ 2 6 ℎ 2 36 ℎ 3 ] { } (5)