PSI - Issue 28

Carlos D.S. Souto et al. / Procedia Structural Integrity 28 (2020) 146–154 Carlos D.S. Souto et al. / Structural Integrity Procedia 00 (2020) 000–000

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procedure requires that the finite element mesh is composed of well oriented elements, in such a way that the contour Γ is defined along ξ = const., where ξ is one of the element’s natural coordinates. This means that the elements must be orientated in a “polar” way around the crack tip, as shown in Figure 2b. This requirement is not always easily guaranteed, so this procedure, in practice, has limited applications. Another procedure to numerically calculate J is the so-called Equivalent Domain Integral (EDI), where a contour integral is transformed into an area integral through Green’s theorem. This more practical procedure (depicted in Figure 2c) is used, for example, in the commercial finite element software Abaqus (2017). In this article, the procedure evaluated and integrated in Omicron to calculate the J-integral is the one presented by Monteiro (1984): ( W i + W i + 1 ) · ( y i + 1 − y i ) − ( σ xx , i xx , i + σ xx , i + 1 xx , i + 1 ) · ( y i + 1 − y i ) − ( τ xy , i γ xy , i + τ xy , i + 1 γ xy , i + 1 ) · · ( y i + 1 − y i ) − ( σ yy , i yy , i + σ yy , i + 1 yy , i + 1 ) · ( y i + 1 − y i ) + ( τ xy , i + τ xy , i + 1 ) · ( u i + 1 − u i ) − ( σ yy , i + σ yy , i + 1 ) · · ( v i + 1 − v i )] (9) Where, for the node k , W k is the strain energy density, σ i j , k and i j , k are, respectively, the stress and strain tensors, u k and v k are, respectively, the displacements along x and y ( xy is a global referential). This means that the presented expression approximates the value of J through the sum of nodal quantities, where the contour Γ is defined by n nodes of the finite element mesh. With this approach, some approximations are made since in a finite element model the stress and strain tensors are obtained in the element’s integration points and not on the finite element mesh nodes (an extrapolation is made between the integration points and the mesh nodes), by consequence, the strain energy density at the nodes is also an approximation. This approach also assumes that the quantities involved in the expression vary linearly along a contour element, i.e. between 2 nodes of the contour. However, this procedure is the most simple to implement in a finite element software and lets the user easily choose an integration contour simply by selecting the finite element mesh nodes that define it. J = n − 1 i = 1 1 2

3. Implementation of the numerical approaches in the Omicron software

As mentioned above, the presented procedures for the numerical determination of stress intensity factors were implemented in Omicron (2019), a finite element analysis software developed by C. Souto.

Fig. 3: Omicron’s user interface.