PSI - Issue 2_A

Sascha Hell et al. / Procedia Structural Integrity 2 (2016) 2471–2478 S. Hell and W. Becker / Structural Integrity Procedia 00 (2016) 000–000

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0.2 real part of eigenvalue Re( λ i ) [-] SBFEM enrSBFEM 0.4 0.6 0.8

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a) c) Fig. 5. Exemplary boundary mesh of left part of the “post-cracked” configuration with n e = 12 elements along the long edge: (a) front view, (b) back view. Red lines / surfaces mark crack faces and red circles mark enriched nodes. (c) Comparison of the convergence of the first three eigenvalues λ which are not associated to rigid body motions for the standard and the enriched formulation of the SBFEM. b) number of elements along edge n e [-]

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10 relative displacement error || u || / || u ext || − 1 [-] SBFEM enrSBFEM −4 10 −3 10 −2 10 −1

10 relative strain energy error Π i / Π ext i − 1 [-] SBFEM enrSBFEM −4 10 −3 10 −2 10 −1

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a) c) Fig. 6. (a) Exemplary assembled boundary mesh of the “post-cracked” configuration with n e = 12 (back view). Comparison of the convergence of (b) the displacements || u || 2 at the upper corner opposite to the symmetry planes and the strain energy Π i for a homogeneous vertical loading. Fig. 5a,b show the employed boundary mesh of such a part with n e = 12 elements along the long edge of the body from the front and the back respectively. The boundary faces being coincident with the ones of the “pre-cracked” model are identically meshed. Again, the red lines mark crack faces and the red circles mark enriched nodes. This configuration yields 3 stress singularities with the following extrapolated singularity exponents Re( λ 1 − 1) = − 0 . 5574, Re( λ 2 − 1) = − 0 . 2983 and Re( λ 3 − 1) = − 0 . 1569. In Fig. 5c, the values obtained from the standard and enriched formulation of the SBFEM are given and again the excellent convergence properties of the enr SBFEM are confirmed. In the next step, we want to check the convergence properties of a complete boundary value problem. Accordingly, two of the introduced parts (fig. 5a,b) are assembled as depicted in fig. 6a. Please note that, in contrast to the former illustrations, we now only look at the considered body from the back. The boundary sti ff ness matrices are calculated and assembled as well and the resulting body is subjected to a simple vertical load. This would only have led to a simple homogeneous stress field in the “pre-cracked” configuration, but here, naturally leads to a crack opening of the added triangular crack extension. Fig. 6b illustrates the convergence of the magnitude of the displacements || u || 2 at the upper corner opposite to the symmetry planes. In Fig. 6c, the convergence of the strain energy Π i is shown in a double logarithmic plot. Again, it becomes evident that the enriched formulation of the SBFEM clearly excels the standard one, usually by at least an order of magnitude. Please note that already for only two elements along the diagonal of the crack extension faces ( n e = 12), i.e. as soon as there is at least 1 row of fully enriched elements around the crack extension front, the error in both displacement and strain energy is about 0.1% or less when using the enr SBFEM. Finally, we are going to calculate the incremental energy release rates for two special cases. In the first one, the boundary displacements and enrichment function coe ffi cients of deformation mode co1 (“pre-cracked” configuration) number of elements along edge n e [-] b) number of elements along edge n e [-]