Comments to the paper “Mesh sensitivity effects on fatigue crack growth by crack-tip blunting and re-sharpening” by V. Tvergaard [Int. J. Solids Struct. 44 (2007) 1891–1899

Abstract

Commented paper by Tvergaard (2007) addressed the mesh sensitivity in relation to the disagreement between his finite element (FE) simulations of fatigue cracks and those by Levkovitch et al. (2005) which predicted striations-like folded crack profile. Tvergaard (2007) found out no major effect of mesh refinement and attributed the disparity with Levkovitch et al. (2005) to the smoothness of meshes created by distinct remeshing procedures implemented by cited authors. To contrast these data, the results of extensive analysis of mesh sensitivity taking account of the FE size, aspect ratio, type and technology together with the role of constitutive model, are outlined here and accompanied with deductions about its origins. Plane-strain mode I crack under small scale yielding was simulated. The test-piece geometry and loading were the same as in previous analyses (Toribio and Kharin, 1998, 1999, 2000, 2006), where initial crack of the length a had parallel faces and semicircular tip of the width b0 = a/15,000. Within conventional J2 elastoplasticity guided by the power-law stress-strain relation, the Ziegler’s kinematic hardening as well as isotropic one and ‘‘equivalent’’ perfect plasticity were explored as outlined elsewhere (Toribio and Kharin, 2000). The reference mesh M1 was mainly the same as that with which the mesh-convergence was apparently achieved for non-hardening solid (Toribio and Kharin, 1998, 1999). Mesh sensitivity was analysed reducing the near-tip FE size to the half and quarter of the reference one (respective meshes M2 and M3), as well as varying FE aspect ratio, type and technology – interpolation order, shape (quadrilaterals Q and triangles T) and full and reduced integration (FI and RI). Fourand eight-node elements Q4 and Q8, and the crossed linear triangles arrangement Q4T3 were tried without remeshing. Loading route comprised up to 10 cycles. In all cases, prominent mesh sensitivity appeared with the commencement of bifurcating shear localisations, which were revealed using the plastic-strain rate as sensitive detector. For the same material model, finer meshes manifested earlier, sharper and more numerous localisations, whereas with the same mesh, FE technologies giving stiffer elements (e.g., Q4FI vs. Q4RI o Q8FI) reduced the propensity to shear banding. Regarding FE aspect ratio, earlier and sharper banding was observed if deformed elements became more equilateral at approaching the shear instability. For the same mesh, kinematic hardening favoured bifurcations more intensively than isotropic hardening, all bringing strong mesh sensitivity earlier or later along the loading route (Toribio and Kharin, 2000, 2006). Fig. 1 exemplifies solution bifurcations for perfect plasticity and kinematic hardening, where in the former