PSI - Issue 28

L.V. Stepanova et al. / Procedia Structural Integrity 28 (2020) 1781–1786 Author name / Structural Integrity Procedia 00 (2019) 000–000

1782

2

corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials. © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo Keywords: crack in perfect plastic materials; sectorial solution; statically determinate problem; power law hardening material; mixity parameter; analytical solution; asymptotic solution . 1. Introduction Determination of crack-tip fields in perfect plastic solids under mixed mode loading is the classical problem of nonlinear fracture mechanics which is analyzed in a number of works (Shih (1973), Shih (1974), Dong and Pan (1990a), Symington et al. (1990), Loghin and Joseph (2003), Stepanova and Yakovleva (2015), Stepanova and Yakovleva (2016), Loghin and Joseph (2020)). The study of the near-crack -tip stress fields has received a great deal of attention for fully yielded materials surrounding the crack tip and for the occurrence of elastic sectors in the vicinity of the crack tip. Shih (1974) introduced the concept of the near crack-tip mode-mixity parameter p M . The mixity parameter p M varies from a value of 0 for pure mode II to a value of 1 for pure mode I. As it is noted by Loghin and Joseph (2020) since the work of Shih (1974) several authors have reconsidered these plane strain solutions and noted interesting behavior near mode I. Symington et al. (1990) performed a computational asymptotic study with extensive results. Their approach was used to obtain converged solutions for a range of hardening exponents from 0 p M  , up to about 0.83. p M  Shih (1974) pointed out that a high gradient in the radial stress develops near mode I, where convergence becomes difficult and/or an upper limit to these mixed mode solutions occurs. The mixed mode solutions presented by Loghin and Joseph (2003) were in direct agreement with the Symington et al. (1990) solutions and this behavior. Shlyannikov (2003) extended the range for a given hardening exponent, but did not report solutions up to unity. For example, for a hardening exponent of n = 3, solutions up to 0.94 p M  were presented. In the computational solution, Shlyannikov (2003) replaced the mode-mix constraint equation using p M with an angle at which the shear stress reaches its maximum. As it is pointed out by Loghin and Joseph (2020) a more recent addition to the literature of mixed mode loading in plane strain, specifically for power law creep, is by Dai et al. (2019) who obtained higher order terms beyond the leading order mixed mode term. This study also provides an excellent summary of the connections between power law plasticity and the more recent activity in power law creep. there have been mixed mode fracture studies for non-hardening and hardening material behavior. For plane stress conditions Shih (1973) and Dong and Pan (1990a) considered perfectly plastic material behavior, while Dong and Pan (1990b) and Rahman and Hancock (2006) have considered elastic, perfectly plastic material behavior. Nomenclature ij  stress tensor components , r  in-plane coordinates p M mixity parameter k yield stress under shear Y  yield stress under tensile load (0) ij  the zero approximation of stresses j  angles separating sectors in analytical solution Loghin and Joseph (2020) reconsidered a classic problem in nonlinear fracture mechanics. Their solution is limited to the following key assumptions: plane stress (so three-dimensional effects are neglected), fully yielded material around the crack tip (so linear elastic sectors are not considered in the asymptotic analysis), and small-scale yielding. While it is clear that more realistic assumptions are important, the correct solution of this classic problem is also important. On the important matter of three dimensional effects, the pure mode I summary provided by Chao (1993) on plane stress, plane strain and three-dimensional effects is interesting. A key point is that the theory of plane stress for mode I fracture is useful, but it is understood that plane stress is for very thin specimens, i.e., an important limiting © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo