PSI - Issue 66

Yu.G. Matvienko et al. / Procedia Structural Integrity 66 (2024) 148–152 Author name / Structural Integrity Procedia 00 (2025) 000 – 000

149

2

local criterion that gives the line of the crack propagation. Most widely used incremental criteria for determining the trajectory of a crack are based on an analysis of energy and stress-strain fields in the vicinity of the crack tip starting from the pioneer papers of Erdogan and Sih (1963), Sih (1974). The following criteria for predicting both onset and direction of fracture especially under complex mixed mode loading conditions have been currently employed, namely, the maximum tangential stress (MTS) criterion and the generalized maximum tangential stress criterion taking into account the nonsingular T xx -stress, maximum tangential strain criterion and the extended maximum tangential strain criterion, the maximum energy release rate criterion, criterion of the minimum energy consumption for fracture, criterion of the strain energy density, the J-integral concept and etc. Possible fracture mechanics criteria for a search of the crack path in some examples can be found in the following review given by Pook (2016). It is important to note the inclusion of nonsingular T-stresses into local fracture criteria when predicting the crack trajectory in the framework of two-parameter fracture mechanics as it was demonstrated by Matvienko (2020). At the same time, when predicting the trajectory of a crack, only two limiting cases of the stress state are usually considered, namely, plane strain and plane stress. As a result, only the component of nonsingular Т xx -stresses is involved into consideration as it was firstly proposed by Aliha et al. (2010) in the generalized maximum tangential stress criterion. In the case of three-dimensional solids, the nonsingular T xx - and T zz -stresses characterize two dimensional constraint in the transverse ( Т xx ) and longitudinal ( Т zz ) directions along the crack front and should be included in the fracture criterion and the crack path prediction for a mixed mode I/II loading conditions in three dimensional solids.

Nomenclature K

stress intensity factor

K Ic fracture toughness T xx , T zz components of nonsingular T-stress α crack inclined angle θ* crack propagation angle σ Y stress yield σ θ tangential crack tip stress

2. Model and crack path criterion The tangential crack tip stress σ θ in polar coordinate under mixed mode I/II loading conditions is described by the series solution including the singular and the nonsingular terms according to Williams (1957)

1

3

2 



  

  

(1)

  , r    

2

2 sin .    T

cos

cos

sin

K

K



I

II

xx

2 2

2

r



where r and θ are the crack tip coordinates. The singular terms are stress intensity factors K I and K II for mode I and mode II, respectively. It should be noted that the T-stress can be considered as a constraint parameter for different configurations of solids and type of loading. The following the maximum tangential stress criterion can be written to search the crack path



(2)

0.







*

|



This criterion assumes that a crack propagates when the maximum tangential stress reaches its critical value in the fracture process zone in the vicinity of the crack front. It means that the direction of crack propagation corresponds to the direction of the maximum tangential stress.