PSI - Issue 5

M. Dabiri et al. / Procedia Structural Integrity 5 (2017) 385–392 M. Dabir et al. / Structural Integrity Procedia 00 (2017) 000 – 000

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= ( + 2 ) (1+ )

(9) The Poisson’s ratio is defined by , and the subscript stands for the plastic component of strain. Using this modified stress-strain curve along with the set of original equations of previous section, the notch analysis can be performed under plane strain conditions.

4.3. Elastic-plastic local strain analysis and the TCD concept

Performing elastic-plastic analysis that broadly utilizes the finite element method is a promising technique for capturing the local strain values at the notch root, especially in Low Cycle Fatigue (LCF) in the presence of notch root plasticity. The accuracy level, however, strongly depends on the way the material response is defined and the modelling techniques. In addition, it has been widely accepted that the average stress over a small area around the stress raiser (effective stress) is responsible for the fatigue failure. That is, fatigue will occur within this volume. This clarifies the fact that although the plasticity allows stress field redistribution, using only the maximum strain at the notch root via finite element analysis could lead to inaccurate (over-conservative) fatigue life predictions. Taylor (Taylor, 2007) developing the concept of critical distance has defined this distance by introducing a characteristic material length constant, . This constant holds different values depending on the material, load types and ratios in medium/high cycle fatigue (Susmel, 2008), and its value decreases with an increasing number of cycles. However, in a low cycle regime, provided that the elastic-plastic response of the material in question is explicitly considered, the material characteristic length value shows dependency on neither the number of cycles nor the load spectrum (constant and variable) and notch configurations (Susmel & Taylor, 2010) (Susmel & Taylor, 2015). One of the main advantages of this proposed method is the ability to eliminate the conservatism that occurs when using the finite element method in the elastic- plastic strain analysis of sharp notches. In addition, the method’s negligence in consideration of inherent multiaxiality (e.g. biaxiality at the notch root) can be covered by the suggested modifications (Susmel, et al., 2011). However, due to the proportional variation of the stress/strain states defining the process zone under nominal uniaxial fatigue loading, it has been proven that the degree of multiaxiality of the local stress/strain fields can be neglected with little loss of accuracy (Susmel, et al., 2011). It has been shown (Susmel & Taylor, 2010) (Susmel & Taylor, 2015) that TCD in the form of the point method (PM) yields the closest results to experiments in notch analysis of round specimens under LCF. Therefore, the same approach was utilized in this study, which assumes that the desired point is located /2 from the notch root. Experimental calibration tests are required, combined with numerical techniques (finite element model), to obtain the value of . This value is considered constant regardless of the notch configuration. 5. Numerical modelling The CSSC of the material in Ramberg-Osgood form was used as the material behaviour input for the FE model to analyse the non-linear response of the notched specimens. The cyclic softening is considered in the model by introducing the stabilized behaviour as the tabular pair values of true stresses versus true plastic strains. The average value for the cyclic Young’s modulus, E' = 185 MPa, and a Poisson's ratio of = 0.3 define the elastic response of the material. Taking advantage of the axial symmetry of the specimen, an axisymmetric model with axisymmetric stress elements was used. This type of modelling allows the inhibition of plastic flow due to the multiaxial stress state that occurs in a circumferentially grooved bar without the complexity of a 3-dimensional model to be considered. Second order reduced integration quadrilateral elements were used for stress concentration calculations and strain analysis (CAX8R: 8-node biquadratic axisymmetric quadrilateral, reduced integration element). Because the stress and strain values are influenced by element size, an attempt was made to utilize enough elements of appropriate size, especially in the notch root. Based on the recommendations of Fricke (Fricke, 2012), the element size should be less than one quarter of the notch root radius, or at least six quadratic elements per 90 degrees should be used around a circular notch. A finer mesh size on the order of 10 -2 mm 2 is used in this study because of the non-linearity in the root of the notch.