Creep behavior of V-notched components

Geometrical discontinues such as notches play a significant rule in structural integrity of the components, especially when the component is subjected to very severe conditions, such as the high temperature fatigue or creep. In this paper, a generalized form of the existing notch tip creep stressstrain analysis method developed by Nuñez and Glinka, is developed and extended to a wide variety of blunt V-notches. Assuming the generalized Lazzarin-Tovo solution that allows a unified approach to the evaluation of linear elastic stress fields in the vicinity of both cracks and notches is the key in getting the extension to blunt V-notches. Numerous cases have been analysed and the stress fields obtained according to the proposed method were compared with proper finite element data, showing a very good agreement.


INTRODUCTION
ue to the complexities of the geometry and loading conditions in modern structural components, it is essential to be able to predict the behavior of components including geometrical discontinuities that generate localized high stress concentration zones [1,7].They become even more important when, in operating conditions, the component is subjected to very demanding conditions such as high temperature fatigue or creep loading.Various methods have been proposed by researchers to evaluate the behavior of structural components under various loading conditions [8][9][10][11][12].The structural components show a nonlinear stress-strain response such as creep (visco-plasticity) under applied load in a high temperature environment.In presence of geometric discontinuities such as notches, localized-creep takes place in a small region near the notch root.On the other hand, non-localized (or gross) creep condition refers to situations in which the far stress field also experiences some creep which may lead to more intense creeping in the vicinity of the notch tip.

D
To the best of the authors' knowledge only a few solutions related to the localized time-dependent creep-plasticity problems are available in literature.Nuñez and Glinka [13] proposed a solution for non-localized creep strains/stresses at the notch root, based on the linear-elastic behavior of the material, the constitutive law and the material creep model.The formulation was derived by using the total strain energy density rule proposed by Neuber [14].Considering the U-notched specimens (2α=0 and ρ≠0) very good results were obtained using this method.The main aim of the current paper is to extend the method proposed by Nuñez and Glinka to blunt V-notches.For this aim, the Creager and Paris [15] equations were substituted with the Lazzarin and Tovo [16] equations.Finally an approach for fast evaluation of the stresses/strains at notches under non-localized creeping condition is proposed which doesn't require any complex and time-consuming FE non-linear analyses.Output of the proposed approach can be used as input for creep life prediction models based on local approaches.

NOTCHES
uñez and Glinka [13] presented a method for the estimation of stress and strain at U-notch tip, subjected to non-localized creep.The method was based on the Neuber [14] concept extended to time dependent plane stress problems and on the introduction of KΩ parameter introduced by Moftakhar et al. [17].It can be assumed in fact that the total strain energy density changes occurring in the far field produce magnified effects at the notch tip.For this reason, the total strain energy density concentration factor is introduced in order to magnify the energy at the notch tip.The introduction of this parameter and of the far field stress and strain contribution in the Neuber's time dependent formulation is the main difference within the non-localized and localized creep formulation that, instead, can be easily derived directly by extending the Neuber's rule.Details about the original formulation can be found in the original works Nuñez and Glinka [13] and in Gallo et al. [18].The key to extend the Nuñez-Glinka method to blunt Vnotches is the assumption of the Lazzarin and Tovo [16] equations to describe the early elastic state of the system.The Lazzarin-Tovo equations, in the presence of a traction loading, along the bisector (x axis), can be expressed as follows, as a function of the maximum stress (see Fig. 1): N where σmax can be expressed as a function of stress concentration factor Kt (evaluated through linear elastic finite element analysis) and the applied load σnom, Employing the more general conformal mapping of Neuber [19] that permit a unified analysis of sharp and blunt notches, the notch radius, ρ, and the origin of the coordinate system, r 0 , are related by the following equation on the basis of trigonometric considerations: where The main steps to extend the method to blunt V-Notches can be summarised as follows:  Assumption of Lazzarin-Tovo equations to describe the stress distribution ahead the notch tip instead of Creager-Paris equations;  Calculation of the origin of the coordinate system, r 0, as a function of the opening angle and notch radius, as described by Eq. ( 3);  Re-definition of the plastic zone correction factor Cp that is a function of plastic zone size rp and plastic zone increment Δr p ; The definition of the parameters Cp, rp, Δrp is very similar to that clearly reported by Glinka [20], except for the assumption of different elastic stress distribution equations.Definition of these variables is briefly reported hereafter.Referring to Fig. 2, considering the Von Mises [21] yield criterion in polar coordinate: and introducing Eqs.(1) into Eq.( 4), a first approximation of rp that can be solved numerically is obtained.Once rp is known, the force F 1 can be evaluated as follows: The stress σy (rp) is considered to be constant inside the plastic zone, which means elastic-perfectly plastic behavior is assumed.The lower integration limit is r 0 , which depends on the opening angle and notch tip radius.Due to the plastic yielding at the notch tip, the force F 1 cannot be carried through by the material in the plastic zone r p .But in order to satisfy the equilibrium conditions of the notched body, the force F1 has to be carried through by the material beyond the plastic zone r p .As a result, stress redistribution occurs, increasing the plastic zone r p by an increment ∆r p .If the plastic zone is small in comparison to the surrounding elastic stress field, the redistribution is not significant, and it can be interpreted as a shift of the elastic field over the distance ∆rp away from the notch tip.Therefore the force F1 is mainly carried through the material over the distance ∆rp, and therefore the force F2 (represented by the area depicted in the Fig. 1-b) must be equal to F 1 .For this reasons, F 1 = F 2 = σ θ (r p )Δr p , and the plastic zone increment can be expressed as the ratio between F1 and σθ evaluated (through Lazzarin-Tovo equations [16]) at a distance equal to the previously calculated rp: Substituting in Eq. ( 6) the formula given by Eq. ( 5) for F1 and the explicit form of σθ, the expression for the evaluation of Δr p is obtained: The last step consists in the definition of the plastic zone correction factor C p , which is according to Glinka [20] but introducing the Lazzarin-Tovo equations: At this point, the general stepwise procedure to be followed to generate a solution is identical to that proposed by Nuñez and Glinka [13]: 1. Determine the notch tip stress, 22 e  , and strain, 22 e  , using the linear-elastic analysis.

22
Δ n c  , for a given time increment Δtn.The selected creep hardening rule has to be followed.
4. Determine the decrement of stress,

22
Δ n t  , from Eq. ( 10), due to the previously determined increment of creep strain, 22 Δ n c  : 5. For a given time increment Δtn, determine from Eq. ( 11) the increment of the total strain at the notch tip, 22 Δ n t  : 6. Repeat steps from 3 to 5 over the required time period.

RESULTS
he proposed new method has been applied to a hypothetical plate weakened by lateral symmetric V-notches, under Mode I loading; see Fig 1b .The notch tip radius ρ and the opening angle 2α have been varied, while for the notch depth a, a constant value equal to 10 mm has been assumed.Three values of the opening angle 2α have been considered: 60°, 120° and 135°.The notch tip radius assumes for every opening angle three values: 0.5, 1 and 6 mm.The plate has a constant height, H, equal to 192 mm and a width, W, equal to 100 mm.The numerical results have been obtained thanks to the implementation of the new developed method and its equations in MATLAB®.In the same time, a 2D finite element analysis has been carried out through ANSYS.The Solid 8 node 183 element has been employed and plane stress condition is assumed.The material elastic (E, ν, σ ys ) and Norton Creep power law (n, B) properties are reported in Tab. 1.For the sake of brevity, only few examples are reported in Fig. 2 (a-b).All the other cases present the same trend of Fig. 2. The theoretical results are in good agreement with the numerical FE values.All the stresses and strains as a function of time have been predicted with acceptable errors.In detail, maximum discrepancy in modulus of 20% has been found for both quantities, with a medium error about 10%.Considering the examples of the strain prediction depicted in Fig. 2(b), the discrepancy within numerical and predicted solution of the strain is most likely due to different approximations introduced in the theoretical formulation, such as the assumption of the elastic-perfectly plastic behavior of the material inside the plastic zone and the employment of the Irwin's method to estimate the plastic radius.

Figure 1 :
Figure 1: (a) Coordinate system and symbols used for the stress field components in Lazzarin-Tovo equations; (b) coordinate system and symbols used for the elastic stress field redistribution for blunt V-notches.
obtained from the linear elastic analysis χ1 mode I associated constant