Notch approximation methods for components under thermomechanical stresses

In this paper two different kinds of notch approximation methods are presented. The proposed methods are known as the incremental notch approximation method (e. g. Neuber method [1] or ESED [2, 3]) and the structural yield surface approach (SFF) (after Köttgen et al. [3]), while two different approaches of the SFF are used. All methods are capable to handle multiaxial cyclic loading. For a validation, the methods are compared to the results of a finite element analysis (FEA) of a notched component. The presented comparison is made for a non-proportional loading case. One objective of the work is to find out, which is suitable to extend it to thermomechanical loads as a next step.


Introduction
Welded joints represent one of the most important joining techniques in mechanical engineering.Accordingly, there are many areas of application in which welded components are essential, e. g. due to design requirements (e. g. in power plants, aerospace, gas turbines or in automotive).Such welded components are subjected not only to mechanical but also to thermal loads.The loading cases are often highly complex, i. e. multiaxial, and non-proportional.
Since the notches of the welded joints cause stress concentrations that can lead to fatigue failure or even fracture, knowledge of the loading condition at the notch is of central importance for calculating the fatigue life of such components.
A common way to calculate the stress-strain state at the notch is the use of notch approximation methods.A major advantage of these is the short calculation time with sufficient accuracy.In this work, the incremental notch approximation using Neuber's rule [1] and the structural yield surface approach (SFF) according to Köttgen et al. [3] using the pseudo-stress and the pseudo-strain approach, respectively, are compared.The aim is to identify, which method is more suitable to be extended to thermomechanical loads in future.
For the comparison, an elastic-plastic finite element analysis (FEA) on a notched component is used.The elastic-plastic FEA generally offers a very accurate way to determine the stress-strain state at the notch.The disadvantage, however, is an extremely high computing time, as well as immense amounts of data.Here, it offers a good opportunity to validate and compare the results.
The option of the experimental determination of the stress-strain state at the notch root should also be mentioned.The disadvantage of the high costs and time expenditure and the fact that a realization is only possible with simple geometries strongly limits the use of this method.

Notch stress-strain state under multiaxial loading
For a notched component under multiaxial loading, a three-dimensional stress-strain state is induced.For the calculation, a point on the surface of the notch root is considered.In Fig. 1, there is a notched component on the left side with its global coordinate system {, , }.To this component, two different load channels [ 1 ();  2 ()] are applied.On the right side of Fig. 1, an enlargement of the considered point on the surface of the notch root is shown.The local coordinate system {, , } is chosen so that the normal to the free surface is in the direction of the x-axis.This results in a plane stress state.The stress and the strain tensor can be expressed as ). (1) Figure 1.Notched component with two different load channels [ 1 ();  2 ()] applied on and its global coordinate system {, , } on the left side.Stress strain state at a point on the notch root surface with its local coordinate system {, , } on the right side (x-axis as direction of the normal to the free surface).
The proposed notch correction methods in this study require the time history of local stress or strain tensors as input variables, based on linear elasticity theory.For the calculation of the pseudo-stress, the linear elasticity theory applies.Thus, the principle of superposition applies.The pseudo stress tensor can be expressed as . (2) The left-side superscript  indicates the pseudo-elastic solution obtained using linear elasticity theory.This nomenclature remains across this paper.The values of   are not only the stress concentration factors, but also the conversion factors from the load (as local section force) to the nominal stress in the investigated cross section, which can be determined by a linear elastic FEA.Since the stress concentration factors differs for different points on the surface of the notch, it is important to identify the critical point.In case of the plane stress state described above (x-axis perpendicular to the free surface), this leads to seven unknowns which can be found in (1).Namely three unknown stresses   ,   ,   and four unknown strains   ,   ,   ,   .

Proposed approximation methods
The different notch approximation methods compared here are the incremental notch approximation method using Neuber's rule [1] and the structural yield surface approach according to Köttgen et al. [4] using the pseudo-stress and the pseudo-strain approach.A comparison of the methods with their respective advantages and disadvantages is made in chapter 5.

Incremental methods
To anticipate a major advantage of the incremental methods is that no additional set of parameters is required.The elastic-plastic behavior at the notch can be determined with sufficient accuracy using the peusdo-elastic solution obtained from linear elasticity theory.The probably best known incremental notch approximation methods are the Neuber's rule [1] and the equivalent strain energy density (ESED) method of Molski and Glinka [2,3].Over the years there have been many extensions to make the methods applicable to different scenarios.Topper et al.
[5] developed Neuber's rule to predict elasticplastic behavior for fatigue loading.Later Hoffmann et al. [6,7,8] extended it to account for multiaxiality and to deal with non-proportional loading cases.
The original Neuber's rule for the uniaxial case can be written as In Eq. ( 3) the   and   are the pseudo elastic stress and pseudo elastic strain obtained from linear elasticity theory, while  and  are the respective real quantities at the notch root.The multiplication of the stress and strain represents the total strain energy density.The basic idea of these methods is the equality of the total strain energy density in the pseudo-elastic case and in the real case at the notch root.
The graphical interpretation of the uniaxial Neuber's rule can be seen in Fig. 2. In Fig. 2, the blue curve is the Hooke's straight.The material yielding curve in orange is obtained using an appropriate material model like Ramberg-Osgood for this case, which looks as follows: In the uniaxial case, the equation system of two unknowns (stress and strain) can be solved using Eq.
(3) and Eq. ( 4).In Fig. 3, a graphical interpretation of the incremental multiaxial Neuber's rule is shown.The basic idea -equality of the total strain energy density for the pseudo-elastic and the real notch case -still exists.
But now the equality of the increments of the total strain energy density respectively between two loading increments is required.Furthermore, all tensor entries must be considered.The equation system in Voigt notation then looks like    Δ    + Δ       =   Δ   + Δ     . ( where no summation over the indices takes place although these are tensor entries.As material model, now a multiaxial plasticity model is needed.In this study, the kinematic hardening model proposed by Ohno and Wang [9,10] is used.The material model again gives a relationship between the real quantities at the notch root.This can be simplified as Here    represents the elastic-plastic tangent modulus.Thus, a system of seven equations arises (7).
Three equations are obtained from the assumption of energy conservation (7, I-III) and four equations from the material model (7, IV-VII).For the case of a free surface at the notch root there are seven unknowns, marked in red in Eq. ( 7).

Structural yield surface approach
The original version of the structural yield surface approach (SFF) was proposed by Barkey [12].Later, Köttgen et al. [4] extended this approach by two different methods -the pseudo-stress and the pseudostrain approach.Both establish a relationship between the pseudo-elastic and the elastic-plastic i.e., real quantities at the notch root.For both approaches, a structural stress-strain curve for a monotonic loading case must be obtained first.This can easily be done by using a simple uniaxial Neuber method or an elastic-plastic FEA.Then the parameters of the structural model can be determined by using the structural stress strain curve.As next step, the structural model is integrated to determine one of the two real quantities (stress/ strain).The last step is to integrate the material model to determine the respective other quantity.As plasticity model, the one according to Ohno and Wang is used for the SFF, too.Thereby, the assumption of a decomposition of the strain into an elastic and plastic part for small deformations is made: The right-sided superscript  indicates the elastic part of the strain tensor, while the right-sided superscript  indicates the plastic part of the strain tensor.
In this paper the pseudo stress approach and the pseudo strain approach, both get compared to FEA results.Still, just the most important elements will be repeated.For more details, Köttgen et al. [4] is recommended.
In the pseudo-strain approach, the elastic part of the pseudo-strain    is the same, as the elastic part of the elastic-plastic strain   , see Fig 4.This is why the structural stress-strain curve can be defined as In the pseudo-stress approach, the equality of the elastic part of the strain is not given.As soon as yielding starts, there is no more correlation between the total elastic-plastic strain and the pseudo-stress, see Fig. 5.This leads to the following inequality: Therefore, the structural stress strain curve in the pseudo-stress approach is defined as Pseudo-strain approach.Elastic-plastic stresspseudo-strain curve in orange.Material curve in blue.
Pseudo-stress elastic-plastic strain curve in orange.Material curve in blue.

Comparison of all methods to FEA results
In this study, the incremental notch approximation using Neuber's rule, the pseudo-stress approach and the pseudo-strain approach are compared to the results of a FEA.The comparison is done for a nonproportional loading case with variable amplitudes.Fig. 6 shows the notched component which is used for the FEA.In Fig. 7, the mesh of the modelled component with its global coordinate system, boundary conditions and the applied normal force and torsional moment   is shown.Fig. 8 shows a more detailed view of the notch-area with its hotspot and the corresponding local coordinate system.Fig. 9 shows the results of the comparison as follows: The complex loading case can be taken from the load phase at the top left and the load time sequence at the top right.The 90°-out-of-phase non-proportionality and the variable amplitudes are displayed in the load time sequence, too.For the compared results, the first column compares results of the normalstress-plane in the y-direction and the second column compares the results of the yz-shear-stress-plane.A comparison of the normal-stress-plane in the z-direction is omitted in this paper, since these strain components are smaller by a factor of 10 anyway and are therefore less relevant for fatigue failure.
The second row shows a comparison of the incremental method using Neuber's rule in blue with the FE results in black.For both planes a well-known picture for the use of Neuber's rule appears -the overestimation of the maximum and the minimum strains.Nevertheless, the course of the hysteresis fits well.It remains to be noted that the incremental method provides a very good result even for such a complex loading case.
The third row shows a comparison of the pseudo stress approach (   ) in red with the FE results black.It shows for both planes an excellent fit for the hysteresis.The maximum and the minimum strains for this approach are nearly the same as calculated with FEA.The quality of the results is remarkable for such a load case.
The last row shows a comparison of the pseudo strain approach (   ) in yellow with the FE results black.For both planes, the hysteresis are fitting well.In the normal-stress-plane, the maximum strain is a little overestimated.The minimum strain in this plane is almost the same as in the FE calculation.In the shear-stress-plane, however, there is a slight underestimation of the maximum strain -which means an unsafe result regarding component design due to fatigue life.However, the underestimation is relatively small.The minimum strain for this plane fits the results of the FEA.The results for the pseudo strain approach are still excellent.

Conclusion and outlook
Regarding the accuracy of the results for the selected loading cases, all three methods perform very well.The pseudo-stress approach proved to be the most accurate in this study followed by the pseudo-strain approach.As incremental method, the Neuber's rule is used here, which is well known for conservative results.Nevertheless, the results of the incremental method are also very good.
Thus, the structural yield surface (SFF) approaches according to Köttgen et al. [4] is considered more accurate for this study.Another benefit is the efficiency, because with the SFF approach, the differential equations can be integrated directly, which brings numerical stability and speeds up the calculation by a factor of 10 compared to the incremental methods.
Another well-known problem of the incremental methods is the identification of reversal points, which is very difficult, especially dealing with non-proportional loading cases.Using the SFF no identification of the reversal points is required, due to the direct integration.Nevertheless, the SFF needs another set of parameters to create the structural model first.This is a relatively acceptable disadvantage, since the results quality and efficiency are greatly increased and especially for complex and high loading cases, with plastification in the net cross-section, FE calculations must precede the incremental methods to provide parameters, too.The structural model can be provided from such FE calculations directly.
As a conclusion of this study, the structural yield surface approaches are more suitable to be extended to thermomechanical loads.As a next step, the parameters used in the material model should be represented as a function of temperature.If this does not provide sufficient accuracy, a next step would be to use a visco-plasticity model.Comparison of notch approximation methods to FEA results for a non-proportional loading case with variable amplitudes.Neuber: blue; Pseudostress approach: red; Pseudo-strain approach: yellow; FEA: black

Figure 2 .
Figure 2. Graphical interpretation of the uniaxial Neuber's rule.Material curve in orange.Hooke's straight in blue.Neuber hyperbola in black Figure 4.Pseudo-strain approach.Elastic-plastic stresspseudo-strain curve in orange.Material curve in blue.

Figure 7 .
Figure 7. Mesh of the modelled component including boundary conditions in its global coordinate system.

Figure 8 .
Figure 8. Detailed view of the notch with its local coordinate system.