# Finite Element Based Fatigue Life Prediction Biology Essay

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In this paper a methodology is proposed to apply endurance function model with genetic algorithm to estimate fatigue life of notched or smooth components. Endurance function model is based on stress tensor invariants and deviatoric stress invariants. In the proposed methodology the FEA is used to simplify the application of endurance function model. Experimental results from published literature are considered for the case studies to evaluate the proposed methodology. The results show that proposed methodology simplified the application of endurance function model specially reducing the need of notch sensitivity factors and stress invariants can be calculated directly from the stresses at the critical point. The comparison with experimental results shows that with proper calibration of the model it can predict fatigue life accurately.

Key words: Fatigue life, endurance function, stress invariants, finite element analysis, generic algorithm

## INTRODUCTION

Since the investigations by Wohler in 1860, fatigue experiments and predictions have played a major role in mechanical design (Lee et al., 2005; Stephens et al. 2000), researchers investigating the fatigue problem have made huge efforts in order to devise sound methodologies suitable for safely assessing mechanical components subjected to time-variable loadings (Manson and Halford, 2006). It is acknowledged globally that correctly estimating fatigue damage in real components is a complex process involving a high number of different variables that have to be properly taken into account in order to avoid unwanted and dangerous failures (Brighenti and Capinteri, 2012). Any reliable fatigue assessment technique should be able to efficiently and simultaneously model the damaging effect of non-zero superimposed static stresses, degree of multiaxiality of the stress field and the role of stress concentration phenomena (Susmel and Taylor, 2008). Especially in the case of cyclic or random multiaxial loading histories, the fatigue assessment is difficult to be correctly performed since damage accumulation depends on all the components of the stress tensor and their variation during the whole phenomenon (Susmel and Taylor, 2008; Macha and Nieslony, 2012). To make their results close to reality, the calibration of such an engineering fatigue assessment method should be based on pieces of experimental information that can easily be obtained through tests run in accordance with the pertinent standard codes (Brighenti and Capinteri, 2012; Susmel and Taylor, 2008; Atzori et al., 2006; Susmel and Taylor, 2012; Susmel and Tovo, 2011). The stress analysis is conducted to correctly estimate fatigue damage by directly post-processing simple linear elastic finite element (FE) models (Bishop and Sherratt, 2000).

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To deal with fatigue life assessment problem of structural components under multiaxial load histories (proportional or non proportional, cyclic or random), Brighenti and Carpinteri (Brighenti and Capinteri, 2012) proposed an endurance function based fatigue life estimation model, based on continuum damage mechanics formulation (Ottosen et al., 2008). The model does not require any evaluation of a critical plane as the model consider the damage accumulated at a point using stress tensor invariants and deviatoric stress invariants, and invariants are not coordinate system specific quantities. And as per continuum mechanics concept the endurance function is defined to be continuously evolving function with applied loading so there is no requirement for any conventional loading cycle counting algorithm (Brighenti and Capinteri, 2012; Ottosen et al., 2008). The damage (D) is evaluated at a specific point of the structural component through appropriate endurance function (E) and suitable expression of damage increment (dD). The fatigue life is assumed to occur when damage D reaches unity. A genetic algorithm (GA) approach is employed to evaluate numerically the several parameters used for characterization of the damage mechanics approach, once the effects of some experimental complex histories are known (Brighenti and Capinteri, 2012; Brighenti et al., 2006).

In this paper a methodology has been proposed to predict fatigue life using the endurance function model and GA procedures coupled with finite element analysis. Two steel alloys EN3B (cold rolled low carbon steel) and C-40 (Carbon steel) are selected for study. Experimental fatigue life data for tension torsion test from published literature is used (Susmel and Taylor, 2008; Atzori et al., 2006). The determination of stress invariants for endurance function is greatly simplified due to the use of Finite element analysis, and also helped in reducing the number of parameters by one i.e. stress concentration effect can be avoided as we can obtain the exact value of stresses at the notch. The results show that the above mentioned methodology worked well in low and medium cycle range for high cycle the results are highly conservative.

## FINITE ELEMENT MODELING AND ANALYSIS

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Two sets of experimental data of tension torsion fatigue life on steel alloys EN3B and C40 were considered in this paper (Susmel and Taylor, 2008; Atzori et al., 2006). The specimen dimension detail used to obtain the results for each alloy is shown in Figure 1 (a&b). Finite element analysis is performed on both specimen geometries with same respective loading conditions used for experimental testing Table 1 & 2 and FEA model is shown in Figure 2. Load is applied as force and moment which will result in required applied normal and shear stresses at net area.

EN3B speciemen.bmp

(a)

c40 specimen.bmp

(b)

Figure 1: (a) Specimen used for EN3B Testing with notch radius 1.25mm (Susmel and Taylor, 2008) (b) . Specimen used for C40 Testing (Atzori et al., 2006)

Table 1: Experimental loading conditions and fatigue life of EN3B specimen having notch radius 1.25mm (Susmel and Taylor, 2008)

## Normal stress at net area Ïƒa (MPa)

## Shear stress at net area Ï„a (MPa)

## Load Ratio (R)

## Phase Difference (Â°)

## Cycles to Failure (Nf) x106

275

158.8

-1

0

0.046

259.6

155.9

-1

0

0.083

230

132.8

-1

0

0.2

200

115.5

-1

0

0.044

190

109.7

-1

0

1.4

180

103.9

-1

0

2.17

285

164.5

-1

90

0.032 / 0.037

270

155.9

-1

90

0.059

260

150.1

-1

90

0.31

250

144.3

-1

90

0.079

230

132.8

-1

90

0.15 / 0.24

200

115.5

-1

90

2.1

Table 2: Experimental loading conditions and fatigue life of C-40 specimen having notch radius 0.5mm (Atzori et al., 2006)

## Normal stress at net area Ïƒa (MPa)

## Shear stress at net area Ï„a (MPa)

## Load Ratio (R)

## Phase Difference (Â°)

## Cycles to Failure (Nf) Ã-106

221

221

-1

0

0.012

200

200

-1

0

0.027

179

179

-1

0

0.19 / 0.22

160

160

-1

0

0.072

129.75

129.75

-1

0

0.18 / 0.63

118.8

118.8

-1

0

0.44

101

101

-1

0

2.0

199.7

199.7

-1

90

0.011

180

180

-1

90

0.014 / 0.018

160.25

160.25

-1

90

0.041 / 0.047

140

140

-1

90

0.019 / 0.28

129.65

129.65

-1

90

0.11 / 0.69

119.5

119.5

-1

90

0.94

109.3

109.3

-1

90

1.11

99.6

99.6

-1

90

2.0

158

158

0

0

0.026

138.5

138.5

0

0

0.047

119

119

0

0

0.15

99.55

99.55

0

0

0.35

89.4

89.4

0

0

0.43 / 0.49

79.72

79.72

0

0

0.79

67.9

67.9

0

0

2.0

158

158

0

90

0.021 / 0.022

138.75

138.75

0

90

0.033 / 0.036

119.3

119.3

0

90

0.085 / 0.1

99.25

99.25

0

90

0.087 / 0.21

89.55

89.55

0

90

0.34

66.8

66.8

0

90

2.0

Untitled.png

Figure 2: FEA model of specimen

## Mesh Sensitivity Analysis.

Mesh sensitivity analysis has been performed to get the optimum mesh size which will give good balance between accuracy and processor time and storage load (Kamal et al., 2012). Max Principal, von mises and Tresca stresses with number of nodes and elements are monitored for mesh convergence with acceptable processing load. Table 3 and Fig 3(a&b) show the result of mesh sensitivity analysis. As the specimens are of nearly same geometry so mesh sensitivity is not needed for individual specimens. From Fig 3 (a & b) it can be seen that after mesh size of 0.175 mm the value of stresses are not changing in appreciable amount but there is an exponential rise in number of nodes and elements, which will result in increase of processor time and storage requirement without much increase in accuracy of stress results. Hence to get the optimum performance mesh size of 0.175 mm is selected for meshing both specimen models.

Table 3: Mesh sensitivity analysis results

## Mesh Size (mm)

## Von mises stress (MPa)

## Tresca Stress (MPa)

## Max Principal (MPa)

## No. of nodes

## No. of elements

0.1

562.96

575.72

576.27

587762

409838

0.125

562

573.85

574.53

414193

287994

0.15

561.1

572.8

574

260754

181331

0.175

560.78

572.65

575.42

176749

122197

0.2

554.79

565.7

569.95

144013

99252

(a)

(b)

Figure 3: (a) Mesh size vs calculated FEA stresses (b) Mesh size vs no. of nodes and elements of FEA model

## ENDURANCE FUNCTION MODEL

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Examples of our workBrighenti and Carpentri, 2012 proposed an endurance function model on the basis of continuum damage mechanics approach (Ottosen et al., 2008), with assumption that the whole fatigue life is crack nucleation dominated and fatigue life for crack propagation is negligible with respect to total life. For isotropic materials the endurance function can be expressed as below;

(1)

Where, a1 to a5 and ï³0 are the material constants, I1, I2, I3 are stress tensor invariants and J2, J3 deviatoric stress invariants are functions of stress tensor ï³ and effective deviatoric stress tensor Se respectively. Here Se = S - Sb , i.e. S is current applied deviatoric stress tensor and Sb is back stress tensor which measures the endurance function evolution in the stress space.

Damage is assumed to occur when E(ï³,Se) > 0 and no damage occur when E(ï³,Se) â‰¤ 0, and increment in damage will happen when dE (increment in endurance function) > 0. To define dE properly it is specified that if there is a case where stress value at point i result in E(ï³i,Se,i) > 0 and previous point result in E(ï³i-1,Se,i-1) < 0 the quantity E(ï³i-1,Se,i-1) is set equal to zero, which in turn result in keeping dE always greater than zero.

The damage D is evaluated by considering progressive accumulation of damage increments i.e. at each load step damage increment is equal to or greater than zero dD â‰¥ 0 and consequently material damage D is a non decreasing positive function i.e. D â‰¥ 0 (Manson and Halford, 2006). And final collapse occur when D reaches unity (D=1). The damage rate dD is assumed to depend on current value of E as well as dE, the relationship between dD and dE is as followed;

(2)

Where A and B are material constants. The stress gradient effect is taken into account by inserting a reducing factor G in to Eq(2);

(3)

Here,

(4)

Where G is notch gradient correction factor which depends on V (material constant) and stress field parameter ï§, it represents the stress gradient absolute value at notch root. For evolution of deviatoric back stress Sb, it is assumed to follow the relationship below;

(5)

Where C and h are material parameters.

## Application of Genetic Algorithm (GA)

Genetic Algorithm is used to find out the optimum values of these parameters. Such algorithms (random stochastic methods of global optimization) are used to minimize or maximize an objective function chosen for a given problem to be solved. Such an approach can be useful to evaluate the model parameters (in the context of model parameter tuning), once the response of the physical system to a given input is known (Brighenti et al., 2006). The GAs have some advantages with respect to classical techniques, they allow us to handle problems with multiple minima and non-convexity properties, thus avoiding numerical instability and missing of global optimum (Davis, 1991). GA can handle any kind of objective function, and simply operate by using simple concepts (such as random numbers generation, choice, switch and combination of such generated numbers) to get a new 'population' characterized by performance better than that of the previous one. By iteratively repeating the 'evolution procedure' until a given tolerance is attained (Gantovnik et al., 2003). In order to apply the endurance function model, values of 11 parameters are needed to be evaluated which are defined in the equations above Eq.1-Eq.5. If the fatigue life Nf is known for generic multiaxial stress history where damage D reaches unity, a prediction error can be defined as follows;

e = D(a1,a2â€¦.., Nf) - 1 (6)

where D(a1,a2â€¦.., Nf) is the damage evaluated at Nf. The values of model parameters can be found by minimizing this error function using GA procedure (Brighenti and Capinteri, 2012).

## SIMPLIFIED ENDURANCE FUNCTION MODEL

Two materials i.e. EN3B and C40 steels are used in this study with six sets of fatigue life data Table 1 & 2, under cyclic multiaxial in phase and out of phase loadings. Specimens are notched specimens as shown in Fig 1(a&b). Mechanical characteristics of both steel are as follows in Table4 (Brighenti and Capinteri, 2012; Susmel and Taylor, 2008).

Table 4: Mechanical Properties of EN3B and C40 Steels (Brighenti and Capinteri, 2012; Susmel and Taylor, 2008)

## Material Name

## Young's Modulus (GPa)

## Yield Stress (MPa)

## Ultimate tensile strength (MPa)

## Loading Condition

## Fatigue limit (MPa) (@ cycles)

## R

## Phase

En3B

208.5

653

676

-1

0

192.4 (106 cycles)

-1

90

188.2 (106 cycles)

C40

206

537

715

-1

0

101 (2x106)

-1

90

99.6 (2x106)

0

0

67.9 (2x106)

0

90

66.8 (2x106)

Initially linear stress analysis of specimen was conducted at every load step mentioned in Table 1 & 2, and all three principal stresses (ï³1, ï³2, ï³3) are recorded at notch root where highest value of maximum principal stress is occurring (Rahman et al., 2009a, b, 2009), as shown in Figure 4. Neuber elastic plastic correction is applied where stresses are found to be above yield strength (Stephens, 2000; Bishop and Sherratt, 2000, SAEJ1099, 2002).

Stress invariants (I1, I2, I3, J2, J3) are than calculated from following relationships depending on principal stresses after neuber correction is applied where necessary (Socie and Marquis, 2000).

(7)

stress of specimen.jpg

Figure 4: Location at notch root surface (origin of coordinate system shown) used for determining principal stresses.

For the endurance function parameters, where G is notch gradient correction factor, becomes G = 1 due to the fact that FEA gives the value of stress at the notch root directly. This in return nullify the requirement to determine parameter V and stress field parameter ï§. Thus reducing the number of endurance function parameters by one. For both of the considered materials EN3B and C40 steel the cyclic behavior is assumed to be following stable hysteresis loop, this in return let the change in back stress dSb = 0. Thus there is no need to calculate parameter C and h in Eq. 5. So finally from 11 parameters of endurance function the required are reduced to eight, a1, a2, a3, a4, a5, sigma0, A and B.

Now to calculate the value of parameters at calibration points, GA is used where error function which is to be minimized is defined as Eq. 8, with the assumption that as cyclic behavior of material is stable the damage in each cycle is inverse of fatigue life in cycles.

where damage per cycle is calculated from endurance function. These equations are inserted in GA algorithm for optimization of endurance function parameters (Clarich et al., 2011; Perillo et al., 2009, modeFRONTIER, user manual). Fig .5 shows the workflow diagram.

I:\Progress presentation jan 2013\workflow.jpeg

Figure 5: Workflow diagram of optimization model

Initial values table required for GA, is generated randomly from the ranges of coefficients assumed. Range for sigma0 is defined with upper limit as conventional fatigue limit (Brighenti and Capinteri, 2012; Susmel and Taylor, 2008) and lower limit is set to be 25% less than upper limit initially and then modified with ranges of other coefficients until the combinations of coefficients becomes stable which are resulting in minimum error. From stable it is assumed that the sets of coefficients resulting in minimum error have no significant change in parameters. Then top three values of coefficients from each set with respect to minimum error are weighted average on the basis of error (Hariharan et al., 2011), and the resulting weighted average coefficients are characterized as the coefficients of Endurance Function at respective calibration point. Table 5 (a & b) shown the coefficients of calibration points determined using GA.

Table 5: (a) Coefficients of calibration points for Endurance function model obtained from GA for EN3B

## Load (MPa)

## a1

## a2

## a3

## a4

## a5

## sigma0

## A

## B

## ï³

## ï´

## EN3B (R=-1 and phase =0)

275

158.8

0.5107

1.004

0.2313

1.129

1.197

180

5.348x10-9

0.2299

180

103.9

0.4001

0.5233

0.4113

0.7745

0.8194

166.5

1.009x10-9

0.08192

## EN3B (R=-1 and phase = 90Â°)

285

164.5

0.4255

1.19

0.3625

1.076

0.7991

141.4

7.056x10-9

0.2389

200

115.5

0.2708

0.327

0.2189

0.7392

0.4868

149.7

2.882x10-9

0.07962

Table 5: (b) Coefficients of calibration points for Endurance function model obtained from GA for C40

## Load (MPa)

## a1

## a2

## a3

## a4

## a5

## sigma0

## A

## B

## ï³

## ï´

## C40 (R=-1 and phase =0)

200

200

0.1412

0.3676

0.4504

1.404

-0.3757

80.18

8.034x10-9

0.4285

101

101

0.4972

0.7769

0.3194

1.44

0.9721

85.33

1.485x10-10

0.2322

## C40 (R=-1 and phase = 90Â°)

199.7

199.7

0.4358

0.5965

0.6721

1.428

1.165

76.7

2.868x10-8

0.2134

99.6

99.6

0.2615

0.2907

0.4818

1.345

-0.2697

79.02

1.032x10-9

0.1377

## C40 (R= 0 and phase = 0Â°)

158.1

158.1

0.7446

0.2931

0.432

1.731

0.2959

58.4

8.05x10-9

0.224

67.9

67.9

0.05374

0.5927

0.6035

0.9929

0.3145

61.08

1.609x10-9

0.03816

## C40 (R= 0 and phase = 90Â°)

158

158

0.4721

0.4814

0.5829

1.655

0.7878

54.44

7.373x10-9

0.2677

66.8

66.8

0.1452

0.6464

0.572

0.9294

0.6613

61.83

1.198x10-9

0.1313

## RESULTS AND DISCUSSION

Coefficients for all calibration points calculated using GA, as shown in Table 5 for both materials and their respective load sets, are used to determine coefficients for other load points by interpolation between the calibration points, and with respective values of stress and deviatoric stress invariants (calculated earlier), fatigue life is estimated using the endurance function model. Predicted fatigue life is reported in Table 6 (a & b);

Table 6: (a) Predicted fatigue life of EN3B steel using Endurance Function model

## Load (MPa)

## Predicted cycles to failure (Np)x106

## Normal Stress Ïƒ (MPa)

## Shear Stress Ï„ (MPa)

## For R=-1 and phase = 0

275

158.8

0.042

259.6

155.9

0.068

230

132.8

0.18

200

115.5

0.59

190

109.7

0.99

180

103.9

1.83

## For R=-1 and phase = 90

285

164.5

0.032

270

155.9

0.054

260

150.1

0.076

250

144.3

0.11

230

132.8

0.24

200

115.5

0.97

Table 6: (b) Predicted fatigue life of C40 steel using Endurance Function model

## Load (MPa)

## Predicted cycles to failure (Np)x106

## Normal Stress Ïƒ (MPa)

## Shear Stress Ï„ (MPa)

## For R=-1 and phase = 0

221

221

0.026

200

200

0.027

179

179

0.032

160

160

0.045

129.75

129.75

0.11

118.8

118.8

0.18

101

101

1.75

## For R=-1 and phase = 90

199.7

199.7

0.0092

180

180

0.014

160.25

160.25

0.023

140

140

0.051

129.65

129.65

0.069

119.5

119.5

0.14

109.3

109.3

0.31

99.6

99.6

1.2

For R=0 and phase = 0

158

158

0.025

138.5

138.5

0.048

119

119

0.1

99.55

99.55

0.23

89.4

89.4

0.38

79.72

79.72

0.65

67.9

67.9

0.14

For R=0 and phase = 90

158

158

0.02

138.75

138.75

0.035

119.3

119.3

0.066

99.25

99.25

0.14

89.55

89.55

0.21

66.8

66.8

0.85

First thing to notice from the fatigue life prediction results is that, the fatigue life values calculated at the calibration points itself are not the same as experimental life (Table 1,2,5 & 6), which should be theoretically same as these points are used to calculate coefficients for endurance function model. The reason lies in the fact that coefficients calculated at each calibration point are the weighted average of three sets of coefficients resulting in minimum error determined from GA, which in turn let the estimation of fatigue life at the calibration point to stray away a little. But author suggest that this weighted average should be taken and not just rely on one set with minimum error, so to better capture the trend of coefficients from the three sets from which weighted average is being take from. The interpolation of coefficients of endurance function for load points other than calibration points is introduced which is a better approximation to estimate fatigue life than the proposed method of calculating only one set of coefficients (Brighenti and Capinteri, 2012). Also the idea of taking weighted average of coefficients calculated from different load values (Brighenti and Capinteri, 2012) is in turn creating a bias on the basis of error towards the coefficients with minimum error, which should not be there as all calibration load points are experimental values and have equal weight. This also prove the applicability of interpolation between coefficients as this keep weight of coefficients at calibration load points same and estimate at other points following the trend of changing coefficients with load values as shown in Fig 6.

Figure 6: Trend of coefficients for calibration points for EN3B R=-1 phase=0

Stress life curves from the predicted and experimental life data for EN3B is shown in Figure 7 (a & b) and for C40 steel in Fig 8(a-d). From results it can be seen that endurance function with the proposed application methodology show good agreement with experimental data for both materials in case of in phase loading. In out of phase case for both materials results are in good agreement in low cycle region, but turn to conservative side as we move towards high cycle region one reason of this is that the more number of experimental data points are in the low cycle region which leads to calibration of model biased towards low cycle region. Also there is scatter in experimental data as visible in Fig 7(b) and 8(b), which can lead to error in the prediction of fatigue life.

(a) (b)

Figure 7:.S-N curve for EN3B steel (a) for R= -1 and Phase = 0Â° (b) for R= -1 and Phase = 90Â°

(a) (b)

(c) (d)

Figure 8: S-N curve for C40 steel (a) for R= -1 and phase = 0Â° (b) for R= -1 and phase = 90Â° (c) for R= 0 and phase = 0Â° (d) for R= 0 and phase = 90Â°

One simple solution to improve the prediction accuracy is to increase the number or calibration points, which will lead to the better capture of the change in experimental S-N curve trend, as shown in Figure 9. Here one more calibration point is included with load Ïƒ & Ï„ = 140 MPa, which clearly resulted in improved agreement with experimental S-N curve. So it can be deduced from here that data set required for endurance model to work properly should have more data points and in both low and high cycle regions, so more calibration points can be used to fit the endurance model with the experimental data. The data set used in this study is small in size so only one extra calibration point is included to check this hypothesis and results show that the idea of more calibration point will indeed work well. So from these results we can conclude that the methodology defined in this study can lead us to a model which works well in low as well as high cycle region.

Figure 9: S-N curves for C40 steel for R= -1 and phase 90 with Predicted curve from one more calibration point

## CONCLUSION

A methodology has been proposed to apply the endurance function model with GA to estimate fatigue life. In the proposed methodology application of FEA was included which in turn simplified the endurance function model by reducing one coefficient for notch gradient correction factor. Also application of FEA resulted in simplified method for determining the stress tensor and deviatoric stress invariants. Interpolation technique is introduced in the proposed methodology to estimate the coefficients at each load point using calibration point coefficients which resulted in better representation of fatigue behavior from endurance function model. The results show that endurance function model is in good agreement with experimental fatigue life data for in-phase loading and also to some extent for out of phase loading but due to less data points in experimental data high cycle region is not accurately predicted. Scheme to use more than two calibration points for one data set shows improved prediction in fatigue life. From the study it is concluded that the proposed methodology for endurance function model is resulted in accurate prediction of fatigue life for stable fatigue life behavior of material, further development is needed to accommodate the non linear behavior of material to increase the application region of endurance function model with proposed methodology, to variable amplitude and random loading cases.