High-Cycle Fatigue Analysis of a Cylindrical Test Specimen

This is a benchmark model for the Fatigue Module. A cylindrical test specimen is subjected to nonproportional loading. Three stress based models — Findley, Matake, and Normal stress — are compared to analytical values and to each other. The nonsmooth behavior of the Matake model is captured and discussed.

Model Definition

A cylindrical test specimen with a geometry according to Figure 1 is subjected to fatigue testing. A combination of an axial force and a twisting moment is applied in such a way that the stress scenario shown in Figure 2 is affecting the central thin part of the specimen. The maximum shear stress is present at the outer radius of the bar.

Figure 1: Geometry of the test specimen.

Figure 2: Stress history at the outer radius in the central part of the specimen.

This stress state can be simulated with a cyclic normal force of 15708 N and a cyclic twisting moment of 22.672 Nm.

The specimen is made from mild steel with a Young’s modulus E of 210 GPa and a Poisson’s ratio ν of 0.3.

The examined fatigue criteria are Findley, Matake, and Normal stress. Material parameters are calculated from uniaxial fatigue tests. Results from two tests are available.

In the first test, the material is tested in reversed tension-compression, which means that the stress amplitude is alternating around zero stress, and the endurance limit is found to be

MPa.

In the second test, the material is tested in pure tension, where the stress is pulsating between zero to two times the stress amplitude. The endurance limit is then

MPa. The subscript of the endurance limit shows the R-value of the test.

Findley parameters are related to the fatigue test data via

and can thus be computed as fF = 213 MPa and kF = 0.20.

Matake parameters are related to the fatigue test data via

which gives fM = 223 MPa and kF = 0.27. In the Normal stress model, the stress limit equals twice the stress amplitude of the fatigue test. The model does not take into account the R-value dependency, and therefore two different limits are obtained. In order to give conservative results, the lower limit,

,is chosen and thus the model parameter is fN = 576 MPa. This actually indicates that normal stress may not be a good criterion for this material.

Results and Discussion

The load cycle is obtained in four load steps. The resulting stress at the outer radius of the thin section follows the specification, see Figure 3.

Figure 3: Resulting stresses at the outer radius of the smallest section of the specimen.

The three fatigue criteria evaluated on the boundary of the specimen are presented in Figure 4, Figure 5, and Figure 6.

In this example, it is the central part of the specimen which is in focus for the evaluation, even though the maximum fatigue usage is slightly higher in the fillets.

The Findley model shows that the highest fatigue usage factor is found at the transition between the thin cylinder and the fillet, with the value being 0.99 against 0.95 in the center. A smooth transition in the results indicates that the specified search resolution for the critical plane is sufficient to correctly capture the fatigue response.

Figure 4: Fatigue usage factor Findley criterion.

The Matake criterion gives less smooth results. The reason can be found in the definition of the criterion, which selects the critical plane using the shear stresses only and then adds the normal stress. This is discussed in more detail on page 10. The analysis of the test specimen predicts a fatigue usage factor in the range from 0.79 to 0.99 in the central part, as compared to 1.03 at some points at the transition from the thin cylinder to the notch.

Figure 5: Fatigue usage factor Matake criterion.

The Normal Stress criterion predicts yet another fatigue usage factor. Again, the highest fatigue usage factor is found on the transition from the thin section to the notch, 0.93, as compared to the center of the specimen, where it is 0.88.

Figure 6: Fatigue usage factor Normal stress criterion.

The stress state on the surface in the central part of the specimen can be evaluated using analytical expressions. As there are no tractions on the boundary, it is in a state of plane stress. Therefore, the search for the critical plane can be reduced to two dimensions. In that case, the normal and shear stresses on a plane can be obtained as

(1)

where σa and σb are orthogonal normal stresses, and τab is the shear stress. The orientation angle

is calculated from the axis of σa toward σb. In the current example, the σa is the axial stress, and τab as the shear stress caused by the twisting moment.

In the current example, a search resolution of Q = 11 is used. This gives a 9° angular increment in the critical plane evaluation. Using Equation 1, the normal and shear stresses evaluated on discrete planes are given in Table 1 and Table 2. The number in the subscripts indicates the load case.

Table 1: Normal stress on discrete planes.
(°)	σn,1(MPa)	σn,2(MPa)	σn,3(MPa)	σn,4(MPa)	σn,max(MPa)	Δσn(MPa)
0	200	200	-200	-200	200	400
9	231	159	-231	-159	231	462
18	249	113	-249	-133	249	498
27	252	65	-252	-65	252	504
36	241	21	-241	-21	241	481
45	215	-15	-215	15	215	431
54	179	-41	-17	41	179	358
63	135	-52	-135	52	135	269
72	87	-49	-87	49	87	174
81	41	-31	-41	31	41	81
90	0	0	0	0	0	0
99	-31	41	31	-41	41	81
108	-49	87	49	-87	87	174
117	-52	135	52	-135	135	269
126	-41	179	41	-179	179	358
135	-15	215	15	-215	215	431
144	21	241	-21	-241	230	481
153	65	252	-65	-252	252	504
162	113	249	-113	-249	249	498
171	159	231	-159	231	231	462

Table 2: Shear stress on discrete planes.
(°)	τ1 (MPa)	τ2 (MPa)	τ3 (MPa)	τ4 (MPa)	Δτ (MPa)
0	115	-115	-115	115	230
9	79	-141	-79	141	281
18	35	-152	-35	152	304
27	-13	-149	13	149	298
36	-59	-131	59	131	262
45	-100	-100	100	100	200
54	-131	-59	131	59	262
63	-149	-13	149	13	298
72	-152	35	152	-35	304
81	-141	79	141	-79	281
90	-115	115	115	-115	231
99	-79	141	79	-141	281
108	-35	152	35	-152	304
117	13	149	-13	-149	298
126	59	131	-59	-131	262
135	100	100	-100	-100	200
144	131	59	-131	-59	262
153	149	13	-149	-13	298
162	152	-35	-152	35	304
171	141	-79	-141	79	281

The Findley criterion searches for a critical plane where a combination between the shear stress range and the normal stress is highest. This stress is shown in the second column of Table 3, where it can be seen that 202 MPa is found at planes oriented at 18° and 162°. This results into a usage factor of 202/213 = 0.95, which is in good agreement with the computed results.

Table 3: Fatigue stress.
(°)	Δτ/2+0.20σn (MPa)	Δτ/2+0.27σn (MPa)	Δσn(MPa)
0	155	169	400
9	187	203	462
18	202	219	498
27	199	217	504
36	179	196	481
45	143	158	431
54	167	179	358
63	176	185	269
72	170	175	174
81	149	152	81
90	115	115	0
99	149	152	81
108	170	175	174
117	176	185	269
126	167	179	358
135	143	158	431
144	179	196	481
153	199	217	504
162	202	219	498
171	187	203	462

The Matake criterion selects the critical plane as the one with the largest shear stress range. This occurs at orientations 18°, 72°, 108°, and 162° where the range is 304 MPa, see Table 2. The maximum normal stress on those planes is either 87 MPa or 249 MPa, see Table 1 and the Matake stress is either 219 MPa or 175 MPa, see Table 3. Since the Matake criterion does not contain the normal stress in the selection of the critical plane, the fatigue criteria is calculated with either one of them. In Figure 5, this feature is demonstrated by the nonsmooth results. The analytical values at the critical planes for the Matake usage factor are 219/223 = 0.98 and 175/223 = 0.78, which is in good agreement with the computed results.

The Normal stress criterion considers the plane with the largest normal stress range. The last column of Table 3 shows that this is found on planes with orientations 27° and 153° where the range is 504 MPa. This results into a fatigue usage factor of 504/576 = 0.88, which is in good agreement with the computed results.

Notes About the COMSOL Implementation

In the critical plane evaluation, a search resolution Q indicates the number of evaluation point along 90° of a unit circle. The angle between two evaluation points is then at most (90°)/(Q − 1).

Fatigue_Module/Verification_Examples/cylindrical_test_specimen

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select .

Click .

Click

In the tree, select .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
Moment	22.672 [N*m]	22.672 N·m	Twisting moment
Force	15708 [N]	15708 N	Normal force

Geometry 1

Work Plane 1 (wp1)

In the toolbar, click

In the window for , click

Work Plane 1 (wp1)>Plane Geometry

In the window, click .

Work Plane 1 (wp1)>Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

In the text field, type 5 5 0 0 10 10 6.01.

In the text field, type 20 0 0 50 50 30 30.

Click the

button in the toolbar.

Work Plane 1 (wp1)>Quadratic Bézier 1 (qb1)

In the toolbar, click

and choose .

In the window for , locate the section.

In row , set to 6.01.

In row , set to 5.25.

In row , set to 5.

In row , set to 30.

In row , set to 25.

In row , set to 20.

Work Plane 1 (wp1)>Convert to Solid 1 (csol1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , click

Work Plane 1 (wp1)>Fillet 1 (fil1)

In the toolbar, click

On the object , select Point 5 only.

In the window for , locate the section.

In the text field, type 4.

Work Plane 1 (wp1)>Mirror 1 (mir1)

In the toolbar, click

and choose .

Select the object only.

In the window for , locate the section.

In the text field, type 0.

In the text field, type 1.

Locate the section. Select the check box.

Click

Click the

button in the toolbar.

Work Plane 1 (wp1)>Union 1 (uni1)

In the toolbar, click

and choose .

Click in the window and then press Ctrl+A to select both objects.

In the window for , locate the section.

Clear the check box.

In the toolbar, click

Revolve 1 (rev1)

In the window, under right-click and choose .

In the window for , locate the section.

Clear the check box.

Click

Click the

button in the toolbar.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Young’s modulus	E	210e9	Pa	Young’s modulus and Poisson’s ratio
Poisson’s ratio	nu	0.3	1	Young’s modulus and Poisson’s ratio
Density	rho	7800	kg/m³	Basic

Solid Mechanics (solid)

Fixed Constraint 1

In the window, under right-click and choose .

Select Boundaries 3, 4, 25, and 27 only.

Rigid Connector 1

In the toolbar, click

and choose .

Select Boundaries 11, 12, 47, and 48 only.

Applied Force 1

In the toolbar, click

and choose .

In the window for , locate the section.

Specify the F vector as


0	x
Force	y
0	z

In the toolbar, click

and choose .

Rigid Connector 1

In the window, click .

Applied Moment 1

In the toolbar, click

and choose .

In the window for , locate the section.

Specify the M vector as


0	x
Moment	y
0	z

In the toolbar, click

and choose .

Global Definitions

Load Group: Force

In the window, under click .

In the window for , type Load Group: Force in the text field.

In the text field, type lgF.

Load Group: Moment

In the window, under click .

In the window for , type Load Group: Moment in the text field.

In the text field, type lgM.

Create a load cycle consisting of four load cases.

Study 1

Step 1: Stationary

In the window, under click .

In the window for , click to expand the section.

Select the check box.

Click four times.

In the table, enter the following settings:


Load case	lgF	Weight	lgM	Weight
Load case 1	√	1.0	√	1.0
Load case 2	√	1.0	√	-1.0
Load case 3	√	-1.0	√	-1.0
Load case 4	√	-1.0	√	1.0

In the toolbar, click

Results

Verify stresses at the center of the test specimen.

Stresses

In the toolbar, click

and choose .

In the window for , type Stresses in the text field.

Point Graph 1

Right-click and choose .

Select Point 20 only.

It might be easier to select the correct point by using the window. To open this window, in the toolbar click and choose . (If you are running the cross-platform desktop, you find in the main menu.)

In the window for , click in the upper-right corner of the section. From the menu, choose .

Locate the section. From the list, choose .

Click to expand the section. Select the check box.

From the list, choose .

In the table, enter the following settings:


Legends
normal stress

Point Graph 2

Right-click and choose .

In the window for , locate the section.

In the text field, type solid.SGpXY.

Locate the section. In the table, enter the following settings:


Legends
shear stress

Point Graph 3

Right-click and choose .

In the window for , locate the section.

In the text field, type solid.misesGp.

Locate the section. In the table, enter the following settings:


Legends
von Mises

Stresses

In the window, click .

In the window for , click to expand the section.

From the list, choose .

Locate the section.

Select the check box. In the associated text field, type Stress (MPa).

In the toolbar, click

Add Physics

In the toolbar, click

to open the window.

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

Fatigue Findley

In the window for , type Fatigue Findley in the text field.

Stress-Based 1

Right-click and choose the boundary evaluation .

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Add Physics

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

Fatigue Matake

In the window for , type Fatigue Matake in the text field.

Right-click and choose the boundary evaluation .

Stress-Based 1

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Add Physics

Go to the window.

In the tree, select .

Find the subsection. In the table, clear the check box for .

Click in the window toolbar.

In the toolbar, click

to close the window.

Fatigue Normal Stress

In the window for , type Fatigue Normal Stress in the text field.

Right-click and choose the boundary evaluation .

Stress-Based 1

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Materials

Material 2 (mat2)

In the window, under right-click and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Normal stress sensitivity coefficient	k_Findley	0.20	1	Findley
Limit factor	f_Findley	213[MPa]	Pa	Findley
Normal stress sensitivity coefficient	k_Matake	0.27	1	Matake
Limit factor	f_Matake	223[MPa]	Pa	Matake
Limit factor	f_NormalStress	576[MPa]	Pa	Normal stress