fatigue failure criteria

8/9/2019 Fatigue Failure Criteria

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Fatigue Failure Criteria

The simplified failure envelopes for composite materials are not

derived from physical theories of failure, in which the actual physical

processes that cause failure on a microscopic level are integrated to

obtain a failure theory. We, instead, deal with phenomenological theories

in which we ignore the actual failure mechanisms and concentrate on the

gross macroscopic events of failure. Phenomenological theories are based

on curve fitting, so they are failure criteria and not theories of any kind

(the term theory implies a formal derivation process). Phenomenological

theories are based on curve fitting, so they are failure criteria and not

theories of any kind (the term theory implies a formal derivation process).

Unfortunately, with curve fitting, the ability to determine the failure mode

is lost. That is, curve-fit failure criteria are generally disassociated with

knowledge of precisely how the material fails, only the occurrence of

failure is predicted and not the actual mode of failure. For conventional

engineering metals, the curve fitting process works fairly well; however,

the curve fitting process is less challenged for metals than for orthotropic

materials because metals are isotropic, so they do not have strengths in

different directions. Table (1) shows the well-known failure criteria for

orthotropic materials under plane stress state.

Bernasconi A. et al., conducted tensile and fatigue tests on glass

fiber reinforced polyamide specimens with different fiber orientations,

their results showed decreasing values of elastic modulus, ultimate tensile

stress and fatigue strength for increasing values of the orientation angle.

The Tsai-Hill criterion allowed to predict the dependence of the ultimate

tensile strength values upon the orientation angle with good accuracy.

Ming-Hwa R. Jen and Lee C.-H., because it was difficult for them

to measure the shear strength of unidirectional specimens in the


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laboratory they indirectly used the Tsai-Hill failure criterion with

measurements of tensile and compressive strengths of [45o]16 specimens

to estimate the shear strength.

Owen M. J. and Rice D. J. studied both the static and fatigue strength

of woven-roving GFRP under uniaxial and biaxial stress conditions using

both flat laminates and thin-walled tubes. The fatigue tests were done

under tension-tension and tension-compression conditions. Their work

was mainly concerned with different failure criteria and validating them

for woven-roving GFRP under the previously mentioned loading

conditions. They came to the following conclusions:

1- Woven-roving tubes appear to be stronger than flat

specimens under both tension-tension and tension-

compression. The reason for that was the absence of free

edges in tubular specimens.

2- The fit of an individual failure criterion varies from one type

of material to another.

3- Failure criteria disregard failure processes and hence, cannot

allow for interacting failure mechanisms. Apparently, the

degree to which a criterion fits the test results deteriorates

with increasing fatigue life.

4- For the tested cases, the modified Fischer criterion is the best

one that suits the static work, while the Norris-Distortional

Energy criterion provides the best fit for fatigue results. If

the same criterion is required for static and fatigue results,

then the Norris-Distortional energy is considered to be the

best-fit overall.


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Table (1): Failure criteria

No. Name Mathematical formula

1 Max. stress11

F ; 22 F ; 66 F

2 Max. strain2

12

1F

1 ;

1

1E

2E12

2

F2

;66

F

3 Hillcriterion 1

F

F

F

1

F

1

F

2

6

62

2

2212

22

1

2

1

1

4 TsaiHill

1F

F

F

F

2

6

62

2

2

21

212

1

1

5 Norrisinteraction 1

F

F

F

2

6

62

2

22

1

1

6 Norris

distortionalenergy

1F

F

FF

F

2

6

62

2

2

21

212

1

1

or 1F

2

1

1

or 1F

2

2

2

7 Hoffman1

F

FF

F-F

FF

F-F

FF

FF

2

6

62

2c2t

2t2c1

1c1t

1t1c

2c2t

22

1c1t

2121

8 ModifiedMarin 1

F

FF

F-F

FF

F-F

FF

FF

k2

6

62

2c2t

2t2c1

1c1t

1t1c

2c2t

22

1c1t

21221

where: K2is a floating constant.

9 TsaiWu

1

F

2H

FF

FF

F

1

F

1

F

1

F

12

6

62112

2c2t

22

1c1t

21

22c2t

11c1t

and the following condition must be fulfilled, for stability:

0HFFFF

1 212

2c2t1c1t


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Table (1) continue: Failure criteria.

No. Name Mathematical formula

10 Ashkenazi

12F

F

F

F

2112

2

6

62

2

22

1

1

26

22

21

2x

12F

1

F

1

F

1

40.5F where: xis the global stress of

45oin tension

11 Tsai -

Hahn

The same formula as TsaiWu, but H12takes the form:

2c2t1c1t12

FFFF

10.5-H

12 Cowin The same formula as TsaiWu, but H12takes the form:

262c2t1c1t

122F

1

FFFF

1H

13 Fischer

1F

F

F

kF

2

6

62

2

22

1

212

1

1

; where:

122121

122211

11EE2

1E1Ek

Where:

- (1) and (2) are the local stress components in directions (1)

and (2), respectively.

- (6) is the local shear stress component.

- (F1t) & (F1c) and (F2t) & (F2c) are the local tension and

compression strength components in directions (1) and (2),

respectively.

- (F6) is the local shear strength component.

- (12) & (21) represent Poissons ratios in the local directions.

- (E1)&(E2) are the local modullii of elasticity in directions (1)

and (2), respectively.

-

(H12) Normal interaction component of a strength tensor


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Owen M. J. et al., tested thin-walled tubes under combined axial

loading and internal pressure, both for static and dynamic loading. A

limited degree of agreement between failure criteria and results could be

obtained except those criteria that involve complex stress properties. For

static results, Norris-Interaction failure criterion provided the most

acceptable prediction and was thus useful when complex stress data is not

available. In some selected ratios of (y / x), where y and x are the

normal global stress components, Hoffman and Norris criteria gave better

fit with static results, and the Tsai-Wu criterion was strictly invalid since

the magnitude of H12 violated the imposed stability condition shown in

Table (1). They concluded that Norris-Interaction failure criterion was

reasonable for both static and fatigue rupture if xy = 0, but it was

unacceptable if xy 0 in fatigue loading. They also concluded that failure

criteria do not pay any heed to damage mechanisms or failure modes.

Yang G., used the Tsai-Hill criterion successfully for his

investigation of strength in the case of glass and carbon reinforced

composite plates subjected to off-axis tension and compression. He

reported that the application of this model for predicting the off-axis

shear stress vs. fiber angle was not useful.

The mean goal of the study of Fuchs C. et al. was to check the

extent of Tsai-Hill criterion applicability to polymer-polymer microfibril

reinforced composites for characterizing their mechanical behavior, and

they found that samples with such a structure provide a good scope for

verifying the application of Tsai-Hill criterion to polymer-polymer

composites.

Atcholi K. E. et al. concluded that, using Hill's function could give

good results if the fatigue cycle is symmetric (zero mean stress) and


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therefore, the smaller difference between tensile and compressive

strength has to be accounted for. This is true for unidirectional glass fiber

reinforced epoxy. For other materials with more complex structures,

complicated functions, as Tsai-Wu criterion may be required.

Guess T. R., tested thin-walled filament winding epoxy matrix

composites under uniaxial tensile or biaxial stress tests with two types of

winding orientation. The combined stress state of tension and hoop stress

due to internal pressure was performed. They mentioned that the

analytical prediction of elastic constants and strength of highly

anisotropic composites is complex and always requires experimental

verification.


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Applicability of Failure Criteria:

To evaluate the validity of the failure criteria, we shall consider the right

hand side of the equations representing the failure criteria as a relative

damage. The relation between the relative damage (R.D.) with the

number of cycles to failure (N) is constructed for different criteria. In

these curves as much as they are close to unity, this means the validity of

the particular criterion to the test conditions. If it less than unity, then the

criterion is predicting a specimen life more than the actual life of the

experiment.

Selecting Suitable Failure Criteria:

Many works had been done considering the suitability of failure criteria

to similar materials. Considering these works, we found that the most

widely used and suitable criteria for GFRP under different loading

conditions were the following criteria:

1- Hill

2- Tsai - Hahn

3-Norris - Distortional Energy

4- Tsai - Hill

5- TsaiWu

6-Norris & McKinnon

Case Study:

We used the local stress components of some fiber orientations [0,90o]2s,

[30o,-60

o]2s, [45

o]2s and [60

o,-30

o]2s specimens to be substituted in the

selected six criteria. This substitution has shown that all failure criteria

have the same form for the [0,90o]2s specimens, as shown in Table (2).

But for other fiber orientations there are different equations according to

the failure criterion.


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Table (2): Selected failure criteria.

Failure Criterion [30o,-60

o]2s, [45

o]2sand [60

o,-30

o]2s [0,90

o]2s Specimens

1- Hill

1221

21

2

6

6

2

2

2

2

1

1

FFFFF

1

2

6

6

2

1

1

FF

2- Tsai-Hahn

121

21

2

6

6

2

2

2

2

1

1

FFFFF

3- Norris-Distortional

4- TsaiHill

5- TsaiWu

1221

21

2

6

6

2

2

2

2

1

1

FFFFF

6- Norris & McKinnon

1

2

6

6

2

2

2

2

1

1

FFF

The Relative Damage for [0, 90o]2sSpecimens:

The relative damage (R.D.) were calculated, according to the

pervious failure criteria, for the [0, 90o]2s specimens under completely

reversed pure bending, completely reversed pure torsion and combined

bending and torsion fatigue loading with different negative and positive

stress ratios ( R = -1, -0.75, -0.5, -0.25, 0, 0.25, 0.5). Figure (6-23)

represents the relative damage for the [0, 90o]2s specimens against the

number of cycles to failure. The values of R.D. is far from unity. This

means that, the available different failure criteria are not suitable under

theses conditions and must be modified to best suit the studied case.


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The Relative Damage for [30o,-60

o]2s, [45

o]2sand [60

o,-30

o]2s

Specimens:

The figures represent the relative damage for the [30o,-60

o]2s,

[45o

]2s and [60o

,-30o

]2s specimens under completely reversed purebending, completely reversed pure torsion and combined bending and

torsion fatigue loading with different negative and positive stress ratios (

R = -1, -0.75, -0.5, -0.25, 0, 0.25, 0.5) against the number of cycles to

failure. From these Figures, it can be noticed that these faliure criteria are

not valid and must be modified.

Cycles to failure (N)

R.D. of all failure criteria for the [0, 90o]2sspecimens

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5Pure torsion R = - 1Pure bending R = - 1

R.D.


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Relative damage (R.D.) applying Hill failurecriterion for the [30o,-60

o]2sspecimens


Relative damage (R.D.) applying Tsai-Hahn failure criterion for the [30o,-60

o]2sspecimens

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

3.3

3.5

3.8R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5Pure torsion R = - 1Pure bending R = - 1

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

3.3

3.5

3.8


.

.


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Relative damage (R.D.) applying Tsai-Wu failure criterion for the [30o,-60

o]2sspecimens.

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

R = - 1

R = - 0.75R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5Pure torsion R = - 1Pure bending R = - 1

.

.


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Relative damage (R.D.) applying Norris & McKinnon failure criterion for the [30o,-60

o]2s

specimens


Relative damage (R.D.) applying Hill failure criterion for the [45o]2sspecimens

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5

Pure torsion R = - 1Pure bending R = - 1

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0.25R = 0.5Pure torsion R = - 1Pure bending R = - 1

R..

.

.


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Relative damage (R.D.) applying Tsai-Hahn failure criterion for the [45o]2sspecimens

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6


.

.


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Relative damage (R.D.) applying Tsai-Wu failure criterion for the [45o]2sspecimens


Relative damage (R.D.) applying Norris & McKinnon failure criterion for the [45

o

]2sspecimens

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0.25R = 0.5


1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8


.

.

.

.


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Relative damage (R.D.) applying Hill failure criterion for the [60o,-30

o]2sspecimens.

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0


R.D.


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Relative damage (R.D.) applying Tsai-Hahn failure criterion for the [60o,-30

o]2sspecimens


Relative damage (R.D.) applying Tsai-Wu failure criterion for the [60

o

,-30

o

]2sspecimens.

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

R = - 1R = - 0.75R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5


1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0

3.3

3.5

3.8


.

.

R.

.


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Relative damage (R.D.) applying Norris & McKinnon failure criterion for the [60o,-30

o]2s

specimens.

1E+2 1E+3 1E+4 1E+5 1E+6

0.0

0.3

0.5

0.8

1.0

1.3

1.5

1.8

2.0

2.3

2.5

2.8

3.0R = - 1R = - 0.75

R = - 0.5R = - 0.25R = 0R = 0.25R = 0.5Pure torsion R = - 1Pure bending R = - 1

.

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