Critical Plane Approach in Stage I and Stage II of Fatigue Under Multiaxial Loading A. KAROLCZUK E. MACHA Opole University of Technology, Department of Mechanics and Machine Design, POLAND XIII INTERNATIONAL COLLOQUIUM, “MECHANICAL FATIGUE OF METALS” September 25-28, 2006, Ternopil, Ukraine
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Critical Plane Approach in Stage I and Stage II of Fatigue Under Multiaxial Loading
A. KAROLCZUKE. MACHA
Opole University of Technology, Department of Mechanics and Machine Design, POLAND
XIII INTERNATIONAL COLLOQUIUM, “MECHANICAL FATIGUE OF METALS”September 25-28, 2006, Ternopil, Ukraine
2
Plan of the presentation
• Introduction
• A brief review of some multiaxial fatigue failure criteria based on the critical plane approach
- The Findley Criterion- The Matake Criterion- The Maximum Normal Stress Criterion on the Critical Plane
• Damage Degree Accumulation and Fatigue Life Calculation
• Fatigue tests
• Stress and strain computations
• Evaluation and discussion
• Conclusions
3
Introduction
• The critical plane approach (Stanfield 1935)
This approach is based upon the experimental observation that fatigue cracks initiate and grow on certain material planes.
Assumption:Only stress or/and strain components acting on the critical plane are responsible for the material fatigue failure.
The critical plane criteria define different functions that combine the shear and normal stress or/and strain components on a plane into one equivalent parameter.
4
Introduction
Problems:- orientation of the critical plane- orientation of the fracture plane
- It is commonly accepted that depends on the test conditions (loading level, temperature, material type, state of stress, ect.) material generally forms one of the two types of cracks - shear cracks or tensile cracks.
- The shear cracks are formed on the maximum shear stress plane and Forsyth called this process as Stage I.
- The tensile cracks are formed in Stage II that is predominated by the maximum normal stress component.
Either under multiaxial and uniaxial fatigue tests the cracks may initiate and propagate on different planes – contradictory to the one critical plane orientation.
5
Multiaxial fatigue failure criteria
The Findley Criterion
fk nans max,,
The critical plane orientation coincides with the plane orientation where the maximum value of this linear combination occurs.
m
fafnans N
Nk
1
max,,
where af, m are the fatigue limit and the exponent of the S-N curve for fully reversed (R=-1) torsion loading, respectively; Nf is the considered number of cycles to failure; N is the number of cycles corresponding to the fatigue limit af for fully reversed torsion loading.
6
Multiaxial fatigue failure criteria
The Findley criterion adapted to random loading:
)()()( tktt nnseq
The equivalent shear stress history eq(t) at observation time T is then used as the cyclic counting variable.
7
The Matake Criterion
Multiaxial fatigue failure criteria
m
fafanans N
Nk
1
,,
The critical plane orientation coincides with the maximum shear stress amplitude
12 af
afk
The Matake criterion adapted to random loading:
)()()( tktt nnseq
8
The Maximum Normal Stress Criterion on the Critical Plane (max{n})
Multiaxial fatigue failure criteria
m
fafan N
N1
,
where m is the exponent of the S-N curve for fully reversed (R=-1) push-pull loading; N is the number of cycles corresponding to the fatigue limit af.
For random loading the equivalent stress history is as follows
)()( tt neq
9
Damage Degree Accumulation
For the variable-amplitude loading, the linear damage accumulation hypotheses of Sorensen-Kogayev is used.
j
i
iDp
D1
)(1
where damage degree is computed by the general equation as follows
afiaeq
afiaeqi
f
i
i
aFFfor
aFFforN
n
D)(,
)(,)(
)(
)(
0
where F is the generalised fatigue damage parameter (for the Findley criterion: F=), n(i) is the number of cycles assigned into the i-th stress level, a is a coefficient allowing to include amplitudes below Faf in the damage accumulation, is a computed
number of cycles to failure (S-N curve) for the i-th stress level
)(ifN
Nf
Fa
S-N curve)(,iaeqF
)(ifN
10
Damage Degree Accumulation
For Serensen-Kogayev p is calculated according to the following equations:
max
,
1
)()(,
afaeq
j
iaf
iiaeq
aFF
aFfFp
j
i
i
ii
n
nf
1
)(
)()(
where f(i) is the frequency of the i-th stress/strain level, is the maximum amplitude of the generalised fatigue damage parameter (F=, , or )
max,aeqF
Accumulated damage degree D at observation time T is used to estimate the fatigue life according to the following expression
)(TD
TTcal
11
Fatigue tests
Fatigue tests were performed on the round full cross-section specimen made of 18G2A (A 765-94 ASTM) steel under constant- and variable-amplitude combined bending and torsion moments (bending: Mb(t), torsion: Mt(t))
No δ Mb,max M
rad Nm - [o] [o]
Constant-amplitude loading
1 0 8.0; 10.0; 10.3 0.68 18.1 17.1 19.0
2 0 6.4; 7.4; 8.2; 9.8 0.96 21.9 20.0 23.8
3 0 5.3; 6.2; 7.2; 1.44 26.5 23.8 29.2
4 π/2 8.9; 9.2; 9.6; 10.3 0.68 12.3 9.1 15.5
5 π/2 8.3 0.98 8.4 7.3 9.5
6 π/2 6.4; 7.2 1.42 10.2 6.4 13.9
Variable-amplitude loading
7 - 18.4 (torsion) ∞ 43.6/86.3 42.245.0/82.3 90.2
8 - 16.3 (bending) 0 1.5 0.82.2
exp̂ maxmin ˆˆ
26 28 30 32 34 36-1
0
1Parts of the primary loading course
25 25.2 25.4 25.6 25.8 26 26.2-0.5
0
0.5
time, s
12
Fatigue tests
(i) one general crack orientation at the macroscale was observed under all the investigated constant-amplitude loadings (=zx,max/zz,max0.71) and under the variable-amplitude bending;
(i) (ii)
(ii) two crack orientations were observed for the specimens subjected to variable-amplitude torsion. The first orientation with a crack length of around 0.3-0.6 mm is parallel to the specimen axis. The other orientation comes from branching of the primary crack, and these branching directions are inclined to the specimen axis by around 45o
Two fatigue crack behaviours were noticed:
13
Stress and strain computations
Stress and strain histories in an arbitrary point (x, y) of the specimen cross-section were computed from bending and torsion moments Mb(t), Mt(t) considering the plastic strains.
The Chu multi-surface plasticity model of material behaviour with kinematic hardening was applied to determine the strain-stress relation and the influence of loading history on the strain state at each point of the specimen cross-section.
It is assumed that the material in some regions of the specimen cross-section could be in the elastic–plastic state. However, it was also assumed, according to hypothesis that the beam cross-section remains flat, that the strains change along the specimen cross-section in a linear way.
14
For every increment of bending Mb(t) and torsion Mt(t) moments the following quasi-static equilibrium equations were solved by the trust-region method
Stress and strain computations
0)(),,( tMydAtyx b
A
zz 0)(),( tMdAt t
A
z
Quarter of specimen cross-section and mesh
15
Exemplary stress courses
16
Evaluation and discussion
Evaluation concerns:
• Orientation of the critical plane versus fracture plane orientation• Fatigue lives
Analysis was performed with the use of local approach, hot spot: y=R, x=0
hot spot
17
Evaluation and discussion
Orientation of the critical plane versus fracture plane orientation
The following error parameters were used to compare the experimental macroscopic fracture plane orientation with the critical plane orientation
)()(exp
)( ˆ ical
iiE
N
i
im E
NE
1
)(,
1
0
5
10
15
20
25
30
Fatigue criterion
E
,m, %
Matake
Findley
max(n)
k=0.2k=0.4
k=0.8
k=1.6
18
Evaluation and discussion
Fatigue lives
The following errors parameters were applied for the fatigue life verification:
)(exp
)()( log
i
icali
T
TE
N
i
im E
NE
1
)(1
N
im
istd EE
NE
1
2)(
1
1
The mean error parameter Em reflects the general results conformity.
The standard deviation error parameter Estd is the superior parameter since it reflects the scatter of the results and therefore gives us the information about the failure criterion ability to correlate the different kind of multiaxial stress states and the equivalent damage parameter.
19
Evaluation and discussion
102
104
106
108
102
103
104
105
106
107
108
Texp
, s
Tca
l, s
3
3
Em =-0.4
Estd
=0.56
=0, =0.29
=0, =0.44
=0, =0.68
=/2, =0.50
=/2, =0.62
=/2, =0.71
var.amp.-torsionvar.amp.-bending
102
104
106
108
102
103
104
105
106
107
108
Texp
, s
Tca
l, s
3
3
Em =0.05
Estd
=0.69
=0, =0.29
=0, =0.44
=0, =0.68
=/2, =0.50
=/2, =0.62
=/2, =0.71
var.amp.-torsionvar.amp.-bending
Comparison between the experimental fatigue lives Texp and the calculated fatigue lives Tcal: (a) the maximum normal stress criterion,
(a) (b)
(b) the maximum shear stress criterion (the Findley criterion for k = 0)
20
Evaluation and discussion
102
104
106
108
102
103
104
105
106
107
108
Texp
, s
Tca
l, s
3
3
Em =-1.14
Estd
=0.46
=0, =0.29
=0, =0.44
=0, =0.68
=/2, =0.50
=/2, =0.62
=/2, =0.71
var.amp.-torsionvar.amp.-bending
102
104
106
108
102
103
104
105
106
107
108
Texp
, s
Tca
l, s
3
3
Em =-0.58
Estd
=0.32
=0, =0.29
=0, =0.44
=0, =0.68
=/2, =0.50
=/2, =0.62
=/2, =0.71
var.amp.-torsionvar.amp.-bending
(c) (d)
Comparison between the experimental fatigue lives Texp and the calculated fatigue lives Tcal: (c) the Matake criterion, k=0.4,
It was assumed that selection of fatigue criterion could be made by the maximum damage degree computed by two simple criteria, i.e. the maximum normal stress criterion and the maximum shear stress criterion. For each specimen, the damage degree on the critical plane is computed by these two criteria (max{n,ns}) and than the fatigue life Tcal is determined by the highest damage degree.
(d) the max{n,ns} criterion
21
Evaluation and discussion
)(exp
)()( log
i
icali
T
TE
N
i
im E
NE
1
)(1
N
im
istd EE
NE
1
2)(
1
1
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
Fatigue criterion
Em
, %
Matake
Findley
max(n)
max(n,
ns)
k=0.0 k=0.2 k=0.4 k=0.8
0
0.5
1
1.5
Fatigue criterion
Est
d, %
Matake
Findley
max(n)
max(n,ns)
k=0.0
k=0.2 k=0.4
k=0.8
22
Conclusions
The following general conclusion appears:
1. The critical plane and the fracture plane notions must be separated. The critical plane is simply a plane that used in the fatigue life assessment. The fracture plane at the microscale/macroscale is a plane where material cohesion is lost.
2. Depending on loading levels, state of stress etc. the critical plane and fracture plane orientations may or not coincide. We postulate that the critical plane approach may be successfully used in the fatigue life estimation under different test conditions but the proposed damage parameter should be equivalent to the uniaxial one not only in term of the total fatigue life but also in term of the macroscopic fracture plane behaviour.
Related Documents
