Accumulation of Fatigue Damage Using Memory of the Material

Available online at www.sciencedirect.com

ScienceDirect Procedia Materials Science 3 (2014) 2 – 7

20th European Conference on Fracture (ECF20)

Accumulation of Fatigue Damage using Memory of the Material Ewelina Böhma, Marta Kureka, Grzegorz Junakb, Marek Cieślab, Tadeusz Łagodaa* a

Opole University of Technology, Department of Mechanics and Machines Design, ul. Mikołajczyka 5, 45-271 Opole, Poland b Silesian University of Technology , Department of Mechanics of Materials, ul. Krasińskiego 8, 40-019 Katowice, Poland

Abstract This work proposes a new model for cumulation of fatigue failures based on material memory, developed on the grounds of memorising senseless material. Cumulation of fatigue failures (D) is based on employing time history of the trajectory and cumulation of fatigue failures with the help of a coefficient dependent on material memory. Moreover, the work presents comparison of experimental and numerical studies for P91 and P92 steels using the model developed by authors. Theoretical assumptions have been confirmed by obtained results of numerical computations. © Ltd. This is an openby access article under the CC BY-NC-ND license ©2014 2014Elsevier The Authors. Published Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department Structural Engineering. of Structural Engineering Keywords: cumulation of failures, Palmgren-Miner, material memory;

Nomenclature a b c d i m

memorising rate forgetting rate asymptote 1/b – the reverse of forgetting rate given in number of cycles given load cycle the volume of memorised information

* Corresponding author. Tel.: +48 77 449 8415; fax: +48 77 449 9934. E-mail address: [email protected]

2211-8128 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Selection and peer-review under responsibility of the Norwegian University of Science and Technology (NTNU), Department of Structural Engineering doi:10.1016/j.mspro.2014.06.002

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Ewelina Böhm et al. / Procedia Materials Science 3 (2014) 2 – 7

mi N Nfi t εa λ ψ

forgetting rate total number of cycles until failure number of cycles determined using Basquin characteristic time strain amplitude non-volatile memory level forgetting rate

1. Introduction Fatigue is a combination of phenomena occurring in material and permanently affecting the time of its safe use. It is caused by loads changing in time, which are dependent on their duration and size [15]. Loads responsible for fatigue phenomenon are called fatigue loads (they may be either constant-amplitude or with random trajectories). These loads are cyclic in their nature, but their amplitudes may differ considerably. An essential element of fatigue durability assessment is to evaluate material or structural element failure. Fatigue cracks occur as a result of reaching by material certain defect level - in this case we speak of cumulation of failures. The rate of this phenomenon occurrence depends to a large extent on material microstructure and characteristics, and range and intensity of stresses increasing with number of cycles. These three factors are most often taken into consideration in failure cumulation process. The literature contains great number of linear and non-linear hypotheses on fatigue failure cumulations. The most commonly used linear hypothesis for summing up these failures was formulated by Palmgren (1924) [13], who examined ball bearings. It assumes that for constant-amplitude stress each cycle contributes to the occurrence of defects in the same way [15]. Due to its simplicity and historic connotations, currently it is the most frequently used hypothesis, both by engineers and in applications for fatigue computations. As presented in the earlier publications by the authors [8,9]. Many models, also from the nonlinear group, are often not used for materials and loads, for which they would work better than currently applied Palmgren hypothesis modified by Miner, which is commonly called in the circles the Palmgren-Miner hypothesis (1945) [12]. This model was also analysed by authors in their former works [14]. We distinguish several sample nonlinear failure cumulation methods. Among other, they include: Corten – Dolan hypothesis [3], Marco and Starkey hypothesis [4], Mason Halford hypothesis [11], Henry hypothesis [5]. While analysing Polish and foreign literature, we see the trend to create models based on statistical relations referring to natural phenomena linked with appropriate probability distributions. The work of Yung Li et al. [16] presents an approach, which takes into account also the development of fatigue crack resulting from failure, and failure impact on nucleus nucleation effect (nucleus is the point in material, where fatigue will be initiated). Authors of this work have attempted to propose methodology making use of theoretical assumptions taken from the field of psychology. This work presents a hypothesis for cumulation of fatigue defects using the so-called material memory. The work discusses extensive low-cycle experimental research studies carried out for P91 and P92 steels, involving block loads. Computations have been performed, which use the model developed by authors. Moreover, the model is shown against a background of results obtained using the Palmgren-Miner hypothesis. 2. The functions of forgetting and memorising The terms of memorising and forgetting have existed in psychology for a long time now. Generally, in the literature there are two models describing curves characterising both forgetting and memorising functions. Roughly, the Ebbinghaus forgetting curve [1] may be defined as an exponential function:

m

a  c e bt  c

(1)

where: m – the volume of memorised information, a – memorising rate, b – forgetting rate, c – asymptote, t – time.

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The above functions have been proposed to describe human memory when memorising the so-called senseless material. It seems that much the same approach may be applied when determining fatigue durability of structural components subject to loads characterised by variable amplitudes. Similar approach has been shown e.g. in works [7,10]. Moreover, in the proposed model it is required to switch from time function to the number of cycles, which is used for the purposes of fatigue failure cumulation. Considering this, we will introduce a new coefficient m, which is the weight determining the impact of information on ultimate material memory: m

a  c e



N d

c ,

(2)

where: N – total number of cycles until failure, d=1/b – the reverse of forgetting rate given in number of cycles. Finally, failure degree after time T o may be determined using the following formula: n

D'

1 , N fi

¦m

i

i 1

(3)

where: i – given load cycle, Nfi - the number of cycles determined from Basquin or Wöhler characteristic, mi – forgetting rate determined using formula (3). Formula (2) shows that for the purposes of the proposed model it is required to determine three material constants: a, c and d. The above model also has to be correct for cyclic loads. Considering this, we insert formula (2) in (3) and write in integral form: Nf

D'

³ o

where Nf

N º  1 ª «a  c e d  c » dN , Nf « »¼ ¬

¦n

i

(4)

- is total number of cycles until failure.

i

Assuming that summary failure degree S(T)=D=1, we receive: Nf

³ o

N º  1 ª «a  c e d  c » dN Nf « »¼ ¬

1

(5)

After having integrated formula (5), we obtain:

a

1  c N f Nf §  d ¨¨1  e d ¨ ©

· ¸ ¸¸ ¹

c.

In simplest case, taking asymptote c=0 in formula (6), we receive:

(6)

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Ewelina Böhm et al. / Procedia Materials Science 3 (2014) 2 – 7

a

Nf Nf §  ¨ d ¨1  e d ¨ ©

· ¸ ¸¸ ¹

.

(7)

Finally, on the basis of formulae (2) and (7), and taking c = 0, we obtain:

m

Nf

e



N d

Nf d §  ¨ d ¨¨1  e ©

(8)

· ¸ ¸¸ ¹

3. Experimental studies Experimental data is required in order to verify assumptions presented in previous paragraph. Data taken from work by Junak and Cieśla [6] was used for this purpose. Samples made of P91 and P92 steels were used in lowcycle studies. These steels are used in chemical and power industries e.g. to make pipelines or components of conventional boiler or tank batteries. Table 1 presents chemical constitution of these steels, whereas Table 2 contains their basic mechanical properties [6]. Table 1. Chemical constitution of P91 and P92 steels [6]. Content of an element, wt. %, Fe the rest Steel

C

Si

Mn

P

S

Ni

Cr

Mo

P91

0.10

0.36

0.42

0.017

0.004

0.13

8.75

0.96

W -

P92

0.11

0.30

0.60

0.017

0.004

0.20

9.50

0.50

1.90

Table 2. Basic mechanical properties of P91 and P92 steels [6]. Steel

Rm [MPa]

R0,2 [MPa]

A5 [%]

Z [%]

Rz100000/600C

P91 P92

694.4 705.7

544.6 529.7

23.8 25.4

71.4 67.5

98 132

The tests were carried out at room temperature for controlled deformation. In case of block loads they were performed for two strain ranges: Δεt=0.6 and 1.2%. For P91 steel the studies were carried out for 3780 and 720 cycles, while for P92 steel - for 3304 and 640 cycles. Table 3 shows fatigue characteristics computed using Manson-Coffin relation:

Ha

V'f (

˜ (2 ˜ N f ) b  H ' f ˜ (2 ˜ N f ) c

(9)

Table 3. Fatigue characteristics according to Manson-Coffin . Steel

V 'f /(

ܾ

ߝ௙ᇱ

ܿ

P91 P92

0.065 0.063

-0.028 -0.028

1.02 1.01

-0.74 -0.64

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4. Verification of the proposed cumulation model According to the Palmgren - Miner model assumptions, the sum of failures should be equal to one. Independently of block working sequence, the results will remain near that value. Whereas, assumptions of the authors’ model are based on the principle that the material remembers the loads it has been subject to, and this has measurable impact on cumulation of failures. We will receive different results for the cases, where firstly we have block with higher load amplitude, and then with lower one, and other results for the opposite case. Below, there are cumulative tables with results obtained through computations performed using Palmgren-Miner model: j

1

¦N

D

i 1

(10)

fi

and the model proposed by the authors, formula (3), using (8) and (10) in final form. j

Nf



N d

1 (11) d § · N fi  i 1 ¸ ¨ d ¸¸ ¨¨1  e ¹ © Table 4 contains computations for P91 steel in the combination: high amplitude (H) – low amplitude (L), and Table 5 the other way round, that is for the combination: low amplitude – high amplitude. Tables 6 and 7 present computations for P92 steel analogically as for P91 steel, also in both combinations. D'

¦

e

Nf

Table 4. Comparative table for P91 steel in combination high amplitude (H) – low amplitude (L). N1/N1f

N2/N2f

0.10 0.83 0.18 0.75 0.48 0.32 0.60 0.25 0.72 0.15 Average damage degree

Palmgren-Miner (10)

(11) d=N

(11) d=N/2

(11) d=2N

0.93 0.93 0.80 0.85 0.87 0.88

1.05 1.10 0.93 0.96 0.94 1.00

0.79 0.86 0.74 0.74 0.70 0.77

1.24 1.26 1.07 1.12 1.12 1.16

(11) d=N 1.01 1.06 1.15 1.19 1.13 1.11

(11) d=N/2 0.57 0.60 0.66 0.71 0.71 0.65

(11) d=2N 1.39 1.46 1.56 1.61 1.50 1.50

(11) d=N 1.07 1.00 1.05 0.95 0.92 0.97 0.90 0.97 0.98

(11) d=N/2 0.80 0.77 0.83 0.77 0.73 0.75 0.67 0.72 0.76

(11) d=2N 1.26 1.15 1.19 1.08 1.06 1.14 1.08 1.21 1.15

Table 5. Comparative table for P91 steel in combination low amplitude (L) – high amplitude (H). N1/N1f

N2/N2f

0.28 0.95 0.37 0.92 0.54 0.84 0.66 0.75 0.81 0.47 Average damage degree

Palmgren-Miner (10) 1.23 1.29 1.38 1.41 1.28 1.32

Table 6. Comparative table for P92 steel in combination high amplitude (H) – low amplitude (L). N1/N1f

N2/N2f

0.10 0.85 0.15 0.70 0.24 0.63 0.32 0.47 0.47 0.32 0.63 0.24 0.70 0.14 0.85 0.10 Average damage degree

Palmgren-Miner (10) 0.95 0.85 0.87 0.79 0.79 0.87 0.84 0.95 0.86

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Ewelina Böhm et al. / Procedia Materials Science 3 (2014) 2 – 7 Table 7. Comparative table for P92 steel in combination low amplitude (L) – high amplitude (H). N1/N1f

N2/N2f

0.22 0.92 0.45 0.78 0.59 0.63 0.70 0.47 0.85 0.23 Average damage degree

Palmgren-Miner (10) 1.14 1.23 1.22 1.17 1.08 1.17

(11) d=N 0.95 1.02 1.04 1.03 1.01 1.01

(11) d=N/2 0.53 0.59 0.62 0.63 0.65 0.60

(11) d=2N 1.30 1.39 1.40 1.36 1.30 1.35

5. Conclusions and observations The following observations and conclusions may be formulated following results analysis: x x x x

Combination of theories widely applied in psychology with scientific field of material fatigue gave extremely advantageous results in form of a new model for cumulation of failures. Comparison of obtained results to the results of experimental studies allowed proving rightness of the proposed model for the assessment of fatigue durability for materials where, according to the PM, failure level is different than 1, and for the proposed model it approaches expected value of 1. Change in the value of coefficient d significantly affects the failure cumulation process. Completion of a series of experimental studies confirming the model assumptions will affect the degree of its reliability and usability for wider group of materials and load types.

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