Study of the size effect for non-alloy steels S235JR, S355J2+C and acid-resistant steel 1.4301 Tomasz Tomaszews omaszewski ki,, and Przemysław Strzelecki
Citation: AIP Citation: AIP Conference Proceedings Proceedings 1780, 020008 (2016); doi: 10.1063/1.4965940 View online: http://dx.doi.org/10.1063/1.4965940 View Table of Contents: http://aip.scitation.org/toc/apc/1780/1 Published by the American the American Institute Institute of Physics
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Application of Weibull distribution distribution to describe S-N curve with with using small number number specimens AIP Conference Proceedings 1780, 020007 (2016); 10.1063/1.4965939
Study of the Size Effect for Non-Alloy Steels S235JR, S355J2+C and Acid-Resistant Steel 1.4301 Tomasz Tomaszewski 1
1, a)
and Przemysław Strzelecki
1, b)
University of Science and Technology in Bydgoszcz, al. Prof. S. Kaliskiego 7, 85-791 Bydgoszcz, Poland a)
[email protected] [email protected]
b)
Abstract. A change in th e dimensions of a specimen vs vs.. the reference size can affect fatigue properties of a material. This phenomenon is referred to to as the “size effect”. Th e study concerned checked how this effect can influence non-alloy steels S235JR, S355J2+C and acid-resistant steel 1.4301. Two specimen sizes and two load types were tested. Selected models of the size effect were verified to estimate fatigue strength for a cross-sectional area other than the one obtained experimentally.
INTRODUCTION Failing to address the size effect in engineering computations can lead to major size estimation errors. It is particularly important for non-homogeneous materials [3]. The size effect is a complex phenomenon depending on material structure, specimen shape, load type and component manufacturing processes [6]. As a consequence, we do not have any generic analytical or numerical models for estimating the effect. The magnitude of influence of the size effect on fatigue properties depends on material type and its local structural features (grain size, micro-cracks, inclusions, discontinuities, dislocations and other flaws) [2]. Generally, it is assumed that fatigue resistance of a material is reversely proportional to the object size. This change is typically non-linear, observed up to a certain cut-off value [9]. See Fig. 1 for a schematic representation of this relation (the “O” labels stand for cut-off values of change in K and S o). K , e zi s n oi t c e s s
1 s o r c f o
K oo K t n ei ci
smaller specimen f f e o
standard specimen
real-life object
C
S on S on
S oo S
Cross-sectional area, area, S S oo FIGURE 1. Schematic presentation of relation between coefficient of cross-sectional area K and the cross-sectional area of the object; own study based on [9]
Fatigue Failure and Fracture Mechanics XXVI AIP Conf. Proc. 1780, 020008-1–020008-8; doi: 10.1063/1.4965940 Published by AIP Publishing. 978-0-7354-1442-6/$30.00
020008-1
The study aimed to check consequences of the size effect for selected construction materials (acid-resistant steel 1.4301 and non-alloy steels S235JR, S355J2+C). Its monotonic and fatigue tests tested 2 specimen sizes and 2 specimen shapes. The fatigue tests were carried out within the high-cycle fatigue range using axial load ( R R = = -1) and rotary bending.
SIZE EFFECT The size effect materializes as an increase in fatigue strength/life following reduction of the cross-sectional area of the object. This claim applies to geometrically similar specimens made of the same material. Maintaining --N consistent parameters of specimen production is a requirement. N characteristic for geometrically similar = const; Fig. 2). The cross-sectional area specimens stay parallel within the high-cycle fatigue range ( coefficient K coefficient K , defined as the quotient of strengths of the test and reference specimens, determines the magnitude of fatigue strength difference. F F
a
σ
g o l g o l
S on
σ a,m
S om
σ a,n
1 1
β β
F
N
1/4
m n
log N
F
FIGURE 2. High-cycle fatigue characteristic for specimens with different cross-sectional area [3]
Paper [5] proposes a division of the size effect according to the following three criteria: statistical, geometrical and technological. The statistical size effect is based on statistical distribution of defects (heterogeneity, inclusions, cracks, material flaws) per material surface unit. The crack originates from the unit containing the most dangerous defect. The larger specimen surface, the more likely cracking initiation and connecting cracks. This results in a higher probability of initiation of cracks in larger specimen. The geometrical size effect is related to non-linear distribution of stress with high stress concentration (notched specimens) or load type (bending, torsional). The gradient of stress in the notch is smaller in reference specimen than in smaller specimen. For large differences, this is correlated to a change in the material material properties. properties. The technological size effect addresses the process of production of the object (e.g., surface roughness, top layer thickness, residual stress, micro-structure changes). For an object produced by a sophisticated process (e.g., welded joints), this can contribute to a modification of the material properties depending on specimen size.
Statistical Approach Model The statistical theory of the weakest link using the Weibull distribution of probability of destruction is the generally known model of size effect seen from the statistical point of view [14]. Fatigue life tests use the function of two-dimensional distribution of probability of destruction depending on the number of cycles ( P P (log N N )). Fatigue life at a given stress level features a certain scatter, the distribution of which is described on the logarithmic scale [8] and expressed by the following relation:
P P ( N N ) 1 exp =
−
−
log ( N N ) log ( N N 0 )
020008-2
m m
(1)
where: N 0 – reference fatigue life (probability 0.63) for a specific stress level, N m N . m – – distribution shape coefficient corresponding to a fatigue life scatter bandwidth bandwidth N If the distribution of results obtained for the tested material can be described with relation (1), then, it is possible to relate results for specimens with different cross-sectional areas. It is assumed that the distribution shape parameter m N 0 are material constants independent of the specimen size and stress distribution. For identical m and scale N probability of dest ruction and stress distribution and for two different specimen sizes, the following relation can be applied: 1
N N 2 N N 1
S m S o1 S o 2 S m
=
(2)
o
o
where: N 1 – fatigue life for a specimen with known cross-sectional area S N S oo1, N S oo2. N 2 – estimated fatigue life for a specimen with determined cross-sectional area S
Geometrical Approach Model The effect of non-uniform distribution of stress and specimen size on the fatigue properties of a material is described using the volumetric method. The method addresses the local concentration of stress for objects of V nn%), for which the probability of initiation of a different size. The model is based on a highly stressed volume ((V crack or of enlargement of an existing flaw is higher. Quantity V V nn% is defined as the volume of material exposed to at nn% = n% max n% least n n% % of the maximum stress ( max). The percentage of the whole specimen volume takes the value of approx. 95% [6] or 90% [9]. The model determines fatigue strength depending on specimen size and applied load type. Highly stressed volume of the notched specimen is determined based on the distribution of maximum stress on the notch root (Fig. 3). For specimens subjected to pure bending (Fig. 4a) or rotary bending (Fig. 4b), volume V V nn% is the area more distant from the bending axis than at least n n%. %. A A
ρ
A A
ρ
F F
x
n
% % a n
m
σ σ σ
F F d d d d
V V nn%%
FIGURE 3. Stress distribution in the notch tip taking account of highly stressed volume; own study based on [6] σ max σ max
V V nn%% r
%
V V nn%%
r
n
M. M.
ω ω
M. M. -σ max -σ max
(a)
(b)
FIGURE 4. Stress distribution and highly stressed volume in the round specimen during: a) pure bending, b) rotary bending [4]
This method describes the correlation of fatigue strength and volume V V nn%. It is described in the form of a linear relation between the logarithm of local stress amplitude and the logarithm of highly stressed volume (Fig. 5). In its basic form, the model is expressed as [6]:
020008-3
σ a AV AV n%v =
(3)
−
where: A, v – parameters dependent on the material, V n% – highly stressed volume for n = 95, 90. For any relation V n%, equation (3) can be noted as follows:
σ a,1 σ a, 2
v
=
V n%,1 V n%,2
(4)
where: a,1, a,2 – fatigue strength of specimen for volume V n%,1, V n%,2. Other Model Other Model
The monofractal approach assuming material structure damage is the main law describing consequences of the size effect. The cross-sectional area of an object shows material weakening resulting from heterogeneity, cracks and other flaws. Damage to a ligament of a heterogeneous solid object is modeled by gaps of sets of fractals analogical to the mathematical Cantor set. The system contains dimensions smaller than the total area. New mechanical properties are derived from physical dimensions depending on fractal dimension [3]. Application of the monofractal approach within the high-cycle fatigue range is based on the Basquin equation. Assuming that -N characteristic are parallel for specimens of different size and the parameter of constant C depends on dimension D, the following relation can be derived [3]:
C B
=
D C A A D B
−
d f f β
(5)
where: D A, D B – characteristic dimensions for geometrically similar specimens A and B assuming D B > D A, C A, C B – constant parameters of a -N characteristic, d f f – slope of a curve. EXPERIMENTAL EXPERIMENTAL RESULTS RESULTS
Sensitivity of a material to the size effect was studied for acid-resistant steel 1.4301 and for non-alloy steels S235JR, S355J2+C. For steel S355J2+C, the study attempted to verify the influence of the geometric size effect within the scope of studies on the effect of stress gradient occurring in the rotary bending test. The specimens were made from a drawn bar with 10 mm diameter. The tests were performed on a specimen with a cross-sectional area conforming to most of the material fatigue 2 standards [1, 7] and on a smaller specimen (S (S o = 3.1; 3.5 mm ). The purpose of testing smaller specimens ensued from described in paper [11] benefits from testing minispecimens. The specimens featured constant form theoretical stress concentration factor k . Production process parameters were fixed. See Fig. 5 and Table 1 for geometries of round specimens used in the fatigue tests. Fig. 6 compiles -N characteristic within the high-cycle fatigue range for axial loading ( R = R = -1) and for rotary bending.
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100 50 5 0
0. O
±
0
Ra 1.25
1
R
d
Dimensions of the round specimens [13] mm mm22 d mm R S d mm R mm S o mm 5 25 19.6 2 12.5 3.1
TABLE 1. 1. TABLE
O 9 O
9 O
30
8
FIGURE 5. FIGURE 5. Geometry
of the round specimen for fatigue testing made of steel S355J2+C [13], cut from aa drawn drawn bar bar with with 10 10 mm mm diameter d iameter 600 S o = 19.6 mm
a P M
S o = 3.1 mm
600
2
2
2
a P M
500
a a
S o = 3.1 mm 500
a a
σ σ
, e d u t i l p m a s s e r t S
2
S o = 19.6mm
σ σ
, e d u t i l p m a s s e r t S
400
Load: tension-compression, R = -1 , f = 7 Hz 300 10
3
10
4
10
5
10
6
10
400
Load: rotary bending, f = 28.5 Hz 300
7
300
3
10
Number of cycles, N cycles
10
5
10
6
10
7
Number of cycles, N cycles
(a) FIGURE 6. FIGURE 6. Fatigue
4
10
(b)
characteristic for steel S355J2+C determined by: a) tension-compression ( R = ( R = -1), b) rotary bending
The statistical size effect (for axial loads) was tested for non-alloy steel S235JR and for acid-resistant steel 1.4301. The flat specimens (Fig. 7, Table 2) were made from 4 mm thick plate. As in the case of the round --N specimens, two specimen sizes were tested. Fig. 8 compiles N characteristic within the high-cycle fatigue range for steels S235JR and 1.4301. --N Experimental points for the proposed N characteristic were approximated to the form of linear regression in bilogarithmic scale and to the form of the Basquin equation (Table 3). The tests aimed to identify materials sensitive to the size effect. The minimum number of points for the initial tests was set at 7 [10]. The mutual relation of the courses of each of the straight lines was evaluated statistically using the test of --N parallelism of slopes (a). (a). The test was performed for N characteristic, for a specimen with cross-sectional area 2 S S oo = 19.6; 28 mm . It was demonstrated that all the straight lines in each of the diagrams are parallel one to another. l R
5 0 . 0
Ra 0.32
± 1
w
Dimensions of the flat specimens mm mm22 R R mm l mm l mm S S o mm 7 25 100 28 2.5 18 35 3.5
TABLE 2. 2. TABLE t mm t mm 2
w
t
4 1.4
FIGURE 7. FIGURE 7. Geometry
of the flat specimen for fatigu e testing made of steel 1.4301 and S235JR, cut from 4 mm thick plate
020008-5
mm w1 mm
280
300 2
2
S o = = 28 mm mm a P M a a
σ σ
, e d u t i l p m a s s e r t S
S o = = 28 mm mm
2
250
S o = = 3.5 mm
a P M
2
S o = = 3.5 mm
260
a
200
σ
, e d u t i l p m a s s e r t S
150 Load: tension-compression, R = -1 , f R = f = = 7 H Hzz
100 10
3
10
4
10
5
10
6
10
7
Number of cycles, N cycles
240
220 Load: tension-compression, R = R = -1 , f = f = 7 Hz
200
200
3
10
4
10
5
10
6
7
10
10
Number of cycles, N cycles
(a)
(b)
FIGURE 8. Fatigue characteristic for steel: a) S235JR, b) 1.43 01
Material
S355J2+C
S235JR 1.4301
TABLE 3. Fatigue characteristic parameters for various specimen sizes Linear regression line Basquin relation 2 log a = a log N + b C = N ( a) Type of load S o mm a b C 3.1 -0.1276 3.2761 5.56·1025 Rotary 7.86 bending 19.6 -0.1008 3.1601 2.26·1031 9.92 3.1 -0.1003 3.0981 7.96·1030 9.97 Axial load, 39 R = R = -1 19.6 -0.0755 2.9869 3.46·10 13.24 3.5 -0.1103 2.8520 7.24·1025 9.07 Axial load, R = R = -1 28 -0.1173 2.8808 3.57·1024 8.53 3.5 -0.0476 2.651 5.61·1055 21.03 Axial load, R = R = -1 28 -0.0399 2 .578 2.78·1064 25.00
Correlation coefficient, R2
0.956 0.901 0.989 0.970 0.965 0.967 0.895 0.983 2
The coefficient of cross-sectional area size K size K ( K K S S – minispecimen’s tensile strength (S (S oo = 3.1; 3.5 mm ) to the K HC tensile strength of the specimen with cross-sectional area S S oo = 19.6; 28 mm2;; K – minispecimen’s fatigue strength HC 2 2 S oo = 3.1; 3.5 mm ) to fatigue strength of the specimen with cross-sectional area S ((S S oo = 19.6; 28 mm ) was used as a measure of material sensitivity to change of the cross-sectional area. See Table 4 for calculated values of coefficient K . K High consistency of fatigue life values obtained for specimens of different different cross-sectional cross-sectional areas areas (ranging (ranging from from 2 3.1 to 19.6 mm ) was observed for steels S355J2+C and S235JR. This conclusion applies to the both load types (for steel S355J2+C). The values of coefficient K coefficient K close to 1 are the proof. For acid-resistant steel 1.4301, results of fatigue property testing with axial load were varied. The size effect is evident, which is suggested by the high value of coefficient K coefficient K (1.086). TABLE 4. Compilation of values of cr cross-s oss-sectional ectional area coefficient K coefficient K within the material variability range, load type 2 Material Type of load S o mm K K S HC 3.1 0.974 Rotary bending 19.6* 1 S355J2+C 3.1 1.014 0.976 Axial load, R = R = -1 19.6* 1 1 3.5 1.008 1.012 Axial load, S235JR R = R = -1 28* 1 1 3.5 1.069 1.086 Axial load, 1.4301 R = R = -1 28* 1 1 *equivalent point A in point A in Fig. 1
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VERIFICATION OF SELECTED ANALYTICAL MODELS Selected size effect models were used for estimating fatigue strength for other than the original cross-sectional area of the specimen (used in the study). Due to the differences in results for the tension-compression load (statistical effect size), the verification used model based on the weakest link theory and on the monofractal approach. The acid-resistant steel 1.4301 was studied because this material was sensitive to the size effect (Fig. 8b). 2 S oo = 3.5 mm ). Computations were done based on the fatigue characteristic obtained for the minispecimen ((S --N Fatigue strength of the object with cross-sectional area S S oo = 28 mm2 was determined based on this N characteristic. 2 The regression line provided by experimental testing of specimen S S oo = 28 mm was used as the reference characteristic. The standard deviation of remainders (mean estimation error) and the coefficient of remainder variability were used to determine the degree of matching between the estimated straight lines (based on models) and the experimental data for the reference specimen. See Fig. 9 and Table 5 for results of the experimental tests and results provided by the implemented selected size effect models. Paper [12] presents a method for implementing the studied models for aluminium alloy AW-6063. 300 steel 1.4301
2
a P M a a
σ σ
280
2
Experimental points - S o = 3.5 mm
3
1
2
Experimental points - S o = 2 8 m m
4
260
, e d u t i l 240 p m a s s e r 220 t S
5
Load: tension-compression, R = -1, f = 7 Hz
200 10
3
4
10
5
10
6
10
10
2
1
Linear regression line - S o = 3.5 mm
4
Linear regression line - S o = 28 mm
2
Analytical - Weakest link theory
3
Analytical - Coefficient K S
5
Analytical - Monofractal approach approach
2
7
Number of cycles, N cycles FIGURE 9. Fatigue characteristic determined based on exp erimental tests and analytical models TABLE 5. Values of parameters of the linear regression line, standard error of the estimate and coefficient of residual variation Linear regression line Standard error of Coefficient of log log + b the estimate for residual variation a a = a log N Data a b S e MPa V e% Minispecimen S S oo = 3.5 mm2 -0.0476 2.651 3.3 1.3 2 Specimen S -0.0399 2.577 2.0 0.8 S oo = 28 mm Weakest link theory -0.0561 2.678 16.9 7.0 Monofractal approach -0.0435 2.578 10.2 4.2 Coefficient K Coefficient K S -0.0476 2.622 6.4 2.6 S from monotonic test
--N All the models in use enable setting off the base N characteristic in the right direction, which means that they estimate smaller values of strength. It is consistent with the results of the experimental tests. The model based on the weakest link theory uses statistical distribution parameters. The small number of measurements performed at a given stress amplitude level and the failure to address material constants translates directly into accuracy of results that are biased by an error of 7%. For the monofractal approach, it is necessary to determine the mean value of the fractal dimension based on documented experimental test results (for at least two different cross-sectional areas) for the given group of materials. The remainder variability coefficient is approx. 4.2%. The estimated fatigue strength values are on the safe side (below the regression line for the experimental points).
020008-7
The easiest to use among the analytical models verified is the weakest link model, the application of which requires determination of just parameters of distribution of experimental data for -N characteristic. -N characteristic by the K S coefficient is biased by a small error of 2.6%. The bias The parallel offset of the requires further verification for other construction material groups. SUMMARY SUMMARY
Fatigue properties of construction steels S355J2+C and S235JR were insensitive to change in the cross-sectional 2 2 area within the 3.1 ÷ 19.6 mm and 3.5 ÷ 28 mm ranges, respectively. The initial tests of acid-resistant steel 1.4301 identified dependence of fatigue strength and tensile strength on cross-sectional area. Coefficient K ( K S = 1.069, K HC = 1.086) was used to describe the result discrepancy. Completed studies of verification from the point of view of fatigue strength demonstrated that the fatigue characteristic obtained within the high-cycle range were correct for all the size effect models studied. The smallest mean estimation error determined for stress amplitude was obtained for the monofractal model. Given the insensitivity of material S355J2+C (in the axial loading and rotary bending tests) to the size effect, the model based on highly stressed volume was not studied. Accuracy of implementation of this model will be verified in further papers by the authors. REFERENCES REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
13. 14.
ASTM E466. Standard practice for conducting force controlled constant amplitude axial fatigue tests of metallic materials. Z. P. Bažant, “Size effect in blunt fracture concrete, rock, metal”, in Journal of Engi neering Mechanics ASCE 110 (1984), pp. 518-535. A. Carpinteri, A. Spagnoli, S. Vantadori, “Size effect in S-N curves: A fractal approach to finite-life fatigue strength”, in International Journal of Fatigue 31 (2009), pp. 927-933. H. P. Gaenser, “Some notes on gradient, volumetric and weakest link concepts in fatigue”, in Computational Materials Science 44 (2008), pp. 230-239. K. H. Kloos, A. Buch, D. Zankov, “Pure geometrical size effect in fatigue tests with constant stress amplitude and in programme tests”, in Materialwissenschaft und Werkstofftechnik 12 (1981), pp. 40-50. R. Kuguel, “A relation between the theoretical stress concentration factor and the fatigue notch factor deduced from the concept of highly stressed volume”, in Proc. ASTM 61 (1961), pp. 732-748. PN-74/H-04327 The study of metal fatigue. The test of axial tension - compression at constant cycle of external loads [in Polish]. J. Schijve, “Fatigue of Structures and Materials”, in Springer (2009). C. M. Sonsino, G. Fischer, “Local assessment concepts for the structural life of complex loaded components”, in Materialwissenschaft und Werkstofftechnik 36 (2005), pp. 632-641. P. Strzelecki, J. Sempruch, K. Nowicki, “Comparing guidelines concerning construction of the S-N curve within limited fatigue life range”, in Polish Maritime Research 22/3 (2015), pp. 67-74. T. Tomaszewski, J. Sempruch, “Verification of the fatigue test method applied with the use of mini specimen”, in Key Engineering Materials 598 (2014), pp. 243-248. T. Tomaszewski, J. Sempruch, T. Pitkowski, “Verification of selected models of size effect based on highcycle fatigue testing on mini specimens made of EN AW-6063 aluminum alloy”, in Journal of Theoretical and Applied Mechanics 52(4) (2014), pp. 883-894. T. Tomaszewski, P. Strzelecki, J. Sempruch, “Geometric size effect in relation to the fatigue life of S355J2+C steel under variable bending conditions”, in Engineering Mechanics (2016), pp. 554-557. W. Weibull, “A statistical representation of fatigue failures in solids”, in Transaction of the Royal Institute of Technology 27 (1949).
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