loading, an approximate method is put forward for the determination of maximum strain hardening levels in the case of non-proportional low-cycle loading with ...
Strength of Materials, Vol. 38, No. 2, 2006
AN APPROXIMATE METHOD OF DETERMINATION OF MAXIMUM STRAIN HARDENING LEVELS IN METALS UNDER NONPROPORTIONAL LOW-CYCLE LOADING UDC 539.43
M. V. Borodii
Using the findings of analysis of deformation curves for metallic materials under static and cyclic loading, an approximate method is put forward for the determination of maximum strain hardening levels in the case of non-proportional low-cycle loading with strain monitoring. Based on the correlation between strain hardening data obtained from the static and proportional and nonproportional cyclic deformation curves, an approximate analytical relationship is built up which allows for predicting maximum strain hardening levels under nonproportional low-cycle loading. Keywords: static and cyclic deformation curves, low-cycle fatigue, nonproportional deformation, strain hardening. Introduction. Most structural components when in operation are subjected to combined loads. However, even if the external loads are uniaxial a local multiaxial stress state is realized in many local zones in complex-shaped structures. Many materials in this state, which is close to deformation with a strain control, under cyclic deformation in the region of elastoplastic strains exhibit cyclic instability that shows up in an extra cyclic strain hardening. The level of this cyclic hardening and the strain amplitude level constitute an important characteristic involved in modern models for predicting the kinetics of cyclic plasticity and life time. For the majority of such models that use energy-based parameters of damage on a critical plane or modified strain parameters [1], to know the strain hardening level for an arbitrary cycle path is a prerequisite for adequate prediction of life time. Some papers [2–6] addressing the construction of cyclic plasticity models propose the nonproportionality parameters Φ which represent the cycle shape and make it possible to establish, with a given accuracy, relationships between the strain hardening level σ and the cycle path shape. For this purpose, the following expression is used: σ (Φ) = (σ npr − σ pr ) Φ + σ pr ,
(1)
where σ pr and σ npr are respectively the basic maximum values of equivalent stresses in proportional and nonproportional cyclic deformation. If we use the known parameter that represent a degree of additional hardening α (it is also called the nonproportionality sensitivity factor), equation (1) can be re-arranged as follows: σ (Φ) = (1 + α Φ) σ pr ,
(2)
where α is the parameter given by α=
σ npr − σ pr σ pr
=
σ npr σ pr
− 1.
(3)
Pisarenko Institute of Problems of Strength, National Academy of Sciences of Ukraine, Kiev, Ukraine. Translated from Problemy Prochnosti, No. 2, pp. 29 – 38, March – April, 2006. Original article submitted February 21, 2005. 128
0039–2316/06/3802–0128 © 2006 Springer Science + Business Media, Inc.
For defining specific forms of relationship (1) or (2), the basic values used were the experimental strain hardening data obtained from two basic experiments for a proportional cyclic path in tension–compression or alternate torsion and for a nonproportional path with a constant strain intensity value (a circular path). In most cases where the peak-to-peak amplitude of strains, the lowest σ pr level and the largest σ npr level correspond to the first and second paths, respectively, and therefore are used as basic characteristics. Cyclic deformation curves for the first group of paths (i.e., for the proportional deformation) have been reported elsewhere for a fairly wide class of structural materials. Also, many leading laboratories and companies publish appropriate data sheets and schedules, e.g., [7, 8], providing new experimental data on cyclic properties of materials under uniaxial loading. Similar information can be also found in on-line data bases. Therefore, it is quite easy to determine the σ pr parameter. This is not the case with σ npr . It turns out that experimental data (cyclic stress strain curves) for nonproportional deformation and, furthermore, for a maximum-hardening path (a circular one) are very scarce. First, this is attributable to the fact that more stringent requirements are imposed on test equipment thus making the experiments more expensive. Second, there are no systematized procedures for nonproportional loading tests since they are not regulated by any applicable codes or standards, as distinct from the experiments of the first group. Under the circumstances, to find alternative methods for getting information on maximum hardening levels in nonproportional deformation (from the standard test results) will remain an important task for many years to come. In this context, the objective of the present work has been to verify, using a larger body of experimental data proposed in [9], the method of predicting the maximum strain hardening levels in cyclic nonproportional deformation, σ npr , and find the limits of its applicability. The most widespread available data on the standard mechanical characteristics of materials and cyclic stress–strain curves in proportional low-cycle deformation will be used as reference characteristics. Metals’ Tendency to Additional Hardening in Nonproportional Deformation. The attempt to assess metals’ tendency to additional strain hardening as observed during the nonproportional low-cycle deformation was discussed in [10]. An approach that enables both the qualitative and quantitative assessment of this tendency was put forward later [9], although rather briefly. The approach is fully phenomenological and applies to the analytical determination of maximum hardening levels for biaxial cyclic paths as formed under combined loading by tension–compression and alternate torsion. In order to define its applicability limits, let us study it in more detail using a larger body of representative data. Recall that experimental data obtained for 11 materials that differ in cyclic properties (steels Kh16N9M2 [11], 08Kh18N10T [12], 45 [13], SS304 [14], SS310 [15], SS316 [16], SS316L [17], SNCM630 [18] and alloys 800H [19], VT9 [12], AA6061 [14]) were analyzed in [9] for the purpose of constructing an analytical relationship for the prediction of maximum hardening levels. In addition to the above-mentioned data, we will study here the experimental data for steels SS316 [20, 21], 1Cr18Ni9Ò³ [22], S460N, SS347, AA5083 [23], 45 [24, 25], 42CrMo [26], In 718 [27], AISI 1045 [28] and alloys D16 [25], 2CrNiMoV [11]. Also, we will analyze some of the 11 materials studied earlier, in a wider range of strain amplitude values in cyclic deformation. Table 1 below summarizes mechanical characteristics of the materials and their cyclic stress–strain curve characteristics for the cases of proportional and nonproportional loading by tension–compression and alternate torsion with a symmetrical loading cycle. It gives also the data on the degree of additional strain hardening as observed during the static and cyclic loading. The degree of strain hardening under static loading is expressed in terms of a dimensionless parameter β which is defined as a ratio of material’s ultimate strength σ u to yield stress σ Y : β=
σu − 1. σY
(4)
The degree of additional strain hardening under nonproportional cyclic deformation is represented by the dimensionless parameter α given by formula (2). Recall that it is defined as a ratio of the strain hardening under nonproportional cyclic deformation to that under uniaxial cyclic deformation. 129
TABLE 1. Mechanical Characteristics and Levels of Strain Hardening under Static and Cyclic Loading Material
Mechanical characteristics, degree of hardening
Characteristics of stress–strain curves, degree of hardening
σY , MPa
σ u , MPa
β
εà , %
AA6061
320
350
0.090
42CrMo
868
955
0.100
VT9 AISI 1045 SNCM630 2CrNiMoV In 718 S460N
865 505 951 600 1172 500
973 585 1103 710 1407 643
0.120 0.160 0.160 0.180 0.200 0.290
Steel 45 [24] Steel 45 [13] D16 Steel 45 [25] 1Cr18Ni9Ti
377 352 240 340 310
623 599 420 610 605
0.650 0.700 0.750 0.794 0.950
AA5083
169
340
1.010
SS316 [21]
260
530
1.040
Kh16N9M2 08Kh18N10T SS347
280 320 230
600 690 565
1.140 1.160 1.460
SS316L
230
565
1.460
SS347 800H
246 200
615 530
1.500 1.520
SS316 [16] SS316 [20]
230 230
600 600
1.610 1.610
SS304
260
690
1.650
SS310
195
560
1.870
0.250 0.400 0.600 0.900 0.470 0.600 1.200 0.700 – 0.600 0.300 – 0.104 0.144 0.173 0.231 0.250 0.400 0.470 0.750 1.000 (0.200) 0.300 0.400 1.000 (0.231) 0.346 0.200 0.400 0.600 0.500 0.600 0.231 0.346 0.577 0.470 0.600 0.800 0.577 0.100 0.200 0.300 0.400 0.600 0.200 0.530 0.585 0.250 0.400 0.200
σ pr , MPa 184 260 280 293 554 580 644 – – 751 – – 172 217 244 270 270 400 404 – – (350) 448 472 570 (161) 221 281 342 388 – – 226 237 252 283 300 350 328 250 300 320 350 375 290 340 370 265 315 260
σ npr , MPa 200 247 290 318 623 656 729 – – – – – 215 290 321 400 350 475 354 – – (365) 501 620 703 (164) 246 349 538 600 – – 338 448 607 400 440 530 620 355 450 500 530 540 490 580 660 505 620 –
Note: Asterisk indicates the degree of hardening directly by the available published data. 130
α 0.087 − 0.052 0.042 0.087 0.124 0.131 0.131 0.080* 0.110* 0.050* 0.060* 0.100* 0.250 0.339 0.315 0.481 0.296 0.187 − 0.124 0.160* 0.200* (0.043) 0.116 0.313 0.230 (0.018) 0.113 0.242 0.573 0.546 0.250* 0.300* 0.495 0.892 1.411 0.413 0.466 0.514 0.890 0.420 0.500 0.562 0.514 0.440 0.689 0.706 0.783 0.906 0.968 1.040*
In Table 1 the materials are arranged in order of increasing β. This reveals a clear trend: as the degree of static hardening grows so does the degree of additional cyclic hardening. Some deviations that are due to natural scatter of data or objective factors to be discussed below do not upset the basic pattern. With a high probability we can state that the degree of maximum additional hardening up to 10% is observed for the materials whose static hardening does not exceed 20%. These materials can be arbitrarily classified as nonsensitive or weakly sensitive to additional cyclic hardening. The materials with a degree of static hardening ranging from 20 to 100% should be assigned to a medium group characterized by the additional cyclic hardening up to 35%. Finally, the materials whose static hardening exceeds 100% should be considered as strongly sensitive to the additional strain hardening or, in other words, to the cycle nonproportionality. For them, consideration must necessarily be given to the factor of considerable hardening which adversely affects the low-cycle fatigue life [11, 15], namely: the latter is significantly reduced. Prediction of Additional Cyclic Hardening and Analysis Thereof. Since there is a clear correlation between the degrees of hardening under static and cyclic loading, of interest is to define its analytical form. Processing of experimental data for various classes of metallic materials has enabled us to establish the pattern of this relationship. Various variants were discussed, and the linear law for the case of semilogarithmic representation of data (Fig. 1) was found to be the best suited one. For the first time [9], this relationship was analytically written as: log | α| = 0. 705 β − 1. 22.
(5)
The coefficients in expression (5) were obtained by the least-squares approximation of experimental data for the group of materials [11–19] which were studied earlier in [9]. Figure 1 shows relationship (5) as a solid line. It gives also the values (in the form of dots) of the degree of additional hardening as determined from the experimental data (see Table 1). In this case, the bracketed experimental data in Table 1 were ignored since they are beyond the reasonable limits of error. An attempt to analyze this inconsistency came up with a simple explanation. It turns out that these experimental data were obtained in cyclic deformation with strain amplitude levels which were exclusively elastic with almost no plastic component. Therefore, this case corresponded to the region of high-cycle fatigue. Under these circumstances, almost no effect of additional cyclic hardening was observed in nonproportional deformation. Based on the aforesaid, we just have the first important limitation on applicability of relationship (5). For the determination of maximum hardening levels, expression (5) can be considered to hold good for cyclic deformation with plastic strain amplitude levels exceeding some conventional lower limit ε ap which is about 0.02%. It is this fact which may explain a considerable difference between the experimental value of the α parameter and that predicted for ε a = 0. 345% for aluminum alloy AA5083 because in this case we have ε ap = 0. 021% which is very close to the above-mentioned lower limit. As the strain amplitude is raised, the α parameter can be expected to somewhat grow as well. It is obvious that expression (5) is an approximate one for it neglects the dependence of the degree of cyclic hardening on the strain amplitude level. Specifically, the tabulated data show that for some materials, e.g., for steels 1Cr18Ni9Ti and SS347, there is a significant variation of the α depending on the strain amplitude level. It is conceivable that there might exist also an upper limitation on the application of expression (5) depending on the level of amplitude of cyclic strains. Based on the experimental data given herein, ε a can be determined to be 1%. Thus, the “fluctuations” of the α parameter depending on the strain amplitude do occur but for most materials they are not significant and can be ignored (within the engineering accuracy) versus the benefits opened up for researchers, i.e., prediction of strain hardening in a fairly easy way. However, it should be remembered that such prediction can be considered as some first approximation. Figure 2a illustrates schematically the assessment of efficiency of the additional hardening prediction by comparing the calculated α calc (5) and experimental α exp (3) parameters of additional hardening. They do not appear to agree well. However, it should be kept in mind that we compare relative values. For instance, if additional hardening is 20 MPa a prediction error of about 100% will give the value of additional hardening 40 MPa. Taking into account that additional hardening is reckoned from a certain reference level of stresses, which for the majority of materials ranges from 250 to 700 MPa, the error in absolute values of hardening may amount to a few percent. 131
Fig. 1. Relationship between static and cyclic hardening. σ calc , MPa
α calc
α exp a
σ exp , MPa b
Fig. 2. A comparison between the calculated and experimental values of the additional hardening parameter α (a) and the absolute values of strain hardening (b). (Symbols are the same as in Fig. 1.) To illustrate these considerations, the same results but in absolute values are analyzed in Fig. 2b. The maximum levels of hardening were determined from the following expression: σ npr = (1 + α ) σ pr ,
(7)
where either α calc or α exp was used depending on which parameter was to be evaluated. The representation of data in this manner greatly improves perception of the results. For the majority of the materials analyzed, the prediction error does not exceed ±10%. However, for some of them, such as steels S360N, SS347, SS316 [21], the method outlined above turns out to be inadequately efficient. The maximum error for these steels was 26, 36, and 15%, respectively. 132
As to actual engineering calculations, in view of a possible prediction error, a more conservative assessment (Fig. 1, dashed line) can be applied which corresponds to an increase in the α parameter by 30%. Clearly, in doing so we significantly overestimate the level of additional hardening for a large class of materials, but this will go into the structure’s safety margin. Steel 45 [13] considerably differs in its cyclic properties from all the other materials studied. The level of strain hardening in this steel under proportional cyclic deformation turned out to be higher than that under nonproportional deformation. Then, formula (3) gives a negative value of the hardening parameter α. To allow for this aspect the α parameter in the linear relationship (5) is taken in its magnitude. This is just a slight refinement of the expression for predicting maximum hardening levels [9]. This modification brings up another question: with which sign should the α parameter be used in predicting strain hardening by formula (6)? Conceivably, the answer might only be found through some specialized metalographic studies of material structure in view of peculiarities of the production process employed. However, it can be stated that in the overwhelming majority of cases the α parameter will be positive. CONCLUSIONS 1. A correlation has been established between standard characteristics of materials and the degree of additional hardening under nonproportional deformation. 2. It has been for the first time that an analytical expression was put forward to describe this correlation. The expression has been demonstrated to be suitable for engineering calculations. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
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