1、STP-PT-056EXTEND STRESS-STRAIN CURVE PARAMETERS AND CYCLIC STRESS-STRAIN CURVES TO ALL MATERIALS LISTED FOR SECTION VIII, DIVISIONS 1 AND 2 CONSTRUCTIONSTP-PT-056 EXTEND STRESS-STRAIN CURVE PARAMETERS AND CYCLIC STRESS -STRAIN CURVES TO ALL MATERIALS LISTED FOR SECTION VIII, DIVISIONS 1 AND 2 CONSTR
2、UCTION Prepared by: Wolfgang Hoffelner RWH consult GmbH Date of Issuance: April 5, 2013 This report was prepared as an account of work sponsored by ASME Pressure Technology Codes S = So + h (Su-So) (ep)0.5/1 + h (ep0.5) This equation has the characteristic of a proportional limit: So when ep is zero
3、 and an upper limit of Su when ep is very large. (When h (ep0.5) is large compared to 1 the term cancels with the term in the numerator and one is left with So + Su - So or Su, the ultimate strength. The hardening parameter, h, may be calculated by inserting Sy for S and 0.2% for strain into the equ
4、ation and solving for h. One gets h = (So-Sy)/(Sy-Su)(0.2).5 In our work on alloy 800H we noticed that So was approximately 0.72Sy. The equation only works for engineering strains to a few percent but that is all that is needed for the isochronous curves and buckling analyses in III-NH. Other austen
5、itic alloys may have different values for the ratio of proportional limit to 0.2% yield strength. The attractive feature of the simple approach is that one only needs the Y-1 value and the U value given in II-D. We recently used this equation for alloy 253 with some success. A more elaborate model u
6、sing the rational polynomial was published by Joe Hammond and Vinod Sikka in 1977. See J. P. Hammond and V. K. Sikka, “Predicted Strains in Austenitic Stainless Steels at Stresses above Yield,“ pp. 309-322 in Effects of Melting and Processing Variables on the Mechanical Properties of Steel, MPC-6, A
7、SME, New York, 1977. This approach gives certainly very good values for austenitic materials but they were not more accurate than the procedure described above using YS and UTS. A general application of this method needed determination of S0-values for other classes of materials which could be done
8、but it needed an additional effort and further parameterization work. More information can be found in the literature 67. A critical comparison of the results gained with the rational polynomial and with the Ramberg-Osgood fit for IN 800 H is given in Appendix II. 2.2 True Stress-Strain Curves Many
9、attempts were made to get a relation similar to equation (1) for the true stress-strain curves. Hollomon 3 proposed an approach similar to equation (3) also for the true stress strain curve. Cofie et al. made a similar attempt to get a one-power law fit of true stress-strain data specifically tuned
10、to ASME needs 8 and they proposed a power law of the form for the true stress-strain curve particularly with respect to austenitic materials. The idea of this approach was the demand that the yield strength as well as the ultimate tensile strength must be points of the curve. This allowed the determ
11、ination of a and n in equation (10) and consequently an analytical expression for the true stress-strain curve. The results of this approach were compared with others and a very good agreement amongst them was found. However, the agreement with real measurements turned out to be less convincing whic
12、h can already be expected from the results shown in Figure 3. The results shown in Figure 3 clearly demonstrate that a simple power-law type of fit cannot lead proper results, because there is no simple power law relationship between the true plastic strain and the true plastic stress. This is to so
13、me extent due to the structure of the relationship itself as quickly demonstrated in the following. Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves STP-PT-056 7 Let us assume for the engineering stress strain curve the validity of the following equation: Where s is engineering stress
14、, e is engineering strain and E is Youngs modulus. In case of a true stress-strain curve the power law expression would become something like As is only for norming purpose it has not been converted further. Equation (2) can be written as: The second term is a polynomial with rational exponent which
15、 can be developed into a series which gives (taking only the first order term): In other words the conversion from engineering to true stress adds an additional element to the power law describing the engineering curve which can be seen also from Figure 3. It is important to notice that a single pow
16、er law fit to true stress-strain data is still used in current code documents as a possible alternative to the MPC 2-power law description. As long as this remains confined to small stresses only (e.g. cyclic stress-strain curves) this might be valid, but for the whole strain range (up to UTS) highl
17、y non-conservative assessments can be obtained. This discrepancy between parameterization of engineering vs. true stress-strain curve is not at all new and it has been several times discussed in the literature. Equations like the Ludewik equation (12) or the similar Swift equation (13) were proposed
18、 and deeper analysis of them can be found in the literature e.g. 24591011. A general representation of a true stress-strain curve would be a multiple power law expression. Usually, two power law terms as shown in equation (17) are sufficient. Such an approach is currently used for the determination
19、of the true stress-strain curves in several ASME (e.g. Sect VIII /2) procedures. It allows a construction of true stress-strain curves from yield stress and ultimate tensile stress both given in tables Y-1 and U (MPC-method). Due to its importance in current code procedures the method shall be descr
20、ibed and analyzed in more depth. The general expression is given as STP-PT-056 Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves 8 (18) With the two power law expressions and (19) for the low stress portion and for the high stress portion. The exponent 1/m2and the coefficient A2are mai
21、nly determined by the demand that the point (UTS,UTS) must be part of the curve. The expression m2is therefore an expression for the true ultimate tensile strain which is obviously based on earlier experimental findings. This is a critical point also for the determination of the engineering stress-s
22、train curves (mentioned already above) which will discussed separately later. For the low stress portion 1is demanded that the yield stress (more precisely the 0.2% proof stress) and the point when significant plasticity happens (p) are points of the curve. The discrimination between low and high st
23、rain portion (H) is based on the difference between the actual true strain and a function H which depends on yield stress and ultimate tensile stress shown in equation (20). (20) To gain better insight into what numerically happens during this procedure we analyzed the behavior for 304L at room temp
24、erature where a set of measured data exists 14. A yield stress of 258 MPa and an ultimate tensile stress of 617 MPa were assumed for the calculations. Additionally, Youngs modulus of 196,000 GPa was used. The results can be seen from Figures 4 and 5. Gamma 1 and gamma 2 are the low stress related st
25、rain and the high stress related strain, respectively. The functions (1.0 tanhH) describe a smoothened step function between the two strains (Figure 4). The situation becomes better visible in Figure 5 where different results were plotted. It can be seen that the two strains eps_1 and eps_2 are real
26、ly two different power functions which meet far above the calculated stress when eps_1 changes to eps_2. It is obvious that a discontinuity occurs when changing from eps_1 to eps_2 control as reflected in the curve called MPC_value. Figure 4Shape o f (1+tanhyp(H) and (1-tanhyp(H) for 304 L Stainless
27、 Steel at Room Temperature Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves STP-PT-056 9 Figure 5Comparison of Different Parameterizations of a True Stress-strain Curve for an Austenitic Steel at Room Temperature Eps_1 and eps_2 are the two power law functions of the MPC approach. MPC
28、 represents the final results of the MPC procedure. RO-eng represents a Ramberg-Osgood fit performed in the engineering stress-strain frame from which the true stress-strain data were obtained and experiment means experimental data. The real interesting comparison is the one with experimental data (
29、experiment) which shows that in this case the agreement between MPC-values and experiment is good up to about 350 MPa (eps_1) and for stresses above 600 MPa (eps_2). But between 350 and 600 MPa the agreement is not good at all. However, a really good prediction of the experimental data is found with
30、 the RO-eng approach which was based on the engineering stress-strain curve from which the true stress-strain values were determined according to equations (2). Similar analyses of experimental data from for different materials led to the same result. The MPC-procedure could well describe the low st
31、rain and the high strain regime, but there was always a portion of the stress-strain curve where at the change from low to high strain a discontinuity occurred. It became also evident that the RO-eng curve determined for the engineering stress-strain curve and converted into a true stress-strain cur
32、ve gave the best agreement with the experimental data. In the following improvements of the MPC-procedure to smoothen the difference between the low strain and the high strain portion will be discussed. One problem is the definition of the function H which is exclusively responsible for the switch.
33、Using the notation of ASME VIII/2 with t, true stress, ys, engineering yield stress and uts, engineering ultimate tensile stress, H is defined as: (21) As K is also defined as a polynomial in ys/utsthe whole expression depends only on the difference between tand a constant (for given ysand uts). Thi
34、s term does not provide any possibility for modifications. As smoothening approach it was tried to determine the point where the slopes of the low strain and the high strain curves are the same: STP-PT-056 Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves 10 Let us start from the assum
35、ption that we have two power laws describing different portions of one curve: The stress where the slopes of both curves are the same, , can be determined by setting the derivatives of the two curves to equal which leads to: In a next step the curve describing the higher stress portion is shifted by
36、 the difference between the strains at towards lower strain. The resulting curve is shown in Figure 6 as “equal slope.” The other curves shown are the same as the ones shown already in Figure 4 and Figure 5. The agreement with the experimental data is now considerably better up to a strain of about
37、0.1. However, there is now a higher deviation from the measured data at higher strains. Figure 6Comparison of a Modified MPC True Stress-strain Curve (Equal Slope) with the Other Stress-strain curve parameterizations shown in Figures 4 and 5 Extend Stress-Strain Parameters and Cyclic Stress-Strain C
38、urves STP-PT-056 11 Figure 7Comparison of R esults from MPC and RO-eng Results Omitting Data-points at the Transition from Low Strain to High Strain for the MPC Approach Figure 8Comparison of Different True Stress-strain Parameterizations In Figure 8, MPC Ben and RO Ben Refer to the ASME FFS-1 Proce
39、dure Provided in 15, MPC Wolf represents the own evaluation according to ASME VIII/2 which is the same as FFS-1. RO Wolf is the Ramberg-Osgood fit starting from the engineering stress-strain curve, equal slope stands again for the MPC-procedure modified by fitting low and high strain regimes at equa
40、l slopes of the two power laws. The second smoothening procedure was just to omit the data in the cross-over regime and draw a smooth curve. The result is compared in Figure 7 with experimental data and the engineering stress-strain based Ramberg-Osgood curve. This leads to an extremely good agreeme
41、nt of all three curves. The only problem remains the smoothening procedure which is a bit of an arbitrary procedure. Similar results were also obtained with 2.25 Cr-1Mo in a comparison with the ASME FFS-1 evaluation provided by Ben Hantz 15 which was summarized in Figure 8. STP-PT-056 Extend Stress-
42、Strain Parameters and Cyclic Stress-Strain Curves 12 It becomes obvious that the MPC-Ben solution gives exactly the same results as MPC Wolf which means that the ASME FFS-1 MPC procedure fully agrees with the MPC-procedure adopted for this report. RO Wolf is the RO-parameterization based on the engi
43、neering stress-strain curve (usually called RO-eng) whereas RO Ben is a single power law fit of the true stress-strain curve similar to the Hollomon or Cofie-approach (see equ. 9 and 10). As highlighted in equations 11-13 a single power law fit is normally not able to map the true stress strain curv
44、e properly and it leads in almost all cases to an extended arc between YS and UTS, leading to exaggerated stresses. Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves STP-PT-056 13 3 THE DILEMMA OF FINDING THE (USUALLY NOT AVAILABLE) ULTIMATE TENSILE STRAIN Before recommendations concer
45、ning applicability of the different methods will be given another critical issue must be highlighted; the determination of the ultimate tensile strain. 3.1 Rasmussen Procedure The discussion will commence with the ideas of Rasmussen 16. He analyzed the ultimate tensile strain for austenitic, ferriti
46、c and duplex steels and he obtained the best correlation with the parameterization (see also Figure 9): Where uis ultimate engineering tensile strain, 0.2is engineering proof stress, uis engineering ultimate tensile stress. Figure 9Correlation between Yield Stress, Ultimate Tensile Stress and Ultima
47、te Tensile Strain (replotted from Rasmussen 16) The high scatter of data is not surprising for experimentally determined mechanical values. However, my own trials with other experimental data from the literature did not show the behavior anticipated by Rasmussen. Also the ultimate tensile strains of
48、 ferritic and duplex materials in Figure 9 seem to be rather independent form 0.2/u. 3.2 The MPC-approach From relation (24), which is given in the MPC procedure, the true ultimate tensile stress can be obtained using STP-PT-056 Extend Stress-Strain Parameters and Cyclic Stress-Strain Curves 14 with
49、 material dependent values given in Table 1. Using the usual conversion between true stress/strain and engineering stress strain (equ 1) one obtains with uts,t = suts(1+e) for euts: or This means that under the assumption of the validity of Table 1 the strain values at ultimate tensile stress can be determined which allows the construction of engineering stress-strain curve
