1、DEVELOPMENT OF AVERAGE ISOCHRONOUS STRESS-STRAIN CURVES AND EQUATIONS AND EXTERNAL PRESSURE CHARTS AND EQUATIONS FOR 9CR-1MO-V STEELSTP-PT-080STP-PT-080 DEVELOPMENT OF AVERAGE ISOCHRONOUS STRESS-STRAIN CURVES AND EQUATIONS AND EXTERNAL PRESSURE CHARTS AND EQUATIONS FOR 9Cr-1Mo-V STEEL Prepared by: M
2、AAN JAWAD, Ph.D., P.E. Global Engineering at other conditions tertiary creep is important. The model must in some cases be predictive of conditions for which there are no available data, specifically estimating creep strains at very low stresses, high temperatures, and long times. In describing the
3、model, we try to maintain a distinction between terms such as condition, parameter, constant, and coefficient. The conditions are the inputs to the model: stress, temperature, and time. The parameters of the model are the values that are used to describe the shape of the creep curve at a specific se
4、t of conditions. For example, the stress exponent, n, is a parameter, and the time to rupture, tr, may also be considered a parameter. The model coefficients are used in describing the parameters as functions of stress and temperature. The term constant is only used for specific coefficients that ta
5、ke on a special role in a time-temperature parameterization. In this report, the term constant is exclusively used for the Larson-Miller constant. It is highly desirable to keep the number of parameters low to minimize the effort of determining the coefficients. Many have come to view the classic th
6、ree stage description of creep as the result of a primary stage where hardening mechanisms result in diminishing creep rates and a tertiary creep stage where damage and aging mechanisms produce an increasing creep rate. The second stage, where creep rate appears to be constant, is simply the transit
7、ion between the two stages. Primary-tertiary forms for creep models often involve four parameters, two each for the primary and tertiary stages. To determine these parameters, three approaches are possible. The first is to fit the entire curve. This can be quite difficult depending on the creep mode
8、l since it involves non-linear regression. The second is to fit either the tertiary creep or primary creep and then make adjustments for the missing component. The third is to fit each separately and look for a method to combine the curves. The model proposed below seeks to use information contained
9、 within the tertiary creep portion of the curve to provide an estimate of the primary creep strain. 1.2 Logarithmic Creep Rate Formulation Description of Tertiary Creep The model expression for tertiary creep is1 303lnln (1.1) where is the creep strain, is the creep strain rate,03 represents the ini
10、tial creep rate at zero strain and 3 provides the dependence of the strain rate on the creep strain. In the present paper, we refer to this form of the creep law as the logarithmic-rate form. 1 The use of the term is intentional and is used to distinguish the resultant values in the present approach
11、 from those values tabulated for aged material in ASME FFS-1 / API-579. STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 2 Upon integration, assuming there is no initial strain or other stages of creep, equation (1.1) becomes2 t30333 1ln1 (1
12、.2) Then, the time to reach the rupture strain is 303exp1 rrt (1.3) Which for any combination, 33 r (1.4) can be approximated within 5% simply by the limit, 3031rt (1.5) At some conditions, the initial creep rate and the actual minimum creep rate are of the same order. In these situations, equation
13、(1.5) provides an estimate of the value of from minimum creep rate and time to rupture; furthermore, in such situations, 1/ 3 can be regarded as the Monkman-Grant strain. Description of Primary Creep and Combined Creep Strain It is recognized that primary creep is important under many conditions of
14、practical interest. Neglecting the early part of the creep curve could lead to significant errors, since the difference between the initial creep rate as derived from the latter stages of the creep curve and the actual minimum creep rate measured in a test can be orders of magnitude. A candidate exp
15、ression for the primary creep rate may be expressed in a similar form3: 101lnln (1.6) which leads to an expression for creep as t10111 1ln1 (1.7) In this case, the creep rate decreases with time and strain accumulation. Treating the tertiary and primary creep terms as independent contributions to th
16、e total creep strain leads to the following creep model: ttc 30331011 1ln11ln1 (1.8) 2 An early example of this equation can be found in Sandstrom, R. and Kondyr, “Model for Tertiary Creep in Mo- and Cr-Mo-Steels,” pp. 275-284 in Mechanical Behavior of Metals, Vol.2 Pergamon Press, New York, NY, 197
17、6. 3 Such a procedure was proposed in Cleh, J-P. “An extension of the omega method to primary and tertiary creep of lead-free solders,” Electronic components and technology conference, 2005. STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 3
18、 1.3 Ellis Creep Form Ellis took a related but subtly different approach to fitting a creep model. Starting with equation (1.1) he expressed the creep strain in the tertiary region as4: BtFA ln (1.9) where, 1A (1.10) and 0AB (1.11) which, when combined with equation (1.10) leads to 0B (1.12) Other t
19、han the integration constant, F, Elliss form of the equation is identical to equation (1.7). As will be shown, this integration constant is quite important in matching the tertiary creep parameters to the time to rupture. Ellis linearized the equation as BtFA /exp (1.13) The parameters are determine
20、d by adjusting the parameter A to minimize the R2 value of a linear regression to the tertiary portion of the creep curve. As might be expected, Ellis found that the rupture life corresponds very closely to the ratio F/B. To show this, consider failure to occur at a finite strain, then equation (1.1
21、0) becomes rr BtFA ln (1.14) and AFBt rr ex p1 (1.15) which, upon substitution of the equivalent tertiary creep parameters is identical to equation (1.3) if F = 1. As A greatly exceeded the rupture strain, equation (1.15) becomes BFtr (1.16) Or, from equation (1.12) 0Ftr (1.17) Thus this new paramet
22、er, F, appears in the rupture calculation. Using equation (1.16) allowed Ellis to make the substitution rtFB / (1.18) rtFtFA /ln (1.19) and, finally, 4 Originally, Ellis used the symbol C for the integration constant, but here, to avoid confusion with the Larson-Miller constant, we use the symbol F.
23、 STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 4 rttAFA /1lnln (1.20) Ellis then approximates the tertiary strain as rttA /1ln3 (1.21) Though the first term in equation (1.20), FAln , is constant, it can be replaced by the primary creep c
24、omponent. Ellis then used the Andrade form of the creep equation to characterize the primary creep: rc ttAKt /1ln3/1 (1.22) This form has the limitation of predicting an infinite strain rate at time=0. This model has three parameters, K, A, and tr; but rupture time has already been measured. Using t
25、he known time to rupture, Ellis fit the parameters, K and A, to all the actual creep curves using a least squares regression and then derived parametric expressions for each using the Dorn parametric fit. It is important to note that the constant, A, though conceptually the same in equations (1.9) a
26、nd (1.22) are in practice different because they are determined in different ways. The A-parameter in equation (1.22) is affected by the primary creep term, and as such it does not necessarily produce the best possible fit to the tertiary portion of the curve. Ellis did not attempt to determine a se
27、parate fit to the primary portion of the curve, but if he had, it should be expected that the value of K would also be different. Non-linear combination model Both equations (1.8) and (1.22) represent linear combinations of primary and tertiary creep where the two creep strain functions do not inter
28、act. In the case of the Ellis approach, the whole curve is fit at once. In the case of the logarithmic rate model, the tertiary creep and primary creep parameters are derived separately. The non-linear combination approach uses the form of the tertiary logarithmic rate relationship with a different
29、method for accounting for primary creep and its effect on rupture life. From equation (1.17) rtF 0 (1.23) or, 3rrttF (1.24) where 3rt is calculated from equation (1.5). Thus one way to consider the parameter F is that it is the ratio of the actual rupture time to the rupture estimate that is derived
30、 from estimating the life based on the tertiary portion of the curve. Ellis recorded the F parameter from as little as 0.31 to nearly 0.82 which provides a rough indication of the fraction of life in tertiary creep. F is also related to the computed initial strain that comes as a result of a fit to
31、the tertiary portion of the curve. And in this sense it has the most relevance to the problem of isochronous curve generation. From equations (1.9) and (1.10), at time zero: Fln10 (1.25) STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 5 Of
32、course the initial strain is actually zero. If, instead of considering “F” as a fixed value, but instead as a consequence of the primary creep, we can use equation (1.25) to derive a time-dependent value of F. For example, using the logarithmic rate description of primary creep Ft ln11ln1 0111 (1.26
33、) or, using an Andrade type form: Fkt p ln1 (1.27) In most cases, either equation is sufficient, both employ two parameters, but equation (1.26) has a strategic advantage in its consistency with the tertiary creep form and a practical advantage in that the influence of the primary creep at long time
34、s is much less at relatively long times. Using (1.26), the F parameter becomes 10111 tF (1.28) and then substituting into equation (1.20) and replacing the constants, A, , and B with their logarithmic rate equivalents leads to tt 300113 131ln1 (1.29) As before, when determining the rupture time from
35、 the expressions for creep, it follows that: rr tt 30011 131 (1.30) This equation provides a relation between all the model parameters required to match the observed rupture time, tr. 1.4 Determination of Model Parameters Preliminary comparison of Logarithmic Rate and Power-Law forms for Creep In or
36、der to derive the parameters, we first consider the shape of the creep curve. The NIMS datasheets for grade 91 provide high quality data to perform data analysis. We first note that power law behavior appears linear on (a) Logarithmic strain versus logarithmic time, (b) Logarithmic strain rate versu
37、s logarithmic time, and (c) Logarithmic strain rate versus logarithmic strain, since pkt (1.31a) tpk logloglog (1.32a) 1 pkpt (1.31b) tpkp log1loglog (1.32b) ppp pk 11 (1.31c) STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 6 log1loglog 1 p
38、ppk p (1.32c) The Andrade form is a particular case of the power law equation with p = 1/3. In regions where the creep behavior obeys the Logarithmic Rate form Equations (1.1) and (1.7) a plot of the logarithm of the creep rate versus linear strain would appear linear. Inspection of the creep curve
39、shapes would seem to be a simple way of distinguishing between power law and logarithmic rate-type behavior. The published NIMS datasheets provide creep curves in different formats for a variety of heats designated as MGA, MGB, MGC, MgC. Over a particular range of either time or strain, curve fits f
40、rom either form may provide very reasonable results. It can be observed for the NIMS curves that the power law form for the primary creep works well for most curves over several orders of magnitude in time (Figure 1.1(a) and (b), but at very small strains it tends to break down as a limit to the cre
41、ep rate is reached (Figure 1.1(c). The logarithmic rate-type formulation generally holds for the tertiary portion of the curve, but some curves tend to bend down slightly (Figure 1.1(d). In primary creep, it is difficult to find an appropriate region to perform a regression for logarithmic rate para
42、meters. STP-PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 7 (a) (b) (c) (d) Figure 1.1. Examples of formats of creep data from NIMS Attempts to fit Primary and Tertiary Separately Best Fits for Primary Creep Fits for the primary creep region
43、were performed from digitization of the log creep-rate versus log time NIMS charts using a power law formulation and the standard trendline option in Microsoft Excel. An example is shown in Figure 1.2. The coefficients and R2 value for the trendline of each material and condition were recorded. STP-
44、PT-080: Isochronous Stress-Strain Curves and External Pressure Charts and Equations for 9Cr-1Mo-V Steel 8 Figure 1.2. Example fit of a power-law equation to the primary creep portion of the curve Equations to describe the Primary-Creep Parameters There are two parameters to describe power law creep,
45、 k and p. The first, k, can be described adequately through a Larson-Miller expression as shown in Figure 1.3. The second (Figure 1.4), the time exponent, shows significant scatter and can only be roughly described as linear in stress. The effect of the temperature appears to be to shift the stress
46、range. These trends are visible, but the inconsistency of the test results does not allow for complex models to be winnowed from the available data. In order to describe the p-parameter in terms of stress and temperature, it is first assumed that the stress dependency is the same for all temperature
47、s. Or, aTbp (1.33) Best fit linear equations are developed for the p-vs-stress curve at each temperature for which there is sufficient data and then the slope, weighted and averaged over all the temperatures, is used. Next the stress dependency portion of the curve is subtracted out, to find a best-
48、fit expression for the temperature dependency: apTb (1.34) In this manner the experimental data is fit in a completely phenomenological manner. A comparison of the temperature function, b(T), and the data used to generate it is shown in Figure 1.5. The scatter in the data is about +/- 0.1 about the mean. The equation (1.33) is expected to have limited valid
