A method for predicting oxidation-fatigue crack growth rate in nickel-based superalloys

Abstract

The present invention discloses a method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloys, comprising the following steps: establishing a cycle-dependent crack growth rate prediction model based on fatigue crack growth rate tests; expressing the time-dependent fatigue crack growth rate as a correlation function of the size parameter s of the environmental influence zone; comparing the cycle-dependent crack growth rate prediction model with the oxidation rate constant k in the parabolic model of actual oxidation kinetics. p Combined with the actual temperature, material parameters and oxidation rate constant k, the present invention takes into account the actual temperature, material parameters and oxidation rate constant k. p The fatigue crack growth rate at different temperatures can be better predicted by the method, which effectively reduces the experimental cost and time required to test the actual crack growth rate of nickel-based high-temperature alloys at various temperatures.

Description

Technical Field

[0001] The present invention belongs to the field of damage tolerance design of metallic material structures, and particularly relates to a fatigue crack growth rate prediction method capable of considering oxidation-promoted load-holding fatigue crack growth at different temperatures. Background Art

[0002] As turbine inlet temperatures continue to rise both domestically and internationally to meet the demands of developing high thrust-to-weight and high power-to-weight ratio aircraft engines, the impact of oxidation damage on fatigue crack propagation in high-temperature alloys used in turbine disks and on the surface of turbine disks has become increasingly significant, significantly increasing the risk of component fracture failure. This poses new challenges to the existing damage tolerance design of aircraft engine turbine disks for fatigue life. A core aspect of damage tolerance design is the need for accurate and reliable fatigue crack growth models. Currently, the Paris model and its various evolution models are commonly used for the long crack growth stage. However, aircraft engine turbine disks undergo a variety of cycle-related and time-related damage mechanisms (oxidation, creep) under service conditions. These mechanisms, influenced by actual parameters such as grain size, frequency, and hold time, can directly or indirectly affect crack propagation in turbine disks. Predicting crack growth rate while fully considering these parameters has been a pressing challenge for predictive models in recent years. While research on the fatigue properties of various nickel-based superalloys under different conditions has yielded considerable results, there are relatively few patents for models predicting fatigue crack growth rates under high-temperature, sustained-load conditions. Currently, only patent application number CN201710712352.3, titled "A Method for Predicting Fatigue Crack Growth in Metallic Materials," considers this under varying loads at room temperature. Furthermore, due to the cost and time associated with testing, it is not possible to exhaustively enumerate the actual crack growth rates at all temperatures. Consequently, a method for effectively predicting fatigue crack growth rates at various temperatures is urgently needed. Therefore, the development of models for predicting metal fatigue crack growth rates under varying temperatures is of great practical significance. Summary of the Invention

[0003] The purpose of the present invention is to provide a method for predicting the oxidation-fatigue crack growth rate of nickel-based high-temperature alloys, so as to accurately predict the fatigue crack growth rate of nickel-based high-temperature alloys at different temperatures, and serve and support the damage tolerance design of components such as aircraft engine turbine disks.

[0004] To achieve the above object, the present invention adopts the following technical solutions:

[0005] A method for predicting oxidation-fatigue crack growth rate of a nickel-based high-temperature alloy comprises the following steps:

[0006] Step 1: Based on the fatigue crack growth rate test, a cycle-dependent crack growth rate prediction model is established;

[0007] Step 2: Express the time-dependent fatigue crack growth rate as a function of the size parameter s of the environmental influence zone;

[0008] Step 3: Compare the time-dependent fatigue crack growth rate prediction model with the oxidation rate constant k in the parabolic model of actual oxidation kinetics. p Combined;

[0009] Step 4: The oxidation rate constant k related to the change in oxide mass per unit area per unit time pm Converted into the oxidation rate constant k per unit time considering the oxidation length pl , by combining the length-dependent oxidation rate constant k pl , which is assigned as the environmental impact zone size parameter s to the improved combined oxidation rate constant k in step three p The time-dependent fatigue crack growth rate prediction model is finally obtained to describe the time-dependent fatigue crack growth rate prediction model at different temperatures;

[0010] Step 5: Establish the final fatigue crack growth model. The overall crack growth model is divided into two parts: cycle-related and time-related. The fatigue crack growth model is used to predict the oxidation-fatigue crack growth rate of nickel-based high-temperature alloys.

[0011] In the first step, a fatigue crack growth rate test is performed using a standard CT specimen under the conditions of stress ratio R and prefabricated crack length a0, and the fatigue crack growth rate test data of crack length a, loading time t, and number of cycles N are recorded until the standard specimen breaks and fails; at the same time, the crack growth rate is divided into two parts, one is cycle-dependent crack growth, and the other is time-dependent crack growth; based on the fatigue crack growth model, a cycle-dependent crack growth rate prediction model is established:

[0012]

[0013] Where ΔK is the stress intensity factor range, K th is the fatigue crack growth threshold, A and m are material constants, and the fatigue crack growth rate data [ΔK i ,(da / dN) i ], perform the least squares linear fitting on the above formula (1) to determine the A and m parameters corresponding to the nickel-based high-temperature alloy.

[0014] In the step 1, the loading condition of the cyclically correlated crack growth test is required to be a high-frequency sine wave / triangular wave loading waveform at room temperature.

[0015] In the step 1, the temperature of the time-dependent fatigue crack growth test is above 600° C., and the holding time is above 20 seconds and below 600 seconds.

[0016] In the second step, based on the CHAN's fatigue crack growth model of the Southwest Research Institute of the United States considering the alloy microstructure, the time-dependent fatigue crack growth rate is expressed as a function of the size parameter s×d of the crack tip environmental influence zone:

[0017]

[0018] Where E is the elastic modulus, σ y is the yield stress, t o is the oxidation time, is the plastic strain at the crack tip, and n is the formula constant.

[0019] In step 3, the original time-dependent fatigue crack growth model of CHAN's is compared with the oxidation rate constant k in the parabolic model of actual oxidation kinetics. p Combined, k p The expression is as follows:

[0020]

[0021] Where ΔM is the changed mass of the oxide, and A is the unit area;

[0022] Oxidation rate constant k p The expanded form of is represented by the exponential expression, namely:

[0023]

[0024] Where Q is the activation energy, T is the absolute temperature, R is the gas constant, and C' is a constant.

[0025] In the step 3, the oxidation rate constant k p Assigned to the fatigue crack growth rate prediction model, the specific process is as follows:

[0026] Based on the crack tip oxidation behavior, there is a formula for the change in oxide mass per unit area:

[0027]

[0028] Oxidation time t o There is a relationship dt = t o , then k p ×t oThat is, the crack extension length per unit time. Combined with the original oxidation crack growth rate model, it is believed that in the environmental influence zone of s×d size at the crack tip, the change law of the environmental influence zone parameter s perpendicular to the loading direction caused by unit oxidation time conforms to the parabolic law of oxidation kinetics, that is:

[0029] s=k p (t o ) 1 / 2 (7)

[0030] Then we have:

[0031]

[0032]

[0033] That is, the final:

[0034]

[0035] In step 4, the oxidation rate constant k related to mass per unit time needs to be pm Converted into a length-dependent oxidation rate constant k per unit time pl The specific process is as follows:

[0036]

[0037] In step 5, the fatigue crack growth model is described by the following formula:

[0038]

[0039] in is the total fatigue crack growth rate, is the cycle-dependent fatigue crack growth rate, Time-dependent fatigue crack growth rate, f is the current cyclic loading frequency, ρ oxide is the oxide density, m is the constant of the cycle-dependent fatigue crack growth part, γ,n is the formula constant of the time-dependent fatigue crack growth part, d o The height of the material reference crack tip unit is used. By substituting the material-related parameters, the predicted crack growth rate under different ΔK is finally obtained.

[0040] In step 5, it is necessary to obtain the model parameters n and oxidation time t of the time-dependent fatigue crack growth part. o Function curve related to temperature T, using temperature T parameter to obtain the corresponding parameter t at different temperatures o , n, and thus predict the fatigue crack growth rate at different temperatures, and the fitting functions are:

[0041] nFitting function:

[0042] lg(n)=A2 exp(T / t2)+y1 (12)

[0043] t o Fitting function:

[0044] ln((t o ) 1 / 2 )=A1 exp(-ln(T) / t1)+y0 (13)

[0045] Where A2, t2, y1 are n fitting function parameters, A1, t1, y0 are t o Fitting function parameters.

[0046] Beneficial effects: The present invention provides a prediction model for the oxidation-fatigue crack growth rate of nickel-based high-temperature alloys. This method takes into account the influence of actual environmental parameters, material parameters and oxidation rate constants, and realizes the accurate prediction of the fatigue crack growth rate of nickel-based high-temperature alloys at different temperatures. It can serve and support the damage tolerance design of components such as aircraft engine turbine disks. BRIEF DESCRIPTION OF THE DRAWINGS

[0047] Figure 1 is a schematic diagram of the fatigue crack growth test specimen used;

[0048] Figure 2 is the ln((t o ) 1 / 2 )-ln(T) relationship curve;

[0049] Figure 3 logarithm obtained from fatigue crack growth test data of coarse-grained FGH4098 alloy 10 (n)-T relationship curve;

[0050] Figure 4 The following are the prediction results of the fatigue crack growth rate model for coarse-grained FGH4098 alloy, where (a) is the prediction effect of the da / dN-ΔK curve model at 700°C; (b) is the prediction effect of the da / dN-ΔK curve model at 650°C, 750°C, and 800°C. DETAILED DESCRIPTION

[0051] The present invention will be further explained below with reference to the accompanying drawings.

[0052] A method for predicting oxidation-fatigue crack growth rate of a nickel-based high-temperature alloy according to the present invention comprises the following steps:

[0053] Step 1: Based on the fatigue crack growth rate test, a cycle-dependent crack growth rate prediction model is established;

[0054] A fatigue crack growth rate test was conducted on a standard CT specimen under the conditions of stress ratio R and prefabricated crack length a0. The fatigue crack growth rate test data of crack length a, loading time t, and number of cycles N were recorded until the standard specimen fractured and failed. The crack growth rate was divided into two parts: one is cycle-dependent crack growth, and the other is time-dependent crack growth. Based on the fatigue crack growth model, a cycle-dependent crack growth rate prediction model was established:

[0055]

[0056] Where ΔK is the stress intensity factor range, K th is the fatigue crack growth threshold, A and m are material constants, and the fatigue crack growth rate data [ΔK i ,(da / dN) i ], perform the least squares linear fitting on the above formula (1) to determine the A and m parameters corresponding to the nickel-based high-temperature alloy.

[0057] The cyclic-dependent crack growth test requires a high-frequency sinusoidal / triangular wave loading waveform at room temperature. The time-dependent fatigue crack growth test requires a temperature above 600°C and a holding time of at least 20 seconds and less than 600 seconds.

[0058] Step 2: Express the time-dependent fatigue crack growth rate as a function of the size parameter s of the environmental influence zone;

[0059] Based on the CHAN's fatigue crack growth model of the Southwest Research Institute of the United States, which takes into account the alloy microstructure, the time-dependent fatigue crack growth rate is expressed as a function of the size parameters s×d of the crack tip environmental influence zone:

[0060]

[0061] Where E is the elastic modulus, σ y is the yield stress, t o is the oxidation time, is the plastic strain at the crack tip, and n is the formula constant.

[0062] Step 3: Compare the time-dependent fatigue crack growth rate prediction model with the oxidation rate constant k in the parabolic model of actual oxidation kinetics. p Combined;

[0063] The oxidation rate constant k in the parabolic model of the actual oxidation kinetics is compared with the original CHAN's model of time-dependent fatigue crack growth.p Combined, k p The expression is as follows:

[0064]

[0065] Where ΔM is the changed mass of the oxide, and A is the unit area;

[0066] Oxidation rate constant k p The expanded form of is represented by the exponential expression, namely:

[0067]

[0068] Where Q is the activation energy, T is the absolute temperature, R is the gas constant, and C' is a constant.

[0069] The oxidation rate constant k p Assigned to the fatigue crack growth rate prediction model, the specific process is as follows:

[0070] Based on the crack tip oxidation behavior, there is a formula for the change in oxide mass per unit area:

[0071]

[0072] Oxidation time t o There is a relationship dt = t o , then k p ×t o That is, the crack extension length per unit time. Combined with the original oxidation crack growth rate model, it is believed that in the environmental influence zone of s×d size at the crack tip, the change law of the environmental influence zone parameter s perpendicular to the loading direction caused by unit oxidation time conforms to the parabolic law of oxidation kinetics, that is:

[0073] s=k p (t o ) 1 / 2 (7)

[0074] Then we have:

[0075]

[0076]

[0077] That is, the final:

[0078]

[0079] Step 4: The oxidation rate constant k related to the change in oxide mass per unit area per unit time pm Converted into the oxidation rate constant k per unit time considering the oxidation length pl, by combining the length-dependent oxidation rate constant k pl , which is assigned as the environmental impact zone size parameter s to the improved combined oxidation rate constant k in step three p The time-dependent fatigue crack growth rate prediction model is finally obtained to describe the time-dependent fatigue crack growth rate prediction model at different temperatures;

[0080] The oxidation rate constant k related to mass per unit time needs to be pm Converted into a length-dependent oxidation rate constant k per unit time pl The specific process is as follows:

[0081]

[0082] Step 5: Establish the final fatigue crack growth model. The overall crack growth model is divided into two parts: cycle-related and time-related. The fatigue crack growth model is used to predict the oxidation-fatigue crack growth rate of nickel-based superalloys.

[0083] The fatigue crack growth model is described by the following equation:

[0084]

[0085] in is the total fatigue crack growth rate, is the cycle-dependent fatigue crack growth rate, Time-dependent fatigue crack growth rate, f is the current cyclic loading frequency, ρ oxide is the oxide density, m is the constant of the cycle-dependent fatigue crack growth part, γ,n is the formula constant of the time-dependent fatigue crack growth part, d o The height of the material reference crack tip unit is used. By substituting the material-related parameters, the predicted crack growth rate under different ΔK is finally obtained.

[0086] The model parameters n and oxidation time t of the time-dependent fatigue crack growth part need to be obtained o Function curve related to temperature T, using temperature T parameter to obtain the corresponding parameter t at different temperatures o , n, and thus predict the fatigue crack growth rate at different temperatures, and the fitting functions are:

[0087] nFitting function:

[0088] lg(m)=A2 exp(T / t2)+y1 (12)

[0089] t o Fitting function:

[0090] ln((t o )1 / 2 )=A1 exp(-ln(T) / t1)+y0 (13)

[0091] Where A2, t2, y1 are n fitting function parameters, A1, t1, y0 are t o Fitting function parameters.

[0092] The present invention will be further described below with reference to the embodiments.

[0093] Example

[0094] Step 1: In this example, the nickel-based high-temperature alloy coarse-grained FGH4098 is selected, and the CT sample size is as follows: Figure 1 As shown, the thickness B = 12.7mm, length 31.8mm, width 30.5mm, and pre-crack length a0 = 1mm. The test was carried out in accordance with ASTM E647-08 standard. The fatigue crack growth test was carried out under 1s-90s-1s-1s trapezoidal wave load (holding load 90s), F max The test was conducted under loading conditions of 4154 N and R = 0.1. The crack growth rate was measured using the direct current potential drop (DCPD) method. The stress intensity factor (SIF) at the crack tip was calculated after the test: the SIF near the crack tip during crack growth was determined using the SIF calculation formula for CT specimens in ASTM E647-08.

[0095] Step 2: Determine the cycle-related crack growth parameters: The fatigue-related crack growth model is established based on the Paris model:

[0096]

[0097] Where ΔK is the stress intensity factor range, ΔK th is the fatigue crack growth threshold, and A and m are material constants. In this model, the ΔK-dadN data of the corresponding material at room temperature and high-frequency sine waves are used for parameter fitting to obtain the corresponding A and m parameters. In this example, the fatigue crack growth rate of coarse-grained FGH4098 alloy at room temperature and 5Hz sine waves is used as the data basis for the cyclic-dependent crack growth model. The fitting parameters are shown in Table 1:

[0098] Table 1 Parameters of the fitting function for cycle-dependent crack growth of coarse-grained FGH4098 alloy

[0099]

[0100] Step 3: Determine the oxidation rate constant k for the time-dependent crack growth parameter p :

[0101] The model is compared with the actual oxidation rate constant k pBy combining the oxidation rate constant, a time-dependent crack growth model that actually describes oxygen-induced damage is obtained. In this example, coarse-grained FGH4098 is taken as an example, and its oxidation rate constant k p have:

[0102]

[0103] At the same time, the oxidation rate constant is converted into the oxidation rate constant k considering the oxidation length pl , as shown below:

[0104]

[0105] where k pm To consider the oxide density ρ oxied The mass-dependent oxidation rate constant, k pl To consider the oxide density ρ oxied In this example, the oxidation rate constant k of coarse-grained FGH4098 at different temperatures can be obtained. pl , as shown in Table 2:

[0106] Table 2 Oxidation rate constant k related to oxidation length pl

[0107]

[0108] Step 4: At the same time, based on the addition of the oxidation rate constant k p The microstructure oxidation model of the coarse-grained FGH4098 alloy is constructed by substituting the oxidation rate constant k pl , total crack growth rate (da / dN) dwell Expressed as:

[0109]

[0110] The coarse-grained FGH4098 model parameters are shown in Tables 3 and 4:

[0111] Table 3 Some model parameters of coarse-grained FGH4098

[0112]

[0113] Table 4 Elastic modulus E of coarse-grained FGH4098 alloy

[0114]

[0115] Step 5: Calculate and simulate the n-parameter function required by the model through actual data. The fitting function is shown in the following formula. The fitting function parameters are shown in Table 5. The fitting function curve is shown in Figure 2 shown.

[0116] nFitting function:

[0117] lg(n)=A2 exp(T / t2)+y1

[0118] Table 5n parameter fitting function parameters (all dimensionless parameters)

[0119]

[0120] Step 6: Calculate and simulate the t required by the model using actual data o The parameter function and the fitting function are shown in the following formula. The fitting function parameters are shown in Table 6. The fitting function curve is shown in Figure 3 shown.

[0121] t o Fitting function:

[0122] ln((t o ) 1 / 2 )=A1 exp(-ln(T) / t1)+y0

[0123] Table 6t o Parameter fitting function parameters

[0124]

[0125] Step 7: Through steps 5 and 6, we finally get two parameters t o ,n corresponding functional relationship. In order to extrapolate the fatigue crack growth curve at 700℃ to verify the reliability of the model, the crack growth rate at 700℃ was obtained by extrapolating the data at 650℃, 750℃, and 800℃ in the actual test, as shown in Figure 4 As shown in (a). The predicted fatigue crack rate at 700℃ is well compared with the actual crack growth rate. The model predicts the crack growth rate in the range of ΔK = 30-50 MPa·m 1 / 2 The above results are within 2 orders of magnitude of the actual crack growth rate. At the same time, the existing data at 650℃, 750℃, and 800℃ are also reversed, and the results are as follows: Figure 4 As shown in (b), the crack growth rate at each temperature is well predicted, and the predicted fatigue crack growth rate in each ΔK segment is within 2 orders of magnitude compared with the actual one.

[0126] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principles of the present invention. These improvements and modifications should also be regarded as within the scope of protection of the present invention.

Claims

1. A method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloys, characterized by: The following steps are involved: Step 1: Based on the fatigue crack growth rate test, a cycle-dependent crack growth rate prediction model is established; Step 2: Express the time-dependent fatigue crack growth rate as a function of the size parameter s of the environmental influence zone; Step 3: Compare the time-dependent fatigue crack growth rate prediction model with the oxidation rate constant k in the parabolic model of actual oxidation kinetics. p Combined; In the third step, the original time-dependent fatigue crack growth model of the fatigue crack growth model of the Southwest Research Institute of the United States, CHAN's, which takes into account the alloy microstructure, is combined with the oxidation rate constant k in the parabolic model of the actual oxidation kinetics. p Combined, k p The expression is as follows: Where ΔM is the changed mass of the oxide, and A is the unit area; Oxidation rate constant k p The expanded form of is represented by the exponential expression, namely: Where Q is the activation energy, T is the absolute temperature, R is the gas constant, and C' is a constant; The oxidation rate constant k p Assigned to the fatigue crack growth rate prediction model, the specific process is as follows: Based on the crack tip oxidation behavior, there is a formula for the change in oxide mass per unit area: Oxidation time t o There is a relationship dt = t o , then k p ×t o That is, the crack extension length per unit time. Combined with the original oxidation crack growth rate model, it is believed that in the environmental influence zone of s×d size at the crack tip, the change law of the environmental influence zone parameter s perpendicular to the loading direction caused by unit oxidation time conforms to the parabolic law of oxidation kinetics, that is: s=k p (t o ) 1 / 2 (7) Then we have: That is, the final: Step 4: The oxidation rate constant k related to the change in oxide mass per unit area per unit time pm Converted into the oxidation rate constant k per unit time considering the oxidation length pl , by combining the length-dependent oxidation rate constant k pl , which is assigned as the environmental impact zone size parameter s to the improved combined oxidation rate constant k in step three p The time-dependent fatigue crack growth rate prediction model is finally obtained to describe the time-dependent fatigue crack growth rate prediction model at different temperatures; Step 5: Establish the final fatigue crack growth model. The overall crack growth model is divided into two parts: cycle-related and time-related. The fatigue crack growth model is used to predict the oxidation-fatigue crack growth rate of nickel-based high-temperature alloys.

2. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 1, characterized in that: In the step 1, a fatigue crack growth rate test is performed using a standard CT specimen under the conditions of a stress ratio R and a prefabricated crack length a0, and the fatigue crack growth rate test data of the crack length a, loading time t, and number of cycles N are recorded until the standard specimen breaks and fails; and the crack growth rate is divided into two parts, one of which is cycle-dependent crack growth and the other is time-dependent crack growth; Based on the fatigue crack growth model, a cycle-dependent crack growth rate prediction model is established: Where ΔK is the stress intensity factor range, K th is the fatigue crack growth threshold, A and m are material constants, and the fatigue crack growth rate data [ΔK i ,(da / dN) i ], perform the least squares linear fitting on the above formula (1) to determine the A and m parameters corresponding to the nickel-based high-temperature alloy.

3. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 2, characterized in that: In the step 1, the loading condition of the cyclically correlated crack growth test is required to be a high-frequency sine wave / triangular wave loading waveform at room temperature.

4. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 2, characterized in that: In the step 1, the temperature of the time-dependent fatigue crack growth test is above 600° C., and the holding time is above 20 seconds and below 600 seconds.

5. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 1, characterized in that: In the second step, based on the CHAN's fatigue crack growth model of the Southwest Research Institute of the United States considering the alloy microstructure, the time-dependent fatigue crack growth rate is expressed as a function of the size parameter s×d of the crack tip environmental influence zone: Where E is the elastic modulus, σ y is the yield stress, t o is the oxidation time, is the plastic strain at the crack tip, and n is the formula constant.

6. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 1, characterized in that: In step 4, the oxidation rate constant k related to mass per unit time needs to be pm Converted into a length-dependent oxidation rate constant k per unit time pl The specific process is as follows:

7. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 1, characterized in that: In step 5, the fatigue crack growth model is described by the following formula: in is the total fatigue crack growth rate, is the cycle-dependent fatigue crack growth rate, Time-dependent fatigue crack growth rate, f is the current cyclic loading frequency, ρ oxide is the oxide density, m is the constant of the cycle-dependent fatigue crack growth part, γ,n is the formula constant of the time-dependent fatigue crack growth part, d o The height of the material reference crack tip unit is used. By substituting the material-related parameters, the predicted crack growth rate under different ΔK is finally obtained.

8. The method for predicting oxidation-fatigue crack growth rate of nickel-based high-temperature alloy according to claim 7, characterized in that: In step 5, it is necessary to obtain the model parameters n and oxidation time t of the time-dependent fatigue crack growth part. o Function curve related to temperature T, using temperature T parameter to obtain the corresponding parameter t at different temperatures o , n, and thus predict the fatigue crack growth rate at different temperatures, and the fitting functions are: nFitting function: lg(n)=A2 exp(T / t2)+y1 (12) t o Fitting function: ln((t o ) 1 / 2 )=A1 exp(-ln(T) / t1)+y0 (13) Where A2, t2, y1 are n fitting function parameters, A1, t1, y0 are t o Fitting function parameters.

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