Calibration method for the brittle fracture assessment parameters for materials based on the Beremin model

Abstract

A calibration method for brittle fracture assessment parameters for pressure vessel materials based on the Beremin model includes selecting at least two types of specimens of different constraints, and calculating the fracture toughness values K.sub.0 corresponding to 63.2% failure probability for each type of specimens at a same calibration temperature by using the respective fracture toughness data. The method proceeds by obtaining the stress-strain curve of the material at the calibration temperature, generating finite element models for each type of specimens, and calculating the maximum principal stress and element volume of every element at K=K.sub.0 in each model. A series of values of m are assumed to compute a group of σ.sub.u values for each type of specimens, and then m˜σ.sub.u curves are plotted for each type of specimens. Brittle fracture assessment parameters are then determined for the material according to the coordinates of the intersection of the m˜σ.sub.u curves.

Claims

1. A calibration method for the brittle fracture assessment parameters for materials based on the Beremin model, the method comprises the following steps: (1) Selecting at least two types of specimens made of a same material but with different constraints, and calculating the fracture toughness value K.sub.0 corresponding to 63.2% failure probability for each type of specimens at a same calibration temperature by using the respective fracture toughness data; (2) Constructing finite element models for each type of specimens using the stress-strain curve of the material measured at the same calibration temperature, and calculating the maximum principal stress σ.sub.1,i and element volume Vi of each element at K=K.sub.0 in each model, where K is a stress intensity factor that describes the intensity of far field loading on the crack front, and i is an order number of elements; (3) Assuming a series of values of the Weibull slope m and calculating a set of values of the Weibull scale parameter σ.sub.u for each type of specimens according to the following equation, and plotting the Beremin's parameters characteristic curves for each type of specimens with the curves representing the relationship between m and σ.sub.u for each type of specimens; ${\sigma }_u=\sqrt[m]{\underset{i}{\overset{n}{.Math.}}{\left({\sigma }_{1,i}\right)}^m\frac{V_i}{V_0}}$ wherein, n represents the number of elements in the fracture process region, V.sub.0 represents a reference volume; (4) Determining the brittle fracture assessment parameters for the material according to the coordinates of the intersection of the Beremin's parameters characteristic curves; (5) using the brittle fracture assessment parameters in safety engineering.

2. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 1, wherein, in the step (1), fracture toughness tests on each type of specimens are carried out at the same calibration temperature to obtain the fracture toughness data.

3. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 1 wherein, in the step (1), fracture toughness tests are carried out on each type of specimens at different temperatures to obtain the fracture toughness data, and calculating the fracture toughness value K.sub.0 corresponding to 63.2% failure probability at the same calibration temperature by using the predetermined master curve.

4. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 3 wherein, in the step (1), the same calibration temperature is the lowest of the different temperatures.

5. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 1 wherein, in the step (2), the uniaxial tensile test is carried out at the same calibration temperature to obtain the stress-strain curve.

6. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 5 wherein, in the step (3), the values of m are taken as integers larger than 5 and less than 40.

7. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 6 wherein, in the step (3), the fracture process region is defined as the volume inside the loci σ.sub.1,i≧λσ.sub.ys, wherein λ is a constant, σ.sub.ys is the yield strength of the material at the calibration temperature.

8. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 7 wherein, in the step (3), the value of λ is 1 or 2.

9. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 1 wherein, in the step (1), at least six fracture toughness data are obtained for each type of specimens are required.

10. The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model according to claim 9 wherein, in the step (1), at least fifteen fracture toughness data are obtained for each type of specimens are required.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) These and other features and advantages of the present invention will become more readily appreciated when considered in connection with the following detailed description and appended drawings, wherein:

(2) FIG. 1 is the flow diagram of the calibration method according to this invention;

(3) FIG. 2 is the schematic drawing for the calibration method based on the m˜σ.sub.u curves intersection;

(4) FIG. 3 shows the Beremin model's parameters for 16MnR steel according to the example 1 of this invention;

(5) FIG. 4 shows the Beremin model's parameters for 16MnR steel according to the RGD calibration method;

(6) FIG. 5 shows the Beremin model's parameters for A508-3 forging according to the example 2 of this invention; and

(7) FIG. 6 shows the Beremin model's parameters for A508-3 forging according to the RGD calibration method.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

(8) The flow diagram of the calibration method for the brittle fracture assessment parameters for materials based on the Beremin model is shown in FIG. 1, and the details are described in the following:

(9) (1) Select at least two types of specimens made of a same material but with different constraints such as high constraint specimen A and low constraint specimen B. Perform fracture toughness test using specimen A and specimen B in the ductile-to-brittle transition region to obtain two sets of fracture toughness data, K.sub.Jc(k),A and K.sub.Jc(j),B wherein k and j are the testing order numbers. Generally speaking, the more data in each set, the greater the accuracy of the brittle fracture assessment parameters, m and σ.sub.u, from the calibration method. Therefore, each set preferably has at least 6 fracture toughness data and more preferably has at least 15 fracture toughness data. The fracture toughness values K.sub.0(A) and K.sub.0(B) corresponding to 63.2% failure probability can be determined respectively for specimens A and B at a same calibration temperature T by using the fracture toughness data.

(10) (1.1) If the fracture toughness data K.sub.Jc(k),A and K.sub.Jc(j),B for specimens A and B are measured at the calibration temperature T=T.sub.A=T.sub.B, the fracture toughness values K.sub.0(A) and K.sub.0(B) corresponding to 63.2% failure probability at the calibration temperature can be calculated directly.

(11) (1.2) If specimens A and B are tested at different temperatures T.sub.A≢T.sub.B to generate fracture toughness data K.sub.Jc(k),A and K.sub.Jc(j),B, master curve for the material can be determined in accordance with ASTM E1921 proposed by the American Society for Testing and Materials, which can make the estimates of K.sub.0 corresponding to 63.2% failure probability for the specimens at the calibration temperature T.

(12) It should be noted that the estimation of fracture toughness of the two types of specimens using master curve is conducted under the assumption that brittle fracture occurs. Low constraint specimen B should be generally tested at lower temperature T.sub.B, while fracture toughness test on high constraint specimen A can be performed at higher temperature T.sub.A at which specimen B may exhibit significant ductile tearing prior to cleavage fracture. Therefore, it is suggested that the fracture toughness data for high constraint specimen A should be converted to those tested at temperature T.sub.B as specimen B. According to the requirements in ASTM E1921, it is also suggested that fracture toughness test on high constraint specimen A should be performed to establish the master curve for the material such that the estimated value of K.sub.0(A) can be obtained at the calibration temperature T=T.sub.B.

(13) (2) Uniaxial tensile testing is carried out at the same calibration temperature T mentioned above to obtain the tensile property of the material. Perform finite element analyses for high constraint specimen A and low constraint specimen B, and then export the maximum principal stress σ.sub.1,i and element volume V.sub.i of each element at K=K.sub.0 in each model, where K is a stress intensity factor that describes the intensity of far field loading on the crack front, and i is an order number of elements.

(14) (3) Beremin model adopts a two-parameter Weibull distribution to predict the cumulative failure probability of cleavage fracture, P.sub.f, for structures, as follows:

(15) $\begin{array}{cc}{P}_f\left({\sigma }_w\right)=1-\exp \left[-\left(\frac{{\int }_{V_{\mathrm{pl}}}{\sigma }_1^{m}dV}{{\sigma }_u^m{V}_0}\right)\right]=1-\exp \left[-{\left(\frac{{\sigma }_w}{{\sigma }_u}\right)}^m\right]\mathrm{where}{\sigma }_w=\sqrt[m]{{\int }_{\mathrm{pl}}^{}{\left({\sigma }_l\right)}^m\frac{\mathrm{dV}}{V-0},}& \left(1\right)\end{array}$
named Weibull stress, is a driving force for cleavage fracture; the Weibull slope, m, describes the scatter in the microcracks distribution and its value quantifies the degree of scatter of experimental failure data; the scale parameter of the Weibull distribution, σ.sub.u, is related to the microscale material toughness and corresponds to the σ.sub.w value at P.sub.f=63.2%.

(16) Therefore, at the level of loading K=K.sub.0 corresponding to P.sub.f=63.2%, the Equation (2) is obtained:

(17) $\begin{array}{cc}{\sigma }_u={\sigma }_w=\sqrt[m]{{\int }_{V_{\mathrm{pl}}}{\left({\sigma }_1\right)}^m\frac{dV}{V_0}}=\sqrt[m]{\underset{i}{\overset{n}{.Math.}}{\left({\sigma }_{1,i}\right)}^m\frac{V_i}{V_0}}& \left(2\right)\end{array}$

(18) Where V.sub.pl represents the fracture process region; n denotes the number of elements in the fracture process region; σ.sub.1,i and V.sub.i represent the maximum principal stress and element volume of each element in the fracture process region; V.sub.0 represents a reference volume; V.sub.pl is defined as the region where the maximum principal stress exceeds the yield strength: σ.sub.1,i≧λσ.sub.ys, where λ is a constant factor and is generally taken equal to 1 or 2; σ.sub.ys is the yield strength of the material at the calibration temperature T.

(19) Assuming a series of values of the Weibull slope m=m.sub.1, m.sub.2, m.sub.3 . . . etc. (The values of m are usually taken equal to integers larger than 5 and less than 40) and calculate the σ.sub.w for specimens A and B at K.sub.J=K.sub.0(A) and K.sub.0(B) respectively, based on the Equation (2) using the values of σ.sub.1,i and V.sub.i obtained in step (2). Since the value of σ.sub.u is the value of σ.sub.w at K.sub.J=K.sub.0, two m˜σ.sub.u curves are obtained as illustrated in FIG. 2, namely the characteristic curves for the Beremin model's parameters.

(20) (4) Find the intersection of the two m˜σ.sub.u curves marked with “O” as illustrated in FIG. 2 and determine the values of the brittle fracture assessment parameters (m, σ.sub.u) for the material by the coordinate of the intersection point.

(21) The following is the details of the present invention in specific embodiments. The attention must be paid that the examples are only used for the purpose of illustration, not to limit the scope of the invention.

EXAMPLE 1

(22) The material is a homemade C—Mn steel 16MnR which is widely used for manufacturing pressure vessels in China. Select three-point bend specimen with thickness of 0.5 inches (0.5T-SE(B) specimen) as the high constraint specimen. For 0.5T-SE(B) specimen, the width to thickness ratio W/B is equal to 2. Select the pre-crack Charpy size specimen (PCVN specimen) as the low constraint specimen, which has the width to thickness ratio W/B equal to 1. Both the 0.5T-SE(B) and PCVN specimens have the span to width ratio S/W=4 and the nominal crack depth ratio a.sub.0/W=0.5.

(23) The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model comprises the following steps:

(24) (1) Test the 0.5T-SE(B) and PCVN specimens at T=−100° C. to generate two sets of fracture toughness data which are listed in Tables 1 and 2. The K.sub.0 values for the 0.5T-SE(B) and PCVN specimens at T=−100° C. are calculated as K.sub.0(0.5T)=126.9 MPa√{square root over (m)} and K.sub.0(PCVN)=208.4 MPA√{square root over (m)} respectively, based on the fracture toughness data K.sub.Jc(0.5T) and K.sub.Jc(PCVN) in Tables 1 and 2.

(25) TABLE-US-00001 TABLE 1 Specimen ID K.sub.Jc(0.5 T) (MPa{square root over (m)}) 16MnR0.5T-8 54.5 16MnR0.5T-10 55.6 16MnR0.5T-5 103.3 16MnR0.5T-4 103.6 16MnR0.5T-3 109.1 16MnR0.5T-7 111.7 16MnR0.5T-12 121.1 16MnR0.5T-6 128.4 16MnR0.5T-11 185.6 16MnR0.5T-9 202.7

(26) TABLE-US-00002 TABLE 2 Specimen ID K.sub.Jc(PCVN) (MPa{square root over (m)}) 16MnRPCVN32 85.9 16MnRPCVN10 100.9 16MnRPCVN34 102.7 16MnRPCVN14 150.7 16MnRPCVN31 156.8 16MnRPCVN35 187.9 16MnRPCVN33 192.4 16MnRPCVN36 193.2 16MnRPCVN13 206.5 16MnRPCVN12 211.0 16MnRPCVN38 215.8 16MnRPCVN37 222.0 16MnRPCVN16 236.4 16MnRPCVN15 254.4 16MnRPCVN18 284.4 16MnRPCVN17 288.1

(27) (2) Uniaxial tensile testing is carried out at −100° C. to obtain the stress-strain curve for 16MnR steel. Perform finite element analyses for the 0.5T-SE(B) and the PCVN specimens and then export the maximum principal stress, σ.sub.1,i and element volume V.sub.i of each element in each model at K=K.sub.0. The fracture toughness region is defined as the region where σ.sub.1,i≧λσ.sub.ys with λ=1.

(28) (3) The reference volume V.sub.0 is taken as (50 μm).sup.3 in the example. Assume m=6, 7, 8 . . . , 10 and calculate the σ.sub.w using the data extracted from the fracture process region. Since the value of σ.sub.u is the value of σ.sub.w at K=K.sub.0, two m˜σ.sub.u curves are obtained as illustrated in FIG. 3.

(29) (4) Find the intersection of the two m˜σ.sub.u curves in FIG. 3 and obtain the Beremin model's parameters, m=7.3 and σ.sub.u=6194 MPa, for 16MnR steel, by the coordinate of the intersection point.

(30) RGD calibration method is applied to determine the Weibull slope m. As shown in FIG. 4, the point “A”, whose ordinate is K.sub.0(0.5T)=126.9 MPa√{square root over (m)} and abscissa is K.sub.0(PCVN)=208.4 MPa√{square root over (m)}, falls in the area between the two curves corresponding to m=7 and m=8. Consequently, the Weibull slope m is calculated as 7.3 by interpolation. At K.sub.J=K.sub.0(0.5T) or K.sub.0(PCVN), the calibrated σ.sub.u is calculated as 6194 MPa. The calibration results from the calibration method in this invention are exactly equal to the (m, σ.sub.u) values obtained by the RGD procedure.

EXAMPLE 2

(31) The material is a A508-3 forging for the construction of nuclear pressure vessels. Select three-point bend specimen with thickness of 0.5 inches (0.5T-SE (B) specimen) as the high constraint specimen. For 0.5T-SE(B) specimen, the width to thickness ratio W/B is equal to 2. Select the pre-crack Charpy size specimen (PCVN specimen) as the low constraint specimen, which has the width to thickness ratio W/B is equal to 1. Both the 0.5T-SE (B) and PCVN specimens have the span to width ratio S/W=4 and the nominal crack depth ratio a.sub.0/W=0.5.

(32) The calibration method for the brittle fracture assessment parameters for materials based on the Beremin model comprises the following steps:

(33) (1) Test the 0.5T-SE(B) specimens at three different temperatures −81° C., −60° C. and −40° C., and test the PCVN specimens at −100° C. The fracture toughness data K.sub.Jc(0.5T) and K.sub.Jc(PCVN) are showed in Tables 3 and 4 respectively. The K.sub.0 values for the PCVN specimens at T=−100° C. is calculated as K.sub.0(PCVN)=117.8 MPa√{square root over (m)} by using the fracture toughness data K.sub.Jc(PCVN) in Table 4. The reference temperature T.sub.0 of master curve is determined to be −61° C. using ASTM E1921 multi-temperature analysis procedure for the fracture toughness data for 0.5T-SE(B) specimen in Table 3. The K.sub.0 value for 0.5T-SE(B) specimen at T=−100° C. is estimated to be 76.5 MPa√{square root over (m)} by master curve.

(34) TABLE-US-00003 TABLE 3 Temperature (° C.) Specimen ID K.sub.Jc(0.5 T) (MPa{square root over (m)}) −81 3A17 66.4 3A13 70.1 3A11 78.0 3A15 81.2 3A14 89.6 3A16 104.2 3A12 109.8 −60 2A13 110.8 3A1A 112.6 2A11 113.3 2A12 126.5 2A14 142.7 2A15 147.6 −40 3A19 161.1 3A18 208.6

(35) TABLE-US-00004 TABLE 4 Temperature (° C.) Specimen ID K.sub.Jc(PCVN) (MPa{square root over (m)}) −100 1A1B 73.8 1A1L 92.3 1A1D 93.4 1A15 101.9 1A14 104.6 1A1A 106.2 1A18 107.1 1A19 108.5 1A1C 114.2 1A17 149.2 1A16 153.0

(36) (2) Uniaxial tensile testing is carried out at −100° C. to obtain the stress-strain curve for A508-3 forging. Perform finite element analyses for the 0.5T-SE(B) and the PCVN specimens and then export the maximum principal stress σ.sub.1,i and element volume V.sub.i of each element in each model at K=K.sub.0. The fracture toughness region is defined as the region where σ.sub.1,i≧λσ.sub.ys with λ=1.

(37) (3) The reference volume V.sub.0 is taken as (50 μm).sup.3 in the example. Assume m=10, 11, . . . , 12 and calculate the σ.sub.w using the data extracted from the fracture process region. Since the value of σ.sub.u is the value of σ.sub.w at K=K.sub.0, two m˜σ.sub.u curves are obtained as illustrated in FIG. 5.

(38) (4) Find the intersection of the two m˜σ.sub.u curves in FIG. 5 and obtain the Beremin model's parameters, m=17.7 and σ.sub.u=2486 MPa, for A508-3 forging, by the coordinate of the intersection point. In addition, the data point (m, σ.sub.u) on the region where the two m˜σ.sub.u curves are almost overlapped (16<m<30) can be taken as the equivalent solutions for the calibrated parameters.

(39) RGD calibration method is applied to determine the Weibull slope m. As shown in FIG. 6, the Weibull slope m is calculated as 17.7 by interpolation. At K.sub.J=K.sub.0(0.5T) or K.sub.0(PCVN), the calibrated σ.sub.u is calculated to be 2486 MPa.

(40) The calibration method proposed in the invention eliminates the redundant calculations of the σ.sub.w and the toughness scaling based on equal σ.sub.w values in the case of K.sub.J≢K.sub.0. With a series of assumed m values, the σ.sub.w values are calculate only at K=K.sub.0 in the corresponding specimen to construct m˜σ.sub.u curves for the specimens of different constraints. The calibrated values of m and σ.sub.u are simultaneously obtained through the intersection of the m˜σ.sub.u curves. It can be observed from the above examples that the calibration method in the present invention has much lower computational cost compared with the RGD calibration method and the same calibration accuracy as the RGD calibration method.

(41) Compared with the RGD calibration method (FIG. 4 and FIG. 6), the calibration method of the present invention visually displays the calibration process as illustrated in FIG. 3 and FIG. 5. The solutions for (m, σ.sub.u) can be decided by the different cases of intersection of m˜σ.sub.u curves. FIG. 3 and FIG. 5 show that the pair of m˜σ.sub.u curves for 16MnR are overlapped in a specific range, and so are the pair of m˜σ.sub.u curves for A508-3 forging. According to the argument of the proposed calibration method, it indicates that there are equivalent pairs of (m, σ.sub.u) for toughness scaling across different constraint structures, especially the example 2. The RGD calibration method may neglect the equivalent solutions for (m, σ.sub.u), but only yield the most accurate one.