Evaluation of Strength Properties of In-Service X80 Pipeline Steel Using Small Punch Test and Backpropagation Neural Network

. Evaluating the strength properties of materials of an in-service pipeline without shutting down transportation has been always a challenge. A novel and non-destructive method for determining the true stress-strain curve of pipeline steel based on backpropagation artificial neural network and small punch test is proposed in this study. The elastoplastic mechanical properties of the pipeline steels could be obtained by this method. The load-displacement curves of 2261 groups of different hypothetical materials were obtained by the finite element model of small punch test within Gurson-Tvergaard-Needleman (GTN) damage parameters and used to train the neural network. The relationship between the load-displacement curve of small punch test and the true stress-strain curve of the conventional uniaxial tensile test was established based on the trained neural network. The accuracy and wide applicability of the trained neural network were verified by the experimental data of four types of materials obtained by small punch test and standard tensile test, respectively. The established relationship can be used to predict the true stress-strain curve of the pipeline steels to determine the elastoplastic mechanical properties only by the load-displacement curve of the small punch test without performing the conventional tensile test.


Introduction
Small punch test (SPT) [1][2][3][4] is one of these nearly non-destructive methods for miniaturized specimen techniques. It has been applied for the evaluation of a wide range of mechanical properties, including strength, ductility, fracture toughness, ductile-brittle transition temperature and high temperature mechanical behavior, i.e. creep. Mao correlated the yield load and the maximum load of the SPT with the yield strengths and the ultimate tensile strength of various kinds of materials, and gave the corresponding empirical formula [5]. Chica analyzed the initial stage of the load-displacement curve of the SPT, correlated the slope of the initial stage with the elastic modulus of alloys, and obtained the empirical formula between them [6]. Based on the principle of equivalent energy, Peng [7] analyzed the 4 third stage of load-displacement curve of SPT, and the true stress-strain curve and ultimate tensile strength of the materials were obtained. Baik et al. [8] tested the temper-embrittled Ni-Cr steel and defined the area under the load-displacement curve as the fracture energy of the SPT, and founded the correlation between the Charpy V-notch test results and the SPT, then the ductile-brittle transition temperature (DBTT) was determined. Hurst [9] proposed a J-integral estimation method for annular notched small punch specimens of P91 martensitic steel based on fracture energy. The local analysis and simulation of J-R bending line and fracture toughness JIC were carried out by using SPT. The empirical correlation method was used by the most of researchers to obtain the correlation formula of material strength and fracture toughness based on their own test results. However, due to a wide variety of differences of the nature of material used in the test, the configuration of the testing machine, and the definition methods of yield load on the load-displacement curve of SPT, the empirical formulas proposed by different researchers are also different [10][11][12][13][14][15][16]. These empirical formulas are not comparable to each other, resulting in the consumption of research efforts and research resources. Therefore, it is urgent to find a more reasonable and effective method to correlate the small punch test with the conventional standard tests, and then the accurate elastic-plastic mechanical properties of materials would be extracted.
In this study, a method was proposed to obtain the true stress-strain curve of metallic materials through SPT and backpropagation (BP) neural network [17][18][19], which is an algorithm of artificial neural network and is widely used in the training of feedforward neural networks for supervised learning. The complete elastoplastic mechanical properties of materials could be obtained by this method. The load-displacement curves of SPTs were obtained by the finite element model of SPT with Gurson-Tvergaard-Needleman (GTN) damage parameters. The BP neural network was trained with the load-displacement curves and true stress-strain curves, and the relationship between them was established. The load-displacement curve of X80 pipeline steel obtained by SPT was input to BP neural network, and the true stress-strain curve of X80 pipeline steel was obtained, thus the yield strength and ultimate tensile strength of X80 pipeline steel 5 were also determined. Compared with the conventional empirical correlation method, this method has the advantages of low cost, accuracy and high efficiency.

Experiment and finite element modeling
The chemical composition of X80 pipeline steel used in the test was shown in Table 1. The engineering stress-strain curve of virgin X80 pipeline steel was obtained by conventional uniaxial tensile test and the strain distribution of fractured tensile specimen was measured by digital image correlation (DIC) method.
Three groups of tests were carried out and average values of the load and displacement were used to plot the curve. The strain distribution of fractured tensile specimen and engineering stress-strain curves of  The plastic parameters of X80 pipeline steel required for numerical simulation were obtained from the true stress-strain curve. Hollomon formula were used to describe the elastoplastic constitutive relations of most ductile materials as follows, where σ is true stress, σy is yield strength, ε is true strain, E is elastic modulus, K is strength coefficient and n is strain hardening index. The true stress and true strain of materials can be calculated by Eq. (2) and Eq. (3).
where σE is engineering stress and εE is engineering strain. The true stress-strain curve of material is shown in Fig. 2, the part of the curve between yield strength and ultimate tensile strength was fitted by power function, and the parameters K and n in Hollomon formula were obtained.  In addition, the relationship between engineering stress and real stress-strain can be obtained by deducing the above formulas, which can be expressed as follows: where the e=2.718 is a natural constant. When the engineering stress reaches the maximum value, i.e. σb, the ultimate strain εE equals the strain hardening index n [21], so the ultimate tensile strength can be given as follows [22],

True strain (%)
真实应力-应变曲线 7 from the above relationship, it can be seen that the ultimate tensile strength of different materials can be deduced from the corresponding Hollomon formula. The SPT device was shown in Fig. 3 (a). The punch head was hemispherical, the radius r was 1.25 mm and the lower die diameter D was 4 mm, with 0.5 mm chamfer edge. The thickness t and diameter d of SPT specimen were 0.5 mm and 10 mm, respectively.
The specimen was placed on the lower die and pressed by the upper die to prevent sliding. During the test, the punch was loaded downward at a constant rate of 0.2 mm/min. The center of the specimen was pressed down until the specimen breaks. The load and the displacement of the punch were recorded during the whole test process, and the load-displacement curve of the SPT was obtained, as shown in Fig.   4 (b). The whole curve can be divided into five parts: I. Elastic stage; II. Transition stage from elasticity to plasticity; III. Strengthening stage; IV. Geometric softening and failure stage; V. Fracture stage of specimen. The load-displacement curves of the SPT can be used to obtain the parameters that characterize the elastoplastic behavior of the material, such as the initial slopeini of the first stage, yield load FP and maximum load Fmax, as shown in Fig. 3 where the t is the initial thickness of the small punch specimen, α, β, λ1, and λ2 are the correlation coefficients, which are equal to 17 Therefore, it is necessary to develop a new test method to obtain the mechanical properties of in-service pipeline materials under the condition of ensuring the integrity of pipeline. Martin et al. [23][24] obtained fracture parameters of tough materials by combining neural network with SPT, which has high accuracy.
It showed that the combination of neural network and SPT was suitable for testing mechanical properties of materials. Therefore, in this study, BP neural network combined with SPT was used to obtain the elastoplastic mechanical properties parameters of in-service pipeline materials.
The finite element software ABAQUS was used to simulate the SPT process. The SPT specimen was a circular axisymmetric structure. Therefore, Fig 4 shows the 2D axisymmetric finite element model of 9 SPT specimen. The punch, upper die and lower die were considered as rigid body without considering the deformation of them. The upper die and lower die were fixed in all degrees of freedom, whereas the punch can be moved vertically by a displacement boundary condition. The contact between the specimen and punch, upper die and lower die was modeled, and the friction coefficient μ was 0.15. Four-node quadratic reduction integral element CAX4R was selected for the element. Ten layers of meshes are divided in thickness direction.
where σeq represents the macroscopic effective Von Mises stress, σy denotes the yield strength of matrix 10 material, σH is the macro hydrostatic stress, f * , q1, q2 and q3 are the material damage parameters. Through numerical analysis, the commonly used values of damage parameters given by Tvergaard [26]  qq  。f * is defined as: where f denotes the current void volume fraction; fF and fc define the void volume fraction at failure and at a critical situation where f starts to deviate from f*, respectively; 

Experimental results and continuous damage model
In the finite element model of SPT, the void parameters f0, fN, fc and fF of X80 pipeline steel are unknown, so the values of these parameters should be determined. The specimens with diameter of 10.0 mm and thickness of 0.600 mm were cut from the pipeline by wire cutting. Sandpaper was used to grind the 11 specimens to remove the influence areas of overheating and work hardening, so that there were no cracks and other macro defects in the specimens. In order to ensure the accuracy of the test, three groups of tests were prepared. The final specimen thickness t was 0.495 mm, 0.498 mm and 0.499 mm, respectively.
The load-displacement data of each specimen were recorded and three groups of load-displacement curves of SPT were obtained. The finite element model of multi-group SPT were simulated to ensure that the plastic parameters of the material were unchanged, and only the GTN damage parameters were changed, Finally, compare the simulation results with the test results. Fig. 5 shows the curve with the highest matching degree with test. In this paper, the GTN damage parameters are finally determined as shown in Table 2.

BP neural network modeling
BP neural network can train a lot of data and establish corresponding relationship between two different factors. In this paper, the load-displacement curve of SPT was taken as input set and the true stress-strain curve as output set. The main training scheme of BP neural network is shown in Fig. 6.  In the training process of BP neural network, a large number of data need to be calculated to reduce the error continuously. Fig. 7 shows the calculation process, it is divided into two steps: (1) Firstly, the target error value was set, and a group of load-displacement data and real stress-strain data of SPT of the same material were input. After receiving a set of load-displacement data of the SPT, the activation function f1(e) calculated these data in this layer., and the calculated result y1 was output to the second layer neurons.
The output set y was calculated by the second layer neurons, and the error δ between the result and the input true stress-strain data was obtained; (2) The calculated error δ was fed back to the layers of neurons, and the weight value of each layer of neurons were corrected by formula (9), (10), (11) respectively, and the corrected weight value was used to repeat step (1) until the error was less than the preset target error.  where w is weight value, η is the learning rate, f(e) is the activation function and δ is the calculation error.

Assembling the data base
The training set of BP neural network includes two parts: the load-displacement curve of SPT as input set and the true stress-strain curve of the same material as output set. The true stress-strain curve can be obtained by systematically changing the parameters K and n in Hollomon formula. In order to obtain more accurate training results, the accuracy and comprehensiveness of training set data should be guaranteed. Therefore, the database should contain most of the true stress-strain curves of common metals. The strain hardening index n of most materials ranges from 0.005 to 0.6, and the strength coefficient K ranges from 200 MPa to 2000 MPa. However, when parameter K becomes small, with the decrease of n value, the true stress corresponding to the true strain of 0.002, i.e. the yield strength of material becomes very small, and the yield strength ratio of material becomes very small, while this kind of metal is very rare, so this kind of true stress-strain curve can be deleted to improve the accuracy of the training results of BP neural network. Finally, the range of parameters K and n is determined as shown in Table 3. When BP neural network determines input data, each column matrix is identified as a set of input data, so each curve corresponds to a column of data. The load-displacement curve of SPT and the true stressstrain curve of material are composed of data points, and each data point is confirmed by abscissa and ordinate coordinates. Therefore, each data point on the curve needs to be transformed in the way shown in Table 4 Table 4.

Calculation results of BP neural network
The load-displacement curve of X80 pipeline steel obtained from SPT was input into the trained BP neural network to obtain its true stress-strain curve. Compared with the uniaxial tensile test results, as shown in Fig. 8 Table 5.
In order to further verify the accuracy of the calculation results of the BPNN, load-displacement data of AISI 304L, P92 and DP600 of SPT in references [7] and [27] are plotted in Fig. 9 and substituted into the trained BPNN to obtain the corresponding true stress-strain curves, which were compared with the uniaxial tensile test results given by other researchers as Fig. 9 Shown. 16 Fig. 9 Comparison of true stress-strain curves of materials in reference obtained between BP neural network and uniaxial tensile tests network, and the true stress-strain curve can be obtained, and then the elastoplastic mechanical properties parameters of corresponding material can be obtained.

Conclusions
In this study, a method was proposed to obtain the true stress-strain curve of metallic materials by using SPT, finite element method and BP neural network. Through developing the finite element model of SPT with GTN damage parameters, which was verified by the experimental results, load-displacement curves of different 2261 groups of hypothetical materials were obtained, combined with the corresponding true stress-strain curve, BP neural network was trained to establish the relationship between SPT and conventional uniaxial tensile test. By comparing the results of SPT and uniaxial tensile test of X80 pipeline steel and the test data of different types of materials in the references, the accuracy and wide applicability of this method were verified. The main conclusions can be drawn as follows.
(1) The load-displacement curves obtained by the finite element model of SPT with GTN damage parameters have a very good agreement with the experimental results for the X80 pipeline steel.
(2) The load-displacement curve of SPT of X80 pipeline steel is substituted into the trained BP neural network to obtain its true stress-strain curve. The calculated yield strength y_BPNN = 581 MPa  , ultimate tensile strength b_BPNN = 739 MPa  . The error is -2.2% and 3.6% respectively compared with uniaxial tensile test results.
(3) The developed BP neural network is used to calculate the mechanical properties of materials in the references. Compared with the uniaxial tensile test results, the maximum error is -5.7%, which is lower than that of the conventional empirical correlation method.
The results demonstrate that the true stress-strain curves and elastoplastic properties of metallic materials could be accurately obtained by the method combining SPT and artificial BP neural network without performing conventional tensile test. 18