A procedure for predicting strength properties using small punch test and finite element simulation

Accepted Manuscript

A Procedure for Predicting Strength Properties using Small Punch Test and Finite Element Simulation Jiru Zhong , Tong Xu , Kaishu Guan , Jerzy Szpunar PII: DOI: Reference:

S0020-7403(18)33979-1 https://doi.org/10.1016/j.ijmecsci.2019.01.006 MS 4720

To appear in:

International Journal of Mechanical Sciences

Received date: Revised date: Accepted date:

30 November 2018 1 January 2019 3 January 2019

Please cite this article as: Jiru Zhong , Tong Xu , Kaishu Guan , Jerzy Szpunar , A Procedure for Predicting Strength Properties using Small Punch Test and Finite Element Simulation, International Journal of Mechanical Sciences (2019), doi: https://doi.org/10.1016/j.ijmecsci.2019.01.006

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ACCEPTED MANUSCRIPT Highlights 

A procedure was proposed to determine true stress-strain curves from small punch test results.

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Initial yield stress is not an appropriate parameter for the estimation of yield strength. A new methodology based on data analysis was proposed to predict yield strength and ultimate tensile strength. The obtained strength values are in good agreement with those acquired by a uniaxial tensile test.

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Strength predicted by this method shows good reproducibility and repeatability.

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ACCEPTED MANUSCRIPT A Procedure for Predicting Strength Properties using Small Punch Test and Finite Element Simulation 1 Jiru Zhong , Tong Xu2, Kaishu Guan1,*, Jerzy Szpunar3, * 1 School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai, 200237, China 2 China Special Equipment Inspection & Research Institute, Beijing 100013, China 3 Department of Mechanical Engineering, University of Saskatchewan; Saskatoon, SK S7N5A9, Canada

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Abstract Developing a universal approach for acquiring yield strength and ultimate tensile strength by small punch test (SPT) is a long-standing challenge. In this paper, a methodology is proposed to obtain strength properties of bulk materials from SPT results.

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It involves extracting true stress-strain curves from SPT data using iterative finite element simulation. Acquired curves show that initial yield stress cannot be reproduced. This is attributed to the inhomogeneous deformation of SPT specimens. Finite element

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simulations were then conducted on tensile tests with the extracted true stress-strain

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curves to obtain strength. Results indicate that predicted strength shows a convergent tendency with the times of iterative finite element simulation. An approach is proposed to

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process the predicted strength, based on data analysis. The acquired strength properties

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are in good agreement with those obtained by standard tensile testing. This is an entirely novel procedure that predicts reliable tensile properties via a single SPT run. It can be

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employed with confidence to cases for which standard tensile testing is impractical. Keywords: Small punch test; Finite element simulation; True stress-strain curve; Strength properties

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Corresponding author E-mail address: [email protected] (Kaishu Guan); [email protected] (Jerzy Szpunar)

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1. Introduction Material mechanical properties are important parameters for the assessment of the structural integrity of pressure vessels in the petrochemical and chemical industries, and nuclear power plants [1–3], but materials are decaying during their service lives due to

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high temperatures, irradiation, hydrogen, etc. Inspecting changes in material mechanical properties is of crucial importance to in-service pressure vessels. However, extracting bulk material from in-service components for use as conventional mechanical test samples is difficult or impossible. SPT was developed to meet this need due to the small

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amount of material required. Over 30 years’ development, SPT has been used to predict elastic modulus [4], tensile properties [5–7], ductile-to-brittle transition temperatures [7,8], fracture toughness [9,10] and creep properties [11–13].

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Yield strength and ultimate tensile strength are fundamental properties of materials.

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They are important parameters for the design of pressure vessels and reflect material mechanical states in components [14–16]. Efforts have been made to create a correlation

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between SPT results and strength; as a result, different empirical correlation equations

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have been proposed [5,6,17–23]. Many tests including SPT and tensile tests are

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mandatory for the establishment of such correlation equations. The earliest equation proposed by Mao and Takahashi [17] indicates that yield strength correlates linearly to yield force and ultimate tensile strength has a linear relation with maximum force. After Mao and Takahashi, researchers introduced modifications into the equation [5,6,18–23]. These modifications include correlation constant, the definition of yield load, and the expression for predicting ultimate tensile strength. Regardless of empirical equations 3

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employed to process SPT data, the accuracy of obtained strength, especially yield strength, is not adequate [5,22,23]. Simonovski et al. [23] suggested that it is difficult to obtain a reliable yield strength by empirical correlation. A study by Dymacek et al. [22] indicated that a unified empirical correlation equation is hard to develop. Accurate tensile

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properties are important for the estimation of structural integrity. The inaccuracy of predicted strength restricts the wide spread of empirical correlation methods.

SPT has been recognized as an alternative to conventional tensile testing. In recent

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years, it has become ever more important to develop a unified and reliable method for determining strength this way because as a new characterization technique, SPT is being used in many cases to estimate mechanical properties. Planques et al. [24] characterized the mechanical properties of thermal barrier coatings by SPT. Dobes et al. [25] used SPT

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to estimate the yield strength and ultimate tensile strength of aluminum composite.

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Simulation methodology might be a new approach to explore SPT results. With the development of computational techniques, finite element simulation has been used to

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identify material parameters. Zhao et al. [26], Kamaya and Kawakubo [27] and Joun et al.

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[28] extracted stress-strain curves over a large range of strains from tensile tests. Patel

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and Kalidindi [29] and Dean and Clyne [30] identified plastic parameters using a spherical indentation test and simulation. Simulation methodology has also been employed to SPT. Lotfolahpour et al. [31] extracted GTN model parameters of 304 stainless steel from SPT using a genetic algorithm and neural networks. Husain et al. [32] determined stress-strain curves from SPT. Yang et al. [33] estimated the mechanical properties of Incoloy800H by finite element simulation and SPT. The strategy of such a 4

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method lies in fitting the simulation and experimental results. Material parameters will be determined when simulation results coincide with experimental results. Attempts have been made to determine plastic parameters from SPT data [31–33], but acquiring strength by simulation has not been reported previously. This paper aims to

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propose a universal procedure for predicting strength by SPT and simulation. The repeatability and reproducibility of results has received considerable attention in SPT standardization. To verify the possibility for use in every laboratory, we carried out a

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systematic study to investigate the repeatability and reproducibility of such a procedure. 2. Material and Methodology 2.1 Material and small punch test

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The material used in this work is 3Cr1MoV, which was formed by forging. The mechanical properties of this material are assumed to be isotropic. Three SPT specimens

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with a diameter of 10 mm were cut from a 3Cr1MoV block using a wire electro discharge

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cutting machine and then were polished to a thickness of 0.50 mm. Tests were conducted

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on a servo hydraulic system with a maximum load capacity of 5 kN. Fig. 1a displays the apparatus for fixing and measuring displacement changes of specimens. The SPT

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specimen was clamped by upper and lower dies. In most previous works, the displacements

of

specimens

were

measured

through

an

indirect

method

[5,8,9,16,17,20,21,31–33], i.e. a crosshead movement of the servo hydraulic system or a corrected crosshead movement was assumed as the displacement change of specimens. Compared with indirect measurement, direct measurement has an advantage in avoiding the effect of machine stiffness on measurement accuracy. In this paper, the displacement 5

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or deflection changes were measured directly from the lower surface of samples with a

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ceramic rod (see Fig. 1a).

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Fig. 1 Schematic illustration of SPT: (a) experiment, (b) finite element model 2.2 Finite element simulation

Finite element simulation is crucial to the prediction of yield strength and ultimate

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tensile strength in this work. Simulations involved SPT and standard tensile testing. The

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commercial software used for simulation was ABAQUS/Standard 6.10. Fig. 1b is the SPT model. Due to the axial symmetry of specimens, a 2D model was sufficient for

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describing its deformation during testing. Four-node axisymmetric reduced integration

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(CAX4R) elements were used to mesh specimens. Abendroth [34] reported that the geometry of the lower die had a significant effect on SPT results. To provide the exact

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dimension for the simulation, the chamfer of the lower die was measured with a three-dimensional Profilometer. In addition to the SPT specimen, parts in Fig. 1b were set as rigid bodies. A displacement boundary condition along the Y direction was applied on the rigid ball. The upper and lower clamps were fixed. The position change of Node 1 was measured as the deflection change of specimens. As to the tensile test, a round bar with a diameter of 6 mm, an initial gauge length of 30 mm was used in this paper. An 6

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axial symmetrical finite element model was adopted to simulate tensile tests. The specimen was further simplified by taking advantage of the symmetry of the specimen in the longitudinal direction and meshed with CAX4R elements, as shown in Fig. 2. A prescribed displacement was applied on the right edge of the specimen. The distance

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change between Node 1 and Node 2 was measured. The initial distance between Node 1 and Node 2 was half of the initial gauge length of the tensile specimen. The material parameters required for simulation were Young’s modulus, Poisson’s ratio, friction

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coefficient, and a true stress-plastic strain curve. Young’s modulus is 200GPa; Poisson’s ratio and friction coefficient are 0.3 and 0.2, respectively. The true stress-plastic strain curve is expressed as σ(𝜀𝑝 ) = 𝜎0 + 𝐾𝜀𝑝 𝑛 , where σ is true stress and 𝜀𝑝 is true plastic strain; 𝜎0 represents the initial yield stress corresponding to true strain of 0; 𝐾 is

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strength coefficient; 𝑛 denotes the strain hardening exponent. The true stress-strain

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curves mentioned in this work refer to true stress-plastic strain curves.

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Fig. 2 Schematic illustration of finite element model of tensile tests

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2.3 Procedure for predicting strength Yield strength and ultimate tensile strength were predicted through iterative finite

element simulation. The strategy was to extract a true stress-strain curve from the force-deflection curve of SPT and then to explore the extracted true stress-strain curve. Fig. 3 depicts the procedure for predicting true stress-strain curves. A true stress-strain curve was described with three material parameters: 𝜎0 , 𝐾 and 𝑛. Parameters (𝜎0 , 𝐾 7

ACCEPTED MANUSCRIPT and 𝑛 ) were randomly initialized for simulating SPT. The force-deflection curve, calculated by finite element simulation, was compared with the measured one. Comparison started from a deflection of 0.05 mm up to 1.4 mm. To evaluate the difference between simulation and experimental results, an objective function 𝑓 of 𝜎0 ,

𝑒𝑥𝑝

1

𝐹𝑗𝐹𝐸 (𝜎0 ,𝐾,𝑛)−𝐹𝑗

𝑁

𝐹𝑗

𝑓(𝜎0 , 𝐾, 𝑛) = ∑𝑁 𝑗=1 |

𝑒𝑥𝑝

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𝐾 and 𝑛 was created: | × 100%

(1)

where 𝑓 represents the difference between simulation and experiment; 𝐹𝑗𝑒𝑥𝑝 and 𝐹𝑗𝐹𝐸 are, respectively, force measured from SPT and calculated by finite element simulation at

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a given deflection point 𝑗; 𝑁 is the total number of deflection points involved in the calculation of objective function 𝑓. The procedure would stop when the value of objective function 𝑓 was less than 1%. If 𝑓 was greater than 1%, hybrid particle swarm

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optimization (HPSO) was used to revise the parameters. The details about updating

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parameters through HPSO were introduced in our previous work [35]. During this iterative procedure, true stress-strain curves ( 𝜎0 , 𝐾 and 𝑛 ) were recorded. This

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procedure ran 8 times for the discussion of the repeatability and reproducibility of

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predicted results. To obtain strength, two methodologies were used to process the

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acquired true stress-strain curves. One was to assume initial yield stress as yield strength because yield strength has been similar to initial yield stress in standard tensile tests. The other was to simulate tensile tests with the extracted true stress-strain curves; as a result, load-displacement curves of tensile tests were generated. Yield strength and ultimate tensile strength were then extracted from the load-displacement curves according to the instruction of ASTM standard D638. 8

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Fig. 3 Procedure for acquiring true stress-strain curves 3. Results and Discussion 3.1 Small punch test

Experimental force-deflection curves of SPT are given in Fig. 4. The curves are

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comparable with each other, showing SPT has good reproducibility. Fig. 4 shows that the reaction force of specimen C was between specimens A and B. The force-deflection curve of specimen C may reflect that material’s average mechanical properties. The

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difference between specimens A and C was 2.0%, and it was 2.2% between specimens B

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and C. This difference may be due to the scatter of investigated materials.

Fig. 4 Force-deflection curves of SPT

Specimen C was used to identify true stress-strain curves. After identifications, force-deflection curves with different values of 𝑓 were generated. Fig. 5a shows the 9

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comparison of curves between simulations and experiment. It indicates that the predicted curves agree well with experimental results when 𝑓 is within 3%. However, the corresponding initial yield stress 𝜎0 at 𝑓 values of 0.9, 1.5, 2.0 and 3.0 was significant disparate (see Fig. 5b). 𝜎0 has been close to yield strength in standard tensile tests. Yang

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et al. [33] attempted to acquire the yield strength of Incoloy 800H by assuming that 𝜎0 was equal to yield strength. The repeatability and reproducibility of results has received considerable attention in SPT standardization, while Fig. 5 cannot confirm that 𝜎0 has a repeatable tendency. It is necessary to investigate the reproducibility before considering

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𝜎0 as yield strength.

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Fig. 5 Predicted results with different values of objective function: (a) force-deflection curves, (b) initial yield stress 3.2 Reproducibility of initial yield stress

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The reproducibility of parameters identified by simulation has not been addressed in nearly all previous studies [26–32]. To explore the reproducibility, extra two 𝜎0 at 𝑓 values of 0.9, 1.5, 2.0 and 3.0 were extracted from our simulation data, as shown in Fig 6. It is clear that the difference of 𝜎0 at an 𝑓 value of 0.9 was the smallest among those four different 𝑓 values. However, the maximum difference at 𝑓 of 0.9 was over 100 Mpa, indicating that 𝜎0 cannot be reproduced. To be a general code for predicting 10

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strength is inapplicable for SPT.

Fig. 6 Reproducibility of initial yield stress

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To clarify the scatter of 𝜎0 , we analyzed predicted curves and strain evolution of SPT specimens. Force-deflection curves and the corresponding true stress-strain curves are, respectively, depicted in Figs. 7a–d and e–h. Force-deflection curves show excellent

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reproducibility when 𝑓 is within 2% (see Figs. 7b–d). Curves given in Fig. 7a can be

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reproduced before the deflection of 0.9 mm, while the repeatability and reproducibility is progressively worse with the increase of deflection. Figs. e–h clearly show that

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stress-strain curves, which generate the same 𝑓 values, do not coincide exactly with

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each other, especially when 𝑓 is above 1.5%. The reason for this is that 𝑓 was

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calculated at different deflection intervals for different curves. For instance, curves given in Fig. 7a show that Simulation B is in good agreement with experiment in deflection intervals [0, 0.5], while the fitness in [0.9, 1.4] is not as good as that in [0, 0.5]. In contrast, Simulation C coincides with the experimental curve well in deflection intervals [0.9, 1.4], but the agreement in [0, 0.5] is not good enough. As a consequence, different true stress-strain curves generated the same 𝑓 values. This may explain why 𝜎0 was 11

ACCEPTED MANUSCRIPT scattered. Another factor leading to the nonrepeatability of 𝜎0 is that the force-deflection curves did not extremely depend on 𝜎0 . True stress-strain curves in Fig. 7h display good repeatability and reproducibility, but 𝜎0 varied significantly from one simulation to the next. The reasons are that the deformation of the SPT specimen was inhomogeneous and

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plastic deformation occurred from the start (see Fig. 8). We can see that only a small amount of material was deformed at a deflection of 0.05 mm and more material was involved in plastic deformation with the increasing deflection. The same observation has

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been reported in Ref. [34]. The reaction force was around 100 N at a deflection of 0.05 mm (see Figs. 7a–d) but the maximum plastic strain reached 0.09 mm/mm (see Fig. 8a), indicating that the reaction force depended on three plastic parameters (𝜎0 , 𝐾 and 𝑛) from the start. Thus, an erroneous calculated 𝜎0 may be compensated with alterations in

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𝐾 and 𝑛 .The rapid increase of stress reduced the effects of a low 𝜎0 on the reaction

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force of SPT, as shown in Fig. 7h. It is clear that 𝜎0 was low in Simulation A but stress increased dramatically within a very limited range of strains. The low 𝜎0 was

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compensated by the subsequent high stress. Thus, a true stress-strain curve with a low 𝜎0

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was able to produce a force-deflection curve similar to that generated by a high 𝜎0 . Material gradually evolves in deformation during SPT and no specific force in

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force-deflection curves represents the value of 𝜎0 , which is different from standard tensile tests. The inhomogeneous deformation of SPT specimens caused that the predicted true stress-strain curves were not completely the same and the force-deflection curve was not extremely sensitive to 𝜎0 . In the case of scattered 𝜎0 , a new approach should be proposed to determine yield strength. 12

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Fig. 7 Predicted curves at the same value of objective function: (a–d) force-deflection curves, (e–h) true stress-strain curves 13

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Fig. 8 Strain distribution of SPT specimen at different deflections: (a) 0.05mm, (b) 0.5mm, (c) 1.0mm, (d) 1.4mm 3.3 Strength predicted by simulation

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Extracting true stress-strain curves from SPT has been reported in previous papers [31–33], but few efforts have been made to determine strength using them. The conventional method for determining yield strength and ultimate tensile strength is the

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standard tensile test. Husain et al. [32] and Peng et al. [36] confirmed that true stress-strain curves extracted from SPT data were coincident with those determined by

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standard tensile tests. It provides the support of theory for predicting strength. Strength

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can be predicted by simulating the tensile test or using the method presented in Ref. [37].

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Both methods drew same conclusions. To be easier been understood, we adopted simulation method. Tensile tests were simulated with the true stress-strain curves in Figs.

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7e–h. Predicted load-displacement curves of tensile tests are illustrated in Fig. 9 where the overlap region is shown to increase with the decrease of 𝑓. Curves given in Fig. 9d indicate that force was different when displacement was over 3 mm. This resulted from the difference of the predicted true stress-strain curves. The stress-strain curves given in Fig. 7h were not exactly the same with the result that the difference between curves was obviously at a large displacement. To obtain yield strength and ultimate tensile strength, 14

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curves in Fig. 9 were processed according to ASTM Standard D638. Figs. 10a and 10b depict yield strength and ultimate tensile strength, respectively. Yield strength shows a better reproducibility when 𝑓 is around 0.9, which is different from 𝜎0 . The explanations are that yield strength is defined as the engineering stress at an engineering

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plastic strain of 0.002, also, true stress after 𝜎0 was in good agreement with each other (see Fig. 7h). The result was that the effects of 𝜎0 on yield strength were not obvious.

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This is the same reason for the convergence of ultimate tensile strength.

Fig. 9 Load-displacement curves of tensile tests: (a) f=3.0%, (b) f=2.0%, (c) f=1.5%, (d) f=0.9% Fig. 10 shows that the scatter of predicted strength decreased with the decrease of 𝑓,

indicating strength can be reproduced if 𝑓 is small enough. To apply this approach for predicting strength confidently, true stress-strain curves with an 𝑓 value less than 6% were used to simulate tensile tests; as a result, strength was obtained. Figs. 11a and 11b 15

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Fig. 10 Reproducibility of strength: (a) yield strength, (b) ultimate tensile strength

illustrate calculated yield strength and ultimate tensile strength, respectively. It shows

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that the strength gradually converges with the decrease of 𝑓. This is attributed to the convergence of the true stress-strain curves in Figs. 7e-h. The upper and lower boundaries of strength show linear relationships with 𝑓, and two linear equations are

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used to fit those two boundaries. The intersection point was defined as material strength.

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Table 1 lists the strength obtained by SPT and the standard tensile test. While the yield strength was slightly overestimated, ultimate tensile strength was underestimated. The

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errors of predicted yield strength and ultimate strength are 0.7% and 3.5%, respectively.

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Compared with empirical correlation, this method provides more accurate results. Dymacek et al. [22] reported that errors were usually over 10% by empirical correlation.

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The reasons are that the empirical correlation method lacks the support of theory and the dimension of the lower die, which affects SPT results significantly, was not included in empirical correlation equations. Different from the empirical correlation, the method presented in this work was based on J2 plastic flow theory. In addition, the dimension of the lower die was taken into account when the simulation was performed. Fig. 5a illustrates that curves match experimental results well when 𝑓 is around 3%, 16

ACCEPTED MANUSCRIPT while the predicted strength covers a wide range when 𝑓 is 3% (see Fig. 11). This infers that strength cannot be reproduced even if simulation curves agree with experimental results visually. This problem has not been addressed in previous studies [26–32]. Fig. 11 confirmed that the importance of the quantization of the difference between simulations

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and experiment. In this paper, strength was determined by the convergent trend of simulation results. A large quantity of simulation results were involved in the determination of strength, which avoided reliance on a single result. This approach has

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good repeatability and reproducibility and can be employed with confidence to cases for

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which standard tensile testing is impractical.

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Fig. 11 Relationship between predicted strength and objective function: (a) yield strength, (b) ultimate tensile strength

Table 1 Yield strength and ultimate tensile strength Ultimate tensile strength (MPa)

Small punch test

542.0

642.8

Standard tensile test

538.0

666.0

Error (%)

0.7

3.5

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Yield strength (MPa)

4. Conclusions A new procedure integrating hybrid particle swarm optimization and finite element 17

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simulation is proposed to extract yield strength and ultimate tensile strength from a small punch test. The main conclusions are given below: 1. Initial yield stress is not an appropriate parameter for the estimation of yield strength. Plastic deformation occurs from the start of small punch test (SPT) and

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only a small amount of material evolves with the result that the reaction force is not extremely sensitive to the initial yield stress. The low initial yield stress can be compensated by the high stress, with the result that a low initial yield stress is

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able to generate a force-deflection curve similar to that produced by a high initial yield stress.

2. When the difference (𝑓) between force-deflection curves of simulation and experiment is quantified, the reproducibility of the predicted true stress-strain

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curve and strength increases with the decrease of the difference. For determining

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a true stress-strain curve and strength using SPT and finite element simulation, the difference should be as small as possible.

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3. Predicted strength shows a trend of convergence with 𝑓. A new methodology

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based on data analysis is proposed to predict strength. The obtained strength value

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is in good agreement with that acquired by a standard tensile test. Strength values predicted by this simulation method show good reproducibility and repeatability, indicating it could be developed as a unified method for extracting yield strength and ultimate tensile strength.

4. Predicting strength by finite element simulation is a possibility for used in every laboratory. To be a universal method, many materials will be employed to verify 18

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the reliability of such a method in our future work. To develop SPT as an alternative to standard tensile tests, we will extract reduction of area and percentage plastic extension at fracture from SPT data using finite element simulation in the future.

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Acknowledgement The authors are grateful to the Chinese Scholarship Council and State Bureau of Quality and Technical Supervision of China (201510070) for providing financial support.

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