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Application of small punch test to investigate mechanical behaviours and deformation characteristics of Incoloy800H
Journal of Alloys and Compounds 765 (2018) 497e504
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Application of small punch test to investigate mechanical behaviours and deformation characteristics of Incoloy800H Sisheng Yang, Xiang Ling*, Lin Xue School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing, 211816, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2018 Received in revised form 2 June 2018 Accepted 19 June 2018 Available online 21 June 2018
In this study, mechanical behaviours and deformation characteristics of Incoloy800H were investigated experimentally and analytically by small punch test at various temperatures. First, parameter sensitivity analysis was performed in order to estimate the influence of anisotropy, geometric dimensions, and temperature on material strengths. Then, the mechanical properties of Incoloy800H were investigated. A relatively simple yet reasonable relation between mechanical properties and specimen thickness was obtained. A new prediction model based on the deformation energy was established. Meanwhile, the inverse prediction model based on golden section search algorithm was proposed to estimate the yield strength and constitutive parameters on the basis of iterative calculations. On the other hand, a small punch deformation model was discussed. The hardness, strain, and stress triaxiality were analysed in order to estimate the evolution of plastic damage and deformation character. Finally, an economic, accurate mechanical behaviour evaluation system of Incoloy800H was established on the basis of the small punch test. © 2018 Elsevier B.V. All rights reserved.
Keywords: Small punch test Mechanical properties Deformation characteristics Prediction model
1. Introduction High-Cr-Ni alloys have been the subject of widespread investigation in recent years owing to their excellent high-temperature strength and tribological performance [1e3]. However, severe service environments and high costs impede the sampling volume of in-service components, which is essential for standard measurement techniques in evaluating material properties [4,5]. Thus, studies of the small specimen evaluation technique, which has the merits of convenience and of being nearly nondestructive, have attracted increasing attention in the field of novel materials. Among these techniques, it has been demonstrated that the small punch test holds several advantages over traditional uniaxial tests, and can be used to predict mechanical properties extracted from load-displacement curves through miniature specimens [6]. In the first case, this potential leads to the establishment of different mathematical relationships between material properties and experimental characteristic parameters [7e9]. Maximum load, which plays a key role in the experimental curves, is widely used to assess the tensile strength of various materials [10]. The feasibility
* Corresponding author. E-mail address: [email protected] (X. Ling). https://doi.org/10.1016/j.jallcom.2018.06.243 0925-8388/© 2018 Elsevier B.V. All rights reserved.
of estimating yield strength by elastic-plastic transition load has been recognized in different in-service environments [11]. Different notches were also designed and applied in the prediction of crack propagation and fracture toughness [12,13]. It can be found that these analyses focus primarily on the determination of characteristic parameters. Despite many efforts, apparent errors caused by size effects and complex non-linear conditions still hinder the development of this technique. Researchers have since focused their thoughts on the establishment of empirical correlations for certain kinds of materials, and some new approaches have been proposed. Because of the advantage of simple form, Isselin [14] introduced the elastic deformation energy and established an energy-based prediction method of yield strength. Finite element models were established with different constitutive models in order to analyse the deformation process and evaluate the material parameters. The loaddisplacements curve was split up and used to train neural networks by Abendroth et al. [15]. On the basis of the neural networks, a data base was built and applied in the prediction of constitutive parameters by using a conjugate directions algorithm. In our previous research, an inverse finite element model has been presented and partial strength parameters are predicted [16]. The results also demonstrate that proper analysis of loading condition and deformation characteristics is the foundation of accurate fracture
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strength prediction. Unfortunately, there are few investigations that address this. Meanwhile, due to the influence of parameter sensitivity, direct analysis should be introduced in order to describe the difference caused by sampling direction and geometric dimensions in the further study. Considering the extensive utilization of Incoloy800H in engineering practice, how to establish a synthetic evaluation method by means of small punch test is the critical focus of this paper. As a high temperature alloy, the influence of temperature is another noteworthy problem. Thus, from an analysis of the influence of sampling direction, specimen and temperature, a small punch deformation energy and inverse finite element model were presented and used to estimate mechanical properties. The mechanical strength of Incoloy800H was obtained in order to establish a reasonable mathematical relationship between material strength and small punch parameters under different temperatures. Meanwhile, the deformation and fracture characteristics of Incoloy800H were evaluated by an examination of hardness, strain and stress triaxiality. Finally, a new evaluation system based on experimental analysis, inverse prediction and deformation mechanism evaluation was presented. The mechanical properties and deformation characteristics of Incoloy800H under different temperatures were analysed, which may provide new insights into the small punch test.
2. Experiment The material used in this study was Incoloy800H. In order to investigate the conventional mechanical properties of Incoloy800H, uniaxial tensile tests were completed in the INSTRON5869 universal testing machine. The self-designed experimental device of the small punch test, which can be used at different temperatures, is shown in Fig. 1. For round specimens, the loading and deformation conditions are
Fig. 1. Schematic diagram of the small punch test.
symmetrical. Thus, a round specimen of 10 mm diameter was chosen for the small punch analysis [17]. The specimen, with 0.5 mm thickness, was obtained here by wire-electrode cutting and polishing. Then, the specimen was placed in a fixture consisting of two parts: an upper die and a lower die. Furthermore, in order to manifest the plastic characteristics of materials clearly, a ball with 2.4 mm in diameter was used. Meanwhile, an induction heat device and three thermocouples were used in order to perform the hightemperature tests within 3 C temperature tolerance. During the experiment, the specimen was extruded at a speed of 0.5 mm/min, and the load-displacement curves were obtained until an apparent load decrease occurred. 3. Results and discussion 3.1. Parameter sensitivity analysis It has always been a challenge to the researcher to minimise the influence factors of micro-damage experimental techniques in the mechanical properties analysis. For the small punch test, especially in the evaluation of high-temperature alloys, the key factors to be considered include: 1) sampling direction, 2) specimen thickness and 3) experimental temperature. In this section, the parameter sensitivity analysis was completed in order to establish the small punch evaluation method of Incoloy800H. First, three different sampling directions were chosen as shown in Fig. 2. Fig. 3 shows that the load-displacement curves are similar, including the elastic deformation stage, plastic deformation stage, membrane stretching stage, and plastic instability stage. The fine deviation in the maximum load region might be induced by microdefects, which would lead to failure along with the aggravation of necking. The result clearly proves that sampling direction does not have an apparent effect on the material properties, which may be due to the cubic structure of Incoloy800H. As an austenitic high temperature steel with a face-centred cubic structure, the specimen's anisotropy will not affect the accuracy of the small punch analysis. Different to sampling direction, geometric dimension is an inevitable influence factor in the mechanical property evaluation of Incoloy800H. The previous studies focused primarily on the loaddisplacement curves; in fact, how to establish a reasonable mathematical relation between material strength and character parameters (Py/h20 and Pmax/h0 dmax) is a key problem in the small punch analysis. It has been proved that specimen thickness affects the damage evolution and fracture position [18]. Therefore, five different specimen thicknesses (0.5 mm, 0.6 mm, 0.7 mm, 0.8 mm and 0.85 mm) were used, and load-displacement curves were investigated in this study. Compared to a thinner specimen, an obvious upward trend and a larger fracture displacement can be obtained for the thicker sample. The increase of specimen thickness leads to the enhancement of the specimen's bearing capacity. Thus, the fracture time is retarded and fracture displacement is enlarged with the increase of specimen thickness.
Fig. 2. Sampling direction of Incoloy800H.
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irregular micro-cracks in the specimen and calculation deviation in the analysis, which proves that specimen thickness will not affect the accuracy of small punch test within a proper thickness range. In previous research results, the relation was established on the basis of experimental data at room temperature [19]. However, as a high-temperature alloy, the in-service temperature is a significant influence factor in the mechanical properties evaluation of Incoloy800H. Thus, in this section, small punch tests were completed at different temperatures. Specimens with 0.5 mm thickness were tested at a speed rate of 0.5 mm/min. High-temperature tensile tests were also completed. Meanwhile, the partial high temperature tensile data of Incoloy800H were cited from the study of Yang
Fig. 3. Load-displacement curves of the small punch test with different sampling directions.
In addition, a linear relationship can be obtained between character load and specimen thickness for Incoloy800H. García [19] established the calculation equation by adding two geometric dimensions: thickness and fracture displacement:
Rel ¼ k1
Py h20
(1)
Pmax þ k3 h0 dmax
(2)
Rm ¼ k2
where Py is the elastic-plastic transition load, Rel is the yield strength, Pmax is the maximum load, Rm is ultimate tensile strength, dmax reflects the fracture displacement, h0 is the initial specimen thickness; k1, k2, and k3 represent the parameters of material properties. Based on Equations (1) and (2), the values of Py/h20 and Pmax/ h0dmax are shown in Fig. 4. It can be found that the value of the character parameters is a constant with increasing specimen thickness. The fluctuation of evaluation results may be caused by
Fig. 4. Change in Py/h20 and Pmax/h0dmax with specimen thickness.
Fig. 5. Relationship between the small punch test and conventional tensile test at different temperatures （a）Yield strength（b）Ultimate strength（c）Elongation.
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[20]. Fig. 5 clearly shows that character load is not a material constant, but is strongly temperature-dependent. Higher experimental temperature corresponds to lower value of elastic-plastic transition load. Meanwhile, it is worthy to be noted that the elastic-plastic transition load tends to be a constant value in a temperature interval range from 300 C to 400 C. A similar conclusion can also be obtained in the evaluation of maximum load and fracture displacement. The only difference is that a slight rise can be found when the temperature is higher than 250 C. This means that higher mechanical properties and better ductility can be achieved in this temperature range. However, it should be noted that an apparent decrease of maximum load and fracture displacement would occur when the temperature increased further. For ultimate tensile strength and elongation, the phenomenon may be interpreted by internal diffusion and dislocation motion of material. Within the proper temperature range, the resistance of dislocation motion may increase. Thus, a rise of material strength can be found. On the basis of a global view of the experimental results at different temperatures, there is good agreement between the results of the small punch test and traditional tensile test. This implies that the small punch test can be used to analyse the high temperature properties of Incoloy800H well and precisely. However, the defects of the current evaluation system cannot be neglected according to the foregoing discussion. Especially in the analysis of high-temperature steel, most studies introduce parameter T to describe the influence of temperature. Obviously, this method cannot be applied in the analysis of Incoloy800H. Indeed, because of the minimum specimen size, a minor local defect may result in apparent deviation in the prediction of character parameters. How to establish a more accurate prediction method is a key problem that is discussed in the next section.
the results of yield strength. The result indicates that deformation energy can be applied in the prediction of mechanical properties under different temperatures. Compared to the traditional equation, the effect of temperature is presented in the evolution of elastic deformation energy, which avoids the unnecessary errors caused by the introduction of parameter T. Thus, a more accurate result is obtained here:
Rel ¼ 0:564
Ey þ 82:823 dy h20
where Ey is the elastic-plastic deformation energy.
Rm ¼ 0:046
Emax þ 395:094 dmax h20
(4)
where Emax is the maximum load deformation energy. (2)Inverse prediction Because of the nonlinear deformation process and various parameters in the small punch tests, the mechanical and damage properties of in-service material are difficult to be estimated by means of empirical correlations in some cases. In order to estimate yield strength and constitutive parameters, inverse finite element model was established in this work. It has been proven to be a flexible approach that makes it possible to evaluate material properties by small punch test [16]. The detailed scheme of the identification procedure is shown in Fig. 6. A two-dimensional symmetric finite element model of small punch test was established by ABAQUS software. During experiment, the degree of freedom in the thickness direction of specimen would be restricted by the upper die and the lower die. Thus, to deliver necessary accuracy, the ball, upper die and lower die were
3.2. Mechanical properties An important goal of the research is to estimate mechanical properties such as yield and ultimate strengths. Using the results in the previous section, the influence of temperature on the mechanical properties can be reflected in the change of loaddisplacement curves. Therefore, how to describe the material behaviour of Incoloy800H from the load-displacement curves observed in the small punch test is the main objective of this section. (1) Direct estimation At present, several scholars have attempted to use small punch test to evaluate the mechanical properties based on the idea of empirical fitting. Considering a thin round disc subjected to variable load under a steady-state temperature environment, the irreversible specimen deformation is caused by the mechanical work transmitted by the ball. Isselin [14] attempted to propose elastic deformation energy on the basis of unbounding displacement and proved that yield strength is proportional to the deformation energy and inversely proportional to specimen thickness. Thus, the integration of the load-displacement curve under specific loads was selected and defined as the deformation energy [19]. From the high-temperature experimental data of the small punch test, the elastic-plastic deformation energy and the maximum load deformation energy were obtained in this paper. Meanwhile, the yield strength and ultimate tensile strength were presented based on the uniaxial tensile test. Finally, empirical formulas for yield strength and ultimate strength of Incoloy800H under various temperatures were estimated as shown in Equation (3) and Equation (4). It can also be found that the ultimate strength is proportional to the deformation energy and inversely proportional to specimen thickness and fracture displacement which is similar to
(3)
Fig. 6. Inverse prediction model.
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Fig. 8. Schematic diagram of golden section search algorithm.
Fig. 7. Comparison of results of simulation and experiment for Incoloy800H.
established and defined as rigid body. The specimen was defined as deformable. Considering the deformation characteristics of small punch specimen, symmetric constraints were applied and finer elements were used in the centre region. Meanwhile, 15 elements were utilized in the thickness direction and the mesh was set as CAX4R. Finally, a constant rate of 0.5 mm/min was applied in the ball and the specimen deformation was recorded. The deformation behaviour of specimen is shown in Fig. 7. It can be found that there is high consistency between the results of experiment and simulation in the elastic-plastic deformation phases. Meanwhile, fine difference can also be found at high-load region. The reason may be due to the local thickness reduction and apparent stress concentration along with necking during large non-linear deformation of small punch specimen. The isotropic assumption of simulation is another reason. It should be noted that the load-displacement curve is sensitive to the elastic-plastic parameters such as yield strength and constitutive parameters in the elastic-plastic deformation phase. Meanwhile, membrane stretching and plastic instability deformation stages are apparent affected by the damage parameters presented by different damage models. Thus, the elastic-plastic deformation curve is chosen here in order to predict the material parameter by inverse model. The objective function was defined by the difference between the load-displacement curves of experiment and simulation:
f ¼
ZD EXP ðDÞ f FEA ðDÞdðDÞ f
(5)
are calculated and compared. Then, the search interval is reduced with order 0.618 in each iteration as shown in Fig. 8. When the difference between minimum and maximum value of search interval is less than the set value, the calculation is completed and the mean value is defined as the final optimal result. It should be noted that the value of elastic module, Poisson's ratio and gravity of material change less in the actual engineering. Meanwhile, the maximum value of true stress e true equivalent plastic strain can also be obtained based on direct analysis of small punch test [22]. Thus, they can be defined as the known parameters in our researches. Then, the initial yield stress and power hardening model parameters are predicted in this paper. It should be noted that the initial yield stress is approximate to the yield strength of material. Thus, we believe that the value of initial yield stress can be defined as the yield strength of material in the analysis. Here, it was predicted with three sets of original intervals in order to describe the influence of data range during single parameter calculation. As listed in Table 1, the results indicate that the golden section search algorithm can be used as a powerful method to predict the material parameters. Meanwhile, original interval does not have remarkable influence on the predicted precision. Fine deviation may be induced by the search principle and precision set. Another key point of inverse finite model is the prediction of material constitutive parameters. Here, a simple prediction method was proposed in order to describe the power hardening model of Incoloy800H (For conventional tensile test, the strain hardening exponent n ¼ 0.4, strength coefficient K ¼ 1174.48). Due to the research of Sun [23], the value of K can be calculated by equation (7):
0
Where D is the small punch displacement, f EXP ðDÞ is the function of experimental curve, f FEA ðDÞ is the function of simulation curve. In order to predict the material parameters accurately and rapidly, the golden section search algorithm is introduced in this study. The method is an efficient region elimination method to narrow the range of values [21]. For a single variable, a line segment with length L was divided into two parts with triples of points. A golden ratio was used to describe the ratio of the minor subsegment and the major subsegment as listed here:
L1 L L1 ¼ ¼ 0:618 L L1
(6)
Where L is the length of line, L1 is the major subsegment, respectively. Two points of golden section can be found for a single search interval. During calculation, the objective functions of two points
K¼
sj ε
εj j
(7)
Where sj is the true instability stress, εj is the instability strain. Based on previous researches, the value of sj and εj can be obtained by maximum load and displacement of small punch test as shown in Fig. 3. Here, the K is about 1153. Based on this, the original interval range [0.22,0.56] was set before the calculation of n. Finally, the value is predicted by inverse model and obtained as 0.39. It can be found that reasonable results were obtained by the inverse model for yield strength and constitutive parameters. High prediction precision is proven. By combining the experimental equation and the inverse prediction model, a small-punch evaluation system of Incoloy800H was established. The mechanical properties and constitutive parameters of Incoloy800H can be predicted under different in-service conditions. This may provide some insight into related fields.
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Table 1 Predicted values of yield strength. Tensile result (MPa)
Upper and lower bounds D2½0:001mm; 0:75mm s2½150MPa; 300MPa
s2½100MPa; 350MPa
D2½0:001mm; 1mm s2½150MPa; 300MPa
225.58
219.83
219.05
211.47
D2½0:001mm; 0:75mm
equivalent strain effectively has been a basis problem in the application of small punch test. Based on Chakrabarty model [24], membrane stretch model was presented and shown in Fig. 10. With the increase of small punch displacement, the small punch deformation region can be divided into two parts: the contact region and the non-contact region. The strain of contact boundary is [25]:
" εb ¼ 2 ln
2 þ 2 cos a
# (9)
ð1 þ cos bÞ2
Where εb is the strain at contact boundary, a and b are deformation angles. The centre strain εc can be described as:
1 þ cos a εc ¼ 2 ln 1 þ cos b
(10)
It should be noted that the fracture position exists in the contact boundary. Thus, the equivalent strain is defined as the average value of the centre strain and the strain at the moving contact boundary here:
Fig. 9. Small punch specimen of Incoloy800H.
"
2 þ 2 cos a
#
1 þ cos a 1 þ cos b
3.3. Deformation characteristics
ε ¼ ln
As a complex localized deformation process, reasonable estimation of multiaxial deformation characteristics is the foundation of small punch analysis. The basic principles of the small punch test are well known: a localized elastic-plastic deformation leads to severe necking and failure fracture of the thin disc as shown in Fig. 9. Essentially, two main features can be distinguished: non-linear and localizing. For the biaxial planar stress-strain process, the equivalent strain can be defined by:
The values of angle a and b can be defined by the following geometric form:
εp ¼ lnðh0 =hT Þ
(8)
where εp is equivalent strain, h0 is the initial specimen thickness and hT is the current specimen thickness. Considering the variational thickness, how to estimate the
ð1 þ cos bÞ2
þ ln
r sin a ¼ sin2 b b
(12)
Similar to equivalent strain, the small punch deflection can also be written as expression of deformation angles and geometric parameters [25]:
H ¼ r sin2 b ln
tanðb=2Þ þ rð1 cos bÞ tanða=2Þ
(13)
During small punch test, the necking and failure occur in the membrane stretch and plastic instability phases corresponding to the displacement about 1 mme2.4 mm. Thus, a simple mathematical relationship between the equivalent strain and the displacement in this deformation phases is obtained based on equation (11e13)and listed as follow:
ε ¼ 0:8996H 0:4909
Fig. 10. Model of small punch specimen deformation.
(11)
(14)
It can be found that a similar linear relationship can be established between equivalent strain and displacement in the membrane stretch and plastic instability phases. With the increase of centre displacement, the equivalent strain increases and the specimen thickness reduces. When the equivalent strain is larger than 0.8, obvious logarithmic relationships between hardness and equivalent strain have been proved by Kim in the Erichsen test [26]. A similar analytic relationship was also established by Sonmez for cold formed parts [27]. Thus, the deformation characteristics of Incoloy800H are also discussed by considering hardness in this paper. For the fracture specimen, the hardness was measured as shown in Fig. 11.
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Finally, the triaxiality factor of the top surface (TFt) can be defined as
sxx þ syy þ szz 3
TFt ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i. sxx syy 2 þ syy szz 2 þ ðsxx szz Þ2 2
TFt ¼
2 4F 3 pd2 seq
(18)
(19)
Unlike the top surface, the value of szz is zero at the bottom. Thus:
TFB ¼
Fig. 11. Micro hardness profiles with distance from the specimen centre.
It should be noted that there is no apparent deformation in the specimen when the distance is 3 mme5 mm away from specimen centre. In this region, the disc was placed between the upper die and lower die. Thus, the change of the hardness value is not apparent. With the depth of the ball, an obvious rise of hardness can be found at a position which is approximately 1.5 mme2.5 mm away from the centre of the specimen. Meanwhile, the hardness value on the upper surface is clearly larger than that of the specimen centre and lower surface in this region, which means a higher strain state can be found on the upper surface. This is the action of deformation and extrusion, which will be discussed by stress triaxiality in the following section. When the position was approximately 0.6 mme1mm away from centre, obvious homogeneous deformation occurred, and the maximum value of hardness was obtained. In this region, plastic damage becomes severe from top to bottom in the thickness direction, which leads to the local necking and induces fracture finally. From the Gurson-Tvergaard-Needleman model, the evolution of void volume fraction was discussed, and a similar strain distribution and failure position is obtained, which proves the foregoing analysis [18]. In the case of the multi-axial deformation state, the stress state evaluation is another important issue to be solved in the fracture mechanism analysis for the small punch test. Here, we defined sxx , syy , szz as the principal stress of three directions. It should be noted that szz is the compression stress of the top surface and can be described as [28,29]:
4F
szz ¼ 2 pd
(15)
where d is the diameter of the indentation impression, and F is the load. As a symmetric deformation process, sxx , syy can be described as:
sxx ¼ syy
(16)
Thus, the equivalent stress seq is
1 h
seq ¼ pffiffiffi sxx syy 2 ¼ sxx szz
2
i1=2 2 þ syy szz þ ðsxx szz Þ2 (17)
sm 23sxx ¼ ¼ 2=3 seq sxx
(20)
Equation (19) indicates that the value of the stress triaxiality factor of the top surface changes with the increase in load. Meanwhile, the shear state is shown and a large deformation can be found at the upper surface because of the action of shear and compression deformations. The hardness value of the upper surface is larger than that of the lower surface when the position is about approximately 1.5mme2.5 mm away from the centre as shown in Fig. 11. By contrast, when the specimen deformation turns into the membrane elongation stage, homogenous deformation and apparent increase of strain are achieved at the bottom of the specimen with tensile deformation. Compared to the upper surface, the stress feature at the bottom surface is approximate to the tensile deformation state. For local failure, the fracture strain under multiaxial stress state is proposed by the ASME strain limit criterion [30]:
asl 1 TF εf ¼ εc exp 3 1þm
(21)
Where εf is the multiaxial strain limit, εc is strain limit under uniaxial stress state which can be described by the equivalent strain as discussed above, m and asl are material constants. With the increase in stress triaxiality factor, the multiaxial strain limit decreases. This may be used to explain the phenomenon that failure often occur at the bottom of specimen in the small punch tests. In this section, the hardness, strain and stress triaxiality were discussed in order to estimate the localized deformation of small punch specimen. Then, the damage and fracture region of Incoloy800H was obtained. In different deformation phases, different deformation behaviors of small punch specimen were proven. This means that the small punch deformation can not be described by a single model. It proves that the deformation energy model is a new insight, which can avoid the parameter calculation error introduced by different character load calculation models. Meanwhile, reasonable analysis of deformation mechanism may help us to estimate the complex deformation state of small punch test under multiaxial stress state in the further study. 4. Conclusion To estimate the mechanical properties and deformation characteristics of Incoloy800H accurately and economically, a new evaluation method of the small punch test was established in this paper. Both experimental investigation and mechanism analysis were carried out. The main conclusions in this paper can be drawn as follows: 1. Experimental estimation, inverse prediction and deformation mechanism analysis, which can be used to accurately estimate
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the mechanical properties of Incoloy800H under different conditions, were accomplished in this study. Considering the influence of temperature, anisotropy and geometric dimensions, parameter sensitive analysis was performed. The results show that elastic-plastic transition load and maximum load increase linearly with the increase in specimen thickness. Both different thicknesses and sampling direction have no apparent effect on the accurate evaluation of the mechanical properties of Incoloy800H. A new prediction equation based on deformation energy was proposed that can solve the difficulty in the evaluation of material strengths at different temperatures. Furthermore, the inverse prediction model was estimated and the deformation parameters were obtained. The hardness, strain and stress triaxiality were analysed in order to estimate the deformation characteristics of Incoloy800H. An approximate shear process can be found on the upper surface, whereas a tensile process occurs on the lower surface, which leads to an obvious change in hardness and strain in the various deformation regions. When the position was about 0.6 mme1 mm away from sample centre, obvious homogeneous deformation occurred. In this region, plastic damage becomes severe from top to bottom in the thickness direction and induces fracture finally.
Acknowledgements The authors would like to thank the China Postdoctoral Science Foundation (2017M621738) for support of this study. References [1] X.H. Shi, W.D. Zeng, Q.Y. Zhao, W.W. Peng, C. Kang, Study on the microstructure and mechanical properties of Aermet 100 steel at the tempering temperature around 482 C, J. Alloys Compd. 679 (2016) 184e190. [2] K. Natesan, P.S. Shankar, Uniaxial creep response of Alloy 800H in impure helium and in low oxygen potential environments for nuclear reactor applications, J. Nucl. Mater. 394 (2009) 46e51. [3] K.K. Tang, Fracture control over thermalemechanical creep and fatigue crack growth in near-alpha titanium alloy, Eng. Fract. Mech. 110 (2013) 300e313. [4] T. Nambu, K. Shimizu, Y. Matsumoto, R. Rong, N. Watanabe, H. Yukawa, M. Morinaga, I. Yasuda, Enhanced hydrogen embrittlement of Pd-coated niobium metal membrane detected by in situ small punch test under hydrogen permeation, J. Alloys Compd. 446e447 (2007) 588e592. ~ eda, I.I. Cuesta, I. Pen ~ uelas, A. Díaz, J.M. Alegre, Damage [5] E. Martínez-Pan modeling in small punch test specimens, Theor. Appl. Fract. Mech. 86 (2016) 51e60. [6] Y.W. Ma, K.B. Yoon, Assessment of tensile strength using small punch test for transversely isotropic aluminum 2024 alloy produced by equal channel angular pressing, Mater. Sci. Eng. A 527 (2010) 3630e3638. [7] Y.S. Chang, J.M. Kim, J.B. Choi, Y.J. Kim, M.C. Kim, B.S. Lee, Derivation of ductile fracture resistance by use of small punch specimens, Eng. Fract. Mech. 75 (2008) 3413e3427.
[8] D.T. Blagoeva, R.C. Hurst, Application of the CEN (European Committee for Standardization) small punch creep testing code of practice to a representative repair welded P91 pipe, Mater. Sci. Eng. A 510e511 (2009) 219e233. [9] R. Alizaden, R. Mahmudi, M.J. Esfandyarpour, Shear punch creep test: a novel localized method for evaluating creep properties, Scripta Mater. 64 (2011) 442e445. [10] D. Finarelli, F. Carsughi, P. Jung, The small ball punch test at FZJ, J. Nucl. Mater. 377 (2008) 65e71. rdenas, F.J. Belzunce, C. Betego n, Me[11] C. Rodríguez, J. García Cabezas, E. Ca chanical properties characterization of heat-affected zone using the small punch test, Weld. J. 88 (2009) 188e192. [12] I.I. Cuesta, J.M. Alegre, Determination of the fracture toughness by applying a structural integrity approach to pre-cracked Small Punch Test specimens, Eng. Fract. Mech. 78 (2011) 289e300. [13] K.S. Guan, L. Hua, Q.Q. Wang, X.H. Zou, M. Song, Assessment of toughness in long term service CrMo low alloy steel by fracture toughness and small punch test, Nucl. Eng. Des. 241 (2011) 1407e1413. [14] J. Isselin, T. Shoji, Yield strength evaluation by small-punch test, J. Test. Eval. 37 (2009) 531e537. [15] M. Abendroth, M. Kuna, Identification of ductile damage and fracture parameters from the small punch test using neural networks, Eng. Fract. Mech. 73 (2006) 710e725. [16] S.S. Yang, Y. Cao, X. Ling, Y. Qian, Assessment of mechanical properties of Incoloy800H by means of small punch test and inverse analysis, J. Alloys Compd. 695 (2017) 2499e2505. [17] J. Kameda, X. Mao, Small-punch and TEM-disc testing techniques and their application to characterization of radiation damage, J. Mater. Sci. 27 (1992) 983e989. [18] S.S. Yang, J.X. Zhou, X. Ling, Z. Yang, Effect of geometric factors and processing parameters on plastic damage of SUS304 stainless steel by small punch test, Mater. Des. 41 (2012) 447e452. rez, Estimation of the mechanical [19] T.E. García, C. Rodríguez, F.J. Belzunce, C. Sua properties of metallic materials by means of the small punch test, J. Alloys Compd. 582 (2014) 708e717. [20] J.M. Yang, Application and performance analysis of Incoloy 800H alloy, Pet. Refin. Eng. 32 (2002) 18e20 (in Chinese). [21] J.M. Cai, D. Han, C.X. Chen, S.Y. Chen, Application of the golden section search algorithm in the nonlinear isoconversional calculations to the determination of the activation energy from nonisothermal kinetic conversion data, Solid State Sci. 12 (2010) 829e833. [22] S.P. Singh, S. Bhattacharya, D.K. Sehgal, Evaluation of high temperature mechanical strength of Cr-Mo grade steel through small punch test technique, Eng. Fail. Anal. 39 (2014) 207e220. [23] D.J. Sun, J.P. Lin, R.T. Liu, Q.S. Hu, Research on strength coefficient in power hardening model of sheet metal, J. Plasticity Eng. 16 (2009) 149e152 (in Chinese). [24] J. Chakrabarty, A theory of stretch forming over hemispherical punch heads, Int. J. Mech. Sci. 12 (1970) 315e325. [25] Z. Yang, Z.W. Wang, Relationship between strain and central deflection in small punch creep specimens, Int. J. Press Ves. Pip. 80 (2003) 397e404. [26] H. Kim, S.M. Lee, T. Altan, Prediction of hardness distribution in cold backward extruded cups, J. Mater. Process. Technol. 59 (1996) 113e121. [27] F.O. Sonmez, A. Demir, Analytical relations between hardness and strain for cold formed parts, J. Mater. Process. Technol. 186 (2007) 163e173. [28] T.S. Byun, J.H. Hong, F.M. Haggag, K. Farrell, E.H. Lee, Measurement of throughthe-thickness variations of mechanical properties in SA508 Gr.3 pressure vessel steels using ball indentation test technique, Int. J. Press Ves. Pip. 74 (1997) 231e238. [29] T.S. Byun, E.H. Lee, J.D. Hunn, K. Farrell, L.K. Mansur, Characterization of plastic deformation in a disk bend test, J. Nucl. Mater. 294 (2001) 256e266. [30] ASME BPVC SECTION VIII Pressure Vessels Division 2 Alternatie Rules, 2007.
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Application of small punch test to investigate mechanical behaviours and deformation characteristics of Incoloy800H Sisheng Yang, Xiang Ling*, Lin Xue School of Mechanical and Power Engineering, Nanjing Tech University, Nanjing, 211816, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 March 2018 Received in revised form 2 June 2018 Accepted 19 June 2018 Available online 21 June 2018
In this study, mechanical behaviours and deformation characteristics of Incoloy800H were investigated experimentally and analytically by small punch test at various temperatures. First, parameter sensitivity analysis was performed in order to estimate the influence of anisotropy, geometric dimensions, and temperature on material strengths. Then, the mechanical properties of Incoloy800H were investigated. A relatively simple yet reasonable relation between mechanical properties and specimen thickness was obtained. A new prediction model based on the deformation energy was established. Meanwhile, the inverse prediction model based on golden section search algorithm was proposed to estimate the yield strength and constitutive parameters on the basis of iterative calculations. On the other hand, a small punch deformation model was discussed. The hardness, strain, and stress triaxiality were analysed in order to estimate the evolution of plastic damage and deformation character. Finally, an economic, accurate mechanical behaviour evaluation system of Incoloy800H was established on the basis of the small punch test. © 2018 Elsevier B.V. All rights reserved.
Keywords: Small punch test Mechanical properties Deformation characteristics Prediction model
1. Introduction High-Cr-Ni alloys have been the subject of widespread investigation in recent years owing to their excellent high-temperature strength and tribological performance [1e3]. However, severe service environments and high costs impede the sampling volume of in-service components, which is essential for standard measurement techniques in evaluating material properties [4,5]. Thus, studies of the small specimen evaluation technique, which has the merits of convenience and of being nearly nondestructive, have attracted increasing attention in the field of novel materials. Among these techniques, it has been demonstrated that the small punch test holds several advantages over traditional uniaxial tests, and can be used to predict mechanical properties extracted from load-displacement curves through miniature specimens [6]. In the first case, this potential leads to the establishment of different mathematical relationships between material properties and experimental characteristic parameters [7e9]. Maximum load, which plays a key role in the experimental curves, is widely used to assess the tensile strength of various materials [10]. The feasibility
* Corresponding author. E-mail address: [email protected] (X. Ling). https://doi.org/10.1016/j.jallcom.2018.06.243 0925-8388/© 2018 Elsevier B.V. All rights reserved.
of estimating yield strength by elastic-plastic transition load has been recognized in different in-service environments [11]. Different notches were also designed and applied in the prediction of crack propagation and fracture toughness [12,13]. It can be found that these analyses focus primarily on the determination of characteristic parameters. Despite many efforts, apparent errors caused by size effects and complex non-linear conditions still hinder the development of this technique. Researchers have since focused their thoughts on the establishment of empirical correlations for certain kinds of materials, and some new approaches have been proposed. Because of the advantage of simple form, Isselin [14] introduced the elastic deformation energy and established an energy-based prediction method of yield strength. Finite element models were established with different constitutive models in order to analyse the deformation process and evaluate the material parameters. The loaddisplacements curve was split up and used to train neural networks by Abendroth et al. [15]. On the basis of the neural networks, a data base was built and applied in the prediction of constitutive parameters by using a conjugate directions algorithm. In our previous research, an inverse finite element model has been presented and partial strength parameters are predicted [16]. The results also demonstrate that proper analysis of loading condition and deformation characteristics is the foundation of accurate fracture
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strength prediction. Unfortunately, there are few investigations that address this. Meanwhile, due to the influence of parameter sensitivity, direct analysis should be introduced in order to describe the difference caused by sampling direction and geometric dimensions in the further study. Considering the extensive utilization of Incoloy800H in engineering practice, how to establish a synthetic evaluation method by means of small punch test is the critical focus of this paper. As a high temperature alloy, the influence of temperature is another noteworthy problem. Thus, from an analysis of the influence of sampling direction, specimen and temperature, a small punch deformation energy and inverse finite element model were presented and used to estimate mechanical properties. The mechanical strength of Incoloy800H was obtained in order to establish a reasonable mathematical relationship between material strength and small punch parameters under different temperatures. Meanwhile, the deformation and fracture characteristics of Incoloy800H were evaluated by an examination of hardness, strain and stress triaxiality. Finally, a new evaluation system based on experimental analysis, inverse prediction and deformation mechanism evaluation was presented. The mechanical properties and deformation characteristics of Incoloy800H under different temperatures were analysed, which may provide new insights into the small punch test.
2. Experiment The material used in this study was Incoloy800H. In order to investigate the conventional mechanical properties of Incoloy800H, uniaxial tensile tests were completed in the INSTRON5869 universal testing machine. The self-designed experimental device of the small punch test, which can be used at different temperatures, is shown in Fig. 1. For round specimens, the loading and deformation conditions are
Fig. 1. Schematic diagram of the small punch test.
symmetrical. Thus, a round specimen of 10 mm diameter was chosen for the small punch analysis [17]. The specimen, with 0.5 mm thickness, was obtained here by wire-electrode cutting and polishing. Then, the specimen was placed in a fixture consisting of two parts: an upper die and a lower die. Furthermore, in order to manifest the plastic characteristics of materials clearly, a ball with 2.4 mm in diameter was used. Meanwhile, an induction heat device and three thermocouples were used in order to perform the hightemperature tests within 3 C temperature tolerance. During the experiment, the specimen was extruded at a speed of 0.5 mm/min, and the load-displacement curves were obtained until an apparent load decrease occurred. 3. Results and discussion 3.1. Parameter sensitivity analysis It has always been a challenge to the researcher to minimise the influence factors of micro-damage experimental techniques in the mechanical properties analysis. For the small punch test, especially in the evaluation of high-temperature alloys, the key factors to be considered include: 1) sampling direction, 2) specimen thickness and 3) experimental temperature. In this section, the parameter sensitivity analysis was completed in order to establish the small punch evaluation method of Incoloy800H. First, three different sampling directions were chosen as shown in Fig. 2. Fig. 3 shows that the load-displacement curves are similar, including the elastic deformation stage, plastic deformation stage, membrane stretching stage, and plastic instability stage. The fine deviation in the maximum load region might be induced by microdefects, which would lead to failure along with the aggravation of necking. The result clearly proves that sampling direction does not have an apparent effect on the material properties, which may be due to the cubic structure of Incoloy800H. As an austenitic high temperature steel with a face-centred cubic structure, the specimen's anisotropy will not affect the accuracy of the small punch analysis. Different to sampling direction, geometric dimension is an inevitable influence factor in the mechanical property evaluation of Incoloy800H. The previous studies focused primarily on the loaddisplacement curves; in fact, how to establish a reasonable mathematical relation between material strength and character parameters (Py/h20 and Pmax/h0 dmax) is a key problem in the small punch analysis. It has been proved that specimen thickness affects the damage evolution and fracture position [18]. Therefore, five different specimen thicknesses (0.5 mm, 0.6 mm, 0.7 mm, 0.8 mm and 0.85 mm) were used, and load-displacement curves were investigated in this study. Compared to a thinner specimen, an obvious upward trend and a larger fracture displacement can be obtained for the thicker sample. The increase of specimen thickness leads to the enhancement of the specimen's bearing capacity. Thus, the fracture time is retarded and fracture displacement is enlarged with the increase of specimen thickness.
Fig. 2. Sampling direction of Incoloy800H.
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irregular micro-cracks in the specimen and calculation deviation in the analysis, which proves that specimen thickness will not affect the accuracy of small punch test within a proper thickness range. In previous research results, the relation was established on the basis of experimental data at room temperature [19]. However, as a high-temperature alloy, the in-service temperature is a significant influence factor in the mechanical properties evaluation of Incoloy800H. Thus, in this section, small punch tests were completed at different temperatures. Specimens with 0.5 mm thickness were tested at a speed rate of 0.5 mm/min. High-temperature tensile tests were also completed. Meanwhile, the partial high temperature tensile data of Incoloy800H were cited from the study of Yang
Fig. 3. Load-displacement curves of the small punch test with different sampling directions.
In addition, a linear relationship can be obtained between character load and specimen thickness for Incoloy800H. García [19] established the calculation equation by adding two geometric dimensions: thickness and fracture displacement:
Rel ¼ k1
Py h20
(1)
Pmax þ k3 h0 dmax
(2)
Rm ¼ k2
where Py is the elastic-plastic transition load, Rel is the yield strength, Pmax is the maximum load, Rm is ultimate tensile strength, dmax reflects the fracture displacement, h0 is the initial specimen thickness; k1, k2, and k3 represent the parameters of material properties. Based on Equations (1) and (2), the values of Py/h20 and Pmax/ h0dmax are shown in Fig. 4. It can be found that the value of the character parameters is a constant with increasing specimen thickness. The fluctuation of evaluation results may be caused by
Fig. 4. Change in Py/h20 and Pmax/h0dmax with specimen thickness.
Fig. 5. Relationship between the small punch test and conventional tensile test at different temperatures （a）Yield strength（b）Ultimate strength（c）Elongation.
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[20]. Fig. 5 clearly shows that character load is not a material constant, but is strongly temperature-dependent. Higher experimental temperature corresponds to lower value of elastic-plastic transition load. Meanwhile, it is worthy to be noted that the elastic-plastic transition load tends to be a constant value in a temperature interval range from 300 C to 400 C. A similar conclusion can also be obtained in the evaluation of maximum load and fracture displacement. The only difference is that a slight rise can be found when the temperature is higher than 250 C. This means that higher mechanical properties and better ductility can be achieved in this temperature range. However, it should be noted that an apparent decrease of maximum load and fracture displacement would occur when the temperature increased further. For ultimate tensile strength and elongation, the phenomenon may be interpreted by internal diffusion and dislocation motion of material. Within the proper temperature range, the resistance of dislocation motion may increase. Thus, a rise of material strength can be found. On the basis of a global view of the experimental results at different temperatures, there is good agreement between the results of the small punch test and traditional tensile test. This implies that the small punch test can be used to analyse the high temperature properties of Incoloy800H well and precisely. However, the defects of the current evaluation system cannot be neglected according to the foregoing discussion. Especially in the analysis of high-temperature steel, most studies introduce parameter T to describe the influence of temperature. Obviously, this method cannot be applied in the analysis of Incoloy800H. Indeed, because of the minimum specimen size, a minor local defect may result in apparent deviation in the prediction of character parameters. How to establish a more accurate prediction method is a key problem that is discussed in the next section.
the results of yield strength. The result indicates that deformation energy can be applied in the prediction of mechanical properties under different temperatures. Compared to the traditional equation, the effect of temperature is presented in the evolution of elastic deformation energy, which avoids the unnecessary errors caused by the introduction of parameter T. Thus, a more accurate result is obtained here:
Rel ¼ 0:564
Ey þ 82:823 dy h20
where Ey is the elastic-plastic deformation energy.
Rm ¼ 0:046
Emax þ 395:094 dmax h20
(4)
where Emax is the maximum load deformation energy. (2)Inverse prediction Because of the nonlinear deformation process and various parameters in the small punch tests, the mechanical and damage properties of in-service material are difficult to be estimated by means of empirical correlations in some cases. In order to estimate yield strength and constitutive parameters, inverse finite element model was established in this work. It has been proven to be a flexible approach that makes it possible to evaluate material properties by small punch test [16]. The detailed scheme of the identification procedure is shown in Fig. 6. A two-dimensional symmetric finite element model of small punch test was established by ABAQUS software. During experiment, the degree of freedom in the thickness direction of specimen would be restricted by the upper die and the lower die. Thus, to deliver necessary accuracy, the ball, upper die and lower die were
3.2. Mechanical properties An important goal of the research is to estimate mechanical properties such as yield and ultimate strengths. Using the results in the previous section, the influence of temperature on the mechanical properties can be reflected in the change of loaddisplacement curves. Therefore, how to describe the material behaviour of Incoloy800H from the load-displacement curves observed in the small punch test is the main objective of this section. (1) Direct estimation At present, several scholars have attempted to use small punch test to evaluate the mechanical properties based on the idea of empirical fitting. Considering a thin round disc subjected to variable load under a steady-state temperature environment, the irreversible specimen deformation is caused by the mechanical work transmitted by the ball. Isselin [14] attempted to propose elastic deformation energy on the basis of unbounding displacement and proved that yield strength is proportional to the deformation energy and inversely proportional to specimen thickness. Thus, the integration of the load-displacement curve under specific loads was selected and defined as the deformation energy [19]. From the high-temperature experimental data of the small punch test, the elastic-plastic deformation energy and the maximum load deformation energy were obtained in this paper. Meanwhile, the yield strength and ultimate tensile strength were presented based on the uniaxial tensile test. Finally, empirical formulas for yield strength and ultimate strength of Incoloy800H under various temperatures were estimated as shown in Equation (3) and Equation (4). It can also be found that the ultimate strength is proportional to the deformation energy and inversely proportional to specimen thickness and fracture displacement which is similar to
(3)
Fig. 6. Inverse prediction model.
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Fig. 8. Schematic diagram of golden section search algorithm.
Fig. 7. Comparison of results of simulation and experiment for Incoloy800H.
established and defined as rigid body. The specimen was defined as deformable. Considering the deformation characteristics of small punch specimen, symmetric constraints were applied and finer elements were used in the centre region. Meanwhile, 15 elements were utilized in the thickness direction and the mesh was set as CAX4R. Finally, a constant rate of 0.5 mm/min was applied in the ball and the specimen deformation was recorded. The deformation behaviour of specimen is shown in Fig. 7. It can be found that there is high consistency between the results of experiment and simulation in the elastic-plastic deformation phases. Meanwhile, fine difference can also be found at high-load region. The reason may be due to the local thickness reduction and apparent stress concentration along with necking during large non-linear deformation of small punch specimen. The isotropic assumption of simulation is another reason. It should be noted that the load-displacement curve is sensitive to the elastic-plastic parameters such as yield strength and constitutive parameters in the elastic-plastic deformation phase. Meanwhile, membrane stretching and plastic instability deformation stages are apparent affected by the damage parameters presented by different damage models. Thus, the elastic-plastic deformation curve is chosen here in order to predict the material parameter by inverse model. The objective function was defined by the difference between the load-displacement curves of experiment and simulation:
f ¼
ZD EXP ðDÞ f FEA ðDÞdðDÞ f
(5)
are calculated and compared. Then, the search interval is reduced with order 0.618 in each iteration as shown in Fig. 8. When the difference between minimum and maximum value of search interval is less than the set value, the calculation is completed and the mean value is defined as the final optimal result. It should be noted that the value of elastic module, Poisson's ratio and gravity of material change less in the actual engineering. Meanwhile, the maximum value of true stress e true equivalent plastic strain can also be obtained based on direct analysis of small punch test [22]. Thus, they can be defined as the known parameters in our researches. Then, the initial yield stress and power hardening model parameters are predicted in this paper. It should be noted that the initial yield stress is approximate to the yield strength of material. Thus, we believe that the value of initial yield stress can be defined as the yield strength of material in the analysis. Here, it was predicted with three sets of original intervals in order to describe the influence of data range during single parameter calculation. As listed in Table 1, the results indicate that the golden section search algorithm can be used as a powerful method to predict the material parameters. Meanwhile, original interval does not have remarkable influence on the predicted precision. Fine deviation may be induced by the search principle and precision set. Another key point of inverse finite model is the prediction of material constitutive parameters. Here, a simple prediction method was proposed in order to describe the power hardening model of Incoloy800H (For conventional tensile test, the strain hardening exponent n ¼ 0.4, strength coefficient K ¼ 1174.48). Due to the research of Sun [23], the value of K can be calculated by equation (7):
0
Where D is the small punch displacement, f EXP ðDÞ is the function of experimental curve, f FEA ðDÞ is the function of simulation curve. In order to predict the material parameters accurately and rapidly, the golden section search algorithm is introduced in this study. The method is an efficient region elimination method to narrow the range of values [21]. For a single variable, a line segment with length L was divided into two parts with triples of points. A golden ratio was used to describe the ratio of the minor subsegment and the major subsegment as listed here:
L1 L L1 ¼ ¼ 0:618 L L1
(6)
Where L is the length of line, L1 is the major subsegment, respectively. Two points of golden section can be found for a single search interval. During calculation, the objective functions of two points
K¼
sj ε
εj j
(7)
Where sj is the true instability stress, εj is the instability strain. Based on previous researches, the value of sj and εj can be obtained by maximum load and displacement of small punch test as shown in Fig. 3. Here, the K is about 1153. Based on this, the original interval range [0.22,0.56] was set before the calculation of n. Finally, the value is predicted by inverse model and obtained as 0.39. It can be found that reasonable results were obtained by the inverse model for yield strength and constitutive parameters. High prediction precision is proven. By combining the experimental equation and the inverse prediction model, a small-punch evaluation system of Incoloy800H was established. The mechanical properties and constitutive parameters of Incoloy800H can be predicted under different in-service conditions. This may provide some insight into related fields.
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Table 1 Predicted values of yield strength. Tensile result (MPa)
Upper and lower bounds D2½0:001mm; 0:75mm s2½150MPa; 300MPa
s2½100MPa; 350MPa
D2½0:001mm; 1mm s2½150MPa; 300MPa
225.58
219.83
219.05
211.47
D2½0:001mm; 0:75mm
equivalent strain effectively has been a basis problem in the application of small punch test. Based on Chakrabarty model [24], membrane stretch model was presented and shown in Fig. 10. With the increase of small punch displacement, the small punch deformation region can be divided into two parts: the contact region and the non-contact region. The strain of contact boundary is [25]:
" εb ¼ 2 ln
2 þ 2 cos a
# (9)
ð1 þ cos bÞ2
Where εb is the strain at contact boundary, a and b are deformation angles. The centre strain εc can be described as:
1 þ cos a εc ¼ 2 ln 1 þ cos b
(10)
It should be noted that the fracture position exists in the contact boundary. Thus, the equivalent strain is defined as the average value of the centre strain and the strain at the moving contact boundary here:
Fig. 9. Small punch specimen of Incoloy800H.
"
2 þ 2 cos a
#
1 þ cos a 1 þ cos b
3.3. Deformation characteristics
ε ¼ ln
As a complex localized deformation process, reasonable estimation of multiaxial deformation characteristics is the foundation of small punch analysis. The basic principles of the small punch test are well known: a localized elastic-plastic deformation leads to severe necking and failure fracture of the thin disc as shown in Fig. 9. Essentially, two main features can be distinguished: non-linear and localizing. For the biaxial planar stress-strain process, the equivalent strain can be defined by:
The values of angle a and b can be defined by the following geometric form:
εp ¼ lnðh0 =hT Þ
(8)
where εp is equivalent strain, h0 is the initial specimen thickness and hT is the current specimen thickness. Considering the variational thickness, how to estimate the
ð1 þ cos bÞ2
þ ln
r sin a ¼ sin2 b b
(12)
Similar to equivalent strain, the small punch deflection can also be written as expression of deformation angles and geometric parameters [25]:
H ¼ r sin2 b ln
tanðb=2Þ þ rð1 cos bÞ tanða=2Þ
(13)
During small punch test, the necking and failure occur in the membrane stretch and plastic instability phases corresponding to the displacement about 1 mme2.4 mm. Thus, a simple mathematical relationship between the equivalent strain and the displacement in this deformation phases is obtained based on equation (11e13)and listed as follow:
ε ¼ 0:8996H 0:4909
Fig. 10. Model of small punch specimen deformation.
(11)
(14)
It can be found that a similar linear relationship can be established between equivalent strain and displacement in the membrane stretch and plastic instability phases. With the increase of centre displacement, the equivalent strain increases and the specimen thickness reduces. When the equivalent strain is larger than 0.8, obvious logarithmic relationships between hardness and equivalent strain have been proved by Kim in the Erichsen test [26]. A similar analytic relationship was also established by Sonmez for cold formed parts [27]. Thus, the deformation characteristics of Incoloy800H are also discussed by considering hardness in this paper. For the fracture specimen, the hardness was measured as shown in Fig. 11.
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Finally, the triaxiality factor of the top surface (TFt) can be defined as
sxx þ syy þ szz 3
TFt ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h i. sxx syy 2 þ syy szz 2 þ ðsxx szz Þ2 2
TFt ¼
2 4F 3 pd2 seq
(18)
(19)
Unlike the top surface, the value of szz is zero at the bottom. Thus:
TFB ¼
Fig. 11. Micro hardness profiles with distance from the specimen centre.
It should be noted that there is no apparent deformation in the specimen when the distance is 3 mme5 mm away from specimen centre. In this region, the disc was placed between the upper die and lower die. Thus, the change of the hardness value is not apparent. With the depth of the ball, an obvious rise of hardness can be found at a position which is approximately 1.5 mme2.5 mm away from the centre of the specimen. Meanwhile, the hardness value on the upper surface is clearly larger than that of the specimen centre and lower surface in this region, which means a higher strain state can be found on the upper surface. This is the action of deformation and extrusion, which will be discussed by stress triaxiality in the following section. When the position was approximately 0.6 mme1mm away from centre, obvious homogeneous deformation occurred, and the maximum value of hardness was obtained. In this region, plastic damage becomes severe from top to bottom in the thickness direction, which leads to the local necking and induces fracture finally. From the Gurson-Tvergaard-Needleman model, the evolution of void volume fraction was discussed, and a similar strain distribution and failure position is obtained, which proves the foregoing analysis [18]. In the case of the multi-axial deformation state, the stress state evaluation is another important issue to be solved in the fracture mechanism analysis for the small punch test. Here, we defined sxx , syy , szz as the principal stress of three directions. It should be noted that szz is the compression stress of the top surface and can be described as [28,29]:
4F
szz ¼ 2 pd
(15)
where d is the diameter of the indentation impression, and F is the load. As a symmetric deformation process, sxx , syy can be described as:
sxx ¼ syy
(16)
Thus, the equivalent stress seq is
1 h
seq ¼ pffiffiffi sxx syy 2 ¼ sxx szz
2
i1=2 2 þ syy szz þ ðsxx szz Þ2 (17)
sm 23sxx ¼ ¼ 2=3 seq sxx
(20)
Equation (19) indicates that the value of the stress triaxiality factor of the top surface changes with the increase in load. Meanwhile, the shear state is shown and a large deformation can be found at the upper surface because of the action of shear and compression deformations. The hardness value of the upper surface is larger than that of the lower surface when the position is about approximately 1.5mme2.5 mm away from the centre as shown in Fig. 11. By contrast, when the specimen deformation turns into the membrane elongation stage, homogenous deformation and apparent increase of strain are achieved at the bottom of the specimen with tensile deformation. Compared to the upper surface, the stress feature at the bottom surface is approximate to the tensile deformation state. For local failure, the fracture strain under multiaxial stress state is proposed by the ASME strain limit criterion [30]:
asl 1 TF εf ¼ εc exp 3 1þm
(21)
Where εf is the multiaxial strain limit, εc is strain limit under uniaxial stress state which can be described by the equivalent strain as discussed above, m and asl are material constants. With the increase in stress triaxiality factor, the multiaxial strain limit decreases. This may be used to explain the phenomenon that failure often occur at the bottom of specimen in the small punch tests. In this section, the hardness, strain and stress triaxiality were discussed in order to estimate the localized deformation of small punch specimen. Then, the damage and fracture region of Incoloy800H was obtained. In different deformation phases, different deformation behaviors of small punch specimen were proven. This means that the small punch deformation can not be described by a single model. It proves that the deformation energy model is a new insight, which can avoid the parameter calculation error introduced by different character load calculation models. Meanwhile, reasonable analysis of deformation mechanism may help us to estimate the complex deformation state of small punch test under multiaxial stress state in the further study. 4. Conclusion To estimate the mechanical properties and deformation characteristics of Incoloy800H accurately and economically, a new evaluation method of the small punch test was established in this paper. Both experimental investigation and mechanism analysis were carried out. The main conclusions in this paper can be drawn as follows: 1. Experimental estimation, inverse prediction and deformation mechanism analysis, which can be used to accurately estimate
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the mechanical properties of Incoloy800H under different conditions, were accomplished in this study. Considering the influence of temperature, anisotropy and geometric dimensions, parameter sensitive analysis was performed. The results show that elastic-plastic transition load and maximum load increase linearly with the increase in specimen thickness. Both different thicknesses and sampling direction have no apparent effect on the accurate evaluation of the mechanical properties of Incoloy800H. A new prediction equation based on deformation energy was proposed that can solve the difficulty in the evaluation of material strengths at different temperatures. Furthermore, the inverse prediction model was estimated and the deformation parameters were obtained. The hardness, strain and stress triaxiality were analysed in order to estimate the deformation characteristics of Incoloy800H. An approximate shear process can be found on the upper surface, whereas a tensile process occurs on the lower surface, which leads to an obvious change in hardness and strain in the various deformation regions. When the position was about 0.6 mme1 mm away from sample centre, obvious homogeneous deformation occurred. In this region, plastic damage becomes severe from top to bottom in the thickness direction and induces fracture finally.
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