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Three-dimensional numerical simulation on plastic damage in small punch specimen of Zirconium
International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Three-dimensional numerical simulation on plastic damage in small punch specimen of Zirconium Ruomei Hu, Xiang Ling* School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 210009, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 March 2009 Received in revised form 17 July 2009 Accepted 20 October 2009
Small punch test (SPT) technique was used to evaluate the mechanical properties of Zirconium in this paper. The dimension of the disc specimen is 4 10 0.5 mm. Plastic damage in small punch specimen of Zirconium was investigated both experimentally and numerically, because it has great influence on small punch specimen. In order to simulate the plastic damage in the small punch specimen of Zirconium, the 3D finite element model incorporated with Gurson–Tvergaard–Needleman (GTN) plastic damage constitutive equation was established. Void growth and initiation of ductile crack of the small punch specimen were predicted. Results show that damage occurs on the bottom side of the specimen and grows across the specimen until complete failure, which has good agreement with the observation in the experiment. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.
Keywords: Small punch GTN model Plastic damage Numerical simulation Zirconium
1. Introduction Zirconium is one of the corrosion-resistant materials [1]. It is a rare metal and costs much to gain the mechanical properties using the routine testing techniques, also it is impossible to get the conventional sample for the equipment in-service. So using the miniature test technique-small punch test [2–4] technique to study the mechanical properties of commercially pure zirconium under room temperature is significant. In the present paper, small punch test of R60702 Zirconium was carried out and a 3D finite element model incorporated with Gurson–Tvergaard–Needleman plastic damage constitutive equation was established. Then the small punch test process of the specimens of R60702 Zirconium was simulated by using the finite element software, ABAQUS.
2. Experimental 2.1. Uniaxial tensile test Tensile test was performed at room temperature. And it was carried out with 5800 testing machine at a constant crosshead velocity of 0.5 mm per minute. The specimen of uniaxial tensile test
* Corresponding author. Tel.: þ86 25 83587321; fax: þ86 25 83600956. E-mail addresses: [email protected] (R. Hu), [email protected] (X. Ling).
is shown in Fig. 1. Nominal stress and strain curve shown in Fig. 2 is acquired from the tensile test. When defining plasticity data in ABAQUS, true stress and true strain must be used. ABAQUS requires these values to interpret the data in the input file correctly. However, quite often material test data are supplied using values of nominal stress and strain. In such situations the plastic material data of nominal stress and strain have to be converted to true stress and strain. The true stress and strain curve is shown in Fig. 3. 2.2. Small punch test The small punch tests are performed using the device as seen in Fig. 4. The SP test basically consists of a disk specimen holder; a pushing rod and a steel ball. In the SPT, specimens in the form of disks are punched in their center by a steel ball, which leads at the beginning of loading to a biaxial stress state of the specimen. The diameter of the steel ball is 2.38 mm. The punch driven by the crosshead plate of the testing machine deforms the specimen centrically at a constant velocity, Fig. 5 shows the deformed specimen after failure. The disk specimens are extracted from the tensile specimen by cutting it perpendicularly to the gauge length. And then the specimens were polished on both sides using a series of diminishing grit size silicon paper down to 1000 m grit size so that the deviation of the thickness is accurate to 0.01 mm [5,6]. The final thickness of the specimen is 0.5 mm. The results of this experiment are the
0308-0161/$ – see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2009.10.008
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R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 1. Specimen of uniaxial tensile test.
load–displacement curve of the punch and a deformation field as can be seen in Figs. 6 and 7. 3. Finite element simulation Fig. 3. Real stress and strain curves.
3.1. GTN model The material model, based on the constitutive damage law developed by Gurson, Tveergard and Needleman (GTN) [7–10], is widely-used to describe the micromechanical effects of damage in ductile metals. Gurson deduced a flow potential for void growth in an ideal-plastic material [11], which was extended by Tvergaard and Needleman [12], who introduced additional parameters (q1, q2, q3) and a modified damage variable f*. The void volume fraction f is used to measure the damage of the material. And the significant part of the model is the yield function
f¼
P 2 e
s
P 3 þ2q1 f * cos h q2 h ð1 þ q3 Þf *2 ¼ 0 s 2
(1)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P0 P0 ffi 3
P Where ¼ 2 ij ij denotes the V. Mises stress and P Pe 1 h ¼ 3 kk the hydrostatic stress. q1, q2, q3are parameters to weight the different terms of the yield function. The isotropic hardening behavior of the matrix material is denoted with s. And the damage variable is denoted with f*, which can be calculated by the following function
( *
f ðf Þ ¼
f; f * f fc þ fu f c ðf fc Þ f
c
f fc fc f ff
(2)
the material completely fails. When f reaches fc, the voids aggregate. Then the load-carrying ability of the material weakens rapidly. The change in void volume fraction is
f_ ¼ f_ growth þ f_ nucleation
Where f_ growth is the change due to void growth and f_ nucleation due to nucleation of the new voids. Based on the law of conservation of mass, the void growth can be given by
f_ growth ¼ ð1 f ÞDpkk
(4)
And the nucleation of voids follows the strain controlled relationship
f_ nucleation ¼ Að3p Þs_ M p Dkk
(5)
Where denotes the macroscopic bulk strain increment, and sM the matrix stress rate. Moreover, the nucleation of voids obeys the normal distribution which has a mean value 3N and a standard deviation sN
In the above relationship, fc is a critical value of the void volume fraction and ff is the ultimate value of void volume fraction where
Fig. 2. Nominal stress and strain curves.
(3)
Fig. 4. Small punch testing apparatus.
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 5. Deformed specimen after failure.
A ¼
" # fN 1 3p 3N 2 pffiffiffiffiffiffiffi exp 2 sN hsN 2p
Fig. 7. Deformation field of R60702 Zirconium specimen.
(6)
Where fN denotes the volume fraction of void seeds, and 3p the equivalent plastic stress of the matrix material. 3.2. Finite element modeling The GTN model has been applied in the simulation using the ABAQUS software. The material is assumed to be homogeneous, elastoplastic with isotropic hardening. The small punch tests are simulated by means of the rigid-plastic finite element method. Researches have been performed on small punch creep tests [13,14] using planar models. But in this paper a three-dimensional finite element model for SPT was established using ABAQUS 6.5-1 explicit code. The model is shown in Fig. 8. Die and down-holder are fixed in all degrees of freedom, whereas the punch can be moved vertically by a displacement boundary condition. The contact between specimen and punch, die and down-holder is modeled including
Fig. 6. Load–displacement curve of SPT.
815
friction whereas the friction coefficient m can be varied. The mesh is composed of 10 388 linear reduced integration elements. It is biased toward the disc center in order to make the size of the elements smaller there. And their aspect ratio improved. The material parameters obtained from the uniaxial tensile test of R60702 Zirconium are E ¼ 99 200 MPa, m ¼ 0.3, ss ¼ 321 MPa. According to literature [15], the parameters of GTN material model are q1 ¼1.5, q2 ¼ 1, q3 ¼ q21 ¼ 2.25, ff ¼ 0.2, fc ¼ 0.17, fN ¼ 0.04, 3N ¼ 0.3, SN ¼ 0.1. 3.3. Simulation results and damage analysis The load–displacement curve of FE simulation is compared to that of the SP tests, which is shown in Fig. 9. It is obvious that the
Fig. 8. 3D FE model for the small punch specimen.
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R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 10. Simulating result with different friction coefficients. Fig. 9. Comparison of load & displacement curves of R60702 Zirconium.
curves agree well except for the deviation during the final failure stage. It is related to the localization of damage and the crack propagation. So, load–displacement curve yielded from experiment and FE model with ductile damage constitutive equations are in good agreement, which implies that the FE model created in this paper is reasonable. The effect of friction coefficient on SPT is simulated; results show that the max load is different with varied friction coefficient, the smaller the coefficient is, the greater the max load will be (Fig. 10).
Fig. 11(a) w (d) shows evolution of the void volume fraction (damage) in the small punch specimen. Damage occurs on the bottom side of the specimen and grows across the specimen until complete failure, which has good agreement with the observation in the experiment. Initial failure occurs at the bottom surface 0.78–0.85 mm away from the center, which agrees well with the observation from the test. The damage parameters are composed of f_ growth andf_ nucleation , which are denoted by VVFG and VVFN respectively. Fig. 12 displays the contour of void growth fraction of
Fig. 11. Evolution of the void volume fraction (damage) in a small punch specimen.
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 12. Contour of void growth fraction of R60702 Zirconium (VVFG).
the specimen, while void nucleation fraction of R60702 Zirconium is shown in Fig. 13.
4. Conclusions Simulation on the small punch test process has been presented in this paper, also plastic damage in small punch specimen of Zirconium is investigated. The results are summarized as follows: (1) The 3D finite element model incorporated with GTN constitutive equation has been applied to simulate the plastic damage of Zirconium successfully. (2) The load–displacement curve of FE simulation has good agreement with that of the small punch tests. (3) Damage occurs on the bottom side of the specimen and grows across the specimen until complete failure. (4) Initial failure occurs at the bottom surface 0.78–0.85 mm away from the center, which agrees well with the observation from the test. (5) Friction coefficient has some effects on simulation results, the smaller the coefficient m is, the greater the max load will be.
Acknowledgments The authors wish to acknowledge the financial support provided by National Natural Science Foundation of China (No. 50275072).
817
Fig. 13. Contour of void nucleation fraction of R60702 Zirconium (VVFN).
References [1] [2] Bulloch JH. Toughness losses in low alloy steels at high temperatures: an appraisal of certain factors concerning the small punch test. Int J Press Vessels Piping 1998;75:791–804. [3] Bulloch JH. The small punch toughness test: some detailed fractographic information. Int J Press Vessels Piping 1995;63:177–94. [4] Milicka Karel, Dobes Ferdinand. Small punch testing of P91 steel. Int J Press Vessels Piping 2006;83:625–34. [5] Ule B, Sustar T. The effect of initial hot plastic deformation on creep behavior of 12 Cr steel specimens in small punch tests. Mater High Temperatures 2001;18(3):163–70. [6] [7] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part IdYield criteria and flow rules for porous ductile materials. J Eng Mater Technol 1977;99:2–15. [8] Gurson AL. Porous rigid-plastic materials containing rigid inclusionsyield function, plastic potential and void nucleation. Fracture 1977;2:357–64. [9] Tvergaard V. Influence of voids on shear band instabilities under plane strain conditions. Int J Fracture Mechanics 1981;17:389–407. [10] Version 6.2ABAQUS/Explicit users manual, vol. II. USA: Hibbitt, Karlsson & Sorensen, Inc; 2001. [11] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part Idyield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99:2–15. [12] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [13] Ling Xiang, Zheng Yangyan, You Yingjun, Chen Yuxin. Creep damage in small punch creep specimens of type 304 stainless steel. Int J Press Vessels Piping 2007;84(5):304–9. [14] Yang Z, Wang ZW. Relationship between strain and central deflection in small punch creep specimens. Int J Press Vessels Piping 2003;80:397–404. [15] Scheider I, Brocks W. Simulation of cup-cone fracture using the cohesive model. Eng Fracture Mechanics 2003;(70)::1943–61.
Contents lists available at ScienceDirect
International Journal of Pressure Vessels and Piping journal homepage: www.elsevier.com/locate/ijpvp
Three-dimensional numerical simulation on plastic damage in small punch specimen of Zirconium Ruomei Hu, Xiang Ling* School of Mechanical and Power Engineering, Nanjing University of Technology, Nanjing 210009, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 24 March 2009 Received in revised form 17 July 2009 Accepted 20 October 2009
Small punch test (SPT) technique was used to evaluate the mechanical properties of Zirconium in this paper. The dimension of the disc specimen is 4 10 0.5 mm. Plastic damage in small punch specimen of Zirconium was investigated both experimentally and numerically, because it has great influence on small punch specimen. In order to simulate the plastic damage in the small punch specimen of Zirconium, the 3D finite element model incorporated with Gurson–Tvergaard–Needleman (GTN) plastic damage constitutive equation was established. Void growth and initiation of ductile crack of the small punch specimen were predicted. Results show that damage occurs on the bottom side of the specimen and grows across the specimen until complete failure, which has good agreement with the observation in the experiment. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.
Keywords: Small punch GTN model Plastic damage Numerical simulation Zirconium
1. Introduction Zirconium is one of the corrosion-resistant materials [1]. It is a rare metal and costs much to gain the mechanical properties using the routine testing techniques, also it is impossible to get the conventional sample for the equipment in-service. So using the miniature test technique-small punch test [2–4] technique to study the mechanical properties of commercially pure zirconium under room temperature is significant. In the present paper, small punch test of R60702 Zirconium was carried out and a 3D finite element model incorporated with Gurson–Tvergaard–Needleman plastic damage constitutive equation was established. Then the small punch test process of the specimens of R60702 Zirconium was simulated by using the finite element software, ABAQUS.
2. Experimental 2.1. Uniaxial tensile test Tensile test was performed at room temperature. And it was carried out with 5800 testing machine at a constant crosshead velocity of 0.5 mm per minute. The specimen of uniaxial tensile test
* Corresponding author. Tel.: þ86 25 83587321; fax: þ86 25 83600956. E-mail addresses: [email protected] (R. Hu), [email protected] (X. Ling).
is shown in Fig. 1. Nominal stress and strain curve shown in Fig. 2 is acquired from the tensile test. When defining plasticity data in ABAQUS, true stress and true strain must be used. ABAQUS requires these values to interpret the data in the input file correctly. However, quite often material test data are supplied using values of nominal stress and strain. In such situations the plastic material data of nominal stress and strain have to be converted to true stress and strain. The true stress and strain curve is shown in Fig. 3. 2.2. Small punch test The small punch tests are performed using the device as seen in Fig. 4. The SP test basically consists of a disk specimen holder; a pushing rod and a steel ball. In the SPT, specimens in the form of disks are punched in their center by a steel ball, which leads at the beginning of loading to a biaxial stress state of the specimen. The diameter of the steel ball is 2.38 mm. The punch driven by the crosshead plate of the testing machine deforms the specimen centrically at a constant velocity, Fig. 5 shows the deformed specimen after failure. The disk specimens are extracted from the tensile specimen by cutting it perpendicularly to the gauge length. And then the specimens were polished on both sides using a series of diminishing grit size silicon paper down to 1000 m grit size so that the deviation of the thickness is accurate to 0.01 mm [5,6]. The final thickness of the specimen is 0.5 mm. The results of this experiment are the
0308-0161/$ – see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.ijpvp.2009.10.008
814
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 1. Specimen of uniaxial tensile test.
load–displacement curve of the punch and a deformation field as can be seen in Figs. 6 and 7. 3. Finite element simulation Fig. 3. Real stress and strain curves.
3.1. GTN model The material model, based on the constitutive damage law developed by Gurson, Tveergard and Needleman (GTN) [7–10], is widely-used to describe the micromechanical effects of damage in ductile metals. Gurson deduced a flow potential for void growth in an ideal-plastic material [11], which was extended by Tvergaard and Needleman [12], who introduced additional parameters (q1, q2, q3) and a modified damage variable f*. The void volume fraction f is used to measure the damage of the material. And the significant part of the model is the yield function
f¼
P 2 e
s
P 3 þ2q1 f * cos h q2 h ð1 þ q3 Þf *2 ¼ 0 s 2
(1)
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P0 P0 ffi 3
P Where ¼ 2 ij ij denotes the V. Mises stress and P Pe 1 h ¼ 3 kk the hydrostatic stress. q1, q2, q3are parameters to weight the different terms of the yield function. The isotropic hardening behavior of the matrix material is denoted with s. And the damage variable is denoted with f*, which can be calculated by the following function
( *
f ðf Þ ¼
f; f * f fc þ fu f c ðf fc Þ f
c
f fc fc f ff
(2)
the material completely fails. When f reaches fc, the voids aggregate. Then the load-carrying ability of the material weakens rapidly. The change in void volume fraction is
f_ ¼ f_ growth þ f_ nucleation
Where f_ growth is the change due to void growth and f_ nucleation due to nucleation of the new voids. Based on the law of conservation of mass, the void growth can be given by
f_ growth ¼ ð1 f ÞDpkk
(4)
And the nucleation of voids follows the strain controlled relationship
f_ nucleation ¼ Að3p Þs_ M p Dkk
(5)
Where denotes the macroscopic bulk strain increment, and sM the matrix stress rate. Moreover, the nucleation of voids obeys the normal distribution which has a mean value 3N and a standard deviation sN
In the above relationship, fc is a critical value of the void volume fraction and ff is the ultimate value of void volume fraction where
Fig. 2. Nominal stress and strain curves.
(3)
Fig. 4. Small punch testing apparatus.
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 5. Deformed specimen after failure.
A ¼
" # fN 1 3p 3N 2 pffiffiffiffiffiffiffi exp 2 sN hsN 2p
Fig. 7. Deformation field of R60702 Zirconium specimen.
(6)
Where fN denotes the volume fraction of void seeds, and 3p the equivalent plastic stress of the matrix material. 3.2. Finite element modeling The GTN model has been applied in the simulation using the ABAQUS software. The material is assumed to be homogeneous, elastoplastic with isotropic hardening. The small punch tests are simulated by means of the rigid-plastic finite element method. Researches have been performed on small punch creep tests [13,14] using planar models. But in this paper a three-dimensional finite element model for SPT was established using ABAQUS 6.5-1 explicit code. The model is shown in Fig. 8. Die and down-holder are fixed in all degrees of freedom, whereas the punch can be moved vertically by a displacement boundary condition. The contact between specimen and punch, die and down-holder is modeled including
Fig. 6. Load–displacement curve of SPT.
815
friction whereas the friction coefficient m can be varied. The mesh is composed of 10 388 linear reduced integration elements. It is biased toward the disc center in order to make the size of the elements smaller there. And their aspect ratio improved. The material parameters obtained from the uniaxial tensile test of R60702 Zirconium are E ¼ 99 200 MPa, m ¼ 0.3, ss ¼ 321 MPa. According to literature [15], the parameters of GTN material model are q1 ¼1.5, q2 ¼ 1, q3 ¼ q21 ¼ 2.25, ff ¼ 0.2, fc ¼ 0.17, fN ¼ 0.04, 3N ¼ 0.3, SN ¼ 0.1. 3.3. Simulation results and damage analysis The load–displacement curve of FE simulation is compared to that of the SP tests, which is shown in Fig. 9. It is obvious that the
Fig. 8. 3D FE model for the small punch specimen.
816
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 10. Simulating result with different friction coefficients. Fig. 9. Comparison of load & displacement curves of R60702 Zirconium.
curves agree well except for the deviation during the final failure stage. It is related to the localization of damage and the crack propagation. So, load–displacement curve yielded from experiment and FE model with ductile damage constitutive equations are in good agreement, which implies that the FE model created in this paper is reasonable. The effect of friction coefficient on SPT is simulated; results show that the max load is different with varied friction coefficient, the smaller the coefficient is, the greater the max load will be (Fig. 10).
Fig. 11(a) w (d) shows evolution of the void volume fraction (damage) in the small punch specimen. Damage occurs on the bottom side of the specimen and grows across the specimen until complete failure, which has good agreement with the observation in the experiment. Initial failure occurs at the bottom surface 0.78–0.85 mm away from the center, which agrees well with the observation from the test. The damage parameters are composed of f_ growth andf_ nucleation , which are denoted by VVFG and VVFN respectively. Fig. 12 displays the contour of void growth fraction of
Fig. 11. Evolution of the void volume fraction (damage) in a small punch specimen.
R. Hu, X. Ling / International Journal of Pressure Vessels and Piping 86 (2009) 813–817
Fig. 12. Contour of void growth fraction of R60702 Zirconium (VVFG).
the specimen, while void nucleation fraction of R60702 Zirconium is shown in Fig. 13.
4. Conclusions Simulation on the small punch test process has been presented in this paper, also plastic damage in small punch specimen of Zirconium is investigated. The results are summarized as follows: (1) The 3D finite element model incorporated with GTN constitutive equation has been applied to simulate the plastic damage of Zirconium successfully. (2) The load–displacement curve of FE simulation has good agreement with that of the small punch tests. (3) Damage occurs on the bottom side of the specimen and grows across the specimen until complete failure. (4) Initial failure occurs at the bottom surface 0.78–0.85 mm away from the center, which agrees well with the observation from the test. (5) Friction coefficient has some effects on simulation results, the smaller the coefficient m is, the greater the max load will be.
Acknowledgments The authors wish to acknowledge the financial support provided by National Natural Science Foundation of China (No. 50275072).
817
Fig. 13. Contour of void nucleation fraction of R60702 Zirconium (VVFN).
References [1] [2] Bulloch JH. Toughness losses in low alloy steels at high temperatures: an appraisal of certain factors concerning the small punch test. Int J Press Vessels Piping 1998;75:791–804. [3] Bulloch JH. The small punch toughness test: some detailed fractographic information. Int J Press Vessels Piping 1995;63:177–94. [4] Milicka Karel, Dobes Ferdinand. Small punch testing of P91 steel. Int J Press Vessels Piping 2006;83:625–34. [5] Ule B, Sustar T. The effect of initial hot plastic deformation on creep behavior of 12 Cr steel specimens in small punch tests. Mater High Temperatures 2001;18(3):163–70. [6] [7] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part IdYield criteria and flow rules for porous ductile materials. J Eng Mater Technol 1977;99:2–15. [8] Gurson AL. Porous rigid-plastic materials containing rigid inclusionsyield function, plastic potential and void nucleation. Fracture 1977;2:357–64. [9] Tvergaard V. Influence of voids on shear band instabilities under plane strain conditions. Int J Fracture Mechanics 1981;17:389–407. [10] Version 6.2ABAQUS/Explicit users manual, vol. II. USA: Hibbitt, Karlsson & Sorensen, Inc; 2001. [11] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: part Idyield criteria and flow rules for porous ductile media. ASME J Eng Mater Technol 1977;99:2–15. [12] Tvergaard V, Needleman A. Analysis of the cup-cone fracture in a round tensile bar. Acta Metall 1984;32:157–69. [13] Ling Xiang, Zheng Yangyan, You Yingjun, Chen Yuxin. Creep damage in small punch creep specimens of type 304 stainless steel. Int J Press Vessels Piping 2007;84(5):304–9. [14] Yang Z, Wang ZW. Relationship between strain and central deflection in small punch creep specimens. Int J Press Vessels Piping 2003;80:397–404. [15] Scheider I, Brocks W. Simulation of cup-cone fracture using the cohesive model. Eng Fracture Mechanics 2003;(70)::1943–61.