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Prediction of strength using flat cylindrical indentation method
Theoretical and Applied Fracture Mechanics 46 (2006) 70–74 www.elsevier.com/locate/tafmec
Prediction of strength using flat cylindrical indentation method Z.F. Yue *, B.X. Xu School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China Available online 21 June 2006
Abstract Strength of structural components is predicted. Two cases have been studied to explore the possibility of determining the damage level of the materials by the flat cylindrical indentation method with the help of the finite element method (FEM). The first uses the Gurson model for analyzing the elastic–plastic damage. The second uses the Katchanov–Robotnov law to predict the creep damage. The analytical results show that the damage levels can be determined by the flat cylindrical indentation experimental method. 2006 Elsevier Ltd. All rights reserved. Keywords: Damage prediction; Flat cylindrical indentation; Elastic–plastic damage; Creep damage; Finite element method
1. Introduction Studies of strength of materials have involved different means [1–5]. There are two types of methods used to determine residual life and strength of materials. One of them is the destructive testing method, such as, accelerated creep testing, creep crack propagating testing, materials density testing and so on. The other is the non-destructive testing method, such as metallographic testing. There are disadvantages associated with the destructive testing method although it is accurate. In this connection, non-destructive testing method has attracted much attention [6]. Indentation testing has been as a standard method for material characterization. It is easy to *
Corresponding author. Tel./fax: +86 29 88495540. E-mail address: [email protected] (Z.F. Yue).
apply and it requires only a small section of the sample material. The method has been used extensively for micro-/nano-systems [7–13]. One of the important results is that flat cylindrical indenters are better than other indenter geometries with reference to creep testing. The experimental and finite element analysis all show that there is steady state indentation rate (ratio of indentation depth to creep time) under a constant creep indentation stress. A lot of work can be found in [6,14–17]. This testing method has also been successfully used to determine the properties of one phase material, particle reinforced material, thin film material and fiber interface [18,19]. Recently, the creep damage law has been successfully induced to the indentation technique [20]. This work seeks to determine the damage level of materials with the flat cylindrical indentation method. Two situations are considered: elastic–plastic and creep damage.
0167-8442/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2006.05.003
Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
71
2. Finite element model The elastic–plastic damage model is that of Gurson. It can be directly used for the finite element code ABAQUS [21]. The creep damage model is that of Katchanov–Robotnov [22,23]. It can be written as e_ e ¼
Brne n ð1 xÞ
ð1Þ Fig. 2. The time relationship between S11 and P.
and x_ ¼
D½ar1 þ ð1 aÞre ð1 xÞ
X
U
ð2Þ 3. FEM analysis
where ee and re are equivalent creep strain and stress, respectively. In Eq. (2), r1 is the maximum principal stress and x the damage variable. The terms B, D, n, x, a and U are material parameters. The parameter a describes the relative importance of r1 and re on creep damage and rupture. The creep damage is caused by re for a = 0, and by r1 for a = 1. It is assumed that a = 0, which is also reasonable for most metals [24]. This creep damage law has been implemented the finite element code ABAQUS as a user subroutine ‘CREEP’. The finite element scheme for the two damage models is the same for the two cases studied as shown in Fig. 1. The pre-damage was produced by applying stress S11 in the direction of 11. Indentation with stress P is carried out in the direction of 33 after the stress S11 is unloaded to zero. The relationship of S11 and P is shown in Fig. 2. The flat cylindrical indenter with the radius 2 mm is used and it can be assumed to be rigid. The surface area in contact with the indenter is loaded by the uniform indentation stress with the same indentation depth. This can be realized by a technology of MPC cards [25].
Fig. 1. Model.
3.1. Elastic–plastic case The parameters for the Gurson model are taken from [26]. They are: q1 = 1.5, q2 = 1, q3 = 2.25, f0 = 0.05, eN = 0.3, S = 0.1 and fN = 0.04. Different pre-damage D can be produced by varying S11max. The relationship between the indentation depth d and indentation stress p is shown in Fig. 3. The relationship of d and D under different stress level p is shown in Fig. 4. From Figs. 3 and 4 it is seen that there is an approximately linear relationship among the indentation stress p, indentation depth d and damage D. Obviously, the pre-damage of materials, which is induced by S11max, can be determined if the relationship of indentation depth d and indentation stress P is obtained in advance. Hence, the residual life of materials can be predicted using the elastic– plastic damage approach.
Fig. 3. The relationship of indentation depth d and indentation stress P under the constant of pre-damage D.
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Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
Fig. 4. The relationship of indentation d and pre-damage D under the constant indentation stress P.
Fig. 5. The relationship of indentation d and indentation creep time t for different pre-damage x.
Table 1 Creep damage model parameters for Eqs. (1) and (2) (taken from [26]) B [MPan/s] h [MpaX/s] X a U n
1.00 · 1018 2.00 · 1018 4.5 0 4.5 4.5
3.2. Creep damage case Table 1 gives the parameters for the creep damage model. The applied stress S11 is 500 MPa. The creep pre-damage x can be found from the different creep time. The indentation stress P is 500 MPa and the creep time t is 10,000 s. The indentation depth d with the function of indentation creep time t is shown in Fig. 5. The relationship of indentation depth d and creep time t is almost linear, Fig. 5. _ Moreover, the indentation depth rate dd=dtðdÞ increases with the increasing of x as in Fig. 6. Compared to the Gurson model, indentation creep experiments are easier for determining the damage levels of materials. The relationship between indentation depth rate dd/dt and the pre-damage levels x can be obtained as a standard curve by specimen experiments. The creep pre-damage can be determined directly by comparing results of indentation creep damage experiments with the standard curves.
Fig. 6. The relationship of indentation depth rate d/t and the predamage x.
4. Discussion 4.1. Determination of pre-damage of materials with the flat cylindrical indentation You predict damage levels of materials, only indentation experiment is needed. It suffices to compare indentation testing results with the standard curves which can be, obtained by usual specimen experiments in advance. 4.2. The influence of damage direction The damage analyses are based on using the equivalent stress. But in fact, the damage levels of
Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
materials are affected by the full stress state. As an example for the creep damage model, the predamage levels are S11 = 500 MPa and S11 = S22 = 500 MPa with the creep time of 104 s. The response of the indentation is also the same. The reason is that indentation creep damage is produced by the equivalent stress. This method can also be used only if the damage is produced by the equivalent stress. To determine the residual strength and residual life of the materials and components, the stress states of the materials and components are usually known in advance. Hence, the D–d–P for elastic–plastic damage case and dd/dt-x for creep damage case can be obtained. 5. Conclusion Two cases have been successfully studied to explore the possibility of predicting the pre-damage levels of materials with the flat cylindrical indentation technique by the finite element method. The main conclusions can be drawn as follows: 1. Using the Gurson model, flat cylindrical indentation behavior has been simulated for the predamage materials. The emphasis has been placed on the influence of pre-damage on the response of indentation behavior. The results show that there is an approximately linear relation between the pre-damage and indentation depth under a constant indentation stress. Further studies show that the pre-damage can be predicted with the flat cylindrical indentation testing. 2. The indentation creep damage model has been studied by implementing the Katchanov–Robotnov continuum creep damage law into the finite element code ABAQUS. It is shown that the indentation depth rate increases with increasing of the pre-damage. The pre-damage of the materials was also predicted with the flat cylindrical indentation creep experiments. 3. The pre-damage of the materials and components can be predicted with the flat cylindrical indentation from analyses of elastic–plastic case (Gurson model) and creep case. Since the method is non-destructive, it can be used in the field. 4. For turbine blade application, the D–d–P (elastic–plasticity damage) or dd/dt-x (creep damage) standard curve can be obtained by specimen experiments in advance for the materials and components. Based on these results, the predamage of materials can be predicted by compar-
73
ing the indentation experimental result with the standard curve. Acknowledgements This work is supported by the China National Nature Foundation (50375124, 10472094) and China Aviation Foundation (02C53011, 03B53003). References [1] G.A. Webster, R.A. Ainsworth, High Temperature Component Life Predictment, Chapman & Hall, London, UK, 1994. [2] F.C. Campbell, Non-destructive inspection, Manufacturing Processes for Advanced Composites, Elsevier, 2005. [3] A.K. Ray, Y.N. Tiwari, S. Chaudhuri, Evaluation of mechanical properties and predictment of residual life of a service-exposed water wall tube, Eng. Failure Anal. 7 (2000) 393–402. [4] F. De Backer, V. Schoss, G. Maussner, Investigations on the evaluation of the residual fatigue life-time in austenitic stainless steels, Nucl. Eng. Des. 206 (2001) 201–219. [5] D. Huynh, J. He, D. Tran, Damage location vector: a nondestructive structural damage detection technique, Compos. Struct. 83 (2005) 2353–2367. [6] J.C.M. Li, Impression creep and other localized tests, Mater. Sci. Eng. A 322 (2002) 23–42. [7] M.F. Doerner, W.D. Nix, A method for interpreting the date from depth-sensing indentation measurements, J. Mater. Res. 4 (1986) 601–616. [8] W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (1992) 1564–1583. [9] J.J. Kim, J.H. Jeong, K.R. Lee, D. Kwon, A new indentation cracking method for evaluating interfacial adhesion energy of hard films, Thin Solid Films 441 (2003) 72–179. [10] G.S. Smith, Multiscale simulations of indentation, Harvard University, The thesis for the Doctor Degree, 2000. [11] W. Zhang, Finite element analysis of induced damage due to indentation and scratching on brittle materials, Michigan Technological University, The thesis for the Doctor Degree, 2001. [12] T. Chudoba, F. Richter, Investigation of creep behaviour under load during indentation experiments and its influence on hardness and modulus results, Surf. Coat. Technol. 148 (2001) 191–198. [13] A. Carpinteri, B. Chiaia, S. Invernizzi, Numerical analysis of indentation fracture in quasi-brittle materials, Eng. Fract. Mech. 71 (2004) 567–577. [14] F.Q. Yang, J.C.M. Li, C.W. Shih, Computer simulation of impression using the hyperbolic sine stress law, Mater. Sci. Eng. A 201 (1995) 50–57. [15] F.Q. Yang, J.C.M. Li, Impress and diffusional creep of anisotropic media, J. Appl. Phys. 77 (1) (1995) 110–117. [16] F.Q. Yang, J.C.M. Li, Impression creep of a thin film by vacancy diffusion. I. Straight punch, J. Appl. Phy. 74 (7) (1993) 4382–4389. [17] F.Q. Yang, J.C.M. Li, Impression test of 63Sn–37Pb eutectic alloy, Mater. Sci. Eng. A 201 (1995) 40–49.
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Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
[18] Z.F. Yue, Z.Z. Lu, A numerical determination of the fiber/ matrix interlayer creep properties from the indentation creep testing in fiber reinforced composites, Mater. Sci. Eng. A 352 (2003) 266–272. [19] Z.F. Yue, G. Eggeler, B. Stoeckhert, A creep finite element analysis of indentation creep testing in two phase microstrctures(particle/matrix- and thin film/subtrate-systems), Compos. Mater. Sci. 21 (2001) 37–56. [20] B.X. Xu, J.S. Wan, B. Zhao, Z.F. Yue, Numerical study of creep damage behavior of thin film/substrate systems under indentation, Materialwiss. Werkst. 36 (2) (2005) 75– 79. [21] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. I. Yield criteria and flow rules for
[22] [23] [24]
[25] [26]
porous ductile media, J. Eng. Mater. Technol. 99 (1) (1977) 2–15. L.M. Kachanov, A. Izv, Nauk USSR Otd. Tekt. Nauk., 27, 1958. Y.M. Rabothov, Creep Problems in Structural Members, North-Holland, Amsterdam, 1969. G. Eggeler, C. Wiesner, A numerical study of parameters controlling stress redistribution in circular notched specimens during creep, J. Strain Anal. 28 (1993) 13–22. Hibbit, Karlsson, Sorensen, ABAQUS User’s Manual, Version 5.6, 1998. W.P. Wang, Z.F. Yue, Z.G. Yamg, Damage analysis of tensile specimens of powder metallurgy material including voids, Chin. J. Nonferrous Metals 14 (2004) 949–955.
Prediction of strength using flat cylindrical indentation method Z.F. Yue *, B.X. Xu School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an 710072, PR China Available online 21 June 2006
Abstract Strength of structural components is predicted. Two cases have been studied to explore the possibility of determining the damage level of the materials by the flat cylindrical indentation method with the help of the finite element method (FEM). The first uses the Gurson model for analyzing the elastic–plastic damage. The second uses the Katchanov–Robotnov law to predict the creep damage. The analytical results show that the damage levels can be determined by the flat cylindrical indentation experimental method. 2006 Elsevier Ltd. All rights reserved. Keywords: Damage prediction; Flat cylindrical indentation; Elastic–plastic damage; Creep damage; Finite element method
1. Introduction Studies of strength of materials have involved different means [1–5]. There are two types of methods used to determine residual life and strength of materials. One of them is the destructive testing method, such as, accelerated creep testing, creep crack propagating testing, materials density testing and so on. The other is the non-destructive testing method, such as metallographic testing. There are disadvantages associated with the destructive testing method although it is accurate. In this connection, non-destructive testing method has attracted much attention [6]. Indentation testing has been as a standard method for material characterization. It is easy to *
Corresponding author. Tel./fax: +86 29 88495540. E-mail address: [email protected] (Z.F. Yue).
apply and it requires only a small section of the sample material. The method has been used extensively for micro-/nano-systems [7–13]. One of the important results is that flat cylindrical indenters are better than other indenter geometries with reference to creep testing. The experimental and finite element analysis all show that there is steady state indentation rate (ratio of indentation depth to creep time) under a constant creep indentation stress. A lot of work can be found in [6,14–17]. This testing method has also been successfully used to determine the properties of one phase material, particle reinforced material, thin film material and fiber interface [18,19]. Recently, the creep damage law has been successfully induced to the indentation technique [20]. This work seeks to determine the damage level of materials with the flat cylindrical indentation method. Two situations are considered: elastic–plastic and creep damage.
0167-8442/$ - see front matter 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.tafmec.2006.05.003
Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
71
2. Finite element model The elastic–plastic damage model is that of Gurson. It can be directly used for the finite element code ABAQUS [21]. The creep damage model is that of Katchanov–Robotnov [22,23]. It can be written as e_ e ¼
Brne n ð1 xÞ
ð1Þ Fig. 2. The time relationship between S11 and P.
and x_ ¼
D½ar1 þ ð1 aÞre ð1 xÞ
X
U
ð2Þ 3. FEM analysis
where ee and re are equivalent creep strain and stress, respectively. In Eq. (2), r1 is the maximum principal stress and x the damage variable. The terms B, D, n, x, a and U are material parameters. The parameter a describes the relative importance of r1 and re on creep damage and rupture. The creep damage is caused by re for a = 0, and by r1 for a = 1. It is assumed that a = 0, which is also reasonable for most metals [24]. This creep damage law has been implemented the finite element code ABAQUS as a user subroutine ‘CREEP’. The finite element scheme for the two damage models is the same for the two cases studied as shown in Fig. 1. The pre-damage was produced by applying stress S11 in the direction of 11. Indentation with stress P is carried out in the direction of 33 after the stress S11 is unloaded to zero. The relationship of S11 and P is shown in Fig. 2. The flat cylindrical indenter with the radius 2 mm is used and it can be assumed to be rigid. The surface area in contact with the indenter is loaded by the uniform indentation stress with the same indentation depth. This can be realized by a technology of MPC cards [25].
Fig. 1. Model.
3.1. Elastic–plastic case The parameters for the Gurson model are taken from [26]. They are: q1 = 1.5, q2 = 1, q3 = 2.25, f0 = 0.05, eN = 0.3, S = 0.1 and fN = 0.04. Different pre-damage D can be produced by varying S11max. The relationship between the indentation depth d and indentation stress p is shown in Fig. 3. The relationship of d and D under different stress level p is shown in Fig. 4. From Figs. 3 and 4 it is seen that there is an approximately linear relationship among the indentation stress p, indentation depth d and damage D. Obviously, the pre-damage of materials, which is induced by S11max, can be determined if the relationship of indentation depth d and indentation stress P is obtained in advance. Hence, the residual life of materials can be predicted using the elastic– plastic damage approach.
Fig. 3. The relationship of indentation depth d and indentation stress P under the constant of pre-damage D.
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Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
Fig. 4. The relationship of indentation d and pre-damage D under the constant indentation stress P.
Fig. 5. The relationship of indentation d and indentation creep time t for different pre-damage x.
Table 1 Creep damage model parameters for Eqs. (1) and (2) (taken from [26]) B [MPan/s] h [MpaX/s] X a U n
1.00 · 1018 2.00 · 1018 4.5 0 4.5 4.5
3.2. Creep damage case Table 1 gives the parameters for the creep damage model. The applied stress S11 is 500 MPa. The creep pre-damage x can be found from the different creep time. The indentation stress P is 500 MPa and the creep time t is 10,000 s. The indentation depth d with the function of indentation creep time t is shown in Fig. 5. The relationship of indentation depth d and creep time t is almost linear, Fig. 5. _ Moreover, the indentation depth rate dd=dtðdÞ increases with the increasing of x as in Fig. 6. Compared to the Gurson model, indentation creep experiments are easier for determining the damage levels of materials. The relationship between indentation depth rate dd/dt and the pre-damage levels x can be obtained as a standard curve by specimen experiments. The creep pre-damage can be determined directly by comparing results of indentation creep damage experiments with the standard curves.
Fig. 6. The relationship of indentation depth rate d/t and the predamage x.
4. Discussion 4.1. Determination of pre-damage of materials with the flat cylindrical indentation You predict damage levels of materials, only indentation experiment is needed. It suffices to compare indentation testing results with the standard curves which can be, obtained by usual specimen experiments in advance. 4.2. The influence of damage direction The damage analyses are based on using the equivalent stress. But in fact, the damage levels of
Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
materials are affected by the full stress state. As an example for the creep damage model, the predamage levels are S11 = 500 MPa and S11 = S22 = 500 MPa with the creep time of 104 s. The response of the indentation is also the same. The reason is that indentation creep damage is produced by the equivalent stress. This method can also be used only if the damage is produced by the equivalent stress. To determine the residual strength and residual life of the materials and components, the stress states of the materials and components are usually known in advance. Hence, the D–d–P for elastic–plastic damage case and dd/dt-x for creep damage case can be obtained. 5. Conclusion Two cases have been successfully studied to explore the possibility of predicting the pre-damage levels of materials with the flat cylindrical indentation technique by the finite element method. The main conclusions can be drawn as follows: 1. Using the Gurson model, flat cylindrical indentation behavior has been simulated for the predamage materials. The emphasis has been placed on the influence of pre-damage on the response of indentation behavior. The results show that there is an approximately linear relation between the pre-damage and indentation depth under a constant indentation stress. Further studies show that the pre-damage can be predicted with the flat cylindrical indentation testing. 2. The indentation creep damage model has been studied by implementing the Katchanov–Robotnov continuum creep damage law into the finite element code ABAQUS. It is shown that the indentation depth rate increases with increasing of the pre-damage. The pre-damage of the materials was also predicted with the flat cylindrical indentation creep experiments. 3. The pre-damage of the materials and components can be predicted with the flat cylindrical indentation from analyses of elastic–plastic case (Gurson model) and creep case. Since the method is non-destructive, it can be used in the field. 4. For turbine blade application, the D–d–P (elastic–plasticity damage) or dd/dt-x (creep damage) standard curve can be obtained by specimen experiments in advance for the materials and components. Based on these results, the predamage of materials can be predicted by compar-
73
ing the indentation experimental result with the standard curve. Acknowledgements This work is supported by the China National Nature Foundation (50375124, 10472094) and China Aviation Foundation (02C53011, 03B53003). References [1] G.A. Webster, R.A. Ainsworth, High Temperature Component Life Predictment, Chapman & Hall, London, UK, 1994. [2] F.C. Campbell, Non-destructive inspection, Manufacturing Processes for Advanced Composites, Elsevier, 2005. [3] A.K. Ray, Y.N. Tiwari, S. Chaudhuri, Evaluation of mechanical properties and predictment of residual life of a service-exposed water wall tube, Eng. Failure Anal. 7 (2000) 393–402. [4] F. De Backer, V. Schoss, G. Maussner, Investigations on the evaluation of the residual fatigue life-time in austenitic stainless steels, Nucl. Eng. Des. 206 (2001) 201–219. [5] D. Huynh, J. He, D. Tran, Damage location vector: a nondestructive structural damage detection technique, Compos. Struct. 83 (2005) 2353–2367. [6] J.C.M. Li, Impression creep and other localized tests, Mater. Sci. Eng. A 322 (2002) 23–42. [7] M.F. Doerner, W.D. Nix, A method for interpreting the date from depth-sensing indentation measurements, J. Mater. Res. 4 (1986) 601–616. [8] W.C. Oliver, G.M. Pharr, An improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments, J. Mater. Res. 7 (1992) 1564–1583. [9] J.J. Kim, J.H. Jeong, K.R. Lee, D. Kwon, A new indentation cracking method for evaluating interfacial adhesion energy of hard films, Thin Solid Films 441 (2003) 72–179. [10] G.S. Smith, Multiscale simulations of indentation, Harvard University, The thesis for the Doctor Degree, 2000. [11] W. Zhang, Finite element analysis of induced damage due to indentation and scratching on brittle materials, Michigan Technological University, The thesis for the Doctor Degree, 2001. [12] T. Chudoba, F. Richter, Investigation of creep behaviour under load during indentation experiments and its influence on hardness and modulus results, Surf. Coat. Technol. 148 (2001) 191–198. [13] A. Carpinteri, B. Chiaia, S. Invernizzi, Numerical analysis of indentation fracture in quasi-brittle materials, Eng. Fract. Mech. 71 (2004) 567–577. [14] F.Q. Yang, J.C.M. Li, C.W. Shih, Computer simulation of impression using the hyperbolic sine stress law, Mater. Sci. Eng. A 201 (1995) 50–57. [15] F.Q. Yang, J.C.M. Li, Impress and diffusional creep of anisotropic media, J. Appl. Phys. 77 (1) (1995) 110–117. [16] F.Q. Yang, J.C.M. Li, Impression creep of a thin film by vacancy diffusion. I. Straight punch, J. Appl. Phy. 74 (7) (1993) 4382–4389. [17] F.Q. Yang, J.C.M. Li, Impression test of 63Sn–37Pb eutectic alloy, Mater. Sci. Eng. A 201 (1995) 40–49.
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Z.F. Yue, B.X. Xu / Theoretical and Applied Fracture Mechanics 46 (2006) 70–74
[18] Z.F. Yue, Z.Z. Lu, A numerical determination of the fiber/ matrix interlayer creep properties from the indentation creep testing in fiber reinforced composites, Mater. Sci. Eng. A 352 (2003) 266–272. [19] Z.F. Yue, G. Eggeler, B. Stoeckhert, A creep finite element analysis of indentation creep testing in two phase microstrctures(particle/matrix- and thin film/subtrate-systems), Compos. Mater. Sci. 21 (2001) 37–56. [20] B.X. Xu, J.S. Wan, B. Zhao, Z.F. Yue, Numerical study of creep damage behavior of thin film/substrate systems under indentation, Materialwiss. Werkst. 36 (2) (2005) 75– 79. [21] A.L. Gurson, Continuum theory of ductile rupture by void nucleation and growth. I. Yield criteria and flow rules for
[22] [23] [24]
[25] [26]
porous ductile media, J. Eng. Mater. Technol. 99 (1) (1977) 2–15. L.M. Kachanov, A. Izv, Nauk USSR Otd. Tekt. Nauk., 27, 1958. Y.M. Rabothov, Creep Problems in Structural Members, North-Holland, Amsterdam, 1969. G. Eggeler, C. Wiesner, A numerical study of parameters controlling stress redistribution in circular notched specimens during creep, J. Strain Anal. 28 (1993) 13–22. Hibbit, Karlsson, Sorensen, ABAQUS User’s Manual, Version 5.6, 1998. W.P. Wang, Z.F. Yue, Z.G. Yamg, Damage analysis of tensile specimens of powder metallurgy material including voids, Chin. J. Nonferrous Metals 14 (2004) 949–955.