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A novel damage variable to characterize evolution of microstructure with plastic deformation for ductile metal materials under tensile loading
Engineering Fracture Mechanics 78 (2011) 503–513
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
A novel damage variable to characterize evolution of microstructure with plastic deformation for ductile metal materials under tensile loading Z.M. Shi a,*, H.L. Ma a,b, J.B. Li a,c a
School of Materials Science and Engineering, Inner Mongolia University of Technology, Hohhot 010051, China Department of Mathematics, Taiyuan College of Technology, Taiyuan 030008, China c Department of Mechanics, Taiyuan University of Science and Technology, Taiyuan 030024, China b
a r t i c l e
i n f o
Article history: Received 23 May 2009 Received in revised form 23 November 2009 Accepted 25 December 2009 Available online 4 January 2010 Keywords: Armco iron Mild steel Microstructure Optical microscopy Plastic deformation Damage parameter
a b s t r a c t Damage and fracture of ductile metal materials are greatly associated with evolution of their microstructure under loading, which meso-dimensionally corresponds to grain deformation as well as nucleation, growth, and coalescence of microvoids in later local deformation. The evolution behavior of microstructure is important to realize ductile damage and fracture mechanism of materials. In the present work, a novel damage variable of shape factor of grains was put forward to quantitatively describe the character of microstructure; its evolution with the plastic deformation was built-up by microanalytical and mechanical experiments for Armco iron and mild steel tensile bars. The evolution rule based on the damage variable of shape factor is possible to be extended into all of ductile metal materials. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Ductile metal materials generally show three stages of elastic deformation, plastic deformation, and ultimate fracture under loading. Many researchers have studied their damage and fracture behaviors in terms of macro, meso, and micro scales, in which the mesodamage is the most interesting area. For most metal materials, the ductile fracture undergoes nucleation, extension, and coalescence of voids. McClintock [1] put forward a criterion by the growth of voids for the ductile fracture. Rice and Tracey [2] studied the enlargement behavior of voids in triaxiaI stress fields. Gurson [3] proposed a damage variable of void volume fraction and set up constitutive equations to describe nucleation and growth of existing voids for ductile porous metal materials. The Gurson model has been modified and widely used to study deformation and failure of metal materials containing microvoids. Kameda [4] developed a microscopic constitutive equation related to the growth of voids subjected to a hydrostatic tensile stress in basis of the mobility of dislocations and the thermally activated shear stress. Zhen and Zhou [5] proposed a mesoparameter of void extension ratio to describe the law of nucleation, growth, and coalescence of voids based on substantial experiments for BS4360-50D steel; they found that when the effective plastic strain is less than 0.10, much less microvoids with several micrometer (existing in as-received materials) can be observed; from about 0.14 plastic strain (approaching necking), the nucleation of voids can be found, and the corresponding void volume fraction is about 0.13%; the coalescence of microvoids starts at about 0.6 plastic strain. Bate and Hutchinson [6] found that the area of grain boundary non-linearly increases with the increase of the plastic strain in states of tensile, compression, torsion,
* Corresponding author. Tel./fax: +86 471 6576221. E-mail address: [email protected] (Z.M. Shi). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.12.011
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513
and shear, in basis of an ellipsoid grain model, which is geometrically equivalent to the random equi-axed grain. The increase in area of grain boundary has a significant influence on mechanical properties of materials. The fracture of solid materials is an accumulation of microstructure variation at different measuring scales (micro and meso) such as crystal defects, microvoids, and grains. During the plastic deformation of a ductile metal material, there is a great evolution in its mesoscale, that is, the grains can be elongated, rotated, refined, or orientated [7–11]. In this case, the plastic deformation extent largely depends on the character of materials, such as type of materials, size of grains, as well as the content and distribution of secondary phase and inclusion particle. When the plastic strain arrives at a definite extent, the microvoids can preferentially nucleate around the inclusion particles [5]. Therefore, the plastic deformation directly affects nucleation and extension of microvoids. There is a necessity to clarify the relationship between the evolution of microstructure and plastic deformation so as to systematically describe the evolution procedure of the microstructure from beginning of plastic deformation up to fracture, in which the nucleation, growth, and coalescence of microvoids are certainly included. Consequently, how to define a damage variable that can describe the microstructure and its evolution develops into a significant issue. In this work, a shape factor of grains was selected as the damage variable which can be measured through the quantitative metallographic analysis, and its evolution equation with the plastic strain was established for ductile metal materials of Armco iron and mild steel under tensile loading. 2. Definition of damage variable for microstructure Ductile metal materials are composed of single-phase or multi-phase structures with an order of micrometer magnitude. Fig. 1 shows the sketch map for their microstructure. The shape and size of grains are essential parameters for describing the basic microstructure of materials. In the present work, three kinds of shape factors were adopted.
Fig. 1. Sketch map of microstructure of metal materials; a: matrix; b: grain boundary; c: second phase: (a) single phase and (b) dual phase.
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513 Table 1 The symbol, meaning, and expression of three kinds of shape factors. Symbol
Meaning
U
Shape factor
Expression P L U¼P
Value range
W
Relative shape factor
U0 W ¼ U U0
WP0
D
Normalized shape factor
D ¼ UUfUU00
06D61
U>0
A
2.1. Shape factor U The shape factor U is expressed by
P
U¼P
L A
ð1Þ
where L is the length of boundary of grains, which can be statistically measured for the longitudinal section of samples in microanalytical experiments, and A is the area that is preset by the field of microscope. The boundary length in a definite area should be increased with their elongation under tensile loading. Here, only the total length of boundary of phases, so called internal boundary of materials, was considered. Since the shape factor includes the effect of shape and size of grains, it can overall represent the microstructure character for as-received and deformed microstructure of materials. 2.2. Relative shape factor W The relative shape factor W is defined as
W¼
U U0 U0
ð2Þ
where U is the shape factor at a plastic strain and U0 is the shape factor for as-received materials. The relative shape factor is a dimensionless parameter, which can be used to describe relative increment of the shape factor. 2.3. Normalized shape factor D The shape factor can also be normalized into another dimensionless form
D¼
U U0 Uf U0
ð3Þ
where Uf is the shape factor at the fracture strain. It can be noticed from the following description that D has an explicit physical significance. (1) If U = U0, D ¼ 0, the material is located at the non-damage state (as-received state); (2) If U = Uf D ¼ 1, the material is fractured. (3) If U0 < U < Uf, 0 < D < 1, the material is just located in a plastic deformation state. In consideration of the clarity, three kinds of shape factors are summarized in Table 1. 3. Materials and methods 3.1. Materials Armco iron and mild steel rolled round bars with a diameter of 20 mm were used as raw materials. They were annealed at 800 °C for 3 h to eliminate residual stress and recover the microstructure into equi-axed grains. Armco iron is a kind of soft metal material with a single phase of ferrite microstructure, which is mainly used as magnetic and electronic materials such as relay, anodes of vacuum tube. Mild steel, with a biphase microstructure of ferrite and pearlite, has moderate strength, plasticity, and toughness, as well as excellent welding and hot–cold forming ability, which is widely used to manufacture assemblies such as bolts, pull rods, rockers, shafts, and frameworks. Their chemical compositions are listed in Table 2.
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Table 2 Main chemical composition of materials. Material
Armco iron Mild steel
Chemical composition, wt.% C
Si
Mn
P
S
Fe
<0.040 0.258
<0.003 0.284
<0.070 0.661
<0.008 <0.045
<0.006 <0.045
Balance Balance
Fig. 2. Stress–strain curves of Armco iron and mild steel.
3.2. Specimens The annealed bars were mechanically machined into standard tensile specimens, the working segment is with a diameter of 10 mm and a gauge length of 50 mm. The surface of specimens was polished to eliminate nicks. 3.3. Tensile test The tensile test was conducted on a mechanical machine (CSS-WAWDL type, CRITM), with a displacement loading mode, the displacement rate was controlled to be 0.2 mm/min. An electric extensometer (CBT-17 type, CRITM), with a gage length of 25 mm and measuring range of 10 to +25 mm, was used to detect the displacement. Fig. 2 shows the measured stress–strain curves of Armco iron and mild steel. Table 3 shows mechanical properties of the materials. From Fig. 2 and Table 3, it can be seen that Armco iron has lower Young’s modulus, yield strength, fracture strength, and a larger plasticity than those of mild steel, which are greatly dependant on their microstructure, chemical composition, and heat treatment state. In order to obtain the microstructure photographs at different plastic strains, the specimens were, respectively, loaded and then unloaded at desired strain points, which roughly distribute from yield point to the fracture. 3.4. Observation and analysis of microstructure The unloaded specimens were cut along their longitudinal plane by an electric park machine with a molybdenum wire of 0.2 mm in diameter. The longitudinal plane was polished into metallographic surface. An optical microscope (Leica LM/DM type) equipped with an image analysis software (SISC IAS V8.0) was adopted to observe and analyze the microstructure. For quantitative analysis to the microstructure photographs, it needs to tailor the effective characters such as grain boundary and secondary phase so as to obtain clear images suitable to be recognized and calculated by computer. Therefore, a series of photographic processes were used, such as gray level transfer, image filtering, image splitting, edge strengthening, image binarizing, boundary tailoring, and boundary linearizing.
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513 Table 3 Mechanical parameters of materials. Sample
Armco iron
Mild steel
Young’s modulus, GPa Yield strength, MPa Tensile strength, MPa Longitudinal elongation, % Necking strain
185.45 177.26 405.72 50.02 0.263
208.21 300.54 483.67 34.80 0.161
Because of the random distribution of grains, U is measured and calculated against 5–10 of micrographs. For the uniformly deformed samples, 10 of micrographs were taken along the longitudinal symmetry axis of tensile bars; for the necking samples, 5–6 of micrographs were taken against the field around the necking transverse symmetry plane and the longitudinal axis. 4. Results and discussion 4.1. Microstructure and processed images at different plastic strains The typical microstructure images of two materials at different plastic strains are shown in Figs. 3 and 4. The left and right parts, respectively, are the real and processed ones by the image software. It can be found from Figs. 3 and 4 that, with the increase of plastic strain, the grains are gradually elongated along the tensile direction (longitudinal axis). After necking, the elongation of grains become more remarkable. This is because, Armco iron with single ferrite phase has higher purity. In contrast, the mild steel consisted of ferrite and brittle secondary phase of pearlite (mixture of ferrite and carbide) has relative low purity. The different microstructure and purity result in different damage and fracture procedures. For Armco iron, the nucleation and growth of microvoids around small amounts of inclusions are delayed due to its significant dislocation slide in the soft ferrite grains. In contrast, for the mild steel, due to the existing many inclusions and the relative weak boundary combination between ferrite and carbide phases, the dislocation slide is limited to some extent, so the earlier nucleation and growth of microvoids around inclusions and relative weak boundary are promoted. Once the microvoids start to nucleate and grow, the homogeneous plastic deformation will come to cease and the deformation turns to the local one, i.e, the mode of nucleate, growth, and coalescence of microvoids, indicating appearance of necking and weakening of materials. The difference in damage and fracture procedure directly results in different effects on elongation and refinement of grains. Figs. 5 and 6 show the microstructure images at the fracture edge for Armco iron and mild steel. It can be observed that the grains at fracture edge and near surface are significantly refined by the serve plastic shear deformation, especially for Armco iron. This indicates that the stress state also has a strong effect on the mesoplastic deformation process of microstructure. 4.2. Evolution of damage variables with plastic strain Fig. 7 shows the evolution of shape factor U with plastic strain for two materials. It can be noticed that the shape factors of both materials have a similar change trend with increasing plastic strain. But the difference is that Armco iron has a smaller shape factor and it presents a slow increase, while the mild steel has a larger one and it presents a rapid increase. This is because the mild steel has a smaller granularity and hard phase of carbide than that of Armco iron. Therefore, by using the damage variable of shape factor, the real microstructure character of materials can be quantitatively described; its evolution with plastic strain discloses the response of materials in mesoscale resulted from the macro deformation. Fig. 8 displays the evolution of relative shape factor W with plastic strain for two materials. The variations of the relative shape factor with plastic strain are very similar for two materials. By using the relative shape factor, the relative increments of the shape factor are entirely coincident for both materials. This is why the relative shape factor is adopted. Fig. 9 illustrates the evolution of the normalized shape factor D with plastic strain for two materials. From these figures, it can be concluded that, the mild steel has a shorter and earlier plastic deformation stage and a rapid damage procedure than those of Armco iron. 4.3. Fitting equations to the evolution rules The exponential growth function is used to fit curves against the data in Fig. 9. The fitted equations are shown as follows. For Armco iron
D ¼ 0:065 þ 0:002ee=0:084
ð4Þ
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Fig. 3. Microstructure and processed photographs of Armco iron at different plastic strains; left: microstructure; right: processed photographs. (a) e = 0; (b) e = 0.114; (c) e = 0.263 (necking strain); (d) e = 0.350; (e) e = 0.464; (f) e = 0.474 and (g) e = 0.500.
For mild steel
D ¼ 0:043 þ 0:068ee=0:123
ð5Þ
Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513
509
Fig. 3 (continued)
The earlierabove mentioned equations can be summarized into general form.
D ¼ A þ Bee=C
ð6Þ
where A, B, and C are constants of materials. 4.4. Discussion to the meaning of shape factor 4.4.1. Rationality describing microstructure and its evolution by shape factor The shape factor is adopted to describe the microstructure of metal materials. It can be found that the shape factor mainly concerns with the perimeter of grain boundaries (matrix and secondary phase) in a defined area, and it receives much influence from the size, shape, and the plastic deformation extent. With an increase in plastic strain, the grains are elongated, resulting in an increase in grain boundary. Therefore, the shape factor can overall represent the microstructure character for as-received and deformed materials and can quantitatively reflect evolution rule of the microstructure. Therefore, the shape factor may be the most suitable mesodamage variable for describing microstructure character and its evolution procedure with plastic deformation. 4.4.2. Possibility describing damage evolution equation and constitutive equation by using shape factor as damage variable The ductile metal materials mainly have two kinds of fracture modes such as void’s nucleation-growth-coalescence for porous or pore-containing metals and the large homogeneous plastic deformation for ultrapure, pore-free, and single-phase
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Fig. 4. Microstructure and processed photographs of mild steel at different plastic strains; left: microstructure; right: processed photographs. (a) e = 0; (b) e = 0.050; (c) e = 0.161 (necking strain); (d) e = 0.186; (e) e = 0.254; (f) e = 0.286 and (g) e = 0.328.
metals. The symbol of the void nucleation just corresponds to appearance of the necking (deformation localization). However, the present Armco iron as an industry material can not arrive at the ultrapure level. Therefore, both of the materials used in the test have a mixed fracture mode of above routes; only the earlier plastic deformation period is different. It
Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513
Fig. 4 (continued)
Fig. 5. Microstructure at fracture edge for Armco iron.
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Fig. 6. Microstructure at fracture edge for mild steel.
Fig. 7. Evolution of shape factor U with plastic strain for two materials.
Fig. 8. Evolution of relative shape factor W with plastic strain for two materials.
can be observed from Figs. 5 and 6 that Armco iron appears a large plastic deformation before the necking happens; in contrast, the mild steel appears the necking through a shorter plastic deformation period.
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Fig. 9. Evolution of normalized shape factor D with plastic strain for two materials.
For porous or pore-containing metals, the damage behavior and constitutive equation have been extensively investigated as mentioned earlier. The Gurson model including the damage variable of void volume fraction is the most significant description from the necking to fracture since the microvoids roughly are not active before the necking. On the other hand, the nucleation of microvoids is directly resulted from the earlier plastic deformation and the evolution of microstructure. Unfortunately, the effect of microstructue evolution with plastic deformation on damage and fracture has not yet been paid great attention. By using the shape factor, the evolution behavior of microstructure (grains) can be applied from beginning of the plastic deformation up to fracture, which certainly includes the void’s nucleation-growth-coalescence procedure. Consequently, the damage evolution equation and constitutive equation based on the shape factor may be more accurately and comprehensively reflect the nature of damage and fracture than those based on the theory about nucleation and growth of microvoids. Since the present work only considered a special case of monotonic loading and a damage evolution with the longitudinal strain, this assumption needs to be examined by further investigation into the evolution at different stress states, especially in the triaxial stress state for more metal materials. 5. Conclusions The microstructure of ductile metal materials can be quantitatively characterized by using a mesoparameter of shape factor of grains. Both of Armco iron and mild steel have the same rule in the evolution of shape factor with plastic strain, the shape factor increasing with the increase of plastic deformation. The evolution rule of the normalized shape factor with plastic strain can be expressed by equation of D ¼ A þ Bee=C for both of Armco iron and mild steel. The evolution rule of shape factor with plastic strain can more accurately and entirely describe the microstructure evolution from beginning of the plastic deformation to fracture than that based on nucleation and growth of microvoids. Acknowledgement I would like to appreciate Prof. Huang Yuan, Department of Mechanical Engineering, Bergische Universität Wuppertal, Germany, to discuss the meaning of shape factor and its application in setting up a damage evolution equation. References [1] McClintock FA. A criterion for ductile fracture by the growth of voids. J Appl Mech Trans ASME 1968;35:363–71. [2] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxiaI stress fields. J Mech Phys Solids 1969;17:20l–17. [3] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth. I. Yield criteria and flow rules for porous ductile media. J Engng Mater Tech Trans ASME 1977;99:2–15. [4] Kameda J. A microscopic model for the void growth behavior. Acta Metall 1989;37:2067–76. [5] Zhen C, Zhou L. Mesomechanics and application on ductile fracture of metals. Beijing: Defence Industry Press; 1995. [6] Bate PS, Hutchinson WB. Grain boundary area and deformation. Scripta Mater 2005;52:199–203. [7] Li DS, Garmestan H, Schoenfeld S. Evolution of crystal orientation distribution coefficients during plastic deformation. Scripta Mater 2003;49:867–72. [8] Wadsack R, Pippan R, Schedler B. Structural refinement of chromium by severe plastic deformation. Fusion Engng Des 2003;66–68:265–9. [9] Korbel A, Bochniak W. Refinement and control of the metal structure elements by plastic deformation. Scripta Mater 2004;51:755–9. [10] Balasundaram A, Shan ZH, Gokhale AM, Graham S, Horstemeyer MF. Particle rotations during plastic deformation of 5086 aluminum alloy. Mater Charact 2002;48:363–9. [11] Furukawa M, Horita Z, Nemoto M, Langdon TG. The use of severe plastic deformation for microstructural control. Mater Sci Engng A 2002;324:82–9.
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
A novel damage variable to characterize evolution of microstructure with plastic deformation for ductile metal materials under tensile loading Z.M. Shi a,*, H.L. Ma a,b, J.B. Li a,c a
School of Materials Science and Engineering, Inner Mongolia University of Technology, Hohhot 010051, China Department of Mathematics, Taiyuan College of Technology, Taiyuan 030008, China c Department of Mechanics, Taiyuan University of Science and Technology, Taiyuan 030024, China b
a r t i c l e
i n f o
Article history: Received 23 May 2009 Received in revised form 23 November 2009 Accepted 25 December 2009 Available online 4 January 2010 Keywords: Armco iron Mild steel Microstructure Optical microscopy Plastic deformation Damage parameter
a b s t r a c t Damage and fracture of ductile metal materials are greatly associated with evolution of their microstructure under loading, which meso-dimensionally corresponds to grain deformation as well as nucleation, growth, and coalescence of microvoids in later local deformation. The evolution behavior of microstructure is important to realize ductile damage and fracture mechanism of materials. In the present work, a novel damage variable of shape factor of grains was put forward to quantitatively describe the character of microstructure; its evolution with the plastic deformation was built-up by microanalytical and mechanical experiments for Armco iron and mild steel tensile bars. The evolution rule based on the damage variable of shape factor is possible to be extended into all of ductile metal materials. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Ductile metal materials generally show three stages of elastic deformation, plastic deformation, and ultimate fracture under loading. Many researchers have studied their damage and fracture behaviors in terms of macro, meso, and micro scales, in which the mesodamage is the most interesting area. For most metal materials, the ductile fracture undergoes nucleation, extension, and coalescence of voids. McClintock [1] put forward a criterion by the growth of voids for the ductile fracture. Rice and Tracey [2] studied the enlargement behavior of voids in triaxiaI stress fields. Gurson [3] proposed a damage variable of void volume fraction and set up constitutive equations to describe nucleation and growth of existing voids for ductile porous metal materials. The Gurson model has been modified and widely used to study deformation and failure of metal materials containing microvoids. Kameda [4] developed a microscopic constitutive equation related to the growth of voids subjected to a hydrostatic tensile stress in basis of the mobility of dislocations and the thermally activated shear stress. Zhen and Zhou [5] proposed a mesoparameter of void extension ratio to describe the law of nucleation, growth, and coalescence of voids based on substantial experiments for BS4360-50D steel; they found that when the effective plastic strain is less than 0.10, much less microvoids with several micrometer (existing in as-received materials) can be observed; from about 0.14 plastic strain (approaching necking), the nucleation of voids can be found, and the corresponding void volume fraction is about 0.13%; the coalescence of microvoids starts at about 0.6 plastic strain. Bate and Hutchinson [6] found that the area of grain boundary non-linearly increases with the increase of the plastic strain in states of tensile, compression, torsion,
* Corresponding author. Tel./fax: +86 471 6576221. E-mail address: [email protected] (Z.M. Shi). 0013-7944/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2009.12.011
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513
and shear, in basis of an ellipsoid grain model, which is geometrically equivalent to the random equi-axed grain. The increase in area of grain boundary has a significant influence on mechanical properties of materials. The fracture of solid materials is an accumulation of microstructure variation at different measuring scales (micro and meso) such as crystal defects, microvoids, and grains. During the plastic deformation of a ductile metal material, there is a great evolution in its mesoscale, that is, the grains can be elongated, rotated, refined, or orientated [7–11]. In this case, the plastic deformation extent largely depends on the character of materials, such as type of materials, size of grains, as well as the content and distribution of secondary phase and inclusion particle. When the plastic strain arrives at a definite extent, the microvoids can preferentially nucleate around the inclusion particles [5]. Therefore, the plastic deformation directly affects nucleation and extension of microvoids. There is a necessity to clarify the relationship between the evolution of microstructure and plastic deformation so as to systematically describe the evolution procedure of the microstructure from beginning of plastic deformation up to fracture, in which the nucleation, growth, and coalescence of microvoids are certainly included. Consequently, how to define a damage variable that can describe the microstructure and its evolution develops into a significant issue. In this work, a shape factor of grains was selected as the damage variable which can be measured through the quantitative metallographic analysis, and its evolution equation with the plastic strain was established for ductile metal materials of Armco iron and mild steel under tensile loading. 2. Definition of damage variable for microstructure Ductile metal materials are composed of single-phase or multi-phase structures with an order of micrometer magnitude. Fig. 1 shows the sketch map for their microstructure. The shape and size of grains are essential parameters for describing the basic microstructure of materials. In the present work, three kinds of shape factors were adopted.
Fig. 1. Sketch map of microstructure of metal materials; a: matrix; b: grain boundary; c: second phase: (a) single phase and (b) dual phase.
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513 Table 1 The symbol, meaning, and expression of three kinds of shape factors. Symbol
Meaning
U
Shape factor
Expression P L U¼P
Value range
W
Relative shape factor
U0 W ¼ U U0
WP0
D
Normalized shape factor
D ¼ UUfUU00
06D61
U>0
A
2.1. Shape factor U The shape factor U is expressed by
P
U¼P
L A
ð1Þ
where L is the length of boundary of grains, which can be statistically measured for the longitudinal section of samples in microanalytical experiments, and A is the area that is preset by the field of microscope. The boundary length in a definite area should be increased with their elongation under tensile loading. Here, only the total length of boundary of phases, so called internal boundary of materials, was considered. Since the shape factor includes the effect of shape and size of grains, it can overall represent the microstructure character for as-received and deformed microstructure of materials. 2.2. Relative shape factor W The relative shape factor W is defined as
W¼
U U0 U0
ð2Þ
where U is the shape factor at a plastic strain and U0 is the shape factor for as-received materials. The relative shape factor is a dimensionless parameter, which can be used to describe relative increment of the shape factor. 2.3. Normalized shape factor D The shape factor can also be normalized into another dimensionless form
D¼
U U0 Uf U0
ð3Þ
where Uf is the shape factor at the fracture strain. It can be noticed from the following description that D has an explicit physical significance. (1) If U = U0, D ¼ 0, the material is located at the non-damage state (as-received state); (2) If U = Uf D ¼ 1, the material is fractured. (3) If U0 < U < Uf, 0 < D < 1, the material is just located in a plastic deformation state. In consideration of the clarity, three kinds of shape factors are summarized in Table 1. 3. Materials and methods 3.1. Materials Armco iron and mild steel rolled round bars with a diameter of 20 mm were used as raw materials. They were annealed at 800 °C for 3 h to eliminate residual stress and recover the microstructure into equi-axed grains. Armco iron is a kind of soft metal material with a single phase of ferrite microstructure, which is mainly used as magnetic and electronic materials such as relay, anodes of vacuum tube. Mild steel, with a biphase microstructure of ferrite and pearlite, has moderate strength, plasticity, and toughness, as well as excellent welding and hot–cold forming ability, which is widely used to manufacture assemblies such as bolts, pull rods, rockers, shafts, and frameworks. Their chemical compositions are listed in Table 2.
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Table 2 Main chemical composition of materials. Material
Armco iron Mild steel
Chemical composition, wt.% C
Si
Mn
P
S
Fe
<0.040 0.258
<0.003 0.284
<0.070 0.661
<0.008 <0.045
<0.006 <0.045
Balance Balance
Fig. 2. Stress–strain curves of Armco iron and mild steel.
3.2. Specimens The annealed bars were mechanically machined into standard tensile specimens, the working segment is with a diameter of 10 mm and a gauge length of 50 mm. The surface of specimens was polished to eliminate nicks. 3.3. Tensile test The tensile test was conducted on a mechanical machine (CSS-WAWDL type, CRITM), with a displacement loading mode, the displacement rate was controlled to be 0.2 mm/min. An electric extensometer (CBT-17 type, CRITM), with a gage length of 25 mm and measuring range of 10 to +25 mm, was used to detect the displacement. Fig. 2 shows the measured stress–strain curves of Armco iron and mild steel. Table 3 shows mechanical properties of the materials. From Fig. 2 and Table 3, it can be seen that Armco iron has lower Young’s modulus, yield strength, fracture strength, and a larger plasticity than those of mild steel, which are greatly dependant on their microstructure, chemical composition, and heat treatment state. In order to obtain the microstructure photographs at different plastic strains, the specimens were, respectively, loaded and then unloaded at desired strain points, which roughly distribute from yield point to the fracture. 3.4. Observation and analysis of microstructure The unloaded specimens were cut along their longitudinal plane by an electric park machine with a molybdenum wire of 0.2 mm in diameter. The longitudinal plane was polished into metallographic surface. An optical microscope (Leica LM/DM type) equipped with an image analysis software (SISC IAS V8.0) was adopted to observe and analyze the microstructure. For quantitative analysis to the microstructure photographs, it needs to tailor the effective characters such as grain boundary and secondary phase so as to obtain clear images suitable to be recognized and calculated by computer. Therefore, a series of photographic processes were used, such as gray level transfer, image filtering, image splitting, edge strengthening, image binarizing, boundary tailoring, and boundary linearizing.
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Z.M. Shi et al. / Engineering Fracture Mechanics 78 (2011) 503–513 Table 3 Mechanical parameters of materials. Sample
Armco iron
Mild steel
Young’s modulus, GPa Yield strength, MPa Tensile strength, MPa Longitudinal elongation, % Necking strain
185.45 177.26 405.72 50.02 0.263
208.21 300.54 483.67 34.80 0.161
Because of the random distribution of grains, U is measured and calculated against 5–10 of micrographs. For the uniformly deformed samples, 10 of micrographs were taken along the longitudinal symmetry axis of tensile bars; for the necking samples, 5–6 of micrographs were taken against the field around the necking transverse symmetry plane and the longitudinal axis. 4. Results and discussion 4.1. Microstructure and processed images at different plastic strains The typical microstructure images of two materials at different plastic strains are shown in Figs. 3 and 4. The left and right parts, respectively, are the real and processed ones by the image software. It can be found from Figs. 3 and 4 that, with the increase of plastic strain, the grains are gradually elongated along the tensile direction (longitudinal axis). After necking, the elongation of grains become more remarkable. This is because, Armco iron with single ferrite phase has higher purity. In contrast, the mild steel consisted of ferrite and brittle secondary phase of pearlite (mixture of ferrite and carbide) has relative low purity. The different microstructure and purity result in different damage and fracture procedures. For Armco iron, the nucleation and growth of microvoids around small amounts of inclusions are delayed due to its significant dislocation slide in the soft ferrite grains. In contrast, for the mild steel, due to the existing many inclusions and the relative weak boundary combination between ferrite and carbide phases, the dislocation slide is limited to some extent, so the earlier nucleation and growth of microvoids around inclusions and relative weak boundary are promoted. Once the microvoids start to nucleate and grow, the homogeneous plastic deformation will come to cease and the deformation turns to the local one, i.e, the mode of nucleate, growth, and coalescence of microvoids, indicating appearance of necking and weakening of materials. The difference in damage and fracture procedure directly results in different effects on elongation and refinement of grains. Figs. 5 and 6 show the microstructure images at the fracture edge for Armco iron and mild steel. It can be observed that the grains at fracture edge and near surface are significantly refined by the serve plastic shear deformation, especially for Armco iron. This indicates that the stress state also has a strong effect on the mesoplastic deformation process of microstructure. 4.2. Evolution of damage variables with plastic strain Fig. 7 shows the evolution of shape factor U with plastic strain for two materials. It can be noticed that the shape factors of both materials have a similar change trend with increasing plastic strain. But the difference is that Armco iron has a smaller shape factor and it presents a slow increase, while the mild steel has a larger one and it presents a rapid increase. This is because the mild steel has a smaller granularity and hard phase of carbide than that of Armco iron. Therefore, by using the damage variable of shape factor, the real microstructure character of materials can be quantitatively described; its evolution with plastic strain discloses the response of materials in mesoscale resulted from the macro deformation. Fig. 8 displays the evolution of relative shape factor W with plastic strain for two materials. The variations of the relative shape factor with plastic strain are very similar for two materials. By using the relative shape factor, the relative increments of the shape factor are entirely coincident for both materials. This is why the relative shape factor is adopted. Fig. 9 illustrates the evolution of the normalized shape factor D with plastic strain for two materials. From these figures, it can be concluded that, the mild steel has a shorter and earlier plastic deformation stage and a rapid damage procedure than those of Armco iron. 4.3. Fitting equations to the evolution rules The exponential growth function is used to fit curves against the data in Fig. 9. The fitted equations are shown as follows. For Armco iron
D ¼ 0:065 þ 0:002ee=0:084
ð4Þ
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Fig. 3. Microstructure and processed photographs of Armco iron at different plastic strains; left: microstructure; right: processed photographs. (a) e = 0; (b) e = 0.114; (c) e = 0.263 (necking strain); (d) e = 0.350; (e) e = 0.464; (f) e = 0.474 and (g) e = 0.500.
For mild steel
D ¼ 0:043 þ 0:068ee=0:123
ð5Þ
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Fig. 3 (continued)
The earlierabove mentioned equations can be summarized into general form.
D ¼ A þ Bee=C
ð6Þ
where A, B, and C are constants of materials. 4.4. Discussion to the meaning of shape factor 4.4.1. Rationality describing microstructure and its evolution by shape factor The shape factor is adopted to describe the microstructure of metal materials. It can be found that the shape factor mainly concerns with the perimeter of grain boundaries (matrix and secondary phase) in a defined area, and it receives much influence from the size, shape, and the plastic deformation extent. With an increase in plastic strain, the grains are elongated, resulting in an increase in grain boundary. Therefore, the shape factor can overall represent the microstructure character for as-received and deformed materials and can quantitatively reflect evolution rule of the microstructure. Therefore, the shape factor may be the most suitable mesodamage variable for describing microstructure character and its evolution procedure with plastic deformation. 4.4.2. Possibility describing damage evolution equation and constitutive equation by using shape factor as damage variable The ductile metal materials mainly have two kinds of fracture modes such as void’s nucleation-growth-coalescence for porous or pore-containing metals and the large homogeneous plastic deformation for ultrapure, pore-free, and single-phase
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Fig. 4. Microstructure and processed photographs of mild steel at different plastic strains; left: microstructure; right: processed photographs. (a) e = 0; (b) e = 0.050; (c) e = 0.161 (necking strain); (d) e = 0.186; (e) e = 0.254; (f) e = 0.286 and (g) e = 0.328.
metals. The symbol of the void nucleation just corresponds to appearance of the necking (deformation localization). However, the present Armco iron as an industry material can not arrive at the ultrapure level. Therefore, both of the materials used in the test have a mixed fracture mode of above routes; only the earlier plastic deformation period is different. It
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Fig. 4 (continued)
Fig. 5. Microstructure at fracture edge for Armco iron.
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Fig. 6. Microstructure at fracture edge for mild steel.
Fig. 7. Evolution of shape factor U with plastic strain for two materials.
Fig. 8. Evolution of relative shape factor W with plastic strain for two materials.
can be observed from Figs. 5 and 6 that Armco iron appears a large plastic deformation before the necking happens; in contrast, the mild steel appears the necking through a shorter plastic deformation period.
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Fig. 9. Evolution of normalized shape factor D with plastic strain for two materials.
For porous or pore-containing metals, the damage behavior and constitutive equation have been extensively investigated as mentioned earlier. The Gurson model including the damage variable of void volume fraction is the most significant description from the necking to fracture since the microvoids roughly are not active before the necking. On the other hand, the nucleation of microvoids is directly resulted from the earlier plastic deformation and the evolution of microstructure. Unfortunately, the effect of microstructue evolution with plastic deformation on damage and fracture has not yet been paid great attention. By using the shape factor, the evolution behavior of microstructure (grains) can be applied from beginning of the plastic deformation up to fracture, which certainly includes the void’s nucleation-growth-coalescence procedure. Consequently, the damage evolution equation and constitutive equation based on the shape factor may be more accurately and comprehensively reflect the nature of damage and fracture than those based on the theory about nucleation and growth of microvoids. Since the present work only considered a special case of monotonic loading and a damage evolution with the longitudinal strain, this assumption needs to be examined by further investigation into the evolution at different stress states, especially in the triaxial stress state for more metal materials. 5. Conclusions The microstructure of ductile metal materials can be quantitatively characterized by using a mesoparameter of shape factor of grains. Both of Armco iron and mild steel have the same rule in the evolution of shape factor with plastic strain, the shape factor increasing with the increase of plastic deformation. The evolution rule of the normalized shape factor with plastic strain can be expressed by equation of D ¼ A þ Bee=C for both of Armco iron and mild steel. The evolution rule of shape factor with plastic strain can more accurately and entirely describe the microstructure evolution from beginning of the plastic deformation to fracture than that based on nucleation and growth of microvoids. Acknowledgement I would like to appreciate Prof. Huang Yuan, Department of Mechanical Engineering, Bergische Universität Wuppertal, Germany, to discuss the meaning of shape factor and its application in setting up a damage evolution equation. References [1] McClintock FA. A criterion for ductile fracture by the growth of voids. J Appl Mech Trans ASME 1968;35:363–71. [2] Rice JR, Tracey DM. On the ductile enlargement of voids in triaxiaI stress fields. J Mech Phys Solids 1969;17:20l–17. [3] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth. I. Yield criteria and flow rules for porous ductile media. J Engng Mater Tech Trans ASME 1977;99:2–15. [4] Kameda J. A microscopic model for the void growth behavior. Acta Metall 1989;37:2067–76. [5] Zhen C, Zhou L. Mesomechanics and application on ductile fracture of metals. Beijing: Defence Industry Press; 1995. [6] Bate PS, Hutchinson WB. Grain boundary area and deformation. Scripta Mater 2005;52:199–203. [7] Li DS, Garmestan H, Schoenfeld S. Evolution of crystal orientation distribution coefficients during plastic deformation. Scripta Mater 2003;49:867–72. [8] Wadsack R, Pippan R, Schedler B. Structural refinement of chromium by severe plastic deformation. Fusion Engng Des 2003;66–68:265–9. [9] Korbel A, Bochniak W. Refinement and control of the metal structure elements by plastic deformation. Scripta Mater 2004;51:755–9. [10] Balasundaram A, Shan ZH, Gokhale AM, Graham S, Horstemeyer MF. Particle rotations during plastic deformation of 5086 aluminum alloy. Mater Charact 2002;48:363–9. [11] Furukawa M, Horita Z, Nemoto M, Langdon TG. The use of severe plastic deformation for microstructural control. Mater Sci Engng A 2002;324:82–9.