Evolution of cylindrical void and elastoplastic constitutive description

Computational Materials Science 49 (2010) S336–S340

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Evolution of cylindrical void and elastoplastic constitutive description B. Chen a,b,*, Q. Yuan a, J. Luo a, H. Liu a, J.H. Fan a,c a

College of Resources and Environmental Science, Chongqing University, Chongqing 400044, China Key Laboratory for the Exploitation of Southwest Resources and Environmental Disaster Control Engineering, Ministry of Education, Chongqing University, Chongqing 400044, China c School of Engineering, Alfred University, Alfred, NY 14802, USA b

a r t i c l e

i n f o

Article history: Received 10 October 2009 Received in revised form 31 March 2010 Accepted 2 April 2010

Keywords: Pearlitic steel Cylindrical void Void evolution Constitutive equation Elastoplasticity

a b s t r a c t Assuming the voids in a pearlitic steel are cylindrical, a cylindrical void-cell model is presented. From the analysis of the void-cell model a void evolution equation is obtained. Defining a new intrinsic time and the softening function related to the void evolution, and introducing them into an endochronic constitutive equation, the constitutive equation involving void evolution is derived. The corresponding numerical algorithm and the finite element approach are offered, which are applied to analyse the stress and porosity distributions of the unnotched and notched cylindrical specimens of a pearlitic steel. The analytical result of the unnotched specimen shows that the porosity of the specimen increases with the increase of its plastic deformation, which is consistent with the experimental data. The analytical result of the notched specimen reveals that the stress of considering void evolution is larger than that of no considering void evolution. Both the stress and porosity reach their maximum at the notch root. The latter agrees with the experimental result. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Pearlitic steel possesses excellent mechanical properties, which make that the steel has increasedly being used as rail steel [1–3]. In the application of the steel, it has been found that one of the main failure modes of the steel is the nucleation, growth and coalescence of the microvoids in the steel, which are often called as the void evolution or void damage [4–7]. The researches on the rule of the void evolution and the effect of the void evolution on the mechanical behavior of the steel are important to the more rational application and well development of the steel. It is known that the voids usually distribute randomly in a material if it includes voids [4,5]. In order to study the effect of the void evolution on the mechanical properties of materials, a method viable and used frequently is to research the behavior of a simple representative cell of the materials. In such analyses, Gurson [5] made a pioneering contribution. He assumed that the voidmatrix aggregate of a material could be represented with a rigidplastic material cell with a single void at its center, and the void volume fraction (porosity) f of the cell equals that of the aggregate. Then he obtained the constitutive model for the solids containing circular-cylindrical or spherical voids. Needleman [6] presented an elastic–plastic constitutive description and performed a finite element study based on a two-dimensional model for the materials

* Corresponding author at: College of Resources and Environmental Science, Chongqing University, Chongqing 400044, China. E-mail address: [email protected] (B. Chen). 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.04.001

containing voids. Licht and Suquet [7] investigated the growth of a cylindrical void in a nonlinear power law viscous material and obtained a closed solution for the void growth. Tracey [4] derived the upper and lower bounds for the growth rate of the cylindrical voids in a finite volume of a strain-hardening matrix. Pan and Huang [8] considered the effect of circular-cylindrical void growth on the constitutive relations for viscoplastic materials. Reusch et al. [9] extended the Gurson’ model to the case of isotropic ductile damage and crack growth. Mariani and Corigliano [10] built an orthotropic constitutive model for porous ductile media based on the micromechanical analysis of a cylindrical representative volume element. In this paper, a micro–macroscopic combination method is used to explore the constitutive behavior of a pearlitic steel. A void evolution equation was obtained based on the analysis on a cylindrical void-cell model. The obtained evolution equation then was embedded in an elastoplastic constitutive equation through defining a new intrinsic time and the softening function related to the void evolution, the constitutive equation involving void evolution are obtained. The corresponding finite element procedure of the constitutive equation is developed and used to analyse the stress and porosity distributions of a pearlitic steel. It shows that the computed results agree satisfactorily with experimental data. 2. Void evolution equation A representative material element of a pearlitic steel is shown in Fig. 1. The matrix of the material element is assumed homogeneous and incompressible, and many microvoids distribute stochastically

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p

Matrix volume (V m)

f

p Void volume (V v)

f f

p

f

Fig. 1. Material element with voids.

in the element. The volumes of the matrix and the void in the element are denoted by V m and V v , respectively. The total volume of the element V ¼ V v þ V m . Because cylindrical voids, which may result from cylindrical inclusions (e.g., sulfide grain) which decohere from their matrix during deformation, are often found in the steel, it is assumed that the voids in the material element possess cylindrical shape. Furthermore, the material element including the cylindrical voids can be simplified as a cylindrical void-cell model [5], which is a thick-walled cylinder containing a cylindrical void at its center, as shown in Fig. 2. In this void-cell model, the initial and current inner radii of the cell are a0 and a, the initial and current outer radii are b0 and b, the initial and current lengths are l0 and l, and the initial and current radii at an arbitrary point in the matrix are R and r, respectively. From Fig. 2, one has

2pr  dr  l ¼ 2pR  dR  l0 ;

ð1Þ

integrating Eq. (1) gives 2

2

ðb  a2 Þl ¼ ðb0  a20 Þl0 :

ð2Þ

From the definition of the volume fraction of void, the initial and current volume fraction of the void can be expressed as 2

f0 ¼ a20 =b0 ;

2

f ¼ a2 =b :

ð3a; bÞ

Introducing logarithmic strain, one has

b=b0 ¼ expðe11 Þ;

l=l0 ¼ expðe33 Þ:

ð4a; bÞ

Combining Eqs. (2), (3) and (4), one obtains

f ¼ 1  ð1  f0 Þ expðekk Þ:

Eq. (5) is the evolution equation of the void, which shows that the current void volume fraction is related to the initial void volume fraction and volumetric strain. It can be seen in Fig. 3 that the void volume fraction increases if either the initial void volume fraction or the volumetric strain increases.

l0

3. Constitutive description The macroscopic stress and strain in the void-cell model are denoted by Rij and Eij , and the corresponding microscopic stress and strain are rij and eij , respectively. The macroscopic deformation rate of the void-cell model can be defined in terms of the velocity field at the outer surface of the void-cell model [5]

1 E_ ij ¼ V

Z S

1 ðv i nj þ v j ni ÞdS; 2

ð6Þ

where v i is the microscopic velocity field, S is the outer surface of the void-cell, nj is the unit outward normal on S. For the microscopic stress in the cell, the following boundary conditions should be satisfied in a Cartesian reference frame [5,11]

pi ¼ rij nj ¼ Rij nj ; pi ¼ rij nj ¼ 0;

ðr ¼ bÞ;

ð7aÞ

ðr ¼ aÞ;

ð7bÞ

where pi is external traction applied on the outer surface of the model. The microscopic velocity field in the model can be expressed as:

v i ¼ E_ ij xj ;

ðr ¼ bÞ;

ð8Þ

where xj is the local orthogonal principal axes. From Eq. (8) following boundary velocity field in cylindrical coordinate system (r, h, x3) can be obtained as

ð9Þ

Furthermore, the corresponding microscopic strain rate field e_ ij can be expressed as

e_ r ¼ dv r =dr; e_ h ¼ v r =r; e_ 3 ¼ dv 3 =dx3 :

ð10Þ

Considering an axisymmetric deformation, one has

E_ 11 ¼ E_ 22 – 0;

E_ 33 ¼ wE_ 11 ;

E_ ij ¼ 0 ði – jÞ;

ð11Þ

where the parameter w is the strain constraint function which relates the effect of stress triaxiality on void growth, and 0 6 w 6 2 [12]. Making use of these relationships and noticing the condition of matrix incompressibility

x3

x1

Fig. 3. Void volume fraction vs. volume strain under different initial void volume fraction.

m1 jb ¼ E_ 11 b cos h; m2 jb ¼ E_ 22 b sin h; m3 jb ¼ E_ 33 x3 : ð5Þ

e_ 11 þ e_ 22 þ e_ 33 ¼ 0; l

a a0 b b0 r rR

ε

x2

θ

Fig. 2. Cylindrical void-cell model.

ð12Þ

one obtains:

" # 2 1 b w  2 ð2  wÞ E_ 11 ; 2 r " # 2 1 b w þ 2 ð2  wÞ E_ 11 ; ¼ 2 r

e_ r ¼

e_ h e_ z ¼ wE_ 11 :

ð13a; b; cÞ

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Endochronic elastoplastic theory proposed by Valanis [13] provides an alternative approach to the classical plasticity for the description of the inelastic behavior of history-dependent materials. This theory does not require the notion of yield surface and the specification for the loading or unloading rules [13]. Because of these merits, the endochronic elastoplastic theory is to be used to explore the elastoplastic constitutive behavior of the pearlitic steel in this research. The endochronic elastoplastic constitutive equation proposed by Valanis [13] can be written as

s¼

Z

z

qðz  z0 Þ

0

dep 0 0 dz ; dz

trðdrÞ ¼ 3KtrðdeÞ;

ð14a; bÞ

where

qðzÞ ¼

n X

C r ear z ;

dz ¼ df=FðfÞ;

ð15a; bÞ

r¼1

df ¼ kdek;

dep ¼ de  ds=2G;

ð16a; bÞ

r and s denote respectively the tensors of the microscopic stress and the deviatoric component of the stress, whereas e, e and ep the tensors of microscopic strain, deviatoric component and plastic component of the strain, qðzÞ is a the kernel function, G and K are respectively shearing and bulk moduli, C r and ar material constants related to irreversible deformation, FðfÞ the hardening function related to the hardening behavior of the material undergoing plastic deformation, f and z the intrinsic time measure and the intrinsic time scale, respectively, kAk denotes the Euclidean norm of tensor A. Substituting Eq. (13) into Eq. (16a), the increment of intrinsic time measures can be expressed as

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 4 b ð2  wÞ2 þ 3w2 r 4 de11 = 2r 2 :

df ¼

ð17Þ

Considering that the intrinsic time scale is not only related to the irreversible deformation of the material but also to the evolution of void, a new intrinsic time can be defined as

d^z ¼ df=FðfÞgðf Þ;

ð18Þ

where gðf Þ is the softening function related to the softening behavior of the material undergoing void evolution. For simplicity, FðfÞ and gðf Þ are given the following simple forms without considering strain rate effect

FðfÞ ¼ 1 þ b1 fc1 ;

gðf Þ ¼ 1 þ b2 f c2 ;

ð19a; bÞ

where b1 , b2 , c1 , and c2 are material constants. Substituting the incremental form of dep =d^z into Eq. (14) [14], an incremental form of the microscopic constitutive equation can be derived as

Ds ¼

3 X

DsðrÞ ;

DsðrÞ

r¼1

  C r Dep  sðrÞ ð^zn Þ ; ¼ ð1  expðar D^zÞÞ ar D^z

ð20a; bÞ

where sðrÞ ð^zÞ ¼ sðrÞ ð^zn Þ þ DsðrÞ , ^zn is the intrinsic time scale after nth increment of loading and can be determined with Eq. (18). Furthermore, by setting

kr ¼

A¼

1  expðar D^zÞ ; ar D^z n X

kr C r ;

B¼

r¼1

n X

ð21Þ

Ds ¼ 2Gp De þ T p BD^z; where

Tp ¼

1 ; A 1 þ 2G

D^z ¼

Dep : Dep : F ð^zÞg2 ðf ÞD^z

2Gp ¼ AT p ;

ð22a; bÞ

r¼1

one can derive the following incremental constitutive equation

Ds ¼ ADep þ BD^z: Keeping Eq. (16b) in mind, Eq. (23) can be written as

ð23Þ

ð25a; bÞ

ð26Þ

2

Ignoring the effect of voids on the constitutive equation (f = 0), it can prove that the constitutive equations (Eqs. (23) and (24)) reduce to the incremental form of the nonclassical constitutive equation given by Peng and Fan [14]. Moreover, the Chaboche’s constitutive law for back stress can also be obtained as its special case when gðf Þ ¼ 1 and FðfÞ is constant. From the Eqs. (23) and (24) one can derive the following incremental elastoplastic constitutive equation involving microvoids in a matrix form

fDrg ¼ ½Dep fDeg;

ð27Þ

where

½Dep  ¼ ½De  þ

2ðG  Gp Þ ½D2 : H

ð28Þ

For axisymmetric problem, one has

fDrg ¼ ðDrr ; Drh ; Drz ; Drrz ÞT ;

ð29Þ

T

fDeg ¼ ðDer ; Deh ; Dez ; 2Derz Þ ; 3 2 C2 C2 0 C1 6 C1 C2 0 7 7 6 ½De  ¼ 6 7; 4 C1 0 5

ð30Þ

ð31Þ

sym: Gp 4G 2G Tp  þ p ; C2 ¼ K  p ; C3 ¼ C1 ¼ K ; 2 3 3 2GF ðfÞg2 ðf ÞDz Tp Depij Bij ; H ¼1þ 2 2GF ðfÞg2 ðf ÞDz ½D2  ¼ C 3 ðBr ; Bh ; Bz ; Brz ÞT ðDepr ; Deph ; Depz ; Deprz Þ:

ð32Þ ð33Þ ð34Þ

Because the matrix of the porous material is homogeneous, the homogenization principle can be used in the transition between microscopic and macroscopic quantities [5]. Letting U and / be the macroscopic and the microscopic potential functions, respectively, the macroscopic stresses can be expressed as [5]

Rij ¼

@U 1 ¼ @Eij V

Z Vm

@/ 1 dV ¼ @Eij V

Z Vm

skl

@ ekl dV: @Eij

ð35Þ

In the presented constitutive equation, the G and K are the shearing and volumetric moduli of the material, respectively, which can be obtained from engineering handbook. The C r and ar are the material constants from the endochronic elastoplastic theory, which control the stress response related to irreversibly elastoplastic deformation. They can be determined by a simple tensile test [14]. Taking the plastic incompressibility condition, the tensile stress can be expressed in terms of C i ; ai as

r¼

3  pffiffiffiffiffiffiffiffi  pffiffiffiffiffiffiffiffi X Cr  3=2 1  exp  3=2ar ep : r¼1

kr ar sðrÞ ð^zn Þ;

ð24Þ

ar

ð36Þ

Given a set of measured ðri ; epi Þ, we can determine the parameters C r and ar with the least squares approach [14]. The b1 and c1 are the material constants reflecting the material hardening undergoing plastic deformation, which can be determined with the former segment of the tensile curve of the material and the nonlinear curve fitting program. The b2 and c2 are the material constants involving material softening undergoing void evolution

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B. Chen et al. / Computational Materials Science 49 (2010) S336–S340

for the material, which can be confirmed through the latter segment of the tensile curve of the material and the identical nonlinear curve fitting program.

Table 1 The main material parameters in the analysis. G(GPa)

K(GPa)

C1, C2, C3 (GPa)

a1 ; a2 ; a3

b1 ; c1

b2 ; c 2

80

159

47,860, 5328, 465

1620, 274, 26

1.6, 1.2

1.3, 1.1

4. Comparison between the theoretical and experimental results

3.5

Experimental Calculated

3 2.5

f /%

The corresponding finite element algorithm of the presented constitutive model is put forward and used to compute two examples of the cylindrical specimens of a pearlitic steel. One is the porosity–deformation relationship of an unnotched cylindrical specimen and the other is the distributions of the stress and the porosity on the minimum cross section of a notched cylindrical specimen. The geometry of the unnotched cylindrical specimens is 200 mm in length, and 10 mm in diameter. The geometry of the notched cylindrical specimen is shown in Fig. 4. The upper and right quarter of the two kinds of the specimens is taken for the analyses due to the symmetry of the problems. The eight-node isoparametric element with 2  2 Gaussian points is adopted in the finite element algorithm. The finite element meshes and the boundary conditions of the two problems are shown in Figs. 5 and 6, respectively. The axial displacement at the ends of the specimens is prescribed as 0.02 mm per increment, and no radial constraint is applied to the surface of the specimens. The main material parameters adopted in the analyses can be seen in the Table 1. The computed results are compared with experimental data, which are obtained from tensile tests performed on an Instron 1342 servo-hydraulic material testing system. In order to obtain experimentally the relationship between the porosity and deformation, a set of cylindrical specimens is loaded to different levels of stresses and then unloaded. The specimens are then cut off at the minimum cross section and the cut-sections are polished for the observation of the porosity. A photointerpreter as well as the quantitative metallographic method are used to analyse the proportion of the porosity, due to the stochastic distribution of the voids. Fig. 7 shows the relationship between the void volume fraction (porosity) and the plastic deformation obtained by both experimental observation and computation. It can be seen from Fig. 7 that the porosity increases with the increase of the plastic defor-

2 1.5 1 0.5 0

0

0.2

0.4

0.6

ε p /% Fig. 7. Void volume fraction f vs. plastic strain

ep (unnotched specimen).

Fig. 8. Scanning electron micrograph of fractured section (unnotched specimen).

120 R1 10

100 200

Fig. 4. Notched cylindrical specimen (mm).

σz

σz / MPa

12

100 80 60 40 20

No considering void Considering void

0

r

0

Fig. 5. Finite element mesh and boundary condition of unnotched cylindrical specimen.

1

2

3

4

5

r /mm Fig. 9. Stress distribution along the radius of smallest section (notched specimen).

σz

r

Fig. 6. Finite element mesh and boundary condition of notched cylindrical specimen.

mation by an exponential law and that the computed results agree with the experimental data. Fig. 8 is the metallograph at the fracture section of the specimen observed with a scanning electronic microscope, in which many dimples can be observed. These dimples are the roots of voids, indicating the ductile fracture of the material. Fig. 9 shows the computed stress distributions along

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B. Chen et al. / Computational Materials Science 49 (2010) S336–S340

4 3.5 3

f /%

2.5 2 1.5 1

Experimental

0.5 0

Calculated 0

1

2

3

4

5

r /mm Fig. 10. Porosity distribution along the radius of the smallest section (notched specimen).

the radius of the smallest section of the notched cylindrical specimen considering or no considering void evolution. It shows that the stress of considering void evolution is larger than that of no considering void evolution. The stress reaches its maximum at the root of the notch and decrease rapidly toward the center of the specimen. Fig. 10 shows the computed and experimental results of the distribution of the porosity for the notched specimen. It can be seen from Fig. 10 that the porosity also reaches its maximum at the root of the notch and that the computed and the experimental results are well consistent. 5. Conclusions Assuming the voids in pearlitic steel are cylindrical, a cylindrical void-cell model is presented. Based on the analysis on the void-cell model, a void evolution equation is obtained, which reflects that the current void volume fraction increases with the increase of the initial void volume fraction or volumetric strain. Define a new intrinsic time and the softening function related to the void

evolution, the incremental form of the endochronic constitutive equation involving void evolution is derived. The corresponding numerical algorithm and the finite element approach are offered, which are applied to analyse the stress and porosity distributions of the unnotched and notched cylindrical specimens of a pearlitic steel. The analytical result on the unnotched specimen shows that the porosity of the specimen increases with the increase of its plastic deformation by an exponential law, which is consistent with the experimental data. The analytical result on the notched cylindrical specimen indicates that the stress of considering void evolution is larger than that of no considering void evolution. The stress reaches its maximum at the root of the notch. The analytical result also reveals that the porosity of the notched specimen also reaches its maximum at the root of the notch, which agrees with the experimental data. Acknowledgement The authors gratefully acknowledge the financial supports to this work from the Natural Science Foundations of China (Grant No. 10872221). References [1] J. Toribio, E. Ovejero, Materials Science and Engineering A A234 (1997) 579– 582. [2] M. Zelin, Acta Materialia 50 (2002) 4431–4447. [3] G. Sheng, J. Fan, X. Peng, B. Jiang, Iron and Steel 33 (1998) 35–39. [4] D.M. Tracey, Engineering Fracture Mechanics 3 (1971) 301–316. [5] A.L. Gurson, ASME Journal of Engineering Materials and Technology 99 (1977) 2–15. [6] A. Needleman, Journal of Applied Mechanics 39 (1972) 964–970. [7] C. Licht, P. Suquet, Archives of Mechanics 40 (1988) 741–757. [8] K. Pan, Z. Huang, International Journal of Damage Mechanics 3 (1994) 87–106. [9] F. Reusch, B. Svendsen, D. Klingbell, Computational Material Science 26 (2003) 219–229. [10] S. Mariani, A. Corigliano, International Journal of Solids and Structures 38 (2001) 2427–2451. [11] Z.P. Wang, Q. Jiang, Journal of Applied Mechanics 64 (1997) 503–509. [12] Z. Gao, B. Chen, J. Fan, Journal of CQU 3 (1996) 1–8. [13] K.C. Valanis, Archives Mechanics 25 (1971) 517–551. [14] X. Peng, J. Fan, Computers and Structures 47 (1993) 313–320.