Growth and coalescence of non-spherical voids in metals deformed at elevated temperature

Available online at www.sciencedirect.com

International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

Growth and coalescence of non-spherical voids in metals deformed at elevated temperature Helmut Kl%ockera , Viggo Tvergaardb;∗ a

b

Ecole Nationale Superieure des Mines, 158 Cours Fauriel, 42000 Saint-Etienne, France Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, Building 404, Nils Koppels Alle, DK-2800 Kgs. Lyngby, Denmark Received 20 June 2002; received in revised form 22 September 2003; accepted 13 October 2003

Abstract Void growth and coalescence in ductile materials deformed at elevated temperatures is analyzed by 1nite element calculations and by an analytical approach. The void growth model is based on an extension of Gurson’s model to elastic–viscoplastic materials. Coalescence is treated as transition to uniaxial straining in the intervoid ligament by micromechanical analysis. A very simple closed form for the stress in the intervoid ligament at coalescence is given. This closes the micromechanical analysis. Finally, a heuristic estimate (based on the 1nite element analysis) for the overall strain at void coalesce is given. ? 2003 Elsevier Ltd. All rights reserved.

1. Introduction When metals are deformed at high temperatures the active fracture mechanism depends strongly on the rate of deformation, as has been discussed by Ashby et al. [1]. At high strain rates and high stress levels, failure occurs by transgranular creep or ductile fracture. Voids are 1rst nucleated by decohesion or cracking of second-phase particles and then grow until they eventually coalesce to form a macroscopic crack [2]. Early models describe the growth of a single void in an in1nite elastic plastic solid [3,4] or an in1nite viscoplastic solid [5,6]. Finite void volume fractions were addressed 1rst by the plane strain analysis of Needleman [26] representing a square array of cylindrical voids. Gurson [7] determined an analytical expression for the stress potential of a spherical volume of elasto-plastic material containing a concentric spherical void, and Tvergaard [8] used an axisymmetric cell model to represent a periodic array of spherical voids. Leblond et al. [9] used an extension of the Gurson ∗

Corresponding author. Tel.: +45-4525-4273; fax: +45-4593-1475. E-mail address: [email protected] (V. Tvergaard).

0020-7403/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijmecsci.2003.09.018

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model to viscoplastic materials containing spherical voids, while the behaviour of an ellipsoidal volume of perfectly plastic material containing an confocal ellipsoidal void was analyzed by Gologanu et al. [10–12]. Garajeu [13] determined the overall behaviour of a viscoplastic (Norton-like) material containing prolate ellipsoids. A modi1ed Gurson model for porous ductile materials was used by Tvergaard and Needleman [14] to show that the void coalescence implies the localization of plastic How in the intervoid ligament and a transition to uniaxial straining of the unit cell. Koplik and Needleman [15] used similar axisymmetric 1nite element cell-model calculations to determine the critical void volume fraction fc , at which the coalescence process starts. Cell-model calculations were also used by Steglich and Brocks [16] to study elasto-plastic materials, and by Brocks et al. [17] for elasto-viscoplastic damaged materials. All of the above void coalescence models are based on 1nite element calculations and on a modi1cation of the Gurson model to account for void coalescence. Zavaliangos and Anand [18] analyzed the thermo-elastoviscoplasticity of isotropic porous metals. Pan et al. [19] analyzed the strain localization in a rate-sensitive porous plastic solid. They showed the retarding eLect of material rate sensitivity on localization for a void nucleation and growth model. Thomason [20] compared the stress in the intervoid ligament before and after localization for elasto-plastic non-strain hardening materials. The material behaviour before the onset of void coalescence is well described by a Gurson model. After the onset of How localization in the intervoid ligament, a completely independent approach was used. Pardoen and Hutchinson [21] generalized Thomason’s approach to elastoplastic strain hardening materials containing spheroidal voids. Before the onset of void coalescence the Gurson model generalized to non-spherical voids by Gologanu et al. [10–12] was used. The stress in the intervoid ligament after plastic localization was estimated by a heuristic extension of Hill’s [22] slip line analysis for perfectly plastic materials to strain hardening materials. For strain hardening or strain rate hardening materials there is no easy estimate of the stress in the intervoid ligament, nor of the overall stress at the cell after the onset of plastic localization. In the present work, a modi1ed Gurson model (hollow shell) is used to describe void growth in an elastic–viscoplastic material before the onset of void coalescence. The overall material behaviour before the onset of void coalescence (i.e. the equivalent stress–strain curves) is well predicted by a modi1ed Gurson model, but the corresponding velocity 9eld is too poor for predicting void shape changes exactly. Therefore, 1nite element cell calculations were used to 1t the equivalent stress– strain curves and the void shape change. For each current geometry predicted by the Gurson model modi1ed in this manner, localization is tested with a new velocity 1eld respecting the symmetry conditions of a cylindrical cell. The paper is organized as follows. Section 2 is dedicated to the problem formulation. After a short description of the material model and the cell model (Sections 2.1 and 2.2), the extension of Gurson’s model to elastic–viscoplastic damaged materials is described. This part is largely based on previous work by Gologanu et al. [10–12]. But, a new description of the void shape change was adopted. In Section 2.4, the new localization analysis based on a new velocity 1eld is discussed in detail. Section 3 presents the main results of the present analysis. First cell-model results with particular attention on void shape changes are discussed. Section 3.2 compares m-dependent porous material model results to the cell-model results. The m-dependent porous material model gives results very close to the cell-model. But, the computing time may be rather expensive. Therefore, Section 3.3 gives a simple interpretation of the localization stress. Several models were used, corresponding to diLerent levels of approximation. Section 3.4 summarizes the diLerent approaches.

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2. Problem formulation 2.1. Material model The matrix material is described as an isotropically hardening elastic–viscoplastic solid. The matrix How stress M is given by the following empirical relation [15]:

m N n N˙ 0 N ; g() N = 0 1 + ; 0 = : M = g() (2.1) ˙0 0 E  Here, N = N˙ dt, and g() N represents the eLective stress vs. eLective strain response in a tensile test carried out at a strain rate such that N˙ = ˙0 . During hot working most metal alloys follow a similar behaviour with small values of the strain hardening exponent n and with values of m varying with the temperature. Throughout this work a constant value of 0.01 has been considered for n. Four diLerent values of the strain rate sensitivity m are considered 0.01, 0.07, 0.10 and 0.15. The lowest m value corresponds to room temperature. The ratio between the initial yield stress 0 and Young’s modulus E are kept constant throughout this paper (E = 5000 ), corresponding to most material behaviours at working temperature. 2.2. Cell-model and numerical implementation A cylindrical cell of initial length B0 and radius A0 (Fig. 1(a)) containing an ellipsoidal hole of radius a0 and half-length b0 is considered. The initial void volume fraction f0 is given by f0 =

2a20 b0 : 3A20 B0

(2.2)

The initial cell geometry is characterized by the cell aspect ratio 0 = B0 =A0 , the void volume fraction f0 and the void shape W0 = b0 =a0 . The cell is subjected to homogeneous axial and radial displacements u3 and u1 . The macroscopic principal strains and the eLective strain, respectively, are

C L

C L u3

B0

B

B0

b0 A a0 (a)

u3

E

D

C

b0

u1

u1

ρ

a0

A0

A0

(b)

Fig. 1. (a) Cell-model geometry and geometry used for the bifurcation analysis. (b) Geometry corresponding to the modi1ed Gurson model.

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Fig. 2. Finite element meshes for f0 = 0:0013; 0 = 1 and diLerent initial void shapes W0 . (a) W0 = 14 , (b) W0 = 1 and (c) W0 = 6.

given by E11 = E22 = ln




A A0




;

E33 = ln

B B0

 ;

Eeq =

2 |E33 − E11 |: 3

(2.3)

The corresponding macroscopic stresses 11 = 22 and 33 are the forces on the boundaries divided by the current areas. The macroscopic equivalent stress, eq , the mean stress m and stress triaxiality T are given by eq = |33 − 11 |;

m =

(211 + 33 ) ; 3

T=

m : eq

(2.4)

During the loading history the overall axial strain rate E˙ 33 is imposed and the triaxiality ratio T is kept constant, corresponding to a constant ratio between the axial and radial stress. The ABAQUS 1nite element code with 480 axisymmetric hybrid 4-node elements is used. Fig. 2 shows a typical 1nite element mesh with 20 elements in the radial direction and 24 elements in the circumferential direction for three diLerent initial void aspect ratios W0 . 2.3. An m-dependent porous material model Gologanu et al. [10–12] extended the Gurson model to prolate and oblate voids in a plastic material. The stress potential proposed by Gologanu et al. corresponds to an ellipsoidal volume of perfectly plastic material containing an confocal ellipsoidal void (Fig. 1(b))
 
eq + h 2 h =C − q2 (g + f)2 = 0; + 2q(g + 1)(g + f) cosh  (2.5) 0 0 
11 + 22 @ + (1 − 22 )33 ; E˙ pij = ˙ ; (2.6) h = 22 2 @ij

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where f is the current void volume fraction. The parameters C, ; g;  and 2 depend on the current void and cell shape (Appendix A). Gologanu, Leblond and Devaux [11] estimated the evolution of the void shape factor S = ln(W ), based on the micro-mechanical model (2.5), (2.6) used to extend the Gurson model S = ln(W ) = ln(b=a); S˙ = (1 + hS hT hf )(E˙ 33 − E˙ 11 ) + hSf E˙ kk :

(2.7)

Here, hS ; hT ; hf and hSf are four corrective functions depending, respectively, on S; T and f, introduced to represent the results predicted by cell-model calculations (Appendix A). Pardoen and Hutchinson [21] used the stress potential (2.5), (2.6) and expression (2.7) for void shape changes to study void coalescence in an elasto-plastic strain hardening material. These authors used two diLerent expressions for the correction factor hT at a strain hardening of n=0:1 and 0.3. The micro-mechanical analysis seems to give satisfactory results for the overall cell behaviour, i.e. the overall equivalent stress–strain curves. But a precise prediction of the void shape evolution needs the introduction of four diLerent correction functions. Here a similar stress potential to the one proposed by Gologanu et al. [10–12] is used

 eq + h 2 h =C + 2q1 (g + 1)(g + f)pm q2  M M 2 
(1 − m) f = 0: (2.8) − q1 g + (1 + m) Parameters q1 and q2 were introduced in Gurson’s model by Tvergaard [8,23]. The function pm depends on the strain rate hardening parameter m. For elasto-plastic materials pm tends to the cosh function used by Gurson [7] and Gologanu et al. [10–12] pm (x) = hm (x) +

(1 − m) 1 ; (1 + m) hm (x)

hm (x) = [1 + mx(1+m) ]1=m

(2.9)

while ˙ and f˙ are determined by an equality for plastic dissipation and by material conservation, respectively, M ˙pM = (1 − f)ij E˙ pij ;

(2.10)

f˙ = (1 − f)E˙ pkk :

(2.11)

The void shape change will be approximated in terms of a linear relation between the void axial dimension and the void radius

 b a −1 =k −1 ; (2.12) b0 a0 where the value of the void shape change factor k is estimated on the basis of the cell-model results. The variation in matrix How stress M is predicted by the consistency relation ˙ (; M ; f; W ) = 0:

(2.13)

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As the geometric functions in  are rather complex, the consistency relation is solved numerically by a Newton–Raphson method. 2.4. Localization analysis The 1nite element model shows a signi1cant strain localization within the matrix material. This strain localization depends on the cell shape. Here, an approximate method is proposed to be able to obtain estimates of the localization stress relating to the m-dependent porous material model. To approximately describe the overall stress–strain response after localization, a velocity 1eld respecting the symmetry conditions of the cylindrical unit cell (Fig. 1(a)) is considered ˜u˙ = {D˜u˙ (1) + (1 − D)˜u˙ (2) }=2:

(2.14)

Each incompressible velocity 1eld respects the boundary and symmetry conditions of the cylindrical unit cell ˜u˙ (k) (' = 0; z) = 0;

˙e ' ˜u˙ (k) (' = A; z) = A˜

(k = 1; 2);

(2.15)

˜u˙ (k) ('; z = 0) = 0;

˙e z ˜u˙ (k) ('; z = B) = B˜

(k = 1; 2);

(2.16)

where ('; *; z) is a cylindrical coordinate system centred on the cell and the void, ˜e ' and ˜e z are the unit vectors in radial and axial direction, respectively. For k = 1 the velocity 1elds are   ' ' 1 ' 1 (1) − − + + u˙ ' =  ('2 + z 2 )3=2 (A2 + z 2 )3=2 '('2 + B2 )1=2 A2 (A2 + B2 )1=2 B' B˙ ' ; 2 B 

− u˙(1) z

=

z 2z z 2z + 2 2 − 2 − 2 2 2 2 3=2 2 2 3=2 1=2 (' + z ) A (A + z ) (' + B ) A (A + B2 )1=2

(2.17) 

z + B˙ : B

(2.18)

The 1rst term corresponds to a spherically symmetric expansion 1eld. The following terms are the opposite of the expansion 1eld at the cell wall (' = A) and the top of the cell (z = B). 1=B' ˙ corresponds to pure radial straining of the cell, whereas Bz=B corresponds to a homogeneous axial deformation. For k = 2 the velocity 1elds are   + ' ' ' 1 1 (2) − ; (2.19) − 2 − 2 + u˙ ' =  2 2 3=2 2 3=2 2 1=2 (' + z ) (A + z ) (' + B ) B' 2 B   z z 2z z (2) (2.20) ++ ; u˙ z =  + 2 2 − 2 2 2 3=2 2 1=2 2 3=2 (' + z ) A (A + z ) (' + B ) B √ 2 2 ˙ A A + (A A2 + A2 − AB)B˙ −2B √  = ABA˙ + : (2.21) B˙ + = 2 A A2 + A 2 From a critical value of the equivalent strain the radial strain E11 remains constant. Thus the cell radius remains constant and further deformation takes place by overall uniaxial straining. Therefore, the real localization condition, based on the cell-model results, is given by a constant radial strain

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[24]. But this localization behaviour is not predicted by the m-dependent porous material model alone. ˙ = 0). The In the following analysis, only the onset of uniaxial straining is searched for (E˙ 11 = A=A evolution of the cell and void geometry are determined by the m-dependent porous material model. For the current geometry predicted by the m-dependent material model, the axial stress (dependent on D) is determined from the new velocity 1elds (2.14)–(2.22) with the condition of rigid cell walls (A˙ = 0), by requiring equality of external and internal work. Among these stress values, the axial stress to be used and the optimum value of D are determined by the minimization in Eq. (2.22)  (m+1)


˙ 0 ˙0 (D; N A˙ = 0) loc: ˙ 33 B = min dv : (2.22) D ,A2 Vcell ˙0 In the m-dependent porous material model, localization is taken to occur when the axial stress ˙ = 0) becomes smaller than the axial stress predicted by predicted by Eq. (2.22) (satisfying E˙ 11 = A=A the m-dependent porous material model. The value of the optimization factor D varies slightly with current cell geometry. A single value of D for each initial cell geometry and loading was retained. After the onset of localization, the rigid cell model, with the previously determined coeTcient D, is used to predict the overall stress. The condition of zero radial strain leads to the evolution equation for the void volume fraction f˙ = 3 (1 − f)E˙ eq : (2.23) 2

After localization, the void shape change is predicted by the new velocity 1eld (2.19)–(2.21). 3. Results The current cell geometry is described by the cell aspect ratio =B=A, the void aspect ratio W =b=a and the initial void volume fraction f0 . Two diLerent initial void volume fractions (f0 = 0:0013 and 0.0104), two diLerent cell aspect ratios ( = 1 and 4) and four diLerent void aspect ratios (W = 0:25, 0.5, 1.0 and 6) have been considered. The initial void radius a0 is given by the following relation: 2a3 W0 f0 = 0 : 30 For each particular cell geometry, three diLerent values of the triaxiality parameter (T = 1, 2 and 3) and four diLerent values of the strain rate hardening parameter (m = 0:01, 0.07, 0.10 and 0.15) are considered. When nothing else is indicated, the overall axial strain rate E˙ 33 is taken equal to ˙0 . 3.1. Cell-model results 3.1.1. Overall behaviour: stress–strain curves and void volume fraction evolution Fig. 3 shows cell-model results corresponding to an initial void volume fraction f0 = 0:0013 for an initially spherical (W0 = 1) and an initially prolate void (W0 = 6) in a cell with 0 = 1 and for an initially spherical void in an elongated cell (0 = 4), for four diLerent values of the strain rate hardening m = 0:01, 0.07, 0.10 and 0.15. Figs. 3(a)–(c) correspond to stress triaxility ratios of T = 1, 2 and 3, respectively. Fig. 4 shows cell-model results corresponding to the same initial void volume fraction f0 , for an initially spherical void (W0 = 1) and an initially oblate void (W0 = 14 ) in a cell with 0 = 1. Figs. 4(a)–(c) correspond to stress triaxility ratios of T = 1, 2 and 3, respectively.

0.10

T=1

1.0

0.15

0.2

0.04 0.01

0.01

0.4

0.15

0.01 0.07 0.10 0.15

f

0.06

0.6

0.15

0.08

0.8

0.02 0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

Eeq

(a)

Eeq

1.2

0.10 T =2

T =2

1.0

0.0 0.0

0.02

0.1

0.2

0.00 0.0

0.3

1.2

0.2

0.10

T=3

0.15 0.10

0.06

0.07

0.15

0.10

0.04

0.15

0.01

0.4

0.07

0.01

f

0.6

0.0 0.00

T=3

0.08

0.8

0.2

0.3

Eeq

1.0

Σeq σ0

0.1

Eeq

(b)

(c)

0.07 0.10 0.15

0.04

0.07 0.10

0.01

f

0.2

0.15

0.01

0.4

0.01

0.06

0.6 0.07 0.10

Σeq σ0

0.8

0.01

0.08

0.07 0.10

Σeq σ0

T=1

0.01 0.07 0.10 0.15 0.01

1.2

0.15

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

0.01

1290

0.01

0.02

0.05

0.10

Eeq

0.15

0.00 0.00

0.05

0.10

0.15

Eeq

Fig. 3. Cell model results for f0 = 0:0013, 0 = 1, W0 = 1(+); 0 = 1, W0 = 6(•) and 0 = 4; W0 = 1(4). Macroscopic eLective stress vs. eLective strain response and void volume fraction vs. macroscopic eLective strain for diLerent stress triaxialities T . (a) T = 1, (b) T = 2 and (c) T = 3.

The equivalent stress eq increases rapidly to a maximum value followed by a smooth decrease. At a certain critical value Ec of the overall eLective strain Eeq , depending on the stress triaxility, the m value and the initial cell geometry, a rapid drop in equivalent stress occurs. The critical strain Ec corresponding to the stress drop, increases with decreasing stress triaxility T and increases with increasing m values. At small values of the stress triaxiality, prolate voids (W0 ¿ 1) lead to larger critical strains and oblate voids (W0 ¡ 1) lead to smaller critical strains than spherical voids (W0 = 1). At higher values of the stress triaxility (T = 2 and 3) the inHuence of the initial void shape on the critical strain Ec is very small. Elongated cells (0 ¿ 1) lead to smaller critical strains Ec than

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 0.10

T =1

1.0

0.06 f

0.15

0.01

0.04 0.01

0.4

0.07 0.10 0.15

0.2

0.02

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Eeq

Eeq

(a) 1.2

0.10

T =2

1.0

0.07 0.10

0.06

0.01

0.10 0.15

0.07

0.4

0.04 0.02

0.2 0.2

0.00 0.0

0.3

0.1

0.2

Eeq

0.3

Eeq

1.0

T=3

0.08

0.8

0.15

0.10

T=3

0.07 0.10

0.1

(b) 1.2

0.15

f

0.6

0.01

Σeq σ0

0.8

0.0 0.0

T =2

0.08

0.01

Σeq σ0

0.6

0.01

0.08

0.8

0.06

0.6

f

Σeq σ0

T =1 0.01 0.15 0.07 0.10 0.15

1.2

1291

0.15

0.01

0.07 0.10

0.04 0.4

0.02

0.2 0.0 0.00

(c)

0.05

0.10

Eeq

0.15

0.00 0.00

0.05

0.10

0.15

Eeq

Fig. 4. Cell-model results for f0 = 0:0013 and 0 = 1. W0 = 1 (solid curves), W0 = 1=4 (dashed curves). Macroscopic eLective stress vs. eLective strain response and void volume fraction vs. macroscopic eLective strain for diLerent stress triaxialities T . (a) T = 1, (b) T = 2 and (c) T = 3.

equiaxed cells (0 = 1) at all stress triaxility ratios and all m values. The slope of the stress–strain curve after How localization depends on the void shape and the cell geometry, but is independent of the strain rate hardening parameter m. Independent of the cell and void geometry an increase in strain rate sensitivity, m, leads to slower void growth. Moreover the void volume fraction reached at the critical equivalent strain increases with increasing m values. At small values of the stress triaxility ratio (T =1), prolate voids (W0 ¿ 1) grow slower than spherical voids (W0 =1) and oblate voids grow faster. But the void volume fraction reached at the critical equivalent strain increases when the void shape factor W0 is increased. At

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H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 0.8

1.0

T =1 0.76

T =1

0.8

0.6 0.60

0.6

0.45

0.4

0.2 0.20

0.2

0.4

0.88

0.59 0.00

0.00

0.0 0.0

0.96

0.2

0.4

0.6

(a)

0.8

0.0 0.0

0.2

0.4

0.6

0.8

1.0

(b)

Fig. 5. Cell-model results for f0 = 0:0013, 0 = 1 and T = 1. Void shapes at diLerent overall equivalent strains (solid curves) and corresponding ellipsoid (dashes curves). (a) W0 = 1=4, (b) W0 = 1.

higher values of the stress triaxility parameter (T = 2 and 3) the void shape factor W0 has very small inHuence on the void growth rate. Cell elongation (0 ¿ 1) leads to an acceleration of void growth compared to the cell with 0 = 1, for all values of the stress triaxiality ratio, but the void volume fraction reached at the critical strain decreases if the cell shape parameter 0 is increased. 3.1.2. Void shape change Fig. 5 shows cell-model results corresponding to an initial void volume fraction f0 = 0:0013 for an initially spherical (W0 = 1) and an initially oblate void (W0 = 14 ) in an equiaxed cell (0 = 1), submitted to a remote stress triaxiality of T = 1 for a strain rate hardening value of m = 0:10, at diLerent levels of the remote equivalent strain Eeq . Before the drop of the equivalent stress (at strains smaller than Ec ), the void shape may be approximated by an ellipsoid. After the stress drop, the deformation is localized in the minimum section of the cell. Then the ligament between the void and cell wall becomes very small with a more or less “conical” void shape, and the ellipsoidal approximation of the void shape no longer holds. Fig. 6 shows the evolution of the void axial dimension vs. void radius. The void dimensions correspond to points A and B in Fig. 1(a). The solid curves represent the cell-model results, whereas the dashed curves correspond to a linear approximation. Before the onset of How localization, a linear relation between the void dimensions gives a good approximation.

 b a −1 =k −1 : (3.1) b0 a0 The shape change factor k may be described by the following expression, dependent on the matrix strain rate hardening parameter m, the remote stress triaxiality T and the initial void shape W0 =b0 =a0 k = km m + k 0 ;


+0 km = 0 + W0

(3.2) 




+1 + 1 + W0




 1 1 +2 + 2 + ; T W0 T 2

(3.3)

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 5

8 0.15 0.10

7 6

0.15

4

0.07

5

0.10

0.01

b b0

b b0

1293

4

3

0.07 0.01

3

2

2 1 1.0

1 1.0 1.5 2.0 2.5 3.0 3.5 4.0

(a)

a a0

(b)

1.5

2.0

2.5

3.0

a a0

Fig. 6. Cell model results (solid curves) and linear approximation for f0 = 0:0013, W0 = 1 and T = 1, with strain rate hardening exponents m = 0:01, 0.07, 0.10 and 0.15. Void axial dimension vs. void radius for two initial cell shapes 0 . (a) 0 = 1, (b) 0 = 4.



 
1 0 1 1 2 + 2 + + 1 + k0 =  0 + : W0 W0 T W0 T 2

(3.4)

The numerical coeTcients (i ; +i ; i ; i ), obtained by optimization based on the cell-model results, are given in Appendix A. This expression is valid for strain rate hardening parameters in the range 0.01– 0.25, stress triaxialities T in the range 0.9 –in1nity and initial void shapes in the range 18 –10. Figs. 7(a) and (b) show the variation of the shape change parameter k with the strain rate hardening parameter m and the stress triaxiality parameter T . The symbols correspond to cell-model results, whereas dashed curves correspond to the expressions (3.2)–(3.4). At a stress triaxiality T = 1, void growth is more important in the axial than in the radial direction (k ¿ 1). Independent of the void or cell geometry an increase of the strain rate hardening parameter m leads to an increase in the axial deformation of the void, i.e. k increases. The void shape parameter k decreases if the stress triaxiality T is increased. At high values of the stress triaxility, the voids tend towards a slightly oblate shape (W ¡ 1). Figs. 7(c) and (d) show the variation of the shape change parameter k with the initial void volume fraction f0 and the initial void shape W0 . A variation of the initial void volume fraction f0 leads to negligible changes of the void shape change parameter k. An increase of the initial void shape parameter W0 leads to a decrease of the void shape change parameter k. Void shape change is more signi1cant for oblate voids than for prolate voids. 3.1.3. Strain distribution in the matrix material Fig. 8 shows the distribution of axial strain 33 along section DE (Fig. 1(a)) for an elongated cell (0 = 4) and an equiaxed cell (0 = 1) containing an initially spherical void, at diLerent values of the overall equivalent strain Eeq . At equivalent strains Eeq smaller than Ec (before the overall equivalent stress drop), a homogeneous strain distribution is observed. At overall equivalent strains larger than Ec (after the overall equivalent stress drop), the axial strain 33 is con1ned to a small ligament around the minimum section of the cell. The height of this ligament (x3 =B) depends on the initial cell elongation 0 .

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

k

14 13 f0 = 0.0013, T = 1 12 11 10 9 W0 = 0.25 8 1.00 7 6.00 6 5 4 3 2 1 0 0.00 0.05 0.10 0.15 (a) m

5

f0 = 0.0013, W0 = 1 4

m = 0.01 0.07 0.10 0.15

3 k

1294

2 1

(b)

0 1.0

1.5

2.0 T

2.5

3.0

15

3.90

f0 = 0.0013

W0 = 1

3.80 3.70

10

T =1 m = 0.01 0.07 0.10 0.15

k

3.60 3.50

k

m = 0.10 T = 1.00

5

3.40 3.30 3.20 0.000 (c)

0 0.005

0.010

0.015

0.020

f0

0

(d)

1

2

3

4

5

6

W0

Fig. 7. Cell-model results (symbols) and analytical approximation (lines) for the void shape change factor k for 0 = 1:0. (a) k vs. strain rate hardening parameter m, (b) k vs. stress triaxiality T , (c) k vs. initial void volume fraction f0 and (d) k vs. initial void shape W0 . 2.0 1.8 1.6 1.4 1.2 1.0 0.96 0.88 0.8 0.79 0.6 0.59 0.4 0.39 0.19 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (b) x3 B

ε33

ε33

2.0 1.8 1.6 1.4 1.2 0.56 1.0 0.8 0.6 0.54 0.48 0.4 0.31 0.2 0.13 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 (a) x3 B

Fig. 8. Cell-model results for a spherical void (W0 = 1) with f0 = 0:0013, T = 1 and a strain rate hardening exponent m = 0:1 for several values of the overall equivalent strain. Axial strain vs. axial coordinate for several values of the cell height. (a) 0 = 4 and (b) 0 = 1.

3.2. m-Dependent porous material model with localization analysis 3.2.1. Overall behaviour: stress–strain curves and void volume fraction evolution Figs. 9–11 show the comparison between cell-model calculations and the m-dependent porous material model with strain localization possibility. The current geometry is predicted by the m-dependent

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 0.10

T=1

1.0

0.08 0.06 f

0.6 0.4

0.04

0.15

0.01 0.07 0.10

0.02

0.2 0.4

0.6

0.8

0.00 0.0

1.0

0.4

0.01

0.06 0.04

0.10 0.15

0.07

0.01

f

0.6 0.4

0.02

0.2 0.1

0.2

0.00 0.0

0.3

0.1

Eeq

(b)

0.3

0.10 T=3

T=3

1.0

0.06 f

0.6

0.15

0.01

0.08

0.8

0.15

0.01

0.10

0.04

0.4

0.07

Σ eq σ0

0.2

Eeq

1.2

0.02

0.2

(c)

1.0

T=2 0.08

0.8

0.0 0.00

0.8

0.07 0.10 0.15

T=2

1.0

0.0 0.0

0.6

Eeq 0.10

1.2

Σ eq σ0

0.2

Eeq

(a)

0.10

0.2

0.07

Σ eq σ0

0.8

0.0 0.0

T =1 0.01 0.07 0.10 0.15

1.2

1295

0.05

0.10

Eeq

0.15

0.00 0.00

0.05

0.10

0.15

Eeq

Fig. 9. m-dependent porous material model predictions compared to cell-model results for f0 = 0:0013, W0 = 0:25, 0 = 1. Macroscopic eLective stress vs. eLective strain response and void volume fraction vs. macroscopic eLective strain for diLerent stress triaxialities T . (a) T = 1, (b) T = 2 and (c) T = 3.

model as long as no localization occurred. The localization-model calculates the axial stress for each geometry predicted by the m-dependent model. Strain localization is taken to occur, if the overall axial stress 33 predicted by the localization model becomes smaller than the overall axial stress predicted by the m-dependent model. The void shape change before the onset of localization is predicted by the void shape change factor k based on the cell-model results. The parameters q1 and q2 have been chosen in order to get equivalent stress–strain curves and void volume fractions predicted by the m-dependent model as close as possible to the cell-model calculations before the onset of localization. The parameter q1 changes with the initial void shape

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 1.2

0.10

T =1

1.0

0.01 0.07 0.10 0.15

0.08

0.8

0.06

0.6

f 0.04 0.01 0.07 0.10 0.15

0.02

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Eeq 1.2

Eeq 0.10

T =2

T =2

0.08

0.15

1.0

0.06 0.10

0.4

0.07

f

0.6

0.01

0.04 0.02

0.2 0.1

0.2

0.00 0.0

0.3

0.10

T=3

1.0

T=3

0.06 f

0.6 0.10 0.15

0.01

0.07

0.04

0.4

0.02

0.2 0.0 0.00

0.3

0.08

0.8

Σeq σ0

0.2

Eeq

0.15

1.2

(c)

0.1

Eeq

(b)

0.10

0.0 0.0

0.01

Σeq σ0

0.8

0.15

(a)

0.10

0.2

0.07

0.4

0.01

Σeq σ0

T =1

0.07

1296

0.05

0.10

Eeq

0.15

0.00 0.00

0.05

0.10

0.15

Eeq

Fig. 10. m-dependent porous material model predictions compared to cell-model results for f0 = 0:0013, W0 = 1:0 and 0 = 1. Macroscopic eLective stress vs. eLective strain response and void volume fraction vs. macroscopic eLective strain for diLerent stress triaxialities T . (a) T = 1, (b) T = 2 and (c) T = 3.

only q1 =1:5−0:06|W0 −1|. This leads to 1.5 for a spherical void, close to the value of 1.47 determined theoretically by Perrin and Leblond [25] for the Gurson model. A constant value of q2 = 0:96 [24] gives satisfactory results for the equivalent stress–equivalent strain curves. The parameter D in Eq. (2.14) was obtained by minimizing the diLerence between cell-model results and analytical results after the onset of strain localization. Fig. 12 shows cell-model and m-dependent porous material model results for 0 =1. Figs. 12(a) and (b) correspond an initially spherical void in a unit cell submitted to several remote stress triaxialities. Figs. 12(c) and (d) show results for a unit cell submitted to a remote stress triaxiality T = 1, with

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 0.08

T=1

T=1 0.01 0.07 0.10 0.15

1.2

1297

1.0 0.06

f

0.6

0.01 0.07 0.10 0.15

Σeq σ0

0.8 0.04

0.4 0.02 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Eeq

Eeq 0.10

1.00

T= 2 0.01

T=2 0.15

1.20

0.08

0.07 0.10 0.15

(a)

0.06 0.40

0.07 0.10

f

0.60 0.01

0.04 0.02

0.20

0.10

0.20

(b)

0.00 0.00

0.30

0.10

0.20

0.10

T=3

1.0

T=3

0.08 0.06 0.04

0.15

0.10

0.07

0.4

f

0.6 0.01

Σeq σ0

0.8

0.02

0.2 0.0 0.00

(c)

0.15

1.2

0.30

Eeq

Eeq

0.07 0.10

0.00 0.00

0.01

Σeq σ0

0.80

0.05

0.10

Eeq

0.15

0.00 0.00

0.05

0.10

0.15

Eeq

Fig. 11. m-dependent porous material model predictions compared to cell-model results for f0 = 0:0013, W0 = 6:0, 0 = 1. Macroscopic eLective stress vs. eLective strain response and void volume fraction vs. macroscopic eLective strain for diLerent stress triaxialities T . (a) T = 1, (b) T = 2 and (c) T = 3.

voids of diLerent initial shapes. Figs. 12(e) and (f) show results for diLerent initial void volume fractions. A very good correspondence between the cell-model results and the m-dependent material model is observed. The void volume fraction at the onset of the strain localization may be predicted very closely with the present model. All above results were obtained with a rather simple expression of the optimization coeTcient D, as speci1ed in Appendix A. The optimization coeTcient D is independent of the initial void volume fraction f0 , the initial cell shape 0 . D depends on the remote stress triaxiality T , the strain rate hardening parameter m and the initial void shape W0 (Appendix A). The new velocity 1eld gives a good approximation of the local strain rate and stress throughout the matrix material.

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 1.2

0.10

f0 = 0.0013, W0 = 1

f0 = 0.0013, W0 = 1

1.0

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Eeq

Eeq

(b) 0.10

f0 = 0.0013, T = 1

f0 = 0.0013, T = 1

1.0

0.4

0.25 0.4

1 2

4 6

f

0.4

0.06 1 2

0.125 0.25

0.6

0.125

0.08

0.8

0.04

4 6

0.2

0.02

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.00 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0.14

1.0

0.10

0.8

0.0013

W0 = 1, T = 1

0.12

0.0025

W0 = 1, T = 1

0.0037

1.2

Eeq

(d)

0.0104

Eeq

(c)

0.0200

Σeq σ0

0.9

0.02

1.2

0.08 f

0.6

0.0 0.0

0.2

0.4

Eeq

0.6

0.8

0.0013

0.0025

0.0200

0.2

0.0037

0.06 0.4

0.0104

Σeq σeq

1.0

0.04

0.2

(a)

(e)

1.2

f 0.9

1.0

1.2

1.5

1.8

2.0

0.4

1.5

0.06

0.6 3.0

Σeq σ0

0.8

2.0 1.8

0.08 3.0

1298

0.04 0.02 0.00 0.0

1.0

(f)

0.2

0.4

0.6

0.8

1.0

Eeq

Fig. 12. m-dependent porous material model predictions compared to cell-model results for 0 = 1 and m = 0:1. (a and b): for several values of the stress triaxiality T . (c and d): for several values of void shape parameter W0 . (e and f): for several values of the initial void volume fraction f0 .

3.2.2. Strain rate in the matrix material: post localization analysis The new “rigid-cell wall” model not only predicts the overall post localization cell behaviour (i.e. the equivalent stress–equivalent strain curves and the void volume fraction evolution) but also a very good estimate of the local strain rate is given. Fig. 13 shows the axial strain rate throughout the matrix after the onset of strain localization. Fig. 13(a) shows the variation of the axial strain rate close to the minimum intervoid ligament, whereas Fig. 13(b) shows the variation of the axial strain rate all along the x3 -axis. The strain rate is con1ned to a small region around the minimum intervoid ligament.

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

1299

7 ρ=a ρ=a

6 5

ρ =A

1.0

ε 33 ε0 ε 33 ε0

x3

4 0 1 2 3

0.5

1

4

(a)

0 ρ =A -1 0.0 0.5 1.0 1.5 2.0 2.5 3.0

5

0.0 0.0

3 2

0.5

(b)

x1

x3

Fig. 13. Variation of axial strain rate ˙33 after the onset of bifurcation to uniaxial straining. (a) iso-values of ˙33 around the minimum intervoid ligament, (b) variation of ˙33 along x3 -axis.

3.3. Closed form for the localization stress The m-dependent porous material model with localization analysis gives results very close to the cell model. But, the localization analysis based on Eq. (2.22) needs an optimization for each current geometry predicted by the m-dependent porous material model. In the present section, a closed loc form expression for the axial stress 33 depending only on the size of the intervoid ligament d, the initial yield stress 0 , the strain rate hardening parameter m and the remote stress triaxiality T is given loc 33 = 6(1 + 3m)0




d d0

[T (m+1)]

;

(3.5)

where d is the size of the intervoid ligament d=(A−a) is given by the m-dependent porous material loc as function of the equivalent model. Expression (2.22) gives the value of the localization stress, 33 loc remote Eeq . First, ln(33 ) vs. ln(d=d0 ) was plotted for each value of the cell shape factor 0 , the void shape factor W0 , the strain rate hardening parameter m and the stress triaxiality T . The linear loc relation between ln(33 ) and ln(d=d0 ) was determined by least-squares method and the following power law was obtained: loc 33


= (m; W0 ; 0 ; T )0

d d0

[T (m+1)]

:

Expression (3.5) was determined by linear regression over the diLerent values of . The variation of the void radius a depends on the void volume change and the shape change parameter k. Relation (3.1) between (a=a0 ) and (b=b0 ) leads to ˙v a˙  kk  = a 2+k a = b a0 b0

(3.6)

1300

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308 1.2 1.0

0.2

0.15

0.01

T=2

0.15

0.07 0.10

0.01

0.4

0.15

0.6 0.01

Σeq σ0

0.8

T=1

T=3

0.0 0.0

0.2

0.4

0.6

(a)

0.8

1.0

E eq 1.2 1.0

0.15

0.01

0.2

T=3

0.0 0.0

0.15

0.4

T=2

0.01 0.07 0.10 0.15

0.6 0.01

Σeq σ0

0.8

T=1

0.2

0.4

0.6

(b)

0.8

1.0

1.2

E eq 1.2 1.0

T=3

T=2

0.01 0.07 0.10 0.15

0.15

0.4

0.01

0.6 0.01 0.15

Σeq σ0

0.8

T=1

0.2 0.0 0.0

0.2

0.4

(c)

0.6

0.8

1.0

1.2

1.4

E eq

Fig. 14. Overall stress vs. overall deformation. Cell-model results (solid bold curves), m-dependent porous material model (dashed bold curves), closed form for the localization stress according to (3.5) (thin solid curves).

where ˙vkk is the void mean strain rate. Before the onset of strain localization, the void shape is ellipsoidal and the mean void strain rate is given by ˙vkk =

V˙ void 2a˙ b˙ + = (2E˙ 11 + E˙ 33 )=f; = Vvoid a b

(3.7)

where Vvoid and V˙ void are, respectively, the current void volume and its time derivative. Fig. 14 shows macroscopic equivalent stress vs. macroscopic equivalent strain curves for an initial void volume fraction of f0 = 0:0013 in an initially equiaxed cell. Figs. 14(a) – (c) correspond, respectively, to initial void shapes of W0 =0:25, 1 and 6. The bold solid curves represent the cell-model results and the

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

0.01

1.0 0.8 0.6

0.07 0.10 0.15

Σ eq / σ0

d d0

-0.1

0.8

-0.2 0.6 0.4

0.4

0.0

1.0

E 11

1.2

0.01

0.07 0.10 0.15

0.2

(a)

E eq 4.8

0.15

(b)

loc Σ eq (2.22)

loc Σ eq (3.5)

1.2

1.0 0.9

a, b

0.8

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.7 0.8

(d)

loc Σ eq (2.22)

1.1

A, B

E eq

loc Σ eq (3.5)

1.3

1.6

(c)

E eq 1.4

Σ eq σ0

Σ eq σ0

3.2

0.01 0.07

-0.7 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

cell model results m-dependent porous material model (2.22) (3.5)

4.0

0.8

-0.4

-0.6

0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

2.4

-0.3

-0.5

0.2

1301

A, B a, b cell model results m-dependent porous material model (2.22) (3.5)

1.0

1.2

1.4

E eq

Fig. 15. Cell-model results for an elongated void (W0 = 6) in a unit cell with 0 = 1 and T = 1: Equivalent stress and intervoid ligament size vs. equivalent strain (a). Radial strain vs. equivalent strain (b). DiLerent stresses: cell-model, m-dependent material model, localization model and approximation (3.5) (c and d).

bold dashed curves correspond to the localization analysis with optimum value of D (Appendix A). The thin solid curves correspond to expression (3.5) for the localization stress. A very good agreement between cell-model results, localization analysis and closed form (3.5) for the localization stress is obtained. Expression (3.5) leads to a very nice physical interpretation of the “rigid-cell wall” model. The onset of strain localization is controlled essentially by the size of the intervoid ligament. This closed form is also of major importance for practical purposes. The check for localization is reduced to a comparison of the axial strain predicted by the m-dependent porous material model and the axial strain predicted by expression (3.5) based on the intervoid ligament size predicted by the m-dependent porous material model. In fact, before the onset of strain localization the m-dependent porous material operates. Strain localization occurs when the localization stress (3.5) becomes smaller than the stress predicted by the m-dependent material model. 3.4. Summary of the di=erent approaches Fig. 15 summarizes the diLerent approaches. Figs. 15(a) and (b) show cell-model results for an initially elongated void (W0 = 6) embedded in a unit cell with 0 = 1 submitted to a remote stress

1302

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

triaxiality of unity. Fig. 15(a) shows equivalent stress vs. equivalent strain and normalized intervoid ligament size (d=d0) vs. equivalent strain curves. Fig. 15(b) shows the variation of the radial strain E11 with equivalent von Mises strain. Localization of the strain to the intervoid ligament appears if the overall radial strain remains constant. This condition is ful1lled exactly by the localization analysis (2.14)–(2.23). The localization analysis can stand alone and gives very satisfactory results. But, the localization analysis needs to determine the minimum of integral in Eq. (2.22) for each current geometry predicted by the m-dependent porous material model as shown in Figs. 15(c) and (d). Thus, an additional, heuristic approach was proposed in Section 3.3. This approach is based on the observation of the variation of the intervoid ligament size d with the overall strain (Fig. 15(a)). A steep decrease of the intervoid ligament size at the onset of strain localization is observed. As the m-dependent porous material model (with the adequate void shape change parameter) predicts very well the intervoid ligament size, the heuristic approach (3.5)–(3.7) was possible. Figs. 15(c) and (d) show cell-model results, m-dependent porous material predictions, the localization stress (2.22) and the value corresponding to expression (3.5). Expression (3.5) gives very close results to the real localization stress (2.22). Expression (2.22) extends the localization criterion based on the void volume fraction used in Ref. [24] for spherical voids to nonspherical voids. 4. Discussion The present numerical study of void growth to coalescence is analogous to those of Koplik and Needleman [15] and Pardoen and Hutchinson [21], but in the present investigation the focus is on deformation at elevated temperatures corresponding to hot working. In hot working situations the rates of deformation are high enough to neglect surface diLusion eLects on void growth. In real hot working situations voids nucleate from second-phase particles, and the rate of nucleation strongly aLects the strain at 1nal failure. The model does not account for the delays due to later nucleation. The present work studies the eLect of hot working process parameters, i.e. the temperature (through the strain rate hardening parameter m) and the stress triaxiality, as well as the material characteristics (through the cell geometry), on the void growth to coalescence. Void growth to coalescence has been investigated by three models. Cell-model calculations lead to deeper insight in the overall cell behaviour and the stress and strain distribution throughout the matrix without any particular assumption on the velocity 1eld. The m-dependent porous material model is based on an extension of Gurson’s micro-mechanical approach to spheroidal voids, using the 1nite element cell-model calculations to predict void shape change. Similarly, the “rigid-cell wall model” used a micro-mechanical approach based on two kinematically admissible velocity 1elds. For both the m-dependent material model and the “rigid-cell wall model”, the micro-mechanical approach was used to de1ne the structure of the stress 1eld, whereas the 1nite element cell calculations were used to determine the exact value of an optimization parameters (i.e. the shape change factor k or the parameter D). The closed form for the localization stress makes the present work very simple to use in practical situations. Cell-model results: The cell-model calculations show a delay in fracture strain if the remote stress triaxiality decreases and if the strain rate hardening parameter (the temperature) increases. Cell

H. Kl/ocker, V. Tvergaard / International Journal of Mechanical Sciences 45 (2003) 1283 – 1308

1303

elongation leads to a decrease of the strain at fracture. Initially, prolate voids lead to larger strains to fracture than initially spherical or oblate voids. The inHuence of the initial void shape is important at small values of the remote stress triaxiality. m-dependent porous material model with localization analysis: Before the onset of strain localization, the void shape is more or less ellipsoidal and the modi1ed Gurson model seems to be a good approximation to the overall behaviour. After the onset of void coalescence, strain localization in the minimum intervoid ligament leads to “conical” void shapes and the Gurson-type model assuming confocal ellipsoidal voids clearly fails. To predict the onset of strain localization and to describe the post-localization behaviour a “rigid-cell wall” model is used. This allows to predict precisely the void volume fraction at the onset of the strain localization. The m-dependent material model and the localization model depend only on the initial cell geometry. The model may seem more complex than the previous ones, but it includes void shape change eLects and a very precise prediction of the onset of void coalescence as well as the post coalescence behaviour. Two correction functions q1 and k were introduced in the modi1ed Gurson model. These functions do not use more numerical constants (resulting from optimization on the 1nite element cell-model calculations) than the previous models, but the stress–strain curves predicted by the new model are much closer to the 1nite element results. Numerical implementation: The closed form expression (3.5) for the localization stress leads to very simple numerical implementation of the model. In fact, before the onset of strain localization the m-dependent porous material model operates. Strain localization occurs when the localization stress (3.5) becomes smaller than the stress predicted by the m-dependent material model. After the onset of localization, the same expression for the localization stress is still valid but the radial strain rate is set equal to zero. Relation to previous models: The localization model needs the introduction of the additional loc function 33 (3.5) varying with the strain rate hardening parameter m and the initial cell geometry. Previous analyses [15,17,24] used a critical void volume fraction to determine the onset of void coalescence. The variation of the void volume fraction aLects the Gurson-type model before the onset of void coalescence, whereas the new “rigid-cell wall” model has no inHuence on the void growth before the onset of strain localization (strains smaller than Ec ).

Appendix A.

A.1. The void growth model (Gurson-type model) A.1.1. Geometric functions in the stress potential The modi1ed Gurson model is based on the work of Gologanu et al. [12], who analyzed the growth of a spheroidal void in a 1nite perfectly plastic solid with an outer confocal surface under homogeneous straining (Fig. 1(b)). The geometric functions C; ; g;  and 2 used in Eqs. (2.5) and (2.6) are expressed in terms of the inner and outer eccentricity e1 and e2 of the shell considered in Fig. 1(b).

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Prolate √ 2 2 √b − a =b B2 − A2 =B 0 4 (1 + e22 )=(3 √ + e2 ) ( 3 − 2) ln 31 + ln√332 + ln 33 1 √ + ln(f) 3 31 = e1 =e2

3 + e22 + 2 3 + e24

32 = 3 + e12 + 2 3 + e14

√ 3 + 3 + e14

33 = √ 3 + 3 + e24

e1 e2 g 2 1=K



(g +

1)2

+

q22 (g

g1 = g=(g + 1) 34 =

4 − gf5=2 − g15=2 3

q2 (1 − f)(g + 1)(g + f)sh + f)2 + 2q2 (g + 1)(g + f)[(2 − 1 )sh − ch]

sh = sinh[2(1 − 2 )]

ch = cosh[2(1 − 2 )]

[ − e1 (1 − e12 ) + 1 − e12 sin−1 (e1 )] 2e13

[e1 − (1 − e12 ) tanh−1 (e1 )] 2e13

1

Oblate √ 2 2 √a − b =a 2 − B2 =B A 3 e2 = 1 − e22 (1 − e22 )(1 − 2e22 )=(3 − 6e22 + 4e24 ) 2 (gf − g1 ) + 25 (gf5=2 − g15=2 )34 2 + 3 3 ln(gf =g1 ) gf = g=(g + f)

q2 (g + 1)(g + f)sh [1 − f + 2(1 − 1 )]

C

A.1.2. Shape change Shape change in the original model for a perfectly plastic solid. Gologanu et al. [12] introduced four correction functions to get the void shape change predicted by cell-model calculations for a perfectly plastic solid. The 1rst function, hs , depends on the current void shape hs =

9(1 − 1G ) ; 2(1 − 31 )

where 1G = 1=(3 − e12 ) for a prolate and 1G = (1 − e12 )=(3 − 2e12 ) for an oblate void. The second function, hf , depends on the current void volume fraction and the third function, hsf , depends on current void shape and void volume fraction

hf = (1 − f)2 ; hsf =






1 − 31 + 32 − 1 f

and hT is a polynomial function depending on the stress triaxiality ratio. For a strain hardening solid containing an ellipsoidal void, Pardoen and Hutchinson [21] adopted two diLerent functions hT

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Table 1 Constants used in the shape change parameter k in Eqs. (3.2)–(3.4)

1 2 3



+





0.73 −4.74 1.75

0.25 −2.58 13.74

0.09 −1.19 1.75

1.23 −2.85 3.07

for n = 0:1 and 0.3, respectively hT = 1 − 0:555T 2 − 0:045T 4 + 0:002T 6 hT = 1 − 0:54T 2 − 0:034T 4 − 0:00124T 6

for n = 0:1; for n = 0:3:

Shape change for the elasto-viscoplastic solid. In present work, the shape change is described using the 1nite element cell-model calculations to estimate the linear relation between the void axial dimension and the void radius. The void shape factor is given by relations (3.2)–(3.4). The constants are given in Table 1. A.2. Strain localization analysis A closed form expression may be given for the optimization parameter D in the localization model. On the macroscopic scale, strain localization will be searched for with the m-dependent porous material model and the closed form (3.5) for the localization stress. But the value of the optimization parameter D is very interesting in order to analyze the strain rate distribution throughout the matrix material before and after the onset of strain localization.   1   * ˜ ln(W0 ) + F (T )M ˜; M ˜ =  m ; D = ˜B(T )M   m2 ˜B(1) = [ − 0:023; −1:18; 1:25];

˜ F(1) = [ − 0:1; 0:36; 1:25];

˜B(2) = [ − 0:071; −1:05; 3:69];

˜ F(2) = [ − 0:33; 3:88; 3:69];

˜B(3) = [ − 0:056; 0:277; 0:76];

˜ F(3) = [ − 0:95; 11:49; 0:76];

where m and T are, respectively, the strain rate hardening parameter and the remote stress triaxiality. A.3. Numerical optimization procedure A.3.1. Cell-model results and numerical values Critical strain Ec . The exact values of Ec were determined by the cell model. For all cell shape factors 0 , all void shape factors W0 , all values of the strain rate hardening m and all values of the

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remote stress triaxiality T , the radial strain E11 was plotted against the overall equivalent strain Eeq . The critical strain Ec corresponds to the value of Eeq satisfying the following condition:    E11 (i + 1) − E11 (i)   6 2 × 10−4 → Ec = Eeq (i);    E33 (i) where i and i + 1 correspond to the sequence numbers of successive points in the plot E11 vs. Eeq . Void shape change factor k. The exact values of k were determined by the cell model. For all cell shape factors 0 , all void shape factors W0 , all values of the strain rate hardening m and all values of the remote stress triaxiality T , Y = (b=b0 ) − 1 was plotted against X = (a=a0 ) − 1 (Fig. 6). For strains smaller than Ec , the void shape change factor k was determined by minimizing the distance (least-squares method) the line Y = kX . A similar procedure was used for strains larger than Ec . Numerical constant D in the kinematically admissible velocity 9eld. The value of D changes slightly with the current cell and void geometry. Finally a single value of D was retained for given initial void and cell geometry and loading. For all cell shape factors 0 , all void shape factors W0 , loc: all values of the strain rate hardening m and all values of the remote stress triaxiality T , eq vs. Eeq was plotted for a given value of D. The intersection with the m-dependent model results was loc: determined. The value of D was changed until abscissa of the intersection between eq vs. Eeq curves and the eq vs. Eeq curves was as close as possible to Ec . A.3.2. Closed form expressions The same optimization procedure was used for determining the closed form expression of the void shape change factor k and the critical strain Ec and the optimization factor D. First the variation with each parameter has been determined by 1xing all others. The procedure is illustrated for the void shape change factor k. Similar procedures were used to determine a closed form for the critical strain Ec , the optimization factor D and the closed form (3.5) k = am (T; W0 ; 0 ; f0 ) + bm (T; W0 ; 0 ; f0 )m; k = aT (m; W0 ; 0 ; f0 ) + bT (m; W0 ; 0 ; f0 )=T + cT (m; W0 ; 0 ; f0 )=T 2 ; k = af (m; W0 ; 0 ; T ) + bf (m; W0 ; 0 ; T ) ln(f0 =fref ); k = a (m; W0 ; f0 ; T ) + b (m; W0 ; f0 ; T )(1=0 − 1) + c (m; W0 ; f0 ; T )((1=0 )2 − 1) loc in Eq. (3.5)) was plotted k (cell-model results for k and Ec , optimization results for D and 33 against m, 1=T , ln(f0 ) and 1=0 . The distance between numerical results and the assumed closed form expression was minimized using the least-squares method. ac ; bc ; aT ; : : : are the optimized coeTcients. This 1rst optimization was used to get a clear idea of the functions to be used. The 1nal optimization was based on the functions previously determined



 f0 C 1 B 1 am = A + ln 1+ 1+D −1 +E − 1 T fref W0 0 02 



 G 1 f0 1 F 1+ 1+H −1 +I −1 ; + 2 ln T fref W0 0 02

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