An ellipsoidal void model for simulating ductile fracture behavior

Accepted Manuscript An ellipsoidal void model for simulating ductile fracture behavior Kazutake Komori PII: DOI: Reference:

S0167-6636(13)00009-4 http://dx.doi.org/10.1016/j.mechmat.2013.01.002 MECMAT 2081

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Mechanics of Materials

Received Date: Revised Date:

31 August 2012 8 December 2012

Please cite this article as: Komori, K., An ellipsoidal void model for simulating ductile fracture behavior, Mechanics of Materials (2013), doi: http://dx.doi.org/10.1016/j.mechmat.2013.01.002

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An ellipsoidal void model for simulating ductile fracture behavior

Kazutake Komori

Department of Integrated Mechanical Engineering, School of Engineering, Daido University 10-3 Takiharu-town, Minami-ward, Nagoya-city, Aichi-prefecture 457-8530, Japan Tel: +81-52-612-6111; Fax: +81-52-612-5623; E-mail: [email protected]

Abstract: An ellipsoidal void model, which is based on a parallelogrammic void model, is proposed for simulating ductile fracture behavior. It is used to analyze ductile fracture behavior in three plastic deformation modes: plane strain tension, uniaxial tension, and simple shear. The relationship between the fracture strain and the initial void volume fraction in uniaxial tension calculated using the void model agrees with that calculated using a finiteelement void cell and agrees reasonably well with experimentally determined relationships in previous studies. For a specified initial void volume fraction, plane strain tension and simple shear respectively have the smallest and largest nominal fracture strains of the three plastic deformation modes.

Keywords: Ductile fracture, Micromechanics, Void coalescence, Uniaxial tension, Simple shear.

1

Nomenclature a, b

major and minor diameters of void, respectively

C

right Cauchy−Green deformation tensor

E

ratio of energy dissipation rate of internal necking to energy dissipation rate of homogeneous deformation

f , f

void volume fraction and fraction rate of material, respectively

f0

initial void volume fraction

k

shearing yield stress of material

2l1 , 2l2

lengths of velocity discontinuity lines

L, L ′

rectangle dimensions in x’- and y’-directions, respectively

n

strain hardening exponent

p

distance between two neighboring voids

R

orthogonal rotation tensor

r0

radius of cylindrical, spherical, or toroidal void

S0 , S f

initial and final cross-sectional areas, respectively

u , v, w

displacements in x- or r-, y-, and z-directions, respectively

u*, v*, w *

displacements in x-, y-, and z-directions, respectively

U

right stretch tensor

u′, v′

material velocities in x’- and y’-directions, respectively

V ,V0

volumes of material and matrix, respectively

Δv1 , Δv2

velocity discontinuities

xv , y v

x- and y-coordinates of neighboring void, respectively

ε

equivalent strain rate

εf

logarithmic fracture strain

εM

strain of matrix

θ1 , θ 2

angles between maximum principal stress and velocity discontinuity lines

2

λmax , λ min

maximum and minimum principal values of C , respectively

σ

equivalent stress

σ*

imposed hydrostatic stress

σM

tensile yield stress of matrix

φ

angle between line connecting two neighboring voids before deformation and x-axis

φa

angle between direction of major diameter of void a and x-axis

φmax

angle between maximum principal direction of C and x-axis

φR

angle of rotation due to R

Φ, Φ ′

original and approximate Gurson yield functions, respectively

Ψ, Ψ ′

original and approximate Gurson−Tvergaard yield functions, respectively

1. Introduction Ductile fracture, which occurs when a material is subjected to a large plastic deformation, is problematic in metal-forming processes. Consequently, it has been considerably investigated (Dodd and Bai, 1987). Microscopically, ductile fracture occurs through nucleation, growth, and coalescence of voids. Modeling void growth has been investigated by the finite-element method, which revealed how voids deform (Needleman, 1972; Tvergaard, 1981). However, void nucleation (Argon and Im, 1975; Goods and Brown, 1979; Le Roy et al., 1981) and coalescence are insufficiently understood. Several analytical studies have investigated void coalescence on a microscopic level (Thomason, 1990; Melander and Stahlberg, 1980; Koplik and Needleman, 1988; Padoen and Hutchinson, 2000; Benzerga, 2002; Ragab, 2004; Bacha et al., 2008). Of these studies, the series of studies by Thomason (1990) is the best known. Although Thomason proposed threedimensional void models (Thomason, 1985a, 1985b), two-dimensional void models (Thomason, 1968a, 1981, 1982) are still beneficial. Two-dimensional modeling of internal 3

necking of voids (Thomason, 1968a) is particularly effective since it employs a simple void model and is based on upper-bound theory (Avitzur, 1968), which is a method for analyzing metal-forming processes. The relationship between the fracture strain and the void volume fraction calculated using this void model agrees reasonably well with that obtained experimentally by Edelson and Baldwin (1962). Recently, there have been many experimental studies of void coalescence on a microscopic level (Worswick et al., 2001; Tinet et al., 2004; Narayanasamy and Narayanan, 2006; Weck and Wilkinson, 2008a; Weck et al., 2008b). The Thomason model (Thomason, 1968a) and the Melander and Stahlberg model (Melander and Stahlberg, 1980) (which was derived from the Thomason model) assume that voids are rectangular and that the longitudinal direction of a void coincides with the direction of maximum principal stress. In other words, these models assume that the principal strain direction remains constant during plastic deformation. However, these models cannot be utilized to simulate metal-forming processes since the principal strain direction varies during plastic deformation in these processes. In a previous study (Komori, 1999), we proposed a void model based on these models that can be utilized to simulate metal-forming processes. It assumes that voids are parallelograms and that the longitudinal direction of a void differs from the direction of maximum principal stress. In other words, it assumes that the direction of principal strain varies during plastic deformation. It is identical to the Melander and Stahlberg model in uniaxial tension. Hence, in uniaxial tension, the relationship between the fracture strain and the void volume fraction calculated using our void model is identical to that calculated using the Melander and Stahlberg model, while it is close to that calculated using the Thomason model. Our void model (Komori, 1999), the Thomason model (Thomason, 1968a), and the Melander and Stahlberg model (Melander and Stahlberg, 1980) all suffer from two problems. The first problem is that the fracture strain is calculated to be zero when the void volume fraction of the material exceeds 10%. However, the fracture strain obtained experimentally by 4

Edelson and Baldwin is nonzero even when the void volume fraction of the material exceeds 20%. The second problem is that the calculated relationship between the fracture strain and the void volume fraction has not been demonstrated for plastic deformation modes except for uniaxial tension. To overcome these problems, this study proposes an ellipsoidal void model based on our earlier parallelogrammic void model for simulating ductile fracture behavior. It is used to analyze ductile fracture behavior in three plastic deformation modes: plane strain tension, uniaxial tension, and simple shear. From a macroscopic point of view, ductile fracture is highly dependent on the stress triaxiality of a material (Bridgman, 1952), which is zero in simple shear. Hence, it is not easy to predict ductile fracture in simple shear from a macroscopic point of view (Pardoen, 2006; Barsoum and Faleskog, 2007a, 2007b). It is thus particularly valuable to obtain the relationship between the fracture strain and the void volume fraction for simple shear.

2. Analysis method 2.1. Overview of whole analysis Figure 1 shows an overview of the whole analysis procedure at each time step. Macroscopic analysis and microscopic analysis are performed alternately (Zhang and Niemi, 1995; Zhang et al., 2000; Komori, 2006a, 2006b, 2008). First, in macroscopic analysis, the deformation of the material is analyzed by the conventional rigid–plastic finite-element method. The displacement gradient rate and the void volume fraction rate of the material calculated by macroscopic analysis are utilized in the subsequent microscopic analysis. Next, in microscopic analysis, fracture of the material is evaluated by our void model. The microscopic analysis determines whether the material fractures and this information is utilized in the macroscopic analysis of the next time step.

5

Plane strain deformation or axisymmetric deformation is assumed in the macroscopic analysis, whereas plane strain deformation is assumed in the microscopic analysis. The reasons for these assumptions are explained below. When a void lies on the axisymmetric axis, it is appropriate to assume that the void is axisymmetric. However, when a void does not lie on the axisymmetric axis, it is inappropriate to assume that the void is axisymmetric, because there are no toroidal voids. Hence, when voids coalesce, it is not appropriate to assume axisymmetric deformation in the microscopic analysis. Furthermore, the analytical result by Thomason (1968a) agrees fairly well with the experimental result of a uniaxial tensile test of a bar conducted by Edelson and Baldwin (1962). Plane strain deformation is assumed in the analysis, whereas axisymmetric deformation occurs in the experiment. Therefore, in this study, plane strain deformation is assumed to occur when voids coalesce in the microscopic analysis. This assumption needs to be validated by further analysis and experiments in the future. The shape of a void is experimentally found to be three-dimensional. Hence, a spheroidal void should be assumed, whereas an ellipsoidal void is assumed. The reason for this assumption is explained below. Because it is not easy to assume a spheroidal void, Thomason (1985a, 1985b) assumed a square-prismatic void. Furthermore, Thomason (1985a, 1985b) assumed that the direction of the principal strain coincides with the direction of the axis of the square-prismatic void. In other words, the direction of the principal strain is assumed to remain unchanged during deformation. However, because the direction of the principal strain changes during deformation in metal forming processes, this assumption is not acceptable. In metal forming processes, plane strain deformation often appears before ductile fracture occurs. Metal forming processes are divided into two categories: sheet metal forming processes and bulk metal forming processes.

6

In sheet metal forming processes, localized necking of a sheet appears under biaxial stress before ductile fracture occurs. The material neither elongates nor contracts in the length direction of the localized necking region. In other words, plane strain deformation occurs in the plane perpendicular to the length direction. The theory of localized necking of a sheet by Hill (1952) supports plane strain deformation. In bulk metal forming processes, upsetting of a cylinder is a fundamental material test for evaluating ductile fracture. Many experimental studies (Kudo and Aoi, 1967; Thomason, 1968b; Kobayashi, 1970) showed that with decrease in material height, the axial strain decreases whereas the hoop strain increases at the equatorial plane of the material surface. However, accurate strain measurements show that radial strain remains constant before ductile fracture occurs, regardless of decrease in material height (Kuhn and Lee, 1971). In other words, plane strain deformation occurs. Therefore, in this study, an ellipsoidal void is assumed because plane strain deformation often appears before ductile fracture occurs. This assumption needs to be validated by further analysis and experiments in the future.

2.2. Overview of macroscopic analysis The deformation of the material is simulated by the conventional rigid–plastic finiteelement method (Kobayashi et al., 1989). The yield function Φ proposed by Gurson (1977) is adopted:

Φ=

⎛σ ⎞ 3 σ ij′ σ ij′ ⋅ 2 + 2 f cosh⎜⎜ kk ⎟⎟ − 1 − f 2 = 0, 2 σM ⎝ 2σ M ⎠

(1)

where σ M is the tensile yield stress of the matrix and f is the void volume fraction of the material. Since the yield function Φ is not a function of the second power of stress, it is not easy to perform rigid–plastic analysis using Equation (1). Hence, cosh x is approximated by

7

1 + x 2 2 (Tomita, 1990; Ragab and Saleh, 1999). The approximated yield function Φ′ used in this study is

Φ′ =

3 σ ij′ σ ij′ f ⎛ σ kk2 ⎞ 2 ⋅ 2 + ⋅ ⎜⎜ 2 ⎟⎟ − (1 − f ) = 0. 2 σM 4 ⎝σM ⎠

(2)

In metal-forming processes, a crack-free material will have a low maximum stress triaxiality, whereas a material with a crack will have a high maximum stress triaxiality. Hence, the accuracy of the hyperbolic cosine approximation is high in the simulation of crack initiation in the material. However, the accuracy of the hyperbolic cosine approximation is not high when simulating crack propagation in the material. The Gurson yield function (Gurson, 1977) is derived based on the assumption that the void is spherical, whereas the void is assumed to be spheroidal in this study. Since the yield function developed by Gologanu et al. (1993, 1994) is derived from the assumption that the void is spheroidal, it should be employed in this study. However, it is not easy to perform rigid–plastic analysis using the Gologanu yield function. Hence, the approximate Gurson yield function of Equation (2) is used in this study. According to the plastic potential theory and the flow rule, the relationship between the stress and the strain rate is given by (Tomita, 1990) 3 ε ⎛ f ⎞ ⎜ σ ij′ + σ kk δ ij ⎟, 2 σ⎝ 6 ⎠

(3)

⎞ 2 σ⎛ 2 ⎜⎜ εij′ + εkkδ ij ⎟⎟,  3 ε ⎝ 3f ⎠

(4)

εij = ⋅

σ ij = ⋅

where σ is the equivalent stress and ε is the equivalent strain rate. σ and ε are defined as (Tomita, 1990) 3 2

σ 2 = σ ij′ σ ij′ +

f 2 σ kk , 4

(5)

8

2 3

2⎛ 2 ⎞ 2 − 1⎟⎟εkk . 9⎝ f ⎠

ε 2 = εijεij + ⎜⎜

(6)

Although a void was assumed to nucleate in our previous studies (Komori, 1999, 2006a, 2006b, 2008), a void is not assumed to nucleate in this study for the sake of simplicity. We define the volume of matrix that does not contain voids as V0 and the volume of material that contains voids as V . Because the volume of voids is V − V0 , the void volume fraction of the material is f = (V − V0 ) V . By differentiating both sides with respect to time, the following evolution equation, which represents void growth, is obtained: V V V f = 0 2 = (1 − f ) = (1 − f )εkk . V V

(7)

2.3. Overview of microscopic analysis

An overview of the microscopic analysis at each time step, from obtaining the displacement gradient rate and the void volume fraction rate to determining whether the material fractures, is given below: (1) The displacement gradient rate ∂u ∂X and the void volume fraction rate f are obtained from the macroscopic analysis. (2) The deformation gradient ∂x ∂X is calculated. (3) The void shape and configuration are calculated. (4) The ratio of the energy dissipation rate of internal necking to that of homogeneous deformation E is calculated. (5) Whether the material fractures or not (i.e., whether the ratio E is less than unity or not) is used in the macroscopic analysis. The details of each of the above steps are explained in the following sections. The occurrence of fracture is determined based on the void volume fraction and the deformation gradient. Since these are nondimensional values, the void size is not required. In 9

other words, voids are assumed to be much smaller than the material itself. Furthermore, there are assumed to be no clusters of voids so that voids are uniformly distributed throughout the material.

2.4. Assumption about void shape and configuration

In our void model (Komori, 1999, 2006a, 2006b, 2008), we assume that voids are square or parallelogrammic. Since this fundamental assumption is unrealistic, we modified our void model by assuming that voids are circular or ellipsoidal. In other words, in the modified model, a square void becomes a circular void and a parallelogrammic void becomes an ellipsoidal void. Figure 2(a) shows the void shape and configuration before deformation and Figure 2(b) shows the void shape and configuration after deformation. The model makes the following assumptions about the void shape: • The void shape before deformation is a circle. • The void shape after deformation is calculated by assuming that the displacement gradient of the void is identical to that of the material. The model makes the following assumptions about the void configuration: • The void configuration before deformation is identical to the centroid configuration of the hexagonal grid. • The void configuration after deformation is calculated by assuming that the displacement gradient of the void is identical to that of the material. Analytical (Gologanu et al., 1993, 1994) and numerical studies (Fleck et al., 1989, Tvergaard, 1989) of void shape evolution have revealed that the deformation gradient of the voids differs from that of the material. However, the void shape calculated analytically is not necessarily accurate and calculating the void shape numerically requires long computation times. Furthermore, in our previous studies (Komori, 1999, 2006a, 2006b, 2008), it is 10

assumed that the deformation gradient of the voids equals that of the material, and the result of the simulation of ductile fracture behavior in metal forming processes agrees fairly well with the experimental result. Hence, for simplicity, we assume that the deformation gradient of the voids equals that of the material. This assumption simplifies the calculation of the void shape, thereby reducing the computation time. However, as is clarified in Section 3 by comparing the analytical result using our void model with the simulation result using a finiteelement void cell, this assumption is not necessarily appropriate and should be improved in the future.

2.5. Void shape and configuration

Figure 3 shows the void shape and configuration before deformation. When the area of the hexagonal grid is taken to be unity, the area of the triangle denoted by the dashed lines will be one half. If the distance between two neighboring voids is assumed to be p , the area of the triangle will be

3 p 2 4 . Hence, p = 4 4 3 . When the initial void volume fraction is

expressed as f 0 , the area of a circle that denotes a void will be f 0 . Figures 4(a) and (b) show the void shapes and configurations of two neighboring voids before and after deformation, respectively. Two neighboring voids before deformation are still adjacent after deformation. The angle between the line connecting two neighboring voids before deformation and the x-axis is denoted by φ (see Figure 4(a)). We consider void O in Figure 3 and refer to it hereafter as the central void. Voids A, B, C, D, E, and F are adjacent to the central void. Whether the fracture criterion is satisfied between the central void and the adjacent voids should be determined. Since the coordinate axes are undefined in Figure 3, φ will have an arbitrary value between 0 and 2π . The void shape after deformation is calculated by polar decomposition of the deformation gradient ∂x ∂X (Mase, 1970). The right polar decomposition is written as

11

∂x = R ⋅U , ∂X

(8)

where R is the orthogonal rotation tensor and U is the right stretch tensor. To perform right polar decomposition, the right Cauchy−Green deformation tensor C , which is defined as

(∂x

∂X ) ⋅ (∂x ∂X ) , has to be calculated in advance. T

The major diameter of void a and the minor diameter of void b are calculated using a=

4

λmax f ⋅ λmax λmin π

λmin f b= ⋅ 4 λ π max λmin

,

(9)

where λmax and λmin are respectively the maximum and minimum principal values of C . The angle between the direction of the major diameter of void a and the x-axis is denoted by φa in Figure 4(b) and is calculated using

φa = φmax + φR ,

(10)

where φmax is the angle between the maximum principal direction of C and the x-axis and φR is the angle of rotation due to R . The void configuration after deformation is calculated by the deformation gradient ∂x ∂X . The x-coordinate of the neighboring void xv and the y-coordinate of the neighboring void yv are calculated using ⎛ ∂x ⎞ ⎛ ∂x ⎞ p cos φ ⋅ ⎜ ⎟ + p sin φ ⋅ ⎜ ⎟ ∂X ⎠ ∂Y ⎠ ⎝ ⎝ xv = ⎛ ∂x ⎞ det⎜ ⎟ ⎝ ∂X ⎠ ⎛ ∂y ⎞ ⎛ ∂y ⎞ p cos φ ⋅ ⎜ ⎟ + p sin φ ⋅ ⎜ ⎟ ∂X ⎠ ∂Y ⎠ ⎝ ⎝ yv = ⎛ ∂x ⎞ det⎜ ⎟ ⎝ ∂X ⎠

.

(11)

Our void model can be applied to axisymmetric deformation in the macroscopic analysis as well as to plane strain deformation in the macroscopic analysis. In plane strain deformation in

12

the macroscopic analysis, the area of the hexagonal grid does not change since the out-ofplane strain is zero. However, in axisymmetric deformation in the macroscopic analysis, the area of a hexagonal grid changes since the out-of-plane strain is not zero. The determinant of the deformation gradient det (∂x ∂X ) is proportional to the area of the hexagonal grid. Since the area of the hexagonal grid is assumed to be unity, the x- and y-coordinates of the neighboring void are divided by

det (∂x ∂X ) to ensure that the out-of-plane strain does not

change the area of the hexagonal grid.

2.6. Determining energy dissipation rates of internal necking and homogeneous deformation and estimating material fracture

Figure 5 shows the void shape and configuration and the velocity field. Plane strain deformation is assumed. The material is first assumed to deform homogeneously. Figure 5(a) shows the void shape and configuration and the velocity field for homogeneous deformation. The y’-direction is made to coincide with the direction of the maximum principal stress and the x’-direction is made to coincide with the direction of the minimum principal stress. The rectangle indicated by the dotted lines is considered. The length of the rectangle in the x’-direction is L , which is assumed to be the distance between two neighboring voids, and the length of the rectangle in the y’-direction is L′ . The material velocity in the y’-direction is v′ and the material velocity in the x’-direction is u′ . u ′ = −(L L ′)v ′ because of the incompressibility condition. Hence, using the yield criteria, the energy dissipation rate in the material due to homogeneous deformation (i.e., the energy dissipation rate of homogeneous deformation) is expressed by

2v′Lσ max + 2u′L′σ min = 2v′Lσ max − 2(L L′)v′L′σ min = 2v′L(σ max − σ min ) = 4kv′L ,

(12)

where k denotes the shearing yield stress of the material. In calculating the energy dissipation rate of homogeneous deformation, the void volume fraction of the material is assumed to be 13

zero. However, in reality, the void volume fraction of the material is not zero but f . Since the energy dissipation rate in a material whose void volume fraction is zero is 4kv′L , the energy dissipation rate in the material whose void volume fraction is f is assumed to be

(1 − f )4kv′L ; this expression is used in the place of 4kv′L

below.

Next, the material is assumed to deform heterogeneously; i.e., voids are assumed to coalesce by internal necking. Figure 5(b) shows the void shape and configuration and the velocity field for internal necking. The dotted lines indicate the velocity discontinuity line; it is assumed to coincide with the tangent of the two neighboring voids. θ1 and θ 2 denote the angles between the maximum principal stress and the velocity discontinuity line. 2l1 and 2l2 denote the lengths of the velocity discontinuity line. The velocity field is shown as a hodograph. Δv1 and Δv2 denote the velocity discontinuities. The velocity discontinuities are calculated using the law of sines: Δv1 Δv2 2v′ . = = sin θ1 sin θ 2 sin (π − θ1 − θ 2 )

(13)

From Equation (13), the velocity discontinuities are given by Δv1 =

sin θ1 2v ′ sin (θ1 + θ 2 )

sin θ 2 2v ′ Δv2 = sin (θ1 + θ 2 )

.

(14)

Hence, the energy dissipation rate on the velocity discontinuity line when two neighboring voids coalesce by internal necking (i.e., the energy dissipation rate of internal necking) is expressed by k ⋅ Δv1 ⋅ 2l2 + k ⋅ Δv2 ⋅ 2l1 = 4kv′

l1 sin θ 2 + l2 sin θ1 . sin (θ1 + θ 2 )

(15)

Fracture is assumed to occur when the energy dissipation rate of internal necking is lower than that of homogeneous deformation. Hence, using Equations (12) and (15), the fracture criterion can be expressed by 14

(1 − f )L ≥ l1 sin θ 2 + l2 sin θ1 . sin (θ1 + θ 2 )

(16)

The ratio of the energy dissipation rate of internal necking to the energy dissipation rate of homogeneous deformation E is defined as E=

l1 sin θ 2 + l2 sin θ1

(1 − f )L sin (θ1 + θ 2 )

.

(17)

The fracture criterion is satisfied when E is less than unity. Hence, the angle between the line connecting two neighboring voids before deformation and the x-axis (i.e., φ in Figure 4(a)) is optimized by minimizing E . The fracture criterion is satisfied when E is less than unity for a certain value of φ ; in other words, we assume that the material fractures. In the Thomason model (Thomason, 1968a, 1985a, 1985b), the Melander and Stahlberg model (Melander and Stahlberg, 1980), and our void model, which are based on the upperbound theory (Avitzur, 1968), homogeneous deformation and heterogeneous deformation (deformation for internal necking) are assumed. Furthermore, voids are assumed to coalesce when the energy dissipation rate of heterogeneous deformation equals the energy dissipation rate of homogeneous deformation. The energy dissipation rate divided by both the rigid-body velocity of the material and the equivalent stress of the material is the plastic constraint factor. In the three-dimensional Thomason model (1985a, 1985b), the numerical calculation of the plastic constraint factor requires substantial computational time when the shapes of the material and void are specified. It is essential to expend little computational time in calculating the plastic constraint factor, in other words, evaluating whether voids coalesce, because ductile fracture occurs through not only void coalescence but also void nucleation and void growth. Hence, approximate closedform equations that express the relationship between the plastic constraint factor and the shapes of the material and void have been proposed by Pardoen and Hutchinson (2000) and

15

Benzerga (2002), respectively, to calculate the plastic constraint factor using little computational time. In the two-dimensional Thomason model (Thomason, 1968a), the Melander and Stahlberg model (Melander and Stahlberg, 1980), and our void model, the relationship between the plastic constraint factor and the shapes of the material and void are expressed in closed form, respectively. Hence, the plastic constraint factor is calculated, in other words whether voids coalesce is evaluated, using little computational time. As the material deforms, the plastic constraint factor of heterogeneous deformation decreases, whereas the plastic constraint factor of homogeneous deformation increases and voids are assumed to coalesce when the plastic constraint factor of heterogeneous deformation coincides with the plastic constraint factor of homogeneous deformation. According to the upper-bound theory (Avitzur, 1968), the deformation occurs in which the plastic constraint factor is the smallest among all the types of deformation. Hence, homogeneous deformation occurs before void coalescence, whereas heterogeneous deformation occurs after void coalescence. Furthermore, the plastic constraint factor becomes the maximum at void coalescence (Thomason, 1985b). Hence, Bandstra and Koss (2008) assume that the plastic constraint factor becomes the maximum at void coalescence in the simulation of void clusters using the finite-element method. In most of the finite-element simulation using the Gurson-Tvergaard yield function (Tvergaard, 1981), a void is assumed to coalesce when the void volume fraction reaches a critical value. However, the physical meaning of the critical value is not clear. Hence, Zhang and Niemi (1995) evaluated void coalescence using the Thomason model (1985a, 1985b) in the finite-element simulation with the Gurson-Tvergaard yield function. The shearing yield stress of the material k is assumed to be constant in the microscopic analysis, whereas k is not assumed to be constant in the macroscopic analysis. Hence, it is not necessary for k to be constant in the macroscopic analysis. 16

3. Analytical results

This study principally discusses the characteristics of our void model. Hence, in some cases, analysis is not performed by finite-element analysis since no void growth is assumed. The deformation is assumed to be uniform so that no necking deformation occurs and a simple stress−strain relationship, σ M = const. , is assumed, except in Sections 3.4 and 3.5 where the effect of strain hardening is considered. To evaluate our void model, simulations using finite-element void cells are described in Sections 3.1, 3.2, and 3.3. Furthermore, analytical results are compared with experimental results in the literature in Section 3.5.

3.1. Plane strain tension

Figure 6 shows the relationship between the initial void volume fraction and the nominal fracture strain for plane strain tension. Figure 6(a) shows the coordinates and the notation used. Since the length of the analysis region in the y-direction is unity, the displacement in the y-direction v * coincides with the nominal strain. The solid and dotted lines denote the

material and void shapes before and after deformation, respectively. Figure 6(b) shows the relationship between the initial void volume fraction and the nominal fracture strain. When no void growth is assumed, it is not necessary to perform the macroscopic analysis, so that only the microscopic analysis is performed. Figure 6(b) also shows the relationship for the rectangular void calculated using our void model. The void shape before deformation is assumed to be square. When the initial void volume fraction exceeds 12%, the nominal fracture strain of a rectangular void is zero, whereas that of an ellipsoidal void is not zero. When the initial void volume fraction is specified, the nominal fracture strain of an ellipsoidal void with void growth is smaller than that of an ellipsoidal void without void growth. However, the initial

17

void volume fraction below which the nominal fracture strain is zero for an ellipsoidal void with void growth is the same as that for an ellipsoidal void without void growth. Figure 7 shows the final void shape and configuration for plane strain tension. The final state is the state that satisfies the fracture criterion. The velocity discontinuity lines are indicated in the figures depicting the final state. Figure 7(a) shows the final void shape and configuration when the initial void volume fraction is 5%. Since the direction of the minimum principal strain coincides with the xdirection, the angle between the line connecting two neighboring voids before deformation and the x-axis (i.e., φ in Figure 4(a)) is equal to zero. The void volume fraction of the final state of an ellipsoidal void when void growth occurs is larger than that when no void growth occurs. The angles between the two velocity discontinuity lines when void growth does and does not occur are 81° and 84°, respectively. The angle between the two velocity discontinuity lines of a rectangular void is 57°. Figure 7(b) shows the final void shape and configuration when the nominal fracture strain v * is zero. When the nominal fracture strain v * is zero, the fracture criterion is satisfied

when the nominal strain is zero. The state in which the nominal strain is zero is the initial state. Hence, the final void shape and configuration are identical to the initial void shape and configuration. The void volume fractions of the ellipsoidal and rectangular voids are 31.5% and 12%, respectively. The angles between the two velocity discontinuity lines for ellipsoidal and rectangular voids are 72° and 51°, respectively. These angles increase with increasing void volume fraction when the void configuration does not change. The difference between the void volume fractions of the ellipsoidal and rectangular voids results in a difference between the angles for the ellipsoidal and rectangular voids. A simulation using a finite-element void cell is performed to validate our void model. The deformation of the material is simulated by the conventional rigid–plastic finite-element

18

method (Kobayashi et al., 1989). The incompressibility of the material is satisfied by the penalty method. Figure 8 shows the finite-element void cell for plane strain tension. Figure 8(a) shows the coordinates and the notation used. Although the shape of the void cell should be hexagonal, it is not easy to choose suitable boundary conditions for the displacement. Furthermore, the rectangular finite-element void cell is widely used; hence, the shape of the void cell is assumed to be rectangular. r0 denotes the radius of the cylindrical void, which is equal to f 0 π , where f 0 denotes the initial void volume fraction. The following boundary

conditions on the displacement are assumed, u = 0 on

x = 0 and

u = const. on

x =1 2

v = 0 on

y = 0 and

v = const. on v = 1 2

(18) where u and v denote the displacements in the x- and y-directions, respectively. Hence, the rectangular void cell remains rectangular after the displacement. Figure 8(b) shows the initial finite-element mesh when the initial void volume fraction is 1%. The simulation is performed until the displacement in the x-direction u on the outer side of the void cell at y = 0 is negligible compared with the displacement in the x-direction u on the inner side of the void cell at y = 0 . The flow of the material is then assumed to be localized in the plane y = 0 . Hence, the material is assumed to fracture. Figure 8(c) shows the final finite-element mesh when the initial void volume fraction is 1%. The void elongates mainly in the y-direction. Since the void length in the x-direction is not a maximum on y = 0 , the void shape is not elliptical. Since the void shape is assumed to be elliptical in our void model, the void shape in the finite-element void cell differs from the void shape in our void model. Since the deformation of the material near the void is heterogeneous, remeshing is performed at each time step. Remeshing is difficult to perform in the elastic–plastic finiteelement method since the nodal force at each node needs to be redistributed after remeshing.

19

However, redistribution of the nodal force at each node is not required after remeshing in the rigid–plastic finite-element method so that remeshing is easy to perform. Figure 9 shows the simulation results using the finite-element void cell and our void model for plane strain tension. Figure 9(a) shows the relationship between the initial void volume fraction and the nominal fracture strain. With increasing initial void volume fraction, the nominal fracture strain calculated by the finite-element void cell and that calculated by our void model decrease. The initial void volume fraction above which the nominal fracture strain is zero using our void model is almost the same as that using the finite-element void cell. When the initial void volume fraction is larger than 20%, the nominal fracture strain calculated by our void model is slightly larger than that calculated by the finite-element void cell, whereas when the initial void volume fraction is smaller than 10%, the nominal fracture strain calculated by our void model is larger than that calculated by the finite-element void cell. Figure 9(b) shows the relationship between the nominal strain and the y-coordinate of point A. The y-coordinate of point A is equal to half the void length in the y-direction. Since the direction of the maximum principal stress coincides with the y-direction, the void length in the y-direction is essential in our void model. With increasing nominal strain, the y-coordinate of point A calculated by our void model and that calculated by the finite-element void cell increase. However, the y-coordinate of point A calculated by our void model is smaller than that calculated by the finite-element void cell. The following discussion of our void model is based on Figure 9. When the initial void volume fraction is smaller than 10%, the nominal fracture strain is relatively large. Hence, the difference between the y-coordinate of point A calculated by our void model and that calculated by the finite-element void cell is relatively large. Therefore, the difference between the nominal fracture strain calculated by our void model and that calculated by the finiteelement void cell is relatively large. When the initial void volume fraction is larger than 20%, 20

the nominal fracture strain is relatively small. Hence, the difference between the y-coordinate of point A calculated by our void model and that calculated by the finite-element void cell is relatively small. Therefore, the difference between the nominal fracture strain calculated by our void model and that calculated by the finite-element void cell is relatively small. The assumption regarding the void shape in our void model, namely that the displacement gradient of the void is identical to the displacement gradient of the material, should be improved in future.

3.2. Simple shear Figure 10 shows the relationship between the initial void volume fraction and the nominal fracture strain for simple shear. Figure 10(a) shows the coordinates and the notation used. Since the length of the analysis region in the y-direction is equal to unity, the displacement in the x-direction u * coincides with the nominal strain. The solid and dotted lines denote the material and void shapes before and after deformation, respectively. Figure 10(b) shows the relationship between the initial void volume fraction and the nominal fracture strain. According to Equation (3), the volumetric strain rate εkk = (3 4 ) ⋅ (ε σ )⋅ fσ kk . Since the stress triaxiality (σ kk 3σ ) is zero in simple shear, the volumetric strain rate εkk is zero. Hence, no void growth is assumed to occur. It is thus not necessary to perform the macroscopic analysis and only the microscopic analysis is performed. In plane strain tension, since the direction of the minimum principal strain coincides with the x-direction, the angle between the line connecting two neighboring voids before deformation and the x-axis (i.e., φ in Figure 4(a)) is zero. Hence, it does not appear to be necessary to optimize φ . However, it is necessary to optimize φ in simple shear since the direction of the minimum principal strain varies during deformation. Furthermore, to demonstrate the effect of optimization, it is beneficial to fix φ rather than to optimize φ . Hence, φ is fixed for comparison. φ is set to − π 4 , since the angle between the direction of 21

the minimum principal strain and the x-axis is equal to − π 4 when the nominal strain is infinitesimally small. The nominal fracture strain for fixed φ is almost the same as that for optimized φ when the initial void volume fraction exceeds 25%. However, the nominal fracture strain for fixed

φ differs greatly from that for optimized φ when the initial void volume fraction is smaller than 22%. Figure 11 shows the final void shape and configuration for simple shear. The final state is the state that satisfies the fracture criterion. The velocity discontinuity lines are indicated in the figures depicting the final state. Figure 11(a) shows the final void shape and configuration when the initial void volume fraction is 22%. It shows both the final state for fixed φ and the final state for optimized φ . The central void (i.e., void O in Figure 3) coalesces with one of the adjacent voids (i.e., void A, B, C, D, E, or F) for both fixed and optimized φ . φ = −45° and −38° for fixed and optimized φ , respectively. Figure 11(b) shows the final void shape and configuration when the initial void volume fraction is 16%. It shows both the final state for fixed φ and the final state for optimized φ . The central void coalesces with one of the adjacent voids for optimized φ . However, the central void does not coalesce with one of the adjacent voids for fixed φ ; instead nonadjacent voids (e.g., voids A and D in Figure 3) coalesce. φ = −45° and −35° for fixed and optimized φ , respectively. The optimized value of φ increases with decreasing initial void volume fraction. A simulation using the finite-element void cell is performed to validate our void model. Figure 12 shows the finite-element void cell for simple shear. Figure 12(a) shows the coordinates and the notation used. r0 denotes the radius of the cylindrical void; it is equal to f 0 π , where f 0 denotes the initial void volume fraction.

22

Periodic boundary conditions are applied (Barsoum and Faleskog, 2007b; Scheyvaerts et al., 2011), which is not straightforward to do. Periodic boundary conditions are required when two neighboring void coalesce. However, the simulation is not performed until two neighboring voids coalesce. Hence, the following boundary conditions for the displacement are assumed u = y on

x = ± 1 2 and

y = ±1 2

(19) v = 0 on

x = ± 1 2 and

y = ±1 2

where u and v denote the displacements in the x- and y-directions, respectively. Hence, a rectangular void cell becomes parallelogrammic after the displacement. Figure 12(b) shows the initial finite-element mesh when the initial void volume fraction is 10%. Since the angle between the line connecting two neighboring voids before deformation and the x-axis (i.e., φ in Figure 4(a)) is optimized in simple shear, this angle is not known in advance. Hence, the location of the neighboring void is not known in advance. It is thus impossible to perform the simulation until two neighboring voids coalesce; in other words, until the flow of the material is localized. The simulation is then performed until the nominal strain reaches a certain value. Figure 12(c) shows the final finite-element mesh when the initial void volume fraction is 10%. The final state is the state in which the nominal strain becomes unity. The void elongates in the direction of the diagonal of the parallelogrammic void cell. Since the deformation of the material near the void is heterogeneous, remeshing is performed at each time step. Figure 13 shows the simulation results obtained using the finite-element void cell and our void model for simple shear. Figure 13(a) shows the relationship between the nominal strain and the x-coordinate of point A. The simulation is performed until the nominal strain becomes two when the initial void volume fraction is 1%, whereas the simulation is performed until the nominal strain becomes unity when the initial void volume fraction is 10%. With increasing

23

nominal strain, the x-coordinate of point A calculated by the finite-element void cell and that calculated by our void model increase. However, the x-coordinate of point A calculated by our void model is smaller than that calculated by the finite-element void cell. In other words, the void rotates relative to the material. This rotation of the void in simple shear is simulated by employing periodic boundary conditions (Scheyvaerts et al., 2011). Figure 13(b) shows the relationship between the nominal strain and the y-coordinate of point A. When the nominal strain is increased, the y-coordinate of point A calculated by our void model remains constant, whereas the y-coordinate of point A calculated by the finiteelement void cell changes slightly. The y-coordinate of point A calculated by our void model almost coincides with that calculated by the finite-element void cell. The square root of the sum of the square of the x-coordinate of point A and the square of the y-coordinate of point A are approximately equal to half the maximum void length. As is shown in Figure 11, the maximum void length is essential in our void model. Figure 13 shows that the maximum void length calculated by our void model is smaller than that calculated by the finite-element void cell. Hence, the assumption regarding the void shape in our void model, namely that the displacement gradient of the void is identical to the displacement gradient of the material, should be improved in the future. It is well known that under some stress and strain conditions, deformation is localized in a shear band between two neighboring voids (Cox and Low, 1974; Haynes and Gangloff, 1998; Weck and Wilkinson, 2008a). Because the velocity discontinuity lines shown by dotted lines in Figure 5(b) are assumed, both sides of a void expand during the deformation of internal necking. However, according to the result of the simulation using the finite-element void cell (Tvergaard, 1989), one side of a void expands whereas the other side shrinks during the deformation of shear localization. Hence, the deformation of shear localization differs from the deformation of internal necking in this study. The velocity field, which represents the deformation of shear localization, should be incorporated into our void model in the future. 24

Ductile fracture depends not only on the stress triaxiality but also on the Lode parameter (Zhang et al., 2001; Wierzbicki et al., 2005). The deformation of shear localization is related to the Lode parameter (Nahshon and Hutchinson, 2008). Hence, the Lode parameter should be incorporated into our void model in the future.

3.3. Uniaxial tension Figure 14 shows the relationship between the initial void volume fraction and the nominal fracture strain for uniaxial tension. Axisymmetric deformation of the material is assumed in the macroscopic analysis of uniaxial tension. Figure 14(a) shows the coordinates and the notation used. Since the length of the analysis region in the z-direction is equal to unity, the displacement in the z-direction w * coincides with the nominal strain. Figure 14(b) shows the relationship between the initial void volume fraction and the nominal fracture strain. The relationship for when there is no void growth for plane strain tension is also shown for comparison. When the initial void volume fraction is specified, the nominal fracture strain when there is no void growth in uniaxial tension is larger than that when there is no void growth in plane strain tension. The reason for this difference is that the strain in the out-ofplane direction is not zero in uniaxial tension. When the initial void volume fraction is specified, the nominal fracture strain when there is void growth in uniaxial tension is smaller than that when there is no void growth in uniaxial tension. Interestingly, the relationship when there is void growth in uniaxial tension almost coincides with that when there is no void growth in plane strain tension. The simulation using the finite-element void cell is performed to validate our void model. Figure 15 shows the coordinates and notation in the finite-element void cell for uniaxial tension. Figure 15(a) shows the void cell containing a spherical void. r0 denotes the radius of the spherical void. Since in our void model, the void volume fraction is defined in a plane, the void volume fraction is defined in the rz-plane in uniaxial tension. Hence, r0 = 25

f 0 π , where

f 0 denotes the initial void volume fraction. The following boundary conditions are assumed

on displacement u = 0 on r = 0 and

u = const. on r = 1 2

(20) w = 0 on

z = 0 and

w = const. on

z =1 2

where u and w denote the displacements in the r- and z-directions, respectively. Hence, the cylindrical cell remains cylindrical after the displacement. Simulations using a void cell containing a spherical void have been performed in previous studies (Koplik and Needleman, 1988; Tvergaard, 1982). The void cell in these studies is the same as that in this study, which is shown in Figure 15(a). The boundary conditions in these studies are the same as those in this study, which are represented by Equation (20). In these studies, it is assumed that the void configuration in the plane z = 0 is identical to the centroid configuration of the hexagonal grid, and that there are neighboring voids to coalesce with the spherical void. However, since an axisymmetric deformation of a cylindrical void cell containing a spherical void is assumed, it is clear that there are no neighboring voids to coalesce with the spherical void. Figure 15(b) shows a void cell containing a spherical void and a toroidal void. r0 is the radius of the spherical and toroidal voids; r0 =

f 0 π . The following boundary conditions

are assumed on displacement u = 0 on r = 0 and

u = const. on r = 1

w = 0 on

w = const. on

(21) z = 0 and

z =1 2

so that the cylindrical cell remains cylindrical after the deformation. When axisymmetry is assumed, an infinite number of void cells can be assumed to exist in the z-direction, whereas only one void cell can be assumed to exist in the r-direction. The reason for this is as follows. When axisymmetry is assumed, a spherical void whose center is located on the z-axis can be assumed. However, a spherical void whose center does not lie on the z-axis cannot be assumed. Although there are no toroidal voids in reality, a toroidal void

26

whose center lies on the z-axis must be assumed. When a toroidal void is assumed, it is clear that only one void cell can be assumed to exist in the r-direction. The simulation is performed until the displacement in the r-direction u on the outer side of the void cell at z = 0 is negligible compared with the displacement in the r-direction u on the inner side of the void cell at z = 0 . The flow of the material is then assumed to be localized in the plane z = 0 . Hence, the material is assumed to fracture. In Koplik and Needleman (1988), the flow of the material is assumed to be localized when the radius of the void cell remains constant. Hence, it is clear that the condition for assuming the flow localization in this study is virtually the same as that in the study of Koplik and Needleman (1988). Figure 16 shows the final finite-element meshes in the finite-element void cell for uniaxial tension. The initial finite-element mesh containing a spherical void and a toroidal void is made to coincide with twice the initial finite-element mesh containing a spherical void. Since the deformation of the material near the void is heterogeneous, remeshing is performed at each time step. Figure 16(a) shows the final finite-element mesh containing a spherical void and a toroidal void when the initial void volume fraction is 1%. Although the length of the spherical void in the r-direction differs from the length of the toroidal void in the r-direction, the length of the spherical void in the r-direction and the length of the toroidal void in the r-direction are largest on z = 0 . Figure 16(b) shows the final finite-element mesh containing a spherical void and the final finite-element mesh containing a spherical void and a toroidal void when the initial void volume fraction is 20%. The void cell containing a spherical void has a much larger deformation than the void cell containing a spherical void and a toroidal void. In the void cell containing a spherical void, the spherical void elongates in the z-direction becoming needle-like; this has been theoretically predicted by other researchers for a viscous material (Budiansky et al., 1982).

27

Figure 17 shows the simulation results obtained using the finite-element void cell and our void model for uniaxial tension. Figure 17(a) shows the relationship between the initial void volume fraction and the nominal fracture strain. The nominal fracture strain decreases with increasing initial void volume fraction. When the initial void volume fraction is specified, the nominal fracture strain calculated by the void cell containing a spherical void is much larger than that calculated by our void model. This is because there are no neighboring voids to coalesce with the spherical void. When the initial void volume fraction is specified, the nominal fracture strain calculated by the void cell containing a spherical void and a toroidal void is almost equal to that calculated by our void model. This is because the toroidal void coalesces with the spherical void. It is clear that the relationship between the initial void volume fraction and the nominal fracture strain for uniaxial tension calculated by the simulation using the axisymmetric finite-element void cell containing a spherical void is inappropriate. Hence, a simulation using a three-dimensional finite-element void cell containing a spherical void (Worswick and Pick, 1990) is required to obtain the relationship between the initial void volume fraction and the nominal fracture strain for uniaxial tension. Figure 17(b) shows the relationship between the nominal strain and the z-coordinate of point A. Since the direction of the maximum principal stress coincides with the z-direction, the void length in the z-direction is essential in our void model. With increasing nominal strain, the z-coordinate of point A calculated by our void model and that calculated by the finite-element void cell increase. Furthermore, the z-coordinate of point A calculated by our void model is almost equal with that calculated by the finite-element void cell. Figure 17 demonstrates that the assumption about the void shape in our void model, namely that the displacement gradient of the void is identical to the displacement gradient of the material, is appropriate for uniaxial tension.

28

3.4. Effect of strain hardening

Figure 18 shows the relationship between the initial void volume fraction and the nominal fracture strain for various strain hardening exponents. The stress−strain relationship of the matrix is assumed to be

σ M = (ε M + 0.01)n ,

(22)

where ε M denotes the strain of the matrix and n denotes the strain hardening exponent. The relationship obtained in the previous section is supplemented for the case when necking does not occur since the material is assumed to deform uniformly. The relationship between the initial void volume fraction and the nominal fracture strain depends very little on the strain hardening exponent. Figure 19 shows the finite-element meshes for various strain hardening exponents. The initial void volume fraction is 1%. When n = 0, necking occurs if ε M is positive. When n = 0.5, necking occurs if ε M > 0.5. Hence, the finite-element mesh of the final state when n = 0 differs from that when n = 0.5 . However, the cross-sectional area of the final state when n = 0 is approximately the same as that when n = 0.5 . The logarithmic fracture strain ε f is

calculated using the formula ε f = ln (S 0 S f ), where S 0 and S f are the initial and final crosssectional areas, respectively. Hence, ε f when n = 0 (i.e., 1.09) is approximately the same as

ε f when n = 0.5 (i.e., 1.16).

3.5. Comparison with experimental results

Figure 20 shows the experimental and simulation results using our void model for uniaxial tension. Figure 20(a) shows the experimental and simulation results for copper. The relationship calculated using our void model is compared with that calculated using the Thomason model (Thomason, 1968a) and that obtained experimentally by Edelson and Baldwin (1962). The relationship obtained experimentally (Edelson and Baldwin, 1962) was compared with the relationship calculated in other studies (Ragab, 2000, 2004); the calculated 29

relationship did not agree with the experimental relationship. Since the strain hardening exponent of the matrix was 0.5 in the experiment, we set n = 0.5 in our void model. The experiment is described below. The details of experimental procedure are given in the appendix of the study (Edelson and Baldwin, 1962). Tensile specimens were prepared by powder metallurgical methods. Tensile tests were performed for two-phase copper-base alloys with various second-phase particles, including metals, nonmetals, and voids with a 20-fold range in particle size and with volume fractions as high as 0.24. The logarithmic fracture strain ε f was calculated using the formula ε f = ln (S 0 S f ), where S 0 and S f are the original and fracture specimen cross-sectional areas, respectively. Since the strain hardening exponent was not zero, necking deformation was conjectured to occur. However, the deformation was assumed to be uniform in the fracture specimen cross section. When the initial void volume fraction exceeds 10%, the nominal fracture strain calculated using the Thomason model is zero, whereas that calculated using our void model is not zero. The nominal fracture strain obtained experimentally by Edelson and Baldwin is not zero, even when the initial void volume fraction exceeds 20%. The nominal fracture strain calculated using our void model agrees with that obtained experimentally by Edelson and Baldwin when the initial void volume fraction is below 5%. However, the nominal fracture strain calculated using our void model is slightly larger than that obtained experimentally by Edelson and Baldwin when the initial void volume fraction is above 5%. Figure 20(b) shows the experimental and simulation results for iron. The relationship calculated using our void model is compared with that calculated using the Thomason model (Thomason, 1968a) and that obtained experimentally by Spitzig et al. (1988). Since the strain hardening exponent of the matrix was 0.45 in the experiment, we set n = 0.45 in our void model. The experiment is described below. The iron used in this study was prepared from water atomized iron powder. Compacts were prepared from the powder using varying compacting 30

pressures to achieve different densities. After the iron bars has been molded, they were sintered in dry hydrogen. The logarithmic fracture strain ε f was calculated using

ε f = ln (S 0 S f ), where S 0 and S f are the initial and final cross-sectional areas of the specimen, respectively. The relationship calculated using our void model agrees with that obtained experimentally by Spitzig et al.

3.6. Effect of yield function

The Gurson yield function Φ is derived based on the assumption of spherically symmetric deformation of the material around a spherical void. Hence, the effect of the presence of surrounding voids on the material deformation around a central void was not considered. It is of considerable interest to estimate the effect of the presence of surrounding voids on the material deformation around a central void. Tvergaard (1981) proposed a modified Gurson yield function, such that the simulation result using the modified Gurson yield function approaches the simulation result using a finite-element unit cell model. The Gurson−Tvergaard yield function Ψ is Ψ=

⎛σ 3 σ ij′ σ ij′ ⋅ 2 + 3 f cosh⎜⎜ kk 2 σM ⎝ 2σ M

⎞ ⎟⎟ − 1 − (1.5 f )2 = 0, ⎠

(23)

and the approximated yield function Ψ ′ used in this study is Ψ′ =

3 σ ij′ σ ij′ 3 f ⋅ + 2 σ M2 8

⎛σ 2 ⋅ ⎜⎜ kk2 ⎝σM

⎞ ⎟⎟ − (1 − 1.5 f )2 = 0. ⎠

(24)

Figure 21 shows the relationship between the initial void volume fraction and the nominal fracture strain for various yield functions. The nominal fracture strain calculated by the Gurson−Tvergaard yield function is slightly smaller than that calculated by the Gurson yield function for a specified initial void volume fraction. However, the relationship between the initial void volume fraction and the nominal fracture strain calculated by the 31

Gurson−Tvergaard yield function agrees reasonably well with that calculated by the Gurson yield function.

3.7. Effect of stress triaxiality

Figure 22 shows the relationship between the initial void volume fraction and the nominal fracture strain for various stress triaxialities. Plane strain tension is assumed. Figure 22(a) shows the coordinates and the notations used. Because the length of the analysis region in the y-direction is unity, the displacement in the y-direction v * coincides with the nominal strain.

σ * denotes the imposed hydrostatic stress. The solid and dotted lines denote the material and void shapes before and after deformation, respectively. Figure 22(b) shows the relationship between the initial void volume fraction and the nominal fracture strain. The relationship in the case that the imposed stress triaxiality (σ * σ ) is zero, coincides with the relationship for the ellipsoidal void and void growth in Figure 6. Because plane strain tension is assumed, the stress triaxiality (σ kk 3σ ) is 1

3 in the case

that the imposed stress triaxiality (σ * σ ) is zero. When the initial void volume fraction is specified, the nominal fracture strain decreases with increase in the imposed stress triaxiality

(σ * σ ) . The fact that the stress triaxiality strongly affects ductile fracture is well known from the research of Bridgman (Bridgman, 1952).

3.8. Effect of deformation

Figure 23 shows the relationship between the initial void volume fraction and the nominal fracture strain for the three plastic deformation modes. Void growth is assumed. The relationships are given in the previous sections. When the initial void volume fraction is specified, plane strain tension and simple shear respectively have the smallest and largest nominal fracture strains of the three deformations.

32

The reason why the nominal fracture strain in plane strain tension is smaller than that in uniaxial tension is conjectured to be as follows. The stress triaxiality (σ kk 3σ ) in plane strain tension ( 1

3 ) is larger than that in uniaxial tension ( 1 3 ). The strain in the out-of-plane

direction is zero in plane strain tension, whereas that in the out-of-plane direction is not zero in uniaxial tension. The reason why the nominal fracture strain in simple shear is the largest of the three deformations is conjectured to be as follows. The stress triaxiality (σ kk 3σ ) in simple shear, which is zero, is the smallest of the three deformations. The direction of the maximum principal stress does not coincide with the direction of the maximum principal strain in simple shear, whereas it does coincide in plane strain tension and in uniaxial tension.

4. Conclusions

An ellipsoidal void model, which is based on a parallelogrammic void model developed by the author, is proposed and applied to analyze ductile fracture behavior in three plastic deformation modes: plane strain tension, uniaxial tension, and simple shear. The following results were obtained: (1) The relationship between the fracture strain and the initial void volume fraction in uniaxial tension agrees with that calculated using the finite-element void cell and agrees reasonably well with experimental relationships found in previous studies. (2) When the initial void volume fraction is specified, plane strain tension and simple shear have the smallest and largest nominal fracture strains of the three plastic deformation modes.

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Ragab, A.R., 2004a. Application of an extended void growth model with strain hardening and void shape evolution to ductile fracture under axisymmetric tension. Eng. Fract. Mech. 71, 1515-1534. Ragab, A.R., 2004b. A model for ductile fracture based on internal necking of spheroidal voids. Acta Mater. 52, 3997-4009. Scheyvaerts, F., Onck, P.R., Tekoglu, C., Pardoen, T., 2011. The growth and coalescence of ellipsoidal voids in plane strain under combined shear and tension. J. Mech. Phys. Solids 59, 373-397. Spitzig, W.A., Smelser, R.E., Richmond, O., 1988. The evolution of damage and fracture in iron compacts with various initial porosities. Acta Metall. 36, 1201-1211. Thomason, P.F., 1968a. A theory of ductile fracture by internal necking of cavities. J. Inst. Metals 96, 360-365. Thomason, P.F., 1968b. The use of pure aluminium as an analogue for the history of plastic flow, in studies of ductile fracture criteria in steel compression specimens. Int. J. Mech. Sci. 10, 501-518. Thomason, P.F., 1981. Ductile fracture and the stability of incompressible plasticity in the presence of microvoids. Acta Metall. 29, 763-777. Thomason, P.F., 1982. An assessment of plastic-stability models of ductile fracture. Acta Metall. 30, 279-284. Thomason, P.F., 1985a. Three-dimensional models for the plastic limit-loads at incipient failure of the intervoid matrix in ductile porous solids. Acta Metall. 33, 1079-1085. Thomason, P.F., 1985b. A three-dimensional model for ductile fracture by the growth and coalescence of microvoids. Acta Metall. 33, 1087-1095. Thomason, P.F., 1990. Ductile fracture of metals, Pergamon Press, Oxford. Tinet, H., Klocker, H., Coze, J.L., 2004. Damage analysis during hot deformation of a resulfurised stainless steel. Acta Mater. 52, 3825-3842. 37

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38

Figure and table captions

Figure 1 Overview of whole analysis procedure at each time step. Figure 2 Void shape and configuration. (a) Before deformation (b) After deformation Figure 3 Void shape and configuration before deformation. Figure 4 Void shape and configuration of two neighboring voids. (a) Before deformation (b) After deformation Figure 5 Void shape and configuration and velocity field. (a) Homogeneous deformation (b) Internal necking Figure 6 Relationship between initial void volume fraction and nominal fracture strain for plane strain tension. (a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 7 Final void shape and configuration for plane strain tension. (a) Initial void volume fraction = 0.05 (b) Nominal fracture strain v* = 0 Figure 8 Finite-element void cell for plane strain tension. (a) Coordinates and notation (b) Initial finite-element mesh (c) Final finite-element mesh Figure 9 Simulation results using finite-element void cell and our void model for plane strain tension. (a) Relationship between initial void volume fraction and nominal fracture strain (b) Relationship between nominal strain and y-coordinate of point A Figure 10 Relationship between initial void volume fraction and nominal fracture strain for simple shear. (a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 11 Final void shape and configuration for simple shear. 39

(a) Initial void volume fraction = 0.22 (b) Initial void volume fraction =0.16 Figure 12 Finite-element void cell for simple shear. (a) Coordinates and notation (b) Initial finite-element mesh (c) Final finite-element mesh Figure 13 Simulation results using finite-element void cell and our void model for simple shear. (a) Relationship between nominal strain and x-coordinate of point A (b) Relationship between nominal strain and y-coordinate of point A Figure 14 Relationship between initial void volume fraction and nominal fracture strain for uniaxial tension. (a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 15 Coordinates and notation in finite-element void cell for uniaxial tension. (a) Spherical void (b) Spherical void and toroidal void Figure 16 Final finite-element meshes in finite-element void cell for uniaxial tension. (a) Initial void volume fraction =0.01 (b) Initial void volume fraction =0.2 Figure 17 Simulation results using finite-element void cell and our void model for uniaxial tension. (a) Relationship between initial void volume fraction and nominal fracture strain (b) Relationship between nominal strain and z-coordinate of point A Figure 18 Relationship between initial void volume fraction and nominal fracture strain for various strain hardening exponents. Figure 19 Finite-element meshes for various strain hardening exponents. (initial void volume fraction = 0.01) (a) Initial state (b) Final state ( n = 0 ) (c) Final state ( n = 0.5 ) Figure 20 Experimental and simulation results using our void model for uniaxial tension. (a) Copper (b) Iron 40

Figure 21 Relationship between initial void volume fraction and nominal fracture strain for various yield functions. Figure 22 Relationship between initial void volume fraction and nominal fracture strain for various stress triaxialities. (a) Coordinates and notations (b) Relationship between initial void volume fraction and nominal fracture strain Figure 23 Relationship between initial void volume fraction and nominal fracture strain for three plastic deformation modes.

41

Macroscopic Analysis (Analyze Deformation of Material by Rigid-Plastic Finite-Element Method)

Displacement Gradient Rate Void Volume Fraction Rate

Whether or not Material Fractures

Microscopic Analysis (Evaluate Fracture of Material by our Void Model) Figure 1 Overview of whole analysis procedure at each time step.

1

(a) Before deformation (b) After deformation Figure 2 Void shape and configuration.

2

A C

B O

D

p E

F

Figure 3 Void shape and configuration before deformation.

3

p sin  det x X 

y y

yv

 2 f0 

0

a

x

xv

0

p cos  det x X 

b

x

a

(a) Before deformation (b) After deformation Figure 4 Void shape and configuration of two neighboring voids.

4

L v

L v

v  v1 u

L

 2 1

u

l1 y

y

x

v

x

2

v2  1 l2

0 v

v

(a) Homogeneous deformation (b) Internal necking Figure 5 Void shape and configuration and velocity field.

5

Nominal fracture strain v*

y

v* 1

2.0

Ellipsoidal void (no void growth) Ellipsoidal void (void growth) Rectangular void (no void growth)

1.5 1.0 0.5 0.0 0.0

0

1

x

0.1

0.2

0.3

Initial void volume fraction

(a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 6 Relationship between initial void volume fraction and nominal fracture strain for plane strain tension.

6

y

y x

y x

x

Ellipsoidal void Ellipsoidal void Rectangular void (no void growth) (void growth) (no void growth) (a) Initial void volume fraction = 0.05

y

y x

x

Ellipsoidal void Rectangular void (b) Nominal fracture strain v* = 0 Figure 7 Final void shape and configuration for plane strain tension.

7

y 12 A

0

r0 12

x

(a) Coordinates and notation (b) Initial finite-element mesh (c) Final finite-element mesh Figure 8 Finite-element void cell for plane strain tension.

8

y-coordinate of point A

Nominal fracture strain

This study (initial v.v.f. = 0.01) This study (initial v.v.f. = 0.10) Void cell (initial v.v.f. = 0.01) Void cell (initial v.v.f. = 0.10)

1.0 This study Void cell

0.5

0.3 0.2 0.1 0.0

0.0 0.0

0.1

0.2

0.0

0.3

0.2

0.4

0.6

0.8

Nominal strain

Initial void volume fraction

(a) Relationship between initial void volume (b) Relationship between nominal strain fraction and nominal fracture strain and y-coordinate of point A Figure 9 Simulation results using finite-element void cell and our void model for plane strain tension.

9

Nominal fracture strain u*

y u*

1

3.0

2.5 2.0

Optimized φ Fixed φ

1.5 1.0 0.5 0.0 0.0

0

1

x

0.1

0.2

0.3

Initial void volume fraction

(a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 10 Relationship between initial void volume fraction and nominal fracture strain for simple shear.

10

y

y x

x

Fixed  Optimized  (a) Initial void volume fraction = 0.22

y

y x

x

Fixed  Optimized  (b) Initial void volume fraction =0.16 Figure 11 Final void shape and configuration for simple shear.

11

y

12

A 1 2

r0 0

12

x

1 2 (a) Coordinates and notation

(b) Initial finite-element mesh (c) Final finite-element mesh Figure 12 Finite-element void cell for simple shear.

12

This study (initial v.v.f. = 0.01) This study (initial v.v.f. = 0.10) Void cell (initial v.v.f. = 0.01) Void cell (initial v.v.f. = 0.10)

This study (initial v.v.f. = 0.01) This study (initial v.v.f. = 0.10) Void cell (initial v.v.f. = 0.01) Void cell (initial v.v.f. = 0.10)

0.20

y-coordinate of point A

x-coordinate of point A

0.3

0.2

0.1

0.15

0.10 0.05 0.00

0.0 0.0

0.5

1.0

1.5

0.0

2.0

0.5

1.0

1.5

2.0

Nominal strain

Nominal strain

(a) Relationship between nominal strain (b) Relationship between nominal strain and x-coordinate of point A and y-coordinate of point A Figure 13 Simulation results using finite-element void cell and our void model for simple shear.

13

Nominal fracture strain w*

z

w* 1

3.0

No void growth

2.5

Void growth

2.0

Plane strain tension (no void growth)

1.5

1.0 0.5 0.0 0.0

0

1

r

0.1

0.2

0.3

Initial void volume fraction

(a) Coordinates and notation (b) Relationship between initial void volume fraction and nominal fracture strain Figure 14 Relationship between initial void volume fraction and nominal fracture strain for uniaxial tension.

14

z

z 12

12

A

r0

0

12

r

r0

0

r0 1

r

(a) Spherical void (b) Spherical void and toroidal void Figure 15 Coordinates and notation in finite-element void cell for uniaxial tension.

15

(a) Initial void volume fraction =0.01 (b) Initial void volume fraction =0.2 Figure 16 Final finite-element meshes in finite-element void cell for uniaxial tension.

16

Nominal fracture strain

2.0 1.5

Spherical void Spherical void & toroidal void This study

1.0 0.5

z-coordinate of point A

This study (initial v.v.f. = 0.01) This study (initial v.v.f. = 0.10) Void cell (initial v.v.f. = 0.01) Void cell (initial v.v.f. = 0.10)

0.3 0.2 0.1 0.0

0.0

0.0

0.0 0.1 0.2 0.3 0.4 0.5 Initial void volume fraction

0.5

1.0

1.5

2.0

Nominal strain

(a) Relationship between initial void volume (b) Relationship between nominal strain fraction and nominal fracture strain and z-coordinate of point A Figure 17 Simulation results using finite-element void cell and our void model for uniaxial tension.

17

Nominal fracture strain

2.0

No necking (n=0) n=0 n=0.5

1.5

1.0 0.5 0.0 0.00

0.10

0.20

0.30

Initial void volume fraction Figure 18 Relationship between initial void volume fraction and nominal fracture strain for various strain hardening exponents.

18

(a) Initial state

(b) Final state ( n  0 )

(c) Final state ( n  0.5 ) Figure 19 Finite-element meshes for various strain hardening exponents. (initial void volume fraction = 0.01)

19

1.5

Nominal fracture strain

Nominal fracture strain

Edelson & Baldwin (copper-holes) Edelson & Baldwin (copper-chromium) Edelson & Baldwin (copper-alumina) Thomason

1.0 This study (n=0.5)

0.5 0.0 0.0

0.1

0.2

0.3

2.5 2.0

Spitzig, Smelser & Richmond (iron-holes) Thomason This study (n=0.45)

1.5 1.0 0.5 0.0

0.00 0.05 0.10 Initial void volume fraction

Initial void volume fraction

(a) Copper (b) Iron Figure 20 Experimental and simulation results using our void model for uniaxial tension.

20

Nominal fracture strain

2.0

Gurson Gurson-Tvergaard

1.5

1.0 0.5 0.0 0.0

0.1

0.2

0.3

Initial void volume fraction Figure 21 Relationship between initial void volume fraction and nominal fracture strain for various yield functions.

21

v* 1

*

Nominal fracture strain v*

y

2.0

Imposed stress triaxiality - 0.8 Imposed stress triaxiality - 0.4 Imposed stress triaxiality 0.0 Imposed stress triaxiality +1.0 Imposed stress triaxiality +2.0

1.5

1.0 0.5 0.0 0.0

0

1

x

0.1

0.2

0.3

Initial void volume fraction

(a) Coordinates and notations (b) Relationship between initial void volume fraction and nominal fracture strain Figure 22 Relationship between initial void volume fraction and nominal fracture strain for various stress triaxialities.

22

Nominal fracture strain

2.0

Plane strain tension Uniaxial tension Simple shear

1.5

1.0 0.5 0.0 0.0

0.1

0.2

0.3

Initial void volume fraction Figure 23 Relationship between initial void volume fraction and nominal fracture strain for three plastic deformation modes.

23

Nominal fracture strain

2.5 2.0

Spitzig, Smelser & Richmond (iron-holes) Thomason This study (n=0.45)

1.5 1.0 0.5 0.0 0.00 0.05 0.10 Initial void volume fraction

73

*An ellipsoidal void model is proposed.

*Fracture strain for uniaxial tension agrees with finite-element void cell’s value.

*Fracture strain for uniaxial tension agrees reasonably well with experimental value.

*For simple shear, fracture strain is calculated.

74