Predicting ductile fracture in pure metals and alloys using notched tensile specimens by an ellipsoidal void model

Engineering Fracture Mechanics 151 (2016) 51–69

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Predicting ductile fracture in pure metals and alloys using notched tensile specimens by an ellipsoidal void model 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

a r t i c l e

i n f o

Article history: Received 23 July 2015 Accepted 20 November 2015 Available online 29 November 2015 Keywords: Ductile fracture Micromechanics Nonferrous metals Notch tensile testing Metal forming

a b s t r a c t The ductile fracture of nonferrous pure metals and alloys during notch tensile testing was predicted using an ellipsoidal void model. Simulated and experimental tensile tests were performed using four types of nonferrous sheets and bars. Two magnitudes of prestrain were induced in the sheets by rolling and in the bars by drawing. Six notched sheet specimens and thirteen notched bar specimens with different notch-root radii were prepared. A void configuration and void shape for pure metals and those for alloys were assumed. The effects of the prestrain and notch-root radius on the reduction in area calculated agreed reasonably well with those obtained experimentally. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Ductile fracture, which occurs when a material is subjected to large plastic deformation, is troublesome to deal with during metal-forming processes. Although numerous ductile fracture criteria for various materials have been proposed, a ductile fracture criterion applicable to all metal-forming processes is yet to be discovered [1,2]. Because ductile fracture occurs by nucleation, growth, and coalescence of voids [3], it is a microscopic phenomenon. However, the ductile fracture criteria that are widely used for metal-forming processes [4–7] are derived from a macroscopic viewpoint, because these ductile fracture criteria are expressed by the definite integrals of the integrands that are functions of the stress and strain components along the strain path. Hence, improving the prediction accuracy for microscopic ductile fracture phenomena using macroscopic ductile fracture criteria is challenging. Although nucleation and growth of voids are simulated in the Gurson model [8], the coalescence of voids cannot be simulated intrinsically. Hence, for instance, the coalescence of voids is assumed to occur when the void volume fraction reaches a critical value [9]. However, this assumption is inappropriate, because the critical void volume fraction depends on the stress state [10,11]. Hence, the coalescence of voids should be evaluated using a model with a definite physical meaning. Thomason [12] proposed a three-dimensional void model using the upper bound method, in which void coalescence is assumed to occur when the energy dissipation rate of internal necking is less than that of homogeneous deformation. Because the expression of the condition of void coalescence is extremely large and complex, it was expressed in a closed form using an empirical relation [12,13]. Because the accuracy of the empirical relation was not high, the empirical relation was modified by several researchers [14,15] by comparing the empirical relation with the numerically obtained relation. However, the empirical relation was originally obtained under uniform deformation wherein the principal axes of stress and strain coincide. Hence, the empirical relation was not applicable to metal forming processes in which the principal axes E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2015.11.012 0013-7944/Ó 2015 Elsevier Ltd. All rights reserved.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

Nomenclature A0 A
0 A1 A
1 B1 B
1 C C2 C3 d; d0 E f I k 2l1 ; 2l2 L; L0 n p r R RðxÞ t; t 0 v0; v Dv 1 ; Dv 2 x0 ; y0 e_

e0 eM

h1 ; h2

r r0 rM rmax ; rmin

/ d/ U; U0 ðnÞ

f ðiÞ

material constant defined in Eq. (3) material constant defined in Eq. (5) material constant defined in Eq. (4) material constant defined in Eq. (5) material constant defined in Eq. (4) material constant material constant defined in Eq. (13) material constant defined in Eq. (14) material constant defined in Eq. (15) diameters of bars used in tensile tests and annealed bars, respectively ratio of energy dissipation rate of internal necking to energy dissipation rate of homogeneous deformation void volume fraction unit dyadic shearing yield stress of matrix lengths of line of velocity discontinuity lengths of rectangle in x0 - and y0 -directions, respectively material constant defined in Eq. (13) distance between two neighboring voids before forming radius of minimum cross section radius of notch root ramp function thicknesses of sheets used in tensile tests and annealed sheets, respectively material velocities in x0 - and y0 -directions, respectively amounts of velocity discontinuity coordinates in directions of minimum and maximum principal stresses, respectively equivalent strain rate material constant defined in Eq. (13) strain of matrix angles between maximum principal stress and line of velocity discontinuity equivalent stress material constant defined in Eq. (13) tensile yield stress of matrix maximum and minimum principal stresses, respectively angle between axis of elongation and line connecting two neighboring voids range of / in which ratio E for void after forming is less than unity original and approximate Gurson yield functions, respectively volume fraction of void that nucleates in ith step and that exists after nth step

ðnÞ

ð@x=@XÞalloy

deformation gradient for alloy after nth step

ðnÞ ð@x=@XÞpure metal

deformation gradient for pure metal after nth step

ðnÞ ð@x=@XÞðiÞ

deformation gradient that equals I after ith step and that exists after nth step

of stress and strain differ. These empirical relations [13–15] were compared only with the experimental results of the uniaxial tension using notched or smooth round bars [16–19]. Other three-dimensional void models [20,21] were also compared with the experimental uniaxial tension results obtained using notched or smooth round bars. Recently, the author has attempted to predict ductile fracture during sheet metal-forming processes from a microscopic viewpoint [22,23]. The author’s proposed model of void coalescence is based on the two-dimensional void model proposed by Thomason [24] and that proposed by Melander and Ståhlberg [25], which were also derived from a microscopic viewpoint. Both the Thomason model and Melander and Ståhlberg model assume that the void is rectangular and the direction of the major axis of the void coincides with the direction of the maximum principal stress. In contrast, the author’s proposed model assumes that the void is ellipsoidal and makes no assumptions about the relationship between the direction of the major axis of the void and the direction of the maximum principle stress. Hence, the author’s void model can be used for simulating metal-forming processes. In the preceding papers [22,23], the simulation and experiment of the hole-expansion test were performed and the effect of prestrain on the hole-expansion ratio was demonstrated. The simulation results calculated using the author’s void model agreed with the experimental results, whereas the simulation results calculated using most of the conventional ductile

K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

53

fracture criteria [5–7] differed from the experimental results. Because prestrain was induced by rolling in which stress triaxiality was negative, prestrain hardly influenced the hole-expansion ratio in most of the conventional ductile fracture criteria [5–7]. Furthermore, the validities of the void configuration and void shape for an alloy and a pure metal were confirmed. Although the hole-expansion test is a type of material test used to evaluate ductile fracture in sheet metal-forming processes, a fundamental material test that is widely used in metal-forming processes should also be used to examine the applicability of the author’s void model. Furthermore, the validities of the void configuration and void shape for an alloy and a pure metal should be confirmed in other metal-forming processes to validate the author’s void model. In this study, the ellipsoidal void model previously proposed by the author [22,23] was evaluated using a fundamental material test, i.e., notch tensile testing. Simulated and experimental tensile tests were performed using two types of nonferrous pure metal bars, nonferrous alloy bars, nonferrous pure metal sheets, and nonferrous alloy sheets. During notch tensile testing, the stress components of the cross section of the neck are well known to depend on the radius of the curvature of the neck [26–28]; i.e., the stress components of the cross section at a notch root depend on the curvature of the notch root. Hence, the simulated and experimental tensile tests are performed using notched specimens with different notch-root curvatures to obtain various stress triaxialities. The validity of the ellipsoidal void model was confirmed by comparing the simulation results with the experimental results and the results calculated using conventional ductile fracture criteria. 2. Simulation method The simulation method used in this study, which is the same as that used in the preceding studies [22,23], is described briefly, whereas the new simulation method is described in detail. 2.1. Outline A multiscale simulation of the material was performed until it fractured. The material deformation was simulated macroscopically by the rigid-plastic finite-element method, and the material fracture was evaluated using the ellipsoidal void model by microscopic simulation. The deformation gradient and void volume fraction calculated in the macroscopic simulation were used in the microscopic simulation. 2.2. Outline of macroscopic simulation Material deformation was simulated using the conventional rigid-plastic finite-element method [29]. Axisymmetric state was assumed during the simulations of the drawing and tensile testing of bars. Plane-strain state was assumed during the simulation of the rolling of sheets, and plane-stress state was assumed during the simulation of the tensile testing of sheets. The yield function proposed by Gurson [8] was adopted:

3 2

U¼ 


 r0ij r0ij rkk 2  1  f ¼ 0; þ 2f cosh 2r M r2M

ð1Þ

where rM is the tensile yield stress of the matrix and f is the void volume fraction of the material. Because the yield function

U is not a function of the second power of stress, rigid-plastic simulation is not easily performed using Eq. (1). Hence, cosh x was approximated as 1 þ x2 =2 [30]. Therefore, the approximated yield function U0 used in the present study is

U0 ¼

3 rij rij f r2kk 2  þ  2  ð1  f Þ ¼ 0: 2 r2M 4 rM 0

0

ð2Þ

The stress triaxiality rkk =3rM at the center of the minimum cross section of the notched bar specimen is approximately equal to 1=3 þ lnð1 þ r=2RÞ [26], where R is the radius of the notch root and r is the radius of the minimum cross section, which is 4.5 mm in this study. When R is 1 mm, the stress triaxiality at the center of the minimum cross section of the notched bar specimen is 1.5, whereas when R is 5 mm, this value is 0.7. Although the accuracy of the replacement of cosh x with 1 þ x2 =2 in Eq. (1) is not necessarily high when R is 1 mm, it is sufficiently high when R is 5 mm. Furthermore, upon increasing the deformation of the notched bar specimen, R increases, rkk =3rM decreases, and the accuracy of the replacement of cosh x with 1 þ x2 =2 in Eq. (1) increases. rkk =3rM at the center of the minimum cross section of the notched sheet specimen is smaller than 0.7 regardless of the value of R. Hence, the accuracy of the replacement of cosh x with 1 þ x2 =2 in Eq. (1) is sufficiently high regardless of the value of R. As will be described in Section 2.5, the material is assumed to be a rigid plastic in macroscopic and microscopic simulations. Therefore, the approximated yield function U0 is used in this study to ensure consistency between the two simulations. The following two types of evolution equations that denote the change in the void volume fraction are assumed:

f_ ¼ ð1  f Þe_ kk þ A0 e_ and r  kk  B1 e_ ; f_ ¼ ð1  f Þe_ kk þ A1 R  3r

ð3Þ ð4Þ

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

 is the equivalent stress, e_ is the equivalent strain rate, and A0 , A1 , and B1 are the material constants. Function RðxÞ in where r Eq. (4) denotes the ramp function. The first and second terms on the right-hand side of Eqs. (3) and (4) denote void growth and void nucleation, respectively. The reason underlying the use of Eqs. (3) and (4) can be explained as follows. The following evolution equation [31] has been widely used by numerous researchers:

r_ _ þ A
1 kk ; f_ ¼ ð1  f Þe_ kk þ A
0 r 3

ð5Þ

where A
0 and A
1 are the material constants. The second and third terms on the right-hand side of Eq. (5) denote void nucleation. First, on the basis of the experimental result [32], it is assumed that A
0 –0 and A
1 ¼ 0. This indicates that the fraction of the fracture of cementite particles in hyper-eutectoid steel is proportional to the equivalent strain. Second, A
0 ¼ A
1 is assumed with reference to the experimental result [33], which indicates the formation of cavities at the matrix-particle interface when the maximum principal stress reaches a certain value.  =deÞe_ in the _ is equal to ðdr However, Eq. (5) is not necessarily appropriate for the following reasons. First, although r _ _  in Eq. (5) should be replaced by e, with reference to the experimental result [32]; hence, stress–strain relationship, r  þ rkk =3 [31] Eq. (3) is obtained. Second, the assumption that the maximum principal stress rmax is approximated to be r  þ rkk =3 for uniaxial tension, whereas it is ð1=3Þr  þ rkk =3 for uniaxial compresis inappropriate because it is equal to ð2=3Þr  þ rkk =3. Furthermore, A
1 r_ max in Eq. (5) should be replaced by sion. Hence, rmax should be approximated to ð1=2Þr

_
 to be A1 Rðrmax  B1 Þe, where B1 is the material constant, with reference to the experimental result [33]. Approximating r  þ rkk =3. constant results in Eq. (4) because rmax is equal to ð1=2Þr Void nucleation occurs through either inclusion cracking or inclusion–matrix separation [3]. In Eq. (5), the assumption A
0 –0 and A
1 ¼ 0 corresponds to the occurrence of inclusion cracking, whereas the assumption A
0 ¼ A
1 corresponds to the occurrence of inclusion–matrix separation. However, Eq. (5) is derived on the basis of the experimental result of hypereutectoid steel, which is not used in this study. Hence, it is not necessarily appropriate to assume that inclusion cracking occurs when Eq. (3) is used and that inclusion–matrix separation occurs when Eq. (4) is used. 2.3. Outline of microscopic simulation At each step, the following microscopic simulation was performed. First, the void volume fraction f and the deformation gradient @x=@X were calculated by the macroscopic rigid-plastic finite-element simulation. Second, the void configuration and void shape were calculated, as explained in Section 2.4. Third, the ratio of the energy dissipation rate of internal necking to the energy dissipation rate of homogeneous deformation E was calculated and it was determined whether the material had fractured, as explained in Section 2.5. The nucleation of voids conventionally occurs through inclusion cracking or inclusion–matrix separation [3]. In this study, voids were assumed to nucleate through inclusion–matrix separation, whereas in my other study, voids were assumed to nucleate through inclusion cracking [34]. 2.4. Void configuration and void shape For simplicity, the deformation gradient of a void was assumed to be identical to the deformation gradient of the material that surrounds the void. In the conventional nonsteady finite-element simulation of metal-forming processes, the entire step is divided into several infinitesimal steps. After the nth step, a hexagonal grid and a circle that are embedded in the material after the ith step become a deformed ðnÞ

hexagonal grid and a deformed circle, respectively. ð@x=@XÞðiÞ is defined as the deformation gradient that equals I, which is the unit dyadic, after the ith step and that exists after the nth step. The condition after the zeroth step implies the condition before forming. 2.4.1. Pure metals A pure metal does not contain inclusions around which voids nucleate. Beevers and Honeycombe [35] performed the uniaxial tensile test of a bar using pure aluminum and indicated that in the absence of inclusions, cracks were initially observed to form only in some grains and on one slip system in each grain. They [35] also postulated that cracks can form by a dislocation mechanism in the heavily distorted region of the neck of inclusion-free metals and alloys. Hence, the void configuration and void shape defined in my previous paper [36] were assumed for pure metals. Fig. 1 shows the void configuration and void shape for a pure metal. Voids were assumed to nucleate isotropically in the material at each step because cracks appeared in the absence of inclusions. A hexagonal grid and a circle were embedded in the material after each step. At void nucleation, the void configuration was assumed to be the centroid of a hexagonal grid and the void shape was assumed to be circular for convenience because the initial crack shape was unknown. Because the void configuration and void shape calculated under the above assumption are not singular, a single void configuration and

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

0th step (∂x

( ) ∂X )(0 ) = I 0

ith step (i )

(∂x

∂X )(0 )

(∂x

∂X )(0 )

(∂x

(i )

∂X )(i ) = I

nth step (n )

(∂x

(∂x

(n )

∂X )(i )

(∂x

(n )

∂X )(n ) = I

(n )

∂X )pure metal

Fig. 1. Void configuration and void shape for a pure metal.

void shape were calculated by considering the weighted average to determine whether the material fractured, as will be described in Section 2.5. The deformation gradient for a pure metal after the nth step is expressed as follows:

Pn ðnÞ

ð@x=@XÞpure metal ¼

ðnÞ ðnÞ i¼0 f ðiÞ ð@x=@XÞðiÞ Pn ðnÞ i¼0 f ðiÞ

;

ð6Þ

ðnÞ

where f ðiÞ is defined as the volume fraction of a void that nucleates in the ith step and that exists after the nth step. 2.4.2. Alloys An alloy contains inclusions around which voids nucleate. Hence, the void configuration and void shape defined in my previous paper [22] were assumed for alloys. Fig. 2 shows the void configuration and void shape for an alloy. Voids were assumed to nucleate around an inclusion at each step. A hexagonal grid and a circle were embedded in the material after the zeroth step. At void nucleation, the void

0th step (∂x

(0 )

∂X )(0 ) = I

ith step (∂x

(i )

∂X )(0 )

nth step (∂x

(n )

(n )

∂X )(0 ) = (∂x ∂X )alloy

Fig. 2. Void configuration and void shape for an alloy.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

configuration was assumed to be the centroid of the deformed hexagonal grid and the void shape was assumed to be the deformed circle. The deformation gradient for an alloy after the nth step is expressed as follows: ðnÞ

ðnÞ

ð@x=@XÞalloy ¼ ð@x=@XÞð0Þ :

ð7Þ

2.5. Determination of material fracture In the macroscopic simulation, an axisymmetric state was assumed for the bars, whereas a plane-strain state or planestress state was assumed for the sheets. However, a plane-strain state was assumed when the material fracture was determined [37] regardless of the assumed state in the macroscopic simulation. The reason for assuming the plane-strain state is explained below. In metal forming processes, the plane-strain state often appears before ductile fracture occurs. In sheet metal forming processes, localized necking of a sheet appears under biaxial stress before ductile fracture occurs. In instability processes, the plane-strain state appears until the material fractures [38]. The theory of localized necking of a sheet by Hill [39] supports the plane-strain state at the localized necking. In bulk metal forming processes, localized strain instability appears during axial compression of a cylinder before ductile fracture occurs [40,41]. In instability processes, at the equatorial plane of the material surface, the hoop strain increases and axial strain decreases while the radial strain remains constant. In other words, the plane-strain state appears until the material fractures. Therefore, in this study, the plane-strain state was assumed when the material fracture was determined. This assumption must to be validated by further analysis and experiments. Fig. 3 shows the velocity field. First, the material was assumed to deform homogeneously. Fig. 3(a) shows the velocity field for homogeneous deformation. The y0 - and x0 -directions coincided with the directions of the maximum and minimum principal stresses, respectively. The rectangle indicated by the dotted lines is considered. The length of the rectangle in the x0 -direction is L, which was assumed to be the distance between two neighboring voids, and the length of the rectangle in the y0 -direction is L0 . The material velocity in the y0 -direction is v , and that in the x0 -direction is v 0 , which is equal to ðL=L0 Þv because of the incompressibility condition. Hence, considering the yield criteria and void volume fraction f , the energy dissipation rate in the material due to homogeneous deformation is expressed by

ð1  f Þð2v Lrmax þ 2v 0 L0 rmin Þ ¼ ð1  f Þ4kv L;

ð8Þ

where k denotes the shearing yield stress of the matrix. Next, the material was assumed to neck internally. Fig. 3(b) shows the velocity field for internal necking. The line of velocity discontinuity, which is indicated by dotted lines, was assumed to coincide with the tangent of the two neighboring voids. In the figure, h1 and h2 denote the angles between the maximum principal stress and the line of velocity discontinuity, whereas 2l1 and 2l2 denote the lengths of the line of velocity discontinuity. The velocity field is shown as a hodograph. Moreover, Dv 1 and Dv 2 denote the amounts of velocity discontinuity. On the basis of the law of sines, the energy dissipation rate on the line of velocity discontinuity when two neighboring voids coalesce by internal necking is expressed by

L v

L v

Δv1

v

θ 2 θ1 v'

L'

v' l1

y'

y'

x'

v (a) Homogeneous deformation

θ2

Δv2 θ 1

0

l2

v

Hodograph

x'

v

(b) Internal necking

Fig. 3. Velocity field.

K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

57

dφ

p

φ

Fig. 4. Two neighboring voids and a void coalescence region.

k  Dv 1  2l2 þ k  Dv 2  2l1 ¼ 4kv

l1 sin h2 þ l2 sin h1 : sinðh1 þ h2 Þ

ð9Þ

Fracture was assumed to occur when the energy dissipation rate of internal necking was less than that of homogeneous deformation. Hence, on the basis of Eqs. (8) and (9), the fracture criterion was expressed by

ð1  f ÞL P

l1 sin h2 þ l2 sin h1 : sinðh1 þ h2 Þ

ð10Þ

The ratio of the energy dissipation rate of internal necking to that of homogeneous deformation E was defined as

E¼

l1 sin h2 þ l2 sin h1 : ð1  f ÞL sinðh1 þ h2 Þ

ð11Þ

The criterion of void coalescence was satisfied when E was less than unity. Although the criterion of void coalescence is microscopically satisfied for a specified angle between the coordinate axis and the line connecting two neighboring voids, the material does not necessarily develop macroscopic fractures. The criterion of microscopic void coalescence is related to macroscopic material fracture, as shown below. Fig. 4 shows the two neighboring voids and the void coalescence region. The distance between the two neighboring voids pffiffiffiffiffiffiffiffi before forming is p, which is equal to 4 4=3 [37]. Variable / is defined as the angle between the axis of elongation and the line connecting the two neighboring voids. Because the direction of the elongation axis is unknown, / assumes an arbitrary value between 0 and 2p. Furthermore, because the void configuration is assumed to be macroscopically isotropic before forming, the probability density function of / becomes constant regardless of the value of /. When / is specified, the ratio E can be calculated for a void after forming. Hence, the ratio E for a void after forming was calculated for all values of /. d/ was defined as the range of / in which the ratio E for a void after forming was less than unity. The void coalescence region is defined as a sector whose central angle is d/. Six other voids surrounded a given void, as shown in Figs. 1 and 2. When d/ was p=3ð¼ 2p=6Þ, the probability that the ratio E was less than unity for a specified / was unity. Because the void configuration was assumed to be macroscopically isotropic before forming, the macroscopic probability of void coalescence was unity. Similarly, when d/ was p=6ð¼ ð2p=6Þ=2Þ, the probability that the ratio E was less than unity for a specified / was half. Hence, the macroscopic probability of void coalescence was half, which was assumed to be a reasonable criterion for material fracture. Therefore, when the following equation was satisfied, the material was assumed to be macroscopically fractured:

d/ ¼

p 6

:

ð12Þ

2.6. Prestrain Prestrain was considered in a similar manner to ordinary strain, i.e., the strain induced by drawing or rolling was considered in a similar manner to that induced by the tensile tests. 3. Experimental results 3.1. Rolling and drawing experiments Prior to the tensile tests, sheets were rolled and bars were drawn to induce prestrain in the sheet and bar specimens. The following four types of nonferrous sheets were used: JIS A1050, which is a pure aluminum sheet; JIS A5052, which is an aluminum alloy sheet and is equivalent to ISO AlMg2.5; JIS C1100P, which is a tough pitch copper sheet; and JIS C2801, which is a brass sheet and is equivalent to EN CW612N. The sheets of A1050, A5052, C1100P, and C2801 were annealed

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

at temperatures 350 °C, 350 °C, 400 °C, and 450 °C, respectively, for 1 h and then cooled in an electric furnace. The following four types of nonferrous bars were used: JIS A1070, which is a pure aluminum bar; JIS A5056, which is an aluminum alloy bar and is equivalent to ISO AlMg5Cr; JIS C1100B, which is a tough pitch copper bar; and JIS C3604, which is a free cutting brass bar and is equivalent to EN CW606L. The bars of A1070, A5056, C1100B, and C3604 were annealed at temperatures 350 °C, 350 °C, 400 °C, and 450 °C, respectively, for 1 h and then cooled in an electric furnace. Fig. 5 shows the micrographs of the longitudinal sections of the smooth specimens with no prestrain. The horizontal direction coincided with the tensile direction. The specimens of A1050, A1070, A5052, and A5056 were etched using Keller’s etch [42], and the specimens of C1100P, C1100B, C2801, and C3604 were etched using Iron(III) chloride. As expected, inclusions were almost uniformly distributed in A5052, A5056, C2801, and C3604, and no anisotropy was observed in the micrographs of any of the materials. Hence, all the materials were used to evaluate the validity of the ellipsoidal void model discussed in Section 2. Table 1 shows the relationship between the thicknesses of annealed sheets t0 and the thicknesses of the sheets used in tensile tests t. The roll diameter was 180 mm. The reduction in thickness during each rolling was between 10% and 20%.

10μm

10 μm

(a) A1050

(b) A5052

25μm

25μm

(c) C1100P

(d) C2801

10 μm

(e) A1070

10 μm

(f) A5056

25μm

25μm

(g) C1100B

(h) C3604

Fig. 5. Micrographs of the longitudinal sections of smooth specimens with no prestrain.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69 Table 1 Relationship between the thicknesses of the annealed sheets and the thicknesses of the sheets used in tensile tests. Material

t0 (mm)

t (mm)

A1050, A5052, C1100P, C2801

1 2 3

1 1 1

However, the reduction in thickness during the last rolling was determined such that the elongation in length at the central part of the sheet in the width direction became t0 =t. Conventional spindle oil was used for lubricating the sheets of A1050 and A5052, and conventional machine oil was used for lubricating the sheets of C1100P and C2801. Specimens for the tensile test were produced using the central part of the sheet in the width direction. The rolling direction of the sheet was made to coincide with the length direction of the specimen for the tensile test. Table 2 shows the relationship between the diameters of the annealed bars d0 and the diameters of the bars used in tensile tests d. The bars were drawn such that the reduction in area in each die was 15% and that the die angle was equal to 15°. Conventional spindle oil was used for lubricating the bars of A1070 and A5056, and conventional machine oil was used for lubricating the bars of C1100B and C3604. The drawing direction of the bar was made to coincide with the length direction of the specimen for the tensile test. 3.2. Tensile test experiments Tensile tests were performed on smooth sheet or bar specimens with no prestrain by using a hydraulic tension and compression testing machine with a crosshead speed of approximately 0.05 mm/s. Load was measured using a load cell in the testing machine, and elongation was measured using an extensometer of the strain gauge type, which was attached to the specimen. The following stress–strain relationship was determined experimentally and subsequently used in the simulations:

rM ¼ maxðr0 ; C  ðeM  e0 Þn Þ;

ð13Þ

where eM was the strain of the matrix and r0 , C, e0 , and n were material constants. Table 3 shows the material constants used in the stress–strain relation. Fig. 6 shows the micrographs of the fractured surfaces of the smooth specimens with no prestrain. Dimples that were yielded from the voids were observed in all the materials. Hence, the ellipsoidal void model discussed in Section 2 was confirmed to be applicable to all materials. The tensile test experiments of the sheets and bars were performed using smooth and notched specimens. The width of the smooth sheet specimen was 20 mm, and the diameter of the smooth bar specimen was 13 mm. Fig. 7 shows the shapes of the notched sheet and bar specimens. Table 4 shows the dimensions of the notched sheet and bar specimens. Six types of notched sheet specimens with different notch-root radii were used, and thirteen types of notched bar specimens with different notch-root radii were used.

Table 2 Relationship between the diameters of the annealed bars and the diameters of the bars used in tensile tests. Material

d0 (mm)

d (mm)

A1070, A5056, C1100B, C3604

13 16 22

13 13 13

Table 3 Material constants used in the stress–strain relation. Material

r0 (MPa)

C (MPa)

e0

n

A1050 A5052 C1100P C2801 A1070 A5056 C1100B C3604

40 110 80 200 30 140 80 0

130 350 460 830 98 550 460 790

0.01 0.02 0.02 0.005 0.01 0.01 0.02 0.02

0.22 0.20 0.34 0.40 0.20 0.30 0.34 0.40

60

K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

(a) A1050

(b) A5052

(c) C1100P

(d) C2801

(e) A1070

(f) A5056

(g) C1100B

(h) C3604

Fig. 6. Micrographs of fractured surfaces of smooth specimens with no prestrain.

Figs. 8, 9, 10, 11, 12, 13, 14, and 15 show the tensile strength and the reduction in area in the cases of A1050, A5052, C1100P, C2801, A1070, A5056, C1100B, and C3604, respectively. The curvature of the notch root on the horizontal axis was the reciprocal of the radius of the notch root. The tensile strength along the vertical axis was defined as the maximum load divided by the minimum cross section of a smooth or notched specimen. The reduction in area along the vertical axis was calculated from the cross section of the fractured specimen using a digital microscope equipped with the necessary software. Three specimens were used for each

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

20 Notched depth Notch-root radius

φ13 Notched depth Notch-root radius

(a) Sheet

(b) Bar

Fig. 7. Shapes of the notched sheet and bar specimens.

Table 4 Dimensions of the notched sheet and bar specimens.

Notched depth (mm) Radius of notch root (mm)

0 Exp. 0.80 Exp. 1.27 Exp.

Bar

5 1, 2, 3, 5, 7, 9

2 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 13, 26, 39

0 Sim. 0.80 Sim. 1.27 Sim.

150

100

50

0.0

0.2

0.4

0.6

0.8

0 Exp. 0.80 Exp. 1.27 Exp.

90

Reduction in area (%)

Tensile strength (MPa)

200

Sheet

85

80

75

70

1.0

0 Sim. 0.80 Sim. 1.27 Sim.

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

Curvature of notch root (1/mm)

(a) Tensile strength

(b) Reduction in area

500

0 Exp. 0.80 Exp. 1.27 Exp.

0 Sim. 0.80 Sim. 1.27 Sim.

400 300 200 100 0.0

100

Reduction in area (%)

Tensile strength (MPa)

Fig. 8. Tensile strength and reduction in area in the case of A1050.

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Tensile strength

0 Exp. 0.80 Exp. 1.27 Exp.

0 Sim. 0.80 Sim. 1.27 Sim.

90 80 70 60 50 0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(b) Reduction in area

Fig. 9. Tensile strength and reduction in area in the case of A5052.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

0 Exp. 0.80 Exp. 1.27 Exp.

500 400 300 200 100 0.0

0.2

0.4

0.6

0.8

0 Exp. 0.80 Exp. 1.27 Exp.

90

Reduction in area (%)

Tensile strength (MPa)

600

0 Sim. 0.80 Sim. 1.27 Sim.

80 70 60 50 40 0.0

1.0

0 Sim. 0.80 Sim. 1.27 Sim.

Curvature of notch root (1/mm)

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Tensile strength

(b) Reduction in area

Fig. 10. Tensile strength and reduction in area in the case of C1100P.

0 Exp. 0.80 Exp. 1.27 Exp.

800

500

200

0.0

0.2

0.4

0.6

0.8

0 Exp. 0.80 Exp. 1.27 Exp.

90

Reduction in area (%)

Tensile strength (MPa)

1100

0 Sim. 0.80 Sim. 1.27 Sim.

80 70 60 50 40 0.0

1.0

0 Sim. 0.80 Sim. 1.27 Sim.

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

Curvature of notch root (1/mm)

(a) Tensile strength

(b) Reduction in area

Fig. 11. Tensile strength and reduction in area in the case of C2801.

0 Exp. 0.42 Exp. 1.05 Exp.

160 140 120 100 80 60 40

0 Exp. 0.42 Exp. 1.05 Exp.

100

Reduction in area (%)

Tensile strength (MPa)

180

0 Sim. 0.42 Sim. 1.05 Sim.

0 Sim. 0.42 Sim. 1.05 Sim.

90

80

70 0.0

0. 2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Tensile strength

0.0

0. 2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(b) Reduction in area

Fig. 12. Tensile strength and reduction in area in the case of A1070.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

0 Exp. 0.42 Exp. 1.05 Exp.

800 600 400 200 0.0

0.2

0.4

0.6

0.8

0 Exp. 0.42 Exp. 1.05 Exp.

70

Reduction in area (%)

Tensile strength (MPa)

1000

0 Sim. 0.42 Sim. 1.05 Sim.

60 50 40 30 20 10 0.0

1.0

Curvature of notch root (1/mm)

0 Sim. 0.42 Sim. 1.05 Sim.

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Tensile strength

(b) Reduction in area

Fig. 13. Tensile strength and reduction in area in the case of A5056.

0 Exp. 0.42 Exp. 1.05 Exp.

700 500 300 100

0.0

0.2

0.4

0.6

0.8

0 Exp. 0.42 Exp. 1.05 Exp.

80

Reduction in area (%)

Tensile strength (MPa)

900

0 Sim. 0.42 Sim. 1.05 Sim.

70 60 50 40 30 20

1.0

0 Sim. 0.42 Sim. 1.05 Sim.

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

Curvature of notch root (1/mm)

(a) Tensile strength

(b) Reduction in area

Fig. 14. Tensile strength and reduction in area in the case of C1100B.

0 Exp. 0.42 Exp. 1.05 Exp.

1300 1100 900 700 500 300 0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Tensile strength

0 Exp. 0.42 Exp. 1.05 Exp.

80

Reduction in area (%)

Tensile strength (MPa)

1500

0 Sim. 0.42 Sim. 1.05 Sim.

0 Sim. 0.42 Sim. 1.05 Sim.

60 40 20 0 0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(b) Reduction in area

Fig. 15. Tensile strength and reduction in area in the case of C3604.

experimental condition, and the average tensile strength and average reduction in area were plotted. For each experimental condition, the difference between the tensile strength for an arbitrary specimen and the average tensile strength was smaller than 2% of the average tensile strength, whereas the difference between the reduction in area for an arbitrary specimen and the average reduction in area was smaller than 1.5%.

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

The experimental results are plotted using open symbols. The figures in the legend indicate the magnitude of the prestrain. With increasing the curvature of the notch root, the tensile strength increased, whereas the reduction in area decreased. With increasing prestrain, the tensile strength increased, whereas the reduction in area decreased. 4. Simulation results 4.1. Simulation of rolling and drawing The simulation of the rolling of a sheet was performed using the conventional rigid-plastic finite-element method [29], in which a plane-strain state was assumed. The reduction in thickness during each rolling was not constant in the experiments. Nevertheless, it was assumed to be 15% for simplicity. However, the reduction in thickness during the last rolling was assumed such that the elongation in length became t0 =t afterward. The friction between the material and the roll was assumed to obey the Coulomb’s law, and the coefficient of friction was assumed to be 0.1. The initial void volume fraction was assumed to be 0.001 [43]. The volume fraction of the void that existed initially was assumed to constant during the simulation of rolling, because it was assumed to be its minimum value. The simulation of the drawing of a bar was performed using the conventional rigid-plastic finite-element method [29], in which an axisymmetric state was assumed. The friction between the material and the die was assumed to obey the Coulomb’s law, and the coefficient of friction was assumed to be 0.1. The initial void volume fraction was assumed to be 0.001 [43]. 4.2. Simulations of tensile tests The simulations of the tensile tests of sheets were performed using the conventional rigid-plastic finite-element method [29], in which a plane-stress state was assumed. The finite-element meshes during the simulations of the rolling of a sheet were not compatible with the finite-element meshes during the simulations of the tensile tests of sheets. Hence, the material properties such as the void volume fraction during the simulation of the rolling of a sheet were equalized in the thickness direction and then used during the simulations of the tensile tests of sheets. Because localized necking, whose direction is not perpendicular to the tensile direction, appeared in the experiments, line symmetry should not be assumed. However, line symmetry was assumed for simplicity. The radius of the notch root for the smooth sheet specimens was infinite. However, necking did not occur on the axis of line symmetry when the simulation was performed using the smooth sheet specimens with the radius of the notch root being infinite because an initial geometrical imperfection was not assumed. Hence, the radius of the notch root was assumed to be 1 m for the smooth sheet specimens such that necking occurred on the axis of line symmetry. The simulation was performed until the specimen ruptured [44]. The simulations of the tensile tests of bars were performed using the conventional rigid-plastic finite-element method [29], in which an axisymmetric state was assumed. The finite-element meshes during the simulations of the drawing of bars were made to coincide with the finite-element meshes used during the simulations of the tensile tests of bars. In spite of the fact that cup-and-cone rupture was observed during the experiments, line symmetry was nonetheless assumed for simplicity. The radius of the notch root for the smooth bar specimens was infinite. However, necking did not occur on the axis of line symmetry when the simulation was performed using the smooth bar specimens with the radius of the notch root being infinite because an initial geometrical imperfection was not assumed. Hence, the radius of the notch root was assumed to be 1 m for the smooth bar specimens so that necking occurred on the axis of line symmetry. The simulation was performed until the specimen ruptured [44]. The simulations were performed using either Eq. (3) or (4). B1 was assumed to be zero for simplicity. The material constants A0 and A1 were determined such that the reduction in area calculated from the simulation of the specimen with no prestrain and a notch-root radius of 5 mm agreed with that obtained experimentally. The selection of Eq. (3) or (4) was determined on the basis that the effect of prestrain on the reduction in area calculated using Eq. (3) was larger than that calculated using Eq. (4), and that the effect of the notch-root radius calculated using Eq. (4) was larger than that calculated using Eq. (3). Table 5 shows the material constants used in the evolution equations. Eq. (3) was used in the cases of A5052, C1100P, and C1100B, and Eq. (4) was used in the cases of A1050, C2801, A1070, A5056, and C3604. Although the chemical composition of C1100P was the same as that of C1100B, the material constant A0 of C1100P was smaller than that of C1100B. The reason for the difference in A0 was conjectured to be as follows. As shown in Fig. 6, the magnitude of a void at the rupture of C1100P was larger than that of C1100B. Hence, the void nucleation rate of C1100P was smaller than that of C1100B. In the case of A1070, the material constant A1 could not be determined such that the reduction in area calculated from the simulation for the specimen with no prestrain and a notch-root radius of 5 mm agreed with that obtained experimentally for the following reason. When A1 was equal to zero, the flattening of a void calculated from the simulation of the specimen with a prestrain of 1.05 was larger than 0.99 and either h1 or h2 in Fig. 3(b) became smaller than zero, in which internal necking did not occur. Hence, a minimum value of A1 , at which both h1 and h2 were larger than zero and internal necking occured, was chosen.

K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

65

Table 5 Material constants used in evolution equations.

A1050 A5052 C1100P C2801 A1070 A5056 C1100B C3604

Void configuration

A0

A1

Pure metal Alloy Pure metal (Alloy) Alloy (Pure metal) Pure metal Alloy Pure metal Alloy

(0.005) 0.001 0.02 (0.011) – – – – 0.03 –

0.010 – – – 0.026 (0.05) 0.002 0.12 (0.06) 0.13

Figs. 8, 9, 10, 11, 12, 13, 14, and 15 show the tensile strength and the reduction in area in the cases of A1050, A5052, C1100P, C2801, A1070, A5056, C1100B, and C3604, respectively. The simulation results are plotted using solid symbols. The figures in the legend indicate the magnitude of prestrain. By increasing the notch root curvature, the tensile strength increased, whereas the reduction in area decreased. By increasing prestrain, the tensile strength increased, whereas the reduction in area decreased. The reductions in area calculated from the simulations agreed reasonably well with those obtained experimentally, except in the case of C2801. When the notch-root radius was equal to 5 mm, the effect of prestrain on the reduction in area calculated from the simulation agreed with that obtained experimentally, except in the case of C2801. The effect of prestrain on the reduction in area calculated using Eq. (4) was smaller than that calculated using Eq. (3). In the case of C2801, although Eq. (4) was used, the effect of prestrain on the reduction in area calculated from the simulation was larger than that obtained experimentally. The reason for the difference in the effect of prestrain on the reduction in area was considered to be as follows. Because the mass percent of zinc of C2801 was approximately 40%, the metallographic structure of C2801 was composed of alpha and beta phases [45], which are intermetallic compounds. Virtually no chemical constituents except copper and zinc are present in C2801. The mass percent of the alpha phase is not much larger than that of the beta phase. Hence, the decohesion of the interface between the beta phase and surrounding alpha phase is scarcely likely to occur. Therefore, the inclusions around which voids were assumed to nucleate in C2801 hardly existed. The decohesion of the interface between an inclusion and the surrounding matrix was assumed in C2801 because it is an alloy. Therefore, the assumption that C2801 is an alloy is inappropriate in terms of ductile fracture. Although the fracture mechanism of brass from a microscopic viewpoint has been proposed [46], it should be clarified in the future. The chemical composition of C3604 was almost the same as that of C2801. However, in this case, the effect of prestrain on the reduction in area calculated from the simulation agreed with that obtained experimentally. The reason for the agreement in the effect of prestrain on the reduction in area was considered to be as follows. Because the mass percent of zinc in C3604 was approximately 40%, the metallographic structure of C3604 was composed of the alpha and beta phases [45], which were intermetallic compounds. However, lead (the mass percent of which was approximately 3%) existed in C3604 along with copper and zinc. Hence, inclusions of lead, around which voids were assumed to nucleate in an alloy, existed [47]. The effect of the notch-root curvature on the reduction in area calculated from the simulation was smaller than that obtained experimentally in the cases of A1050, A5052, and C2801. The effect of the notch-root curvature on the reduction in area calculated using Eq. (4) was larger than that calculated using Eq. (3). In the cases of A1050 and C2801, although Eq. (4) was used, the effect of the notch-root curvature on the reduction in area calculated from the simulation was smaller than that obtained experimentally. Hence, the void nucleation term in the evolution equation, which denotes the change in the void volume fraction, should be improved in the near future. Fig. 16 shows the finite-element meshes at rupture. The radius of the notch root of the notched specimen was equal to 1 mm, and the radius of the notch root of the smooth specimen was equal to 1 m. The finite-element meshes for notched specimens were the same as those for smooth specimens, and the finite-element meshes for bars were the same as those for sheets. Because the part of the specimen located away from the notch root was hardly deformed, the length of the finite-element meshes in the tensile direction was appropriate. Fig. 17 shows the reduction in area calculated using the other evolution equation. Table 5 shows the material constants used in the other evolution equation (indicated in parentheses). On comparing Fig. 17(a) with Fig. 8(b), it is clear that Eq. (4) was more appropriate than Eq. (3) in the case of A1050. Moreover, on comparing Fig. 17(b) with Fig. 14(b), it is clear that Eq. (3) was more appropriate than Eq. (4) in the case of C1100B. Fig. 18 shows the reduction in area calculated using the other void configuration. Table 5 shows the material constants used in the other void configuration (indicated in parentheses). On comparing Fig. 18(a) with Fig. 10(b), it is clear that the void configuration for a pure metal was more appropriate than that for an alloy in the case of C1100B. In contrast, on comparing Fig. 18(b) with Fig. 11(b), when the notch-root radius was equal to 5 mm, the effect of prestrain on the reduction in area using the void configuration for a pure metal rather than that for an alloy agreed with that obtained experimentally. Hence, it is reasonable to believe that using the void configuration for a pure metal was more appropriate than that for an alloy in the case of C2801.

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(a) C1100P (Notch)

(b) C1100P (Smooth)

(c) C1100B (Notch)

(d) C1100B (Smooth)

Fig. 16. Finite-element meshes at rupture.

0 Exp. 0.80 Exp. 1.27 Exp.

85 80 75 70 65 60

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) A1050

0 Exp. 0.42 Exp. 1.05 Exp.

80

Reduction in area (%)

Reduction in area (%)

90

0 Sim. 0.80 Sim. 1.27 Sim.

0 Sim. 0.42 Sim. 1.05 Sim.

70 60 50 40 30 20

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(b) C1100B

Fig. 17. Reduction in area calculated using the other evolution equation.

4.3. Simulations using conventional ductile fracture criteria The empirical relations [13–15] that represent the condition of void coalescence were derived under the assumption that the principal axes of stress and strain coincide. However, in metal forming processes, the principal axes of stress and strain differ. Furthermore, the empirical relations were obtained under the assumption that a void does not nucleate during forming. However, in pure metals and alloys, a void nucleates during forming. Hence, the simulation results calculated from the empirical relations [13,14] were compared with the experimental results in uniaxial tensile testing of a smooth bar containing a pre-existing three-dimensional void array [18,19]. However, the comparison of the simulation results calculated from the empirical relations and the experimental results obtained using pure metals and alloys in metal forming processes is inappropriate. Simulations were performed using conventional ductile fracture criteria. The conventional von Mises yield function and the conventional Levy–Mises constitutive equation were used. The conventional rigid-plastic finite-element method [29] was also used, in which the incompressibility condition was satisfied by the penalty method. In the ellipsoidal void model, the number of material constants, i.e., A0 or A1 , is one. Hence, a conventional ductile fracture criterion, in which the number of material constants is one, should be used. Therefore, the Cockcroft and Latham fracture

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

0 Exp. 0.80 Exp. 1.27 Exp.

80 70 60 50 40 30 20 0.0

0.2

0.4

0.6

0.8

0 Exp. 0.80 Exp. 1.27 Exp.

90

Reduction in area (%)

Reduction in area (%)

90

0 Sim. 0.80 Sim. 1.27 Sim.

80 70 60 50 40

1.0

0 Sim. 0.80 Sim. 1.27 Sim.

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

Curvature of notch root (1/mm)

(a) C1100P

(b) C2801

Fig. 18. Reduction in area calculated using the other void configuration.

criterion [5] and the fracture criterion proposed by Brozzo et al. [6] were used in this study. The non-dimensional Cockcroft and Latham fracture criterion [5] can be expressed by the following equation

Z

rmax _ edt P C 2 ; r

ð14Þ

where C 2 is the material constant. The fracture criterion proposed by Brozzo et al. [6] can be expressed as

Z

2rmax e_ dt P C 3 ; 3ðrmax  rkk =3Þ

ð15Þ

where C 3 is the material constant. Table 6 shows the material constants used in the conventional ductile fracture criteria. The material constants C 2 and C 3 were determined such that the reduction in area calculated from the simulation for the specimen with no prestrain and a notch-root radius of 5 mm was in good agreement with that obtained experimentally. Although the chemical composition of C1100P was the same as that of C1100B, the material constants C 2 and C 3 of C1100P were larger than those of C1100B. The reason for this difference was conjectured to be as follows. As shown in Table 5, the material constant A0 of C1100P was smaller than that of C1100B. Hence, the rupture strain of C1100P was larger than that of C1100B. Fig. 19 shows the reduction in area calculated using the conventional ductile fracture criteria in the case of C1100P. With an increase in the curvature of the notch root, the reductions in area calculated using the Cockcroft and Latham fracture criterion and the fracture criterion proposed by Brozzo et al. decreased. However, the reductions in area calculated using the Cockcroft and Latham fracture criterion and the fracture criterion proposed by Brozzo et al. hardly changed, with the increase in prestrain. The reason for such small change with the increase of prestrain can be explained as follows. Although plane-strain rolling was simulated in the prestraining process, it could be approximated for convenience as a plane-strain compression, in which the maximum principal stress rmax is equal to zero. Hence, the left-hand side of Eqs. (14) and (15) hardly changed during the prestraining process. Fig. 20 shows the reduction in area calculated using the conventional ductile fracture criteria in the case of C1100B. With an increase in the curvature of the notch root, the reductions in area calculated using the Cockcroft and Latham fracture criterion and the fracture criterion proposed by Brozzo et al. decreased. With an increase in prestrain, the reductions in area calculated using the Cockcroft and Latham fracture criterion and the fracture criterion proposed by Brozzo et al. decreased. However, the effect of prestrain on the reductions in area calculated using the Cockcroft and Latham fracture criterion and the fracture criterion proposed by Brozzo et al. was slightly smaller than that obtained experimentally.

Table 6 Material constants used in conventional ductile fracture criteria.

C1100P C1100B

C2

C3

1.6 1.4

1.7 1.4

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K. Komori / Engineering Fracture Mechanics 151 (2016) 51–69

0 Exp. 0.80 Exp. 1.27 Exp.

0 Cock. 0.80 Cock. 1.27 Cock.

0 Exp. 0.80 Exp. 1.27 Exp. 90

Reduction in area (%)

Reduction in area (%)

90

0 Brozzo 0.80 Brozzo 1.27 Brozzo

80 70 60 50 40 0.0

0.2

0.4

0.6

0.8

80 70 60 50 40 0.0

1.0

Curvature of notch root (1/mm)

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Cockcroft and Latham

(b) Brozzo et al.

Fig. 19. Reduction in area calculated using conventional ductile fracture criteria in the case of C1100P.

0 Exp. 0.42 Exp. 1.05 Exp.

0 Cock. 0.42 Cock. 1.05 Cock.

0 Exp. 0.42 Exp. 1.05 Exp. 80

Reduction in area (%)

Reduction in area (%)

80 70 60 50 40 30 20

0 Brozzo 0.42 Brozzo 1.05 Brozzo

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(a) Cockcroft and Latham

70 60 50 40 30 20

0.0

0.2

0.4

0.6

0.8

1.0

Curvature of notch root (1/mm)

(b) Brozzo et al.

Fig. 20. Reduction in area calculated using conventional ductile fracture criteria in the case of C1100B.

Hence, a comparison of Fig. 19 with Fig. 10(b) and a comparison of Fig. 20 with Fig. 14(b) revealed the validity of the ellipsoidal void model. 5. Conclusions Tensile tests of four types of nonferrous bars and nonferrous sheets were simulated using the ellipsoidal void model proposed by the author, which evaluated ductile fracture in the simulation of metal-forming processes. The following results were obtained: (1) The effect of prestrain on the reduction in area calculated using the ellipsoidal void model agreed with that obtained experimentally. (2) The effect of the notch-root radius on the reduction in area calculated using the ellipsoidal void model agreed reasonably well with that obtained experimentally. (3) The validity of the ellipsoidal void model has been confirmed by a comparison of the simulation results with the experimental results and the results calculated using conventional ductile fracture criteria.

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