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An investigation of the plastic failure of spheroidized steels Fei Zhou a,*, J.N. Wang b, J.S. Lian c a

School of Materials Science and Engineering, Jiangsu Uni6ersity of Science and Technology, Zhenjiang 212013, People’s Republic of China b School of Materials Science and Technology, Shanghai Jiao-Tong Uni6ersity, Shanghai 200030, People’s Republic of China c School of Materials Science and Engineering, Jilin Uni6ersity, Changchun 130022, People’s Republic of China Received 11 December 2000; received in revised form 29 June 2001

Abstract The mechanical properties, fracture behaviors and micro-mechanism of damage have been studied for three spheroidized steels, respectively. It is shown that void formation and growth induced by decohesion at the interface between the ferrite matrix and non-metallic particles or by particle cracking are the main reasons for the failure of these materials during plastic deformation. Experimental results show that the void volume fractions of these materials increase as strain increases, with the increasing rate being larger for the material with more carbide particles. The strain -hardening exponent and fracture strain decrease with the volume fraction of second-phase particle increasing. Based on the concepts of effective stress and strain softening induced by void evolution during plastic deformation, analytical expressions of strain hardening exponent and fracture strain for plastically damaged materials with second-phase particles are derived, respectively, and shown to be in good agreement with experimental data. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Spheroidized steels; Void-damage; Plastic deformation; Ductile fracture

1. Introduction Ductile fracture in steels is a progressive process with void nucleation, growth and coalescence [1]. Under the combined influences of plastic deformation and stress, voids grow, coalesce and finally form a continuous fracture surface and dimples. The dimples show the evidence of void initiation at second-phase particles [2]. In spheroidized steels, owing to carbide particles being un-deformable, stress or strain concentrations can occur around these particles during uniaxial tensile testing. As a result, voids could be nucleated at the interface between the ferrite matrix and the non-metallic particles by decohesion or by particle cracking [3–8]. Therefore, the amount, size and distribution of carbide particles and the ductility of the matrix have strong influences on the nucleation, growth and coalescence of voids and the final fracture of the materials.

* Corresponding author. Present address: Department of Mechanical Engineering, Kyungpook National University, Sankeugdong, Pukgu, Taegu 702-701, South Korea. E-mail address: [email protected] (F. Zhou).

Much progress towards better understanding of the effects of second-phase particles and voids on uniform elongation and final fracture strain has been made in the last two decades. The first descriptive expression for the effect of second-phase particles on uniform strain was derived by Su and Gurland [9], who suggested that the uniform strain of the dual-phase steel be regarded as a force-weighed average of the in situ necking strains of the components. Lian and Chen [10] proposed the concept of the strain softening effect induced by void evolution during deformation, and derived an analytical expression. There are many different analytical expressions for fracture strain [9–11]. Because these expressions only consider the effect of voids on fracture strain, without considering the effect of the secondphase particles, they could not be used for describing the deformation and ductile fracture of spheroidized steels completely. In the present paper, the effect of carbide particles on void volume fraction and the effects of voids and second-phase particles on fracture strain are studied. Based on experimental results and the consideration of the strain softening induced by void or cavity evolution during plastic deformation, analytical expressions of

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F. Zhou et al. / Materials Science and Engineering A332 (2002) 117–122

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Table 1 Heat treatment processes of the present steels Materials

45 T8 GCr15

Quenching

Quenchant

Temperature (°C)

Time (h)

860 780 830

1 1 0.5

Tempering

Salty water Water Water

strain hardening exponent and fracture strain for plastically damaged materials with second-phase particles are derived and compared with experimental data.

Temperature (°C)

Time (h)

700 700 720

24 17 6

2.1. Heat treatment and tensile test Three spheroidized steels, containing different levels of C (0.44, 0.793, 1.06 wt.%), were used. They are designated as steel 45, T8 and GCr15, respectively. Table 1 lists the heat treatment processes that these steels underwent. Three heat-treated steels were machined to 15 standard tensile specimens of ¥ 10 × 50 mm2, respectively. Uniaxial tensile testing was conducted in an AG-10T electronic universal tension machine at room temperature and a strain rate of 1 × 10 − 4 s − 1.

2.2. Measurement of microstructures for second-phase particles The second-phase particles were measured by VIDA 2.0 image analysis, which is equipped with scanning electron microscopy. We use the intercept method to obtain 100 data of particle diameter. If we choose a confidence level of 95%, we could decide the normal distribution of the mean particle diameter based on the table of t-distribution. In order to measure the volume fraction of second-phase particles, a point counting technique was used on 100 micrographs at 1000× at different locations for each specimen. The spacing between the points was greater than the maximum intercept length on the second-phase particle to be measured. A square grid of 0.25 cm2 was used for the

Furnace-cooled Furnace-cooled Furnace-cooled

magnification selected in this test. The volume fraction of second-phase particles is given by: fp =

2. Experimental procedure

Cooling method

Np , N

(1)

where Np is the number of intersections of the grid fall in the second-phase particle, N is the total number of points laid down. The second-phase particle volume fraction fp in Table 2 was the mean value in 100 measurements, whose statistic methods are the same as those of the particle diameter. The probability of measuring second-phase particle volume fraction is p=

Nf , 100

(2)

where, Nf is the number of measurements for fpi between ( fp − x) and ( fp + x), x is the standard deviations. If we obtained the second-phase particle mean volume fraction and the mean particle diameter, another parameters were determined by using the equations given by Leroy et al. [6].

2.3. Measurement of 6oid 6olume fraction The fracture surfaces were observed under a scanning electron microscope. Metallographic examination of the deformed specimens was carried out on both longitudinal and transverse sections. An infiltration method, similar to that described by Clausing [12], was used to preserve the true void shape after sectioning. In order to produce equivalent surfaces, the preparation scheme was: (1) polishing with 0.05 mm alumina; (2) ultrasonic cleaning for 1 2 min; (3) etching with 4% nitric acid alcohol solution; (4) ultransonic cleaning for 1 2 min. Errors of section location were less than 0.05 mm.

Table 2 Microstructural features of the second-phase particles Materials

Volume fraction fp

Probability of fp

Average radius r (mm)

Average distance Average center–center distance up (mm) lp (mm)

Average free path u (mm)

45 T8 GCr15

0.0689 0.0013 0.1389 0.0025 0.1589 0.0032

0.53 0.55 0.58

0.5690.02 0.6390.023 0.67 9 0.025

2.19 1.42 1.34

15.34 7.87 7.14

4.66 3.68 3.66

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Table 3 Mechanical properties of the present materials Materials

|0.2 (MPa)

|b (MPa)

n

mf

lf (%)

|f (MPA)

45 T8 GCr15

210.78 224.21 199.18

544.87 604.98 614.97

0.24 0.21 0.18

1.19 0.87 0.74

28.53 26.00 28.00

1219.37 1167.09 1136.48

Table 4 Void volume fraction and its probabilities for three steels Materials

Strain

Void volume fraction (%)

Probability of measuring void volume fraction

45 45 45 T8 T8 T8 GCr15 GCr15 GCr15

0.20 0.80 1.17 0.20 0.48 0.85 0.20 0.49 0.72

0.249 0.015 1.529 0.025 3.529 0.025 0.489 0.013 1.289 0.023 4.849 0.024 0.809 0.015 2.00 9 0.024 5.20 9 0.025

0.33 0.44 0.57 0.36 0.45 0.62 0.34 0.48 0.66

The void volume fraction was measured using VIDA.2.0 image analyses too. Due to the ductility of specimens in the tensile tests is dependent on the void volume fraction [2], a point counting technique was used, which is the same as that of second-phase particles measuring. The tensile specimen was cut into 8 10 transverse sections from the uniform region to that near the necking zone, each of which had been subjected to the same plastic strain. The variation of void volume fraction with strain was determined on transverse sections. The void volume fraction obtained was the mean value in 100 micrographs at 1500× at different locations in each section of the tensile specimen. If we choose a confidence level of 95%, we could decide the normal distribution of the mean void volume fraction based on the table of t-distribution. The probability of measuring void volume fraction is p=

Nv , 100

(3)

where Nv is the number of measurements for fvi between ( fv − x) and ( fv +x), x is the standard deviations.

3. Experimental results After heat treatment, the three steels contained carbide particles of different volume fractions ( fp =0.068– 0.158%) with the particle size remaining the same (2r = 1.2 mm) (as seen in Table 2). The probability of fp shows that the second-phase particles are distributed uniformly. Results of mechanical testing are presented in Table 3, with |0.2 being the yield stress at 0.2%

strain, |b the ultimate tensile stress, n the strain hardening exponent, mf the fracture strain, mf the total elongation, |f the fracture stress. The nucleation and growth of voids occur simultaneously when the plastic deformation reaches a certain extent. Therefore, the degree of the plastic damage may be represented by the increase of void volume fraction fv. As seen in Table 4, it is known that the void volume fraction and the probability of void volume distribution increase with the strain increasing for three steels, respectively. When the strain is lower than 0.2, it was difficult to find a void in 100 micrographs. As the strain is higher than 0.2, the voids was found at the interface between coarse particle and matrix or at the middle of coarse particles [4]. The results show that the nucleation of voids is occurred by interface decohesion or by particle cracking. When the strain approaches the fracture strain, namely the location of section is near fracture surface, the void volume fraction and probability of void volume distribution approach maximum value. It is indicated that the voids grow in necking zone and the distribution of void is uniform. Fig. 1 shows the data of mean void volume fraction fv as a function of strain for the three steels. It is shown that fv increases with increasing strain. Additionally, at a fixed strain, fv also increases with fp increasing, the volume fraction of the second-phase particle, Thus, among the three steels, the void growth rate of steel GCr15 is the highest while that of steel 45 is the lowest. Based on the plasticity-controlled void growth model, the void volume fraction fv can be expressed as a function of strain as [13]: fv = fv0exp(im),

(4)

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The effects of particle volume fraction on materials properties are illustrated in Fig. 2. It is shown that the strain hardening exponent n and the fracture strain mf decrease as the volume fraction of particles fp increases. The reasons for these may be twofold. First, the hardening second-phase particles that do not undertake plastic deformation would lead to less strain hardening in the presence of more such particles. Second, the nucleation of void induced by decohesion of the particle-matrix interfaces or by particle cracking during plastic deformation could result in less strain hardening and earlier fracturing.

4. Discussion

4.1. Plastic instability of materials with second-phase particles under uniaxial tension

Fig. 1. Void volume fraction as a function of plastic strain for spheroidized steels.

where fv0 is the initial void volume fraction, i the void growth rate, and m the plastic strain. For this expression, it is assumed that the volume fraction of voids be identical to the area fraction. This assumption is reasonable for the present case that spherical void distributed almost uniformly in the matrix. Eq. (4) can be rewritten to ln fv = ln fv0 +im.

(5)

From this equation, the void growth rates i for GCr15, T8 and 45 steels can be determined, being 3.31, 3.30 and 3.05, respectively.

The development of voids or cavities during deformation reduces the true area in a specimen under uniaxial tension. The effective area, which carries the external load P, is only a fraction of the apparent area S: S( =S(1−fv).

(6)

Thus the effective stress of the composite is related to the true stresses in the constituents by [14,15] |¯ =

|¯ = |m(1−fp)+ |p fp, 1− fv

(7)

where | is the apparent stress (P/S), |m and |p are the true stress in the matrix and the second-phase particle, respectively. Since the plastic strain of the particle mp is negligible, the total true plastic strain of the composite m can be related to the strain of the matrix mm by:

Fig. 2. Effects of particles volume fraction on (a) strain hardening exponent and (b) fracture strain.

F. Zhou et al. / Materials Science and Engineering A332 (2002) 117–122

m =mm(1−fp)+ mp fp :mm(1 − fp).

(8)

On the other hand, the plastic deformation of the composite and the matrix obeys the following constitutive laws: |= km n,

(9)

|m =k m , nm m m

(10)

where n and nm are the strain hardening exponents of the composite and the matrix, respectively, K and Km are the corresponding strength constants. The strain-hardening exponent is generally obtained from the stress–strain constitutive equation. If the hardening of the composite follows a power law, the strain-hardening exponent may be determined using the strain at the maximum load. According to the criterion of plastic instability [16], instability occurs at the strain when d| =|. dm

(11)

Combination of Eq. (9) with Eq. (11), the maximum uniform strain of the composite is mu = n.

(12)

n

d| #|m #mm #| #m df = (1−fp) + p pfp (1 − fp) −|¯ v. dm #mm #m #mp #m dm (13) Because mp =0, combining Eqs. (7)– (10) and Eq. (4) with Eq. (13) results in

n

(14)

From Eqs. (11) and (12), necking is predicted to occur when d| n = = 1. |dm mu

(15)

Combining Eq. (14) with Eq. (15), the uniform strain of the composite is found as: mu = n =

nm(1− fp)(1 −fv) , [1+fp(h−1)][1 +fv(i − 1)]

ticles without voids, fv = 0 and Eq. (16) is the same expression as Su and Gurland [9], who pointed out that the uniform strain of the dual-phase steel may be regarded as a force-weighed average of the in situ necking strains of the components. Therefore, Eq. (16) always gives mu B nm for damaged composite materials, indicating that the existence of voids and particles has an accelerating effect on plastic instability.

4.2. Fracture strain model based on strain softening effect For materials with second-phase particles, the failure initiation is due to the sudden loss in load-bearing cross-section caused by the linking-up of voids in the neck of a tensile specimen. Thus, the condition of failure initiation is [9] d| =0. dm

(17)

When Eqs. (17) and (14) are combined, there is:

n

|mnm (1−fp)(1− f*) − |¯ if*= 0, v v mf

(18)

where strain m is the replaced by fracture strain mf, fv* is void volume fracture near the fracture surface. Substituting Eq. (7) into Eq. (18) gives

Taking the differential of Eq. (7), there is

| n d| = m m (1− fp)(1 − fv) −|¯ ifv. dm m

121

(16)

where h = |p/|m is a ratio of the ultimate tensile stress of the hard particle and that of the matrix. For materials without voids and second-phase particles, fv = fp =0, and Eq. (16) becomes mu =n, which is the classical criterion for plastic instability [15]. For plastically damaged materials without second-phase particles, fp =0 and Eq. (16) is identical with the case proposed by Lian and chen [10], who considered the effect of internal deformation damage (voids or cavities) on plastic instability can be equivalent to strain softening. For materials containing non-deforming par-

mf =

fp) nm(1−f*)(1− v . fp(h−1)] if*[1+ v

(19)

This equation suggests that the volume fractures of the second-phase particle fp and void damage fv* have strong influences on the fracture strain mf for spheroidized steels.

4.3. Comparison with experiments The analytical expressions of strain hardening exponent and fracture strain for plastically damage materials with second-phase particles are applied to spheroidized steels with non-deforming particles. For spheroidized steels, owing to the void volume fraction is very low at the necking strain, the values of bulk properties in Eq. (16) are determined by using the present experimental values: fv0 = 0.003, i=3.0, and previous ones: nm = 0.35 [4] and h=4.8 [9]. The prediction of the strain hardening exponent n is shown in Fig. 2(a) to be in good agreement with the present experimental data and those of Leroy et al. [6]. The strainhardening exponent of spheroidized steel is a non-linear function of the strain-hardening exponent of ferrite. Because of low effect of second-phase particles, the strain-hardening exponent of spheroidized steel appears to be strongly influenced by the strain-hardening exponent of ferrite.

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F. Zhou et al. / Materials Science and Engineering A332 (2002) 117–122

Application of Eq. (19) to spheroidized steels is shown in Fig. 2(b), by assuming that the void volume fraction at the fracture site fv* is equal to the experimental values of 0.060.07. Some previous data are also included for comparison. It can be seen that there is good agreement between the theoretical prediction and experimental results. The comparison suggests that the analytical expressions of strain-hardening exponent and fracture strain for plastically damaged materials with second-phase particles may better describe the deformation behavior of spheroidized steels.

5. Conclusions (1) For the present three spheroidized steels, the void growth during plastic deformation follows the plasticity-controlled model. The void growth rate varies with the volume fraction of the second-phase particle. (2) With increasing the volume fraction of the second-phase particles, the volume fraction of voids increases, and the strain-hardening exponent and fracture strain for the spheroidized steels decrease. (3) For plastically damaged materials with secondphase particles, the analytical expressions for strain-

hardening exponent and fracture strain are derived and shown to be in good agreement with both present and previous experimental results.

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