- Email: [email protected]

A model for the dynamic fracture toughness of ductile structural steel Hai Qiu

a,*

, Manabu Enoki b, Yoshiaki Kawaguchi a, Teruo Kishi

c

a

b

Japan Society for the Promotion of Science, 6 Ichibancho, Chiyoda-ku, Tokyo 102-8471, Japan Department of Materials Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan c National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-Shi, Ibaraki 305-0047, Japan Received 24 February 2002; received in revised form 22 March 2002; accepted 16 April 2002

Abstract We assume in this paper that the dynamic fracture toughness KId of ductile structural steels is dependent on void nucleation and void growth. The void nucleation-induced dynamic fracture toughness KIdn and the void growthinduced dynamic fracture toughness KIdg were obtained by modifying the void nucleation-induced and void growthinduced static fracture toughness models, respectively, considering the eﬀect of strain rate and local temperature. By the relationship between the void nucleation-induced dynamic fracture toughness KIdn and the void growth-induced dynamic fracture toughness KIdg ððKId Þ2 ¼ ðKIdn Þ2 þ ðKIdg Þ2 Þ dynamic fracture toughness KId could be quantitatively evaluated. With this model the dynamic fracture toughness of two structural steels (X65 and SA440) was assessed, and the causes for the diﬀerences between the static and dynamic fracture toughness were also discussed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Dynamic loading; Strain rate; Dynamic fracture toughness; Ductile fracture; Structural steel

1. Introduction The critical condition for the onset of ductile fracture under static loading was suggested to be straincontrolled [1–3]. At the critical point of ductile fracture, the local equivalent plastic strain must exceed a critical fracture strain over a characteristic distance. Based on this model, static fracture toughness (KIC ) of ductile structural steels was divided into two parts: void nucleation-induced (KICn ) and void growthinduced fracture toughness (KICg ) in a companion paper [4]. The expressions for the two parts were given by the approach of Thomason and Garrison, respectively. With these equations, the fracture toughness of ductile structural steels can be quantitatively evaluated. Crack initiation under rapid loading is assessed by dynamic fracture toughness KId [5]. It is reasonable to regard that as long as the ductile fracture mode of materials remains constant the critical fracture condition

* Corresponding author. Address: Welding Metallurgy Group, Steel Research Centre, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba-Shi, Ibaraki 305-0047, Japan. Tel.: +81-298-592135; fax: +81-298-592101. E-mail address: [email protected] (H. Qiu).

0013-7944/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 2 ) 0 0 0 8 0 - 2

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for high rate loading is still strain-controlled. Therefore, we could also evaluate KId by void nucleationinduced and void growth-induced dynamic fracture toughness like we did for KIC in Ref. [4]. Thomason gave a KId model for void nucleation-controlled ductile fracture by modifying his KIC model considering the eﬀect of temperature and strain rate on the behavior of materials [6]. In this paper, we use his approach to modify the equations for KICn and KICg to make them ﬁt for the determination of void nucleation-induced and void growth-induced dynamic fracture toughness, respectively, and thus developing a KId model for ductile structural steels. With the developed model the dynamic fracture toughness of structural steels was quantitatively evaluated, and the diﬀerences between KIC and KId were also investigated. 2. Development of dynamic fracture toughness model for ductile structural steel Ductile fracture is regarded being strain controlled under whether static or high speed loading, thus dynamic fracture toughness KId for ductile fracture could be assessed by void nucleation-induced and void growth-induced dynamic fracture toughness as we did for KIC [4]. By the same method as we did in Ref. [4], KId can be expressed by 2 ðKId Þ2 ¼ ðKIdn Þ2 þ KIdg ð1Þ where KIdn and KIdg are the void nucleation-induced and the void growth-induced dynamic fracture toughness, respectively. The determination method of KIdn and KIdg will be brieﬂy summarized as follows. 2.1. Prediction of KIdn For rapid loading, the two primary eﬀects have to be taken into account: (i) The eﬀect of temperature (below 0.4Tm , where Tm is the melting temperature) on the elastic moduli E, l, of the material and (ii) The eﬀect of temperature (below 0.4Tm ) and strain rate on the basic material ﬂow stress and the elevated local stresses at sub-micron sized microstructural particles. Frost and Ashby [7] gave an experimental expression for the eﬀect (i), and can be expressed in the form for ferrous alloys M

E l ¼ ¼ f1 0:81½ðT 300Þ=1810 g E 0 l0

ð2Þ

where M is a correction factor for the eﬀect of temperature on the elastic moduli, T is the absolute temperature in K, and E0 and l0 are the Young’s modulus and elastic shear modulus, respectively, at 300 K. The correction factor, denoted as N here, for the eﬀect of temperature and strain rate on the plastic-ﬂow stress of a material, containing small microstructural particles, was discussed by Thomason [6] with the low temperature rate equation for plasticity limited by discrete obstacles, and it can be expressed as 2Tk 6 _ 1þ ln e =10 1 rs S1 l 0 b3 N ¼ ¼ ð3Þ 600k 8 rs0 S10 ln 10 1þ l0 b 3 where rs and S1 are, respectively, the shear-ﬂow stress and the maximum deviatoric principal stress at temperature T in K, rs0 and S10 are, respectively, the shear ﬂow stress and the maximum deviatoric principal

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591

Fig. 1. Plastic strain and void nucleation strain distributions ahead of a crack.

stress at a quasi-static shear (tensile) strain rate 102 s1 and the normal ambient temperature of 300 K, k is Boltzmann’s constant, b is the Burger’s vector, and e_1 is the maximum principal strain rate. For static loading, the plastic strain ey and the void nucleation strain eny ahead of a crack (as shown in Fig. 1) are given by Eqs. (4) and (5), respectively [4] ey

0:3KI2 ½1 ð x=qÞ=3:81

qrys E

pﬃﬃﬃmþ1 m 0:5 H eny ¼ rc 2= 3 C eny ½0:5 þ ln ð1 þ x=qÞ

ð4Þ

ð5Þ

where KI is the stress intensity factor for mode I, q is the radius of a crack, rys is the yield stress, E is the Young’s modulus, rc is the critical interfacial strength of particle/matrix, H is a material constant related to the particle size, and deﬁned by [8] pﬃﬃﬃﬃ H ¼ ðrc rm Þ= eny ð6Þ where rm is the mean normal stress. Material constant, C, and work hardening exponent, m, obeys the relation r ¼ Cem . High speed loading increases modulus and stress by a factor of M (Eq. (2)) and N (Eq. (3)), respectively, therefore, Eq. (4) could be modiﬁed in the form ey

0:3KI2 ½1 ð x=qÞ=3:81

MN qrys E

ð7Þ

Eq. (7) is only valid over the range 0 < x=q < 3:81. The parameter H in Eq. (5) is a function of modulus and local stress, and is magniﬁed by MN times at high speed loading [6]. The second term in the right side of Eq. (5) is, in fact, the mean normal stress, and could be modiﬁed by multiplying that term by N. rc is magniﬁed by an arbitrary factor L, thus by introducing M and N, Eq. (5) could be rewritten as pﬃﬃﬃmþ1 m 0:5 MNH eny ¼ Lrc N 2= 3 C eny ½0:5 þ ln ð1 þ x=qÞ

ð8Þ It should be noted that Eq. (8) is only ﬁt for the materials in which dislocations have a strong inﬂuence on void nucleation (dislocation model is usually valid when the radius of microstructural particles is less than 1 lm [9]).

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Eqs. (7) and (8) give the plastic strain ey and the void nucleation strain eny ahead of a crack under dynamic loading, respectively. As load increases, ey increases according to Eq. (7). When ey and eny satisfy the following conditions voids begin to initiate: ey ¼ eny

and

dey deny ¼ dx dx

ð9Þ

The critical point derived from Eqs. (7)–(9) is the critical point of void nucleation, and the KI with respect to it is regarded to be the void nucleation-induced dynamic fracture toughness KIdn . 2.2. Prediction of KIdg The void growth-induced fracture toughness KICg for static loading has been given in a companion paper, and the expression is in the form [4] rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rys E x0 ðRv =RI Þ ð10Þ KICg 0:6 where x0 is the distance between particles, and (Rv =RI ) is the radius, Rv , of a void divided by the radius, RI , of the particle initiating the void. The parameters in Eq. (10) which will be aﬀected by temperature and strain rate are rys , E and (Rv =RI ). Because (Rv =RI ) is directly obtained from the fracture surface, (Rv =RI ) need not be corrected by M and N, and retains the same form. Thus, introducing M and N into Eq. (10) gives rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ MN rys E x0 ðRv =RI Þ ð11Þ KIdg 0:6

3. Evaluation of the dynamic fracture toughness of structural steels 3.1. Experimental SN490, X-65 and SA440 which were investigated in a companion paper [4] were also used in this work. Their chemical compositions, microstructures and mechanical properties have been given elsewhere [4]. In this work, we measured the dynamic fracture toughness of the three steels by an instrumented Charpy test at an impact velocity of 4.9 mm/s on the fatigue pre-cracked specimens with length 55 mm, width 10 mm and thickness 10 mm. All tests were conducted at room temperature. Lin et al. [10] have calculated the tensile stress distribution ahead of a crack under static and dynamic loading (5.53 m/s) by FEM, and found that at the same loading level, no signiﬁcant diﬀerences of tensile stress existed at diﬀerent loading velocities. Thus, we can assume that the stress ﬁeld ahead of a crack before crack propagation under static loading is the same as that under dynamic loading in this work. On this assumption the test procedure (see Fig. 2) is determined as: (1) attach a strain gage to the specimen near the fatigue pre-crack tip; (2) measure the relations between load and strain taken from the strain gage at static loading; (3) carry out impact test at high speed loading, and ﬁnd out the crack initiation point (peak point) from the impact curve; (4) determine the load, Pm , corresponding to the crack initiation point from the load–strain calibration curve obtained in (2), and substitute it into the following expression given by ASTM E399 standard [11] to get KId K¼

PS f ða=W Þ BW 3=2

ð12Þ

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593

Fig. 2. The principle of KId measurement.

Fig. 3. Impact curves: (a) small-scale yield condition, and (b) general-scale condition.

where P is the applied load, S is the pin span, B is the thickness, W is the width, a is the crack length, and the values of f ða=W Þ are given in ASTM E399 standard. Eq. (12) is only for three-point bending specimen. Fig. 3 gives two types of impact curves, (a) and (b). A small-scale yield zone ahead of a crack is produced in Fig. 3(a) while large-scale yielding occurs in Fig. 3(b). Eq. (12) is only ﬁt for the small-scale yield condition; thus, it is not suitable to Fig. 3(b). For large-scale yield condition, KId is calculated using the equivalent energy method [12]. Brieﬂy, this concept states that the energy (area under the straight line) up to e is equivalent to the energy (area under the impact curve) up to emax (see Fig. 3). P corresponding to e is regarded as the load with respect to the critical cracking point of the specimen having suﬃcient thickness to avoid general yielding prior to fracture. According to this method, P can be obtained. By substituting P into Eq. (12) the value of KId for general yield condition is achieved. 3.2. Results The fracture surfaces of KId specimens were observed by SEM. X65 and SA440 fractured in ductile mode, and SN490 fractured in cleavage mode. Fracture surface examination indicates that voids in X65 are initiated by Ca inclusions, large voids and small voids in SA440 are formed by MnS inclusions and carbides, respectively. We could conclude from the results obtained in this work and elsewhere [4] that the initiation sources of voids in X65 and SA440 retain constant under both static and high speed loading. Fifteen big voids ahead of the crack tip on the fracture surface for X65 and SA440 were selected, and their values of ðRv =RI Þ were measured. The average values of ðRv =RI Þ of the 15 voids for the steels are listed

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Table 1 The average values of ðRv =RI Þ ðRv =RI Þ (static) [4] ðRv =RI Þ (dynamic)

X65

SA440

12 5.3

3.5 1.7

in Table 1. For convenience of comparison, the values of ðRv =RI Þ of KIC specimens are given simultaneously. Table 1 shows that strain rate decreases the values of ðRv =RI Þ, i.e., ductility degrades with strain rate. It should be noted that the values of ðRv =RI Þ of KIC and KId specimens for SA440 in Table 1 are the ones of MnS inclusion-induced voids. The KId values of SN490, X65 and SA440 are 46, 78, 90 MPa m1=2 , respectively. 3.3. Discussion Under high rate loading, local temperature is elevated by the plastic deformation at a crack tip. The increment of temperature is related to the crack propagation speed or the loading time from zero to the maximum load [13]. The dependence of the rise in temperature on loading times or propagation velocities for steel was given by Rice and Levy [13]. The experimental results show that the loading time of the impact tests in this paper is in the order of 104 s. According to Rice and Levy’s numerical calculation, the temperature rise is about 50 °C for stress intensity factor ¼ 85 Pa m1=2 and loading time ¼ 104 s. We assume that the temperature near the crack tip in this paper raised about 50 °C. Since the impact tests were conducted at room temperature in this work, the local temperature (T) at a crack tip should be about 350 K considering the local temperature increment by plastic deformation. The strain rate near the crack tip was measured to be about 107 s1 . Substituting the experimental results into Eqs. (2) and (3) leads to M ¼ 0:98 and N ¼ 1:35. It has been known [4] that carbides bond much more strongly with the matrix than the MnS inclusions in SA440, therefore, we could assume that the nucleation of carbide-induced voids determines the KIdn , and the growth of MnS inclusion-induced voids determines the KIdg for SA440. The parameters C and m in Eq. (8) are 1161.9 MPa, and 0.212, respectively, for SA440 [14], and other parameters except for M, N and ðRv =RI Þ in Eqs. (7), (8) and (11) have been given elsewhere [4]. The eﬀect of strain rate on rc for the used steels was not clear, for simplicity of calculation the parameter L in Eq. (8) was taken 1 in this work. The x=q and ey with respect to the critical point of void nucleation were determined by Eqs. (7)–(9) as shown in Fig. 4(a) and (b), and their values are summarized in Table 2. The void nucleation strains eny in Fig. 4(a) and (b) are of the Ca inclusion-induced and carbide-induced voids, respectively. The critical values of x=q and ey were substituted into Eq. (7) to obtain KIdn . KIdg was assessed by Eq. (11), and the theoretical dynamic fracture toughness, KIdt , was given by Eq. (1). The KIdn , KIdg , KIdt and the experimental dynamic fracture toughness, KIde , are summarized in Table 3. The KIdn and KIdg for SA440 in Table 3 are carbide-induced and MnS inclusion-induced dynamic fracture toughness, respectively. The KIdt is compared with the KIde in Fig. 5. The ﬁgure indicates that the developed KId model works well. The proportions of KIdn and KIdg in KIdt are shown in Fig. 6, and it is found that both void nucleation and void growth contribute greatly to KId for X65 and SA440. The dynamic fracture toughness loading rate, K_ Id , was determined by dividing the experimental value of KId by the time from zero to the maximum load. The K_ Id for the three steels in this work is in the order of 105 MPa m1=2 /s. The experimental values of KId and KIC for SN490, X65 and SA440 are given in Fig. 7. The data for KIC are cited from Ref. [4]. The ﬁgure shows that KIC is larger than KId for the three steels. At static loading SN490 fractures in ductile mode [4], while it fractures in cleavage mode by impacting. The change of

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595

Fig. 4. Determination of KIdn for (a) X65 and (b) SA440.

Table 2 The critical values of ey and x=q corresponding to the KICn and KIdn ey (static) [4] x=q (static) [4] ey (dynamic) x=q (dynamic)

X65

SA440

0.247 1.4 0.0955 1.61

0.109 1.78 0.036 1.85

Table 3 Values of KIdn , KIdg , KIdt and KIde (MPa m1=2 ) SN490 X65 SA440

KIdn

KIdg

KIdt

KIde

– 51 38

– 55 92

– 75 100

46 78 90

Fig. 5. Comparison of the theoretical dynamic fracture toughness with the experimental results.

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Fig. 6. Composition of the theoretical dynamic fracture toughness.

Fig. 7. Comparison of the experimental static fracture toughness with the experimental dynamic fracture toughness.

fracture mechanism from ductile to cleavage results in the decrease of fracture toughness of SN490. X65 and SA440 remain fracture in ductile mode under both static and dynamic loading, therefore the variations in fracture toughness with strain rate is not associated with fracture mode. To investigate the reasons for the fact that KIC is larger than KId for X65 and SA440, KIdn and KIdg are compared with KICn and KICg , respectively, in Fig. 8. Fig. 8 reveals that KICn and KICg are larger than KIdn and KIdg , respectively, for the two steels. Voids in practical materials are generally initiated by the separation of particle/matrix interfaces [8]. When the local stress at a particle exceeds the critical interfacial strength, rc , the decoherence of particle/ matrix begins to take place. Table 2 shows that the critical ey of KId specimen for void nucleation is smaller than that of KIC specimen. This is because high-speed loading stores up dislocations around particles and thus elevating the local stress at particles. Therefore, it is easy to attain the rc under dynamic loading, that is, at smaller plastic strains the voids can initiate. According to Eq. (7), the decrease of ey due to strain rate decreases the void nucleation-induced fracture toughness, that is, the decrease of ey is the reason for why KICn is larger than KIdn . According to Eq. (11), x0 , ðRv =RI Þ and rys inﬂuence KIdg . The factors, which aﬀect x0 , (Rv =RI ) and rys , will inﬂuence KIdg . For X65 and SA440, strain rate degrades their ductility (see Table 1), but increases their yield strength by a factor N ¼ 1:35. Although the degradation of ductility decreases the void growth-

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597

Fig. 8. Comparison of: (a) KICn with KIdn , and (b) KICg with KIdg .

induced fracture toughness according to Eq. (11), the increase of yield strength compensates its eﬀect and makes KICg larger than KIdg .

4. Conclusions (1) A dynamic fracture toughness model for ductile structural steel is developed based on the suggestion that dynamic fracture toughness is determined by void nucleation and void growth. The theoretical dynamic fracture toughness by the model is in good agreement with the experimental results. (2) Both void nucleation and void growth aﬀect the dynamic fracture toughness of X65 and SA440 greatly. (3) KIC is greater than KId for SN490, X65 and SA440. The reasons for this are the change of fracture mechanism from ductile to cleavage for SN490, and the decrease of void nucleation strain and the increase of yield strength by strain rate for X65 and SA440.

Acknowledgements The authors wish to acknowledge the ﬁnancial support of both Kawatetsu 21st Century Foundation and the Iron and Steel Institute of Japan, and the providing of steels by NKK Company of Japan.

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