Fracture Toughness for Brittle Fracture of Elastic and Plastic Materials

Yoshikazu Tanabe

National Museum of Nature and Science, Tsukuba 305-0005, Japan

The crack energy density causes a new expression of fracture toughness for brittle fracture, together with the concept of Barenblatt’s characteristic distance dc of cohesive zone at the edge of the crack. The Griffith’s formula for fracture toughness is modified by the ratio of ·T/T, which ·T is the true fracture stress and T the maximum strength of the material obtainable by ordinary processing. The fracture · toughness of elastic-plastic material, KIc for brittle fracturepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is described by the product of T and the root of the observed absorption energy by corrected Charpy impact test corrected for instrumental effect, Icp . The characteristic distance dc depends linearly on the released elastic energy, which is the physical basis of above-mentioned estimation formula. [doi:10.2320/matertrans.M2012348]

(Received October 18, 2012; Accepted December 10, 2012; Published January 25, 2013) Keywords: fracture toughness, crack energy density, true fracture stress, corrected Charpy impact value, Barenblatt’s characteristic distance

1. Introduction elastic CED, ¾e is released after unloading of the external force P. In the case of linear elastic materials, ¾ ¼ ¾e ¼ ðð ¯2Þ Þ 2 ¼ ¯ ’ For steels showing the unstable fracture (brittle fracture), 1 =E KI G where is the Poisson s ratio, KI the Oda, Nakamura and Kawakami have reported before ca. 40 stress intensity factor, G the energy release rate. Only the 1,2) years that the fracture toughness KIc is related to the two mode I (opening mode) of crack surface displacement is observed values, the true fracture stress ·T and the Charpy considered and discussion is made based on the two- impact value Iobs as dimensional elasticity and the condition of plane strain, cp sffiffiffiffiffiffiffiffi hereafter. Iobs ¼ · cp ð Þ KIc k T 1 E 2. Brittle Fracture of Perfectly Elastic Materials where E is the Young’s modulus and k a nondimensional constant.13) Equation (1) is important at two points, 1) that The problem of singularity at each crack tip in stress field the fracture toughness KIc is described by the true fracture is overcome by introducing the cohesive forces (or attraction) · obs stress T and Charpy impact value Icp which are easily acting in small edge regions of crack, which is previously measured without any restriction of specimen size, and discussed by Barenblatt8) and Goodier9) in detail. 2) that this equation can be used for the brittle fracture of Tensile stress ·y normal to the surface of a crack and materials with wide range of plastic deformation such as the associated vertical displacement v of the crack surface perfectly elastic, small scale yielding, large scale yielding and (y direction, see Fig. 1), are given by general yielding. T. Oda et al. have explained eq. (1) based K þ K 1 pI ffiffiffiffiffiffiffiffiffiffi0 2 upon the dimensional analysis. Trying to make clear the ·y ¼ þ gð0ÞþOs ð2Þ 2³s 1 physical meaning of eq. (1) needs careful consideration 1 rffiffiffiffiffiffi 2 1 ¯ s 3 because the well-known fracture mechanics is essentially ¼ ð þ Þ 2 þ 2 ð Þ v 4 KI K0 Os2 3 based on elasticity not only in linear fracture mechanics but E 2³ also in J-integral for larger scale yielding. In J-integral, the where « refers to upper and lower faces of displacement. outside of integral contour must be elastic state. J-integral can The stress intensity factors K and K are rffiffiffiffi ZI 0 be used in perfectly elastic state, small scale yielding and a ð Þ ¼ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiT0 t ð Þ large scale yielding states but not in general yielding state. KI 2 ³ dt 4 0 a2 t2 The method of analysis including general yielding is not rffiffiffiffi Z 2 d gð²Þ known. This paper starts from the concept of crack energy K ¼ pffiffiffi d² ð5Þ 47) 0 ³ density (CED) proposed by K. Watanabe. The CED can 0 ² be defined in any state of plastic deformation including and T0(t) is the distribution of applied loads, g(²) the general yielding. distribution of cohesive forces near the crack tip, 2a the The crack energy density is defined to be the area density length of crack, x = a ¹ t where t is a running coordinate on of (strain) energy representing the work (per unit area) given the x axis, s1 = x ¹ a, s2 = a ¹ x and d is the distance for at the point of crack edge from the outside of the crack in acting cohesive forces (see Fig. 1). The stress singularity 1 2 the plane including its point of the crack edge. The CED can occurs from the term s1 in eq. (2), and the postulate 8) 9) be experimentally evaluated from the load-displacement KI + K0 = 0 by Barenblatt and Goodier removes its curves as ¾ ¼ð@U=@aÞ, where total strain energy is singularity. The remaining main terms in eq. (3) are of order R 3 ð Þ¼ 2 U a; u Pdu, the load P, the displacement u and the s2, which is demonstrated in Fig. 1. crack length 2a. When the cohesive force distribution exists near the crack In elastic-plastic materials, CED is denoted by ¾ = ¾e + ¾p edge (tip), the crack energy density ¾ at one end of crack is where ¾e and ¾p are the parts of CED preserved elastically and the work done near the crack edge (W) during an infinitesimal plastically, respectively, during plastic deformation.6) The extension of the crack. Fracture Toughness for Brittle Fracture of Elastic and Plastic Materials 315

σ rffiffiffiffiffiffiffiffiffiffiffiffiffi EG K ¼jK j¼ c ð11Þ Ic 0c 1 ¯2 y 1 s 2 The critical energy release rate or the surface energy depends 2 3 2 on the mechanisms of breaking(rupture) and reflects the s2 v x number of effective bonding which depends on very small voids, cracks, defects, impurities, dislocations and so on. s1 To compare with experimental results, the information of d d distribution of T (t) and g(²) is necessary. The simplest case is s 0 2 the uniform distribution of external force (applied load) and a a T0(t) = · (·: external tensile stress). The distribution of g(²) is not yet known, and the limiting cohesivepffiffiffiffiffiffiffiffi R force distributionpffiffiffi σ ² ¼ ³ d ð²Þ ²d² gc( ) which is given by K0c 2= 0 gc = ,is assumed as g (²) = · where · is the tensile strength of fi c T T Fig. 1 Vertical displacement of the crack faces under the condition of nite the perfectly elastic materials. This assumption seems to be stress at the edge of crack (crack tip). reasonable as the zeroth approximation of gc(²) because the crack extends if atomic or molecular attraction cannot hold Z d ¯2 d ð²Þ 2 bonding. It means cohesive forces cannot become larger than ¾ ¼ W ¼ 2 1 gpffiffiffi d² ð Þ · d ³ 6 T and the critical value of cohesive forces is approximated a E 0 ² by ·T. Of course, this is very rough estimation and only the which can be derived from Goodier’s result by exchanging starting point of discussion of g(²). gm(²) for g(²), where gm(²) is the limiting cohesive force From T0(t) = ·, distribution. (Goodier9) used G for g) The distance d is the pffiffiffiffiffiffi K ¼ · ³a ð12Þ size of the cohesive zone. Using eq. (5), I ¯2 which is the well-known expression of stress intensity 1 2 1113) ¾ ¼ K ð7Þ factor. From gc(²) = ·T E 0 pffiffiffiffiffiffiffiffi j j¼· ³ ð Þ Barenblatt suggests that cracks in perfectly elastic materials K0 T dc 13 extend, when all the work of the cohesive forces is equal to and £ new surface energy 2 . Here the critical value of energy 8 £ d ¼ d ð14Þ release rate Gc is used instead of 2 . c ³2 1 ¯2 ¾ ¼ K2 ¼ G ð8Þ where d is the characteristic distance of cohesive zone. c E 0c c c Equation (13) is the same form as eq. (12) by using dc. where ¾c and K0c is the critical value of ¾ and K0, respectively. For a kind of perfectly elastic material with the largest When the distribution of external force (applied load) T0(x) strength after various plastic processing (a kind of extreme ² exists without the cohesive force distribution g( ), the crack state), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi energy density ¾ is calculated as the apparent work done pffiffiffiffiffiffiffiffi high on the body. The crack 2a is assumed to extend as ¤a high EGc jK j¼T ³dc ¼ ð15Þ infinitesimally, 0c 1 ¯2 Z ¤ where is the maximum strength of the material. As 1 a 1 1 ¯2 T ¾ ¼ · vdx ¼ K2 ð Þ Griffith has previously shown,10) glass has the extreme state ¤ 2 y I 9 a 0 2 E high £ and Gc in eq. (15) is equal to the surface energy 2 s. For which is the well-known result with Griffith10) and many steels, the realization of extreme state is very difficult by the other authors.1113) In Griffith’s theory, the critical energy ordinary processing method such as forging and pressing. ¾c for perfectly elastic materials is assigned to the surface For the perfectly elastic material after ordinary processing energy 2£s. (not extreme but ordinary state), rffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¯2 pffiffiffiffiffiffiffiffi low ¾ ¼ 2 ¼ £ ð Þ low EGc c KIc 2 s 10 jK j¼· ³d ¼ ð16Þ E 0c T c 1 ¯2 low where KIc is usually called fracture toughness. where the critical energy release rate Gc is smaller than + high Barenblatt and Goodier has made the postulate KI Gc . For example, if very small cracks exist within the K0 = 0 in order to remove the stress singularity at the crack distance dc, this sample is perfectly elastic and has lower 8,9) edges. This also shows eqs. (7) and (9) give the same ¾, tensile strength ·T than T. and it is natural that ¾c in eqs. (8) and (10) gives the same If eq. (15) is assumed to be the reference state of the surface energy 2£s or critical energy release rate Gc. These material with the same geometry (same dc), eqs. (15) and + = relations including the postulate KI K0 0 are self- (16) derive sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi consistent in the framework of linear elasticity for the · EGhigh perfectly elastic materials. And the following relation is noted Klow ¼jKlowj¼ T c ð17Þ Ic 0c 1 ¯2 using Gc instead of 2£s, T 316 Y. Tanabe pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where · = ¼ Glow=Ghigh. Equations (12) and (17) give the result that the larger elastic energy state appears by T T c c sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi unloading just before the break(rupture). · EGhigh · ¼ T c ð Þ The fracture toughness KIc appearing in eq. (20), is related 2 18 T ³ð1 ¯ Þa to the linear part of preserved elastic CED just before which is the breaking stress with a crack of 2a. This equation breaking(rupture). Here, it is implicitly assumed that the is different in the term of ·T/T frompffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi the usual Griffith’s breaking of materials such as metals occurs only at the 1013) · ¼ low high formula. The factor T=T Gc =Gc causes the condition of perfectly elastic state. In other words, metals difference of fracture toughness between well-processed undertake the plastic deformation until the perfectly elastic extreme state and less processed ordinary one. The factor state appears. The appearance of perfectly elastic state after ·T/T describes the difference of the stored elastic energy at the plastic deformation, causes the Griffith’s breaking. The the break in perfectly elastic materials. reason why the perfectly elastic state appears after plastic deformation, is that the plastic deformation continues until Ã 3. Fracture Toughness KIc of Elastic-Plastic Solids the moving units in materials vanish. The state that the moving units vanish, is the perfectly elastic one. Breaking The CED of elastic-plastic materials for the mode I of occurs only in the perfectly elastic state. crack surface displacement is Equation (23) is directly derived from eq. (19). This is well-known result as Griffith10)Irwin14)Orowan15) formula. 1 ¯2 ¾ ¼ ¾e þ ¾p ¼ K2 þ ¾p ð19Þ Compared with eq. (22), eq. (23) has no effect of the factor E I ·T/T. The difference of stored elastic energy causes the e p where ¾ and ¾ are the elastic and plastic parts of CED, factor ·T/T and does not affect the plastic part of 2£p. ¾e ¼ðð ¯2Þ Þ 2 ¼ respectively, and 1 =E KI G where G is the energy release rate. The nonlinear part of ¾e is neglected. 4. Estimation of Absorbed Energy by Experimentally If the critical value of ¾ in the right-hand side of eq. (19) Observable Parameter ¾ ¼ðð ¯2Þ Þ 2 is denoted by c 1 =E KIc , KIc is a kind of extended fracture toughness since ¾c is the energy per unit The exact method to observe the absorbed energy relating area for breaking(rupture) of the crack edge. The parameter to the static fracture toughness is not yet known. The surface fi £ fi KIc is the rede ned fracture toughness containing the effect energy 2 s discussed by Grif th and many authors is of plastic deformation. In perfectly elastic materials, KIc is effective concept only in single crystals and amorphous the same as the fracture toughness KIc and related to the materials. The absorbed energy can be approximately fi energy release rate G. The rede ned fracture toughness KIc observed by impact tests under some conditions. The is given as observed Charpy impact value Iobs contains Ghigh þ 2£ cp c p E together with the stored energy in the instrument and the toss K2 ¼ K2 þ 2£ ð20Þ Ic Ic ¯2 p energy of broken specimen. The absorption energy is related 1 high þ £ ¼ ¡ obs ¼ corrected to Charpy impact value as Gc 2 p Icp Icp ¾p ¼ £ 1113) ¡ ¡ ¡ where c 2 p as described in various publications. where the correction parameter is 0 < < 1 and must be Using eq. (17), sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi determined from experimental system. If the observed impact value Iobs contains only the absorption energy by deforma- · E 2 cp K ¼ T Ghigh þ T 2£ ð21Þ tion and fracture, ¡ ¼ 1. Usually, ¡ includes other effects Ic ¯2 c · p T 1 T such as elastic deformation of testing machine and toss high 16,17) where Gc is the stored elastic energy in the material after energy of broken specimen. Kobayashi et al. have plastic deformation, andpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi is also the released energy. In shown the method to measure the correction parameter ¡ ¼ð· Þ ð high ð ¯2ÞÞ ¡ eq. (21), KIc T=T EGc = 1 is related to the by the instrumented impact testing method, and report that state of materials just before breaking(rupture) after plastic is about 0.8 for steel and 0.5 for cast iron.16) Brittle materials deformation finishes (see eq. (17)). Equation (21) is appli- like ceramics reveal rather lower values of ¡ as 0.40.6.17) cable for all elastic-plastic materials independent of degree of It must be noted that the observed Charpy impact value plasticity. generally reflects the mechanism of dynamic fracture tough- When Glow 2£ (or Ghigh ðT Þ22£ ), in which brittle ness and is not always related to the mechanism of static c p c ·T p fracture occurs, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi fracture toughness discussed in the above-mentioned sec- tions. This means that the absorbed energy for static fracture · E K ¼ T Ghigh ð22Þ toughness would be different from the dynamic case. In the Ic ¯2 c T 1 low-velocity (low blow) impact tests in the instrumented and when Glow 2£ (or Ghigh ðT Þ22£ ), in which Charpy impact testing system, the dynamic fracture tough- c p c ·T p ductile fracture occurs, ness is nearly equal to the static one for steel within the error sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi of measurement, which suggests ¡ is correctly estimated.18) E E K ¼ ðGlow þ 2£ Þ ¼ 2£ ð23Þ In the high-velocity impact test, other discussion must be Ic ¯2 c p ¯2 p 1 1 needed including the effect of loss of contact and is beyond Equation (22) shows that the structure of the material the reach of this paper. high changes through plastic deformation although the plastic In eq. (21), Gc appears. Even small plastic deformation high energy is very small, and the elastic energy is stored as Gc causes the change of structure of the material and the elastic low fi low high high which is larger than Gc . This is experimentally veri ed by energy Gc increases to Gc . This Gc is observed by Fracture Toughness for Brittle Fracture of Elastic and Plastic Materials 317 low-velocity impact tests in the case of brittle fracture. /mm Impact tests include all energy as elastic and inelastic 2 * Ic T energies until the break occurs, and also reflect the increase K σ of elastic energy through structural change caused by plastic 1 π

£ high c deformation. If inelastic energy 2 p is very small, Gc is d corrected approximated by Icp . low £ When Gc 2 p, in which brittle fracture occurs, sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi · E K ¼ T Icorrected ð24Þ Characteristic Ic 2 cp Distance, T 1 ¯ low and when G 2£ , in which ductile fracture occurs, obs c p sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Observed Charpy Impact Value, /MNm-1 Icp E ¼ corrected ð Þ ’ KIc 2 Icp 25 Fig. 2 The relationship between the Barenblatt s characteristic distance dc 1 ¯ obs and observed Charpy impact value Icp , in which data used are by Oda These equations (eqs. (24) and (25)) are the estimation et al.1,2) formula of fracture toughness by using easily observable · corrected parameters T and Icp . The parameter T is the pcorrectedffiffiffi the instrumental effect of impact test but its effect ¡ obs maximum strength of materials achieved by processing is rather small in steels. It is clear that dc depends on Icp plastically. It is well known that the ordinary deformation linearly. And the obtained slope 12.5 (GPa)¹1 gives ca. 2 GPa high process leads a kind of upper limit of mechanical strength for T, which is reasonable one. This result denotes that Gc % high ð 2 Þ which is about ten of ideal mechanical strength. This upper is described as Gc T=2E dc. The strain energy 2 limit is T and steel shows T as ca. 2 GPa (for example, T=2E per unit volume gathered in the range of dc, gives high piano wire). Ordinary processing techniques such as forging, Gc , which is the elastic energy released by breaking. It is high rolling, pressing and heat treatment cannot reveal the ideal not clear whether the value of Gc would coincide with mechanical strength (ideal cleavage strength) because those the surface energy or not. In addition, the result that T is techniques cannot control the fine structure of materials obtained from the slope in Fig. 2, suggests that the elastic below the size of a few hundreds of nanometer. In other state just before breaking is the one with the tensile strength words, T is the maximum mechanical strength of the T at least in the range of dc. Moreover, the larger dc, the specimen in case that its fine structure above a few hundreds larger the fracture toughness. The eq. (24) is based on that dc 11,12,16) high of nanometer is precisely controlled. depends linearly on Gc . The fracture toughness KIc in eq. (24) includes the two Equation (24) is important because the fracture toughness corrected peffects,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi one is the effect of absorption energy Icp through KIc can be estimated from the observed physical values as ð corrected ð ¯2ÞÞ ’ · ’ EIcp = 1 with Young s modulus E and the true fracture stress T, Young s modulus E and the absorption · · = corrected other is the effect of the true fracture stress T.If T T energy Icp which can be measured easily and without (in the case of perfectly elastic and extremely well-processed using excessively large specimen. materials), eq. (24) is equal to the Griffith’s formula The apparent maximum strength of specimen, T seems to corrected ¼ ¼ £ (eq. (11)) because of Icp Gc 2 s. be a kind of physical parameters because bulk materials such Equation (24) is derived based on the CED consideration as metals and ceramics can be controlled only over the region and the assumption that the break(rupture) of materials of several hundreds of nanometer by ordinary processing 19) occurs at the perfectly elastic state even after the plastic methods. The value of T can be observed on the bulk deformations. Equation (24) can be used not only in perfectly materials with precisely controlled fine structure over the elastic materials but in general yielding because of no range of a few hundreds of nanometer. restriction of degree of plasticity, although the energy The previously proposed estimation formula, eq. (1) has low £ condition Gc 2 p must be maintained. The estimation apparently a different form from eq. (24). Compared with formula of fracture toughness, eq. (24) has physical base and eqs. (1) and (24), the nondimensional constant k is given by pffiffiffi can be used to estimate the fracture toughness from easily E ¡ corrected ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ measurable parameters, absorption energy Icp and true k 27 1 ¯2 T tensile strength ·T. The constant k is represented by the ratio of Young’s modulus 5. Discussion E and the maximum strength of materials T reachable by plastic processing. In the case of steel, E is ca. 200 GPa, ¯ is 15) From eqs. (15) and (16), ca. 0.3, T ca. 2 GPa and ¡ ca. 0.8, and k is ca. 94 which is 1,2) low 2 a similar value as shown by Oda et al. and Nakamura 1 K E 1 3) d ¼ Ic ¼ Ghigh ð26Þ et al. It must be noted that Oda et al. have not corrected the c ³ · ³ð ¯2Þ 2 c pffiffiffi T 1 T observed Charpy impact values but the effect of ¡ is rather This equation means that the characteristic distance dc is small. Equation (24) is better for use than eq. (1) because it high proportional to Gc for brittle materials. The eq. (26) is the has clear physical meaning. And eq. (24) is important in the high same as eq. (22). As described before, Gc is measured by brittle fracture and cannot be used in the case of ductile corrected low · Icp , and dc is calculated from KIc and T. Figure 2 is fracture, as seen from the difference between eqs. (24) the replotting of data by Oda et al.,1,2) although they have not and (25). 318 Y. Tanabe

6. Conclusions Acknowledgments

The concept of crack energy density (CED) is the base Sincere thanks are due to Mr. Makoto Nakamura, of the discussion of fracture phenomena of elastic-plastic (formerly) Mitsubishi Heavy Industries, Ltd. for the valuable solids such as metallic materials. The fracture toughness is discussions during the course of this research and also due to fi ¾ ¼ðð ¯2Þ Þ 2 ¾ rede ned as c 1 =E KIc , where c is the critical Mr. Seiji Fukuda, (formerly) Mitsubishi Heavy Industries, value of CED and the undertaking energy per unit area at the Ltd. for offering a lot of information. crack edge to break. In brittle fracture of perfectly elastic materials, modified REFERENCES Griffith’s formula of fracture toughness (eq. (17)) and breaking stress with a crack 2a (eq. (18)) is obtained by 1) T. Oda, M. Nakamura and H. Kawakami: J. Jpn. Inst. Met. 39 (1975) taking into account of the Barenblatt’s characteristic distance 725 735 (in Japanese). 2) T. Oda, M. Nakamura and H. Kawakami: Mitsubishi Heavy Industries dc at the edge of crack (tip). Technical Rev. 12 (1975) 100107 (in Japanese). The general expression of fracture toughness of elastic- 3) M. Nakamura and Y. Tanabe: Bull. Natl. Mus. Nat. Sci., E 31 (2008) plastic solids is given by eq. (21). Equation (21) is applicable 1524 (in Japanese). to all materials independent of degree of plasticity because 4) K. Watanabe: Trans. Jpn. Soc. Mech. Eng. 51 (1985) 873882 its derivation is based on the energy consideration without (in Japanese). / 5) K. Watanabe: Trans. Jpn. Soc. Mech. Eng. 47 (1981) 406415 expecting detailed mechanisms of elastic and or plastic (in Japanese). deformations and fracture. 6) K. Watanabe: Trans. Jpn. Soc. Mech. Eng. 48 (1982) 12261236 For the brittle fracture of elastic-plastic materials, eq. (22) (in Japanese). is derived and this equation causes the new estimation 7) K. Watanabe: Prep. Jpn. Soc. Mech. Eng. No. 804-1 (1980) 9194 formula (eq. (24)) which can be used for brittle fracture (in Japanese). 8) G. I. Barenblatt: Adv. Appl. Mech. 7 (1962) 55129. regardless of the deformation such as perfectly elastic, 9) J. N. Goodier: Fracture, An Advance Treatise, (ed. H. Liebowitz, small scale yielding, large scale yielding and general yielding Academic Press, 1968), vol. II (Mathematical Fundamentals), Chap. 1. under the condition of larger elastic energy than plastic 10) A. A. Griffith: Proc. 1st Intern. Congr. Appl. Mech., Delft (1924) energy. pp. 5563. The points are that the break(rupture) of bulk materials 11) H. Okamura: Introduction to Linear Fracture Mechanics, (Baifukan Co., Ltd., 2006) (in Japanese). occurs at the perfectly elastic state after the plastic 12) M. Shiratori, T. Miyoshi and H. Matsushita: Computational Fracture deformation, which is described by Griffith’s mechanism, Mechanics, (Jikkyo Shuppan Co., Ltd., 1999) (in Japanese). and the elastic energy stored in the material after plastic 13) T. L. Anderson: Fracture Mechanics ®Fundamental and Applica- deformation is different from the initial one because of tions®, (CRC Press, Taylor & Francis, 2005). structural change. This elastic energy Ghigh can be observed 14) G. Irwin: Fracturing of Metals, (American Society for Metals, 1948) c pp. 147166. by the low-velocity impact test of instrumented Charpy 15) E. Orowan: Fatigue and Fracture of Metals, (ed. W. M. Murray, Wiley, impact test, although some correction is needed for 1950) pp. 139167. instrumental contribution. 16) T. Kobayashi and I. Yamamoto: Bull. Jpn. Inst. Met. 32 (1993) 151 The new estimation formula of fracture toughness, eq. (24) 159 (in Japanese). 17) T. Kobayashi, M. Niinomi, Y. Koide and K. Matsunuma: Trans. JIM 27 is of practical use since it is possible to estimate with easily corrected · (1986) 775 783. measurable parameters such as Icp and T, without using 18) M. Asano, T. Kobayashi, M. Nawa and K. Niihara: J. Jpn. Inst. Met. 60 the remarkably large size specimen. (1996) 12221228 (in Japanese). 19) N. Tsuji: Tetsu-to-Hagane 88 (2002) 359369 (in Japanese).