3076 立式銑床橫向工作臺設計,立式,銑床,橫向,工作臺,設計 Acta Mechanica Solida Sinica, Vol. 22, No. 5, October, 2009 ISSN 0894-9166Published by AMSS Press, Wuhan, ChinaEFFECTS OF POISSON’S RATIO ON SCALING LAW INHERTZIAN FRACTUREJing Liu Xuyue Wang(Shenzhen Graduate School, Harbin Institute of Technology, Shenzhen 518055, China)Received 3 August 2009, revision received 13 September 2009ABSTRACT In this paper the Auerbach’s scaling law of Hertzian fracture induced by a sphericalindenter pressing on a brittle solid is studied. In the analysis, the singular integral equation methodis used to analyze the fracture behavior of the Hertzian contact problem. The results show thatthe Auerbach’s constant sensitively depends on the Poisson’s ratio, and the e?ective Auerbach’sdomain is also determined for a given value of the Poisson’s ratio.KEY WORDS scaling law, Hertzian fracture, Poisson’s ratioI. INTRODUCTIONWith the rapid development of materials science and technology, investigations on the Hertzianindentation/scratch tests have become an interesting project due to their advantage of relative ease inoperation. Many experimental approaches and physical models have been presented for the purposeto measure material parameters such as hardness and fracture toughness. With regard to the fracturetoughness, Frank and Lawn[1] studied the Auerbach’s scaling law[2] by virtue of fracture mechanics.They obtained a fracture scaling law to estimate the surface energy of the material. The major drawbackof their work lies in the following two aspects: on one hand, they used the Green function of a Gri?thcrack to approximate the solution of a cone crack or an initial cylindrical crack, but this approximationmight be inaccurate; on the other hand, they did not give a criterion to determine the situation ofan initial crack. Mouginot and Maugis[3] investigated the same problem using a revised formulationof a penny-shaped crack in an in?nite space. By introducing the maximum strain energy release ratecriterion for determining the starting radius of an axisymmetric crack, they attributed Auerbach’s lawto the competition between the increase in the surface crack size and the decrease in the stress ?eldalong the crack depth. Their numerical simulation showed that an Auerbach range exists during theformation of a shallow ring crack. They also gave the Auerbach constant for the optical glass. However,their model cannot describe the di?erence between a shallow ring crack and a cone crack. Later, Zeng,Breder and Rowcli?e[4] used a similar method to derive an approximate relationship between the stressintensity factor and the indentation force. Furthermore, Warren et al.[5] described the growth of surface?aw by virtue of eigenstrain methods. Finite element and boundary element methods have also beendeveloped for studying di?erent aspects of the Hertzian indentation problem[6–9]. However, the in?uenceof the Poisson’s ratio on the Hertzian fracture remains unclear.In this paper, the attention is centered to evaluating the e?ects of Poisson’s ratio on the fracturescaling law. Based on our previous work[10], the Somigliana ring dislocation will be used to simulateCorresponding author. E-mail: mewxy@hitsz.edu.cnProject supported by the National Natural Science Foundation of China (No. 10772058).z zr rσrr (a, z ? z )f (z )dz + σrr(a, z ? z )g(z )dzσrz (a, z ? z )f (z )dz + σrz (a, z ? z )g(z )dzσrr(a, z) + σrr(a, z) = 0σrz (a, z) + σrz (a, z) = 0σrr (a, z) = √ + √ (1 + ν) u tan?1 √ + (1 ? ν)ρζ 2σrz (a, z) = ? √Vol. 22, No. 5 Jing Liu et al.: E?ects of Poisson’s Ratio on Scaling Law in Hertzian Fracture · 475 ·the initially cracking behavior in the sense that the Somigliana ring dislocation solution is equivalentto the basic solution of the correspondent axisymmetric crack problem.II. MODEL AND METHODConsider the axisymmetric Hertzian fractureproblem of a semi-in?nite body, as shown in Fig.1,where r and z are the cylindrical coordinate, cis the contact radius, a is the pre-existing ringcrack radius, and h is the crack depth. A sphericalindenter with radius R is applied to the surfaceof the semi-in?nite space with a normal load, P .The material is assumed to be isotropic and lin-early elastic, with Young’s modulus E and Pois-son’s ratio ν, while the indenter is assumed to berigid. Fig. 1 Hertzian fracture problem.The relative normal and shear displacements along the ring crack surfaces can be quantitatively de-scribed by the pile-up of Somigliana ring dislocations[10]. Hence, the fracture problem is reduced tosolving a radial edge ring dislocation of unit intensity and an axial glide ring dislocation of unit intensity,as shown in Fig.1. First, we de?ne the dislocation density functions at any point z = z along the cracksurface asf (z ) = ?g(z ) = ??(u+ ? u?)?z?(u+ ? u?)?z(1)(2)The stress ?elds along the ring crack surfaces can be integrated asσrr(a, z) =σrz (a, z) =00hhAA00hhRR(3)(4)where σijA and σijR are the stress ?elds[10] at (a, z) due to an axial glide ring dislocation and a radialring dislocation, with unit Burgers vectors at (a, z ), respectively.Consider the traction-free boundary conditions on the ring crack surfaces, the stress ?elds in Eqs.(3)and (4) need to be equilibrated by the stress ?elds at r = a that are induced by the indentation forceP in the absence of the ring crack[11]. Thus, one haswherePP(0 < z < h)(0 < z < h)(5)(6)P (1 ? 2ν)3ρ2 1 ?ζu3 ζu√ 1uu1+ u + u2 + ζ 2ζ 2? 2 p0(7)Ps u(1 + u) p0 (8)p0 =3P2πc2, ρ = ac, ζ = zc (9)s = [ζ 2 + (ρ ? 1)2] [ζ 2 + (ρ + 1)2] (10)= ?σrr(a, z)= ?σrz (a, z)→h→hh a KIpm ch a KIIpm cc c,h a 3π3Rγ ?1 h a· 476 · ACTA MECHANICA SOLIDA SINICA 2009u = 12(ρ2 + ζ 2 ? 1 + s) (11)Equations (5) and (6) can be further transformed to the following Cauchy singular integral equations:μ2π(1 ? ν) 0h g(z )z ? z dz + 0hQr11(z ? z )g(z )dz +0hQa12(z ? z )f (z )dz P (12)μ2π(1 ? ν) 0h f (z )z ? z dz + 0hQa22(z ? z )f (z )dz +0hQr21(z ? z )g(z )dz P (13)where the Fredholm kernels Qij ( ) are given in Ref.[10].By means of the numerical method developed by Erdogan[12], the stress intensity factors KI andKIIand the strain energy release rate G can be obtained asKI = zlim+KII = zlim+2π(z ? h)σrr(a, z)2π(z ? h)σrz (a, z)(14)(15)G = 1 ? ν 2E (KI2 + KII2 ) (16)In Hertzian fracture tests, crack initiation occurs always outside the contact area, and the crackinitiation radius is signi?cantly dependent on both the surface ?aw size and the indenter-induced stressdistribution near the contact edge. In our model, for a given initial ring crack with size h, we determinethe corresponding crack radius a by using the criterion of maximal strain energy release rate G. Forconvenience, we normalize the stress intensity factors and strain energy release rate as follows:K1K2,c c,c c==√√(17)(18)φh a,c c =π3c3 E4P 2 1 ? ν 2G (19)where pm is the mean stress on the contact area and is expressed bypm =Pπc2(20)Using the relation[11]c3 = 3(1 ? ν 2)4E PR (21)the normalized strain energy release rate in Eq.(19) can be simpli?ed asφh a, =3π3R16P G (22)For a critical indentation force PC at which the crack begins to grow, one hasG = 2γ = 4PC2 (1 ? ν 2)π3c3E φh ac c (23)PC =π3Eγ2(1 ? ν 2) φ?1/2 ,c c c3/2 = 8 φ ,c c (24)where γ is the surface energy of the indented material.Vol. 22, No. 5 Jing Liu et al.: E?ects of Poisson’s Ratio on Scaling Law in Hertzian Fracture · 477 ·III. NUMERICAL RESULTS AND DISCUSSIONSFor instance, numerical calculations have been conducted for an optical glass with the followingelastic constants[3]: E = 8.0 × 1010 Pa and ν = 0.22. In the calculations, we take ν = 0.26, 0.32 and0.36 to evaluate the e?ect of the Poisson’s ratio of the indented material on the fracture scaling law.In the case of ν = 0.22, we have obtained a fracture scaling law in our previous paper[10], asshown in Fig.2, where the Auerbach constant is A = 4.65 × 103γ. The corresponding scaling law,PC = 4.65 × 103Rγ, was demonstrated to be e?ective in the range of 0.02 < h/c < 0.1. Comparingwith Mouginot’s scaling law[3], PC = 6.7 × 103Rγ, we concluded that they underestimated the surfaceenergy by about 30%. For the representative values of the Poisson’s ratio, we obtain the scaling laws,the Auerbach constant, and the corresponding validation scopes asPC = 4.65 × 103Rγ, A = 4.65 × 103γ, 0.02 < h/c < 0.10 for ν = 0.22,PC = 7.28 × 103Rγ, A = 7.28 × 103γ, 0.02 < h/c < 0.08 for ν = 0.26,PC = 18.7 × 103Rγ, A = 18.7 × 103γ, 0.02 < h/c < 0.06 for ν = 0.32,PC = 36.3 × 103Rγ, A = 36.3 × 103γ, 0.02 < h/c < 0.04 for ν = 0.36,as shown in Figs.2-5, respectively. The results above show that there always exists an e?ective Auerbachdomain for a given Poisson’s ratio. The surface energy can be measured by virtue of the fracture scalinglaw which is consistent with empirical Auerbach’s law. As the e?ect of Poisson’s ratio is concerned, itcan be found that the fracture scaling law greatly depends on the magnitude of Poisson’s ratio, and thistendency was also found in Mouginot’s work[3]. On the other hand, the e?ective Auerbach domain willdecrease with the increase in the Poisson’s ratio of the measured material. In summary, the dependenceof the scaling law on Poisson’s ratio indicates that a larger Poisson’s ratio enhances the failure-resistingability of the material by accommodating larger elastic deformation.Fig. 2. Normalized strain energy release rate φ?1/2 (h/c)versus the crack size h/c when the Poisson’s ratio ν = 0.22.Fig. 4. Normalized strain energy release rate φ?1/2 (h/c)versus the crack size h/c when the Poisson’s ratio ν = 0.32.Fig. 3. Normalized strain energy release rate φ?1/2(h/c)versus the crack size h/c when the Poisson’s ratio ν = 0.26.Fig. 5. Normalized strain energy release rate φ?1/2(h/c)versus the crack size h/c when the Poisson’s ratio ν = 0.36.· 478 · ACTA MECHANICA SOLIDA SINICA 2009IV. CONCLUSIONSThe e?ect of Poisson’s ratio on the fracture scaling law in the indentation of brittle materials has beenexamined by using the singular integral equation method. It is found that for a speci?ed Poisson’s ratioof the indented solid, the fracture scaling law is usually in the form of Auerbach’s empirical relation. Wedemonstrate that with the increase in the Poisson’s ratio, the Auerbach constant signi?cantly increasesand correspondingly, the e?ective Auerbach range decreases. This conclusion suggests that the role ofAuerbach’s law will be decreased in measuring the toughness of the material with a larger Poisson’sratio. However, in order to understand the physical mechanisms of failure in Hertzian contact, it isimportant to develop new methods which can be used to describe the measurable relation in Hertzianfracture test.References[1] Lawn,B., Fracture of Brittle Solids. Cambridge: Cambridge University Press, 1993.[2] Auerbach,F., Measurement of hardness. Annual Physical Chemistry, 1891, 43: 61-100.[3] Mouginot,R. and Maugis,D., Fracture indentation beneath ?at and spherical punches. Journal of MaterialScience, 1985, 20: 4354-4376.[4] Zeng,K., Breder,K. and Rowcli?e,D., The Hertzian stress ?eld and formation of cone cracks – I: Theoreticalapproach. Acta Metallurgy Materials, 1992, 40: 2595-2600.[5] Warren,P., Hills,D. and Dai,D., Mechanics of Hertzian cracking. Tribology International, 1995, 28: 357-362.[6] Chen,S.Y., Farris,T.N. and Chandrasekar,S., Contact mechanics of Hertzian cone cracking. InternationalJournal of Solids and Structures, 1995, 32: 329-340.[7] Anderson,M., Stress distribution and crack initiation for an elastic contact including friction. InternationalJournal of Solids and Structures, 1996, 33: 3673-3696.[8] Kocer,C. and Collins,R., Angle of Hertzian cone cracks. Journal of American Ceramic Society, 1998, 81:1736-1742.[9] De Lacerda,L. and Wrobel,L., An e?cient numerical model for contact-induced crack propagation analysis.International Journal of Solids and Structures, 2002, 39: 5719-5736.[10] Wang,X.Y., Li,L.K.Y., Mai,Y.W. and Shen,Y.G., Theoretical analysis of Hertzian contact fracture: Ringcrack. Engineering Fracture Mechanics, 2008, 75: 4247-4256.[11] Maugis,D., Contact, Adhesion and Rupture of Elastic Solids. Heidelberg: Springer-Verlag, 2000.[12] Erdogan,F., Complex Function Technique. New York: Academic Press, 1975.感謝您試用AnyBizSoft PDF to Word。試用版僅能轉(zhuǎn)換5頁文檔。要轉(zhuǎn)換全部文檔，免費獲取注冊碼請訪問http://www.anypdftools.com/pdf-to-word-cn.html