3076 立式銑床橫向工作臺(tái)設(shè)計(jì)
3076 立式銑床橫向工作臺(tái)設(shè)計(jì),立式,銑床,橫向,工作臺(tái),設(shè)計(jì)
Acta Mechanica Solida Sinica, Vol. 22, No. 5, October, 2009 ISSN 0894-9166Published by AMSS Press, Wuhan, ChinaEFFECTS OF POISSONS 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 Auerbachs 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 Auerbachs constant sensitively depends on the Poissons ratio, and the eective Auerbachsdomain is also determined for a given value of the Poissons ratio.KEY WORDS scaling law, Hertzian fracture, Poissons 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 Lawn1 studied the Auerbachs scaling law2 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 Grithcrack 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 Maugis3 investigated the same problem using a revised formulationof a penny-shaped crack in an innite space. By introducing the maximum strain energy release ratecriterion for determining the starting radius of an axisymmetric crack, they attributed Auerbachs 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 dierence between a shallow ring crack and a cone crack. Later, Zeng,Breder and Rowclie4 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 surfaceaw by virtue of eigenstrain methods. Finite element and boundary element methods have also beendeveloped for studying dierent aspects of the Hertzian indentation problem69. However, the inuenceof the Poissons ratio on the Hertzian fracture remains unclear.In this paper, the attention is centered to evaluating the eects of Poissons ratio on the fracturescaling law. Based on our previous work10, the Somigliana ring dislocation will be used to simulateCorresponding author. E-mail: mewxyhitsz.edu.cnProject supported by the National Natural Science Foundation of China (No. 10772058).z zr rrr (a, z z )f (z )dz + rr(a, z z )g(z )dzrz (a, z z )f (z )dz + rz (a, z z )g(z )dzrr(a, z) + rr(a, z) = 0rz (a, z) + rz (a, z) = 0rr (a, z) = + (1 + ) u tan1 + (1 ) 2rz (a, z) = Vol. 22, No. 5 Jing Liu et al.: Eects of Poissons 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-innite 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-innite space with a normal load, P .The material is assumed to be isotropic and lin-early elastic, with Youngs modulus E and Pois-sons ratio , 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 dislocations10. 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 dene 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 asrr(a, z) =rz (a, z) =00hhAA00hhRR(3)(4)where ijA and ijR are the stress elds10 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 crack11. Thus, one haswherePP(0 z h)(0 z h)(5)(6)P (1 2)32 1 u3 u 1uu1+ u + u2 + 2 2 2 p0(7)Ps u(1 + u) p0 (8)p0 =3P2c2, = ac, = zc (9)s = 2 + ( 1)2 2 + ( + 1)2 (10)= rr(a, z)= rz (a, z)hhh a KIpm ch a KIIpm cc c,h a 33R 1 h a 476 ACTA MECHANICA SOLIDA SINICA 2009u = 12(2 + 2 1 + s) (11)Equations (5) and (6) can be further transformed to the following Cauchy singular integral equations:2(1 ) 0h g(z )z z dz + 0hQr11(z z )g(z )dz +0hQa12(z z )f (z )dz P (12)2(1 ) 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 Erdogan12, the stress intensity factors KI andKIIand the strain energy release rate G can be obtained asKI = zlim+KII = zlim+2(z h)rr(a, z)2(z h)rz (a, z)(14)(15)G = 1 2E (KI2 + KII2 ) (16)In Hertzian fracture tests, crack initiation occurs always outside the contact area, and the crackinitiation radius is signicantly 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)h a,c c =3c3 E4P 2 1 2G (19)where pm is the mean stress on the contact area and is expressed bypm =Pc2(20)Using the relation11c3 = 3(1 2)4E PR (21)the normalized strain energy release rate in Eq.(19) can be simplied ash a, =33R16P G (22)For a critical indentation force PC at which the crack begins to grow, one hasG = 2 = 4PC2 (1 2)3c3E h ac c (23)PC =3E2(1 2) 1/2 ,c c c3/2 = 8 ,c c (24)where is the surface energy of the indented material.Vol. 22, No. 5 Jing Liu et al.: Eects of Poissons 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 constants3: E = 8.0 1010 Pa and = 0.22. In the calculations, we take = 0.26, 0.32 and0.36 to evaluate the eect of the Poissons ratio of the indented material on the fracture scaling law.In the case of = 0.22, we have obtained a fracture scaling law in our previous paper10, asshown in Fig.2, where the Auerbach constant is A = 4.65 103. The corresponding scaling law,PC = 4.65 103R, was demonstrated to be eective in the range of 0.02 h/c 0.1. Comparingwith Mouginots scaling law3, PC = 6.7 103R, we concluded that they underestimated the surfaceenergy by about 30%. For the representative values of the Poissons ratio, we obtain the scaling laws,the Auerbach constant, and the corresponding validation scopes asPC = 4.65 103R, A = 4.65 103, 0.02 h/c 0.10 for = 0.22,PC = 7.28 103R, A = 7.28 103, 0.02 h/c 0.08 for = 0.26,PC = 18.7 103R, A = 18.7 103, 0.02 h/c 0.06 for = 0.32,PC = 36.3 103R, A = 36.3 103, 0.02 h/c 0.04 for = 0.36,as shown in Figs.2-5, respectively. The results above show that there always exists an eective Auerbachdomain for a given Poissons ratio. The surface energy can be measured by virtue of the fracture scalinglaw which is consistent with empirical Auerbachs law. As the eect of Poissons ratio is concerned, itcan be found that the fracture scaling law greatly depends on the magnitude of Poissons ratio, and thistendency was also found in Mouginots work3. On the other hand, the eective Auerbach domain willdecrease with the increase in the Poissons ratio of the measured material. In summary, the dependenceof the scaling law on Poissons ratio indicates that a larger Poissons ratio enhances the failure-resistingability of the material by accommodating larger elastic deformation.Fig. 2. Normalized strain energy release rate 1/2 (h/c)versus the crack size h/c when the Poissons ratio = 0.22.Fig. 4. Normalized strain energy release rate 1/2 (h/c)versus the crack size h/c when the Poissons ratio = 0.32.Fig. 3. Normalized strain energy release rate 1/2(h/c)versus the crack size h/c when the Poissons ratio = 0.26.Fig. 5. Normalized strain energy release rate 1/2(h/c)versus the crack size h/c when the Poissons ratio = 0.36. 478 ACTA MECHANICA SOLIDA SINICA 2009IV. CONCLUSIONSThe eect of Poissons 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 specied Poissons ratioof the indented solid, the fracture scaling law is usually in the form of Auerbachs empirical relation. Wedemonstrate that with the increase in the Poissons ratio, the Auerbach constant signicantly increasesand correspondingly, the eective Auerbach range decreases. This conclusion suggests that the role ofAuerbachs law will be decreased in measuring the toughness of the material with a larger Poissonsratio. 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.References1 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 Rowclie,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 ecient 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(fèi)獲取注冊(cè)碼請(qǐng)?jiān)L問http:/www.anypdftools.com/pdf-to-word-cn.html
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