Non-uniqueness of cohesive-crack stress-separation law of human and bovine bones and remedy by size effect tests
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ORIGINAL PAPER
Non-uniqueness of cohesive-crack stress-separation law of human and bovine bones and remedy by size effect tests Kyung-Tae Kim · Zdenˇek P. Bažant · Qiang Yu
Received: 27 July 2012 / Accepted: 15 February 2013 / Published online: 3 April 2013 © Springer Science+Business Media Dordrecht 2013
Z. P. Bažant (B) Northwestern University, 2145 Sheridan Rd. A123, Evanston, IL, USA e-mail: [email protected]
of different sizes. To demonstrate it, tests of√notched bovine bone beams of sizes in the ratio of 1: 6:6 are conducted. To minimize random scatter, all the specimens are cut from one and the same bovine bone, even though this limits the number of specimens to 8. A strong size effect is found, but an anomaly in the size effect data trend is obtained, probably due to random scatter and too small a number of specimens. Further it is shown that the optimum range of size effect testing based on Bažant’s size effect law approximately coincides with the size range of beams that can be cut from one bovine bone. By size effect fitting of previously published data on human bone, it is shown that the optimum size range calls for beam depths under 10 mm, which is too small for the standard equipment of mechanics of materials labs and would require a special miniaturized precision equipment.
K.-T. Kim Northwestern University, 2145 Sheridan Road, CEE Evanston, IL 60201, USA e-mail: [email protected]
Keywords Scaling · Strength · Fracture energy · Quasibrittle failure · Nonlinear fracture mechanics · Orthotropic materials · Bio-materials
Abstract It is shown that if the bilinear stressseparation law of the cohesive crack model is identified from the complete softening load-deflection curve of a notched human bone specimen of only one size, the problem is ill-conditioned and the result is nonunique. The same measured load-deflection curve can be fitted with values of initial fracture energy and tensile strength differing, respectively, by up to 100 and 72.4 % (of the lower value). The material parameters, however, give very different load-deflection curves when the specimen is scaled up or down significantly. This implies that the aforementioned non-uniqueness could be avoided by testing human bone specimens
Present Address: K.-T. Kim Samsung Techwin R&D Center, Bundang-gu, Seongnam-si, Gyeonggi-do 463-400, Republic of Korea e-mail: [email protected] Q. Yu Department of Civil and Environmental Engineering, Swanson School of Engineering, University of Pittsburgh, Pittsburgh, PA 15261, USA e-mail: [email protected]
1 Introduction Bone fracture used to be modeled in terms of linear elastic fracture mechanics (LEFM), in which the fracture process zone (FPZ) is considered to be a point and the material fracture properties are characterized by a single parameter, the fracture energy G F or, equivalently, by fracture toughness K I c = E e f G F (where E e f = effective elastic modulus of orthotropic
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Fig. 1 Illustration of bilinear law parameters; G f , initial fracture energy, G F , total fracture ener
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