# Non-uniqueness of cohesive-crack stress-separation law of human and bovine bones and remedy by size effect tests

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## Abstract

It is shown that if the bilinear stress-separation 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 non-unique. 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 of different sizes. To demonstrate it, tests of notched bovine bone beams of sizes in the ratio of 1:\(\sqrt{6}\):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.

## Keywords

Scaling Strength Fracture energy Quasibrittle failure Nonlinear fracture mechanics Orthotropic materials Bio-materials## Notes

### Acknowledgments

Partial financial support under NSF grant CMMI-1129449 to Northwestern University is gratefully acknowledged. The first author wishes to thank for partial support under W. P. Murphy Fellowship of Northwestern University.

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