A generalized anisotropic quadric yield criterion and its application to bone tissue at multiple length scales

Schwiedrzik, J. J.; Wolfram, U.; Zysset, P. K. (2013). A generalized anisotropic quadric yield criterion and its application to bone tissue at multiple length scales. Biomechanics and Modeling in Mechanobiology, 12(6), pp. 1155-1168. Springer Berlin Heidelberg 10.1007/s10237-013-0472-5

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Nonlinear computational analysis of materials showing elasto-plasticity or damage relies on knowledge of their yield behavior and strengths under complex stress states. In this work, a generalized anisotropic quadric yield criterion is proposed that is homogeneous of degree one and takes a convex quadric shape with a smooth transition from ellipsoidal to cylindrical or conical surfaces. If in the case of material identification, the shape of the yield function is not known a priori, a minimization using the quadric criterion will result in the optimal shape among the convex quadrics. The convexity limits of the criterion and the transition points between the different shapes are identified. Several special cases of the criterion for distinct material symmetries such as isotropy, cubic symmetry, fabric-based orthotropy and general orthotropy are presented and discussed. The generality of the formulation is demonstrated by showing its degeneration to several classical yield surfaces like the von Mises, Drucker–Prager, Tsai–Wu, Liu, generalized Hill and classical Hill criteria under appropriate conditions. Applicability of the formulation for micromechanical analyses was shown by transformation of a criterion for porous cohesive-frictional materials by Maghous et al. In order to demonstrate the advantages of the generalized formulation, bone is chosen as an example material, since it features yield envelopes with different shapes depending on the considered length scale. A fabric- and density-based quadric criterion for the description of homogenized material behavior of trabecular bone is identified from uniaxial, multiaxial and torsional experimental data. Also, a fabric- and density-based Tsai–Wu yield criterion for homogenized trabecular bone from in silico data is converted to an equivalent quadric criterion by introduction of a transformation of the interaction parameters. Finally, a quadric yield criterion for lamellar bone at the microscale is identified from a nanoindentation study reported in the literature, thus demonstrating the applicability of the generalized formulation to the description of the yield envelope of bone at multiple length scales.

Item Type:

Journal Article (Original Article)


04 Faculty of Medicine > Pre-clinic Human Medicine > Institute for Surgical Technology & Biomechanics ISTB [discontinued]

Graduate School:

Graduate School for Cellular and Biomedical Sciences (GCB)

UniBE Contributor:

Schwiedrzik, Johann Jakob, Wolfram, Uwe, Zysset, Philippe


500 Science > 570 Life sciences; biology
600 Technology > 610 Medicine & health
600 Technology > 620 Engineering




Springer Berlin Heidelberg




Johann Jakob Schwiedrzik

Date Deposited:

21 Apr 2014 08:42

Last Modified:

05 Dec 2022 14:28

Publisher DOI:






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