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

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.

Correlation between broiler lameness and anatomical measurements of bone using radiographical projections with assessments of consistency across and within radiographs.

Lameness represents a major welfare and production issue in the poultry industry with a recent survey estimating 27% of birds lame and 3% unable to walk by 40 d of age. A variety of factors may induce lameness and are typically grouped into 2 broad classes on the basis of being infectious or skeletal in nature with the latter accounting for the majority of cases. The current work sought to build upon a large body of literature assessing the anatomical properties of bone in lame birds. Our specific objectives sought to identify relationships between relevant anatomical properties of the tibia and metatarsus using digital quantification from radiographs of legs and a measure of walking difficulty. Resulting output was statistically analyzed to assess 1) observer reliability for consistency in placing the leg during the radiograph procedure and quantification of the various measures within a radiograph, 2) the relationship between the various measurements of anatomical bone properties and sex, bird mass, and gait score, and 3) the relationship between each measurement and leg symmetry. Our anatomical bone measures were found to be reliable (intra-rater and test-retest reliabilities < 0.75) within radiograph for all measures and 8 of the 10 measures across radiographs. Several measures of bone properties in the tibia correlated to difficulty walking as measured by gait score (P < 0.05), indicating greater angulations with increasing lameness. Of the measures that manifested a gait score × bird mass interaction, heavier birds appeared to exhibit less angulation with increasing difficulty walking with lighter birds the opposite. These interactions suggest possibilities for influencing effects of activity or feed intake on bone mineralization with the bone angulation observed. Our efforts agree with that of others and indicate that angulation of the tibia may be related to lameness, though subsequent efforts involving comprehensive measures of bird activity, growth rates, and internal bone structure will be needed if the validity of the measures are to be accepted.