von Mises stress, fatigue, and failure theory BME 615

von Mises stress - σv In an elastic, isotropic body subjected to 3D loads, a complex 3 dimensional system of stresses is developed. At any point there are stresses acting in different directions, and the direction and magnitude of stresses changes from point to point. The von Mises criterion is a formula for calculating whether the stress combination at a given point will cause failure. Note: it is based on the behavior of typical engineering materials and has relatively little to do with any biological tissue behavior People use it because it’s built into FE codes and gives a single value to map (rather than mapping each value in a stress tensor separately)

von Mises stress - σv In ductile engineering materials, von Mises found that, even though no principal stress exceeds the yield stress, it is possible for yielding to result from a combination of stresses. The von Mises criterion is a formula for combining these 3 stresses into an equivalent stress, which is then compared to the yield stress of the material. Yield stress is a known property of the isotropic material, and is usually considered to be the 1D yield stress.

von Mises stress - σv Failure Criterion Material yielding begins when 2nd deviatoric stress invariant reaches critical value (k) Independent of first stress invariant ( does not depend on hydrostatic component)

von Mises stress - σv Ellipse is the locus of all points with the same σν This is a poor metric as a failure criterion for biological tissues Used for convenience

von Mises Stress

Fatigue and Failure Fatigue Failure Miner’s Rule For bone: Strain a better failure parameter than stress (given by σ/E), not as affected by porosity Strain-N (number of cycles) relationship plotted, follows power law behavior Miner’s Rule For bone: n - # of different load levels, i = particular load level, Ni - # of load applications at level i, NFi - # of load applications that would cause failure at the ith load level; A&B empirical constants

Fatigue and Failure Failure Criteria Complex behavior of bone (anisotropic, asymetric) requires complex failure criteria for multiaxial loading Tsai-Wu criterion used for complex composite materials, works for bone Tsai-Wu criterion: Mathematically, quadratic, ellipsoidal equation Coefficients obtained from uniaxial tension compression, planar shear testing, biaxial testing in various planes