Biomechanics Stress: A Comprehensive Guide

1. Basic biomechanics and bioacoustics

2. Stress • ‘intensity of the acting on a specific ‘

3. Stress Normal stress Shear stress

4. Average Normal Stress in an Axially Loaded Bar • Assumptions: • 1. Homogenous isotropic material • 2. • 3. P acts through of cross sectional area • uniform normal stress (no shear)

5. Uniaxial Tensile Testing • Calculate stress from force data. That is, for each measure of force, f: • stress, T = f/ • stress,  = f/ • Current cross-sectional area is usually not measurable, but if one considers the specimen incompressible:

6. Average Shear Stress • Average shear stress, avg, is assumed to be at each point located on section • Internal resultant shear force, V, is determined

7. Shear • Assume bolt not tightened enough to cause friction in Fig. 1-21 a) • In both cases, the force F is balanced by

8. Single Shear • Can you think of any biomechanics problems where single shear is important?

9. Shear • Double shear results when are considered to balance the force F • This results in a shear force V=F/2

10. Double Shear • Can you think of any biomechanics problems where double shear is important?

11. Stress Simple Stress Example

12. What is a Tensor???

13. Measures of Stress stress = , force per deformed cross-sectional area stress = T, force per undeformed cross-sectional area (also called 1st piola Kirchhoff stress) stress = S, no physical interpretation, but often used in *be careful, different texts/sources will use different notation!

14. Small Displacement Stresses When analyzing a tissue or body where the assumption of small deformation is valid ( %):

15. force force arm area Equilibrium of a Volume Element

16. Indicial Notation (aka Einstein summation convention) • Why use it as compared to boldface notation??? • algebraic manipulations are • ordering of terms is unnecessary because AijBkl means the same thing as BklAij, which is not the case for boldface notation, since ABB  A • it makes coding easier!!!

17. Indicial Notation (aka Einstein summation convention) index : Used to designate a component of a vector or tensor. Remains in the equation once the summation is carried out. Kronecker delta index : Used in the summation process. An index which does not appear in an equation after a summation is carried out. Permutation symbol  of unit vectors in a right-handed coordinate system

18. Indicial Notation (aka Einstein summation convention) Question: what are the dummy and free indices in the familiar form for Hooke’s law shown to the right? Answer: Free indicies: Dummy indices: This is one component of the components in the stress tensor!

19. Indicial Notation (aka Einstein summation convention) Useful relations: Substitution of using the Kronecker delta:

20. Indicial Notation (aka Einstein summation convention) Example

21. Cauchy’s Law Configuration T n

22. Cauchy’s Law - example T n P

23. Second Piola Kirchhoff Stress • S, has no physical meaning • Derived from energy principles Deriving a “functional form” of W is often a common problem in biomechanics

24. Relationship Between Large Displacement Stresses We can give the stress (T) in terms of the stress (t): or We can also give the relationship between the 2nd Piola-Kirchhoff stress (S) and the 1st Piola-Kirchhoff stress (T) and the Cauchy stress (t): *Note, here t is used for Cauchy stress instead of 

25. Second Piola Kirchhoff Stress Example – derivation of stresses