Basic biomechanics and bioacoustics Lecture 2

Stress ‘intensity of the acting on a specific ‘ Lecture 2

Stress Normal stress Shear stress Lecture 2

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) Lecture 2

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: Lecture 2

Average Shear Stress Average shear stress, avg, is assumed to be at each point located on section Internal resultant shear force, V, is determined Lecture 2

Shear Assume bolt not tightened enough to cause friction in Fig. 1-21 a) In both cases, the force F is balanced by Lecture 2

Single Shear Can you think of any biomechanics problems where single shear is important? Lecture 2

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

Double Shear Can you think of any biomechanics problems where double shear is important? Lecture 2

Stress Simple Stress Example Lecture 2

What is a Tensor??? Lecture 2

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! Lecture 2

Small Displacement Stresses When analyzing a tissue or body where the assumption of small deformation is valid ( %):                           Lecture 2

Equilibrium of a Volume Element area force force arm Lecture 2

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!!! Lecture 2

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 Lecture 2

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! Lecture 2

Indicial Notation (aka Einstein summation convention) Useful relations: Substitution of using the Kronecker delta: Lecture 2

Indicial Notation (aka Einstein summation convention) Example Lecture 2

Cauchy’s Law Configuration T n Lecture 2

Cauchy’s Law - example T n P Lecture 2

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 Lecture 2

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  Lecture 2

Second Piola Kirchhoff Stress Example – derivation of stresses Lecture 2