Lecture 2

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

• 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:

Uniaxial Tensile Testing

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

'

' 0

zyzy

zyzy yxyx

area

force

yzzy

yzzy yzxzyx

0

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 BB 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.

index : Used in the summation process. An index which does not appear in an equation after a summation is carried out.

Kronecker delta

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?

klijklij C

Answer:

Free indicies:

Dummy indices:

111ji, for

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

n

T vector n

tensor stress

vector T

σ

Configuration

Lecture 2

Cauchy’s Law - example

P at acting vector (force) traction the Find

j2i3n

:is p point atbody the of surface the to normal The

000

080

004

:as given is P point at tensor stress The

σ

jiji nσT

nT

vector normaln

tensor stressCauchy

vector stress or tractionT

or

σ

σ

n

T

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

tFT 1

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):                         

We can give the stress (T) in terms of the stress (t):                         

T11T1 F FFTS

*Note, here t is used for Cauchy stress instead of

or

Lecture 2

Second Piola Kirchhoff Stress

Example – derivation of stresses