1. Basic biomechanics and bioacoustics
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2. Stress
‘intensity of the acting on a specific ‘
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3. Stress
Normal stress
Shear stress
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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)
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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:
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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
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7. Shear
Assume bolt not tightened enough to cause friction in Fig a)
In both cases, the force F is balanced by
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8. Single Shear
Can you think of any biomechanics problems where single shear is important?
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9. Shear Double shear results when are considered to balance the force F
This results in a shear force V=F/2
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10. Double Shear
Can you think of any biomechanics problems where double shear is important?
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11. Stress
Simple Stress Example
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12. What is a Tensor???
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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!
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14. Small Displacement Stresses
When analyzing a tissue or body where the assumption of small deformation is valid ( %):
                         
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15. Equilibrium of a Volume Element
area
force
force
arm
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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!!!
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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
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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!
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19. Indicial Notation (aka Einstein summation convention)
Useful relations:
Substitution of using the Kronecker delta:
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20. Indicial Notation (aka Einstein summation convention)
Example
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21. Cauchy’s Law
Configuration T n
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22. Cauchy’s Law - example T n P
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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
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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 
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25. Second Piola Kirchhoff Stress
Example – derivation of stresses
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