Lecture 6

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.title[
# Lecture 6
]
.subtitle[
## Coarse Materials I
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2025/30/1
]

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# Joints, Levers, and Linkages

<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/font-awesome/5.14.0/css/all.min.css">

.pull-left[
Today we'll ....

- Wrap up with muscls

- What is stress, strain?

- Define stiffness and other material properties

- Start MP2

]

.pull-right[
![](https://upload.orthobullets.com/topic/9062/images/Basic%20Stress%20Strain%20Curve_moved.jpg)
]

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# Muscles generate mechanical work and power.

## How?
 
.center[

<img src="img/sarcomere.jpg" width="400" />

]

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# Muscles generate mechanical work and power.

## How?
 
.center[

<img src="img/filamenttheory.png" width="350" />

]

.footnote[ Sliding filament (crossbridge) theory. Gordon, Huxley & Julian (1966)]

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# What determines amount of force produced by muscle?

.pull-left[
1. Time after activation

]

.pull-right[
<img src="img/latent.jpg" width="450" />
]

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# What determines amount of force produced by muscle?

.pull-left[
1. Time after activation
2. Muscle (sarcomere) length

<br>
<br>

<img src="img/filiments.jpg" width="550" />
]

.pull-right[
<br>
<br>
<br>
<br>

<img src="img/forcelength.png" width="550" />
]

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# What determines amount of force produced by muscle?

.pull-left[
1. Time after activation
2. Muscle (sarcomere) length
3. Shortening velocity

<br>

<img src="img/forcevel.png" width="550" />

]

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# What determines amount of force produced by muscle?

.pull-left[
1. Time after activation
2. Muscle (sarcomere) length
3. Shortening velocity

<br>

<img src="img/forcevel.png" width="550" />

]

.pull-right[

V=? to maximize power?

`$$P=F\times V$$`
    
<br>
<br>

<img src="https://d3i71xaburhd42.cloudfront.net/2b0ce8ae4ce7e28430b71ba8983e33c3bad9c196/3-Figure2-1.png" width="550" />
]

---
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# What determines amount of force produced by muscle?

.pull-left[
1. Time after activation
2. Muscle (sarcomere) length
3. Shortening velocity

<br>

<img src="img/forcevel.png" width="550" />

]

.pull-right[

Formalize F-V relationship... the Hill equation:

`$$\frac{T}{T_o}=T^\prime=\frac{1-\frac{v}{v_{max}}}{1+c\frac{v}{v_{max}}}=\frac{1-v^\prime}{1+cv^\prime}$$`

<img src="https://d3i71xaburhd42.cloudfront.net/2b0ce8ae4ce7e28430b71ba8983e33c3bad9c196/3-Figure2-1.png" width="550" />
.footnote[
`\(c\approx 1.2-4\)`
]

]

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# Metrics of muscle function

## Generate mechanical work and power

.pull-left[
- Work = `\(F\cdot\Delta L\)` (Joules)
- Power = Work/time (Joles/s = Watts)
- Power = Work x Frequency (Joules/s = Watts)
- Power = F x V (Joules/s = Watts)

<br>

]

.pull-right[
    
<br>
<img src="img/musclework.png" width="250" />
.footnote[

]

]

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# Metrics of muscle function

## The work loop technique

.pull-left[
- Work = `\(F\cdot\Delta L\)` (Joules)
- Power = Work/time (Joles/s = Watts)
- Power = Work x Frequency (Joules/s = Watts)
- Power = F x V (Joules/s = Watts)

.center[
<img src="img/muscletest.png" width="60" />
]

]

.pull-right[
    
<br>
<img src="img/workloop.png" width="200" />
.footnote[

]

]

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# Metrics of muscle function

.center[

<img src="img/motorbreakstrut.png" width="600" />
]

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# Lots more to study

We know we don't know a lot about muscles

.center[

<img src="https://s3-eu-west-1.amazonaws.com/s3-euw1-ap-pe-ws4-cws-documents.ri-prod/9780367190811/central/musclefuels/images/gastrocnemius.jpg" width="400" />
Doublet potentiation

]

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# Lots more to study

We know we don't know a lot about muscles

.center[

<img src="https://journals.physiology.org/cms/10.1152/jappl.1997.82.4.1219/asset/images/large/japp0542801.jpeg" width="400" />
Doublet potentiation

]

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# Lots more to study

We know we don't know a lot about muscles

.center[

<img src="https://www.researchgate.net/profile/Stanley-Salmons/publication/11155153/figure/fig5/AS:276867023163394@1443021650100/Mean-nFTIpP-n-6-for-doublets-lower-surface-triplets-intermediate-surface-and.png" width="400" />

]
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# Material and structures

.pull-left[

- Loads and deformations -- basic concepts
- Stress: the distribution of force over an area
- Strain: a dimensionless measure of deformation
- Stiffness: a measure of how materials respond to stress (or strain)
- Loads: the relationship between stress and strain
- Strength: the stress at breaking

]

.pull-right[
![](img/stressbone.jpg)
]
        
   
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# Material and structures

.pull-left[ 
 
- All (biological) materials respond to external forces by deforming
- We seek a way to understand and characterize these responses
- Material behavior as separate from structural behavior
- We will formalize the relationships between force and deformation

]

.pull-right[
![](img/stressbone.jpg)

]
   
 
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# Material properties

## Types of deformation

.center[

]

.center[

![](img/loads.png)
        
 
        ]

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# Material properties

## Types of deformation

.pull-left[

Stress: `$$\sigma=\frac{F}{A}$$`

`\(Pa=N\cdot m^2\)`
           
Strain `$$\varepsilon=\frac{L-L_o}{L_o}$$`

Units?

]

.pull-right[

<img src="img/loads.png" width="350" />

]

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# Material properties 
  
  
.pull-left[

`$$stress(\sigma)=\frac{F}{A}$$`

<img src="img/strainarrows.png" width="350" />

Young's modulus: `\(E=\frac{\sigma}{\varepsilon}\)`
   
   What are units of `\(E\)`?
   
]

.pull-right[

`\(strain(\varepsilon)=\frac{L-L_o}{L_o}\)`
 
 Hook's law: `\(F=K\Delta L\)`
 
<img src="img/stressstrainstiff.png" width="350" />
             
              
]

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# Material properties 
  
  
.pull-left[

`\(\sigma\)`: distribution of force over and area

`\(\varepsilon\)`: dimensionless measure of length of change

stiffness (E): change in stress relative to strain (slope of stress-strain curve)

<img src="img/tendoncurve.png" width="250" />

]

.pull-right[

Most biological materials are non-linearly elastic

<img src="img/bonehaircurve.png" width="250" />

]

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# Material properties 
  
  
.pull-left[

`\(\sigma\)`: distribution of force over and area

`\(\varepsilon\)`: dimensionless measure of length of change

stiffness (E): change in stress relative to strain (slope of stress-strain curve)

<img src="img/tendoncurve.png" width="250" />

]

.pull-right[

Most biological materials are non-linearly elastic

<img src="img/failuretable.png" width="250" />

]

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# Material properties 
  
  
.pull-left[

`\(\sigma\)`: distribution of force over and area

`\(\varepsilon\)`: dimensionless measure of length of change

stiffness (E): change in stress relative to strain (slope of stress-strain curve)

<img src="img/stressstraincurve.jpg" width="350" />

]

.pull-right[

Most biological materials are non-linearly elastic

<img src="img/failuretable.png" width="250" />

]

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# Thanks!

Slides created via the R package [**xaringan**](https://github.com/yihui/xaringan).