Lecture 8

.title[
# Lecture 8
]
.subtitle[
## Coarse Materials III
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2024/2/8
]

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

- Finish up with solid material properties

- Consider fractures and imperfections

- Consider fluids vs. solids

- Consider fluids + solids (viscoelastics)

]

.pull-right[
![](https://www.physio-pedia.com/images/thumb/b/be/Rep_loading-unloading_curve_intechopen.jpeg/357px-Rep_loading-unloading_curve_intechopen.jpeg)

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# Toughness

### The energy imparted is the **mechanical strain energy** that can be returned or be so great as to break the material

]

<img src="img/toughness.png" width="350" />
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**Toughness:** the energy per unit volume required to break a material

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# Toughness

### Some (but not necessarily all) of the mechanical energy can be returned

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<img src="https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20191021053530314-0709:9781108686624:72215fig26_6b.png?pub-status=live" width="350" />
]

**Resilience (R)**=

`$$\frac{\textrm{strain energy returned}}{\textrm{strain energy absorbed}}$$`

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> material </th>
   <th style="text-align:left;"> R </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Silk </td>
   <td style="text-align:left;"> 35% </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Elastin </td>
   <td style="text-align:left;"> 76% </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Abductin </td>
   <td style="text-align:left;"> 80% </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Collagen (tendon) </td>
   <td style="text-align:left;"> 93% </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Resilin (wing hinge) </td>
   <td style="text-align:left;"> 97% </td>
  </tr>
</tbody>
</table>
]

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# Resilience in the vertebrate body

### The cardiovascular system?

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# Resilience in the vertebrate body

### The cardiovascular system?

<img src="https://www.austincc.edu/apreview/NursingPics/VascularPics/Picture8.jpg" width="400" />
]

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# Fracture theory (without all the equations)

### In an ucompromised structure, stress in applied uniformly

### With imperfections, stress is applied nonuniformly
 ]
 
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# Fracture theory (without all the equations)

### In an ucompromised structure, stress in applied uniformly

### With imperfections, stress is applied nonuniformly
 ]

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# Fracture theory (without all the equations)

### In an ucompromised structure, stress in applied uniformly

.pull-left[
.center[
<img src="img/crack.png" width="150" />
]
### With imperfections, stress is applied nonuniformly

`$\textrm{Strain energy}=U=\int\sigma d\varepsilon=\frac{\sigma^2}{2E}$`
 ]

`$\textrm{Strain energy release}=  \frac{\sigma^2}{E} \pi a^2t$`

Strain energy releases by crack

]

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# Viscoelastics

### Materials that show time dependency are **viscoelastic**

<img src="img/viscos.png" width="700" />
 ]
 
.pull-left[
.center[
`$F\sim \Delta L$`

`$\sigma\sim \varepsilon$`

`$\sigma \sim E\varepsilon$`
]]

`$\sigma\sim d\varepsilon/dt$`

`$\sigma \sim d\mu \varepsilon/dt$`

`$\mu=\textrm{viscosity}$`

Newton's third law
]
]

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# Fluids

<img src="img/fluidstrainrates.png" width="400" />
 ]

- Hookean materials (solids): stress vs. strain relationship is linear (constant stiffness)

- Newtonian fluids: stress vs. strain **rate** is linear (constant viscosity)

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# How do we know when we have a fluid?

]

]

`$\sigma = d\mu \varepsilon/dt$`

]
]

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# Methods to reveal a viscoelastic property

### Creep and stress relaxtion tests

Apply a constant stress?

Relax a stress?

What happens to deformation? Why?
]

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# Methods to reveal a viscoelastic property

### Creep and stress relaxtion tests

.center[
<img src="https://ars.els-cdn.com/content/image/3-s2.0-B9780128098318000052-f05-04-9780128098318.jpg" width="450" />

Apply a constant stress? (A)

Relax a stress? (B)

What happens to deformation? Why?
]

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# What if we have both?

]

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Voigt model

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# Why this in a tendon?

.center[
<img src="https://www.physio-pedia.com/images/thumb/b/be/Rep_loading-unloading_curve_intechopen.jpeg/357px-Rep_loading-unloading_curve_intechopen.jpeg" width="450" />

]

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# Thanks!

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