Processing math: 100%

class: center, middle, inverse, title-slide

.title[
# Lecture 8
]
.subtitle[
## Coarse Materials III
]
.author[
### Dr. Christopher Kenaley
]
.institute[
### Boston College
]
.date[
### 2025/2/6
]

---

class: inverse, top
# Material Properties, Failure, and the Viscous World

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

- 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)

]

---
class: top

# Fracture theory (with only a few the equations)

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

.pull-left[
<img src="img/fracturemode.png" width="700" />

### With imperfections, stress is applied nonuniformly
 ]
 
---
class: top

# Fracture theory (with only a few equations)

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

.pull-left[
<img src="img/fracturemode.png" width="700" />

### With imperfections, stress is applied nonuniformly
 ]

.pull-right[
- Why do bones and shells break explosively?
- Why do cracks propagate?
]

---
class: top

# Fracture theory (with only a few 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

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

.pull-right[
- Why do bones and shells break explosively?
- Why do cracks propagate?

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

Strain energy releases by crack

<img src="img/cracklength.png" width="300" />

]

---
class: top

# Viscoelastics

.center[

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

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

`\(\sigma\sim \varepsilon\)`

`\(\sigma \sim E\varepsilon\)`
]]

.center[
.pull-right[
`\(F\sim \Delta L/\Delta t\)`

`\(\sigma\sim d\varepsilon/dt\)`

`\(\sigma \sim d\mu \varepsilon/dt\)`

`\(\mu=\textrm{viscosity}\)`

Newton's third law
]
]

---
class: top

# Fluids

.center[

<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)

---
class: top

# How do we know when we have a fluid?

.center[
<img src="img/creep2.png" width="650" />

]

.pull-left[

.center[
`\(\sigma = E \varepsilon\)`

]

]

.pull-right[
.center[

`\(\sigma = d\mu \varepsilon/dt\)`

]
]

.center[

<img src="img/springdashpot.png" width="600" />

]

---
class: top

# Methods to reveal a viscoelastic property

### Creep and stress relaxtion tests

.center[

Apply a constant stress?

Relax a stress?

What happens to deformation? Why?
]

---
class: top

# 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?
]

---
class: top

# What if we have both?

.center[
<img src="img/mucuscartilage.png" width="650" />

]

.pull-left[

.center[
Maxwell model

]

]

.pull-right[
.center[

Voigt model

]
]

---
class: top

# 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" />

]

---

class: center, middle

# Thanks!

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

Notes for current slide

Notes for next slide

Lecture 8
Coarse Materials III
Dr. Christopher Kenaley
Boston College
2025/2/6
1 / 13

Material Properties, Failure, and the Viscous World

Today we'll ....

Consider fractures and imperfections
Consider fluids vs. solids
Consider fluids + solids (viscoelastics)

2 / 13

Fracture theory (with only a few the equations)

In an ucompromised structure, stress in applied uniformly

With imperfections, stress is applied nonuniformly

3 / 13

Fracture theory (with only a few equations)

In an ucompromised structure, stress in applied uniformly

With imperfections, stress is applied nonuniformly

Why do bones and shells break explosively?
Why do cracks propagate?

4 / 13

Fracture theory (with only a few the equations)

In an ucompromised structure, stress in applied uniformly

With imperfections, stress is applied nonuniformly

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

Why do bones and shells break explosively?
Why do cracks propagate?

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

Strain energy releases by crack

5 / 13

Viscoelastics

Materials that show time dependency are viscoelastic

$F\sim \Delta L$

$\sigma\sim \varepsilon$

$\sigma \sim E\varepsilon$

$F\sim \Delta L/\Delta t$

$\sigma\sim d\varepsilon/dt$

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

$\mu=\textrm{viscosity}$

Newton's third law

6 / 13

Fluids

Hookean materials (solids): stress vs. strain relationship is linear (constant stiffness)
Newtonian fluids: stress vs. strain rate is linear (constant viscosity)

7 / 13

How do we know when we have a fluid?

$\sigma = E \varepsilon$

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

8 / 13

Methods to reveal a viscoelastic property

Creep and stress relaxtion tests

Apply a constant stress?

Relax a stress?

What happens to deformation? Why?

9 / 13

Methods to reveal a viscoelastic property

Creep and stress relaxtion tests

Apply a constant stress? (A)

Relax a stress? (B)

What happens to deformation? Why?

10 / 13

What if we have both?

Maxwell model

Voigt model

11 / 13

Why this in a tendon?

12 / 13

Thanks!

Slides created via the R package xaringan.

13 / 13

Material Properties, Failure, and the Viscous World

Today we'll ....

Consider fractures and imperfections
Consider fluids vs. solids
Consider fluids + solids (viscoelastics)

2 / 13

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