Name_KEY_
Homework 2
BME 615
due: Sept. 22, 2009
If you dont know where you are going, you will wind up somewhere else.
Yogi Berra
1. Quantifying the complex mechanical properties of biological tissues is an important
but difficult task. Often, linearly e
Force Equilibrium Equations
Borellis De Motu Animalium (On the Movement of Animals)
Useful math for mechanics formulations 1
! ! !
e
Let x , ey , ez be unit vectors in the x, y, z directions
Del operator (or nabla)
!
!
!
= , , =
ex +
ey + ez
y
z
x y z
I am deeply superficial
Ava Gardiner
Heavy Math
The book of Nature is
written in mathematical
symbolsGalileo Galilei
Calculus is the
language of God.
.Richard Feynman
Strain Energy Density
Hyperelasticity
BME 615
University of Wisconsin
Review of salient
Mathema'cs has no symbols for
confused ideas.
George S'gler
Calculus is the language of God.
Richard Feynman
BME 615
Strain
Con$nuum Mechanics
Innitesimal strain descrip'on
(with dis
Skin
Tissue Mechanics
BME 615
Carving (for sculpture) is
easy, you just go down to
the skin and stop.
Michelangelo
Function
Insulation
Protection of underlying organs
Production of vitamins D and B
Sensation
Excretion
Heat regulation
Storage of lipids and
LIGAMENT/TENDON MECHANICS
BME 615
University of Wisconsin
Roll of Ligaments and Tendons
Ligaments and tendons are
similar but not the same tissues
Ligaments and tendons are
similar but not the same tissues
There are differences in:
o
o
o
o
Function (leve
Finite Element Analysis
General Concepts
Applications & Examples
Finite Element Method
Numerical technique giving approximate solutions to
boundary value problems (partial differential equations)
General Method: Combine simple element equations from small
Skeletal Muscle
Mechanical Descriptions
BME 615
Ray Vanderby
Molecular Mechanics
Crossbridge
Mechanics
Force measured is 3-4 pN per crossbridge 1 trillion cross-bridges for 3-4 N
Power stoke is only 11 nm!
Suggests molecular models could be built based up
Bone Structure
Cortical (Compact)
Epiphysis (thin shell)
Diaphysis (shaft, thicker)
Cancellous (trabecular,
spongy)
20% by mass, 80% by
surface
5-70% density of cortical
30-90% porosity
Trabeculae
Plates and rods (~5 nm X
5 nm X 40 nm)
http:/dept
Elas%c Moduli
for linearly elas%c behaviors
h5p:/www.comicvine.com/myvine/elas%c_man/
Terminology
http:/www.civilengineeringterms.com/
Tangent Modulus is
slope of stress-strain
curve at any point
Young's modulus is
typically
BME 615: Vectors and Tensors
Vectors 1. Inner product of vector (Fig.1 (a) a and b is defined by
a ib = a b cos
For example, if two vectors a and b are given 2-D space, their inner product is given by (where e1 and e2 are defined as unit vectors in 1, 2
Name_
"If we knew what it was we were doing, it would not be called research, would it?"
Albert Einstein
Homework 4
BME 615
Due: Oct. 18, 2011
1. You are working out at the shell and bored; your mind wanders to viscoelasticity.
You wonder how much your te
BME 615
Homework 6
due: Nov. 8, 2011
"I have never let my schooling interfere with my education" - Mark Twain
Question 1: If the 2D strain energy density function (formulated for Kirchhoff stresses and
Green strain) for a tissue specimen (assumed incompre
If you arent confused by quantum physics, then you havent really understood it.
Neils Bohr
Homework 7
BME 615
Due: Nov 17, 2011
Question 1: Consider a cylindrically shaped, feline carotid artery which is
incompressible, pseudo-elastic, and can be modeled
We are confronted by insurmountable opportunities.
Walt Kelly, from Pogo
Homework 1
BME 615
due: Sept. 20 2011
1. Often, the mechanical behavior of soft biological tissues is described by models
originally developed to describe rubber. Among many availabl
Homework 2
BME 615
due: Sept. 22, 2011
If you dont know where you are going, you will wind up somewhere else.
Yogi Berra
1. Golden Bucky Question: (only for fun and a shot at the Golden Bucky; not required
and no homework points given for answer)
How many
Name_
Homework 3
BME 615
due: September 29, 2009
We are confronted by insurmountable opportunities.
Walt Kelly, from Pogo
.
1. Problems in 3D can sometimes reduced to 2D with reasonable assumptions. For
example, if stress is negligible in the z direction
Homework 4 (Lab Demo 1: Bone)
(50 points total)
BME 615
due: Oct 13, 2009
1. The data acquired in lab have loading and displacement information from all three principal
directions. Calculate apparent engineering stress from the given force data, and calcu
Name_
"If we knew what it was we were doing, it would not be called research, would it?"
Albert Einstein
Homework 5
BME 615
Due: Oct. 20, 2009
1. Starting with the Kelvin model of Fung, i.e. the standard 3 parameter solid for linear
viscoelasticity, deriv
Homework 8
BME 615
Due: Nov 18, 2009
What would life be if we had no courage to attempt anything?
Vincent van Gogh
1: The male camelopardalis retulata (giraffe) can be 5m tall with his heart at a level
halfway between the top of his head and the ground. A
Name_
BME 615 - Tissue Mechanics
Exam 1
October 22, 2009
"A child of five would understand this. Send someone to fetch a child of five."
Groucho Marx
Question 1: Please be concise when answering the following:
a. If trabecular bone is often described as a