The following chapters are taken from:
Mixture Theories for Biological Tissues
(textbook in preparation)
by Gerard Ateshian, PhD
Department of Biomedical Engineering
Columbia University
I. INTRODUCTION
A. The Biphasic Theory for Small Strains and Linearly
Mathematics 4171, Fall 2012
Final Exam, due 18 December at 1PM in Cupples I 207B
Directions: During this exam, you are allowed to use your class notes, homework assignments from class, and the textbook. If you cite theorems or homework problems, please
in
Final Exam Solutions
Problem 1:
Solution.
1. Observe that since f and g are continuous into R, the function F given by F (x) =
f (x) g (x) is continuous into R. cfw_0 is closed in R, so by continuity of F , the set
F 1 (cfw_0) is closed in X , which is pr
Mathematics 4171, Fall 2012
Assignment 6, due 11 October
Problem 1: Consider the space
(an )n= such that
n=
2 (R)
which consists of all two-sided, real-valued sequences
1/2
(an )
2
2
|an |
=
< .
n=
1. Convince yourself that 2 (R) is a vector space (but do
FIBROCARTILAGE:
Meniscus
Anatomy
Page 1
Anatomy
Anatomy
Page 2
Anatomy
Motivation
common sports injury
short term:
knee pain
long term:
joint instability leading to
osteoarthritis
healing response very limited
Page 3
Motivation
Meniscal legions mos
Lecture Outline
BME 564 : Orthopaedic Biomechanics
Why study ligament biomechanics?
Ligament
LIGAMENT
Clinical significance
Structure
Composition
Biology
Biomechanics
Injury and repair
Theory
How can we model ligament mechanical behavior?
Quasilinear
3/29/16
Experimental Methods
Tendon and Ligament
The primary func:on of tendons and ligaments is to
transmit tensile forces; therefore, mechanical tes:ng
typically consists of uniaxial tensile tests
Stress and Strain
To determine the material prop
7. VISCOELASTIC SOLIDS FUNGS QUASILINEAR THEORY
Viscoelastic solids and fluids
We have already recalled that a material has a viscoelastic behaviour if its response to external
actions is time dependent: the relationship between stresses and strains withi
Partnering Engineering & the Clinic
to Improve Treatment of Young
Adult Hip Disorders
Michael D. Harris, PhD
Program in Physical Therapy
Department of Orthopaedic Surgery
In conjunction with University of Denver and University of Utah
Background The Hip J
Lecture Outline
BME 564 : Orthopaedic Biomechanics
Why study the biomechanics of cartilage?
Cartilage
CARTILAGE
Clinical significance
Injury and repair
Structure
Composition
Biomechanics
Tissue engineering
Theory
How can we model cartilage mechanical
Mathematics 4171, Fall 2012
Midterm Exam Solutions
Problem 1: This problem was mostly already done via homework problems or during
lecture with some very minor modications. I included it because I think the proof/ideas
are important to know in the long ru
Mathematics 4171, Fall 2012
Midterm Exam
Due 30 Oct 2012 by beginning of class
Directions: During this exam, you are allowed to use your class notes, homework assignments from class, and the textbook. If you cite theorems or homework problems, please
incl
Math 4171, Fall 2013
Homework 1
0. To do but not to hand in:
i) Which of the following are true when is inserted in the blank space? Which are
true when is inserted?
a)
b)
c)
d)
e)
g _g, g
g _g, g
g_g, g
g_g, g
g_g, g,g
is true for a), d);
is true for
Math 417, Fall 2009
Final Exam Solutions
All your solutions should be written in the test booklet. No references (text, notes, etc.) are
allowed during the exam.
Throughout the exam:
, 8 and all their subsets (such as ,
unless something different is state
Math 4171, Fall 2013 Exam II
Solutions
Take Home: distributed in class Thursday November 7 and due in class Tuesday, November 12.
Write your solutions on 8 " " paper, with a new sheet for each problem. If you want to solve
#
any extra problems, then I'll
Math 4171
Exam I, Fall 2013
Name_
Write all your answers on the test booklet. No consultation, notes, texts, or electronic devices of
any kind are allowed.
are the sets of natural numbers, rationals, irrationals, and reals. and all subsets of
(such as ,
Math 4171, Fall 2013
Homework 2
0. To do but not to hand in:
i. True or false? Explain.
a) if E is infinite and F is countable, then E F
b) if E is infinite and F is countable, then E F
a) False: for example, let E and F
F
E
b) True:
Let # % ' and
" $ &
Math 4171, Fall 2013
Homework 3 Solutions
0. To do but not to hand in:
i) (Part of Exercise E55) Prove or disprove:
a) i- #!
b) if # 7
2i! , then 7i! -
a) is true since # i! b) is true since # 7
2i!
#- i- - - #i! - #!
#i! - 7i! 2i! i! #i! -
ii) (Exercise
Math 4171, Fall 2013
Homework 7
The following problems are to hand in.
Due in class on Tuesday, November 19
Please write up solutions neatly and staple pages together. If you tear pages out of a spiral
notebook, please remove any ragged edges.
1. (Exercis
Math 4171, Fall 2013
Homework 6 Solutions
The following problems are to hand in.
Due in class on Tuesday, November 5
Please write up solutions neatly and staple pages together. If you tear pages out of a spiral
notebook, please remove any ragged edges.
1.
Math 4171, Fall 2013
Homework 5
0. To do but not to hand in:
a) Prove or give a counterexample:
i) For any B in a space \ g B is equal to the intersection of all
open sets containing B.
False: The simplest example is when g is the trivial topology and l\l
Math 4171, Fall 2013
Homework 4 Solutions
0. To do but not to hand in:
i) (parts of Exercise E10) Let \ . be a pseudometric space. Prove or disprove each
statement.
F% B is never a closed set.
This is false: if . is the discrete unit metric and B \ , then
Math 4171, Fall 2013
Homework 8, Solutions
Please write up solutions neatly and staple pages together. If you tear pages out of a spiral
notebook, please remove any ragged edges.
1. (Exercise E15) Suppose that \ . is a nonempty complete metric space and t
Math 417, Fall 2009
Final Exam
Name_
All your solutions should be written in the test booklet. No references (text, notes, etc.) are
allowed during the exam.
Throughout the exam:
, 8 and all their subsets (such as ,
unless something different is stated.
,