1
MATH45061: SOLUTION SHEET1 VI
1.) The Eulerian rate of deformation tensor D is dened, in Cartesian components, to be
DIJ =
1
(VI,J + VJ,I ) .
2
In order to construct the (traceless) deviatoric tenso
1
MATH45061: EXAMPLE SHEET1 I
1.) A coordinate system i is dened such that the position in a global two-dimensional
Cartesian coordinate system (x, y) is given by
x = 2 1,
y = 1.
Find the covariant an
MATH45061
Three hours
THE UNIVERSITY OF MANCHESTER
CONTINUUM MECHANICS
21 January, 2014
14:00 17:00
Answer ALL FOUR questions in section A (21 marks in total).
Answer THREE of the FOUR questions in se
1
MATH45061: EXAMPLE SHEET1 III
1.) A body is loaded by a body force F per unit mass and a surface traction T . Show that
if the linear momentum and the angular momentum about a particular point Z are
1
MATH45061: EXAMPLE SHEET1 II
1.) A deformation map R = (r) is dened in components in a global Cartesian basis
by
X1 = ex1 , X2 = 2x2 x3 , X3 = x2 + 2x3 .
Find the deformation gradient tensor and det
MATH45061
Three hours
THE UNIVERSITY OF MANCHESTER
CONTINUUM MECHANICS
15 January, 2013
09:45 12:45
Answer ALL FOUR questions in section A (21 marks in total).
Answer THREE of the FOUR questions in se
Chapter 4
Physical Conservation and Balance Laws
& Thermodynamics
The laws of classical physics are, for the most part, expressions of conservation or balances of certain
quantities: e.g. mass, moment
Chapter 7
Fluid Mechanics
Fluid mechanics includes the study of liquids and gases and all materials that exhibit uidity:
molecules can easily slip past each other so the materials ow (rather then simp
Chapter 5
Constitutive Modelling
5.1
Introduction
Thus far we have established four conservation (or balance) equations, an entropy inequality and a
number of kinematic relationships. The overall set
1
MATH6/45061: Comments on Exam 2012/13
The majority of students attempted all questions in Section A, and questions B5, B6 and B8. It is
clear that this was a dicult exam and people were under time p
1
MATH6/45061: Comments on Exam 2013/14
The majority of students attempted all questions in Section A, and there was a fairly even split
between questions B5, B6, B7 and B8. It appeared that students
Chapter 2
Kinematics: Deformation and Flow
2.1
Introduction
We need a suitable mathematical framework in order to describe the behaviour of continua. Our
everyday experience tells us that lumps of mat
1
MATH45061: SOLUTION SHEET1 IV
1.) The rate of change of mass of the deformed region must be equal to the total mass
production rate of the region
D
Dt
dVt =
dVt .
t
(1)
t
Using the Reynolds transp
Chapter 3
Forces, Momentum & Stress
3.1
Newtonian mechanics: a very brief rsum
e
e
In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration
when acted on by
1
MATH45061: SOLUTION SHEET1 V
1.) a.) The faces of the cube remain aligned with the same coordinate planes. We
assign Cartesian coordinates aligned with the original cube (x, y, z), where
0 x, y, z 1
1
MATH45061: SOLUTION SHEET1 III
1.) Conservation of angular momentum about a particular point Z requires that
(R Z) V dV = LZ .
The moment about the point Z can be decomposed into torque due to the b
Chapter 0
Preliminaries
These notes cover the course MATH45061 (Continuum Mechanics) and are intended to supplement
the lectures. The course does not follow any particular text, so you do not need to
1
MATH45061: SOLUTION SHEET1 I
1.) The position vector is given by r = 2 1 e1 + 1 e2 , which can be written in component
form in the global Cartesian basis as
21
1
r=
.
Thus covariant base vectors are
Chapter 6
Elasticity
6.1
(Perfect) Thermoelastic Materials
A solid body that undergoes reversible deformations is said to be perfectly elastic. The key feature
of a solid, as opposed to a uid or a gas