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 tensor D, we rst nd the trace of D:
trace(D) = DKK = VK,K =
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 and contravariant base vectors.
Is the coordinate system
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 section B (54 marks in total). If more than THREE
questio
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
both conserved, then the angular momentum about any po
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 determine whether the deformation is
physically admissible
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 section B (54 marks in total). If more than THREE
questio
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, momentum, angular momentum, energy. These are fundamental pos
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 simply deforming). Unlike
solid materials, uids cannot supp
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 of governing equations are collected in Table 5.1.
The
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 pressure. Overall, section A was well
answered, but peop
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 found this exam dicult and were
under some time pressur
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 matter can both move (change in position) and deform
(chan
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 transport theorem we obtain
t
D
+
Dt
R
V dVt = 0;
and by the
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 external inuences which are known as forces 1 . The res
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. The stretched lengths of the three sets of parallel s
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 body
force, F , and surface traction T :
LZ =
(R Z) F dV
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 buy any
text books. The notes should be suciently self-
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 therefore given by
g 1 = r ,1 =
2
1
,
g 2 = r ,2 =
1
0
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, is that a solid body has a natural or rest state in w