Notation
Unfortunately, common usage and the lack of
sufficient letters in the World's
alphabets means that several
symbols have more than one meaning.
Usually the
context will clarify which
is intended. Furthermore, the need to
develop a
notation which is
consistent throughout the whole of the course means that most
quantities are decorated with
various subscripts and superscripts.
There is
indeed a consistent
scheme for these, even if it is not initially apparent.
There is one major departure from the
notation used in most of the literature
and that is that I
have explicitly shown multiplication operators in equations,
thus 
and never 
. This makes the
notation slightly less
compact but has two
benefits. Firstly, it is consistent
with programming
practice. Secondly it allows two or more letters to
indicate a single
quantity, which is a great
relief to the alphabet. So 
is one variable and not the product 
.
Coordinate bases
The default coordinate basis, to be assumed
when no subscript indicates
otherwise, is a global
physical linear rectangular Cartesian (R.C) basis.
The summation convention is used throughout
with summations taken over as many
spatial axes as are
currently active. Unless a non-R.C base
is being used,
and the distinction
between contravariant vectors and covariant
differential
forms must be made, all
spatial indices are written as subscripts.
Text line symbols
"for all values of ..."
("
displacement
gradient
velocity
gradient
deformation
gradient
"there exists ..."
denotes
functional dependence, e.g 
denotes a function 
of the variable 
.
This
differs from modern mathematical notation which would just write 
but in
our notationis a single variable.
denotes
the matrix representation of a quantity.
If the matrix has a
variance of one, it is assumed to be a column matrix.
angle,
inner product
determinant
(gamma) engineering shear strain
(Delta) large(ish) or finite increment
(delta) small(ish) increment, first
variation, residual
Kronecker delta
(epsilon) (small) true strain
elastic strain
Green's strain
inelastic strain
nominal strain
plastic
strain
(xi) position in normalised/parametric coordinates
(Lambda) normalised
length
(lambda) stretch ratio, eigenvalue,
Lamι's constant
(mu) friction coefficient
(nu) Poisson's ratio
(phi) yield function, potential field
(Psi) Helmholtz free energy
density
(theta) angular position, any general tensor
(rho) mass density
(sigma) true stress, Cauchy stress, aggregate stress,
traction vector
mean/hydrostatic
stress
(Von) Mises stress
nominal stress
yield
stress (current, if not otherwise indicated)
1st Piola-Kirchhoff stress
2nd Piola-Kirchhoff stress
matrix
stress
(tau) shear stress, thermodynamic
tension
(omega) (small) angular displacement
(i.e rotation), angular frequency
angular velocity
area, incidence/topology matrix
acceleration
strain-displacement/material-compatibility matrix
acceleration wrt a moving basis
material compliance, continuity class of a function
any constant, damping coefficient, speed of light (3*10(8)
m/s)
flexural/bending rigidity, elastic modulus function, damage
function
D.o.F degree of freedom, plural Ds.o.F
diameter, differential operator
Young's/elastic
modulus
extension, strain deviator
permutation symbol
force
damping force
spring force
inertial force
void volume fraction in a porous medium, cyclic frequency
metric, gravitational acceleration
(elastic) shear modulus, Gibbs free energy density
Hamiltonian
height, enthalpy density
second moment of area, second moment of mass, unit matrix
1st tensor
invariant
2nd tensor invariant
3rd tensor
invariant
Jacobian, J-integral, second moment in polar coordinates
1st deviatoric/reduced tensor invariant
2nd deviatoric/reduced tensor invariant
3rd deviatoric/reduced tensor invariant
bulk modulus, curvature
mode
I stress intensity factor
plane
strain fracture toughness
mode
II stress intensity factor
mode
III stress intensity factor
kinetic
energy
stress
concentration factor
stiffness, yield stress in shear, thermal conductivity
length, Lagrangian
direction cosine
moment
mass
interpolation function
normal
potential
energy
linear momentum
angular momentum
the set of real numbers, the real number line
thermodynamic force
position wrt a fixed basis
entropy
strain
energy
position wrt a moving basis
s.t such that
temperature, transformation mapping
time
internal energy density
translational displacement
translational velocity, any vector
relative velocity
work
w.r.t "with respect to"
position, current position
distance from the neutral axis of a beam
section modulus
Indexes in general
Indexes can appear all around a text-line
symbol. If an index is in
parenthesis, it indicates
position, e.g:
the
displacement at coordinate position p.
If, in addition, the index is a Fortran implicit integer (i - n),
the
position is a finite
element nodal value, eg:
the
displacement at node (i).
If the index is in brackets, it indicates a
finite element elemental value, eg:
the
displacement in finite element [i].
Pre-superscripts
the value of 'a' at time 't', thus
the value of 'a' at 
, i.e the initial value of 'a'.
Post-superscripts
'a'
to the power (m)
the contravariant component of 'a'
in the direction 'i'
(only
used when the distinction from covariant is needed).
the
multiplicative inverse of matrix [a].
the
transpose of matrix [a];
if the matrix has a
variance of one, the T indicates that it is a row matrix.
a virtual quantity 'a'
Pre-subscripts
the
component of 'a' in the 
direction of a basis 'm'.
The latter will often be
the local basis of a finite
element and would then be written
.
Post-subscripts
the covariant component (when the distinction from contravariant is needed)
of 'a' in the
direction 'i'
the
value of 'a' at the position (p) (which would usually be a node in F.E.A)
the
value of 'a' in the domain [m] (which would usually be an element in F.E.A)
Underlining
a vector a or contravariant
vector (when the distinction from covariant is needed)
an nth order tensor (usually 2nd but
no special symbol is used for higher orders)
Overlining
a 1-form a or covariant vector (when the distinction from contravariant is needed)
an n-form (usually a 2-form but no special symbol is used for
higher orders)
Units
Always try to work in basic S.I units, as
listed at http://www.npl.co.uk/reference/ . You may have to convert from other units, e.g http://www.npl.co.uk/reference/international.html or the old British
Imperial units, which are still in use in some countries, in some industries
and in some special applications. Units
named after someone usually start with a capital.
alternative abbreviation for both min and ft
alternative abbreviation for both s and in
degree, common non-S.I angular unit
cc or cubic
centimetre, common non-S.I volume unit = 10(-6) m(3)
centimetre, common non-S.I length unit = 0.01 m
foot, common pre-S.I length unit = 12 » 0.3 m
hour, common non-S.I time unit = 60 min
inch, common pre-S.I length unit » 2.5 cm
gram, derived S.I mass unit = 1/1000 kg
g, unit of gravitational acceleration = 9.81 m/s(2)
» 10 m/s(2)
for rough calcs.
kilogram, basic S.I mass unit (why is it not the gram?)
pound, pre-S.I unit of mass » 0.45 kg
litre, common non-S.I volume unit = 1000 cc = 1/1000 m(3)
Mach x is x times
the speed of sound. The speed of sound
depends on the
elasticity and density of the
medium and is about 300 m/s in air at sea level.
metre, basic S.I length unit
mile
common pre-S.I length unit = 8/5 km
minute,
both a common non-S.I time unit = 60 s and
a common non-S.I
angular unit = 1/60
millimetre,
derived S.I length unit
Pascal, derived
S.I stress or pressure unit = 1 N/m(2)
(
is not an S.I unit)
second, both the basic S.I time unit and
a common non-S.I
angular unit = 1/60 min
tonne, common non-S.I mass unit = 1000 kg and
ton, common pre-S.I mass unit » 1 tonne
thou,
common pre-S.I length unit = 1/1000 in
yard, common non-S.I length unit = 3 » 0.9 m
micron or micrometre, derived
S.I length unit = 10(-6) m
Alphabetic index (under construction)
The addresses are Section numbers, not page
numbers.
acceleration field: 27.7
active transformation:
14.7
bar finite element:
9.1
basis: 14.4
basis invariant
boundary conditions for a
truss: 15.3
bulk modulus: 28.4
characteristic equation: 23.4
conjugates: 28.1
continuity of functions:
24.13
contravariant vector: 14.7
convective derivative: 27.5
covariant v. invariant: 14.7
covariant: 23.4
covariant vector: 14.7
deformation field: 27.2
deformation gradient: 23.5
displacement field: 27.4
deviator: 23.4
differential form: 9.2
displacement gradient: 23.5
displacement gradient,
material: 27.10
displacement gradient, spatial:
27.10
displacement interpolation
function for a bar element: 9.3
displacement method in finite
element analysis: 9.3
dual vector of a
tensor: 14.6
eigenvalue: 23.4
Eulerian fields: 27.2
finite element: 9.
finite strain: 27.11
frame indifference: 27.8
Galerkin's method of weighted
residuals: 9.5
Gauss point: 24.7
Gaussian quadrature:
9.5, 24.7
generalised plane strain: 24.9
geometric nonlinearity: 27.2
geometric shape function:
9.4
gradient operator: 14.6
Green's finite strain: 27.11
Hamilton-Cayley
equation: 23.4
hydrostatic stress: 23.4
hyperelasticity: 28.6
higher-order finite elements:
24.13
interpolation function for a bar
element: 9.2
invariants: 14.4, 23.4
invariant v. covariant: 14.7
isoparametric element: 9.4
Jacobian: 9.4,
Kirchhoff stress: 28.5
Kronecker delta: 14.6
Lagrangian fields: 27.2
Lagrangian finite strain:
27.11
local basis: 10.1
material deformation
gradient: 27.9
material derivative: 27.5
material description of a
field: 27.2
material displacement
gradient: 27.10
material nonlinearity: 28.1
mean stress: 23.4
method of weighted
residuals: 9.2
metric space: 14.1
moment in space: 14.6
monotonic loading: 28.1
multi-point constraints: 15.4
normalised coordinate: 9.4
parametric coordinate: 9.4
parent shape of quad
element: 24.2
passive transformations:
14.7
path dependence: 28.1
perfect elasticity: 28.1
permutation symbol: 14.6
physical basis: 9.3
Piola-Kirchhoff stresses: 28.5
plane (quadrilateral)
finite element: 24.2
plane stress/strain:
23.7
plate finite elements:
24.12
polar decomposition
theorem: 27.9
position field: 27.2
primary variable in finite
element analysis: 9.3
principal values: 23.4
push-forward: 27.9
pullback: 27.9
quadrilateral finite element:
24.2
reference state:28.1
relative displacement
tensor: 23.5
rate of deformation:
27.10
secant modulus: 28.1
secondary variable in finite element
analysis: 9.2
shape function: 9.4
shell finite elements:
24.12
simple elasticity: 28.1
solid finite elements:
24.11
spatial deformation
gradient: 27.9
spatial derivative: 27.5
spatial description of a
field: 27.2
spatial displacement
gradient: 27.10
stiffness matrix for a bar
element: 9.1, 9.5
stiffness matrix for a bar
structure: 10.l
strain-displacement matrix: 9.5
strain tensor (small):
23.5
streamline: 27.7,
stress tensor: 23.3
subscript nontation: 14.4
summation convention: 14.5
symmetric structures: 15.5, 24.10
tangent modulus:
28.1
tensor: 23.1
tensor transformation:
traction vector:
23.1
transformation mapping:
9.4
transformation matrix:
14.7
truss finite element:
15.1
vector: 14.2
vector addition: 14.3
vector multiplication:
14.6
vector transformation:
14.7
velocity field: 27.6
velocity gradient: 27.10
Voigt notation: 23.3
weighting function: 9.2