Section 3.2
3.2 The Momentum Principles
In Parts I and II, the basic dynamics principles used were Newtons Laws, and these are
equivalent to force equilibrium and moment equilibrium. For example, they were used to
derive the stress transformation equation
Section 1.16
1.16 Curvilinear Coordinates
Up until now, a rectangular Cartesian coordinate system has been used, and a set of
orthogonal unit base vectors e i has been employed as the basis for representation of
vectors and tensors. This basis is independ
Section 1.15
1.15 Tensor Calculus 2: Tensor Functions
1.15.1
Vector-valued functions of a vector
Consider a vector-valued function of a vector
a = a(b),
ai = ai (b j )
This is a function of three independent variables b1 , b2 , b3 , and there are nine par
Section 1.14
1.14 Tensor Calculus I: Tensor Fields
In this section, the concepts from the calculus of vectors are generalised to the calculus of
higher-order tensors.
1.14.1
Tensor-valued Functions
Tensor-valued functions of a scalar
The most basic type o
Section 1.13
1.13 Coordinate Transformation of Tensor Components
It has been seen in 1.5.2 that the transformation equations for the components of a vector
are u i = Qij u j , where [Q ] is the transformation matrix. Note that these Qij s are not the
comp
Section 1.12
1.12 Higher Order Tensors
In this section are discussed some important higher (third and fourth) order tensors.
1.12.1
Fourth Order Tensors
After second-order tensors, the most commonly encountered tensors are the fourth order
tensors A , whi
Section 1.11
1.11 The Eigenvalue Problem and Polar Decomposition
1.11.1
Eigenvalues, Eigenvectors and Invariants of a Tensor
Consider a second-order tensor A. Suppose that one can find a scalar and a (non-zero)
normalised, i.e. unit, vector n such that
An
Section 1.10
1.10 Special Second Order Tensors & Properties of
Second Order Tensors
In this section will be examined a number of special second order tensors, and special
properties of second order tensors, which play important roles in tensor analysis. T
Section 1.9
1.9 Cartesian Tensors
As with the vector, a (higher order) tensor is a mathematical object which represents
many physical phenomena and which exists independently of any coordinate system. In
what follows, a Cartesian coordinate system is used
Section 1.8
1.8 Tensors
Here the concept of the tensor is introduced. Tensors can be of different orders zerothorder tensors, first-order tensors, second-order tensors, and so on. Apart from the zeroth
and first order tensors (see below), the second-order
Section 1.7
1.7 Vector Calculus 2 - Integration
1.7.1
Ordinary Integrals of a Vector
A vector can be integrated in the ordinary way to produce another vector, for example
2
cfw_(t t )e
2
1
1
1.7.2
5
15
+ 2t 2 e 2 3e 3 dt = e1 + e 2 3e 3
6
2
Line Integra
Section 1.6
1.6 Vector Calculus 1 - Differentiation
Calculus involving vectors is discussed in this section, rather intuitively at first and more
formally toward the end of this section.
1.6.1
The Ordinary Calculus
Consider a scalar-valued function of a s
Section 1.5
1.5 Coordinate Transformation of Vector Components
Very often in practical problems, the components of a vector are known in one coordinate
system but it is necessary to find them in some other coordinate system.
For example, one might know th
Section 1.4
1.4 Matrices and Element Form
1.4.1
Matrix Matrix Multiplication
In the next section, 1.5, regarding vector transformation equations, it will be necessary
to multiply various matrices with each other (of sizes 3 1 , 1 3 and 3 3 ). It will be
h
Section 1.3
1.3 Cartesian Vectors
So far the discussion has been in symbolic notation1, that is, no reference to axes or
components or coordinates is made, implied or required. The vectors exist
independently of any coordinate system. It turns out that mu
Section 1.1
1.1 Vector Algebra
1.1.1
Scalars
A physical quantity which is completely described by a single real number is called a
scalar. Physically, it is something which has a magnitude, and is completely described
by this magnitude. Examples are tempe
Section 1.17
1.17 Curvilinear Coordinates: Transformation Laws
1.17.1
Coordinate Transformation Rules
Suppose that one has a second set of curvilinear coordinates ( 1 , 2 , 3 ) , with
i = i ( 1 , 2 , 3 ),
i = i (1 , 2 , 3 )
(1.17.1)
By the chain rule, t
Section 1.18
1.18 Curvilinear Coordinates: Tensor Calculus
1.18.1
Differentiation of the Base Vectors
Differentiation in curvilinear coordinates is more involved than that in Cartesian
coordinates because the base vectors are no longer constant and their
Section 3.1
3.1 Conservation of Mass
3.1.1
Mass and Density
Mass is a non-negative scalar measure of a bodys tendency to resist a change in motion.
Consider a small volume element v whose mass is m . Define the average density of
this volume element by th
Section 2.13
Variation and Linearisation of Kinematic Tensors
2.13.1
The Variation of Kinematic Tensors
The Variation
In this section is reviewed the concept of the variation, introduced in Part I, 5.5.
The variation is defined as follows: consider a func
Section 2.12
2.12 Pull Back, Push Forward and Lie Time Derivatives
2.12.1
Push-Forward and Pull-Back
The concepts of pull-back and push-forward have a number of uses, in particular they will be
used to define the Lie derivative further below.
Vectors
Cons
Section 2.11
Convected Coordinates: Time Rates of Change
In this section, the time derivatives of kinematic tensors described in 2.4-2.6 are now
described using convected coordinates.
2.11.1
Deformation Rates
Time Derivatives of the Base Vectors and the D
Section 2.10
2.10 Convected Coordinates
In this section, the deformation and strain tensors described in 2.2-3 are now described
using convected coordinates (see 1.16). Note that all the tensor relations expressed in
&
symbolic notation already discussed,
Section 2.9
2.9 Rigid Body Rotations of Configurations
In this section are discussed rigid body rotations to the current and reference
configurations.
2.9.1
A Rigid Body Rotation of the Current Configuration
As mentioned in 2.8.1, the circumstance of two
Section 2.8
2.8 Objectivity and Objective Tensors
2.8.1
Dependence on Observer
Consider a rectangular block of material resting on a circular table. A person stands and
observes the material deform, Fig. 2.8.1a. The dashed lines indicate the undeformed
ma
Section 2.7
2.7 Small Strain Theory
When the deformation is small, from 2.2.43-4,
F = I + GradU
= I + (gradu )F
(2.7.1)
I + gradu
neglecting the product of gradu with GradU , since these are small quantities. Thus one
can take GradU = gradu and there is
Section 2.6
2.6 Deformation Rates: Further Topics
2.6.1
Relationship between l, d, w and the rate of change of R
and U
Consider the polar decomposition F = RU . Since R is orthogonal, RR T = I , and a
differentiation of this equation leads to
&
&
R RR T
Section 2.5
2.5 Deformation Rates
In this section, rates of change of the deformation tensors introduced earlier, F, C, E, etc.,
are evaluated, and special tensors used to measure deformation rates are discussed, for
example the velocity gradient l, the r
Section 2.4
2.4 Material Time Derivatives
The motion is now allowed to be a function of time, x = (X, t ) , and attention is given to
time derivatives, both the material time derivative and the local time derivative.
2.4.1
Velocity & Acceleration
The velo
Section 2.3
2.3 Deformation and Strain: Further Topics
2.3.1
Volumetric and Isochoric Deformations
When analysing materials which are only slightly incompressible, it is useful to
decompose the deformation gradient multiplicatively, according to
(
)
F = J