Continuum Mechanics
Mathematical Background
Syllabus Overview
Week 1 – Math Review (Chapter 2).
Week 2 – Kinematics (Chapter 3).
Week 3 – Stress & Conservation of mass,
momenta and energy (Chapter 4 & 5).
Week 4 – Constituative Equations &
Linearized Elasticity (Chapter 6 & 7).
Week 7 – Fluid Mechanics, Heat Transfer &
Viscoelasticity (Chapters 8 & 9).
Assessment
Computer Project – 30%
Mid Term (Take Home) – 25%
Final (Take Home) – 30%
Homework - (15%)
Some Examples
Continuum mechanics forms the basis of CFD, Solid Mechanics, Thermal
Profile Modeling.
General Procedure for solving a
continuum mechanics problem
1.
2.
3.
4.
5.
6.
7.
Decide upon the goal of the problem and desired information;
Identify the geometry of the solid to be modeled;
Determine the loading applied to the solid;
Decide what physics must be included in the model;
Choose (and calibrate) a constitutive law that describes the behavior of the
material;
Choose a method of analysis;
Solve the problem.
Vectors
•
•
•
Vectors are important in
continuum mechanics. The
direction and magnitude of many
variables can be most easily
described by vectors.
Unit vector – a vector of unit
length (often denoted by a caret
(pointy hat).
Vector addition – add like
components (A+B=B+A therefore
vector addition is commutative) –
((A+B)+C=A+(B+C) therefore
associative)
A  2i  3j  k
ˆe A  A  2i  3j  1k
A
22  32  12
Vectors 2
•
•
•
•
•
•
Multiplication by a scalar (same as
addition – therefore associative and
commutative.
Scalar (dot product) – multiply like
terms and add to give a scalar result.
Vector product – most easily found
using matrices – vector product is a
vector perpendicular to the plane
containing the vectors multiplied.
(BxA=-AxB).
Triple Scalar product (A.BxC) – a
scalar that is equal to the area of a
parallelepiped defined by the 3
vectors.
Triple Vector Product (Ax(BxC)) a
vector in the same plane as B and C
can be found by:
Ax(BxC)=B(A.C)-C(A.B)
F  d  Fd cos 
F  d  Fd sin eˆ
Matrices
• Matrix methods are
exceptionally useful for
dealing with vectors, (&
tensors).
• To determine the cross
product of two vectors
you can evaluate the
determinant of a matrix
which is made up of the
basis and the two
vectors.
i
j k
Fd  3 2 3
1 3 4
n
det D  dij    1 di1 Di1
i 1
i 1
Where i is row number and j column
number and the last term on the rhs is
the determinant of the n-1 by n-1
matrix when the ith row and first
column of D are removed
Matrices
• Transpose
(interchange rows
and columns)
A 
T T
A
( A  B )T  A T  B T
sym m etric
AT  A
skewsym m etric
aij  a ji
1 4 3
1 3 1
A  3 2 3  AT  4 2 3
1 3 4
3 3 4
Matrices
• Multiplication (scalar
multiplication of rows
by columns)
AB  C
 
n
C  cij   aik bkj
k 1
 5 2
8
7
AB  C  
2
4

 1 9
1
 36
 3  1 2 4 

6 
36



6
3
5
7

  9
3 
6  2 1 
  9
0
 57
5 2
49
28
28
7
39 87
18 39

43 59
Inverse Matrices
1
A 
AdjA
det A
1
The Adjoint (also called adjunct) of a matrix is the transpose of the matrix obtained by
replacing each element by its cofactor.
2 5  1
 cof11 A  cof12 A  cof13 A   29  22 19 

 
12  7 
A  1 4
3  adj( A)  cof21 A  cof22 A  cof23 A    1
cof31 A  cof32 A  cof33 A   11 16
3 
2  3 5 
det A  245  3 3   155   1 3  253  14  74
Tensors
Properties of anisotropic solids can be represented by tensors. For the most part
we will consider rank 1 (vectors) and rank 2 (matrices) tensors.
 11  12  13
σ   21  22  23
 31  32  33
Matrix methods can be used to solve tensor (i.e. anisotropic continuum mechanics
problems).
The trace of a tensor is the sum of its diagonal terms.
Eigenvalues & Eigenvectors
In general a tensor can be considered as an operator that can stretch and rotate
space. It is possible to find components that have no rotation. These special
components are eigenvalues and eigenvectors. How could this be useful?
A.x  x
A  I .x  0
det A  I   0
Example 3.4.4
•
The state of strain at a point in an elastic
body is given by (microstrain). Determine the
principal strains and principal directions of
the strain.
 4  4 0
E    4 0 0
 0
0 3
Good review of vector calculus
• http://www.enm.bris.ac.uk/admin/courses/
EMa2/EMAT20200_index.html