A
.
Vectors, Tensors,
Strains, Stresses
A–1
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–2
§A.1 NOTATIONAL CONVENTIONS
When studying the formulation of full three-dimensional elements, a compact style of notation
is often convenient. This is provided by vector and tensor calculus, the indicial notation and the
summation convention. We shall use the abbreviated style whenever convenient, side by side with
the more conventional engineering notation.
For the present developments only components in a rectangular Cartesian coordinate (RCC) system,
as opposed to a general curvilinear coordinate system, are required. Consequently no distinction
needs to be made between covariant and contravariant components, which eliminates the need
for lower and upper subscripts. Similarly, there is no need to differentiate between tensorial and
physical components.
§A.1.1 Indicial Notation
A point P(x, y, z, ) in a rectangular Cartesian coordinate (RCC) system (see Figure A.1) will be
represented by the shorter notation
P(xi )
i = 1, 2, 3.
(A.1)
where xi means x1 , x2 , x3 .
A linear algebraic equation involving the Cartesian coordinates (x1 , x2 , x3 ) is ordinarily written
using the summation operator as
3
ai j x j = bi
i = 1, 2, 3,
(A.2)
j=1
where ai j and bi are constants. This expands to three equations:
i =1:
3
a 1 j x x = b1 ,
or a11 x1 + a12 x2 + a3 x3 = b1 ,
a 2 j x x = b2 ,
or a21 x1 + a22 x2 + a23 x3 = b2 ,
a 3 j x j = b3 ,
or a31 x1 + a32 x2 + a33 x3 = b3 .
j=1
i =2:
3
(A.3)
j=1
i =3:
3
An even shorter notation is
ai j x j = bi ,
(A.4)
in which the summation symbol has been suppressed. This is the summation convention: an index
repeated in an expression is understood to be summed over the implied range. In three-dimensional
continuum mechanics, the range is 1 to 3.
A–2
A–3
§A.1
NOTATIONAL CONVENTIONS
P(x1 , x2 , x3 )
tor
x3
x1
o
iti
s
po
ec
nv
x2
Figure A.1. Point P in a rectangular Cartesian coordinate system.
EXAMPLE A.1
The expression
aj j
(A.5)
a11 + a22 + a33
(A.6)
ai j x j
(A.7)
means
REMARK A.1
In an expression such as
i is said to be the free index which can take any of the values 1,2,3, whereas j is said to be the dummy index
which must take the values 1,2,3. In other words, the dummy index is the one that must be summed over the
entire range.
Replacing the dummy index by another symbol (one that does not clash with a free index) changes nothing:
ai j x j = aik xk .
(A.8)
§A.1.2 Scalar Functions
A scalar function f (P) of a point P(xi ) is written using the index notation is
f (xi ).
Some important functions of f (xi ) are the linear, bilinear (quadratic) and differential forms.
A–3
(A.9)
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–4
§A.1.3 Linear form
The function f is linear in the coordinates:
f (xi ) = a1 x1 + a2 x2 + a3 x3 =
3
ai xi = ai xi .
(A.10)
i=1
§A.1.4 Bilinear Form
The function f is bilinear in the coordinates xi . Also called quadratic form:
f (xi ) = a11 x1 x1 + a12 x1 x2 + · · · + a33 x3 x3
=
3 3
(A.11)
ai j xi x j = ai j xi x j .
i=1 j=1
REMARK A.2
The coefficients ai j can be collected in a 3 × 3 matrix A = [ai j ], which written in full is
A = [ai j ] =
a11
a21
a31
a12
a22
a32
a13
a23
a33
.
(A.12)
§A.1.5 Differential Quadratic Form
The differential form
ai j d xi d x j ,
(A.13)
represents the distance squared between two neighboring points in a general curvilinear coordinate
system. In a RCC system, the square of the distance is
ds 2 = (d x1 )2 + (d x2 )2 + (d x3 )2
=
3
d xi d xi = d xi d xi .
(A.14)
i=1
REMARK A.3
The coefficients ai j for the RCC are then 1 if i = j else 0, a relation that may be represented compactly with
the Kronecker delta symbol introduced below. The corresponding matrix A is the identity matrix.
A–4
A–5
§A.1
NOTATIONAL CONVENTIONS
§A.1.6 Differentiation and Derivative Notation
The total differential of f (xi ) is
df =
∂f
∂f
∂f
∂f
d x1 +
d x2 + · · · +
d x3 =
d xi .
∂ x1
∂ x2
∂ x3
∂ xi
(A.15)
Introducing another notational convention:
∂f
= f ,i .
∂ xi
(A.16)
That is, a comma denotes partial differentiation with respect to the coordinate that follows. Then
we can write the total differential as
d f = f ,i d xi = d xi f ,i .
(A.17)
EXAMPLE A.2
Examples involving second derivatives:
f ,i j =
f ,ii =
∂2 f
,
∂ xi ∂ x j
(A.18)
∂2 f
∂2 f
∂2 f
∂2 f
=
+
+
= ∇2 f
∂ xi ∂ xi
∂ x1 ∂ x1
∂ x2 ∂ x2
∂ x3 ∂ x3
(Laplacian of f).
(A.19)
§A.1.7 Chain Rule for Differentiation and the Kronecker δ
Let f (xi ) be a given function in which the coordinates are functions of another set, xi = xi (x j ).
The partials with respect to the new set xi are given by the chain rule
∂f
∂ f ∂ x1
∂ f ∂ x2
∂ f ∂ x3
∂ f ∂ xi
.
=
+
+
=
∂ xj
∂ x1 ∂ x j
∂ x2 ∂ x j
∂ x3 ∂ x j
∂ xi ∂ x j
(A.20)
Let f = xk . Then the previous rule gives
∂ xk ∂ xi
∂ xk
=
=
∂ x j
∂ xi ∂ x j
1
0
if
if
j =k
.
j = k
(A.21)
0,
(A.22)
Introducing the Kronecker delta symbol:
δ jk = 1
if
j =k
then we can express the previous relation compactly as
A–5
else
A–6
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
∂ xk ∂ xi
= δ jk .
∂ xi ∂ x j
(A.23)
xk,i
xi, j = δ jk .
(A.24)
Or even more compactly:
EXAMPLE A.3
Evaluate δii . Applying the summation convention:
δii = δ11 + δ22 + δ33 = 3.
(A.25)
Similarly, δi j δi j = 3.
§A.2 BASE VECTORS
We denote the base unit vectors in a rectangular Cartesian coordinate (RCC) system by i1 , i2 , i3 ,
(see Figure B.1). Any vector a can be represented as
a = a1 i1 + a2 i2 + a3 i3 = ak ik
(A.26)
where ak (k = 1, 2, 3) are the components of a.
a
x3
i3
i1
x1
a=
a
i1
1
+
a
i2
2
+
a
i3
3
i2
x2
Figure B.1. The base vectors i1 , i2 and i3 .
A–6
A–7
§A.3
VECTOR OPERATIONS
a×b
in
b
θ
a
Figure A.2. The cross product of two vectors.
§A.3 VECTOR OPERATIONS
§A.3.1 Dot Product
The dot product, scalar product or interior product is defined as the scalar
a · b = |a||b| cos θ
(A.27)
where |a| is the magnitude or length of a and θ the angle between a and b: |b| cos θ is the projection
of b onto a. The dot product of a with itself is the Euclidean norm (also called 2-norm or squared
length) of a:
a · a = |a|2 = a
(A.28)
The base vector dot products satisfy ii i j = δi j , thus
a · b = (a1 i1 + a2 i2 + a3 i3 )(b1 i1 + b2 i2 + b3 i3 )
= (ak ik ) · (b j i j ) = ak b j ik i j = ak b j δk j = ak bk .
(A.29)
§A.3.2 Cross Product
The cross product, vector product or exterior product is defined as the vector
a × b = |a||b| sin θ in ,
(A.30)
where in is the unit normal vector to a and b forming a right-handed system (see Figure B.2) and θ
is the angle between a and b.
A–7
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–8
It follows that
a × b = −b × a
(A.31)
The geometric interpretation is that |a × b| is the area of the parallelogram formed by a and b.
Obviously a × a = 0, the null vector. The cross product of the unit vectors satisfy
i1 × i1 = 0,
i1 × i2 = i3 ,
i2 × i1 = −i3 ,
i2 × i2 = 0,
i2 × i3 = i1 ,
i3 × i2 = −i1 ,
i3 × i3 = 0,
i3 × i1 = i2 ,
i1 × i3 = −i2 .
(A.32)
In condensed form,
i j × ik = e jk i
(A.33)
where ei jk , the permutation symbol or commutator sysmbol, is defined as
ei jk

 0 if all indices are not distinct,
= +1 if all indices are distinct and form a positive cyclic permutation,

−1 if all indices are distinct and form a negative cyclic permutation.
(A.34)
For example, i1 × i2 = e123 i3 = i3 . Carrying out the product
a × b = (a j i j ) × (bk ik ) = a j bk (i j × i ) = a j bk e jk i .
(A.35)
§A.3.3 Triple Scalar Product
The triple scalar product is defined as the scalar
[a, b, c] = a · b × c = im am i (b j ck e jk )
= δm am b j ck e jk = a b j ck e jk = ai b j ck · ei jk
(A.36)
§A.3.4 Gradient of a Scalar
Let f (xi ) be a scalar function. Its gradient is defined as the vector
grad f = ∇ f =
∂f
∂f
∂f
i1 +
i2 +
i3 = f ik ik = ik f ,k
∂ x1
∂ x2
∂ x3
(A.37)
Thus the “Del” (nabla) symbol may be viewed as a vector operator:
∇ = i1
∂
∂
∂
∂
+ i2
+ i3
= ik
= ik (),k .
∂ x1
∂ x2
∂ x3
∂ xk
A–8
(A.38)
A–9
§A.5
NOTATION AND SUMMATION CONVENTION
§A.3.5 Divergence of a Vector
Let a(xi ) be a vector field. The divergence of a is a scalar function obtained by taking the dot
product of ∇ and a:
∂
) · i j a j = a j, j .
(A.39)
div a = ∇ · a = (ik
∂ xk
§A.3.6 Divergence of a Gradient
Let f (xi ) be a scalar function:
div.grad f = ∇.∇ f = ∇ 2 f = f ,kk .
(A.40)
This is the Laplacian of f .
§A.3.7 Curl of a Vector
Let a(xi ) be a vector field. Its curl is defined as the cross product of ∇ and a:
curl a = ∇ × a.
(A.41)
§A.4 TENSORS
This section is an introduction to tensor analysis and tensor notation.
§A.5 NOTATION AND SUMMATION CONVENTION
In tensor analysis one makes extensive use of indices. A set of n variables x1 , x2 , . . ., xn is usually
denoted as xi , i = 1, . . . , n. A set of n variables y 1 , y 2 , . . ., y n is denoted by denoted by y i ,
i = 1, . . . , n. It is emphasized that y 1 , y 2 , . . . here denote independent variables and not the
powers of the variable y. To make the distiction clear when there is possibility of confusion a
power is written by enclosing the base in parenthesis: (y)2 or (x3 )2 mean the squares of y and x3 ,
respectively.
Consider an equation such as
c1 x1 + c2 x2 + c3 x3 = p,
(A.42)
where ci and p are constants. Geometrically (A.42) represents a plane in the three-dimensional
space x1 , x2 , x3 . This equation can be written more compactly as
3
ci xi = p,
(A.43)
i=1
To further compact this equation we shall introduce the summation convention and write (A.43) in
the simpler form
(A.44)
ci xi = p,
A–9
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–10
where the summation sign has been omitted. This convention, attributed to Einstein, is as follows:
The repetition of an index in a term implies summation over that index over its range. The range
of an index i is the set of n integer values 1 to n that the index may assume. A lower index i, as in
xi is called a subscript whereas an upper index i as in y i is called a superscript. An index that is
summed over is called a dummy index. One that is not summed over is called a free index.
Because a dummy index only indicates summation, the symbol used is immaterial. Thus ai xi may
be replaced by a j yj or ak xk as long as these indices have the same range. This is analogous to the
dymmy variable in an integral:
a
b
f (x) d x =
b
f (y) dy.
(A.45)
a
x j = α jk xk ,
(A.46)
which has [α jk ] as transformation matrix. In the case of two co-original RCC systems each entry
of this matrix has an immediate physical meaning: α jk is the director cosine of the angle formed
by x j and xk .
A–10
A–11
§A.6
CARTESIAN TENSORS
Taking partials of x j with respect to xk we obtain the condition
∂ x
∂ xj
=
α
= α j δk = α jk .
j
∂ xk
∂ xk
(A.47)
xk = α jk x j
(A.48)
∂ xk
= α jk
∂ xj
(A.49)
∂ x j ∂ xk
= α jk αk
∂ xk ∂ x
(A.50)
α jk αk = δ j
(A.51)
The inverse transformation is
Multiplying these relations
but the left hand is also δ j . Thus
This is the definition of an orthogonal transformation.
EXAMPLE A.4
If j = = 1,
2
2
2
+ α22
+ α33
=1
α11
If j = 1, = 2
α11 α21 + α12 α21 + α13 α23 = 0
These are the well known orthonormality properties of direction cosines.
REMARK A.4
In matrix form, AAT = I, and A is called an orthogonal matrix. Note that A−1 = AT .
A–11
A–12
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
§A.6 CARTESIAN TENSORS
Let a and b be two vectors in the RCC system xi . Consider the transformation to RCC system x j :
ak = α jk a j ,
bn = αn b .
(A.52)
Thus
ak bn = α jk αn a j b .
(A.53)
= α jk αn C j .
Ckn
(A.54)
Let Ckn
= ak bn and C j = a j b ; then
This is the definition of a second order tensor or a tensor of order 2 (it has two free indices).
In particular: a vector is a tensor of order 1, and a scalar is a tensor of order 0.
A fourth order tensor would be defined as
Ci jk = αmi αn j α pk αq Cmnpq .
(A.55)
In general tensors are functions of position xi e.g. C j = C j (xi ).
The contraction of a tensor is obtained by setting two indices equal and summing. For example if
j =k
C jk → C j j = C11 + C22 + C33
(A.56)
which is a scalar (tensor of order zero).
§A.7 STRAINS
Let u i (x j ) denote the components of the displacement vector field u(x j ). Then the infinitesimal
strains are given by
ei j = 12 (u i, j + u j,i )
These are the components of the strain tensor [ei j ] = e, which written in full is
A–12
(A.57)
A–13
§A.7
∂u 1
∂ x1
∂u 2
=
∂ x2
∂u 3
=
∂ x3
STRAINS
e11 = u 1,1 =
e22 = u 2,2
e33 = u 3,3
e12
e23
e31
∂u 2
∂u 1
=
+ u 2,1 ) =
+
∂ x2
∂ x1
∂u 3
1
1 ∂u 2
= 2 (u 2,3 + u 3,2 ) = 2
+
∂ x3
∂ x2
∂u 1
1
1 ∂u 3
= 2 (u 3,1 + u 1,3 ) = 2
+
∂ x1
∂ x3
1
(u
2 1,2
1
2
(A.58)
§A.7.1 The Linear Strain Tensor
The infinitesimal or linear strain tensor in the xi coordinate system is

e = [ei j ] = 
e11
e12
e22
symm

e13
e23 
e33
(A.59)
In another coordinate system x j related to xi by the transformation
xi = ai j x j
(A.60)
ei j = aim a jn emn
(A.61)
the strain components become
A–13
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–14
§A.7.2 The Engineering Notation
The standard engineering notation uses x, y, z for x1 , x2 , x3 and u x , u y , u z for u 1 , u 2 , u 3 , respectively.
Then the engineering strains are related to the displacements by
∂u x
∂x
∂u y
e yy = e22 =
∂y
∂u z
ezz = e33 =
∂z
∂u y
∂u x
γx y = 2e12 =
+
∂y
∂x
∂u y
∂u z
γ yz = 2e23 =
+
∂z
∂y
∂u z
∂u x
+
γzx = 2e31 =
∂x
∂z
ex x = e11 =
(A.62)
The linear strain tensor in terms of engineering strains is

1
γ
2 xy
e yy
ex x
[e] = 
symm
1
γ
2 xz
1
γ
2 yz


(A.63)
ezz
§A.8 COMPATIBILITY EQUATIONS
The strain tensor e has 6 independent components. The displacement field has 3 independent
components. It follows that there must be 3 independent conditions between the ei j . These
expressions arise from the condition of compatibility of deformation.
In the three-dimensional case these compatibility equations are
ei j,k + ek,i j = eik, j + e j,ik
(A.64)
For the two dimensional case only one equation survives
e11,22 + e22,11 = 2e12,12
(A.65)
∂ 2 e yy
∂ 2 γx y
∂ 2 ex x
+
=
∂ y2
∂x2
∂x ∂ y
(A.66)
which in standard notation is
A–14
A–15
§A.10
;;
;;
;;
THE STRESS TENSOR
t
m
n
P
cu
tti
ng
pl
an
e
A
Figure A.1. Plane cut through a body for defining the interior force resultants at point P.
§A.9 STRESSES
Consider a continuum body and an interior point P(xi ). Make a cut through P with a plane with
exterior normal n, as illustrated in Figure E.1.
The stress vector at P for direction n is defined as
t
,
A→0 A
tn = lim
(A.67)
where A is a differential area surrounding P on the cutting plane (see Figure E.1).
The couple stress vector for direction n is
m
.
A→0 A
mn = lim
(A.68)
It is optional to include mn in the theory of stress. Doing so leads to the so-called polar material
models. In classical continuum mechanics it is generally assumed that mn = 0, which corresponds
to non-polar materials. Polar material models are generally considered only when continua are
subjected to strong electromagnetic fields.
A–15
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–16
§A.10 THE STRESS TENSOR
Consideration of the equilibrium of an elemental tetrahedron at P whose faces are normal to x1 ,
x2 , x3 and n leads to the expression
ti = σi j n j ,
(A.69)
where ti is the component of t in the xi direction, and n j are the components of n. The nine values
σi j are the components of the Cauchy stress tensor

σ11

σ = [σi j ] = σ21
σ31

σ13
σ23  .
σ33
σ12
σ22
σ32
(A.70)
For non-polar materials this tensor is symmetric. That is, σi j = σ ji .
§A.11 EQUILIBRIUM EQUATIONS
Consider the equilibrium of a body of volume V and surface S subject to the following actions
(a) Body force field f of components bi in V
(b) Acceleration field a = d 2 u/dt 2 = ü (t = time) of components ai in V
(c) Stress vectors t of components ti on S
Dynamic equilibrium along any direction xi requires
ti d S +
S
bi d V =
V
ρai d V,
(A.71)
V
where ρ is the body density. Substitute ti = σi j n j in the surface integral:
σi j n j d S +
S
bi d V =
V
ρai d V.
(A.72)
V
To transform the surface integral to a volume integral we use Gauss’ divergence theorem. For any
vector field a:
a.n d S =
S
or in component form
div.a d V.
(A.73)
∂a j
d V.
∂ xj
(A.74)
V
aj n j d S =
S
V
Consequently the equilibrium integral (A.71) may be reduced to
[σi j, j + bi − ρai ] d V = 0,
V
A–16
(A.75)
A–17
§A.11
EQUILIBRIUM EQUATIONS
for an arbitrary volume. Because the volume is arbitrary we must have
σi j, j + bi − ρai = 0.
(A.76)
These are the three differential equations of dynamic equilibrium, which are obtained by setting
the free index i to 1, 2 and 3. These are also called the internal equilibrium equations, or balance
equations. If the medium is at rest or moving uniformly with respect to an inertial frame, the
accelerations vanish and we obtain the equations of static equilibrium
σi j, j + bi = 0.
(A.77)
EXAMPLE A.5
If i = 1 the first static equilibrium equation along axis x1 is
or, written in full
σ1 j, j + b1 = 0
(A.78)
∂σ12
∂σ13
∂σ11
+
+
+ b1 = 0.
∂ x1
∂ x2
∂ x3
(A.79)
In conventional (engineering) notation:
∂τx y
∂τx z
∂σx x
+
+
+ bx = 0.
∂x
∂y
∂z
A–17
(A.80)
Appendix A: VECTORS, TENSORS, STRAINS, STRESSES
A–18
EXERCISE A.1
Show that, when i, j, k, range over 1, 2, 3:
δi jδi j = 3.
(EA.1)
i jk jki = 6
(EA.2)
i jk A j Ak = 0
(EA.3)
δi j delta jk = deltaik
(EA.4)
A–18