2101202 – Mechanics of Materials I By Watanachai Smittakorn VECTORS AND TENSORS “A beautiful story needs a beautiful language to tell. Tensor is the language of mechanics.” scalars tensors vectors matrices VECTORS • defined as a line with magnitude and direction • denoted by , , , , ,... AB PQ a uF u uur r • two vectors are equal if (both magnitude & direction are equal) • a unit vector is (a vector with a unit magnitude) • zero vector, denoted by 0, is (a vector with zero magnitude) • | AB |, |u|, v are magnitudes of AB , u, v - 1 -
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
VECTORS AND TENSORS
“A beautiful story needs a beautiful language to tell. Tensor is the language of mechanics.”
scalars
tensors
vectors
matrices
VECTORS
• defined as a line with magnitude and direction • denoted by
, , , , , ...AB PQ a u Fuuur r
• two vectors are equal if (both magnitude & direction are equal) • a unit vector is (a vector with a unit magnitude) • zero vector, denoted by 0, is (a vector with zero magnitude) • | AB|, |u|, v are magnitudes of AB, u, v
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Vector addition:
• “parallelogram law” • commutative and associative
a + b = b + a
(a + b) + c = a + (b + c) Vector subtraction:
a – b = a + (-b)
Let e1, e2, e3 be the unit vectors in x1, x2, x3 directions,
x1
x2
x3
u
u1
u2
u3
1 2 3 1 2
2 2 21 2 3
( , , )u u u u u u
u u u u
= + + =
= = + +
1 2 3u e e e
u3
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Scalar product (or dot product):
1 1 2 2 3 3
cos (0 )u v u v u v
θ θ π⋅ = + +
⋅ = ≤ ≤
u vu v u v
e.g., F · s Vector product (or cross product):
2 3 3 2 3 1 1 3 1 2 2 1
1 2 3
1 2 3
( ) ( ) (
sin (0 )
u v u v u v u v u v u v
u u uv v v
θ θ π
× = − + − + −
=
× = ≤ ≤
1 2
1 2 3
u v e e ee e e
u v u v
) 3
2
e.g., r × F Properties:
1 3 3
3 3 3 1
( )( )
( )k k k
× = − ×× + = × + ×× =× = × = × =
× = × = × =
× = × = ×
1 2 2
1 2 2 1
u v v uu v w u v u wu u 0e e e e e e 0e e e e e e e e eu v u v u v
HOMEWORK: 2.5, 2.7, 2.9
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
What is mechanics? Why is tensor important?
scalars
tensors
vectors
matrices
INDICIAL NOTATION
x1, x2,…, xn is denoted as xi, i = 1,…,n
i is an index with range of 1 to n
xi vs (xi) or X member vector/matrix
Summation convention:
3
1 1 2 2 3 31
i i i ii
a x a x a x a x a x=
+ + = ≡∑ Note: The repetition of an index in a term will denote a summation with respect to that index over its range. “dummy index” – one that is summed over “free index” – one that is not summed
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Hence, aixi = ajxj = akxk
e.g., 1 2 3
1 2 32 2 2 2 2
1 2 3
1 1 2 2 3 3
i i
i i
u u uv v v
u u u u u uu v u v u v u v
= + +
= + +
= = + + =
⋅ = + + =
1 2 3
1 2 3
u e e ev e e e
uu v
(more examples: aijxj, aijxjk, aijxij, …)
Kronecker delta (δij):
1 if ,
0 if ij
ij
i j
i j
δ
δ
= =
= ≠
or
( )1 0 00 1 00 0 1
ijδ⎛ ⎞⎜ ⎟= ⎜ ⎟⎜ ⎟⎝ ⎠
then 2
i i ij i j
ij i j
u u u u u
a a
δ
δ
= =
=
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Matrices and Determinants Let A be an m×n matrix and B be an n×p matrix.
( ) 1,..., ; 1,...,
( ) 1,..., ; 1,...,ij
jk
a i m j n
b j n k p
= = =
= = =
A
B
the product of A and B is an m×p matrix defined as ( ) 1,..., ; 1,..., ; 1,...,ik ij jka b i m j n k p⋅ = = = =A B
11 12 13
21 22 23
31 32 33
11 22 33 12 23 31 13 21 32
11 23 32 12 21 33 13 22 31
det det( )ij
a a aa a a a
a a aa a a a a a a a aa a a a a a a a a
= =
= + +
− − −
A
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Permutation symbol (εrst):
1 if , , permute as 1,2,3,1,2,3,...1 if , , permute as 3,2,1,3,2,1,...
0 otherwise
rst
rst
rst
r s tr s t
εεε
== −=
i.e.,
123 231 312
213 321 132
111 222 333 112 121 211
1,1,
... 0
ε ε εε ε εε ε ε ε ε ε
= = == = = −= = = = = = =
then,
1 2 3det( )ij rst r s ta a a aε=
rst s t ru vε× =u v e
( ) ( ) ijk i j ka b cε⋅ × = × ⋅ =a b c a b c
HOMEWORK: 2.16, 2.19
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
TRANSLATION AND ROTATION OF COORDINATES 2-D SPACE
• Translation
x
x’
y’y A
h
k
''
x x hy y k
= += +
or ''
x x hy y k
= −= −
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
• Rotation
x
y
x’
y’ A
θ θ
'cos 'sin'sin 'cos
x x yy x y
θ θθ θ
= −= +
or ' cos sin' sin cos
x x yy x y
θ θθ θ
= += − +
Using index notation,
' (i ij jx x i 1, 2)β= = where
cos( ', )ij i jx xβ =
11 12
21 22
cos sin( )
sin cosij
β β θ θβ
β β θ θ⎛ ⎞ ⎛ ⎞
= =⎜ ⎟ ⎜ ⎟−⎝ ⎠⎝ ⎠
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
The inverse transform is
' ( 1,2i ji jx x i )β= =
Note: (βij) or B is an orthogonal matrix:
1( ) ( ) ( )Tji ij ijβ β β −= =
1−⋅ = ⋅ =TB B B B I
Hence,
ik jk ijβ β δ=
i.e., Consider a unit vector along xi’-axis (βi1,βi2)
2 21 2( ) ( ) 1 ( 1,2i i iβ β+ = = )
)
Unit vector along xi’-axis (βi1,βi2) is perpendicular to unit vector along xj’-axis (βj1,βj2) if i≠j
1 1 2 2 0 (i j i j i jβ β β β+ = ≠
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
3-D SPACE
y
x
z
x’
y’
z’
A
In the same manner as 2-D (See proof in pp. 51-52)
'i ij jx xβ= 'i ji jx xβ=
where direction cosines
'ij i jβ ≡ ⋅e e which is a cosine of angle between xi’ and xj axes Law of transformation of any vector A:
'i ij jA Aβ= 'i ji jA Aβ= HOMEWORK: 2.34
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
ANALYTICAL DEFINITIONS OF SCALARS, VECTORS, AND CARTESIAN TENSORS
P
x1
x2
x3
x2’
x1’
x3’
Scalar (or tensor of rank 0)
• has only single component • equal in all frames of reference
e.g., temperature, moisture, mass, …
1 2 3 1 2 3( , , ) '( ', ', ')x x x x x xφ φ=
Vector (or tensor of rank 1) • has 3 components • obeys law of transformation
e.g., force, displacement, velocity, …
1 2 3 1 2 3
1 2 3 1 2 3
'( ', ', ') ( , , )( , , ) '( ', ', ')
i k
k i
u x x x u x x xu x x x u x x x
ik
ik
ββ
==
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
Tensor of rank 2
• has 9 components • obeys law of transformation
e.g., stress, strain, …
1 2 3 1 2 3
1 2 3 1 2 3
'( ', ', ') ( , , )
( , , ) '( ', ', ')ij mn im jn
mn ij im jn
x x x x x x
x x x x x x
σ σ β β
σ σ β β
=
=
Tensor of rank 3
ijke
Tensor of rank 4
ijklC
: Since based on rectangular Cartesian frame
called “Cartesian tensors”
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
PARTIAL DERIVATIVES
1 21 2
,
... nn
ii
i i
f f fdf dx dx dxx x xf dxx
f dx
∂ ∂ ∂= + +
∂ ∂ ∂∂
=∂
=
Comma notation:
,
,
,
ii
ii j
j
ijij k
k
xuux
x
φφ
σσ
∂≡
∂
∂≡
∂
∂≡
∂
HOMEWORK: 2.37, 2.38
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
EIGENVALUE PROBLEM For a symmetric matrix with real elements A=(aij)
2101202 – Mechanics of Materials I By Watanachai Smittakorn
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Next, from stresses τij in xi-system, find stresses τij’ in xi’-system.
From
x1
x2
x3
xk’
x m’
'i ij jx xβ= where β is cosi eij n of angle between xi’ and xj and Cauchy’s formula
i ji jT nτ= where nj is unit normal vector to any plane If n is parallel to xk’ axis, then
jj kn β=
n
T’k
τkm’
2101202 – Mechanics of Materials I By Watanachai Smittakorn
and
'ki ji kjT τ β= The component T’k in xm’ direction which is stress (τkm’) on xk’ face in xm’ direction is
3
'
m
τ 1 1 2 2 3 3
1 1 2 2 3
' ( ' ' ' )
' ' '
'
k k kkm m
k k km m
ki mi
T T T
T T Tβ
T
β β
β
= + + ⋅
= + +
=
e e e e
Hence,
'km ji kj miτ τ β β=
x
y
z
x’
y’
z’
τ τkm’ ij
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
PRINCIPAL STRESSES & PRINCIPAL PLANES (Shames 15.3, 15.4)
rom Cauchy’s formula (for a state of stress at a
j
Fpoint)
T ni jiτ= where nj is unit normal vector to any plane, there exist three mutually perpendicular axes (called principal axes) where there is zero shear stress (i.e., Ti = τni). (See Figure 1.)
Transformation of stress and strain in 2-D can use Mohr’s circle.
small t
τyy
τxy
τxx
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2101202 – Mechanics of Materials I By Watanachai Smittakorn
PLANE STRAIN (Shames 8.1)
A prismatic body constrained not to bend, stretch, or shorten is loaded in a direction normal to the centerline with no variation in the direction of the axis of the prism. Hence,