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MECH 5312 – Solid Mechanics IIDr. Calvin M. Stewart
Department of Mechanical Engineering
The University of Texas at El Paso
Table of Contents
• Preliminary Math
• Concept of Stress
• Stress Components
• Equilibrium
Preliminary Math
• Base Vectors – Describe the coordinates system in 3D space.
• Base Vectors are Orthogonal to each other. (i.e. the normal of the plane formed by two base vectors is always the third base vector.)
• Let’s assume the unit direction vector e1, e2, and e3 are directed along the positive x, y, and z axes.
• Note: base vectors are not always aligned to x, y, z.
1 2 3
1 0 0
0 1 0
0 0 1
e e e
x
y
z
e1e2
e3
Preliminary Math
• Scalar/Dot Product – the produce of the magnitude of one vector and the component of the second vector in the direction of the first.
1 1 2 2 3 3
1 1 2 2 3 3
1 1 2 2 3 3 1 1 2 2 3 3
1 1 2 2 3 3
i i j j
i j i j
i i
u u u
v v v
u u u v v v
u v
u v
u v
u v u v u v
u e e e
v e e e
u v e e e e e e
e e
e e
1 1 2 2 3 3u v u v u v u v
cosu v u v
where θ is the angle between the vectorsyields a scalar quantity.
v
u
Preliminary Math
• Often,
• Thus, where
1 2 3ˆˆ ˆi j k e e e
i j ij e e
1 0 0
0 1 0
0 0 1
ij
ijI Identity Matrix / Kronecker Delta
Preliminary Math
• Tensor Multiplication
• Tensor, A
• Vector, b
• Product of a Tensor and a Vector is a Vector!
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
1
2
3
b
b
b
b
11 12 13 1 21 1 22 2 23 3
21 22 23 2 21 1 22 2 23 3
31 32 33 3 21 1 22 2 23 3
a a a b a b a b a b
a a a b a b a b a b
a a a b a b a b a b
3x3
3x1
3x3 3x1 3x1
Row x Column
Preliminary Math
• Tensor Multiplication (continued)
11 12 13
21 22 23
31 32 33
a a a
a a a
a a a
A
11 12 13
21 22 23
31 32 33
b b b
b b b
b b b
B
11 12 13 11 12 13
21 22 23 21 22 23
31 32 33 31 32 33
11 11 12 21 13 31 11 12 12 22 13 32 11 13 12 23 13 33
21 11 22 21 23 31 21 12 22 22 23 32 21 13 22 23 23 33
3
a a a b b b
a a a b b b
a a a b b b
a b a b a b a b a b a b a b a b a b
a b a b a b a b a b a b a b a b a b
a
AB
1 11 32 21 33 31 31 12 32 22 33 32 31 13 32 23 33 33b a b a b a b a b a b a b a b a b
Example
Example 1
• Given the following orthogonal base vectors.
• Prove
1 2 3
0 01
2 20
2 20
2 2
2 2
e e e
i j ij e e
Example 2
• Show that the dot product of
• are equal to the projected component of each vector in the index directions.
, , u i u j u k
Example 3
• Given
• Find,
1 2 3
4 5 6
7 8 9
A
0
1
1
b
A b
Concept of Stress
Concept of Stress
• Traction Vector, t
Traction can be decomposed into Normal and Shear Components
limA A
Ft
lim n
nA A
Flim s
A A
F
n t n s
Concept of Stress
• Traction relates to Cauchy Stresses when the normal of a surface aligns with a base vector. For example, the normal aligned with x axis.
lim , lim , limyx z
xx xy xzA A AA A A
FF F
Example
Example 4
• Using static equilibrium to prove that the sum of internal forces across the two halves are equal and opposite. (i.e. internal forces disappear when we remove the section!) Cauchy’s Fundamental Lemma
Stress Components
11 12 13
21 22 23
31 32 33
S
or
Cauchy Stress Tensor, S
xx xy xz
yx yy yz
zx zy zz
S
S σ T
Solecki Boresi
Stewart
Notation ->
Stress Components
• If we know the tractions , t(x), t(y), and t(z) on the x, y, and z surfaces of a 3D element.
• We can find the Cauchy stress components as follows
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
, ,
, ,
, ,
x x x
xx xy xz
y y y
yx yy yz
z z z
zx zy zz
i j k
i j k
i j k
t t t
t t t
t t t
( )xt
( )yt
( )zt
Surface Direction
Stress Components
• We can also determine the relationship between traction, t(n) on an arbitrary plane, n and Cauchy stress by invoking Equilibrium!
Derive!
Stress Components
• Thus the relationship between the traction on an arbitrary plane, nand Cauchy stress equates to
( )n t S n
Concept of Stress
Traction can be decomposed into Normal and Shear Components
n t n
1 2
( ) ( ) 2n n
n t t
( )n
n t n s
Example 5
• Select a σyy such that there will be a traction free plane
1 0 1
0 2
1 2 0
yy
σ
Example 62-4 from Solecki
Transformation of Stress Components
• When the reference coordinate system is rotated.
• The state of stress will transform.
• For example, rotating about z
• x, y, z to x’, y’, z’
x
y
z
x'y'
z'
cos , sin , 0
sin , cos , 0
0, 0, 1
x x x y x z
y x y y y z
z x z y z z
n n n
n n n
n n n
0
0
0 0 1
x x y x
x y z x y y y
z z
n n
n n n n n
n
N
Transformation of Stress Components
• Transformation Tensor, N
Transformed stress, S’
cos sin 0
sin cos 0
0 0 1
N
TS N SN
Example 7
Body Forces
• Body Forces are forces that act on every element of a material and hence on the entire volume of the material.
• Example: Gravitational Forces
Force Vector
, Density
, Body Force Vector
Current Configuration
b
V V
b
dV m dV m
t
t
f b b g
f
x
b x
x
Surface Forces
• Surface forces act on the surface of a material. This surface may be either a part or the whole of the boundary surface
Surface Force Vector
, Traction Vector
Surface Increment
s
S
b
dS
t
dS
f t
f
t x
Equilibrium
• When we speak of Equilibrium, we refer to the Newton’s laws of motion and equilibrium given as,
• Equilibrium ONLY EXISTS if the left hand side (LHS) and right hand side (RHS) of the equation ARE EQUAL.
• In the case where a=0 and v≥0 this equation becomes
• This condition is called “Static Equilibrium”.
where represents the sum of all Forces (both surface and body), m is mass, and is acceleration.
s bm or m
F a F F a
F a
0 0s bor F F F
Derive!
Balance of Linear Momentum
3111 21
1
1 2 3
3212 22
2
1 2 3
13 23 33
3
1 2 3
0
0
0
b
b
b
dd df
dx dx dx
dd df
dx dx dx
d d df
dx dx dx
ij b
i i
j
df a
dx
0ij b
i
j
df
dx
Motion
Static Equilibrium
Balance of Angular Momentum
• Proves,
V
M dV I x v ω
0M
Motion
Static Equilibrium
ij ji
Calvin M. StewartAssistant ProfessorDepartment of Mechanical EngineeringThe University of Texas at El Paso500 W. University Blvd.Suite A126El Paso, Texas 79968-0717
Email: [email protected]: http://me.utep.edu/cmstewart/Phone: 915-747-6179Fax: 915-747-5019
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