Common Variable Types in Elasticity Elasticity theory is a mathematical model of material deformation. Using principles of continuum mechanics, it is formulated in terms of many different types of field variables specified at spatial points in the body under study. Some examples include: Scalars - Single magnitude mass density , temperature T, modulus of elasticity E, . . . Vectors – Three components in three dimensions displacement vector Matrices – Nine components in three dimensions stress matrix Other – Variables with more than nine components Chapter 1 Mathematical Preliminaries , e 1 , e 2 , e 3 are unit basis vectors sticity Theory, Applications and Numerics Sadd , University of Rhode Island 3 2 1 e e e u w v u z zy zx yz y yx xz xy x ] [
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Common Variable Types in ElasticityElasticity theory is a mathematical model of material deformation. Using principles of continuum mechanics, it is formulated in terms of many different types of field variables specified at spatial points in the body under study. Some examples include:
Scalars - Single magnitude mass density , temperature T, modulus of elasticity E, . . .
Vectors – Three components in three dimensions displacement vector
Matrices – Nine components in three dimensions stress matrix
Other – Variables with more than nine components
Chapter 1 Mathematical Preliminaries
321 eeeu wvu
zzyzx
yzyyx
xzxyx
][
, e1, e2, e3 are unit basis vectors
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Index/Tensor NotationWith the wide variety of variables, elasticity formulation makes use of a tensor formalism using index notation. This enables efficient representation of all variables and governing equations using a single standardized method.
Index notation is a shorthand scheme whereby a whole set of numbers or components can be represented by a single symbol with subscripts
333231
232221
131211
3
2
1
,aaaaaaaaa
aaaa
a iji
In general a symbol aij…k with N distinct indices represents 3N distinct numbersAddition, subtraction, multiplication and equality of index symbols are defined in the normal fashion; e.g.
333332323131
232322222121
131312121111
33
22
11
,bababababababababa
babababa
ba ijijii
333231
232221
131211
3
2
1
,aaaaaaaaa
aaaa
a iji
332313
322212
312111
bababababababababa
ba ji
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Notation Rules and DefinitionsSummation Convention - if a subscript appears twice in the same term, summation over that subscript from one to three is implied; for example
332211
3
1
332211
3
1
bababababa
aaaaa
iiij
jijjij
iiiii
A symbol aij…m…n…k is said to be symmetric with respect to index pair mn if
kmnijknmij aa ..................
A symbol aij…m…n…k is said to be antisymmetric with respect to index pair mn if
kmnijknmij aa ..................
Useful Identity ][)()(21)(
21
ijijjiijjiijij aaaaaaa
)(21
)( jiijij aaa )(21
][ jiijij aaa . . . symmetric . . . antisymmetric
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Useful in evaluating determinants and vector cross-products 321321
333231
232221
131211
||]det[ kjiijkkjiijkijij aaaaaaaaaaaaaaa
aa
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Coordinate Transformations
v
e
1
e
3e
2
e3
e2e1
x3
x1
x2
x1
x2
x3 To express elasticity variables in different coordinate systems requires development of transformation rules for scalar, vector, matrix and higher order variables – a concept connected with basic definitions of tensor variables. The two Cartesian frames (x1,x2,x3) and differ only by orientation),,( 321 xxx
),cos( jiij xxQ Using Rotation Matrix
3332321313
3232221212
3132121111
eeeeeeee
eeee
QQQQQQQQQ
jiji Q ee
jjii Q ee
ii
ii
vvvvvvvv
eeeeeeeev
332211
332211 jiji vQv
jjii vQv
transformation laws for Cartesian vector components
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Cartesian TensorsGeneral Transformation Laws
Scalars, vectors, matrices, and higher order quantities can be represented by an index notational scheme, and thus all quantities may then be referred to as tensors of different orders. The transformation properties of a vector can be used to establish the general transformation properties of these tensors. Restricting the transformations to those only between Cartesian coordinate systems, the general set of transformation relations for various orders are:
order general
order fourth,
orderthird,
(matrix)ordersecond,
(vector)orderfirst,(scalar)orderzero,
...... tpqrmtkrjqipmijk
pqrslskrjqipijkl
pqrkrjqipijk
pqjqipij
pipi
aQQQQa
aQQQQa
aQQQa
aQQa
aQaaa
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example 1-2 Transformation ExamplesThe components of a first and second order tensor in a particular coordinate frame are given by
423220301
,241
iji aa
Determine the components of each tensor in a new coordinate system found through a rotation of 60o (/6 radians) about the x3-axis. Choose a counterclockwise rotation when viewing down the negative x3-axis, see Figure 1-2.
The original and primed coordinate systems are shown in Figure 1-2. The solution starts by determining the rotation matrix for this case
10002/12/302/32/1
0cos90cos90cos90cos60cos150cos90cos30cos60cos
ijQ
The transformation for the vector quantity follows from equation (1.5.1)2
22/32322/1
241
10002/12/302/32/1
jiji aQa
and the second order tensor (matrix) transforms according to (1.5.1)3
42/33132/32/3314/54/332/34/34/7
10002/12/302/32/1
423220301
10002/12/302/32/1
T
pqjqipij aQQa
x3
x1
x2
x1
x2
x3
60o
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Principal Values and Directions for Symmetric Second Order Tensors
ijij nna
The direction determined by unit vector n is said to be a principal direction or eigenvector of the symmetric second order tensor aij if there exists a parameter (principal value or eigenvalue) such that
0)( jijij na
Relation is a homogeneous system of three linear algebraic equations in the unknowns n1, n2, n3. The system possesses nontrivial solution if and only if determinant of coefficient matrix vanishes
0]det[ 23 aaaijij IIIIIIa
]det[
)(21
3331
1311
3332
2322
2221
1211
332211
ija
ijijjjiia
iia
aIIIaaaa
aaaa
aaaa
aaaaII
aaaaI
scalars Ia, IIa and IIIa are called the fundamental invariants of the tensor aij
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Principal Axes of Second Order Tensors
3
2
1
000000
ija
It is always possible to identify a right-handed Cartesian coordinate system such that each axes lie along principal directions of any given symmetric second order tensor. Such axes are called the principal axes of the tensor, and the basis vectors are the principal directions {n(1), n(2) , n(3)}
333231
232221
131211
aaaaaaaaa
aij
x3
x1
x2
Principal Axes Original Given Axes
n(1)
x1
x2
x3
n(3) n(2)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example 1-3 Principal Value Problem
Determine the invariants, and principal values and directions of
First determine the principal invariants
340430002
ija
50)169(2340
430002
25625630
0234
433002
,2332
a
aiia
III
IIaI
The characteristic equation then becomes
5,2,5
0)25)(2(050252]det[
321
223
ijija
Thus for this case all principal values are distinct For the 1 = 5 root, equation (1.6.1) gives the system
084
042
03
)1(3
)1(2
)1(3
)1(2
)1(1
nn
nn
n
which gives a normalized solution )2(5
132
)1( een
In similar fashion the other two principal directions are found to be 1)2( en )2(
51
32)3( een
It is easily verified that these directions are mutually orthogonal. Note for this case, the transformation matrix Qij defined by (1.4.1) becomes
5/25/10001
5/15/20
ijQ
500020005
ija
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Vector, Matrix and Tensor Algebra
iibabababa 332211ba
ikjijk babbbaaa eeee
ba
321
321
321
iijijiT
ijjjij
aAAa
AaaA
][}{
}]{[
AaAa
aAAaT
ijij
jiij
jkji
kjij
jkij
BAtrtr
BAtr
BA
BA
BA
)()(
)(
]][[
BAAB
AB
BA
AB
BAAB
TT
T
T
Scalar or Dot Product
Vector or Cross Product
Common Matrix Products
TQaQa
pqjqipij aQQaLaw tionTransforma
OrderSecond
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Calculus of Cartesian TensorsField concept for tensor components
)()(),,()()(),,(
)()(),,(
321
321
321
xx
x
ijiijijij
iiiii
i
axaxxxaaaxaxxxaa
axaxxxaa
Comma notation for partial differentiation ,,, ,,, ijk
kijij
jii
i ax
aax
aax
a
3
3
2
3
1
3
3
2
2
2
1
2
3
1
2
1
1
1
,
xa
xa
xa
xa
xa
xa
xa
xa
xa
a ji
If differentiation index is distinct, order of the tensor will be increased by one; e.g. derivative operation on a vector produces a second order tensor or matrix
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Vector Differential Operations
Directional Derivative of Scalar Field fdsdz
zf
dsdy
yf
dsdx
xf
dsdf
n
321 eeendsdz
dsdy
dsdxs of direction in vector normal unit
zyx
321 eee operator aldifferenti vector
zf
yf
xffff
321 eeegrad function scalar of gradient
Common Differential Operations
i,2
i,
,
,2
ji,
i
eu
eu
u
eeue,
kki
jkijk
ii
ii
ji
i
uVectoraofLaplacian
uVectoraofCurl
uVectoraofDivergence
ScalaraofLaplacian
uVectoraofGradientScalaraofGradient
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example 1-4: Scalar/Vector Field ExampleScalar and vector field functions are given by 22 yx 321 eeeu xyyzx 32
Calculate the following expressions, , 2, ∙ u, u, u.
Using the basic relations:
21
321
21
eeeee
ee
yyxxyyzxzyx
xyyzu
zz
yx
ji
)3(32
///
033000232032
02222
,
2
∙ u
u
u
Gradient Vector Distribution
-10 -5 0 5 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Contours =constant and vector distributions of vector field is orthogonal to -contours (ture in general )
x
y
- (satisfies Laplace equation)
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Vector/Tensor Integral Calculus
Divergence Theorem
Stokes Theorem
Green’s Theorem in the Plane
Zero-Value Theorem
dVdSS V u nu dVadSna
S V kkijkkij ,......
S
dS nudruC
)( S rskijrsttkij dSnadxa
C ,......
VfdVf kijV kij 00 ......
CSgdyfdxdxdy
yf
xg )(
C ySC xSdsfndxdy
yfdsgndxdy
xg ,
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Orthogonal Curvilinear Coordinate Systems
Cylindrical Coordinate System (r,,z) Spherical Coordinate System (R,,)
e3
e2 e1
x3
x1
x2
r
z
re
zee
,tan
cos
cos,sinsin,sincos
1
21
23
22
21
31
23
22
21
321
xx
xxx
x
xxxR
RxRxRx
31
2122
21
321
,tan,
,sin,cos
xzxxxxr
zxrxrx
e3
e2e1
x3
x1
x2
R
Re
e
e
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
General Curvilinear Coordinate Systems
233
222
211
2 )()()()( dhdhdhds
iii hhhh
1ˆ1ˆ1ˆ1ˆ3
332
221
11 ieeee
iii
fh
fh
fh
fh
f
1ˆ1ˆ1ˆ1ˆ3
332
221
11 ieeee
i
ii
i uhhhh
hhh321
321
1u
i
ii
i hhhh
hhh 2321
321
2
)(1
i
ij k
kkjkj
ijk huhh
eu ˆ)(
i jji
j
i
uu
h ij
ji
ee
eu
ˆˆ
ˆ
j kjk
j
kii
i
uu
hh kj
jki e
eee
uˆ
ˆˆˆ2
),,(,),,( 321321 mmmm xxxxx
e3
e2 e1
x3
x1
x2
1
1e
3e
2e
3
2
Common Differential Forms
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example 1-5: Polar Coordinates
rhhrddrds 21222 ,1)()()(
From relations (1.9.5) or simply using the geometry shown in Figure
21
21
cossinˆsincosˆ
eeeeee
r0
ˆˆ,ˆ
ˆ,ˆ
ˆ
rrr
rr ee
ee
ee
The basic vector differential operations then follow to be
eeu
eeeeeeeeu
eu
u
ee
ee
r
rrrr
2
ˆ2ˆ2
ˆˆ1ˆˆ1ˆˆˆˆ
ˆ1)(1
11
1)(1
1ˆˆ
1ˆˆ
222
2222
2
2
2
ruu
ru
ruu
ru
uu
ru
urr
uru
ur
rurr
rrr
rr
ur
rurr
rr
rr
rrr
rrr
zr
r
r
r
θrzθr eeeeeu ˆˆˆ,ˆˆ where uur
e2
e1 x1
x2
r
e
re
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island