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Continuum Mechanics Notation

Words Symbolic Traditional Cartesian Cartesian Tensor Curvilinear Tensor

Defining a point in Euclidean space relative to an origin

r

r = xi + yj + zk

r = xie i the summation

i=1

3

! is implied

r = x x1, x2 , x3( )i+ y x1, x2, x3( )j

+ z x1, x2, x3( )k

Base vectors Not applicable

i • i = 1j• j = 1k • k = 1

and

i • j = 0j• k = 0k • i = 0

e i • e j = !ij The Kronecker delta,

!ij = 1 if

i = j and

!ij = 0 if

i ! j .

g i = !r!x i = r, i

g i • g j = !ij

g i • g j = gij

g i • g j = g ij

The Kronecker delta,

!ij = 1 if

i = j and

! ji = 0 if

i ! j .

g i are the covariant base vectors.

g i are the contravariant base vectors. The curvilinear coordinates,

x i always have superscripts.

Scalar - 0th order tensor

a ,

!

a ,

!

a ,

!

a ,

!

Element of displacement

ds = dr = dr • dr

dr = dxi + dyj + dzk

ds = dx2 + dy2 + dz2

dr = dxie i

ds = dxidxi

dr = dx ig i

ds = gijdx idx j

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Vector – 1st order tensor

v

v = vxi + v yj + vzk

v = vie i

v = v ig i = vigi the summation

i=1

3

! is implied.

vi are the covariant components of

v .

v i are the contravariant components of

v .

vi = gijvj

v i = g ijv j

Magnitude of a vector

v

v = vx2 + v y

2 + vz2

v = vivi

v = vivi

= gijviv j = g ijviv j

Scalar product – dot product of two vectors

a • b = b • a = a b cos!a •b = a b cos! where

! is the angle between

a and

b. a + b( )• c= a • c + b • c

a •b = axbx + ayby + azbz

a • b = aibi

a • b = aibi

= gijaib j = g ijaibj

Vector product of two vectors

c = a ! b = b • " •a

! is the permutation pseudotensor

c is perpendicular to both

a and

b and has magnitude equal to

a b sin! where

! is the angle from

a

cx = aybz ! azby

cy = azbx ! axbz

cz = axby ! aybx

ck = !ijk aibj

!123 etc. = 1!321 etc. = "1all other!ijk = 0

ck = !ijk a ib j

ck = ! ijk aibj

g =g11 g12 g13

g21 g22 g23

g31 g32 g33

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to

b looking along

c.

!123 etc. = g

!321 etc. = " gall other!ijk = 0

!123 etc. = 1g

!321 etc. = " 1g

all other

! ijk = 0

Tensor product

ab( ) • c = a b • c( )c • ab( ) = c • a( )b

Sometimes written

a ! b( ) • c = a b • c( )c • a ! b( ) = c • a( )b

ab( ) • c =

axi + a yj + azk( ) bxcx + bycy + bzcz( )

and

c • ab( )= cxax + cyay + czaz( ) bxi + byj + bzk( )

ab( ) • c = aibjcje i

c • ab( ) = ciaibje j

ab( ) • c = aibjcjg i

= aibjcjg i etc.

c • ab( ) = ciaibje j etc.

Second order tensor

Q

Q = Qxxii + Qxyij + Qxzik

+Qyxji + Qyyjj + Qyzjk

+Qyxki + Qzykj + Qzzkk

Q = Qije ie j

Q = Qijg ig j = Qi. jg ig j

= Q . ji g ig

j = Qijgig j

Qij = gikQ . jk

Q . ji = g ikQkj = g jkQ

ik etc.

Trace of a second order tensor

tr Q( )

tr Q( ) = Qxx + Qyy + Qzz

tr Q( ) = Qii

tr Q( ) = Qijgij = Qi. j! j

i = Qi.i

= Q .ii = Qijg

ij

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Symmetric second order tensor

!T = ! The superscript

T means transpose.

! xy = ! yx etc.

!ij = ! ji

! ij = ! ij

! ij = ! ji

! . ji = ! j

.i = ! ji

Antisymmetric second order tensor

!T = "!

!xy = "!yx etc.

!xx = 0 etc.

!ij = "!ji

! ij = "! ij

!ij = !ji

! . ji = "!j

.i

Unit second order tensor

I

I = ii + jj + kk

I = !ije ie j

I = gijgig j

= g ijg ig j = !ijg ig j

Dot product of vector and 2nd order tensor

f = a • !

f x = ax! x + ay" yx + az" zx

etc.

f j = ai!ij

f j = ai!ij = ai! i

. j

f j = ai! ij

Gradient (grad) of a scalar

grad !( ) = "!

!" = !r •#"

grad !( ) = "! = #!#x

i + #!#y

j + #!#z

k

!" = !x #"#x

+ !y #"#y

+ !z#"#z

grad !( ) = "! = #i!e i

!" = !xi#i"

grad !( ) = "! = #i!g i

!" = !x i#i"

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Gradient (grad) of a vector

grad v( ) = !v

!v = !r •"v

!v = "vx

"xii +

"v y

"xij + .....

!vx = !x "vx

"x+ !y "vx

"y+ !z"vx

"zetc.

!v = "iv je ie j

!v = !xi"iv je j

!v = g i "v"x i = g i

" v jg j( )"x i

= g ig j"v j

"x i + g iv j "g j

"x i

= g ig j"v j

"x i + g iv j "g j

"x i • g kg k

= g ig j"v j

"x i + g iv j#ijkg k

= ! ivjg ig j

= ! iv jgig j

! ivj = "v j

"x i + vk#ikj

! iv j ="v j

"x i $ vk#ijk

are the

covariant derivatives.

!ijk =

g j

"x i • g k = "2r"x i"x j • g k are

the Christoffel symbols.

!v = !x i" iv jgj

Divergence (div) of a vector

! • v

! • v = "vx

"x+"v y

"y+ "vz

"z

! • v = "ivi

! • v = ! ivi

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Curl of a vector

! " v

! " v = #vz

#y$#v y

#z

%

& '

(

) * i

+ #vx

#z$ #vz

#x

%

& '

(

) * j +

#v y

#x$ #vx

#y

%

& '

(

) * k

! " v = #ijk$iv jek

! " v = # ijk! iv jg k

Strain rate (or small strain)

! = !( )T=

12

"v + "v( )T# $ % &

' (

!x = "vx"x

etc.

! xy = 12

"vy

"x+ "vx

"y

#

$ % %

&

' ( ( etc. Engineering

shear strain is

! xy ="vy

"x+ "vx

"y

#

$ % %

&

' ( ( etc.

Try to avoid.

! = ! ije ie j

! ij = ! ji =

12"iv j + " jvi( )

! = ! ijgig j

! ij = ! ji

12" iv j + " jvi( )

Vorticity or angular velocity tensor (if the

12

is included)

! =12

"v# "v( )T$ % & '

( )

0 ! z "! y

"! z 0 !x

! y "!x 0

#

$

% % %

&

'

( ( ( Either

! z = 12

"v y

"x# "vx

"y

$

% &

'

( ) or

! z ="v y

"x# "vx

"y

$

% &

'

( ) . Be careful.

! = !ije ie j

!ij = "!ji =

12#iv j "# jvi( )

! = !ijgig j

!ij = "!ji =

12# iv j "# jvi( )

Rate of change as seen by observer moving at velocity

v

DDt

= !!t

+ v •"

DDt

= !!t

+ vx!!x

+ vy!!y

+ vz!!z

DDt

=

!!t

+ vi!i

DDt

=

!!t

+ v i" i

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Acceleration

a = DvDt

= !v!t

+ v •"v

ax = !vx!t

+ vx!vx!x

+vy!vx!y

+ vz!vx!z

etc.

ai =!vi

!t+ v j! jvi

ai =!vi

!t+ v j" jvi

Conservation of mass

!"!t

+ # • "v( ) = 0

!"!t

+ !!x

"vx( ) + !!y

"vy( )

+ !!z

"vz( ) = 0

!"!t

+ !i "vi( ) = 0

!"!t

+ # i "v i( ) = 0

Conservation of momentum

! DvDt

= " • #

! "vx

"t+ vx

"vx

"x+ v y

"vx

"y+ vz

"vx

"z

#

$ %

&

' (

= ") x

"x+"* yx

"y+ "* zx

"z

etc.

!"v j

"t+ vi"iv j

#

$ %

&

' ( = "i) ij

! "v j

"t+ vi# iv

j$

% &

'

( )

= # i*ij

Laplacian or harmonic operator

!2( ) = ! • !( )[ ]

!2

!x2+ !2

!y2+ !2

!z2

"

# $ $

%

& ' ' ( )

!i!i ( )

g ij! i! j ( )

Biharmonic operator

!4( )= !2 !2( )[ ]

!2

!x2+ !2

!y2+ !2

!z2

"

# $ $

%

& ' '

2

( )

!i!i!i!i ( )

g ijg mn! i! j!m!n ( )

Isotropic elastic material

! is (small) strain.

! =" 1+ #( ) $#tr "( )I

E

!x = 1E

" x #$ " y + " z( )( )= 1

E1+ $( )" x #$ " x + " y + " z( )( )

% xy =1+ $( )& xy

E

! ij =" ij 1+ #( ) $#" kk%ij

E

! ij =" ij 1+ #( ) $#" k

k gij

E

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! xy =2 1+ "( )# xy

E=# xy

G Engineering

shear strain

E is Young’s modulus

! is Poisson’s ratio

G is the Shear modulus

Viscous fluid

! = "pI + 2µ# + $tr #( )I

p is the pressure

µ is the dynamic viscosity.

! is the second coefficient of viscosity.

! ij = " p#ij + 2µ$ ij + %tr $( )#ij

& x = "p + 2µ'x + % 'x + ' y + 'z( )etc.! xy = 2µ$ xy

etc.

! ij = " p#ij + 2µ$ ij + %$ kk#ij

! ij = " pg ij + 2µ# ij + $# kk g ij