Topic 3
Lagrangian
Continuum
Mechanics
Variables for
General Nonlinear
Analysis
Contents:
• The principle of virtual work in terms of the 2nd PiolaKirchhoff stress and Green-Lagrange strain tensors
• Deformation gradient tensor
• Physical interpretation of the deformation gradient
• Change of mass density
• Polar decomposition of deformation gradient
• Green-Lagrange strain tensor
• Second Piola-Kirchhoff stress tensor
• Important properties of the Green-Lagrange strain and
2nd Piola-Kirchhoff stress tensors
• Physical explanations of continuum mechanics variables
• Examples demonstrating the properties of the continuum
mechanics variables
Textbook:
Sections 6.2.1, 6.2.2
Examples:
6.5,6.6,6.7,6.8,6.10,6.11,6.12,6.13,6.14
Topic Three 3-3
CONTINUUM MECHANICS
FORMULATION
Transparency
3-1
For
Large displacements
Large rotations
Large strains
Hence we consider a body subjected to
arbitrary large motions,
We use a Lagrangian description.
PC+~'X1, t+~'X2, '+~'X3)
PCX1, 'X2, 'X3)
Transparency
3-2
Configuration
at time 0
Confi~uration
Configuration
at time t + ~t
I
at time t
'Xi = °Xi
t+~'x·I
= ox·
I
+ lUi
X1
+ t+~'u· i = 1 2 3
· -- t+~'u·I - 'u·I
U1
I
"
3-4 Lagrangian Continuum Mechanics Variables
Transparency
3-3
Regarding the notation we need to
keep firmly in mind that
- the Cartesian axes are stationary.
- the unit distances along the Xi-axes
are the same for °Xi, tXi , t+ ~tXi.
Example:
particle at time 0
0X1
/particle at time t
/U1
1----'---·.·
2
1
...
3
4
5
PRINCIPLE OF VIRTUAL
WORK
Transparency
3-4
Corresponding to time t+dt:
I
t+.lt
t+~tlT'lit.. Ut+~t
~
e··It t+~tdV - ;
t+~t(fll
n
v
where
t+~tffi
=
+
t+~tfF OUi t+~tdV
r
)t+.lt
v
r
)t+.lts
t+~tfr OUr t+~tdS
Topic Three 3-5
Transparency
3·5
and
t+Atl'T"..
I
'I'
Cauchy stresses (forces/unit
area at time t + Llt)
1 ( aOUi
ot+Atei} = 2 at+Atx}
aou} )
+ at+Atxi
variation in the small strains
referred to the configuration
at time t + Llt
We need to rewrite the principle of
virtual work, using new stress and
strain measures:
• We cannot integrate over an
unknown volume.
• We cannot directly work with
increments in the Cauchy stresses.
We introduce:
cis = 2nd
6E
=
Piola-Kirchhoff stress tensor
Green-Lagrange strain tensor
Transparency
3·6
3-6 Lagrangian Continuum Mechanics Variables
Transparency
The 2nd Piola-Kirchhoff stress tensor:
3-7
a
t8
a i}
Ot
a
= -P
t tXi,m T mn tX},n
P
The Green-Lagrange strain tensor:
1
t
(t
OE"yo = -2 aU·"
I,t
t
t
t
)
+ au..
~I + aUk"
,I aUk"
,t
where
Transparency
3-8
Note: We are using the indicial notation
with the summation convention.
For example,
a
t
08 11
t
a
= -Pro
t tX1,1 Tn tX1,1
P
+ atX 1 ,1 tT 12 atX 1 ,2
+ ...
+ ~X1,3 tT33 ~X1,3]
Topic Three 3·7
Using the 2nd Piola-Kirchhoff stress
and Green-Lagrange strain tensors,
we have
Transparency
3-9
This relation holds for all times
at, 2at, ..., t, t+ at, ...
To develop the incremental finite
element equations we will use
~vt+~JSt 8t+~JEt °dV = t+~~
• We now integrate over a known
volume, °V.
• We can incrementally decompose t+~JSt
t+~t
.
an d
oEt, I.e.
o ~=ots iJ-+OSiJt+~t
t
OEiJ- = oE~ + oE~
t+~ts
Transparency
3-10
3-8 Lagrangian Continuum Mechanics Variables
Transparency
3-11
Before developing the incremental con­
tinuum mechanics and finite element
equations, we want to discuss
• some important kinematic
relationships used in geometric
nonlinear analysis
• some properties of the 2nd
Piola-Kirchhoff stress and Green­
Lagrange strain tensors
Transparency
3-12
To explain some important properties of
the 2nd Piola-Kirchhoff stress tensor
and the Green-Lagrange strain tensor,
we consider the
Deformation Gradient Tensor
• This tensor captures the straining and the
rigid body rotations of the material fibers.
• It is a very fundamental quantity used in
continuum mechanics.
Topic Three 3-9
The deformation gradient is defined as
atx1 atx1 atx1
aOx1
atx 2
aOX2
aOX1
aOX2
Transparency
3-13
aOXa
in a Cartesian
coordinate
aOXa system
atX2 atx2
atXa atXa
aO X1 aOX2
atXa
aOXa
Using indicial notation,
Another way to write the deformation
gradient:
Jx = (oVJt~T)T
where
oV
the/
gradient
operator
=
Transparency
3-14
3-10 Lagrangian Continuum Mechanics Variables
Transparency
3-15
The deformation gradient describes the
deformations (rotations and stretches)
of material fibers:
The vectors dOx and
dt~ represent the
orientation and length
of a material fiber at
times 0 and t. They
are related by
dtx
JX dOx
Example: One-dimensional deformation
Transparency
3-16
time 0
time t
X1/ /'
11
H
- ------al.
. . - -._
.
1.0
0.5
...I....
1
Topic Three 3-11
Consider a material particle initially at
X1 = 0.8:
Transparency
3-17
f------X1
°X1
1<1
= 0.800
= 1.120
Consider an adjacent material particle:
I
•I
Compute Jx 11 :
d tx 1 1.211 - 1.120
d OX1 = .850 - .800
JX 11 lo
x1=O.8
= 1.82 ~ Estimate
= 1.80
Transparency
3-18
3-12 Lagrangian Continuum Mechanics Variables
Transparency
3-19
Exam~:
Two-dimensional deformation
time t
f
)
( 0 X1, 0
X2 ~
Transparency
3-20
(tX1. tX2 ).·0_
tx
.481
= [ -.385
.667]
.667
Considering dO~,
X2
X1
[
dt~
=
.75]
= [
o
hx
.481
- .385
dO~
.667][.866]
.667 .5
'Ibpic Three 3-13
Considering cf!.
Transparency
X2
3-21
dtx_ =
tx
0_
cfx_
.667][ 0]
[1]1 [.481
-.385 .667 1.5
=
The mass densittes 0 p and t p may be
related using the deformation gradient:
infinitesimal volumes
time 0
time t
//:
::;;:;Ls-
~
tdV
~dt!l
Three material fibers describe each volume.
Transparency
3-22
3-14 Lagrangian Continuum Mechanics Variables
Transparency
3-23
For an infinitesimal volume, we note
that mass is conserved:
tp tdV = 0p 0dV
~~ volume
volume at .----:--time t~
at
- ~time 0
However, we can show that
Hence
Transparency
3-24
Proof that tdV
= det Jx °dV:
dO~1 =[g}S1 ; dO~ =[!}S2
dO~=[~}S3
° - = dS
Hence dV
1
dS2 ds3 •
Topic Three 3-15
Transparency
3-25
and tdV = (dt~1
X dt~2)
. dt~3
det Jx dS 1 dS2 dS3
=
= det Jx °dV
Example: One-dimensional stretching
Transparency
~timeO
x:r ~time t
1
/
,/,
/
//
uniform stretching
plane strain conditions
1.0
Deformation field:
.25
'x,
=
ox, +
O.250x,
0 001
-+ det J~ = 1.25
Deformation
[1.25
gradient: J~ = 0 1
o
Hence 0p
3-26
0
= 1.25tp (tp <
1
0p makes physical sense)
'""""""-----------------",
3-16 Lagrangian Continuum Mechanics Variables
Transparency
3-27
We also use the inverse deformation
gradient:
o .
;;{d ~
=
MATERIAL FIBER
AT TIME 0
Mathematically,
Proof:
0
t
tX d~~
MATERIAL FIBER
AT TIME t
~X = (JX)-1
dO~ = ~
(Jx dO~)
= (~X
=
Transparency
JX) dO~
I dOx
An important point is:
3-28
Polar decomposition of JX:
JR = orthogonal (rotation) matrix
Ju =
symmetric (stretch) matrix
We can always decompose JX in the
above form.
Topic Three 3-17
Example: Uniform stretch and rotation
timet
3-29
~
I:
~
Transparency
3.0
4.0
=
I
I
X1
dU
dR
°
0]
1.154 -0.750] .. [0.866 -0.500] [1.333
[ 0.887
1.299
0.500 0.888
1.500
Using the deformation gradient, we can
describe the (right) Cauchy-Green
deformation tensor
tc - 0_
txT 0_
tx
0_ -
This tensor depends only on the stretch
tensor riU:
T tAT) (tA tU)
tc = (tu
0_ 0_
0_0_
0_
= (riU)2 (since riA is orthogonal)
Hence ric is invariant under a rigid
body rotation.
Transparency
3-30
3·18 Lagrangian Continuum Mechanics Variables
Transparency
Example: Two-dimensional motion
0
3-31
X2
~
time'O
timel+::;U
D
~
time t
rigid body motion,
rotation of 90°
X1
Transparency
3-32
Jx
=
-
[1.5
.5
.~]
HatX = [ -.5
-1 ]
.2
Jc
=
-
[2.5
.8
.8 ]
1.04
HatC _ [2.5
.8 ]
1.04
0_
0_ -
1.5
.8
The Green-Lagrange strain ·tensor
measures the stretching deformations. It
can be written in several equivalent
forms:
1)
JE
=
~ (ric -
I)
From this,
• JE
is symmetric.
• For a rigid body motion between
times t and t+ ~t, H~E = JE .
• For a rigid body motion between
times 0 and t, JE = Q.
Topic Three 3-19
• ~~ is symmetric because ~C is
symmetric
Transparency
3-33
~~ = ~ (~C -1)
• For a rigid body motion from t to
t+~t, we have
t+~tx = R tv
0
0-
t+~tc
0_
=
-
0
to
0
j..
."
t+4t E
0-
= o~t
• For a rigid body motion
~C = 1 =* ~~ = 0
t
_
1 (t
t
2) aEi} - 2 ,aUi,}:- aU}.~
UNEAR IN
DISPLACEMENTS
t
t
)
+, aUk,i
,aUk,} .
NONUNEAR IN
DISPLACEMENTS
where
Important point: This strain tensor is exact and
holds for any amount of
stretching.
Transparency
3-34
3·20 Lagrangian Continuum Mechanics Variables
Transparency
Example: Uniaxial stJlain
(t )2
t
3-35
A 1 A
0[ + 2 0[
J£11 =
1
engi:~ring
87
1.o+----+-+-
t~
-~1-.0--~~--1+-.0°L
Example: Biaxial straining and rotation
Transparency
3-36
rigid body motion,
rotation of 45°
X2
O
~
time 0;7'
I>------l ~
time t?
time t+~t~
+---------X1
IX
=
0_
[1.50.50]
ric
= [2.25
riE
=
-
-
0
[.625
0
l+Jllx
0_
=
[1.06
1.06
-.354]
.354
0]
.25
IHric =
[2.25
0
.~5]
0 ]
-.375
IHriE
[.625
0
-.3~5]
-
-
=
Topic Three 3-21
Example: Simple shear
Transparency
I'
t~
'I"
1.0
3-37
"
X1
For small displacements, dE is
approximately equal to the small strain
tensor.
The 2nd Piola-Kirchhoff stress tensor
and the Green-Lagrange strain tensor
are energetically conjugate:
t'Ti.j- ~hei.j- = Virtual work at time t per unit
current volume
Js~ ()JE~ = Virtual work at time t per unit
original volume
where dSij is the 2nd Piola-Kirchhoff
stress tensor.
Transparency
3-38
3-22 Lagrangian Continuum Mechanics Variables
Transparency
3-39
The 2nd Piola-Kirchhoff stress tensor:
0p o
t
0
tXi,m 'Tmn tX!-n
P
t
OSi~
=-t
o
ts
0_
= ---.e
t
p
0tX t'T °tXT
-
-
INDICIAL NOTATION
MATRIX NOTATION
Solving for the Cauchy stresses gives
t
_
'Ti~ -
t
opP
t
ts
t
OXi,m 0 mn oXj.n -
INDICIAL NOTATION
t
t'T =
Transparency
3-40
opP
tx 0_
ts 0_
txT 0_
MATRIX NOTATION
Properties of the 2nd Piola-Kirchhoff stress
tensor:
• cis
is symmetric.
is invariant under a rigid-body
motion (translation and/or rotation).
• cis
Hence cis changes only when the
material is deformed.
has no direct physical
interpretation.
• cis
Topic Three 3-23
Example: Two-dimensional motion
Transparency
3-41
Xi
lim\Or~. '__--. ~
.....
Cauchy stresses
at time t
tx =
t
1: =
ts _
0_ -
[10
t+~tx =
.2]
1.5
0_
1000]
[0
1000 2000
[
Transparency
At time t + Llt,
At time t,
0_
rigid body
motion, rotation
of 60°
Cauchy stresses
at time t+ dt
-346
733
733 ]
1330
[.5.866
t+ ~t'T = [
-
t+~ts
0_
= [
634
-1370
-346
733
3-42
-1.20 ]
.923
-1370]
1370
733]
1330
MIT OpenCourseWare
http://ocw.mit.edu
Resource: Finite Element Procedures for Solids and Structures
Klaus-Jürgen Bathe
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