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1. Preliminaries and terminology
1.1. Line integral and its geometrical meaning
Fig. 1.
The planar curve k is described by the function y  g (x) , or by its inverse x  g ( y ) .
I k   f ( x, y ) ds .
See Fig. 1.
k
b
I x   f ( x, y) dx   f ( x, g ( x)) dx .
k
Projection into (xz) plane.
a
d
I y   f ( x, y) dy   f ( g ( y), y) dy .
k
Projection into ( yz ) plane.
c
Example 1. Show relations between the above integrals
Fig. 2.
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Let the function f ( x, y ) is defined as follows
z  f ( x, y )  ax  by  c .
From the ‘boundary’ conditions we get
A: 0  a  0  c
B: 00bc
 c  1; b  1; a  1 .
C : 1 0 0 c
Projection I k to ( x,z ) : I x   f ( x , y ) dx .
k
I x  I x
see Fig. 2
F ( x, y )  f ( x, g ( x))   x  x  1  2 x  1,
b
I x   f ( x, y) dx   f ( x, g ( x)) dx  I x .
k
a
Example 2. Calculate the integrals for the following functions.
f ( x, y )  1  ( x 2  y 2 ) ,
g ( x)  x ,
F ( x)  1  2 x 2 .
Fig. 3
1.2. Gradient
Consider a two-dimensional quantity  constituting a scalar field depending on x, y
   ( x, y ) .
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The total differential shows how  differs with position
d 


dx 
dy .
x
y
Let's introduce two column vectors as
  


dx
    x  , dx    .
dy 


 y 
The former vector is called the gradient of a scalar field  ( x, y ) . Then, the total
differential can be rewritten in the form of a scalar product
d    dx,
T
for which we can write
d    dx cos .
The gradient is orthogonal to contour lines since if   const . , then d  0 and
  dx cos  0
only if cos   0 , which we have for  

2
.
Fig. 4
1.3 Gauss Divergence Theorem
For functions    ( x , y ),    ( x , y ) defined over the area A, consider the
integrals
  x , y 
dA ,
A
x

 x , y 
dA .
A
y

Take the second one first in the counterclockwise direction along both boundaries.
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

A y dA   A y dxdy 
b f 2( x )
  
f (x)
a f ( x )  y dy  dx  a  x, y  f12( x ) dx 
1
b
b
  x, f x   x, f x dx =
2
1
a
b
b
b
b
a
a
a
a
   x , f 2 x  dx    x , f1 x  dx   x , f 2 x  dx    x , f1 x  dx
Fig. 5
We can conclude that
 x, y 
dA   x, y  dx .
y
A
c

(1.1)
Let the curve c is defined by two functions f 2 and f1 respectively. Similarly for the other
coordinate and for the other function   x, y  we get
 x , y 
dA    x , y  dy .

x
A
c

(1.2)
Again in the counterclockwise direction.
The equations (1.1), (1.2) form the Green theorem in the simplest case. Sometimes
also called the Green-Ostrogradsky theorem or the theorem of Green-Gauss.
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Fig. 6


Let us rewrite the equations (1.1), (1.2) by introducing vectors dc and n , in such a
way that
  nx 
n  ,
n y 

dc  dx2  dy 2 ,

n  nx2  ny2  1 .
 dx 

 dy 
dc  dx 1     dy 1   
 dx 
 dy 
2
2
if dx  0 or dy  0 .
Observing triangles in the previous figure one can write
sin  
 dx ny
   (due to counterclockwise orientation of the curve),
dc
n

dy n
cos    x , where n  1 .
dc
n
From it follows

dx  ny dc ,

dy  nx dc .
(1.3)
Substituting (1.3) into (1.1), (1.2), we get

 x , y 
dA   x , y  nx dc ,

x
A
c
(1.4)
 x , y 
dA   x , y  ny dc .
y
A
c
(1.5)

Now define an arbitrary vector q by
qx   
   .
q
y
   
q  
(1.6)
Adding (1.4), (1.5) and using (1.6) we get
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 q x
  x

A
q y 
 dA   qx nx  q y n y  dc .
y 
c
The divergence is defined

q
div q  div q  i .
xi
(1.7)
The scalar product can be expressed
qT n  qi ni .
So we can finally write
 div q dA   q ndc .
T
A
(1.8)
c
This is a so called divergence theorem of Gauss.
The Gauss-Green theorem can be seen as a two-dimensional counterpart of the
integration by parts
uv
 u v   u /v .
/
(1.9)
The Gauss divergence theorem could be found in literature in different forms.
 div q dV   q n dS .
T
V
(1.10)
S
The equivalent notations are as follows
  q dV   q n dS , 
T
T
V
S
T


 x1

x2
 
.
x3 
(1.11)
 div q dV   q . n dS ,
V
S
qi
 x
dV   qi ni dS ,
f
dV   f ni dS .
i
V
 x
V
i
(1.12)
S
S
The Gauss divergence theorem for a tensor quantity is defined as follows
Tij
 x
V
i
dV   ni Tij dS .
(1.13)
S
So the Green theorem represents the transformation of a volume integral into a surface
integral (or vice versa) for quantities associated with a considered body having the volume V ,
bounded by the surface S . The outward normal ni is defined at each point of the surface. The
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function appearing under the integral sign is real valued with the first continuous derivative
within the body.
1.4. The generalization of ‘per partes’ integration (integration by parts)
According to Green divergence theorem we can write

 x uv dV   uv n
i
i
V
dS .
(1.14)
S
The left hand side could be rewritten

u
 x uv dV   x
V
i
i
V
dV   u
V
v
dV .
xi
(1.15)
Equalling the right hand sides of the last two equations and rearranging gives a formula
v
 u x
i
V
u
v dV ,
xi
V
dV   uv  ni dS  
S
(1.16)
which reminds the integration by parts
b
b
a
a
b
 uv dx  u v a   u v dx .
(1.17)
It is of interest to remind the Stokes theorem which transforms the integral over the closed
curve in space to the surface integral
 
 n .   q dS   t . q dc ,

S

(1.18)
c
i

 
curl q  rot q 
x
qx
j

y
qy
k

.
z
qz
Fig. 7
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1.5. Flux

Imagine a surface S in space and the continuum flowing by the velocity v through it.
The volume of material flowing through dS in time dt is
m  ... m/s 1sm .
3
v . n dt dS ,
2
The volume flux is defined
 v .n dS   v n
i
S
i


dS , m3 / s .
(1.19)
S
 v n
Similarly the mass flux is
i
i
dS , kg / s .
(1.20)
S
Fig. 8
Kinetic energy flux corresponds to
1
2 v
2
1 2
mv , so
2
v . n dS ,
(1.21)
S
Dimensional check
m2 kg kgm 2 kgm 1
1 1 
 3  2 m  Nm    .
2
s s
s
s
s
s s
Generally
The flux of  through S is
 v n
i
i
dS ,
(1.22)
S
where  is a quantity (defined per unit mass) which is associated with particles.
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Remark
Often we encounter a so called oriented surface defined by


d  dS n
or
dAi  ni dS .
(1.23)
1.6. Material derivative
Imagine that the motion of a particle is defined by the material description in the form
x  xa,t  .
(1.24)
The particle velocity could simply be calculated by taking the partial time derivative with a
variable a held constant

 x 
v  va ,t      xa ,t  .
 t a t
(1.25)
Similarly, the acceleration is

2
 v 
z     va ,t   2 xa ,t   v .
t
 t a t
(1.26)
These are examples of material time derivatives in material description. Notice the
different types of notation.
The material time derivative may be thought of as a time rate of change that would be
masured by an observer travelling with the specific particle under study. The same physical
phenomenon could be described by the spatial description. The velocity fields is
v  vx ,t  .
(1.27)
Note
The studied phenomenon is supposed to be the same regardless of the formulation being
applied, so we are tempted to use the same symbol for the variable describing it, even if it is
defined by a different function. There are authors using different symbols for the same
variables in material and spatial descriptions respectively.
The derivative, with spatial coordinate x held constant

 v 
   vx ,t 
 t  x t
(1.28)
is called the local rate of change of v . It is the rate of change of an ideal velocity meter
located at the fixed place x . This is not the same thing as the acceleration of the particle
passing the place x just now.
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Remark
For example, in a steady state flow the local rate of change is zero everywhere. This
does not imply, however, that the acceleration of all particles is zero everywhere. Even in a
steady state flow the velocity varies in general from point to point and a particle changes its
velocity as it moves from one place of constant velocity to another place, having a different
constant velocity.
If we want to calculate the particle acceleration from knowledge of spatial velocity
description vx ,t  we have to employ the chain rule of calculus
 v   v   v   x 
       .
 t a  t  x  x   t a
(1.29)
 x 
Since    v we can finally write
 t a
 v   v 
 v 
z        v .
 t a  t  x  x 
(1.30)
Another name for the material derivative is the substantial derivative. There are other
notations used in textbooks and references, as
Dv dv
 v 

 v    .
Dt dt
 t a
(1.31)
In vector notation we can write
z
dv v
v

 v .v 
 v .grad v ,
dt t
t
(1.32)
T
  
where   grad  
 .
 x y z 
For scalar  , vector u or tensor T quantities, the formulas of material derivatives are as
follows
 
d 



 v . f 
 vk
,
dt
t
t
xk
u  du  u  v . u  u  v . grad u ,
dt
ui 
t
t
dui ui
u

 vk i ,
dt
t
xk
dT T
T
Tij  ij  ij  vk ij .
dt
t
xk
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2. Conservation laws
2.1. Conservation of mass
Assume that continuum with density  fills the volume V , bounded by surface S .
The total mass contained in V is
m

 dV .
(2.1)
V
It is assumed that the density    xi ,t  is a continuous function of space and time
coordinates and that there is no flux through the surface S .
The mass of the considered body at configuration 0C is equal to that at configuration t C , i.e.

 d 0V 
0
0
t
V
  a
0

j
t
 dtV ,
(2.2)
V
, t 0  d 0V 
  x , t  d V .
t
j
t
V
(2.3)
V
Realizing that xi  xi a j ,t  and substituting after the integral sign allows rewriting the right
hand side of (2.3) in the form
  x , t  d V    x a , t  J
t
j
t
i
0
V
j
d 0V ,
(2.4)
V
where the Jacobian of the transformation is the determinant of the deformation gradient
Fij 
J  det Fij ,
xi
.
a j
(2.5)
Equations (2.3) and (2.4), written in short, give

0
0
 d 0V 
V

0
t
 J d 0V .
(2.6)
V
Since the last equation must be valid for an arbitrary volume we can write
0
  t J .
(2.7)
But J  J , since J  0 .
Proof
The continuum in 0V completely fills the space. The initial density is 0   0 . At the initial
configuration 0C , there is no deformation, so F  I and consequently J  1 which is greater
than zero. The value of J  0 in the process of deformation would mean that at a certain time
 t0 ,t  the value of the Jacobian would become J  0 . For such a case there would be no oneto-one correspondence
xi  xi a j ,t   ai  ai x j ,t 
(2.8)
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which is a contradiction with initial assumption about the physical acceptability of
deformation description.
The consequence of the conservation of mass is known as the continuity equation.
2.2. Lagrangian (material) form of the continuity equation can be written in different
forms
 J  0  const , where J  det Fij ,
or
or
Fij 
xi
,
a j
(2.9a)
D
 J   0 ,
Dt
(2.9b)
 d 0V  t dtV  t J d 0V  const ,
(2.9c)
 d tV

 J  det Fij .
t
 d 0V
(2.9d)
0
0
or
Remark 1
Remember that the condition J  0 is necessary for the equivalence of material and spatial
descriptions xi  xi a j ,t   ai  ai x j ,t  .
If the Jacobian of the above transformations J  det Fij  0 then the inverse function
of xi  xi a j ,t  does not exist.
Also, if detFij  0 , then, either 0   0 , or t    .
Remark 2
Remember that dx  F da , da  F 1 dx . If det F  0 , then F 1 cannot be computed.
2.3. Eulerian (spatial) form of the continuity equation
Again the total mass of a continuous medium of density  filling the volume V at time t
is
M    dV .
V
The time rate of increase of the total mass in the volume V is
M


dV .
t V t
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We assume that no mass is created inside the volume V . Then the time rate of mass
must be equal to the rate of flux of the mass through the surface.
flux = rate of mass outflow =
  v . n dS
S
 
rate of mass inflow =    v . n dS    div v dV
S
So
 t dV   div v dV ,

V
or
V
V

 t
V
dV   
T
  v dV , where 
T
  

.
 x y z 
From it follows
  t  div v  dV  0 .
 

This equation must be valid for any volume, so
 

 div  v  0 ,
t
or

T
   v  0 ,
t
or
   vi 

0
t
xi
(2.10)
There are different forms of continuity equation in spatial description. The last
equation could be rewritten using the rule for the derivative of a product


v
 vi
  i 0.
t
xi
xi
(2.11)
Realizing that the material derivative of density is
D 


 vi
.
Dt
t
xi
The equation (2.11) could be simplified into
D
v
  i  0.
Dt
xi
(2.12a)
The equivalent formulas are
D
  div v  0 ,
Dt
(2.12b)
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 v v v 
D
   x  y  z   0 ,
Dt
 x y z 
(2.12c)
D
 Dii  0 .
Dt
(2.12d)
Remark 1
The velocity gradient is Lij 
Dij 
vi
, the strain rate is its symmetrical part
xi
1
Lij  L ji . From it follows that Lii  Dii .
2
Remark 2
If the material is incompressible, then   const at any particle and
D
0
Dt
so the incompressibility condition is
vi
 div v  Dii  0 .
xi
2.4. Conservation of linear momentum
For a particle of mass m we say that the rate of change of linear momentum is equal
to the resultant force applied to a particle
 
D
mv  F .
Dt
(2.13)
The validity of this principle is postulated in continuum mechanics.
Continuum form
Assume that at time t a given amount of mass is in volume V , bounded by surface S .
Denote bi [N/kg] - body force (per unit mass) and
ti [N/m2] - surface traction, defined per a unit surface.
Based upon Newton's second law the rate of change momentum of a given amount of mass is
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
D
  v dV    t dS   b dV ,


Dt  V
V
 S
 kg m 3 kgm

 m3 s 2 m  s 2  N 
or
t
i
dS   bi dV 
S
V
 N 2
 m 2 m 
(2.14a)
 kg N 3 
 m3 kg m 


D
vi dV .
Dt V
(2.14b)
The relation between the stress vector and stress components is given by so called
Cauchy relation
ti   ji n j ,
where ti and  ij are the stress vector and Cauchy (true) stress tensor components
respectively. The symbol n j stands for the component of a normal.
Substituting the Cauchy relation into the surface integral in (2.14b) and using the
divergence theorem gives
t
i
dS    ji n j dS  
S
S
V
 ji
x j
dV
(2.15)
The former equality is due to Cauchy relation while the latter is due to divergence
theorem of Gauss.
2.5. Interlude
What is the material time derivative of a volume integral in (2.14b)? It can be proved that
D
Dv
vi dV    i dV .

Dt V
Dt
V
(2.16)
Proof in literature is based on Reynold's transport theorem + Gauss divergence theorem +
continuity equation + definition of material derivative of  .
Using (2.15), (2.16) in (2.14b) we have
  ji
  x
V

j

Dv
 bi  dV    i dV .

Dt
V

(2.17)
From the condition that it must hold for an arbitrarily chosen volume V we get
 ji
Dv
  bi   i
xi
Dt
(2.18a)
which is called the Cauchy equation of motion.
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2.6. Cauchy equation of motion in another form is
 ji
x j
  bi   xi .
(2.18b)
If we introduce body forces per unit volume
N/m 
3
f i  bi
 kg N 
 m3 kg 


(2.19)
we obtain still another form of Cauchy relation of motion in the form
 ji
 f i   xi .
x j
(2.18c)
Remarks
Cauchy equations of motion represent 3 PDE for 6 unknown components of stress.
Notice that  ij is symmetric.
These equations are written for a given spatial domain, for a collection of considered
material particles - filling volume V , bounded by surface S , considered at a configuration
t
C . Derivatives are with respect to spatial coordinates.
In special cases the acceleration could be neglected and equations (2.18) reduce to the
equations of equilibrium
 ij
 fi  0 .
x j
(2.20)
These equations do not contain any kinematic variables. They do not generally suffice
to determine the stress distribution since they are only three partial differential equations for
six independent unknown stress components.
Additional equations must be considered, i.e.
a) displacement vs. strain relations - kinematic relations
b) stress vs., strain relations - constitutive equation.
2.7. Equation of motion in the reference state
It was already mentioned that the Cauchy equations of motion apply to the current
deformed configuration t C . The equations of motion could be transformed to referential
configuration 0C by means of the first and second Piola-Kirchhoff stress tensors.
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It should be reminded that Piola-Kirchhoff tensors are useful stress measures which
recalculate the actual stress at t C to the reference state, i.e. to the non-deformed
configuration 0C .
It was already shown that
1st P.-K.:
 
2nd P.-K.:
S
 1
F 

or  ji 
 1
F σ F T

or S ji 
0
0
 1
F  .
 jr ri
0
(2.21)
 1
F  F 1 .
 js sr ir
(2.22)

F  .
 jr ri
(2.23)

F S F ,
 js sr ir
(2.24)
0
The inverse relations are

F

or  ji 

F S FT

or  ji 

0
for 2nd P.-K.:  
0
for 1st P.-K.:
0
 a 
F 1   i  .
 x j 
 x 
F i ,
 a j 
where
0
(2.25)
So the equation (2.15)

S
ti dS    bi dV    xi dV , where ti   ji n j
V
V
could be transformed to the referential configuration followingly

0
S
 ji 0 n j d 0S  
 0bi d 0V  
0
0
V
 xi d 0V ,
0
0
V
(2.26)
where 0b1  bi a j ,t  using xi  xi a j ,t  .
Notice that for the right-hand side we could write

V

Dv
dv
dV   0 0 i d 0V
V
Dt
dt
since  dV   J d 0V  0 d 0V (see 2.9c)), where J is the Jacobian of the transformation
xi  xi a j ,t  .
xi 
Dvi
– material derivative of vi expressed in spatial coordinates vi  vi x j ,t 
Dt
xi 
dvi
– material derivative of v i expressed in material coordinates vi  vi a j ,t  .
dt
Using the divergence theorem

0
S
 ji 0n j d 0S  
 ji
0
V
a j
d 0V
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and realizing that eq. (2.26) must hold for any volume we finally get
 ji 0 0
  bi  0 xi .
a j
(2.27)
This is the Cauchy equation of motion expressed in referential coordinates by means
of the first Piola-Kirchhoff strain tensor.
Using the relation between the first Piola-Kirchhoff and the second Piola-Kirchhoff
 ji  S jr Fir
  S F T or
(2.28)
we could write the Cauchy equation of motion by means of the second Piola-Kirchhoff stress
tensor in the form

S jr Fir   0 0bi  0 xi .
a j
(2.29)
2.8. Conservation of angular momentum
The law of conservation of angular momentum for a particle of the mass m is
 x1 
D
 
m r  v  r  F , where r   x2  .
Dt
x 
 3
  
(2.30)
For continuum we could similarly write


 


D
r   v dV   r  t dS   r   b dV .
S
V
Dt V
(2.31)
For rewriting it into indicial notation we have to realize that the equivalent of vector product
c  a  b is ci  eijk a j bk , where eijk is the Civita-Levi permutation symbol.
Note
eijk  1
for even permutation of indices, i.e.: 1,2,3 – 2,3,1 – 3,1,2,
eijk  1
for odd permutation of indices, i.e.: 3,2,1 – 2,1,3 – 1,3,2,
eijk  0
for repeating indices: 1,1,2 etc.
So,
D
ermn xm  vn dV   ermn xm tn dS   ermn xm  bn dV .
S
V
Dt V
(2.32)
Substituting the Cauchy relation tn   jn n j , using the divergence theorem for the surface
integral and the conclusion (2.16) concerning the material time derivative of a volume
integral, i.e.
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D
Dv
 vi dV    i dV

V
Dt V
Dt
we could rewrite eq. (32) into the form

V
ermn
  x  

D
xmvn  dV  V ermn  m jn  xm  bn  dV .
Dt
 x j

Realizing that
D xm
 vm ,
Dt
xm
  mj ,
x j
 xm jn  xm jn
 jn

  mj jn  xm
  mn ,
x j
x j
x j
we have
   jn


Dvn 



e
v
v

x

d
V

e
x


b





n
mn  dV
V rmn  m n m Dt 
V rmn  m  x j


 


= 0, equation of motion, see eq. (2.18a)
and also
ermnvmvn  0 ,
since ermn is skew-symmetric in m , n .
Example of double product evaluation shows the trick
e111 e112 e113   v1v1 v1v2
e1mnvmvn  e121 e122 e123  : v2v1 v2v2
e131 e132 e133  v3v1 v3v2
0 0 0  v1v1 v1v2
 0 0 1 : v2v1 v2v2
0  1 0 v3v1 v3v2
v1v3 
v2v3  
v3v3 
v1v3 
v2v3   v2v3  v3v2  0 .
v3v3 
And similarly for other indices. So what remains of eq. (2.32) is

V
ermn mn dV  0 for arbitrary volume
(2.33)
 ermn mn  0 only if  mn is symmetric, i.e.
 mn   nm
(2.34)
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This establishes the symmetry of stress matrix without any assumptions of equilibrium or of
uniformity of the stress distribution. The symmetry of the stress matrix is so called Cauchy's
second law of motion.
2.9. Conservation of energy
If mechanical quantities only are considered the principle of conservation of energy
for the continuum may be derived directly from the equation of motion.
Power input
Assume at first that only external surface traction ti per unit area and body forces bi
per unit mass are doing work on the mass instantaneously occupying volume V , bounded by
S . The power input is
Pinput   ti vi dS    bi vi dV ,
S
(2.35)
V
where vi are components of the velocity field. As before we express the components of
tractions by means of stress components
ti   ji n j
and use the Gauss divergence theorem for the transformation of the surface integral into
volume integral and get

S
ti vi dS    ji n j vi dS  
S
V
  jivi 
x j
  ji
v 
dV   
vi   ji i  dV .
V  x
x j 
 j
(2.36)
Realizing that the velocity gradient if defined by
Lij 
vi
x j
(2.37)
we obtain
  ji


Pinput   
 bi vi   ji Lij  dV



V 

 x j
but
 ji
x j
 bi  
Dvi
Dt
(2.38)
by (2.18a)
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The first term of eq. (2.38) on the right hand side could be rewritten as
v
i
V
Dvi
D 1
D 1

dV     vi vi  dV 
 vi vi dV

Dt
D
t
2
D
t
2


V
V


(2.39)

Remember
material derivative
Kinetic energy
dv 2  2v dv
of volume integral
of the system
see eq. (16)
and represents the time rate of the kinetic energy of the system. Realizing that the stress tensor
is symmetric  ij   ji and using (2.39) we could rewrite (2.38) into the form
Pinput 
D
Dt

V
1
 vi vi dV    ij Lij dV .
V
2
(2.40)
The last term of (40) is a double dot product of stress and velocity gradient tensors. The
velocity gradient tensor can be decomposed into symmetric and skew-symmetric parts
Lij  Dij  Wij ,
where, as explained before, Dij and Wij represent rate of deformation and spin tensors
respectively. It could be shown easily that
 ijWij  0 ,
(  ij - symmetric, Wij - skew-symmetric).
From it follows that
 ij Lij   ij Dij .
(2.41)
The final form for the power input expression is
Pinput 
D 1
 vi vi dV    ij Dij dV .
V
Dt V 2
(2.42)
We can conclude that the power input is the sum of two volume integrals. The first
one is the material time derivative of the kinetic energy of the system, while the second one
contributes to the internal energy.
The scalar  : L equals to  : D and is called stress power per unit volume. Stress
power does not contribute to the kinetic energy of the system. This result is due to Stokes
(1851).
If both mechanical and non-mechanical energies are to be considered the principle of
conservation of energy in its most general form must be used.
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In this form the conservation principle states that the time rate of change of kinetic +
internal energy is equal to sum of the rate of work + all other energies supplied to the
continuum per unit time. Such energies may include thermal, chemical, electromagnetic
energies.
In the following only mechanical and thermal energies are considered. Then the
energy principle takes on the form of the first law of thermodynamics.
For our purposes we will consider a thermodynamic system chosen as a closed system
not interchanging matter with surroundings.
The first law of thermodynamics relates the work done on the system and the heat transfer
into the system to the change in energy of the system.
It is assumed that only energy transfers to the system are by
a) – mechanic work done on the system by surface tractions and body forces,
b) – heat transfer through the boundary,
c) – distributed internal heat sources.
Surface tractions and body forces and their contribution to power input to the system were
already treated by previous paragraph are summarized by eq. (2.42).
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3. Finite deformations, incremental decomposition and finite element discretization
Overview
Principle of virtual work relates the work done by internal and external forces due to
prescribed virtual displacements
 t  tU   t  tW .
(3.1)
On the left hand side we have the virtual strain energy of a system at
 t tU   tt
t  t
t  t
V
t  t
t  t
 ij 
ij d t tV   0
V
t  t
0
t  t
C
S  t  0tEij d 0V .
(3.2)
Using the total Lagrangian approach, the incremental decompositions for the second PiolaKirchhoff stress tensor and the Green-Lagrange strain tensor are
t  t
0
Sij 0t Sij  ΔSij ,
(3.3)
t  t
0
Eij 0t Eij  ΔEij .
(3.4)
It was already shown that the increment of Green-Lagrange strain tensor is
E 

 
 
1
1
1
Z  Z T  Z T Z  Z T Z  Z T Z
2
2
2
where Z  Z ij 

 t ui
is the material displacement gradient.
 0x j
We can introduce the following notation which will consistently be used later
E  EL  EN
EL  EL1  EL2


E L1 
1
Z  Z T
2
E L2 
1
Z T Z  Z T Z
2
E N 
1
Z T Z .
2
(3.6)




Variation of eq. (3.4) gives
 t0 t Eij  Eij .
(3.7)
Substituting eqs. (3.7) and (3.3) to (3.2) we get
 t tU  
0
V
 0
S
V
t
0
ij
S
t
0
ij

 Sij Eij d 0V 


 Sij EijN  EijN d 0V 
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 0
V
S
t
0
ij
EijL  0tSij EijN  Sij EijL  Sij EijN d 0V .
(3.8)
The increment of the second Piola-Kirchhoff stress tensor appearing in the third term
of the integrand could be linearized
Sij 0 Cijkl EijL
(3.9)
and the last term in (3.8) could be neglected since it is one order less than other terms, so the
virtual strain energy could be approximated by
 t  tU  U I  U II  U III
(3.10)
where
U I  
0
U II  
V
0
U III  
S EijL d 0V ,
(3.10a)
S EijN d 0V ,
(3.10b)
Cijkl EklL EijL d 0V .
(3.10c)
t
0 ij
V
t
0 ij
0
V
0
This approximation implicitly assumes that the changes between the configurations
t
C and
t  t
C are small.
For finite element implementation of these ideas it is convenient to switch from the tensor
to matrix notation. The process could be summarized in four steps
 
a) Instead of tensor EijL we will use a column array E L defined by
E   E
L T

L
L
L
L
E22
E33
2E12L 2E23
2E31
.
L
11
(3.11)


~
b) Instead of tensor EijN we will use a column array E N defined by
   E
~
 EN
T
N
11

N
N
N
E12N E13N E21
E22
E23
E31N E32N E33N .
(3.12)
c) The second Piola-Kirchhoff stress tensor will have two appearances. Instead of
t
0
Sij we will use either
 S   S
t
0
T
t
t
0 11 0
 S  defined by
t
0

S 22 0tS33 0tS12 0tS 23 0tS31
or a two-dimensional array
 
 0t S

t~
0S   0
 0

 
0
 S
t
0
0
0 

0 
t

0S 
 
(3.13)
 S~  defined by
t
0
,
(3.14)
9 9
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where
 0t S11
t
t
0 S   0 S 21
 0t S31

 
d)
0
S 

S .
S 
t
0 12
t
0 22
t
0 32
t
0 13
t
0 23
t
0 33
S
S
S
(3.15)
Cijkl   0 C  .
In the matrix notation, the virtual strain energy components, equivalent to those appearing in
eq. (3.10), are
U I  
E   S  d V ,
L T
0
U II  
V
U III  
V
0
(3.17b)
E   C  E  d V ,
(3.17c)
t
0
N
L T
0
(3.17a)
E~   S~ E~  d V ,
N T
0
t
0
V
L
0
0
0
And now, the finite element discretization enters the stage. The generalized
displacements within a finite element are usually expressed by means of shape functions
systematically arranged in A and by generalized nodal displacements q in the form
u  A3*LMAX qLMAX*1
(3.18)
where LMAX is the number of D.O.F. for
a particular element. The increments of
displacements are
u  A q
(3.19)
Knowing the shape functions appearing in A we can easily calculate the components
of the material displacement gradient and then to express its increments. It will contain
derivatives of shape functions and will depend on q and q .
t
0
Z ij 
 t ui
 0x j
Z ij 
 ui
 0x j
(3.20)
The results for linear part of strain increments – in accordance with eqs. (3.9) and (3.11) –
could be expressed in the form


E6L*1  EL1  EL2  0 B L1  0 B L2 q  0 B 6L*LMAX q LMAX*1
(3.21)
where obviously
0
B L 0 B L1  0 B L2 .
(3.22)
The lower left hand index zero emphasises that the derivatives appearing in these
matrices are taken with respect to coordinates 0 xi .
25
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Similarly for the nonlinear part of strain increments
~
E9N*1 0 B9N*LMAX qLMAX*1
(3.23)
Now, the three contributions to the virtual strain energy (3.17a), (3.17b), (3.17c) could
be formally rewritten. The first component is
U I  q T 0t F
(3.24)
where a so called vector of internal forces in nodes is
t
0
F  0
V
B 
L T t
0
0
S d 0V .
(3.25)
For the second component, i.e. (3.17b), we can write
U II  q T 0t K N q ,
(3.26)
where a so called non-linear part of incremental stiffness matrix is
t
0
KN  0
V
B 
0
N T t
0
~
S 0 B N d 0V .
(3.27)
The third component of the virtual strain energy, i.e. (3.17c), can be discretized using
(3.9) and (3.21). After some algebraic manipulations we get
U III  q T t0K L q ,
(3.28)
where
t
0
K L  0
B 
L T
V
0
0
C 0 B L d 0V
(3.29)
is the linear part of incremental stiffness matrix.
Using eqs. (3.24), (3.26) and (3.28) and realizing that the virtual work done by
external forces

t  t

R is

 t  tW  q
T t  t

R ,
(3.30)
we can conclude that in agreement with (1) we get


q T 0t F  0t K L q  0t K N q  q T
t  t
R.
(3.31)
This equation must hold for any virtual diplacement, hence finally we have
t
0
K q  t  t R  0t F ,
(3.32)
t
0
K  0t K L  0t K N
(3.33)
where
This system of algebraic equations constitutes the conditions of equilibrium. Solving
the system gives the unknown increments of nodal displacements and the new displacements
in the configuration
t  t
C could be calculated by
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t  t
q  t q  q
(3.34)
The new displacements, however, have to be taken as the first approximation only and
refined subsequently in an iterative process. Introducing the iteration counter (i ) we can
rewrite eq. (3.32) as follows
t
0
K (i ) q (i ) 
t  t
R  t  0t F (i 1)
for i  0, 1, 2 
(3.35)
At the beginning of the process we set
t  t
q ( 0 )  t q,
t  t
0
F ( 0)  0t F and
t
0
K (0)  0 K .
(3.36)
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The Newton Raphson iteration process with total Lagrangian approach could be
implemented as follows.
Let’s assume that from t  0 to t  tmax there are kmax (same) loading steps. The
maximum force, corresponding to final configuration at time tmax is
tmax
Rkmax R . If a
linear increase of force between consecutive loading steps is considered, then the force
corresponding to k  th loading step is
k
R  k*tmax R/kmax.
for k  1 to kmax do
% loop
for loading steps
i  0;
if k  1 then
k
0
K ( 0)  0K;
else
k
0
K (0) 
k
0
F ( 0)  00F  0 ;
k 1
0
K (ilast);
k
0
F (0) 
k
q ( 0)  0q;
k 1
0
F (ilast);
k
q (0) 
k 1 ( ilast)
;
q
end of if
k
intermediate load level is
R  tmax R * k/kmax;
satisfied .false.
while .not. satisfied do
% iteration loop
i  i  1;
solve
k
k
0
K (i 1) q (i )  kR  0kF (i 1) ;  q (i )
q (i )  k q (i 1)  q (i );
calculate
k
0
S (i );
k
0


K (i )  f k q (i ) , 0k S (i ) ;
k
0


F (i )  g 0k S (i ) ;
% Note: f() and g() are functions
ilast i;
 Δq (i )
satisfied   k (ilast)  1 .and .
 q

k
R  0kF (ilast)
k
R

 2 


end of while loop
end of for loop
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4. What is the material derivative of volume integral?
Let  be a function that is sufficiently smooth in a given volume V . Then
D
D
 dV  
 dV .

V
V
Dt
Dt
(*)
Proof: From left-hand side of the equation (*), we can write
D

  dV     dV     v dS .

V
S
Dt
t V
(4.1)
Using Gauss theorem we can transform surface integral to volume integral and get
 
 
D



 dV    dV   div  v dV       div  v  dV

V
V
V
V
Dt
t
 t

(4.2)
For a divergence of product of scalar function  and general vector function k , we can write
 
div  k  grad  . k   div k
(4.3)
Using this expression we can rewrite integrand of integral (2) and we obtain
 
 

   div  v         div  v   v grad  
t
t
t
 
 

 

   v grad     div v  .
 t

 t

(4.4)
The second bracket of expression (4.4) is from equation of continuity equal to zero and first
bracket from this expression is a definition relation for
D
. After substituting back into
Dt
integral (4.2) we finally get
 
  
D

 

 dV      v grad       div  v   dV 

V
Dt V

 t

  t
 D 

D
   
  . 0 dV  
 dV .

V
V Dt
 Dt 

(4.5)
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5. Conjugate stress and strain measures
A stress is called conjugate to the strain if its scalar product with strain gives work.
Stress and strain quantities giving mechanical work as their scalar product are energetically
conjugate. Mechanical work per unit time is power, or rate of work, so we could also relate
stress and strain rate quantities whose scalar product gives power. Such quantities could be
called power conjugate.
The mechanical work of surface tractions and body forces at the current configuration t C is
W   t t ti t ui d tS   t
t
S
t
V
f i t ui d tV .
(5.1)
All quantities are related to the current configuration. Let's omit the upper left index t
for a moment. Using the Cauchy relation and the Gauss theorem we get
W    ji n j ui dS   f i ui dV 
S
V
 u  
   ji ui 

 
 
 f i ui  dV    ji i   ji  fi  ui  dV .


V
V
 x j

 
 x j  x j
(5.2)
The second term in eq. (5.2) is equal to zero, since it is the equilibrium equation.
Exploiting the fact that the stress tensor is symmetric, the mechanical work could be
calculated by a double dot product of Cauchy (true) strain and infinitesimal strain at the
current configuration t C .
W    jiuij dV    ij ij dV 
V
V

V
 : d V .
(5.3)
We can conclude that the true stress and infinitesimal strain constitutes the
energetically conjugate variables.
Remark
Remember that the infinitesimal strain could be calculated exactly, involving no
approximation, from





1
1
Z  ZT  F  FT  I
2
2
since the deformation gradient 0t Fij 
displacement gradient 0t Z ij 
 t xi
could be expressed by means of the material
 0x j
 t ui
in the form
 0x j
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F  ZI.
Similarly for the mechanical power, or the rate of work gives
P  W    ijuij dV    ij Lij dV    ij Dij  Wij dV 
V
V
   ij Dij dV 
V

V
V
 :D dV ,
(5.4)
so the power conjugate quantities are the true stress  ij and the velocity gradient Lij . Using
the fact that the spin tensor Wij is skew-symmetric and its scalar product with symmetric
stress tensor  ij gives zero, we can state that the true stress and the strain rate Dij are also
power conjugate quantities.
Using the definition of the first Piola-Kirchhoff stress tensor we could express the
previous equation in the reference configuration 0C .
Substituting  ji 

V

F jk  ki into eq. (5.4) we get

0
v dV  
ji ij

F jk  ki Lij dV    ki Lij F jk d 0V .
V

V 0
0
(5.5)
But
Lij Fjk  Fik
(5.6)
since
D
D   t xi

Fik 
Fik 
Dt
Dt   0xk
  t xi  t x j  t vi  t x j
  t
 t
 Lij Fjk .
0
0
  x j  xk  x j  xk
So the mechanical power in the reference configuration is expressed by
P   0  ki Fik d 0V   0  ik Fik d 0V 
V
V

0
V
 : F d 0V .
(5.7)
which is a double dot product of the first Piola-Kirchoff stress tensor and the rate of
deformation gradient tensor. These tensors form another suitable couple of power conjugate
quantities.
Similarly for the second Piola-Kirchhoff stress tensor. Substituting



F S F T or  ij  0 Fir Srs Fjs


0
into eq. (5.4) we get
P    ij Dij dV  
V

Fir F js S rs Dij dV   Fir F js S rs Dij d 0V .
V

V 0
0
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But
Fir Fjs Dij  E rs
since for the Green-Lagrange strain tensor
E

1 T
F FI
2

we could express its time rate by




  1 D F T F  1 F T F  F T F .
E
2 Dt
2
Using eq. (5.6) the previous equation can be rewritten into




  1 F T LT L  F T L F  1 F T D  W T F  F T D  W F 
E
2
2
 



1 T T
F D  W T F  F T D  W F 
2

1 T
F D  W F  F T D  W F  F T D F .
2


since D is symmetric and W skew-symmetric. So the stress power in a reference
configuration can also be expressed by
 d 0V
P    ij Dij dV   0 Sij E ij d 0V   0 S : E
V
V
V
giving another couple of suitable quantities. It is obvious that in terms of mechanical wark we
have
W    ij  ij dV   0 Sij Eij d 0V .
V
V
Finally, let's find what role plays the Almansi strain tensor is these considerations. It can be
proved that
P    : D dV    : A  dV ,
V
V
where
A  D is a so called Rivlin-Ericksen rate of Almansi strain A .
Proof
   
2
2
a) The Almansi strain tensor is defined by d t s  d 0s  2 d t x A d t x .
b) The time rate of the previous expression is

     DDt d s 
D t 2
d s  d 0s
Dt
2
t
2
2


D t
d x A dt x 
Dt
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

 d t x  d t x A d t x 
 2 d t x A d t x  d t x A
But d t x  L d t x , d t x T  d t x T LT
so


  A L dt x .
 2 d t xT LT A  A
c) We can also write
 


D t 2 D t T t
D t
ds 
d x d x  2 dt xT
d x  2 dt xT dt x 
Dt
Dt
Dt
 2 d t x T L d t x  2 d t x T D  W  d t x  2 d t x T D d t x
since
2 d t xT W d t x  0 due to the skew-symmetry of W and the symmetry of d t x T d t x .
This way we have proved that
  A L.
D  A  LT A  A
6. Summary for conjugate strain and stress measures
Measure of strain
Measure of stress
Their scalar product
___________________________________________________________________________
 ij - Cauchy (infinitesimal)
 ij - Cauchy (true) stress
work
Eij - Green-Lagrange
S ij - second Piola-Kirchhoff
work
E ij - rate of Green-Lagrange
S ij - second Piola-Kirchhoff
power, rate of work
Fij - rate of deformation gradient
 ij - first Piola-Kirchhoff
power, rate of work
D ij - strain rate
 ij - Cauchy (true) stress
power, rate of work
Lij - velocity gradient
 ij - Cauchy (true) stress
power, rate of work
Aij - Rivlin-Eriksen rate of Almansi  ij - Cauchy (true) stress
power, rate of work
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34