 2001, W. E. Haisler
19
Chapter 3: Conservation of Linear Momentum
Although the conservation of linear momentum equations
may be left in terms of tractions, it is more convenient and
customary to put the tractions in terms of stresses. For this
purpose, consider a two dimensional planar body as shown
below:
F2
F1
b = thickness
f
y
x
z
Make two free bodies by passing a plane with unit normal n
through the body as shown below:
 2001, W. E. Haisler
20
Chapter 3: Conservation of Linear Momentum
F2
F1
b
f
y
x
z
Original Body
F2
F1
t ( n )
t (n )
-n
n
R1
R2
Free Body #1
R3
Free Body #2
f
 2001, W. E. Haisler
21
Chapter 3: Conservation of Linear Momentum
On the cut edge of each free body, we must place equal and
opposite forces. Since these forces act over some area, we write
these forces in terms of the traction t(n ) as shown on the figure.
Now consider a smaller free body taken from FB #2 above by
passing horizontal and vertical cutting planes as shown below:
j
t (n )
t ( j)
f
A ( j)
A (n )
n
y
x
n
Freebody #2


t (n )
A(i )


t (i )
i


The inclined side of the wedge has a unit outward normal n , a
surface area A( n ) and the traction is t( n ) . The normal makes an
 2001, W. E. Haisler
22
Chapter 3: Conservation of Linear Momentum
angle  with the x-axis as shown (positive is CCW from the xaxis) and the unit normal is given by:
n  nx i  ny j  (cos )i + (sin  ) j
= [cos(  +  )]i +[sin(  +  )]j = -(cos )i - (sin  ) j
.
On the i and j surfaces, the area is A( i ) and A( j ) , respectively,
and the tractions are t( i ) and t( j ) , respectively. Each traction
vector may be resolved into its x and y components as shown
below:
 2001, W. E. Haisler
23
Chapter 3: Conservation of Linear Momentum
t( n )  t ( n ) x i  t ( n ) j
t ( j) y
y
t( i )  t ( i ) x i  t ( i ) j
t ( j) x
A (n )

t (n ) x
y
y
A ( j)
A(i)

x
t (i ) x
t( j )  t ( j ) x i  t ( j ) j
y
n  nx i  n y j

n
t (i ) y
t (n ) y
 (cos ) i + (sin  ) j
A ( i )  A ( n ) cos  , A ( j)  A ( n ) sin 
We now apply conservation of liner momentum for the
tractions, 0   Fext , to obtain:
 2001, W. E. Haisler
24
Chapter 3: Conservation of Linear Momentum
x-component: 0  t (n ) x (A (n ) )  t ( i ) A ( i )  t ( j) A ( j)
x
x
y-component: 0  t (n ) y (A (n ) )  t ( i ) A ( i )  t ( j) A ( j)
y
y
Substituting the area relations into the above gives:
x-component: 0  t (n ) x (A (n ) )  t ( i ) A (n ) cos   t ( j) A (n ) sin 
x
x
y-component: 0  t (n ) y (A ( n ) )  t ( i ) A ( n ) cos   t ( j) A ( n ) sin 
y
y
Canceling the area gives:
t (n ) x   t ( i ) cos   t ( j) sin 
x
x
t (n ) y   t ( i ) cos   t ( j) sin 
y
y
Note that the normal vector has components of
nx  cos  cos ; ny  sin    sin 
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
25
Hence the traction on the normal face can be written as
t ( n ) x  nx t ( i ) +n y t ( j)
x
x
t ( n ) y  nx t ( i ) +n y t ( j)
y
y
This last set of equations is called Cauchy's formula. It
provides the relationship between the traction components that
act on a surface with unit outward normal n and the x and y
components of tractions required to keep the free-body in
equilibrium, i.e., satisfy conservation of linear momentum.
It is useful to write to write the tractions on the x and y faces in
terms of Cauchy stresses as follows:
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
 xx  t ( i ) x ,  yx  t ( j) x ,  xy  t ( i ) y ,  yy  t ( j) y
Note the following notation:
 xx is the traction on the x face acting in the x direction,
 xy is the traction on the x face acting in the y direction,
 yx is the traction on the y face acting in the x direction, etc.
1st subscript = face acting on; 2nd = direction of traction.
With this notation, Cauchy's formula becomes:
t (n ) x  nx xx  ny yx
t (n ) y  nx xy  ny yy
26
 2001, W. E. Haisler
27
Chapter 3: Conservation of Linear Momentum
Recall: traction and normal vectors are given by
t(n )  t (n ) x i  t (n ) j
y
and n  nx i  n y j = (cos ) i + (sin  ) j .
In matrix notation: t (n )    t (n ) x

t ( n )  ,  n    nx
y 
n y 
And the Cauchy stress tensor is given by
 xx  xy   t ( i ) x
      
yy 
 yx
 t ( j) x
t (i) y 

t ( j) y 

Thus, Cauchy formula in vector and matrix notation is given
by:
t
(n )
 n   or
t   n   
 (n ) 
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
28
Once again, Cauchy’s formula is nothing more than
Conservation of Linear Momentum applied to an element of
material as shown below and which is subjected to stress
components on each face as shown. In order for the element
to be in static equilibrium (satisfy COLM), the stresses (and
tractions) must satisfy Cauchy’s formula t(n )  n   .
 yy  t ( j) y
 yx  t ( j) x
 xy  t ( i ) y
 xx  t ( i ) x
n
y
x
z
t( n )
 2001, W. E. Haisler
29
Chapter 3: Conservation of Linear Momentum
Suppose the element is chosen at a boundary where the outward
unit normal is n an external traction t(n ) is applied. Then we can
use Cauchy's formula to determine the internal stress state.
 yy
 yx
external
surface
 xy
n
y
t( n )
x
z
body
 xx
 2001, W. E. Haisler
30
Chapter 3: Conservation of Linear Momentum
Consider a material point in the planar body at some point
(x,y,z). At this point, we observe an internal stress state given by
the stress tensor [] as shown below:
 yy
 yx
F2
F1
b = thickness
f
 xx
 xy
y
x
z
 yx
 yy
 xy
 xx
 2001, W. E. Haisler
31
Chapter 3: Conservation of Linear Momentum
In 2-D, a complete picture of the Cauchy stress tensor and
Cauchy's formula is:
 yy
 yx
 xx
 xy
 yx
y
x
z
 yy
 xy
 yy
 yx
 xy
t( n )
 xx
 xx
 xy
 yx
n
n
 yy
 xx
t( n )
Note: 1) on opposite faces of the normal plane, the normal
vector and traction are opposite. 2) On opposite faces of the
square, stresses are in opposite directions. On the +x face,  xy is
positive in + y direction; on the -x face,  xy is positive in the -y
direction.
 2001, W. E. Haisler
32
Chapter 3: Conservation of Linear Momentum
For a 3-dimensional state of stress, Cauchy's formula becomes
t ( n )  σ xx nx  σ yx n y  σ zx nz
x
t ( n )  σ xy nx  σ yy ny  σ zy nz
y
t ( n )  σ xz nx  σ yz n y  σ zz nz
z
and the Cauchy stress tensor is given by:
  xx

    yx
  zx
 xy
 yy
 zy
 xz   t ( i ) x
 
 yz   t ( j) x

 zz   t
 (k ) x
t (i ) y
t ( j) y
t (k ) y
t (i) z 

t ( j) z 

t (k ) z 

In chapter 4, conservation of angular momentum, we will shown
that the stress tensor must be symmetric so that  xy   yx , etc.
 2001, W. E. Haisler
33
Chapter 3: Conservation of Linear Momentum
y
 yy
t(j)
 yx
 xy
 yz
 zy
0
t(k)
k
 xx
j
x z
i
 zx
 zz
z
x
t(i)
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
34
We want to replace the traction terms ( t , etc.) on the right
(i)
side of conservation of linear momentum by equivalent
stress terms (xx, etc). In dyadic notation (a special vector
notation), the stress matrix  can be written as
  ii 
 ji 
 ki 
xx
yx
zx
 ij
 jj
 kj
xy
yy
zy
 ik 
 jk 
 kk 
xz
yz
zz
Thus, we can write the tractions in terms of stresses as
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
35
, t  j  ,
t  i    i   j  k 
j
i
xx
xy
xz
t  k 
k
Thus, we can write the traction terms in linear momentum as:
 ti  t j  tk  (i   )  ( j   )  (k   )





x y z
x
y
z
 

 
i
j
k
    
 y z
 x
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
36
Conservation of linear momentum becomes
v
   (v   )v   g    
t
Note that the “dot product”    results in a vector (not a
scalar since  is a matrix), and is given by





















 xx  yx  zx 
(


)i 
x
y
 z 
 xy  yy  zy
  (


)j
x
y
z
 xz  yz  zz
(


)k
x
y
z















 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
37
Conservation of Linear Momentum has 3 components:
 xx  yx  zx
 vx
 vx
 vx
 vx
x comp :  (
 vx
 vy
 vz
)   gx 


x
y
z
t
x
y
z
 xy  yy  zy
 vy
 vy
 vy
 vy
y comp :  (
 vx
 vy
 vz
) gy 



x

y
z
t
x
y
z
 xz  yz  zz

vz

vz

vz

vz
z comp :  (
 vx
 vy
 vz
)   gz 



x

y
z
t
x
y
z
 2001, W. E. Haisler
38
Chapter 3: Conservation of Linear Momentum
At a given point, the nine components of a stress tensor
(three stresses each acting on the three x, y, z faces of a
cube) can always be resolved into vector components acting
on a face having a different orientation relative to the x,y,z
axes. Consider a unit normal vector
n  nxi  ny j  nzk
representing the unit normal to a surface. The dot product
n   will give the projection of  onto the plane defined
by the unit normal vector n , and yields the traction (stress)
vector t  n   acting on the surface with normal n .
n
NOTE: tn is not a vector normal to the plane but a
vector that acts on the plane with a unit normal
n.
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
z
t (n)
t(-j)
-i
n
-j
-k
x
t
(-k)
39
For a surface described by
f(x,y,z)=0, the gradient of f, f ,
gives the normal to the surface.
The unit normal is then given
t(-i)
f
n
 nx i  n y j  nz k .
f
y
Also, nx  cos x , n y  cos y ,
and nz  cos z where  x is the
angle between n and x-axis ( i ),
 y is angle between n and y-axis
( j ), and  z is angle between n
and z-axis ( k ).
t  t  i   j  k 
i
i
xx
xy
xz
 2001, W. E. Haisler
40
Chapter 3: Conservation of Linear Momentum
t  n    (n i  n j  n k )  
n
x
y
z
 (n i  n j  n k )  (ii   ij  ik 
x
y
z
xx
xy
xz
 ji 
 ki 
yx
zx
 jj
 kj
yy
zy
 in   jn   kn 
x xx
x xy
x xz
 in   jn   kn 
y yx
y yy
y yz
 in   jn   kn 
z zx
z zy
z zz
 jk 
 kk 
yz
zz
)
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
41
Thus, we have
tn  n    i (nx xx  n y yx  nz zx )
 j (nx xy  n y yy  nz zy )
 k (nx xz  n y yz  nz zz )
Note that the traction vector t has components on the normal
n
face which are in the x,y,z directions. This is an equilibrium
statement of forces in x, y and z directions. Often referred to as
the Cauchy tetrahedron.
 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
42
The Conservation of Linear Momentum is often defined in terms
of the deviatoric (or extra) stress tensor
S.
S   1 ( xx  yy  zz)I   PI
3
where P  1 ( xx  yy  zz)
3
P is the average normal (or hydrostatic) compressive stress.
the identity matrix.
I is
 2001, W. E. Haisler


















43
Chapter 3: Conservation of Linear Momentum


















 xx  P


















S xx
S xy
S xz
S yx
S yy
S yz   yx
S zx
S zy
S zz
 zx
 xy
 xz
 yy  P
 yz
 zy
 zz  P
Note: the sum of the diagonal terms of [S] = 0. The above can
be written as [] = [S] - P [I] where [I] = identity matrix.
Hence, Conservation of Linear Momentum becomes:
[ v (v )v]  g S P
t


















 2001, W. E. Haisler
Chapter 3: Conservation of Linear Momentum
44
Conservation of Linear Momentum has 3 components:
 xx  yx  zx
 vx
 vx
 vx
 vx
x comp :  (
 vx
 vy
 vz
)   gx 


x
y
z
t
x
y
z
 xy  yy  zy
 vy
 vy
 vy
 vy
y comp :  (
 vx
 vy
 vz
) gy 



x

y
z
t
x
y
z
 xz  yz  zz

vz

vz

vz

vz
z comp :  (
 vx
 vy
 vz
)   gz 



x

y
z
t
x
y
z
 Sxx  S yx  Szx  P
 vx
 vx
 vx
 vx
(
 vx
 vy
 vz
)   gx 



x
y
z x
t
x
y
z
 Sxy  S yy  Szy
 vy
 vy
 vy
 vy
(
 vx
 vy
 vz
) gy 


 P
x
y
z y
t
x
y
z
 Sxz  S yz  Szz  P

vz

vz

vz

vz
(
 vx
 vy
 vz
)   gz 




x

y

z
z
t
x
y
z