# Presentation BITS Pilani

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```					LESSION 2
Continued
Stress Equilibrium

We will formulate Differential Equations of
Motion for a Deformable body.
Plus the body force components Bx, By Bz will give us
Stress Equilibrium Equations

Multiply stresses
by area of face    σ xx    σ xy    σ xz
                 Bx  0
and sum.           x       y       z
 σ xy    σ yy    σ yz
Multiply Body                             By  0
x       y       z
Forces by volume.  σ       σ yz    σ zz
xz
                 Bz  0
x       y       z
Collect terms and
Simplify.
Cylindrical Coordinates

x1  r cosθ
x2  r sinθ
x3  z
er  e1 cos θ  e 2 sin θ
ˆ    ˆ          ˆ
e   e1 sin θ  e 2 cos θ
ˆ      ˆ          ˆ
ez  e3
ˆ    ˆ
Cylindrical Coordinates
 σ rr 1  σ rθ  σ rz σ rr  σ θθ 
                              Br  0
r     r      z          r
 σ rθ 1  σ θθ  σ θz 2 σ rθ
                       Bθ  0
r     r      z       r
 σ zr 1  σ θz  σ zz σ zr
                     Bz  0
r     r      z     r
Spherical Coordinates

rr 1 r      1 r 2rr       r cot 
                                                Br  0
r   r     r sin               r
r 1          1  3r       cot 
                                               B  0
r    r       r sin              r
r      1  1  3r  2 cot 
                                       B  0
r    r sin        r          r
Plane Polar Coordinates

 rr 1  r  rz  rr   
                           Br  0
r    r     z         r
 r 1  2 r
              B  0
r    r      r
Coordinate Relationships
Eulerian Coordinates     Lagrangian Coordinates
x  x  x,y,z 
*    *

x  x x* ,y* ,z*
y  y  x,y,z 
*     *
y  y  x ,y ,z 
*   *   *

z  z  x,y,z 
*    *
z  z  x ,y ,z 
*   *   *

Functional relationship between location in
undeformed and deformed (*) coordinates
Strain of a Line Element
Deformations

P  x, y,z   P* x* , y* ,z*   


Q  x  dx, y  dy,z  dz   Q* x*  dx* , y*  dy* ,z*  dz*   
PQ  ds  P Q  ds
*      *   *
Engineering Strain

ds  ds
*
εE 
ds

ε E  1
Coordinate Relationships

x  x  x,y,z 
*     *

y  y  x,y,z 
*     *

z  z  x,y,z 
*     *
Total Differentials
x   *
x      x
*         *
dx 
*
dx     dy     dz
x      y      z
y  *
y   *
y     *
dy 
*
dx     dy     dz
x      y      z
z  *
z   *
z     *
dz 
*
dx     dy     dz
x      y      z
Displacements

u  x  x  displacement in x-direction
*

v  y  y  displacement in y-direction
*

w  z  z  displacement in z-direction
*
Rewriting

x  xu
*

y  yv
*

z zw
*
Lengths of Line Segments

ds2
 dx   dy  dz 
2       2      2

ds   dx   dy   dz 
* 2        * 2      * 2     * 2
Magnification Factor
Normal Strains
Shear Strains
Direction cosines of ds
dx
l
ds
dy
m
ds
dz
n
ds
Interpretation

Let ds lie parallel to the x axis.
l=1 m=0 n=0

1 2
M x   Ex     Ex  xx
2
Interpretation
1 2
M x   Ex   Ex   xx
2
1 2
M y   Ey   Ey   yy
2
1 2
M z   Ez   Ez  zz
2
Final Direction of Line
dx      dy        dz
l      m        n
ds      ds        ds

*          *           *
dx          dy        dz
l  *
*
m  *
*
n  *
*
ds          ds        ds
Final Direction of Line
*
dx ds
l 
*
*
ds ds
*
dy ds
m 
*
*
ds ds
*
dz ds
n 
*
*
ds ds
Recall
x *      x *      x*
dx*       dx       dy      dz
x        y        z
y *
y   *
y    *
dy 
*
dx     dy     dz
x      y      z
x  xu
*
z *
z  *
z   *
dz 
*
dx     dy     dz
x      y      z            y  yv
*

z zw
*
Final Direction of Line
Final Direction of Line

ds       1

ds*
1  E
Final Direction of Line
 u  u    u
1  E  l  1   l+ m  n
*

 x  y    z
v    v   v
1  E  m  l+1   m  n
*
x  y     z
w w    w 
1  E  n  l+ m  1   n
*
x y      z 
Shear Strain

PA  ds1        P*A*  ds1   *
 * *
PB  ds2   P B  ds*
 

2
PA  PB   
PA  l1 ,m1 ,n1   PA  l1 ,m1 ,n1
*   * *
 
PB  l2 ,m2 ,n2   PB  l * ,m* ,n*
        2    2 2
Line segment PA and PB before and after deformation
Shear Strain


cos  l1l2 +m1m2 +n1n2  0
2

cos 
*     * *    * *    * *
=l1 l2 +m1 m2 +n1 n2
Final Direction of Line
 u  u    u
1  E  l  1   l+ m  n
*

 x  y    z
v    v   v
1  E  m  l+1   m  n
*
x  y     z
w w    w 
1  E  n  l+ m  1   n
*
x y      z 
Shear Strains
12  1   E1 1   E2  cos *

 2l1l2ε xx +2m1m2ε yy +2n1n2ε zx  2  l1m2  m1l2  ε zx
2  m1n2  m1n2  ε yz  2  l1n2  n1l2  ε xy

12  Engineering Shear Strain
between PA and PB
Shear Strains
Let PA and PB be oriented parallel to  x,y  axes

l1  1 m1  0 n1  0
l2  0 m2  1 n2  0

12   xy  2 xy
Shear Strains

 xy  2 xy
 yz  2 yz
 xz  2 xz
Small Strains and Rotations
 E1  1
 E2  1

 
*
2

12    1   E1 1   E2  cos    *
*
2
Engineering Shear Strain approximately equal
to the change in angle between PA and PB
Strain Transformation

Strain Tensor:
  xx    xy    xz 
                    
  xy    yy    yz 
  xz    yz    zz 
                    
Normal Strain Transformation
Shear Strain Transformation
Principal Strains

 xx  M     xy      xz
 xy   yy  M     yz  0
 xz      yz zz  M

M  I1M  I2 M  I3  0
3     2
Strain Invariants
M 3  I1M 2  I2 M  I3  0

I1   xx   yy  zz

xx        xy          xx    xz        yy    yz
I2                                              
xy         yy         xz   zz         yz   zz
   xy
2
    xz
2
    yz
2
  xx  yy   xx zz   yy zz
Strain Invariants

xx xy  xz
I 3  xy  yy  yz
xz  yz zz
 xx  yy zz    2xy  yz xz   xx  yz
2
  yy xz
2
  zz  xy
2
Strain Invariants

I1  M1  M 2  M 3
I 2   M1M 2  M 2 M 3  M 3M1
I 3  M1M 2 M 3
Principal Directions
l  xx  M   mxy  nxz  0

l xy  m   yy  M   n yz  0

l xz  m yz  n  zz  M   0

l  m  n 1
2     2     2
Small Displacement
Normal Strains
u
ε xx 
x
v
ε yy 
y
w
ε zz 
z
Small Displacement
Shear Strains
1  v u 
ε xy  ε yx    
 x y 
2        
1  w u 
ε xz  ε zx       
2  x z 
1  w v 
ε yz  ε zy   y  z 

2          
M  E
Strain Compatibility
u    2ε xx     3u
ε xx                
x    y 2
xy 2
v      2ε yy
 3v
ε yy              2
y   x 2
x y
1  v u            2ε xy
 3u      3v
ε xy      2                 2
2  x y  xy xy  2
x y

 ε xx
2      2ε yy         2ε xy
      2
y 2
x 2
xy
Strain Compatibility
 ε yy
2
 ε xx
2
 ε xy
2

        2
x   2
y 2
xy
 ε zz  ε xx
2          2
 ε xz
2
       2
x  2
z 2
xz
 ε zz  ε yy     ε yz
2     2         2

       2
y 2
z 2
yz
Strain Compatibility
 ε zz
2
 ε xy
2
 ε yz
 ε zx
2       2
        
xy z  2
zx yz
 ε yy
2
 ε xz  ε xy  ε yz
2       2       2

             
xz    y 2
yz xy
 ε xx  ε yz  ε xz  ε xy
2         2        2      2

            
yz    x 2
xy xz
Shear Strain

2ε xy  γ xy
2ε xz  γ xz
2ε yz  γ yz
Cylindrical Coordinates
u                       1 u v v
ε rr           γ rθ  2 ε rθ        
r                       r  r r
u 1 v                   u w
ε θθ          γ rz  2 ε rz     
r r θ                   z r
w                       v 1 w
ε zz           γ zθ  2 ε zθ     
z                       z r 
Spherical Coordinates
u
ε rr 
r
u 1 v
ε θθ  
r r θ
u v          1 w
ε    cot  
r r       r sin  
Spherical Coordinates
1 u v v
γ rθ  2 ε rθ        
r  r r
1 u w w
γr   2 εr               
r sin   r r
1  w               1 v
γθ   2 εθ         w cot   
r              r sin  
Transformations

If we use tensor strain components
then strain transforms just like stress.
Principal values, principal directions
and invariants are also just like stress.

Mohr’s circle works for strain as well!
Equations
 3 - Stress Equilibrium Equations

 6 - Strain Displacement Equations

 6 – Compatibility Equations (Derived
from Strain Displacement)

End of session 2_2

```
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