theory of elasticity

1. School of Civil Engineering Spring 2007
CE 595:
Finite Elements in Elasticity
Instructors: Amit Varma, Ph.D.
Timothy M. Whalen, Ph.D.

2. Review of Elasticity -2-
Section 1: Review of Elasticity
1. Stress & Strain
2. Constitutive Theory
3. Energy Methods

3. Review of Elasticity -3-
Section 1.1: Stress and Strain
 Stress at a point Q :
0 0 0
lim ; lim ; lim .
y
x z
x xy xz
A A A
F
F F
A A A
  

 
  
  
  
  
   
Stress matrix ( ) ; Stress vector ( ) .
x
y
x xy xz
z
xy y yz
xy
xz yz z
yz
xz
Q Q


  

  

  


 
 
 
 
 
 
   
 
 
 
   
 
 
 
σ σ

4. Review of Elasticity -4-
1.1: Stress and Strain (cont.)
 Stresses must satisfy equilibrium equations in pointwise
manner:
“Strong Form”

5. Review of Elasticity -5-
1.1: Stress and Strain (cont.)
 Stresses act on inclined surfaces as follows:
 
  
   
ˆ
2 2
ˆ ˆ
ˆ
( ) .
ˆ; .
x xy xz x
xy y yz y
xz yz z z
n
Q n
n
Q
Q Q
  
  
  
  
  
  
   
 
  
 

  
n
n n
S
σ n
S n S

6. Review of Elasticity -6-
1.1: Stress and Strain (cont.)
 Strain at a pt. Q related to displacements :
   
     
 
 
 
: , , : , ,
Displacement functions
, , , , , , , ,
defined by:
, , ;
, , ;
, , .
Q x y z Q x y z
u x y z v x y z w x y z
x x u x y z
y y v x y z
z z w x y z
   

  
  
  

7. Review of Elasticity -7-
1.1: Stress and Strain (cont.)
 Normal strain relates to changes in size :
       
   
     
;
, , = , , .
, ,
. Also, ; .
x
D Q
x y z
Q D QD Q D dx
QD dx
Q D x x x dx u x dx y x u x y dx u x dx y u x y
u x dx y u x y u v w
Q Q Q
dx x y z

  
 
   
 
 
              
 
    
    
  

8. Review of Elasticity -8-
1.1: Stress and Strain (cont.)
 Shearing strain relates to changes in angle :
   
           
, ,
= . . .
xy xz yz
v x dx y u x y dy v u w u w v
Q Q Q Q Q Q
dx dy x y x z y z
    
       
        
     

9. Review of Elasticity -9-
1.1: Stress and Strain (cont.)
 Sometimes FEA programs use elasticity
shearing strains :
 Strains must satisfy 6 compatibility equations:
(usually automatic for most formulations)
1 1 1
2 2 2
. . .
xy xy xz xz yz yz
     
  
2 2
2
2 2
E.g.: .
xy y
x
x y y x
 

 

 
   

10. Review of Elasticity -10-
Section 1.2 : Constitutive Theory
 For linear elastic materials, stresses and strains are
related by the Generalized Hooke’s Law :
       
    .
o o
  
σ C ε ε σ
     
11 12 13 14 15 16
12 22 23 24 25 26
13 23 33 34 35 36
14 24 34 44 45 46
15 25 35 45 55 56
16 26 36 46 56 66
; ;
x x
y y
z z
xy xy
yz yz
xz xz
c c c c c c
c c c c c c
c c c c c c
c c c c c c
c c c c c c
c c c c c c
 
 
 
 
 
 
    
    
    
    
   
   
   

   

   

   

   
    
σ ε C
   
;
; .
o o
Elasticity matrix
residual stresses residual strains










 
σ ε

11. Review of Elasticity -11-
1.2 : Constitutive Theory (cont.)
 For isotropic linear elastic materials, elasticity matrix
takes special form:
 
    
 
 
1
2
1
2
1
2
1 0 0 0
1 0 0 0
1 0 0 0
.
0 0 0 1 2 0 0
1 2 1
0 0 0 0 1 2 0
0 0 0 0 0 1 2
= Young's modulus, = Poisson's ratio.
E
E
  
  
  

 




 
 

 
 

  

   
 

 

 
 
C

12. Review of Elasticity -12-
1.2 : Constitutive Theory (cont.)
 Special cases of GHL:
– Plane Stress : all “out-of-plane” stresses assumed zero.
– Plane Strain : all “out-of-plane” strains assumed zero.
     
 
 
2
1
2
1 0
1 0 .
1
0 0 1
; ;
require
Note: d.
1
x x
y y
xy xy
z x y
E
 
 
 

  





 
   
   
 
   
   
   
 
 
  

 





C
σ ε
     
 
2
1
1 0
1
1
1 0 .
1
2
0 0
1
Note
; ;
requir
: ed.
x x
y y
xy xy
z x y
E
 
 
 


 

   


 


   
   
 
   
   
   
 


 
  
 
 

 
 
 

 
σ ε C

13. Review of Elasticity -13-
1.2 : Constitutive Theory (cont.)
 Other constitutive relations:
– Orthotropic : material has “less” symmetry than isotropic case.
FRP, wood, reinforced concrete, …
– Viscoelastic : stresses in material depend on both strain and strain rate.
Asphalt, soils, concrete (creep), …
– Nonlinear : stresses not proportional to strains.
Elastomers, ductile yielding, cracking, …

14. Review of Elasticity -14-
1.2 : Constitutive Theory (cont.)
 Strain Energy
– Energy stored in an elastic material during deformation; can be
recovered completely.
  
  
 
 
Work done during 1 1 :
.
; .
.
.
If all external work is stored,
.
final
o
final
o
x o x o
x x o o
o o x x
o x x
dW F dF dL FdL
F A dL d L
dW d A L
W A L d
U W V d




 
 
 
 


  
 
 
 
 



15. Review of Elasticity -15-
1.2 : Constitutive Theory (cont.)
 Strain Energy Density : strain energy per unit
volume.
 In general,
.
final
o
o x x
Volume
U U V d
U UdV


 
 
 


.
final final final final final final
o o o o o o
x x y y z z xy xy yz yz xz xz
U d d d d d d
     
     
           
     
     

16. Review of Elasticity -16-
Section 1.3 : Energy Methods
 Energy methods are techniques for satisfying equilibrium
or compatibility on a global level rather than pointwise.
 Two general types can be identified:
– Methods that assume equilibrium and enforce displacement
compatibility.
(Virtual force principle, complementary strain energy theorem, …)
– Methods that assume displacement compatibility and enforce
equilibrium.
(Virtual displacement principle, Castigliano’s 1st theorem, …)
Most important for FEA!

17. Review of Elasticity -17-
1.3 : Energy Methods (cont.)
 Principle of Virtual Displacements (Elastic case):
(aka Principle of Virtual Work, Principle of Minimum Potential Energy)
 Elastic body under the action of body force b
and surface stresses T.
 Apply an admissible virtual displacement
– Infinitesimal in size and speed
– Consistent with constraints
– Has appropriate continuity
– Otherwise arbitrary
 PVD states that for any admissible
is equivalent to static equilibrium.
u
e i
W W
 
 u

18. Review of Elasticity -18-
1.3 : Energy Methods (cont.)
 External and Internal Work:
 So, PVD for an elastic body takes the form
  
 
 
ˆ .
.
e
volume surface volume surface
i
volume
x x y y z z xy xy yz yz xz xz
volume
W dV dA dV dA
W U U dV
dV

 
           
       
 
           
   


b δu T δu b δu σ n δu
δu
      
ˆ .
volume surface volume
dV dA dV
    
  
b δu σ n δu σ δε

19. Review of Elasticity -19-
1.3 : Energy Methods (cont.)
 Recall: Integration by Parts
 In 3D, the corresponding rule is:
            .
b b
b
a
a a
f x g x dx f x g x g x f x dx
 
 
 
 
 
           
, , , , , , , , , , , , .
x
volume surface volume
g f
f x y z x y z dV f x y z g x y z n dA g x y z x y z dV
x x
 
 
 
  

20. Review of Elasticity -20-
     
     
+
+
yz yz
yz yz yz y yz z
volume surface volume surface volume
xz xz
xz xz xz x xz z
volume surface volume surface
dV w n dA w dV v n dA v dV
y z
dV w n dA w dV u n dA u
x z
 
       
 
       
 
   
       
   
 
   
 
  
       
 
 
 
    
    volume
dV

 
 

   
     
+ .
xy
xy xy
xy xy xy x xy y
volume surface volume surface volume
v u
x y
dV v n dA v dV u n dA u dV
x y
 

 
       
 
  
 
 
   
       
   
 
   
    
1.3 : Energy Methods (cont.)
 Take a closer look at internal work:
 
    .
x
x x x x x
volume surface volume
u
dV u n dA u dV
x x
 
     
 
 
      
 
 
 
  
 
    z
z z z z z
volume surface volume
w
dV w n dA w dV
z z
 
     
 
 
      
 
 
 
  
 
    y
y y y y y
volume surface volume
v
dV v n dA v dV
y y


     

  
      
 
 
 
  

21. Review of Elasticity -21-
1.3 : Energy Methods (cont.)
  
ˆ
surface
x
xy
x xz
x xy x
xz yz z
yz
xz z
x
y y yz
xy y y
i y
surface
z
d
z
A
z n
W n d
x y z
v
x y z
A
n w
x z
u
y
  


 
   
  
  
 






 
 
 
 
 
    
 
 
      
     
 
 
    
 
 
  



    
  
 

  
  


 

  

 


σ n δu
     
 
ˆ ˆ
for an
volume
i e
surface volume volume surface
volume
dV
W W dA dV dV dA
dV arbit y
u
r
v
ra
w





 
 
 
 
 



         
   
   

   

A
σ n δu A δu b δu σ n δu
A b δu 0 δu
A b 0
• By reversing the steps, can show that
the equilibrium equations imply
• is called the weak form of
static equilibrium.
i e
W W
 

i e
W W
 


22. Review of Elasticity -22-
1.3 : Energy Methods (cont.)
 Rayleigh-Ritz Method : a specific way of implementing
the Principle of Virtual Displacements.
– Define total potential energy ; PVD is then stated
as
– Assume you can approximate the displacement functions as a
sum of known functions with unknown coefficients.
– Write everything in PVD in terms of virtual displacements and
real displacements. (Note: stresses are real, not virtual!)
– Using algebra, rewrite PVD in the form
– Each unknown virtual coefficient generates one equation to
solve for unknown real coefficients.
i e
W W
  
0
i e
W W
  
  
   
1
unknown virtual coefficient * equation involving real coefficients 0
n
i i
i



23. Review of Elasticity -23-
1.3 : Energy Methods (cont.)
 Rayleigh-Ritz Method: Example
Given: An axial bar has a length L, constant modulus of elasticity E, and a
variable cross-sectional area given by the function ,
where β is a known parameter. Axial forces F1 and F2 act at x = 0 and x=
L, respectively, and the corresponding displacements are u1 and u2 .
Required: Using the Rayleigh-Ritz method and the assumed displacement
function , determine the equation that relates the
axial forces to the axial displacements for this element.
 
 
( ) 1 sin x
L
o
A x A 

 
   
1 2
( ) 1 x x
L L
u x u u
  

24. Review of Elasticity -24-
1.3 : Energy Methods (cont.)
Solution :
1) Treat u1 and u2 as unknown parameters. Thus, the virtual
displacement is given by
2) Calculate internal and external work:
   
1 2
( ) 1 x x
L L
u x u u
  
  
   
 
2 1
2 1 2 1
1 1 2 2
1 1
1 2
(no body force terms).
( ) .
* * .
and * .
e
i x x x x
bar bar
u u
L L L
x
u u u u
L L
x x
W F u F u
W dV A x dx
u
u u
x
E
 
  
    

 

 
 
 

     

  
 

25. Review of Elasticity -25-
1.3 : Energy Methods (cont.)
(Cont) :
2)
3) Equate internal and external work:
   
 
 
   
  
    
 
2 1 2 1
2 1 2 1
2 1 2 1
0
2
2 2
2 1
* * * 1 sin
* * * *
* 1 * 1 .
x L
u u u u x
L L L
i o
x
u u u u L
L L o
u u u u
L L
i o o
W E A dx
E A L
W u EA u EA
  
  

 
 
 
  

 

 
 
  
 
    

  
    
 
  
  
 
2 1 2 1
1 2
2 1
2 2
1 1 2 2 2 1
2
1 1 1 1
2
2
2 2
2 2
* 1 * 1 .
For : 1 1 1
1 .
1 1
For : 1
u u u u
L L
o o
u u
L
o
o
u u
L
o
F u F u u EA u EA
u F EA u F
EA
u F
L
u F EA
 
 

 



   


 


    

      
 

  
    
 

      

