MCE 571 Theory of Elasticity

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Linear Elastic Constitutive Solid Model
Develop Force-Deformation Constitutive Equation in the
Form of Stress-Strain Relations Under the Assumptions:
•
•
•
•
•
Elasticity
Solid Recovers Original Configuration When Loads Are Removed
Linear Relation Between Stress and Strain
Neglect Rate and History Dependent Behavior
Include Only Mechanical Loadings
Thermal, Electrical, Pore-Pressure, and Other Loadings Can Also
Be Included As Special Cases
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Typical One-Dimensional Stress-Strain Behavior

Steel
Cast Iron
Tensile Sample
Aluminum

Applicable Region for
Linear Elastic Behavior
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
=E
Linear Elastic Material Model
Generalized Hooke’s Law
 x  C11e x  C12 e y  C13 e z  2C14 e xy  2C15 e yz  2C16 e zx
 y  C 21e x  C 22 e y  C 23 e z  2C 24 e xy  2C 25 e yz  2C 26 e zx
 z  C31e x  C32 e y  C33 e z  2C34 e xy  2C35 e yz  2C36 e zx
 xy  C 41e x  C 42 e y  C 43 e z  2C 44 e xy  2C 45 e yz  2C 46 e zx
 yz  C51e x  C52 e y  C53 e z  2C54 e xy  2C55 e yz  2C56 e zx
 zx  C61e x  C62 e y  C63 e z  2C64 e xy  2C65 e yz  2C66 e zx
or
Elasticity
  x  C11
   C
 y   21
 z   
 
  xy   
  yz   
  
  zx  C61
C12





   C16   e x 
  
   ey 


  
   ez 


  
  2e xy 
  
  2e yz 


   C66   2ezx 
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
ij  Cijklekl
with
Cijkl  C jikl
Cijkl  Cijlk
36 Independent
Elastic Constants
Anisotropy and Nonhomogeneity
Anisotropy -
Differences in material properties under different directions. Materials like
wood, crystalline minerals, fiber-reinforced composites have such behavior.
Typical Wood Structure
(Body-Centered Crystal)
(Hexagonal Crystal)
(Fiber Reinforced Composite)
Note Particular Material Symmetries Indicated by the Arrows
Nonhomogeneity - Spatial differences in material properties.
Soil materials in the
earth vary with depth, and new functionally graded materials (FGM’s) are now being
developed with deliberate spatial variation in elastic properties to produce desirable behaviors.
Gradation Direction
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Isotropic Materials
Although many materials exhibit non-homogeneous and anisotropic
behavior, we will primarily restrict our study to isotropic solids. For
this case, material response is independent of coordinate rotation
Cijkl  QimQ jnQkp Qlq Cmnpq
ij  Cijklekl
Cijkl  ijkl  ik  jl  il  jk
ij  ekk ij  2eij Generalized Hooke’s Law
 x   ( e x  e y  e z )  2e x
 y   ( e x  e y  e z )  2e y
 z   ( e x  e y  e z )  2e z
 xy  2e xy
 yz  2e yz
 zx  2e zx
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
 - Lamé’s constant
 - shear modulus or modulus of rigidity
Isotropic Materials
Inverted Form - Strain in Terms of Stress
eij 
1 

 ij   kk ij
E
E






1
 x   ( y   z )
E
1
e y   y   ( z   x )
E
1
e z   z   ( x   y )
E
1 
1
e xy 
 xy 
 xy
E
2
1 
1
e yz 
 yz 
 yz
E
2
1 
1
e zx 
 zx 
 zx
E
2
ex 
(3  2)
... Young' s modulus or modulus of elasticity



... Poisson' s ratio
2(    )
E
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Physical Meaning of Elastic Moduli
Simple Tension
Pure Shear
Hydrostatic
Compression
p




p

p

  0 0
 ij   0 0 0


 0 0 0
0  0
 ij    0 0


0 0 0
E   / ex
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
   / 2exy
  /  xy
0 
 p 0
 ij   0  p 0    p ij


0  p 
 0
p  kekk  k
k
E
. . . Bulk Modulus
3(1  2)
Relations Among Elastic Constants
E

E,
E

E,k
E
E,
E
E,
E
,k
 ,
 ,
k,
k,
,
k
E
31  2
3k  E
6k
E  2
2
2
ER
E
33  E 
E  3  R
6
3k 1  2 

k
21   

1   1  2 

9k
6k  
9k k   
3k  
3  2


3k  2
6k  2

3k  

2(  )
k
21  
31  2
1   
3
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island

E  3  R
4
3k 1  2 
21   

1  2 
2

E
1  1  2
3k 3k  E 
9k  E
E  2
3  E

3k
1 
2 
1  2

k

2
k 
3
k
3
( k  )
2

3  2
3


R  E 2  92  2E
Elasticity

E
21   
3kE
9k  E
Typical Values of Elastic Moduli for Common
Engineering Materials
E (GPa)

(GPa)
(GPa)
k(GPa)
(10-6/oC)
Aluminum
68.9
0.34
25.7
54.6
71.8
25.5
Concrete
27.6
0.20
11.5
7.7
15.3
11
Cooper
89.6
0.34
33.4
71
93.3
18
Glass
68.9
0.25
27.6
27.6
45.9
8.8
Nylon
28.3
0.40
10.1
4.04
47.2
102
Rubber
0.0019
0.499
0.654x10-3
0.326
0.326
200
Steel
207
0.29
80.2
111
164
13.5
Elasticity
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Hooke’s Law in Cylindrical Coordinates
 r
σ   r

  rz
x3
z
z
z
rz
   ( er  e  e z )  2e
x2
r
x1
d

dr
 z  ( er  e  ez )  2ez
 r  2er
 z  2ez
 zr  2ezr
Elasticity

 z
 rz 
 z 

 z 
 r  ( er  e  ez )  2er
r
r

 r
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
Hooke’s Law in Spherical Coordinates
R

σ    R
  R
x3
R
R
R


 R

 
 R   ( eR  e  e )  2eR
R

    ( eR  e  e )  2e


x1
x2
    ( eR  e  e )  2e
 R  2eR
   2e
 R  2eR
Elasticity
 R 

  
  
Theory, Applications and Numerics
M.H. Sadd , University of Rhode Island
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