MCE 571 Theory of Elasticity
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Transcript MCE 571 Theory of Elasticity
Linear Elastic Constitutive Solid Model
Develop Force-Deformation Constitutive Equation in the
Form of Stress-Strain Relations Under the Assumptions:
•
•
•
•
•
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
Typical One-Dimensional Stress-Strain Behavior
Steel
Cast Iron
Tensile Sample
Aluminum
Applicable Region for
Linear Elastic Behavior
=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
ij Cijklekl
xy C 41e x C 42 e y C 43 e z 2C 44 e xy 2C 45 e yz 2C 46 e zx
with
yz C51e x C52 e y C53 e z 2C54 e xy 2C55 e yz 2C56 e zx
Cijkl C jikl
zx C61e x C62 e y C63 e z 2C64 e xy 2C65 e yz 2C66 e zx
or
x C11
C
y 21
z
xy
yz
zx C61
C12
C16 e x
ey
ez
2e xy
2e yz
C66 2ezx
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
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 ijkl ik jl il jk
ij ekk ij 2eij Generalized Hooke’s Law
x ( e x e y e z ) 2e x
y ( e x e y e z ) 2e y
z ( e x e y e z ) 2e z
xy 2e xy
yz 2e yz
zx 2e zx
- 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
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
/ 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
31 2
3k E
6k
E 2
2
2
ER
E
33 E
E 3 R
6
3k 1 2
k
21
1 1 2
9k
6k
9k k
3k
3 2
3k 2
6k 2
3k
2( )
k
21
31 2
1
3
E
21
3kE
9k E
E 3 R
4
3k 1 2
21
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 92 2E
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
Hooke’s Law in Cylindrical Coordinates
r
σ r
rz
x3
z
z
z
rz
x2
r
x1
d
dr
z
rz
z
z
r ( er e ez ) 2er
r
r
r
( er e e z ) 2e
z ( er e ez ) 2ez
r 2er
z 2ez
zr 2ezr
Hooke’s Law in Spherical Coordinates
R
σ R
R
x3
R
R
R
x1
R
R ( eR e e ) 2eR
R
R
x2
( eR e e ) 2e
( eR e e ) 2e
R 2eR
2e
R 2eR