Linear Elastic Constitutive Solid Model

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29 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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

Tensile Sample

Steel

Cast Iron

Aluminum





Applicable Region for
Linear Elastic Behavior



=
E




Linear Elastic Material Model

Generalized Hooke’s Law

zx
yz
xy
z
y
x
zx
zx
yz
xy
z
y
x
yz
zx
yz
xy
z
y
x
xy
zx
yz
xy
z
y
x
z
zx
yz
xy
z
y
x
y
zx
yz
xy
z
y
x
x
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
e
C
66
65
64
63
62
61
56
55
54
53
52
51
46
45
44
43
42
41
36
35
34
33
32
31
26
25
24
23
22
21
16
15
14
13
12
11
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2





































































































































zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
e
e
e
e
e
e
C
C
C
C
C
C
2
2
2
66
61
21
16
12
11
kl
ijkl
ij
e
C


ijlk
ijkl
jikl
ijkl
C
C
C
C


or

36 Independent
Elastic Constants

with

Anisotropy and
Nonhomogeneity

Anisotropy

-

Differences in material properties under different directions. Materials like
wood, crystalline minerals, fiber
-
reinforced composites have such behavior.


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.

(Body
-
Centered Crystal)

(Fiber Reinforced Composite)

(Hexagonal Crystal)

Gradation Direction

Typical Wood Structure

Note Particular Material Symmetries Indicated by the Arrows


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

mnpq
lq
kp
jn
im
ijkl
C
Q
Q
Q
Q
C

jk
il
jl
ik
kl
ij
ijkl
C









kl
ijkl
ij
e
C


ij
ij
kk
ij
e
e






2
zx
zx
yz
yz
xy
xy
z
z
y
x
z
y
z
y
x
y
x
z
y
x
x
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e






























2
2
2
2
)
(
2
)
(
2
)
(
Generalized
Hooke’s
Law



-

Lamé’s

constant




-

shear
modulus

or
modulus of rigidity

Isotropic
Materials

Inverted Form
-

Strain in Terms of Stress

ij
kk
ij
ij
E
E
e








1






zx
zx
zx
yz
yz
yz
xy
xy
xy
y
x
z
z
x
z
y
y
z
y
x
x
E
e
E
e
E
e
E
e
E
e
E
e










































2
1
1
2
1
1
2
1
1
)
(
1
)
(
1
)
(
1
elasticity

of

modulus
or

modulus

s
Young'
.
.
.
)
2
3
(








E
ratio

s
Poisson'
.
.
.
)
(
2






Physical Meaning of Elastic Moduli














0
0
0
0
0
0
0
0
ij
x
e
E
/
















0
0
0
0
0
0
0
ij
xy
xy
e






/
2
/
ij
ij
p
p
p
p


















0
0
0
0
0
0
Modulus
Bulk
.
.
.
)
2
1
(
3








E
k
k
ke
p
kk
Simple Tension

Hydrostatic

Compression

Pure Shear













p

p

p




Relations Among Elastic Constants


E



k





E,


E







2
1
3
E





1
2
E










2
1
1
E

E,k

E

k
E
k
6
3


k

E
k
kE

9
3



E
k
E
k
k


9
3
3

E,


E




2
2
E



E
E



3
3





E
E





3
2

E,


E

R
E




2

6
3
R
E




4
3
R
E







,k





2
1
3
k



k









1
2
2
1
3
k




1
3
k


,







1
2












2
1
3
1
2






2
1
2


,












2
1
1









3
1







2
2
1



k,





k
k
6
9





2
6
2
3
k
k

k





3
2
k

k,








k
k
k
3
9




k
3

k

)
(
2
3


k




,











2
3

)
(
2





3
2
3














E
E
R
2
9
2
2
Typical Values of Elastic Moduli for Common
Engineering
Materials



E

(
GPa
)




(
GPa
)


(
GPa
)

k
(
GPa
)


⠱0
-
6
/
o
C
)

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


























z
z
rz
z
r
rz
r
r
σ
zr
zr
z
z
r
r
z
z
r
z
z
r
r
z
r
r
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e







































2
2
2
2
)
(
2
)
(
2
)
(



x
3

x
1

x
2

r



z

dr


z


r


r



rz



z

d





Hooke’s Law in
Spherical
Coordinates





























R
R
R
R
R
σ
R
R
R
R
R
R
R
R
R
e
e
e
e
e
e
e
e
e
e
e
e
e
e
e














































2
2
2
2
)
(
2
)
(
2
)
(

R

x
3

x
1

x
2

R






R









R





