slide cơ học vật chất rắn chapter 4 new material behavior linear elastic solid

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ng
co
an
th

cu

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du
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Chapter 4: Material Behavior – Linear
elastic solid

TDT  University  -­‐  2015
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4.1 Material characterization

 

co

ng

4.2 Linear elastic material – Hooke’s law

an

4.3 Physical meaning of elastic module

cu

u

du
o

ng

th

4.4 Thermo-elastic constitutive relations

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4.1 Material characterization
 

co

an

4.3 Physical meaning of elastic module

ng

4.2 Linear elastic material – Hooke’s law

cu

u

du
o

ng

th

4.4 Thermo-elastic constitutive relations

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⎧ ∂σ x ∂τ yx ∂τ zx
⎪ ∂x + ∂y + ∂z + Fx = 0
⎪
⎪∂τ xy ∂σ y ∂τ zy
+
+
+ Fy = 0
⎨
∂
x
∂
y
∂
z
⎪
⎪ ∂τ
∂τ yz ∂σ z
xz

+
+
+ Fz = 0
⎪
∂y
∂z
⎩ ∂x

ng
co
an
th

So far, we studied
∂u
∂v
∂w
⎧
ε
=
,
ε
=
,
ε
=
y
z
⎪ x ∂x
∂y

∂z
⎪
∂u ∂v
∂v ∂w
⎪
γ
=
+
,
γ
=
+
,
⎨ xy
yz
∂
y
∂
x
∂
z
∂
y
⎪
⎪
∂w ∂u
γ
=
+
⎪ zx

∂
x
∂z
⎩

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4.1 Material characterization
 

+ 3 equilibrium equations

du
o

ng

+ 6 strain-displacement equations

cu

u

⎧ ∂ 2 ex ∂ 2 e y
∂ 2exy
∂ 2 ex
∂ ⎛ ∂eyz ∂ezx ∂exy ⎞
=
+

+
+
=
2
;
⎪ 2
⎜−
⎟
2
∂
y
∂
z
∂
x
∂
x
∂
y
∂
z
∂
y
∂
x
∂
x
∂
y
⎝

⎠
⎪ 2
2
2
2
∂ eyz
∂ ey
∂ ⎛ ∂ezx ∂exy ∂e yz ⎞
⎪ ∂ e y ∂ ez
+
=
2
;
=
+
+
⎨ 2
⎜−
⎟
2
∂
z
∂
y
∂
y
∂
z
∂
z

∂
x
∂
y
∂
y
∂
z
∂
x
⎝
⎠
⎪ 2
2
2
2
∂ ⎛ ∂exy ∂e yz ∂ezx ⎞
⎪ ∂ ez + ∂ ex = 2 ∂ ezx ; ∂ ez
=
+
+
⎜−
⎟
⎪ ∂x 2 ∂z 2
∂z∂x
∂
x
∂
y
∂

z
∂
z
∂
x
∂
y
⎝
⎠
⎩
+ 6 compatibility equation
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4.1 Material characterization
 

ng

In which, 6 compatibility equations represent only 3 independent relations, and

these equations are needed only ensure that a given strain field will produce singlevalued continuous displacements. => No need for the general problems

th

an

co

Excluding the compatibility relations, it is found that we have 9 field equations. The
unknowns in these equations include 3 displacement components, 6 components of
strain, and 6 stress components => total 15 unknowns.

cu

u

du
o

ng

So far, 9 equations are not sufficient to solve for 15 unknowns.
•  We need additional field equations
•  The material response => the relationship between the strains and stresses.

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4.1 Material characterization
 

u

du
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ng

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

cu

• 
• 
• 
• 

• 

th

an

co

ng

Mechanical behavior of solids is normally defined by constitutive stress-strain
relations. Commonly, these relations express the stress as a function of the strain,
strain rate, strain history, temperature, and material properties. Here, we use the
Linear Elastic Constitutive Solid Model in which the Stress-Strain Relations are
under the Assumptions:

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σ

co

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4.1 Material characterization
 

an

Typical One-Dimensional

Tensile Sample

Cast Iron
Aluminum

u

du
o

ng

th

Stress-Strain Behavior

Steel

cu

ε
Applicable Region for
Linear Elastic Behavior

σ = Eε

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4.1 Material characterization
 

an

4.3 Physical meaning of elastic module

co

ng

4.2 Linear elastic material – Hooke’s law

cu

u

du
o

ng

th

4.4 Thermo-elastic constitutive relations

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σ ij = Cijkmε km

C1113

C1211

C1212

C1213

C1311

C1312

C1122

C1123

C1221

C1222

C1223

C1321

C1322

C1132

C1133

C1231

C1232

C1233

C1331

C1332

co

ng

C1112

an

C3123 C3221 C3222
C3133 C3231 C3232

du
o

C3122
C3132

ng

th

C2132 C2133 C2231 C2232 C2233 C2331 C2332
C3112 C3113 C3211 C3212 C3213 C3311 C3312
C3223 C3321 C3322

C3233 C3331 C3332

C1313 ⎤
C1323 ⎥⎥
C1333 ⎥
⎥
C2313 ⎥ ⎡ε11 ε12 ε13 ⎤
C2323 ⎥ : ⎢⎢ε 21 ε 22 ε 23 ⎥⎥
⎥
C2333 ⎥ ⎢⎣ε 31 ε 32 ε 33 ⎥⎦
C3313 ⎥
⎥
C3323 ⎥
C3333 ⎥⎦

C1122 C1133 C1112 C1113 C1123 ⎤ ⎡ ε11 ⎤
C2222 C2233 C2212 C2213 C2223 ⎥⎥ ⎢⎢ ε 22 ⎥⎥
C3322 C3333 C3312 C3313 C3323 ⎥ ⎢ ε 33 ⎥
⎥⎢
⎥
C1222 C1233 C1212 C1213 C1223 ⎥ ⎢ 2ε12 ⎥
C1322 C1333 C1312 C1313 C1323 ⎥ ⎢ 2ε13 ⎥
⎥⎢
⎥
C2322 C2333 C2312 C2313 C2323 ⎥⎦ ⎢⎣ 2ε 23 ⎥⎦

u

⎡σ 11 ⎤ ⎡ C1111
⎢σ ⎥ ⎢C

⎢ 22 ⎥ ⎢ 2211
⎢σ 33 ⎥ ⎢C3311
⎢ ⎥=⎢
⎢σ 12 ⎥ ⎢ C1211
⎢σ 13 ⎥ ⎢ C1311
⎢ ⎥ ⎢
⎢⎣σ 23 ⎥⎦ ⎢⎣C2311

( K , M = 1, 2,3, 4,5, 6)

C2112 C2113 C2211 C2212 C2213 C2311 C2312
C2122 C2123 C2221 C2222 C2223 C2321 C2322

cu

⎡σ 11 σ 12
⎢σ
⎢ 21 σ 22
⎢⎣σ 31 σ 32

⎡ C1111
⎢C
⎢ 1121
⎢ C1131
⎢
σ 13 ⎤ ⎢C2111
σ 23 ⎥⎥ = ⎢C2121
⎢
σ 33 ⎥⎦ ⎢C2131
⎢C

⎢ 3111
⎢C3121
⎢C
⎣ 3131

σ K = CKM ε M

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4.2 Linear elastic material – Hooke’s law

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4.2 Linear elastic material – Hooke’s law

with

Cijkl = C jikl ; Cijkl = Cijlk

du
o

( Due to the symmetry of
stress and strain tensors)

ng

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an

co

σ ij = Cijkl ekl

ng

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⎧σ x = C11ex + C12e y + C13ez + 2C14exy + 2C15e yz + 2C16ezx
⎪
⎪σ y = C21ex + C22e y + C23ez + 2C24exy + 2C25e yz + 2C26ezx
⎪σ = C e + C e + C e + 2C e + 2C e + 2C e
31 x
32 y
33 z
34 xy
35 yz
36 zx
⎪ z

⎨
⎪τ xy = C41ex + C42e y + C43ez + 2C44exy + 2C45e yz + 2C46ezx
⎪τ = C e + C e + C e + 2C e + 2C e + 2C e
51 x
52 y
53 z
54 xy
55 yz
56 zx
⎪ yz
⎪⎩τ zx = C61ex + C62e y + C63ez + 2C64exy + 2C65e yz + 2C66ezx

cu

u

⎡σ x ⎤ ⎡ C11 C12
⎢σ ⎥ ⎢C
⋅
⎢ y ⎥ ⎢ 21
⎢σ ⎥ ⎢ ⋅
⋅
or ⎢ z ⎥ = ⎢
⋅
⎢τ xy ⎥ ⎢ ⋅
⎢τ yz ⎥ ⎢ ⋅
⋅
⎢ ⎥ ⎢
⎣τ zx ⎦ ⎣C61 ⋅

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⋅ ⋅ ⋅ C16 ⎤ ⎡ ex ⎤
⋅ ⋅ ⋅ ⋅ ⎥ ⎢ ey ⎥
⎥⎢
⎥
⋅ ⋅ ⋅ ⋅ ⎥ ⎢ ez ⎥
⎥⎢
⎥
⋅ ⋅ ⋅ ⋅ ⎥ ⎢ 2exy ⎥
⋅ ⋅ ⋅ ⋅ ⎥ ⎢ 2eyz ⎥
⎥⎢
⎥
⋅ ⋅ ⋅ C66 ⎦ ⎣ 2ezx ⎦

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36 Independent
Elastic Constants
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4.2 Linear elastic material – Hooke’s law

(Body-Centered
Crystal)

du
o

Typical Wood
Structure

ng

th

an

co

ng

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

(Hexagonal
Crystal)

(Fiber Reinforced
Composite)

cu

u

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
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4.2 Linear elastic material – Hooke’s law

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Isotropic Materials

ng

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

co

Cijkl = QimQ jn Qkp Qlq Cmnpq

(prove)

an

Cijkl = αδ ijδ kl + βδ ik δ jl + γδ ilδ jk

σ ij = ( αδij δkl + βδik δ jl + γδil δ jk ) ekl = αekk δij + βeij + γeij
Generalized Hooke’s Law

du
o

σ ij = λ ekk δ ij + 2µ eij

ng

th

σ ij = Cijkl ekl

cu

u

⎧σ x = λ ( ex + ey + ez ) + 2µex ; σ y = λ ( ex + e y + ez ) + 2µe y
⎪

⎪
⎨σ z = λ ( ex + ey + ez ) + 2µez
⎪
⎪⎩ τ xy = 2µexy ; τ yz = 2µeyz ; τ zx = 2µezx
λ - Lamé’s constant
µ - shear modulus or modulus of rigidity

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4.2 Linear elastic material – Hooke’s law

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Prove

Cijkl = QimQ jn Qkp Qlq Cmnpq

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ng

Cijkl = αδ ijδ kl + βδ ik δ jl + γδ ilδ jk

an

α ′δ ij′δ kl′ + β ′δ ik′ δ ′jl + γ ′δ il′δ ′jk = QimQ jnQkpQlq (αδ mnδ pq + βδ mpδ nq + γδ mqδ np )

th

= α QimQ jmQkpQlp + β QimQ jnQkmQln + γ QimQ jnQknQlm

du
o

ng

= αδ ijδ kl + βδ ikδ jl + γδ ilδ jk

u

′ = α ′δ ij′δ kl′ + β ′δ ik′ δ ′jl + γ ′δ il′δ ′jk
Cijkl

cu

Cijkl = αδ ijδ kl + βδ ik δ jl + γδ ilδ jk

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4.2 Linear elastic material – Hooke’s law

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Isotropic Materials

1
⎡σ x −ν (σ y + σ z ) ⎤⎦
Inverted Form - Strain in Terms of Stress
E⎣
1
⎡σ y −ν (σ z + σ x ) ⎤⎦
e
=
y
σ kk = ( 3λ + 2µ ) ekk
σ ij = λ ekk δ ij + 2µ eij
E⎣
1
ez = ⎡⎣σ z −ν (σ x + σ y ) ⎤⎦
E
⎞
1 ⎛
λ

eij =
σ ij −
σ kkδ ij ⎟
1 +ν
1
e
=
τ
=
τ xy
2 µ ⎜⎝
3λ + 2 µ
⎠
xy
xy
E
2µ
1 +ν
1
e
=
τ
=
τ yz
yz
yz
1 +ν
ν
E
2

µ
eij =
σ ij − σ kkδ ij
E
E
1 +ν
1
ezx =
τ zx = τ zx
E
2µ
µ (3λ + 2µ )
E=
Young’s modulus or modulus of elasticity
λ+µ
λ
ν=
Poisson’s ratio
2(λ + µ )

cu

u

du
o

ng

th

an

co

ng

ex =

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4.1 Material characterization
 

co

ng

4.2 Linear elastic material – Hooke’s law

an

4.3 Physical meaning of elastic module

cu

u

du
o

ng

th

4.4 Thermo-elastic constitutive relations

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4.3 Physical meaning of elastic module

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Simple Tension

an

co

ng

Consider the simple tension with a sample subjected
to tension in the x-direction. The state of stress is
represented by the one-dimensional field
⎡σ
⎤
0
0
⎢E
⎥
⎢
⎥
⎡σ 0 0⎤
ν
eij = ⎢ 0 − σ
0 ⎥
σ ij = ⎢⎢ 0 0 0⎥⎥
⎢
⎥
E
⎢⎣ 0 0 0⎥⎦

⎢
ν ⎥
⎢0
0
− σ⎥
E ⎦
⎣
Slope of the stress-strain curve or
E = σ / ex
Elastic module in the x-direction

du
o
cu

u

σ
 

ng

th

σ
 

ν = − e y / ex
= − ez / ex

Ratio of the transverse strain to
the axial strain

Standard measurement systems can easily collect axial stress and transverse and
axial strain data, and thus through this, one type of test both elastic constants E
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and ν can be determined for material of interest.
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4.3 Physical meaning of elastic module

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Pure Shear

an

th

τ
 

τ / 2µ

⎡ 0
eij = ⎢τ / 2 µ
0
⎢
⎢⎣ 0
0

0⎤
0⎥
⎥
0 ⎥⎦

ng

τ
 
τ
 

⎡0 τ 0 ⎤
σ ij = ⎢⎢τ 0 0 ⎥⎥
⎢⎣0 0 0 ⎥⎦

co

ng

If a thin-walled cylinder is subjected to torsion loading,
the state of stress on the surface of the cylindrical
sample is given by

du
o

τ
 

= τ / γ xy

Shear modulus which is the slope of
the shear stress-shear strain curve

cu

u

µ = τ / 2exy

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4.3 Physical meaning of elastic module

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Hydrostatic Compression

The final example is associated with the uniform compression (or
tension) loading of a cubical specimen. This type of test can be
realizable if the sample was placed in a high-pressure
compression chamber. The state of stress for this case is given by
⎡ 1 − 2ν
⎤
−
p
0
0
p
 
⎢
⎥
E
0 ⎤
⎡− p 0
⎢
⎥
1
−
2
ν
⎥
eij = ⎢
0

−
p
0
σ ij = ⎢⎢ 0 − p 0 ⎥⎥
⎢
⎥
E
⎢⎣ 0
0 − p ⎥⎦
⎢
1 − 2ν ⎥
⎢
0
0
−
p⎥
E
⎣
⎦
p = − kekk = − kϑ
Elastic constant k represents the ratio of pressure
E
to the dilatation (which represents the change in
k=
Bulk Modulus
material volume)
3(1 − 2ν )

th

cu

u

du
o

ng

p
 

an

co

ng

p
 

Note that when Poisson’s ratio approaches 0.5, the bulk modulus becomes
unbounded and the material does not undergo any volumetric deformation and hence
is incompressible.
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4.3 Physical meaning of elastic module
- Our discussion of elastic modulus for isotropic materials has led to the definition

ng

of five constants λ, µ, E, ν and k. However, keep in mind that only two of these are

co

needed to characterize the material.

an

- In can be shown that all five elastic constants are interrelated, and if any two are

th

given, the remaining three can be determined by using simple formulae. Results of

du
o

ng

these relations are conveniently summarized in Table 4.1.
- In addition, nominal values of elastic constants for particular engineering

cu

u

materials are given in Table 4.2.

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4.3 Physical meaning of elastic module

ν

E,k

E

E,µ

E

E,λ

E

ν,k

3k (1 − 2ν)

ν ,µ

2µ(1 + ν )

k,µ
k,λ

µ,λ

k

ng
du
o

ν

u

λ(1 + ν )(1 − 2ν )
ν
9kµ

6k + µ
9k (k − λ )
3k − λ
µ(3λ + 2µ )
λ+µ

cu

ν ,λ

3k − E
6k
E − 2µ
2µ
2λ
E+λ+R

ν
ν

3k − 2µ
6k + 2µ
λ
3k − λ
λ
2(λ + µ)

µE
3(3µ − E )
E + 3λ + R

6
k

2µ(1 + ν )
3(1 − 2ν )
λ (1 + ν )
3ν

µ
E
2(1 + ν )
3kE
9k − E

ng

E

co

E,ν

k
E
3(1 − 2ν )

an

ν

th

E

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Table 4.1: Relations Among Elastic Constants

µ

E − 3λ + R
4
3k (1 − 2ν )
2(1 + ν )

µ
λ (1 − 2ν )
2ν

λ
Eν
(1 + ν )(1 − 2ν )
3k (3k − E )
9k − E
µ(E − 2µ )
3µ − E
λ
3kν
1+ ν

2µν
1 − 2ν

λ

k

µ

2
k− µ
3

k

3
(k − λ)
2

λ

3λ + 2µ
3

µ

λ

R = E 2 + 9λ2 + 2 Eλ
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4.3 Physical meaning of elastic module

Table 4.2: Typical Values of Elastic Moduli for Common Engineering Materials
k(GPa)

α(10-6/oC)

54.6

71.8

25.5

11.5

7.7

15.3

11

33.4

71

93.3

18

27.6

27.6

45.9

8.8

0.40

10.1

4.04

47.2

102

0.499

0.654x10-3

0.326

0.326

200

0.29

80.2

111

164

13.5

µ(GPa)

Aluminum

68.9

0.34

25.7

Concrete

27.6

0.20

Cooper

89.6

0.34

Glass

68.9

0.25

Nylon

28.3

Rubber

0.0019

Steel

207

cu

u

du
o

ng

th

an

co

ν

ng

λ(GPa)

E (GPa)

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4.3 Physical meaning of elastic module

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Hooke’s Law in Cylindrical Coordinates

ng

x3
 

co

z
 

θ
 

dθ
 

dr
 

cu

x1
 

ng
σr
 

u

r
 

τrθ
 

du
o

σθ
 

τrz
 

τ rθ τ rz ⎤
σ θ τ θ z ⎥⎥
τ θ z σ z ⎥⎦

th

an

σz
 

τθz
 

⎡σ r
σ = ⎢τ rθ
⎢
⎢⎣τ rz

x2
 

σ r = λ (er + eθ + ez ) + 2 µ er
σ θ = λ (er + eθ + ez ) + 2 µ eθ
σ z = λ (er + eθ + ez ) + 2 µ ez
τ rθ = 2 µ erθ
τ θ z = 2 µ eθ z
τ zr = 2 µ ezr
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4.3 Physical meaning of elastic module

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Hooke’s Law in Spherical Coordinates

τRθ
 

du
o
u

cu

x1
 

σφ
 

ng

R
 
τφθ
 
 

θ
 

an

th

σθ
 

co

σR
 
τRφ
 

φ
 

⎡ σ R τ Rϕ τ Rθ ⎤
⎢
⎥
σ = ⎢τ Rϕ σ ϕ τ ϕθ ⎥
⎢⎣τ Rθ τ ϕθ σ θ ⎥⎦

ng

x3
 

σ R = λ (eR + eϕ + eθ ) + 2 µ eR
x2
 

σ ϕ = λ (eR + eϕ + eθ ) + 2 µ eϕ
σ θ = λ (eR + eϕ + eθ ) + 2 µ eθ
τ Rϕ = 2 µ eRϕ
τ ϕθ = 2 µ eϕθ
τ θ R = 2 µ eθ R

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4.1 Material characterization
 

co

an

4.3 Physical meaning of elastic module

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4.2 Linear elastic material – Hooke’s law

cu

u

du
o

ng

th

4.4 Thermo-elastic constitutive relations

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4.4 Thermo-elastic constitutive relations

M)

+ eij(

T)

an

eij = eij(

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- It is well known that a temperature change in an unrestrained elastic solid
produces deformation. Thus a general strain field results from both mechanical and
thermal effects. Within the context of linear small deformation theory, the total strain
can be decomposed into the sum of mechanical and thermal components as

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th

- If T0 is taken as the reference temperature and T as an arbitrary temperature, the
thermal strains in an unrestrained solid can be written in the linear form

eij( ) = α ij (T − T0 )

du
o

T

cu

u

where αij is the coefficient of thermal expansion tensor. Notice that it is the
temperature difference that creates thermal strain. If the material is taken as
isotropic, then eij must be an isotropic second-order tensor, and

eij( ) = α (T − T0 )δ ij
T

where α is the coefficient of thermal expansion. Table 4.2 provides typical values of
this constant for some common materials.
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