12.108 Struct. & Prop’s. of Earth Mat. Lecture 9
Single-crystal elasticity
Assigned Reading:
Nye JF (1957) Physical Properties of Crystals. Oxford University Press, Oxford, UK
(Chapters 5 and 6).
Nye, J. F. “Elastic behavior of single crystals: Anisotropy”, pp. 2415-2423, in
Encyclopedia of Materials: Science and Technology, Elsevier Science, 2001
(http://www.elsevier.com/mrwclus/15/show/Main.htt) From MIT campus.
Resource reading (If you would like to brush up on stress,
strain, and elasticity):
Malvern LE (1969) Introduction to the Mechanics of a Continuous Medium. Prentice
Hall, Englewood Cliffs, NJ. (A rigorous introduction to continuum mechanics.)
Oertel G (1996) Stress and Deformation: A handbook on Tensors in Geology. Oxford
University Press, New York. (Many problems on which to practice.)
1
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
2
Elasticity
Elastic and inelastic behavior
For small strains at constant temperatures, the amount of strain is
determined by the magnitude of the stress causing it.
Perfectly Elastic:
Purely elastic behavior implies thermodynamic reversibility. All the work
put into deforming the body is recoverable. Deformation is not timedependent.
The relation between stress and strain may be linear or non-linear.
Anelastic:
Time-dependent behavior in which the material strain is recoverable
Plastic and viscoplastic behavior:
When a material is deformed to large strains, some of the strain is
permanent and not recoverable; this permanent strain is plastic (if time
independent) or viscoplastic (if time dependent).
Elastic
Recoverable TimeLinear
independent
Nonlinear
Inelastic
Anelastic
Part. Recov.
Timedependent
Lin or Non­
lin.
Plastic
Not recover.
Time-indep.
“
Visco-plas.
“
time-dep
“
Rock Physics Le05: Strain/Elasticity, 2003 02 27
200
3
Non-linear
σ ( MPa)
Tensile
compression
Linear Elastic
Recoverable
Recoverable
Elastic Energy
extension
ε
-3
10
Compressive
σ ( MPa)
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Anelastic
Non-elastic
Heat
lost/cycle
-3
10
ε
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
Anisotropic (Single Crystal) Elasticity
Table removed due to copyright reasons
Image removed due to copyright reasons
4
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
5
Coordinate Conventions for Elasticity Tensor
The exact values of the elasticity tensor depend on the definition of the
coordinate system.
The convention used is that the modulii are referred to an othonormal
coordinate systems, x1, x2, x3, with unit vectors, ê1 , …. These axes are
related to the mineral lattice vectors by the following convention
Monoclinic
Tetragonal,
trigonal,
hexagonal
Orthorhombic
and cubic
eˆ 2 || b
eˆ1 || a, eˆ 3 || c,
eˆ1 || a, eˆ 2 || b, eˆ 3 || c,
Linear Elastic Equations
Given the restrictions of reversibility and linearity,
Hooke’s law can be written for strain as a function of stress
ε ij = cijklσ kl ( i, j , k , l = 1,...,3)
ε I = cIJ σ J ( I , J = 1,..., 6 )
σ ij = sijkl ε kl ( i, j, k , l = 1,...,3)
ε I = cIJ σ J ( I , J = 1,..., 6 )
The matrix notation (right-hand set of equations) reduces the number of
subscripts to two, but the equations do not strictly obey tensor
transformation rules and require factors of 2 and 4 to be inserted in the
relations between sijkl and sIJ.
s
Compliance
Modulus
Stress-1
c
Stiffness
Constant
Stress
The reduction from 81 separate components to 36 occurs because the
stress and strain tensors are symmetric.
Because the material behavior is thermodynamically reversible, a
chemical potential energy, the Helmholtz free energy, can be defined to
express the energy of the elastic solid. For this thermodynamic potential to
exist, requires the stiffness and compliance tensors and matrices to be
symmetric. This limits the number of independent components of the most
general elasticity tensor to 21.
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
Symmetry causes further restrictions:
Crystal symmetry requires further reductions in the number of the
stiffnesses and compliances
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Hexagonal
Cubic
Isotropic
21
13
9
7,6
7,6
5
3
2
6
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
8
Stiffness Moduli
Cubic
Gold (Au)
Halite NaCl
Spinel (Mg Al204)
Wustite (Fc0)
Periclase (MgO)
α-Iron (Fe)
Grossular Garnite3
Pyrite FeS2
Diamond
Cii
191
165
283
246
294
230
322
361
1079
C44
42
34
155
45
155
117
105
105
578
C12 (GPa)
162
47
155
149
93
135
91
34
124
Remark
metal
halide
Al204
Oxide
Oxide
metal
Al. Silicate
Sulfide
covalent
ρ
19.3
2.2
3.6
5.73
3.6
7.9
3.6
3.5
Tm (°C)
1063
801
2135
~1400
2800
1408
(~1400)
5.0
(3652
Hexagonal
Graphite (C)
Ice-I (H20)
β−quartz (S:02)
C11
1060
13.5
117
C33
36.5
15
110
C44
0.3
3.0
36
C12
180
65
16
C13
15
6
33
Remark
Sp2bonding
hydrogen
framework
ρ
2.26
0.9
-
Tm (°C)
(3652)
0
1470-
Remark
carbonate
oxide
oxide
framework
ρ
Τm (°C)
(1339°C)
~1550
1840
(1470)
Trigonal
Calcite (CaC03)
Hematite (Fe203)
Corundum (Al203)
α-quartz (Si02)
C11
144
242
497
86.5
C33
84
227
501
107
C44
33.5
85.7
147
58
C12
53.9
54.6
162
7
C13
51
15.4
116
12
C14
-20.5
-125
-22
-18
Orthorhombic
Perovskite (MgSi03)
Olivine (Mg2Si04)
Fayalite (Fe2Si04)
Enstatite (MgSi03)
Ferrosilite (FeSi03)
Wadsleyite4
C11
515
328
266
224
198
360
C22
525
200
168
178
136
383
C33
435
235
232
214
175
273
C44
179
67
32.3
78
59
112
C55
202
81
47
76
58
118
C66
175
81
57
82
49
98
C12
117
69
94
72
84
75
C13
117
69
92
54
72
110
C23
139
73
92
53
55
105
Remark
Clspckd
Isolated
Isolated
Chain
Chain
Chain
Sources: Simmons, M. G., and H. Wang, Single crystal elastic constants and calculated aggregate properties, MIT Press, 1975. Bass, J. D., Elasticity of Minerals, Glasses, and Melts, pp.45-63, in Mineral Physics and crystallography: a handbook of physical constants, edited by T. J. Ahrens, AGU, Washington, DC., 1995.
3
4
(Mg,Fe,Mn)3Al2Si3O12
β−(Mg2SiO4)
ρ
3.20
4.38
3.20
4.00
3.47
12.108 Struct. & Prop’s. of Earth Mat. Lecture 9
9
Tetragonal
C11
424
C33
489
C44
131
C66`
48
C12
70
C13
149
Remark
Zircon (ZrSi 04)
ρ
4.7
Stishovite (Si 02)
453
776
252
302
211
203
Framework
4.20
α-Cristobalite (Si 02)
59.4
42.4
67
26
4
-4.4
Framework
2.335
Tm (°C)
(1540
decp)
high
pressure
1713
Monoclinic
Albite (NaAlSi3O8)
Anorthite (CaAl2Si2O8)
Microcline (KSi3 AlO8)
C11
74
124
67
C22
131
205
169
C33
128
156
118
C44
17
24
14
C55
30
40
24
C66
32
42
36
C12
36
66
45
C13
39
50
27
C23
31
42
20
C15
-6.6
-19
-.2
Labradorite5
Diopside (CaMgSi2O6)
Hedenbergite (CaFeSi2O6)
Hornblende6
Coesite (SiO2)
99
204
222
116
161
158
175
176
160
230
150
238
249
192
232
22
68
55
57
68
35
59
63
32
73
37
71
60
37
59
63
84
69
45
82
49
88
79
61
103
27
48
86
66
36
-25
-19
12
4.3
-36
5
Albite 30-50%, Anorthite 70-50%
6
(Ca,Na)2-3(Mg,Fe,Al)5(Al,Si)8O2(OH)2
C25
-13
-7
12.
3
-11
-20
13
-2.5
2.6
C35
-20
-18
-15
C46
-25
-1
-2
Remark
Framework
Framework
Framework
ρ
2.62
2.76
2.56
Tm (°C)
1100°C
1550°C
1150°C
-12
-34
26
10
-39
-5
-11
-10
-6.2
9.9
Framework
Chain
Chain
Chain
Framework
2.70
3.31
3.65
3.12
2.911
1280°C
1390°C
(1713)
12.108 Struct. & Prop’s. of Earth Mat. Lecture 9
10
Elastic stiffness or compliance in an arbritrary direction
Vector Transformation
Express the components of a vector in a new coordinate system that has
been rotated. The components of the vector in the new system are
Ei' = aij E j
where Ei are the components of the vector in the old system, aij are the
direction cosines between the old and new coordinate systems, and Ej’ are
the components in the new system.
Tensor Transformation:
Tensors of the second rank transform in much the same way as vectors:
Ei'j = aik a jl Ekl
Tensors of the fourth rank (like stiffness or compliance) transform as
'
sijkl
= aim a jn ako alp smnop
Thus, the Young’s modulus (c1111) could be calculated in any arbitrary
direction.
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
12
Relations among moduli, stiffnesses and compliances
In isotropic elastic materials, the most commonly used elastic (moduli,
constants, stiffnesses, compliances) are Young’s Modulus, the Bulk
modulus, the Volumetric compressibility, Poisson’s ratio, and the Lame
parameters.
Young’s Modulus
l
σ1
w
½ dl
½ dw
The relation between the load on a bar and the fractional stretch of a bar
along its long axis is known as Young’s modulus.
σ l = Eε l
εl =
1
σl
E
where the subscript l is used to indicate the direction in which the
measurement is made, but in an isotropic material the direction is
immaterial.
In an anisotropic material, the equivalent measurement depends on the
direction (relative to some fixed axes in the material) in which the
measurement is made).
When the measurement is made along the 1 axis in a single crystal, the
equivalent modulus (stiffness, compliance) is
σ 11 = c1111ε11 equivalently σ 1 = c11ε1 or
ε11 = s1111σ 11
When the Young’s modulus measurement is made in the 1 direction,
E = 1/ s1111
When the measurement is made in another direction D
1/ E = aDi aDj aDk aDl sijkl
where aDi is the direction cosine between D and the axis i.
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
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Some typical values of compressibility and Poisson’s ratio
Rock/Material
Obsidian
Granite
Diorite
Gabbro
Diabase
Dunite
Slate
Limestone
Dolomite
Quartzite
Mica Schist
Feldspathic Gneiss
Amphibolite
Oligoclase
Anorthosite
Rocksalt
Anhydrite
Ice
Steel
Aluminum
Compressibility (10-5 MPa-1)
2.9-3.
1.9-2.0
1.4-1.7
1.1-1.3
1.2-1.4
0.8-0.9
1.9-2.3
1.3-1.4
1.0-1.1
2.4-2.6
2.1-2.3
1.8-2.2
1.1-1.3
1.5-1.6
1.2-1.4
4.3
1.8
12
0.6
1.3
Poisson’s Ratio
0.12-0.16
0.23-0.27
0.26-0.29
0.27-0.31
0.27-0.30
0.26-0.28
0.15-0.20
0.27-0.30
0.30
0.12-0.15
0.15-0.20
0.15-0.20
0.28-0.30
0.29
0.31
0.25
0.30
0.36
0.28-0.29
0.34-0.36
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
15
Poisson’s Ratio
ν ≡
−ε ⊥l
ε l
for an isotropic material σ l = Eε
l
s22111
=
ε 22 ε 22
ν
=
=
=s
σ 11 Eε11
E 21
For rocks, Poisson’s ratio is commonly 0.25. In some materials, it may be
very small or even negative.
Can you think of a common use for a material with a
low or negative Poisson’s ratio?
For an incompressible material, Poisson’s ratio is 0.5.
Why? Is it possible for ν to be even larger than that? Rigidity Modulus
h
µε shear = σ shear
dl
When the body and the crystal axes are aligned, s44 =
1
µ
Lamé Constants
For an isotropic material, the stress strain relations are often written:
σ 1 = ( λ + 2µ ) ε1 + λε 2 + λε 3
σ 2 = λε1 + ( λ + 2µ ) ε 2 + λε 3
σ 3 = λε1 + λε 2 + ( λ + 2µ ) ε 3
From the equations above it is clear that if the crystal axis 1 is aligned
with the exterior axis 1 then
2 µ + λ = c1111 and c1122 = λ
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
Compressibility
Consider imposition of pressure
σ ij = − pδ ij
Then the equation for the strain is
ε ij = − psijkl δ kl
define ∆ ≡ ε ii = − psiikk
and
βT = −
∆
= siikk
p
Write the equation for the isothermal compressibility
for a cubic crystal.
βT cubic = 3 ( s11 + 2s12 )
Does the compressibility depend on the direction of measurement? The bulk modulus is the inverse of the compressibility. 16
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Rock Physics Le05: Strain/Elasticity, 2003 02 27
17
Relations among Various Moduli and Elastic
Constants
Name
Inv. Young’s
Modulus
Poisson’s Ratio
Symbol
1/E
ν,σ
Definition
Normal Strain
Normal Stress
Normal Strain
⊥Norm. Strain
S/C
S1111
S2211
/S1111
Units
Stress-1
λ
2(λ + µ )
µ, G Shear Stress
Shear Strain
C1212
Stress
Lame Const.
λ
Response of
Stress to Volum.Change
C1111
Stress
Dilation/Pressure
λ+µ
µ ( 3λ + 2µ )
---
Shear Modulus
Compressibility B
Relations
Stress-1
E
2(1 + ν )
λ=
Eν
(1 + ν )(1 − ν )
E
µ
=λ +2
3(1 − 2ν )
3
Equivalent formulations for isotropic elastic materials in principal stress representation: 1
ν
ν
σ 11 − σ 22 − σ 33
E
E
E
1
ν
ν
ε 22 = σ 11 − σ 22 − σ 33
E
E
E
1
ν
ν
ε 33 = − σ 11 − σ 22 + σ 33
E
E
E
In general:
ε11 =
ε ij =
1 +ν
ν
σ ij − δ ijσ kk
E
E
σ 1 = ( λ + 2µ ) ε1
+ λε 2 + λε 3
σ 2 = λε1 + ( λ + 2µ ) ε 2 + λε 3
σ 3 = λε1 + λε 2
+ ( λ + 2µ ) ε 3
σ ij = λδ ij ε kk + 2µε ij