Chapter 3
Constitutive Models Relations between Stress and Strain
3.2 Linear elastic material behavior
You
are probably familiar with the behavior of a linear elastic material from
introductory materials courses.
3.2.1 Isotropic, linear elastic material behavior
If
you conduct a uniaxial tensile test on almost any material, and keep the stress
levels sufficiently low, you will observe the following behavior:
The specimen deforms reversibly: If you remove the loads, the solid returns to
its original shape.
The strain in the specimen depends only on the
stress applied to it it doesn’t depend on the rate of loading, or
the history of loading.
For most materials, the stress is a linear
function of strain, as shown in the picture above. Because the strains are small, this is true
whatever stress measure is adopted (Cauchy stress or nominal stress), and is
true whatever strain measure is adopted (Lagrange strain or infinitesimal
strain).
For most, but not all, materials, the material
has no characteristic orientation. Thus,
if you cut a tensile specimen out of a block of material, as shown in the
figure, the the stressstrain curve will
be independent of the orientation of the specimen relative to the block of material. Such materials are said to be isotropic.
If you heat a specimen of the material,
increasing its temperature uniformly, it will generally change its shape
slightly. If the material is isotropic
(no preferred material orientation) and homogeneous, then the specimen will
simply increase in size, without shape change.
3.2.2 Stressstrain relations
for isotropic, linear elastic materials. Young’s Modulus, Poissons ratio and the
Thermal Expansion Coefficient.
Before writing down stressstrain relations,
we need to decide what strain and stress measures we want to use. Because the model only works for small shape
changes
Deformation is characterized using the
infinitesimal strain tensor defined in Section 2.1.7. This is convenient for calculations, but has
the disadvantage that linear elastic constitutive equations can
only be used if the solid experiences small rotations, as well as small shape
changes.
All stress measures are taken to be
equal. We can use the Cauchy stress as the stress measure.
You probably already know the stressstrain relations
for an isotropic, linear elastic solid.
They are repeated below for convenience.
Here, E and are Young’s modulus and Poisson’s ratio, is the coefficient of thermal expansion, and is the increase in temperature of the
solid. The remaining relations can be
deduced from the fact that both and are symmetric.
The inverse relationship can be expressed as
HEALTH
WARNING: Note the factor of 2 in the strain vector. Most texts, and most FEM codes use this
factor of two, but not all. In addition,
shear strains and stresses are often listed in a different order in the strain
and stress vectors. For isotropic
materials this makes no difference, but you need to be careful when listing
material constants for anisotropic materials (see below).
We
can write this expression in a much more convenient form using index
notation. Verify for yourself that the
matrix expression above is equivalent to
The inverse
relation is
The stress-strain relations are often expressed using the elastic modulus tensor or the elastic
compliance tensor as
In terms
of elastic constants, and are
3.2.3 Reduced stress-strain equations for plane
deformation of isotropic solids
For plane strain or
plane stress deformations, some
strain or stress components are always zero (by definition) so the
stress-strain laws can be simplified.
For a
plane strain deformation . The stress strain laws are therefore
In index
notation
where Greek
subscripts can have values 1 or 2.
For a
plane stress deformation
3.2.4 Representative
values for density, and elastic constants of isotropic solids
Most
of the data in the table below were taken from the excellent introductory text
`Engineering Materials,’ by M.F. Ashby and D.R.H. Jones, Pergamon Press. The remainder are from random web pages…
Note the units values of E
are given in ;
the G stands for Giga, and is short for . The units for density are in - that’s Mega grams. One mega gram is 1000 kg.
Material
|
Mass density
|
Youngs Modulus
|
Poisson Ratio
|
Expansion coeft
|
Tungsten Carbide
|
14 17
|
450650
|
0.22
|
|
Silicon Carbide
|
2.5 3.2
|
450
|
0.22
|
|
Tungsten
|
13.4
|
410
|
0.30
|
|
Alumina
|
3.9
|
390
|
0.25
|
|
Titanium Carbide
|
4.9
|
380
|
0.19
|
|
Silicon Nitride
|
3.2
|
320 - 270
|
0.22
|
|
Nickel
|
8.9
|
215
|
0.31
|
|
CFRP
|
1.5-1.6
|
70 200
|
0.20
|
|
Iron
|
7.9
|
196
|
0.30
|
|
Low alloy steels
|
7.8
|
200 - 210
|
0.30
|
|
Stainless steel
|
7.5-7.7
|
190 - 200
|
0.30
|
|
Mild steel
|
7.8
|
196
|
0.30
|
|
Copper
|
8.9
|
124
|
0.34
|
|
Titanium
|
4.5
|
116
|
0.30
|
|
Silicon
|
2.5-3.2
|
107
|
0.22
|
|
Silica glass
|
2.6
|
94
|
0.16
|
|
Aluminum & alloys
|
2.6-2.9
|
69-79
|
0.35
|
|
Concrete
|
2.4-2.5
|
45-50
|
0.3
|
|
GFRP
|
1.4-2.2
|
7-45
|
|
|
Wood, parallel grain
|
0.4-0.8
|
9-16
|
0.2
|
|
Polyimides
|
1.4
|
3-5
|
0.1-0.45
|
|
Nylon
|
1.1 1.2
|
2 4
|
0.25
|
|
PMMA
|
1.2
|
3.4
|
0.35-0.4
|
|
Polycarbonate
|
1.2 1.3
|
2.6
|
0.36
|
|
Natural Rubbers
|
0.83-0.91
|
0.01-0.1
|
0.49
|
|
PVC
|
1.3-1.6
|
0.003-0.01
|
0.41
|
|
3.2.5 Other Elastic
Constants bulk, shear and Lame modulus.
Young’s modulus and Poisson’s ratio are the most common
properties used to characterize elastic solids, but other measures are also
used. For example, we define the shear modulus, bulk
modulus and Lame modulus of an
elastic solid as follows:
A nice table relating
all the possible combinations of moduli to all other possible combinations is
given below. Enjoy!
|
Lame
Modulus
|
Shear
Modulus
|
Young’s
Modulus
|
Poisson’s
Ratio
|
Bulk
Modulus
|
|
|
|
|
|
|
|
|
Irrational
|
|
Irrational
|
Irrational
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
3.2.6 Physical Interpretation of
elastic constants for isotropic solids
It is important to have a feel for the physical significance
of the two elastic constants E and .
Young’s
modulus E is the slope of the stressstrain curve in
uniaxial tension. It has dimensions of
stress ( ) and is usually large for steel, .
You can think of E as a measure of
the stiffness of the solid. The larger the value of E, the stiffer the solid. For
a stable material, E>0.
Poisson’s ratio is the ratio of lateral to longitudinal strain
in uniaxial tensile stress. It is dimensionless and typically ranges from 0.20.49, and is
around 0.3 for most metals. For a stable
material, .
It is a measure of the compressibility of the solid. If ,
the solid is incompressible its volume remains constant, no matter how it
is deformed. If ,
then stretching a specimen causes no lateral contraction. Some bizarre materials have -- if
you stretch a round bar of such a material, the bar increases in diameter!!
Thermal
expansion coefficient quantifies
the change in volume of a material if it is heated in the absence of
stress. It has dimensions of (degrees
Kelvin)-1 and is usually very small.
For steel,
The bulk
modulus quantifies the resistance of the solid to volume changes. It has a large value (usually bigger than E).
The shear
modulus quantifies its resistance to volume preserving shear deformations. Its value is usually somewhat smaller than E.
3.2.7 Strain Energy Density for Isotropic Solids
Note the following
observations
If you
deform a block of material, you do work on it (or, in some cases, it may do
work on you…)
In an
elastic material, the work done during loading is stored as recoverable strain
energy in the solid. If you unload the
material, the specimen does work on you, and when it reaches its initial
configuration you come out even.
The work
done to deform a specimen depends only on the state of strain at the end of the
test. It is independent of the history
of loading.
Based
on these observations, we define the strain
energy density of a solid as the work done per unit volume to deform a
material from a stress free reference state to a loaded state.
To
write down an expression for the strain energy density, it is convenient to
separate the strain into two parts
where, for an isotropic
solid,
represents the strain due
to thermal expansion (known as thermal strain), and
is the strain due to
mechanical loading (known as elastic strain).
Work
is done on the specimen only during mechanical loading. It is straightforward to show that the strain
energy density is
You can also re-write this
as
Observe that
3.2.8 Stress-strain relation for a
general anisotropic linear elastic material the elastic stiffness and compliance tensors
The
simple isotropic model described in the preceding section is unable to describe
the response of some materials accurately, even though the material may deform
elastically. This is because some
materials do have a characteristic orientation.
For example, in a block of wood, the grain is oriented in a particular
direction in the specimen. The block
will be stiffer if it is loaded parallel to the grain than if it is loaded
perpendicular to the grain. The same
observation applies to fiber reinforced composite materials. Generally, single
crystal specimens of a material will also be anisotropic this is important when modeling stress effects
in small structures such as microelectronic circuits. Even polycrystalline
metals may be anisotropic, because a preferred texture may form in the specimen
during manufacture.
A
more general stressstrain relation
is needed to describe anisotropic solids.
The
most general linear stressstrain relation
has the form
Here,
is a fourth order tensor (horrors!), known as
the elastic stiffness tensor, and is the thermal expansion coefficient tensor.
The stress strain relation is invertible:
where
is known as the elastic compliance tensor
At
first sight it appears that the stiffness tensor has 81 components. Imagine having to measure and keep track of
81 material properties! Fortunately, must have the following symmetries
This
reduces the number of material constants to 21. The compliance tensor has the
same symmetries as .
To
see the origin of the symmetries of ,
note that
The
stress tensor is symmetric, which is only possible if
If a
strain energy density exists for the material, the elastic stiffness tensor
must satisfy
The
previous two symmetries imply ,
since and .
To
see that ,
note that by definition
and
recall further that the stress is the derivative of the strain energy density
with respect to strain
Combining
these,
Now,
note that
so
that
These symmetries allow us
to write the stress-strain relations in a more compact matrix form as
where ,
etc are the elastic stiffnesses of
the material. The inverse has the form
where ,
etc are the elastic compliances of
the material.
To
satisfy Drucker stability, the eigenvalues of the elastic stiffness and
compliance matrices must all be greater than zero.
HEALTH WARNING: The shear strain and shear stress components are not
always listed in the order given when defining the elastic and compliance
matrices. The conventions used here are
common and are particularly convenient in analytical calculations involving
anisotropic solids. But many sources use
other conventions. Be careful to enter
material data in the correct order when specifying properties for anisotropic
solids.
3.2.9 Physical Interpretation of the
Anisotropic Elastic Constants.
|
It
is easiest to interpret ,
rather than . Imagine applying a uniaxial stress, say ,
to an anisotropic specimen. In general,
this would induce both extensional and shear deformation in the solid, as shown
in the figure.
The strain induced by the
uniaxial stress would be
All the constants have dimensions . The constant looks like a uniaxial compliance, (like ), while the ratios are generalized versions of Poisson’s ratio:
they quantify the lateral contraction of a uniaxial tensile specimen. The shear terms are new in an isotropic material, no shear strain is
induced by uniaxial tension.
3.2.10 Strain energy density for anisotropic, linear
elastic solids
The
strain energy density of an anisotropic material is
3.2.11 Basis change formulas for anisotropic elastic
constants
|
The
material constants or for a particular material are usually
specified in a basis with coordinate axes aligned with particular symmetry
planes (if any) in the material. When
solving problems involving anisotropic materials it is frequently necessary to
transform these values to a coordinate system that is oriented in some
convenient way relative to the boundaries of the solid. Since is a fourth rank tensor, the basis change
formulas are highly tedious, unfortunately.
Suppose
that the components of the stiffness tensor are given in a basis ,
and we wish to determine its components in a second basis, . We define the usual transformation tensor with components ,
or in matrix form
This
is an orthogonal matrix satisfying .
In practice, the matrix can be computed in terms of the angles between the
basis vectors. It is straightforward to show that stress, strain, thermal
expansion and elasticity tensors transform as
The basis change formula
for the elasticity tensor in matrix form can be expressed as
where the basis change
matrix K is computed as
and the modulo function
satisfies
Although
these expressions look cumbersome they are quite convenient for computer
implementation.
The basis change for the
compliance tensor follows as
where
The
proof of these expressions is merely tiresome algebra and will not be given
here. Ting’s book `Anisotropic
Elasticity: Theory and Applications’ OUP (1996) has a nice clear discussion.
For
the particular case of rotation through an angle in a counterclockwise sense about the axes, respectively, the rotation matrix
reduces to
where
.
The inverse matrix can be obtained simply by changing the sign of
the angle in each rotation matrix. Clearly, applying the three rotations
successively can produce an arbitrary orientation change.
For an isotropic material, the elastic
stress-strain relations, the elasticity matrices and thermal expansion
coefficient are unaffected by basis changes.
3.2.12 The effect of material symmetry on
stress-strain relations for anisotropic materials
A
general anisotropic solid has 21 independent elastic constants. Note that in
general, tensile stress may induce shear strain, and shear stress may cause
extension.
If
a material has a symmetry plane, then applying stress normal or parallel to
this plane induces only extension in direction normal and parallel to the plane
For
example, suppose the material contains a single symmetry plane, and let be normal to this plane.
Then
the components of the elastic stiffnes matrix ( ). (symmetrical terms also vanish, of course).
This leaves 13 independent constants.
Similar
restrictions on the thermal expansion coefficient can be determined using
symmetry conditions. Details are left as
an exercise.
In
the following sections, we list the stress-strain relations for anisotropic
materials with various numbers of symmetry planes.
3.2.13 Stress-strain relations for linear elastic
orthotropic materials
An
orthotropic material has three mutually perpendicular symmetry planes. This
type of material has 9 independent material constants. With basis vectors perpendicular to the
symmetry plane, the elastic stiffness matrix has the form
This
relationship is sometimes expressed in inverse form, in terms of generalized
Young’s moduli and Poisson’s ratios (which have the same significance as
Young’s modulus and Poisson’s ratio for uniaxial loading along the three basis
vectors) as follows
Here the generalized
Poisson’s ratios are not symmetric but instead satisfy (no sums). This ensures that the stiffness
matrix is symmetric.
The engineering constants
are related to the components of the compliance tensor by
or in inverse form
For an orthotropic material thermal expansion
cannot induce shear (in this basis) but the expansion in the three directions
need not be equal. Consequently the
thermal expansion coefficient tensor has the form
3.2.14
Stress-strain relations for linear elastic Transversely Isotropic Material
A special case of an orthotropic solid is one that
contains a plane of isotropy (this implies that the solid can be rotated with
respect to the loading direction about one axis without measurable effect on
the solid’s response). Choose perpendicular to this symmetry plane. Then, transverse isotropy requires that ,
,
,
,
so that the stiffness matrix has the form
The engineering constants must satisfy
and
the compliance matrix has the form
where
. As before the Poisson’s ratios are not
symmetric, but satisfy
The
engineering constants and stiffnesses are related by
For
this material the two thermal expansion coefficients in the symmetry plane must
be equal, so the thermal expansion coefficient tensor has the form
3.2.15 Representative
values for elastic constants of transversely isotropic hexagonal close packed
crystals
|
Hexagonal close-packed crystals are an example of
transversely isotropic materials. The axis must be taken to be perpendicular to the
basal (0001) plane of the crystal, as shown in the picture. Since the plane perpendicular to is isotropic the orientation of and is arbitrary.
A table of values of stiffnesses (taken from Freund
and Suresh, Thin Film Materials, CUP 2003) is listed below. F&S list the original sources for their
data on page 163.
|
(GPa)
|
(GPa)
|
(GPa)
|
(GPa)
|
(GPa)
|
Be
|
292.3
|
336.4
|
162.5
|
26.7
|
14
|
C
|
1160
|
46.6
|
2.3
|
290
|
109
|
Cd
|
115.8
|
51.4
|
20.4
|
39.8
|
40.6
|
Co
|
307
|
358.1
|
78.3
|
165
|
103
|
Hf
|
181.1
|
196.9
|
55.7
|
77.2
|
66.1
|
Mg
|
59.7
|
61.7
|
16.4
|
26.2
|
21.7
|
Ti
|
162.4
|
180.7
|
46.7
|
92
|
69
|
Zn
|
161
|
61
|
38.3
|
34.2
|
50.1
|
Zr
|
143.4
|
164.8
|
32
|
72.8
|
65.3
|
ZnO
|
209.7
|
210.9
|
42.5
|
121.1
|
105.1
|
The
engineering constants can be calculated to be
|
(GPa)
|
(GPa)
|
|
|
|
(GPa)
|
(GPa)
|
Be
|
289.38
|
335.17
|
0.09
|
0.04
|
0.04
|
162.50
|
132.80
|
C
|
903.69
|
30.21
|
0.04
|
0.08
|
2.25
|
2.30
|
435.00
|
Cd
|
83.02
|
30.21
|
0.09
|
0.26
|
0.72
|
20.40
|
38.00
|
Co
|
211.30
|
313.15
|
0.49
|
0.22
|
0.15
|
78.30
|
71.00
|
Hf
|
139.87
|
163.07
|
0.35
|
0.26
|
0.22
|
55.70
|
51.95
|
Mg
|
45.45
|
50.74
|
0.36
|
0.25
|
0.23
|
16.40
|
16.75
|
Ti
|
104.37
|
143.27
|
0.48
|
0.27
|
0.20
|
46.70
|
35.20
|
Zn
|
119.45
|
35.28
|
-0.06
|
0.26
|
0.87
|
38.30
|
63.40
|
Zr
|
98.79
|
125.35
|
0.40
|
0.30
|
0.24
|
32.00
|
35.30
|
ZnO
|
127.30
|
144.12
|
0.44
|
0.32
|
0.28
|
42.50
|
44.30
|
3.2.16 Linear elastic stress-strain relations for
cubic materials
A huge number of materials have cubic symmetry all the FCC and BCC metals, for example. The constitutive law for such a material is
particularly simple, and can be parameterized by only 3 material constants. Pick basis vectors perpendicular to the
symmetry planes, as shown.
Then
or in terms of engineering
constants
This is virtually identical to the constitutive law
for an isotropic solid, except that the shear modulus is not related to the Poisson’s ratio and
Young’s modulus through the usual relation given in Section 3.1.6. In fact, the ratio
provides
a convenient measure of anisotropy. For the material is isotropic.
For
this material the thermal expansion coefficient matrix must be isotropic.
The
relationships between the elastic constants are
3.2.17 Representative values for elastic properties of
cubic crystals and compounds
A
table of elastic constants for various cubic crystals and compounds (modified
from Simmons and Wang ‘Single Crystal Elastic Constants and Calculated
Aggregate Properties’ MIT Press (1970)) is given below
Material
|
|
(GPa)
|
(GPa)
|
(GPa)
|
(GPa)
|
|
(GPa)
|
|
Ag
|
(fcc)
|
124.00
|
46.10
|
93.40
|
43.75
|
0.43
|
46.10
|
3.01
|
Al
|
(fcc)
|
107.30
|
28.30
|
60.90
|
63.20
|
0.36
|
28.30
|
1.22
|
Au
|
(fcc)
|
192.90
|
41.50
|
163.80
|
42.46
|
0.46
|
41.50
|
2.85
|
Cu
|
(fcc)
|
168.40
|
75.40
|
121.40
|
66.69
|
0.42
|
75.40
|
3.21
|
Ir
|
(fcc)
|
580.00
|
256.00
|
242.00
|
437.51
|
0.29
|
256.00
|
1.51
|
Ni
|
(fcc)
|
246.50
|
127.40
|
147.30
|
136.31
|
0.37
|
127.40
|
2.57
|
Pb
|
(fcc)
|
49.50
|
14.90
|
42.30
|
10.52
|
0.46
|
14.90
|
4.14
|
Pd
|
(fcc)
|
227.10
|
71.70
|
176.00
|
73.41
|
0.44
|
71.70
|
2.81
|
Pt
|
(fcc)
|
346.70
|
76.50
|
250.70
|
136.29
|
0.42
|
76.50
|
1.59
|
Cr
|
(bcc)
|
339.80
|
99.00
|
58.60
|
322.56
|
0.15
|
99.00
|
0.70
|
Fe
|
(bcc)
|
231.40
|
116.40
|
134.70
|
132.28
|
0.37
|
116.40
|
2.41
|
K
|
(bcc)
|
4.14
|
2.63
|
2.21
|
2.60
|
0.35
|
2.63
|
2.73
|
Li
|
(bcc)
|
13.50
|
8.78
|
11.44
|
3.00
|
0.46
|
8.78
|
8.52
|
Mo
|
(bcc)
|
440.80
|
121.70
|
172.40
|
343.86
|
0.28
|
121.70
|
0.91
|
Na
|
(bcc)
|
6.15
|
5.92
|
4.96
|
1.72
|
0.45
|
5.92
|
9.95
|
Nb
|
(bcc)
|
240.20
|
28.20
|
125.60
|
153.95
|
0.34
|
28.20
|
0.49
|
Ta
|
(bcc)
|
260.20
|
82.60
|
154.50
|
145.08
|
0.37
|
82.60
|
1.56
|
V
|
(bcc)
|
228.00
|
42.60
|
118.70
|
146.72
|
0.34
|
42.60
|
0.78
|
W
|
(bcc)
|
522.40
|
160.80
|
204.40
|
407.43
|
0.28
|
160.80
|
1.01
|
C
|
(dc)
|
949.00
|
521.00
|
151.00
|
907.54
|
0.14
|
521.00
|
1.31
|
Ge
|
(dc)
|
128.40
|
66.70
|
48.20
|
102.09
|
0.27
|
66.70
|
1.66
|
Si
|
(dc)
|
166.20
|
79.80
|
64.40
|
130.23
|
0.28
|
79.80
|
1.57
|
GaAs
|
|
118.80
|
59.40
|
53.70
|
85.37
|
0.31
|
59.40
|
1.82
|
GaP
|
|
141.20
|
70.50
|
62.50
|
102.85
|
0.31
|
70.50
|
1.79
|
InP
|
|
102.20
|
46.00
|
57.60
|
60.68
|
0.36
|
46.00
|
2.06
|
KCl
|
|
39.50
|
6.30
|
4.90
|
38.42
|
0.11
|
6.30
|
0.36
|
LiF
|
|
114.00
|
63.60
|
47.70
|
85.86
|
0.29
|
63.60
|
1.92
|
MgO
|
|
287.60
|
151.40
|
87.40
|
246.86
|
0.23
|
151.40
|
1.51
|
NaCl
|
|
49.60
|
12.90
|
12.40
|
44.64
|
0.20
|
12.90
|
0.69
|
TiC
|
|
500.00
|
175.00
|
113.00
|
458.34
|
0.18
|
175.00
|
0.90
|