ELASTICITY

1
ELASTICITY

• Elasticity
• Elasticity of Composites
• Viscoelasticity
• Elasticity of Crystals (Elastic Anisotropy)

Elasticity Theory, Applications and
Numerics Martin H. Sadd Elsevier
Butterworth-Heinemann, Oxford (2005)
2
Revise modes of deformation, stress, strain and
other basics (click here) before starting this
chapter etc.
3
Let us start with some observations

• When you pull a rubber band and release it, the
  band regains it original length.
• It is much more difficult (requires more load) to
  extend a metal wire as compared to a rubber
  string.
• It is difficult to extend a straight metal wire
  however, if it is coiled in the form of a spring,
  one gets considerable extensions easily.
• A rubber string becomes brittle when dipped in
  liquid nitrogen and breaks, when one tries to
  extend the same.
• A diver gets lift-off using the elastic energy
  stored in the diving board. If the diving board
  is too compliant, the diver cannot get sufficient
  lift-off.
• A rim of metal is heated to expand the loop and
  then fitted around a wooden wheel (of say a
  bullock cart). On cooling of the rim it fits
  tightly around the wheel.

Click here to know about all the mechanisms by
which materials fail
4
Elasticity

• Elastic deformation is reversible deformation-
  i.e. when load/forces/constraints are released
  the body returns to its original configuration
  (shape and size).
• Elastic deformation can be caused by
  tension/compression or shear forces.
• Usually in metals and ceramics elastic
  deformation is seen at low strains (less than
  103). At higher strains, in addition to elastic
  deformation we get plastic deformation. However,
  other materials can be stretched elastically to
  large strains (few 100).So elastic deformation
  should not be assumed to be small deformation!
• The elastic behaviour of metals and ceramics is
  usually linear (i.e. the stress strain curve is
  linear).

Linear
E.g. Al deformed at small strains
Elasticity
Non-linear
E.g. deformation of an elastomer like rubber
(or elastic deformation)
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Time dependent aspects of elastic deformation

• In normal elastic behaviour (e.g. when small
  load is applied to a metal wire, causing a strain
  within elastic limit) it is assumed that the
  strain is appears instantaneously.
• Similarly when load beyond the elastic limit is
  applied (in an uniaxial tension test Slide 7 ),
  it is assumed that the plastic strain develops
  instantaneously. It is further assumed that the
  load is applied quasi-statically (i.e. very
  slowly).
• However, in some cases the elastic or plastic
  strain need not develop instantaneously.
  Elongation may take place at constant load with
  time. These effects are termed Anelasticity (for
  time dependent elastic deformation at constant
  load) and Viscoelasticity. Anelastic materials
  are a subset of viscoelastic materials. In
  anelastic materials the strain is fully
  recoverable and in dynamic loading, the
  stress-strain response depends on the frequency.
  Viscoelastic response depends on the rate of
  loading (strain rate).
• Creep is one such phenomenon, where permanent
  deformation takes place at constant load (or
  stress). E.g. if sufficient weight is hung at the
  end of a lead wire at room temperature, it will
  elongate and finally fail.

Recoverable
Instantaneous
Elastic
Time dependent
Anelasticity
Deformation
Instantaneous
Plastic
Viscoelasticity
Time dependent
Permanent
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Atomic model for elasticity

• At the atomic level elastic deformation takes
  place by the deformation of bonds (change in bond
  length or bond angle).
• Let us consider the stretching of bonds (leading
  to elastic deformation).
• Atoms in a solid feel an attractive force at
  larger atomic separations and feel a repulsive
  force (when electron clouds overlap too much)
  at shorter separations. (Of course, at very large
  separations there is no force felt).
• The energy and the force (which is a gradient of
  the energy field) display functional behaviour as
  in the equations below. This implies under a
  state of compressive stress the atoms want to
  go apart and under tensile stress they want to
  come closer.

Energy
Force
Repulsive
A,B,m,n ? constantsm gt n
Attractive
The plots of these functions is shown in the next
slide
This is one simple form of interatomic
potentials (also called Lennard-Jones potential,
wherein m 12 and n 6).
7

• The plot of the inter-atomic potential and force
  functions show that at the equilibrium
  inter-atomic separation (r0) the potential energy
  of the system is a minimum and force experienced
  is zero.
• In reality the atoms are undergoing thermal
  vibration about this equilibrium position. The
  amplitude of vibration increases with increasing
  temperature. Due to the slight left-right
  asymmetry about the minimum in the U-r plot,
  increased thermal vibration leads to an expansion
  of the crystal.

Repulsive
Repulsive
r0
Force (F) ?
Potential energy (U) ?
r ?
r ?
Attractive
r0
Attractive
r0
Equilibrium separation between atoms
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• Youngs modulus is the slope of the
  Force-Interatomic spacing curve (F-r curve), at
  the equilibrium interatomic separation (r0). I.e.
  it is the curvature of the energy-displacement
  (interatomic separation) plot.
• In reality the Elastic modulus is 4th rank tensor
  (Eijkl) and the curve below captures one aspect
  of it.

Youngs modulus (Y / E)
Near r0 the red line (tangent to the F-r curve at
r r0) coincides with the blue line (F-r) curve

• Youngs modulus is proportional to the ve slope
  of the F-r curve at r r0

r ?
Force ?
r0
For displacements around r0 ? Force-displacement
curve is approximately linear? THE LINEAR
ELASTIC REGION
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Stress-strain curves in the elastic region

• In metals and ceramics the elastic strains (i.e.
  the strains beyond which plastic deformation sets
  in or fracture takes place) are very small
  (10?3). As these strains are very small it does
  not matter if we use engineering stress-strain or
  true stress strain values (these concepts will be
  discussed later). The stress-strain plot is
  linear for such materials.
• Polymers (with special reference to elastomers)
  shown non-linear stress-strain behaviour in the
  elastic region. Rotation of the long chain
  molecules around a C-C bond can cause tensile
  elongation. The elastic strains can be large in
  elastomers (some can even extend a few hundred
  percent), but the modulus (slope of the
  stress-strain plot) is very small. Additionally,
  the behaviour of elastomers in compression is
  different from that in tension.

Tension
Stress-strain curve for an elastomer
Stress ?
?T due to uncoilingof polymer chains
?C
Due to efficient filling of space
Strain ?
?T
?C
?T
gt
Compression
Brittle ceramics may show no plastic
deformation and may fracture after elastic
deformation.
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Other elastic moduli

• We have noted that elastic modulus is a 4th rank
  tensor (with 81 components in general in 3D). In
  normal materials this is a symmetric tensor (i.e.
  it is sufficient to consider one set of the
  off-diagonal terms).
• In practice many of these components may be zero
  and additionally, many of them may have the same
  value (i.e. those surviving terms may not be
  independent). E.g. for a cubic crystal there are
  only 3 independent constants (in two index
  condensed notation these are E11, E12, E44).
• For an isotropic material the number of
  independent constants is only 2. Typically these
  are chosen as E and ? (though in principle we
  could chose any other two as these moduli are
  interrelated for a isotropic material). More
  explained sooner.
• In a polycrystal (say made of grains of cubic
  crystal), due to average over all orientations,
  the material behaves like an isotropic material.
  More about this discussed elsewhere.
  Mathematically the isotropic material properties
  can be obtained from single crystal properties by
  Voigt or Reuss averaging.

In tensor notation

• E ? Youngs modulus
• G ? Shear modulus
• K ? Bulk modulus
• ? ? Poissons ratio

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Relation between the elastic constants of an
isotropic material

• We have noted that for an isotropic material, the
  elastic properties are described by two
  independent elastic constants (typically, E
  ?).
• The other elastic constants like the bulk
  modulus (K) and the shear modulus (G or ?) can
  be derived from these constants by standard
  formulae (e.g. below).? List of elastic moduli
  for isotropic materials Youngs modulus (Y, E),
  Poissons ratio (?), Shear Modulus (G, ?), Bulk
  modulus (K), Lamés constant (?), P-wave modulus
  (M).
• The relative magnitudes of these constants can be
  seen from the table below.
• Similar to the stress-strain relation we saw
  before, other types of stresses can be related to
  the corresponding strain via the appropriate
  elastic moduli/constants (as below).

Property Magnitude
Youngs modulus (Y, E) 70 GPa
Poissons ratio 0.3
Shear Modulus (G) 26.9 GPa
Bulk Modulus (K) 58.3 GPa
Lamé constant (?) 40.4 GPa
Click here to know how isotropy can arise in a
material
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• When a body is pulled (let us assume an isotropic
  body for now), it will elongate along the pulling
  direction and will contract along the orthogonal
  direction. The negative ratio of the transverse
  strain (?t) to the longitudinal strain (?t) is
  called the Poissions ratio (?).
• In the equation below as B1 lt B0, the term in
  square brackets in the numerator is ve and
  hence, Poissions ratio is a positive quantity
  (for usual materials). I.e. usual materials
  shrink in the transverse direction, when they are
  pulled.
• The value of E, G and ? for some common materials
  are in the table (Table-E). Zero and even
  negative Possions ratio have been reported in
  literature. The modulus of materials expected to
  be positive, i.e. the material resists
  deformation and stores energy in the deformed
  condition. However, structures and
  material-structures can display negative
  stiffness (e.g. when a thin rod is pushed it will
  show negative stiffness during bulking? observed
  in displacement control mode).

Initial configuration is represented by 0 in
subscript
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Elastic properties of some common materials
T1 gives independent elastic constants of a cubic
crystal. T2 lists elastic properties of selected
isotropic materials. F1 shows a stress-strain
plot in the elastic regime of select materials.
T1 Anisotropic Elastic constants for Cubic System 2 Anisotropic Elastic constants for Cubic System 2 Anisotropic Elastic constants for Cubic System 2
T1 E11 (GPa) E44 (GPa) E12 (GPa)
Al 108 28.3 62
Au 190 42.3 161
Cu 169 75.3 122
Fe 230 117 135
Ni 247 122 153
Pd 224 71.6 173
Si 165 79.2 64
F1
T2 Specification E (GPa) ? sy (MPa) ?y (?10?3) Ref.
Metals Aluminum BS 1490 (99 pure) 71 0.345 240 3.4 1
Metals Mild Steel BS 070M20 gt0.2 wt. C 210 0.293 215 1.0 2
Ceramics and Glass Aluminum Oxide (Al2O3) - 350 344.7-408.8 0.21-0.27 255-260.6 (Fracture) 0.7 3
Ceramics and Glass Glass (SiO2) - 73 72.76-74.15 0.166-0.177 69.6-74.53 (Fracture) 1.0 3
Polymers Polyester General purpose grade 4 1.03-4.48 0.33 51.5-55.1 12.5 3
Room temperature data
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Bonding and Elastic modulus

• Materials with strong bonds have a deep potential
  energy well with a high curvature ? high elastic
  modulus.
• Along the period of a periodic table the covalent
  character of the bond and its strength increase ?
  systematic increase in elastic modulus.
• Down a period the covalent character of the
  bonding ? ? ? in Y.
• On heating the elastic modulus decreases on
  heating from 0 K ? M.P there is a decrease of
  about 10-20 in modulus (i.e. Youngs modulus is
  not a sensitive function of T).

Along the period ? Li Be B Cdiamond Cgraphite
Atomic number (Z) 3 4 5 6 6
Youngs Modulus (GN / m2) 11.5 289 440 1140 8
Down the row ? Cdiamond Si Ge Sn Pb
Atomic number (Z) 6 14 32 50 82
Youngs Modulus (GN / m2) 1140 103 99 52 16
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Anisotropy in the Elastic modulus
What is meant by anisotropy (click here Slide
13)?

• In a crystal the interatomic and inter-planar
  distance varies with direction? this aspect can
  be used to intuitively understand the origin of
  elastic anisotropy. We already know that linear
  and planar densities depend on the specific
  choice (i.e. the 100CCP direction has different
  linear density as compared to the 111CCP and
  similarly, the (100)CCP has a different planar
  density as compared to (111)CCP).
• In a elastically anisotropic material, stretching
  along certain directions can lead to shearing.
• Elastic anisotropy is especially pronounced in
  materials with ? two kinds of bonds, e.g. in
  graphite E 10?10 950 GPa, E 0001 8 GPa ?
  Two kinds of ordering along two directions, e.g.
  in Decagonal QC E100000?E000001

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How to determine the elastic modulus?

• The Youngs modulus (Y) of a isotropic material
  can be determined from the stress-strain diagram.
  But, this is not a very accurate method, as the
  machine compliance is in series with the specimen
  compliance. The slope of the stress-strain curve
  at any point is termed as the tangent modulus.
  In the initial part of the s-e curve this
  measures the Youngs modulus. Other kinds of
  mechanical tests can also be used for the
  measurement of Y.
• A better method to measure Y is using wave
  transmission (e.g. ultrasonic pulse echo
  transmission technique) in the material. This is
  best used for homogeneous, isotropic,
  non-dispersive material (wherein, the velocity of
  the wave does not change with frequency). Common
  polycrystalline metals, ceramics and inorganic
  glasses are best suited to this method. Soft
  plastics and rubber cannot be characterized by
  this method due to high dispersion, attenuation
  of sound wave and non-linear elastic properties.
  Porosity and other internal defects can affect
  the measurement.
• The essence of the ultrasound technique is to
  determine the longitudinal and shear wave
  velocity of sound in the material (symbols vl
  vs). Y and Poissons ratio (?) can be
  determined using the formulae as below.

Note again that it is not a good idea to
calculate Youngs modulus for s-e plot
Peak Stress 193.0 MPa
Peak Load 5.83 kN

• ? ? density

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Funda Check
What is meant by anisotropy?

• Anisotropy implies direction dependence of a
  given property. An isotropic material has the
  same value of a property along all directions.
• Neumanns principle states that symmetry of a
  property will be equal to or greater than the
  point group symmetry of the crystal. Cubic
  crystals (with 4/m ?3 2/m) symmetry are isotropic
  with respect to optical properties.
• Amorphous materials are isotropic.
  Polycrystalline materials with random orientation
  of grains are also isotropic. When such a
  polycrystalline material is plastically deformed,
  e.g. by rolling, the material develops
  crystallographic texture (preferential
  orientation of the grains) and this can lead to
  anisotropy.
• Two important origins of anisotropy are (i)
  crystalline, (ii) geometry of sample (shape).
• Couple of effects of crystalline anisotropy (i)
  if an electric field is applied along a
  direction, the current may flow along a different
  direction (ii) if a body is pulled, it may shear
  in addition to being elongated.

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Material dependent
Elastic modulus
Property
Geometry dependent
Stiffness
Elastic modulus in design

• In many applications the elastic properties of
  the material is the main one to be considered.
  However, other factors and properties should also
  be taken into account.
• More than the modulus, it is the stiffness, which
  comes into focus as a design parameter.
• Stiffness (as we have noted) of a structure is
  its ability to resist elastic deformation
  (deflection) on loading ? depends on the geometry
  of the component. Spring steel (e.g. high carbon
  steel with Si, Mn Cr) will be characterized by
  its elastic modulus. In the form of the coil the
  spring (a structure with a certain geometry)
  shows more elongation for the same load and
  stiffness is the relevant property of the spring
  in design.
• In many applications high modulus in conjunction
  with good ductility should be chosen (good
  ductility avoids catastrophic failure in case of
  accidental overloading)
• Covalently bonded materials (e.g. diamond has a
  high E (1140 GPa)) in spite of their high modulus
  are rarely used in engineering applications as
  they are brittle (poor tolerance to cracks- for
  more on this refer to chapter on fracture). Ionic
  solids are also very brittle.

Ionic solids ? NaCl MgO Al2O3 TiC Silica glass
Youngs Modulus (GN / m2) 37 310 402 308 70
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• METALS
• Metals are used in multiple engineering
  applications due to a combination of properties
  they possess (i) reasonable elastic modulus,
  (ii) good ductility, (iii) ability to be alloyed
  to give good combination of properties, (iv)
  amenable to many types of fabrication techniques
  (casting, forging, extrusion, etc.), (v) good
  electrical and thermal conductivity, etc. The
  main issue with metals is often many of them have
  poor oxidation and corrosion resistance and not
  so good tolerance to high temperatures.
• Metals of the First transition series posses a
  good combination of ductility modulus (200
  GPa). Second third transition series have an
  even higher modulus, but their higher density is
  a shortcoming.
• POLYMERS
• Polymers are light weight and have become
  universal. However, they have a low modulus
  dependent and cannot withstand high temperatures.
  They have a poor wear resistance as well.
• The nature of secondary bonds (Van der Walls /
  hydrogen) is responsible for their low modulus.
  Further aspects which determine the modulus of
  polymers are (i) presence of bulky side groups,
  (ii) branching in the chains ? Unbranched
  polyethylene E 0.2 GPa, ? Polystyrene
  with large phenyl side group E 3 GPa, ?
  3D network polymer phenol formaldehyde E 3-5
  GPa(iii) Extent of cross-linking (more
  cross-linking give rise to higher stiffness).

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Increasing the modulus of a material

• The modulus of a metal can be increased by
  suitable alloying.
• E is a structure (microstructure) insensitive
  property and this implies that grain size,
  dislocation density, etc. do not play an
  important role in determining the elastic modulus
  of a material.
• One of the important strategies to increase the
  modulus of a material is by forming a
  hyrbrid/composite with an elastically harder
  (stiffer) material. E.g. TiB2 is added to Fe to
  increase the modulus of Fe.
• COMPOSITES
• A hard second phase (termed as reinforcement) can
  be added to a low E material to increase the
  modulus of the base material. The second phase
  can be in the form of particles, fibres,
  laminates, etc.
• Typically the second phase though harder is
  brittle and the ductility is provided by the
  matrix. If the reinforcement cracks the
  propagation of the crack is arrested by the
  matrix.

Particulate composite
Aligned fiber composite
Laminate composite
Hybrids are designed to improve certain
properties of monolithic materials
Click here to know more about hybrids
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Youngs modulus of a composite

• The Youngs modulus of the composite is between
  that of the matrix and the reinforcement.
• Let us consider two extreme cases.(A) Isostrain
  ? the matrix and the reinforcement (say long
  fibres) are under identical strain. This is
  known as Voigt averaging. (B) Isostress ? the
  matrix and fibre are under identical stress. This
  is known as Reuss averaging.
• Let the composite be loaded in uniaxial tension
  and the volume fraction of the fibre be Vf
  (automatically the volume fraction of the matrix
  is Vm (1? Vf)).

Isostrain
Voigt averaging
This is like resistances in series
These formulae are derived later
Isostress
This is like resistances in parallel
Reuss averaging
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• The modulus of a real composite will lie between
  these two extremes (usually closer to isostrain).
  The modulus of the composite will depend on the
  shape of the reinforcement and the nature of the
  interface (e.g. in a long aligned fibre composite
  having a perfectly bonded interface with no
  slippage will lead to isostrain conditions.
• Purely from a modulus perspective, a larger
  volume fraction will give a higher modulus
  however, ductility and other considerations
  typically limit the volume fraction of
  reinforcement in a composite to about 30.

Voigt averaging

• Under iso-strain conditions ?m ?f ?c
• I.e. resistances in series configuration

• Under iso-stress conditions ?m ?f ?c
• I.e. resistances in parallel configuration
• Usually not found in practice

Reuss averaging
Yf
Isostrain
Yc ?
Isostress
For a given fiber fraction f, the modulii of
various conceivable composites lie between an
upper bound given by isostrain conditionand a
lower bound given by isostress condition
Ym
Volume fraction ?
Vf
A
B
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Derivation of formulae
Isostrain
F
Isostress
Af
Am
L
A
F
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Elasticity of Single Crystals and Anisotropy

• For small stress, strain is proportional
  (linearly varies with) strain ? Hookes law.
• The proportionality constants are 4th order
  (rank) tensors. The stiffness tensor is Eijkl or
  Cijkl. The compliance tensor is denoted as Sijkl.

• Number of components of Eijkl Cijkl (or Sijkl)
• in most general case (in practice the number of
  components reduces considerably)
• 2D 2?2?2?2 16, (i 1,2)
• 3D 3?3?3?3 81, (i 1,2,3)

Tensor form
Hookes law
Usage-1 In short Usage-2 Dimensions
S (or s) Elastic Compliance constant (Elastic) Compliance Elastic Modulus (N/m2)?1 stress?1
C (or c) Y, E Elastic Stiffness constant (Elastic) Stiffness Elastic Constant (Youngs modulus) (N/m2) stress
Youngs modulus is an elastic constant and not
an elastic modulus!
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What does the existence of these Cijkl components
physically mean?

• If we apply just one component of stress (say
  ?11), this in general will produce not only ?11
  but also other components of strain (which other
  components will depend on symmetry of the
  crystal- which we shall consider shortly). A
  glimpse of this we have already seen (in the
  topic on understanding stress and strain)? if we
  pull a cylinder along y-direction, it will
  contract along the x-direction ? ?11 exists
  (apart from elongation in the y-direction ?22).
• But, this may not be the only other strain
  produced.

2D
3D

• This implies that the body may shear also (?12
  ?21 exists). This occurs due to anisotropy in
  the crystal.
• So (in general) a bar ? might shear if pulled
  (in addition to elongating)? may twist if bent
  (in addition to bending)? may bend if twisted
  (in addition to twisting).

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• We saw that a single stress component ?11 may be
  related to many strains.
• Similarly, many stress components may give rise
  to a single strain (?11).

2D
3D
There are 4 such equations (for each component
?ij)
There are 9 such equations
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No of independent components of Cijkl and Sijkl

• The total number of expected components are 2D
  2?2?2?2 16, (i 1,2) 3D 3?3?3?3 81 , (i
  1,2,3). But, not all of these are independent.
• ? Cijkl Cijlk (also, Sijkl Sijlk) ? Cijkl
  Cjikl (also, Sijkl Sjikl)? only 36 of the 81
  components are independent (for even the most
  general crystal).
• From energy arguments it can further be seen
  that Cijkl Cklij (also, Sijkl Sklij) ? of
  the 36 components only 21 survive (even for the
  crystal of the lowest symmetry? i.e. triclinic
  crystal).
• Given that crystals can have higher symmetry than
  the triclinic crystal, the number of independent
  components of Cijkl (or Sijkl) may be
  significantly reduced.
• All cubic crystals (irrespective of their point
  group symmetry) have only 3 independent
  components of Cijkl (or Sijkl)? these are C1111,
  C1122, C2323.
• This is further reduced in the case of isotropic
  materials to 2 independent constants (C1111,
  C1122). Usually, these independent constants are
  written as E ?.
• Hence, for second rank tensor properties (like
  magnetic susceptibility, ?ij) cubic crystals are
  isotropic but, not for forth rank tensor
  properties like Elastic Stiffness.

The reasoning is postponed for later.
28

• The magnitude of anisotropy in cubic crystals is
  quantified using the Zener anisotropy factor (A)
  as defined in the equation below.
• A value of A of one (1) implies that the
  crystal is isotropic.

? 1011 N/m2 ? 1011 N/m2 ? 1011 N/m2
Structure C11 C12 C44 A
BCC Li 0.135 0.114 0.088 8.4
Na 0.074 0.062 0.042 7.2
K 0.037 0.031 0.019 607
DC C 10.20 2.50 4.92 1.3
Si 1.66 0.64 0.80 1.6
Ge 1.30 0.49 0.67 1.7
NaCl type NaCl 0.485 0.125 0.127 0.7
KCl 0.405 0.066 0.063 0.37
RbCl 0.363 0.062 0.047 0.31
A gt 1 ? lt111gt direction stiffest
Zener Anisotropy factory (A)
A lt 1 ? lt100gt direction stiffest