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Final Answers
© 2000-2016   Gérard P. Michon, Ph.D.    

Thermodynamics
of  Elasticity


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On the Thermodynamics of Elasticity


(2013-01-15)   Elastic Deformations
If the stress vanishes, so does the strain  (i.e., the relative elongation).

 Come back later, we're
 still working on this one...

Rubber elasticity   |   Hyperelastic material   |   Cauchy elastic material


(2013-03-08)   Hysteresis
Stored elastic energy is never fully recovered.

 Come back later, we're
 still working on this one...


(2013-01-20)   Elastomers are polymer endowed with viscoelasticity
Vulcanized cross-linked structures main survive strains of nearly  900%.

 Come back later, we're
 still working on this one...


(2008-03-16)   Coefficients of Thermal Expansion   (CTE)
For isotropic substances, the  thermal expansivity  is equal to three times the  linear coefficient of thermal expansion.

Usually, the size of a body increases with temperature.  The  relative  increase is a characteristic of the material  (under given conditions of temperature and pressure)  known as the thermal expansion coefficient.

That coefficient is usually positive  (at constant pressure, an increase in temperature entails an increase in volume).  However, some substances contract when warmed up.  One example is liquid water between  0°C  and  3.98°C  (under the  normal pressure  of  1 atm).  A lesser-known example is bismuth  (which expands  3.32%  upon solidification at  271.40°C).

Unfortunately, there are  two  flavors of this coefficient depending on whether a length (L) or a volume (V) is used to gauge the aforementioned  size.  Because volume is proportional to the cube of length, the volumetric (or cubical) coefficient  b  is exactly  3 times  the linear coefficient  a  for an  isotropic  substance  (HINT:  dV/V = 3 dL/L).  The coefficient  b  is also known as the  thermal expansivity  of the material:

a  =   1   æ
è
L ö
ø
 
 p
  b  =   1   æ
è
V ö
ø
 
 p
   =   3 a
vinculum vinculum vinculum vinculum
L T V T

[ This notation is dominant, but some authors swap the rôles of a and b. ]  A traditional source of confusion is that the  linear coefficient  (a)  is usually tabulated for  solids  whereas the  volumetric coefficient  (b)  is commonly tabulated for fluids.  (What's  really  confusing is that some such tables call "a" the cubical coefficient.)  Here's an  unambiguous  table:

Coefficients of Thermal Expansion  (CTE)
Coefficients of expansion
( in  ppm/K, at 20°C, 1 atm)
Linear
( a = b/3 )
Cubical
( b = 3a )
 
G
A
S
E
S
Ammonia  ( NH 3 )  3573
Carbon Dioxide  ( CO 2 ) 3472
Nitrogen  ( N 2 ) 3426
Dry Air  (in 1842b = 3401)  3425
Oxygen  ( O 2 ) 3420
Ideal Gas  ( b = 1/T )(1137)3411
 
L
I
Q
U
I
D
S
 
&
 
P
O
L
Y
M
E
R
S
Ethanol  (C2H5 OH) 1659
Benzene  (C6H6 ) 1210
Gasoline950
Glycerol   ( C3 H5 (OH)3 )(167)500
Liquid Gallium (30°C)(120)360
Rubber77(231)
Nylon 6-676(228)
Water(68.89)206.66
Mercury (Hg)(60.6)181.8
Benzocyclobutene (BCB)42126
 
S
O
L
I
D
S
Lithium (Li)47(141)
Aluminum (Al)24(72)
Plutonium (Pu)19.84(59.52)
Brass19(57)
Solid Gallium (Ga) 18(55)
Silver  (Ag)18(54)
 Stainless Steel  (18% Cr, 8% Ni) 17.3(51.9)
Copper  (Cu)16.6(50)
Gold  (Au)14.2(43)
Steel  (1% C)11.8(35.4)
Iron  (Fe)11.1(33.3)
Ordinary glass9(27)
Tungsten (W)4.3(13)
Pyrex Glass3.25(9.75)
Silicon2.5(7.5)
Invar  (64% Fe, 36% Ni)0.62(1.9)
Fused quartz  (SiO2)0.59(1.8)

Thermal Expansion of Anisotropic Solids :

The above is only valid for  amorphous  substances or crystals with cubic symmetry.  Other crystals may have different coefficients of thermal expansions along different directions.  (The coefficient is even strongly  negative  along certain axes in feldspars, for example.)

The linear "coefficient" of thermal expansion  (CTE)  is thus really a  tensor  a  which maps the spatial vector  L  separating two points of the crystal to the temperature-derivative  of  L  (which isn't necessarily parallel to  L ):

a L  =   æ
è
L ö
ø
 
 p
vinculum
T

The cubical coefficient of thermal expansion is simply the  trace of that tensor  (the expression of the trace is the same in any coordinate system).

b   =   tr ( a )   =   axx  +  ayy  +  azz

I don't see any reason why the  thermal expansion tensor  should be symmetrical  (which would imply orthogonal principal directions).  Yet, I couldn't find any discussion of the torsional component in the literature...

Wikipedia :   Thermal expansion
 
Crystal Physics  by  Johann Potoschnig   (2011-04-11).
Coefficients of Thermal Expansion  by  Kaye & Laby   (NPL, 2010).
Measuring the CTE of anisotropic materials by ESPI  (Electronic Speckle Pattern Interferometry)
 
Anisotropic thermal expansion in the silicate b-eucryptite:  A neutron diffraction and density functional study
A.I. Lichtenstein,  R. O. Jones,  H. Xu,  and  P.J. Heaney ,  Physical Review B, 58, 10   (Sept. 1998-II).
 
On the Orientation of the Thermal and Compositional Strain Ellipsoids in Feldspars
C. Willaime,  W.L. Brown  and  M.C. Perucaud,  American Mineralogist, 59   (1974).
 
Thermal Expansion of Polytetrafluoroethylene (Teflon) from -190° to +300°C
Richard K. Kirkby,  Journal of Research of the National Bureau of Standards, 57, 2   (1956).
Solid teflon undergoes a phase transition at the temperature where it was polymerized.


(2013-01-16)   Invar Alloy :   36% Ni, 64% Fe (by mass)
A remarkable anomaly.

 Come back later, we're
 still working on this one...

Wikipedia :   Invar (1896)   |   Charles-Edouard Guillaume (1861-1938; Nobel 1920)
 
Invar and Elinvar: Nobel lecture by Charles-Edouard Guillaume (1920)


(2002-06-02)   What's the speed of sound in a solid?

The waves that propagate in the midst of a solid's bulk are called body waves. For an homogeneous isotropic material, there are (only) two types of these.  Although both could possibly qualify as "sound", the term "speed of sound" is best reserved for Vp, the celerity of P-waves (which are always faster than any other mechanical waves for a given solid).  This is also called longitudinal velocity (VL = Vp) in contradistinction with the slower transverse velocity (VT = Vs) of S-waves.

  • P waves of speed Vp (primary waves or pressure waves) are longitudinal compression waves, for which material moves back and forth along the direction of propagation.  They're also called push-pull waves. 
  • S waves of speed Vs (secondary waves or shear waves) are transverse waves for which material moves side-to-side, in a direction perpendicular to the direction of propagation.  The "S" may also be remembered as standing for "slow", "shake" or "shock" (since S-waves often cause the most damage in actual earthquakes because of amplitudes that are often much larger than P-waves).

At the surface of a solid, different propagation conditions prevail, corresponding to surface waves (known to seismologists as Rayleigh waves or Love waves).

In the Lab, it's most convenient to study a given material in the form of a thin rod.  Two types of waves leave invariant the axis of such a rod:

Hammer
Rod
  • Extensional waves have the following speed (Ve): 
    vinculum
    Ve  =  Vs  Ö (3Vp 2 -4Vs 2 ) / (Vp 2 -Vs 2 )
    They are triggered in a thin rod by some longitudinal stress (for example, a hammer hitting one end in the direction of the axis). However, any material with a nonzero Poisson's ratio [that's to say almost any material] will respond to an axial force [or stress] with both an axial and a lateral elongation [or strain], which means that the cross section of the rod varies accordingly and extensional waves thus involve some radial motion.  In materials with a very small Poisson's ratio (like beryllium or cork), these are virtually identical to P-waves.   Applying Torque and 
 Measuring Twist
  • Torsional waves propagate a change in torsion.  They have the same speed (Vs) as shear waves.

Besides the solid's density ( r), the following dynamic quantities are relevant:

     Simeon Poisson 
 1781-1840
  • Poisson's Ratio (n) is the ratio of the lateral shrinking to the longitudinal elongation which occurs in the direction of a pull.
    In a few rare so-called auxetic materials, this may be negative, which indicates the expansion is both longitudinal and lateral.
  • Elasticity (E) is the increase in tension per unit of cross-sectional area or a small relative increase in the elongation of a wire.  E  is called Young's modulus, to distinguish it from other elasticity coefficients.  It was first described in 1807 by Thomas Young (1773-1829).
    A relative increase in length is a dimensionless quantity called strain, and E is thus a "stress to strain ratio".  So are the other elasticity coefficients G, K and l, described below.  All of these are in units of pressure (or stress): pascals (Pa, N/m) or GPa.
  • Rigidity (G) is also called modulus of rigidity, shear modulus, or torsional modulus: ...   Come back later, we're
 still working on this one...
  • Stiffness (K) is also called bulk modulus  or  incompressibility.  It's the increase in pressure for a small relative decrease in volume.  K is the inverse of compressibility (k).
    K   =   - æ
    è
    p ö
    ø
     
    S
            [Subscript "S" is for isentropic, see below.]
    vinculum
    V
  • Lamé modulus (l):   l = K-2G/3   is one of the two Lamé constants sometimes used in basic elasticity theory (the other one is   m = G ).
    Since   l = 3K n / (1+n) ,  the  Lamé modulus  is negative for auxetic materials (i.e., the materials for which  -1 < n < 0 ).
    Both  Lamé constants  are named after Gabriel Lamé de la Droitière (1795-1870; X1814) sixth holder of the original (1794) chair of physics at  Polytechnique,  from 1832 to 1844.
Dynamic (Acoustic) Properties of Selected Isotropic
(or Polycrystalline) Substances (20°C, 1 atm)
Ranked by Vp
(speed of sound)
Vp
(m/s)
Vs
(m/s)
Ve
(m/s)
n E
(GPa)
G
(GPa)
K
(GPa)
r
(g/L)
Ultrahard Fullerite 250009500160000.416 8102861610 3170
"Fused" Diamond 1745311574172230.107 1043471443 3516
Boron 150009460144700.17 490210247 2340
Beryllium 128908820128400.06 306143118 1848
Tungsten Carbide 7002409464490.24 650262417 15630
Aluminum 6435303550000.36 682579 2702
Magnesium 6402309950870.347 4516.749 1739
Titanium 6090312550800.32 11744109 4540
Fused Quartz 599237535759 0.177 733137.7 2201
Iron 596032405205 0.29 21383170 7874
Steel (1% C) 593532205175 0.29 21081168 7850
Granite (LdB) 5820336053130.25 753050 2667
Osmium 5795313550420.29 574222462 22570
Pyrex Glass 5640328051750.24 622541 2320
a-Cristobalite 541540905288 -0.16 65.339.116.4 2335
Tungsten (W) 5212288546140.279 412161311 19350
Basalt (typ.) 5140307048000.22 632538 2740
Crown Glass 5100284045360.28 461834 2240
NS106, CuNi18Zn20 4762238138880.33 13249.5132 8730
Copper 4760232438100.34 13048139 8960
Brass, CuZn30 4725210034850.38 10438141 8600
Zinc 4210243738500.25 1064270 7136
Flint Glass 3980238037200.24 542232 3880
Ice   (0°C) 3760200032280.30 9.63.78.1 917
Gold 3240120520300.42 8028166 19330
Nylon 6-6 2620107017900.40 3.61.35.9 1110
Lead 222070011900.45 165.549 11350
Bismuth 2191109617901/3 31.211.731.2 9747
Rubber 15502543»0.50 0.00180.00062.3 950
Seawater 1520000.50 002.4 1025
Fresh Water 1485000.50 002.2 998
Mercury 1450000.50 0028.5 13546
Cork 500354500»0.00 0.0620.0310.021 250
Dry Air 343000.50 00142 kPa 1.204
  Vp
(m/s)
Vs
(m/s)
Ve
(m/s)
n E
(GPa)
G
(GPa)
K
(GPa)
r
(g/L)
Ideal Rubber V
large
u3
small
u
small
» ½ E
small
E / 3
small
K
large
E / u2
K / V2
Fluid V00 ½ 00KK / V2
Ideal Cork VV2V 0 E E / 2 E / 3 E / V2
Data compiled from various sources.  May have been adjusted slightly for self-consistency.
 Extensional speed
 and Poisson's ratio
 as a function of the
 secondary/primary
 body wave speed ratio.

The following relations hold between the above quantities.  (Do multiply by  109  the values tabulated  in  gigapascals  for  E, G and K.)

    9    =     3  +   1   
vinculum vinculum vinculum
EGK

E  =  2(1+n)G  =  3(1-2n)K  =  9 KG / (3K+G)
(Vp)2 = 3K/r  (1-n) / (1+n)   =  (K + 4G/3) / r
(Vs)2 = G/r
(Ve)2 = E/r      [Note:   Ve > Vs   if   n > -½ ]
vinculum
Ve  =  Vs  Ö (3Vp 2 -4Vs 2 ) / (Vp 2 -Vs 2 )

x + n  =  x n + ½       [where   x = (Vs/Vp)2 ]
1/x  =  4/3 + K/G  =  1 + 3K/E
x  =  (n-½) / (n-1)


(2002-06-02)  Thermodynamics of Elasticity
Isothermal  (static)  and adiabatic  (dynamic)  parameters.

The above elasticity coefficients involved in wave propagation are the isentropic (or adiabatic) ones.  Static measurements of these coefficients may also be done fairly slowly under isothermal conditions.  A correction is to be applied to translate such an isothermal coefficient into an adiabatic one.  For example, the adiabatic bulk modulus (K = KS) is slightly larger than its isothermal counterpart (KT).

At temperature T, we denote by cp the specific heat capacity at constant stress (in J/K per kg) and b the expansivity (the relative change in volume per kelvin, also known as  thermal coefficient of cubical expansion ).  Introducing   W = r cp / Tb2   we prove elsewhere on this site that:

    1     =     1   +   1    
vinculum vinculum vinculum
KT KS W

KS   =   KT / (1 - KT / W )
KT   =   KS / (1 + KS / W )

b  =   1   æ
è
V ö
ø
 
 p
  is  3 times  the relative change in length per kelvin  (a).
vinculum vinculum
V T

Usually, the quantity W is much larger than K and the relative difference between the adiabatic and the (smaller) isothermal coefficient is thus very close to K/W. For example, in the case of iron,  b  is listed at 35.4 10-6/K, whereas Cp is about  448 J/K/kg  and r is 7874 g/L.  At 20°C (T = 293.15 K), this makes W roughly  9600 GPa.  As K itself is about 56 times smaller than that (about 170 Gpa), we obtain a relative difference of 1.8 %, which is of borderline relevance here.

For the record, the isothermal and adiabatic versions of rigidity are identical (G = GS = GT)  but the two flavors of  Young's modulus  (E)  obey a relation similar to what we just described for the  bulk modulus  (K)  except that the  linear  coefficient of expansion  (a = b/3)  takes on the rôle of the  cubical  one  (b)  so the final relation involves  9W  instead of  W :

    1    =     1  +   1   
vinculum vinculum vinculum
ETES9 W

The above identity   9/E  =  3/G  +  1/K   holds in both the adiabatic and isothermal cases.  I'm calling  W  the  (thermal) wring  of the material.

Further Reading, References & Online Data...


 Coat of arms of
 John William Strutt, 
 Lord Rayleigh (2002-06-09)   [ SAW  =  Surface Acoustic Waves ]
What's the speed of a Rayleigh wave?

 John W.S. Rayleigh  Rayleigh waves were described mathematically in 1885 by  Lord Rayleigh  (born  John William Strutt,  1842-1919)  before  they were observed in earthquakes.

A Rayleigh wave is a surface acoustic wave (SAW) which propagates in an homogeneous and isotropic solid at the planar boundary with some medium having little or no inertia (air or vacuum).

In the midst of such a solid, body waves propagate with two different celerities.  The so-called P-waves travel at the longitudinal celerity Vp whereas S-waves travel at the transverse celerity Vs.  We may call x the ratio Vs/Vp  [which is at most ½Ö3 (86.6%) but only exceeds ½Ö2 (70.7%) in the rare case of auxetic substances].  This parameter x is tied to the solid's Poisson's ratio (n):

n + x2  =  n x2 + ½ ;     n  =  (1-2x2 ) / (2-2x2 ) ;     x2  =  (1-2n) / (2-2n)

If we call VR the celerity of Rayleigh waves, we may call h the ratio VR/VS and the following relation holds:

h6 - 8 h4 + 8 (3-2x2 ) h2 - 16 (1-x2 )   =   0         [ VR = h VS ]

As a cubic polynomial of h2, this equation may have other positive roots with a physical meaning (see article by Edouard G. Nesvijski) but there's only one which is less than unity and is relevant to pure Rayleigh propagation.  More precisely, h decreases from about 0.95531250 for rubbery materials  (x » 0)  to 0.87403205 for Ideal Cork  (x = ½Ö2)  or, possibly, even down to 0.68889218 for the most auxetic substances we could dream of  (x » ½Ö3).  An approximative formula for ordinary materials (positive n) was proposed by L. Bergmann  (in 1954):

h   »   (0.87 + 1.12 n) / (1+n)     =     (2.86 - 3.98 x2 ) / (3 - 4 x2 )

 Come back later, we're
 still working on this one...

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