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# Thermodynamicsof  Elasticity

### Related Links (Outside this Site)

Wikipedia Portal  |  PhysLink.com  |  Physics Central
Speed of a Screaming Slingshot Monkey  by  Dmitriy Gekhman (2007).

### Videos

"Young's modulus"   by  Walter Lewin  (MIT 8.01 #26,  12 Nov. 1999)

Physics of Life    by  Pr. J. Scott TurnerSUNY-ESF :
Stress and strain   |   Fun with Young's modulus (hysteresis and resilience)

Stretching Copper Wire (ductility)

A refrigerator with rubber bands  by  Ben Krasnow  (2016-08-23).

## On the Thermodynamics of Elasticity

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

Rubber elasticity   |   Hyperelastic material   |   Cauchy elastic material

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

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

(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 æè ¶L öø p b  = 1 æè ¶V öø p =   3 a 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:

Linear( a = b/3 ) Cubical( b = 3a )  GASES Coefficients of expansion( in  ppm/K, at 20°C, 1 atm) Ammonia  ( NH 3 ) 3573 Carbon Dioxide  ( CO 2 ) 3472 Nitrogen  ( N 2 ) 3426 Dry Air  (in 1842:  b = 3401) 3425 Oxygen  ( O 2 ) 3420 Ideal Gas  ( b = 1/T ) (1137) 3411 Ethanol  (C2H5 OH) 1659 Benzene  (C6H6 ) 1210 950 Glycerol   ( C3 H5 (OH)3 ) (167) 500 Liquid Gallium (30°C) (120) 360 77 (231) 76 (228) (68.89) 206.66 (60.6) 181.8 42 126 47 (141) 24 (72) 19.84 (59.52) 19 (57) Solid Gallium (Ga) 18 (55) 18 (54) Stainless Steel  (18% Cr, 8% Ni) 17.3 (51.9) 16.6 (50) 14.2 (43) 11.8 (35.4) 11.1 (33.3) 9 (27) 4.3 (13) 3.25 (9.75) 2.5 (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  = æè ¶ L öø p ¶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.

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:

• Extensional waves have the following speed (Ve):   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.
• 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:

• 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: ...
• 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   =   -V æè ¶p öø S [Subscript "S" is for isentropic, see below.] ¶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.
Vp(m/s) Vs(m/s) Ve(m/s) n E(GPa) G(GPa) K(GPa) r(g/L) Ultrahard Fullerite Ranked by Vp(speed of sound) 25000 9500 16000 0.416 810 286 1610 3170 17453 11574 17223 0.107 1043 471 443 3516 15000 9460 14470 0.17 490 210 247 2340 12890 8820 12840 0.06 306 143 118 1848 7002 4094 6449 0.24 650 262 417 15630 6435 3035 5000 0.36 68 25 79 2702 6402 3099 5087 0.347 45 16.7 49 1739 6090 3125 5080 0.32 117 44 109 4540 5992 3753 5759 0.177 73 31 37.7 2201 5960 3240 5205 0.29 213 83 170 7874 5935 3220 5175 0.29 210 81 168 7850 5820 3360 5313 0.25 75 30 50 2667 5795 3135 5042 0.29 574 222 462 22570 5640 3280 5175 0.24 62 25 41 2320 5415 4090 5288 -0.16 65.3 39.1 16.4 2335 5212 2885 4614 0.279 412 161 311 19350 5140 3070 4800 0.22 63 25 38 2740 5100 2840 4536 0.28 46 18 34 2240 NS106, CuNi18Zn20 4762 2381 3888 0.33 132 49.5 132 8730 4760 2324 3810 0.34 130 48 139 8960 Brass, CuZn30 4725 2100 3485 0.38 104 38 141 8600 4210 2437 3850 0.25 106 42 70 7136 3980 2380 3720 0.24 54 22 32 3880 3760 2000 3228 0.30 9.6 3.7 8.1 917 3240 1205 2030 0.42 80 28 166 19330 2620 1070 1790 0.40 3.6 1.3 5.9 1110 2220 700 1190 0.45 16 5.5 49 11350 2191 1096 1790 1/3 31.2 11.7 31.2 9747 1550 25 43 »0.50 0.0018 0.0006 2.3 950 1520 0 0 0.50 0 0 2.4 1025 1485 0 0 0.50 0 0 2.2 998 1450 0 0 0.50 0 0 28.5 13546 500 354 500 »0.00 0.062 0.031 0.021 250 343 0 0 0.50 0 0 142 kPa 1.204 Vlarge u/Ö3small usmall » ½ Esmall E / 3small Klarge E / u2K / V2 V 0 0 ½ 0 0 K K / V2 V V/Ö2 V 0 E E / 2 E / 3 E / V2 Data compiled from various sources.  May have been adjusted slightly for self-consistency.

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 E G K

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 ]
 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 KT KS W

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

 b  = 1 æè ¶V öø p is  3 times  the relative change in length per kelvin  (a). 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 ET ES 9 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...

(2002-06-09)   [ SAW  =  Surface Acoustic Waves ]
What's the speed of a Rayleigh wave?

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 )