(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

æ è

¶L

ö ø

_{p}

b =

1

æ è

¶V

ö ø

_{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:

Coefficients of Thermal Expansion (CTE)

Coefficients of expansion ( in ppm/K, at 20°C, 1 atm)

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
tensora
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}

¶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 ) =
a_{xx} +
a_{yy} +
a_{zz}

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...

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^{2 }) 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_{s }).

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.

Data compiled from various
sources. May have been adjusted slightly for self-consistency.

The following relations hold between the above quantities.
(Do multiply by 10^{9} 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 / Tb^{2 } we prove
elsewhere on this site that:

1

=

1

+

1

K_{T}

K_{S}

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).

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

E_{T}

E_{S}

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...

David R. Lide (Editor-in-Chief)
CRC Handbook of Chemistry and Physics
76th Edition (1995-1996)_{ }
[page 14-34; data from 1950 to 1960]

Amir Yeganeh-Haeri,
Donald
J. Weidner,
John B. Parise
(1992).
Elasticity of a-cristobalite:
A silicon dioxide with a negative Poisson's ratio.
Science257: 650-652.

About Utrahard Fullerite :

V.D. Blank, V.M. Levin, V.M. Prokhorov, S.G. Buga, G.A. Dubitskii,
N.R. Serebryanaya Elastic properties of ultrahard
fullerites Journal of Experimental and Theoretical Physics.
October 1998, Volume 87, Issue 4, pp. 741-746._{ }

(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 + x^{2}
=
n x^{2} + ½ ;
n
=
(1-2x^{2 }) / (2-2x^{2 })
;
x^{2}
=
(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:

As a cubic polynomial of h^{2},
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):