Young's modulus, Poisson's ratio

Authier, A.; Zarembowitch, A.

doi:10.1107/97809553602060000630

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 83

Section 1.3.3.4.3. Young's modulus, Poisson's ratio

A. Authier^a ^* and A. Zarembowitch^b

^a Institut de Minéralogie et de la Physique des Milieux Condensés, Bâtiment 7, 140 rue de Lourmel, 75015 Paris, France, and ^bLaboratoire de Physique des Milieux Condensés, Université P. et M. Curie, 75252 Paris CEDEX 05, France
Correspondence e-mail: aauthier@wanadoo.fr

1.3.3.4.3. Young's modulus, Poisson's ratio

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If the applied stress reduces to a uniaxial stress, $[T_{11} = T]$ , the strain tensor is of the form $[S_{\alpha} = s_{1\alpha} T. ]$ In particular, $[S_{1} = s_{11}T\semi \quad S_{2} = s_{12}T. ]$ We deduce from this that Young's modulus (equation 1.3.3.1) is $[E = 1/s_{11}. \eqno(1.3.3.13)]$

The elongation of a bar under the action of a uniaxial stress is characterized by $[S_{1}]$ and the diminution of the cross section is characterized by $[S_{2}]$ and $[S_{3}]$ . For a cubic material, the relative diminution of the diameter is $[S_{2} = S_{3} = s_{12}T. ]$ One deduces from this that $[s_{12}]$ is necessarily of opposite sign to $[s_{11}]$ and one calls the ratio $[\nu = -s_{12}/s_{11}]$ Poisson's ratio.

Putting this value into expression (1.3.3.12) for the coefficient of compressibility in cubic or isotropic materials gives $[\kappa^{-1} = 3 s_{11}(1 - 2 \nu). \eqno(1.3.3.14)]$

As the coefficient of compressibility , by definition, is always positive, we have $[0 \,\,\lt\,\, \nu \,\,\lt\,\, 0.5.]$

In practice, Poisson's ratio is always close to 0.3. It is a dimensionless number. The quantity $[s_{44}/2 (s_{11} - s_{12})]$ represents the departure from isotropy of the material and is the anisotropy factor. It is to be noted that cubic materials are not isotropic for elastic properties. Table 1.3.3.2 gives the values of $[s_{11}]$ , $[s_{12}]$ , $[s_{44}]$ , ν and $[s_{44}/2 (s_{11} - s_{12})]$ for a few cubic materials.

Table 1.3.3.2 | top | pdf |
Elastic compliances of some cubic materials in (GPa)⁻¹ (after Landoldt-Börnstein, 1979)

Material	$[s_{11}]$	$[s_{12}]$	$[s_{44}]$	ν	$[s_{44}/2(s_{11}-s_{12}) ]$
Ag	22.9	−9.8	22.1	0.428	0.338
Al	16.0	−5.8	35.3	0.362	0.810
C (diamond)	1.12	−0.14	1.83	0.125	0.726
Cu	15.0	−6.3	13.3	0.42	0.312
Fe	7.67	−2.83	8.57	0.369	0.408
Ge	9.73	−2.64	14.9	0.271	0.602
Mo	2.71	−0.74	9.00	0.273	1.304
Ni	7.67	−2.93	8.23	0.382	0.388
Pb	93.7	−43.04	68.0	0.459	0.249
Si	7.74	−2.16	12.60	0.279	0.636
W	2.49	−0.70	6.35	0.281	0.995
LiF	11.6	−3.35	15.8	0.289	0.528
MgO	4.01	−0.96	6.47	0.239	0.651
NaCl	23.9	−3.20	78.7	0.133	1.452
GaAs	11.75	−3.66	16.8	0.311	0.545
ZnS	19.7	−7.6	22.6	0.386	0.414
BaTiO₃	8.33	−2.68	9.24	0.322	0.420
Adamantane	240	−79	295	0.329	0.462
Almandine (Fe₃Al₂Si₃O₁₂)	4.036	−1.093	10.77	0.271	1.050
Spinel (MgAl₂O₄)	5.80	−2.05	6.49	0.353	0.413

References

International Tables for Crystallography (2006). Vol. D. ch. 1.3, p. 83