Poisson's Ratio

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Poisson's ratio or Traverse contraction

Definition of Poisson´s ratio

Assuming a slender round test specimen which is in the plane stress state, a measurable reduction in cross-section Δd occurs in addition to the elongation Δl of the test specimen during tensile loading as a result of volume constancy in the case of small deformations. This reduction of the cross-section is also called transverse contraction and results metrologically as follows [1]:

Elongation:

$\Delta l=l-l_{0}\!$

Transverse contraction:

$\Delta d=d_{0}-d\!$

In prismatic specimens, a simultaneous reduction in the thickness and width of the specimen is observed:

Width change:

$\Delta b=b_{0}-b\!$

Thickness change:

$\Delta h=h_{0}-h\!$

From the relative change in the test specimen dimension, the strains in the longitudinal and transverse directions can be calculated, assuming that the relative change in width and thickness are identical:

Longitudinal strain:

$\varepsilon _{l}\,=\,{\frac {\Delta l}{l_{0}}}\!$

Transverse strain:

$\varepsilon _{q}\,=\,{\frac {\Delta b}{b_{0}}}={\frac {\Delta h}{h_{0}}}\!$

The quotient of the transverse and longitudinal strain is called the transverse contraction or Poisson's ratio and is a material-specific elastic material parameter that reflects incompressible material behaviour at the value of 0.5:

Poisson's ratio:

$\mu \,=\,{\frac {\varepsilon _{q}}{\varepsilon _{l}}}\!$

Temperature and velocity dependence of Poisson´s ratio

The Poisson's ratio for plastics depends on the temperature and the strain rate and is known from experience to be in the range of 0.3 to 0.45. Precision extensometers with a resolution of 0.1 µm are required to measure the Poisson's ratio. The Poisson's ratio is determined in the tensile test at a strain rate of 1 mm/min in the linear range of the strain interval of 0.05 % < ε < ε_y.

Examples for the dependence of Poisson´s ratio on temperature

A comprehensive literature analysis on the experimental values of the transverse contraction or Poisson's ratios is given in [2] for the materials ABS, PA 6, PA 66, PA 6/GF, PA 66/GF, PAI, PBT, PC, PE-HD, PE-LD, PEEK, PEEK/GF, PEI, PEKK, PES, PI, PI/GF, PK, PMMA, POM, PP, PPA, PPA/GF, PPS, PPS/GF, PPSU, PS, PS/PPE, PTFE, PUR, PVC, PVC/GF, SAN and SB. The compilation was supplemented by previously unpublished measurement results. In addition, examples of the dependence of the Poisson's ratio on temperature and normative elongation (see: tensile test) are given (Fig. 1).

Fig. 1:

Dependence of Poisson's ratio on temperature for different materials

In [3] the dependence of the Poisson's ratio on the test temperature is tabulated.

Table 1:

Poisson's ratio µ of the materials as a function of the test temperature

Table 1 shows the Poisson's ratios determined as an overview for selected materials and test temperatures. It can be seen that there is a tendency for the Poisson's ratio to increase with increasing temperature. For the materials PMMA and PVC, there is a temperature dependence that can be fitted with a linear regression function. This is not possible for the PP and PA 6 materials, as the glass temperature T_g is in the temperature interval investigated, which occurs in the range of 0 to 10 °C for polypropylene and between 40 and 60 °C for PA 6, depending on the glass fibre content and the conditioning state.

See also

• Ultrasound – Elastic parameters
• Tensile test

References

[1]	Grellmann, W., Seidler, S. (Eds.): Polymer Testing. Carl Hanser Munich (2022) 3. Edition, p. 112 (ISBN 978-1-56990-806-8; E-Book: ISBN 978-1-56990-807-5; see AMK-Library under A 22)
[2]	Bierögel, C., Grellmann, W.: Quasi-static Tensile Test. In: Grellmann, W., Seidler, S.: Mechanical and Thermomechanical Properties of Polymers. Landolt Börnstein. Volume VIII/6A3, Springer Verlag, Berlin (2014) 136–142 (ISBN 978-3-642-55165-9; see AMK-Library under A 16)
[3]	Sirch, C., Bierögel, C., Grellmann, W.: Temperaturabhängige Bestimmung der lokalen Querkontraktionszahl an Kunststoffen mittels Laserextensometrie. In: Grellmann, W., Frenz, H.: Fortschritte in der Werkstoffprüfung für Forschung und Praxis – Werkstoffeinsatz, Qualitätsicherung und Schadenanalyse. 32. Vortrags- und Diskussionstagung Werkstoffprüfung 2014, 4. und 5. Dezember 2014, Berlin, proceedings p. 155–160 (ISBN 978-3-981-45168-9; see AMK-Library under A 17)