# Poisson's constant

Poisson´s constant is the ratio of heat capacity at constant pressure Cp to heat capacity at constant volume ${\displaystyle C_{v}}$.

It is denoted by a Greek letter ${\displaystyle \gamma }$ (gamma) or ${\displaystyle \kappa }$ (kappa)

## Equation:

${\displaystyle \gamma ={\frac {C_{p}}{C_{v}}}={\frac {c_{p}}{c_{v}}}}$

• ${\displaystyle \gamma }$ – Poisson's constant;
• ${\displaystyle C_{p}}$ – heat capacity at constant pressure;
• ${\displaystyle C_{v}}$ – heat capacity at constant volume;
• ${\displaystyle c_{V}}$ a ${\displaystyle c_{P}}$ – relevant specific heat capacities.

Because ${\displaystyle c_{P}}$ is always greater than ${\displaystyle c_{V}}$, Poisson´s constant is always greater than 1.

Poisson's constant makes it possible to describe adiabatic process:
${\displaystyle pV^{\gamma }=konst.}$

• ${\displaystyle p}$ – gas pressure,
• ${\displaystyle V}$ – volume of gas,
• ${\displaystyle \gamma }$ – Poisson´s constant.

## Ratio for an ideal gas

For an ideal gas, the heat capacity does not depend on temperature. Therefore, we can express the enthalpy as ${\displaystyle H=C_{p}\cdot T}$ and internal energy ${\displaystyle U=C_{v}\cdot T}$. Thus, we can say that Poisson´s constant is the ratio of enthalpy to internal energy: ${\displaystyle \gamma ={\frac {H}{U}}}$

On the other hand, we can express heat capacities via Poisson's constant
${\displaystyle C_{p}={\frac {\gamma \cdot R}{(\gamma -1)}}}$

${\displaystyle C_{v}={\frac {R}{(\gamma -1)}}}$, where ${\displaystyle R}$ – universal gas constant.

## Poisson's number

In practice, the inverse value of Poisson's constant, the so-called Poisson's number is more often used. It is denoted by a Greek letter ${\displaystyle \mu }$ (in some sources ${\displaystyle \nu }$). The value is also dimensionless and for most materials takes on values ​​from the interval 0 to 0.5. Applies to:

${\displaystyle \mu ={\frac {1}{\gamma }}=|{\frac {\varepsilon _{x}}{\varepsilon _{y}}}|}$

• ${\displaystyle \nu }$ – Poisson's number,
• ${\displaystyle \gamma }$ – Poisson's constant,
• ${\displaystyle \varepsilon _{x}}$ – relative deformation in the supporting direction (stress direction),
• ${\displaystyle \varepsilon _{y}}$ – relative deformation in the transverse direction (perpendicular to the stress direction).

Poisson's number is independent of the direction of loading for isotropic materials. For anisotropic materials such as wood or composites, the Poisson's number is different along the direction of loading to the structure.

From the above definition, it follows that Poisson's number is always positive, because it represents the absolute value of the share of proportional deformations. Because it is true for most materials that they taper in the transverse direction when stretched, and thus
${\displaystyle \varepsilon _{x}>0}$
${\displaystyle \varepsilon _{y}=0}$

Some sources give a definition of Poisson's number in the form:
${\displaystyle \nu =-{\frac {\varepsilon _{y}}{\varepsilon _{x}}}}$

However, there are modern materials that expand when stretched in the transverse direction. Using the second relation, these materials have a negative Poisson's number.

### Values ​​of Poisson's number for selected materials

Steel 0,27 to 0,30        concrete 0,20           rubber 0,50


### The relationship between modulus of elasticity

For an isotropic material, Poisson's number relates the modulus of elasticity in tension, the so-called Young's modulus, to the modulus of elasticity in shear according to the equation:

${\displaystyle G={\frac {E}{2\cdot (1+\nu )}}}$, where

• ${\displaystyle G}$ – modulus of elasticity in shear,
• ${\displaystyle E}$ – Young's modulus,
• ${\displaystyle \nu }$ – Poisson's number.