Units

An important issue which frequently raises questions concerns units. Finite element programs do not know any units. The user has to take care of that. In fact, there is only one golden rule: the user must make sure that the numbers he provides have consistent units. The number of units one can freely choose depends on the application. For thermomechanical problems you can choose four units, e.g. for length, mass, time and temperature. If these are chosen, everything else is fixed. If you choose SI units for these quantities, i.e. m for length, kg for mass, s for time and K for temperature, force will be in kgm$/$s$^2 =$   N, pressure will be in N$/$m$^2 =$   kg$/$ms$^2$, density will be in kg$/$m$^3$, thermal conductivity in W$/$mK$=$   J$/$smK$=$   Nm$/$smK$=$   kgm$^2/$s$^3$mK$=$   kgm$/$s$^3$K , specific heat in J$/$kgK$=$   Nm$/$kgK$=$   m$^2/$s$^2$K and so on. The density of steel in the SI system is 7800 kg$/$m$^3$.

If you choose mm for length, g for mass, s for time and K for temperature, force will be in gmm$/$s$^2$ and thermal conductivity in gmm$/$s$^3$K. In the $\{$mm$,$g$,$s$,$K$\}$ system the density of steel is $7.8 \times 10^{-3}$ since $7800$   kg$/$m$^3 = 7800 \times 10^{-6}$   g$/$mm$^3$.

However, you can also choose other quantities as the independent ones. A popular system at my company is mm for length, N for force, s for time and K for temperature. Now, since force = mass $\times$ length / time$^2$, we get that mass = force $\times$    time$^2$/length. This leads to Ns$^2/$mm for the mass and Ns$^2/$mm$^4$ for density. This means that in the $\{$mm$,$   N$,$   s$,$   K$\}$ system the density of steel is $7.8 \times 10^{-9}$ since $7800$   kg$/$m$^3 = 7800$   Ns$^2/$m$^4 = 7.8 \times 10^{-9}$   Ns$^2/$mm$^4$.

Notice that your are not totally free in choosing the four basic units: you cannot choose the unit of force, mass, length and time as basic units since they are linked with each other through force = mass $\times$ length / time$^2$.

Finally, a couple of additional examples. Young's Modulus for steel is $210000$   N$/$   mm$^2 = 210000 \times 10^6$   N$/$m$^2 = 210000 \times 10^6$   kg$/$ms$^2 = 210000 \times 10^6$   g$/$mms$^2$. So its value in the SI system is $210 \times 10^9$, in the $\{$mm$,$g$,$s$,$K$\}$ system it is also $210 \times 10^9$ and in the $\{$mm$,$   N$,$   s$,$   K$\}$ system it is $210 \times 10^3$. The heat capacity of steel is $446$   J$/$kgK$= 446$   m$^2/$s$^2$K$= 446 \times 10^6$   mm$^2/$s$^2$K, so in the SI system it is $446.$, in the $\{$mm$,$g$,$s$,$K$\}$ and $\{$mm$,$   N$,$   s$,$   K$\}$ system it is $446 \times 10^6$.

Table 1 gives an overview of frequently used units in three different systems: the $\{$m$,$   kg$,$   s$,$   K$\}$ system, the $\{$mm$,$   N$,$   s$,$   K$\}$ system and the $\{$cm$,$   g$,$   s$,$   K$\}$ system.

Table 1: Frequently used units in different unit systems.

symbol	meaning	m,kg,s,K	mm,N,s,K	cm,g,s,K
E	Young's Modulus	$1 \frac{N}{m^2 } = 1 \frac{kg}{m s^2 }$	$= 10^{-6} \frac{N}{mm^2 }$	$= 1 \frac{g}{mm s^2 }$
$\rho$	Density	1 $\frac{kg}{m^3 }$	$= 10^{-12} \frac{Ns^2}{mm^4 }$	$= 10^{-6} \frac{g}{mm^3 }$
F	Force	$1 N = 1 \frac{kg m}{ s^2 }$	$= 1 N$	$= 10^{6} \frac{g\: mm}{s^2 }$
$c_p$	Specific Heat	$1 \frac{J}{kg K} = 1 \frac{m^2}{ s^2 K}$	$= 10^{6} \frac{mm^2}{s^2 K }$	$= 10^6 \frac{mm^2}{s^2 K }$
$\lambda$	Conductivity	1 $\frac{W}{m K } = 1 \frac{kg m}{s^3 K }$	$= 1 \frac{N}{s K }$	$= 10^6 \frac{g\: mm}{s^3 K}$
h	Film Coefficient	$1 \frac{W}{m^2 K } = 1 \frac{kg}{s^3 K }$	$= 10^{-3} \frac{N}{mm\: s K }$	$= 10^3 \frac{g}{s^3 K }$
$\mu$	Dynamic Viscosity	$1 \frac{N s}{m^2 } = 1 \frac{kg}{m s }$	$= 10^{-6} \frac{N s}{mm^2 }$	$= 1 \frac{g}{mm\: s }$