# 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 kgms   N, pressure will be in Nm   kgms, density will be in kgm, thermal conductivity in WmK   JsmK   NmsmK   kgmsmK   kgmsK , specific heat in JkgK   NmkgK   msK and so on. The density of steel in the SI system is 7800 kgm.

If you choose mm for length, g for mass, s for time and K for temperature, force will be in gmms and thermal conductivity in gmmsK. In the mmgsK system the density of steel is since    kgm   gmm.

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 length / time, we get that mass = force    time/length. This leads to Nsmm for the mass and Nsmm for density. This means that in the mm   N   s   K system the density of steel is since    kgm   Nsm   Nsmm.

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 length / time.

Finally, a couple of additional examples. Young's Modulus for steel is    N   mm   Nm   kgms   gmms. So its value in the SI system is , in the mmgsK system it is also and in the mm   N   s   K system it is . The heat capacity of steel is    JkgK   msK   mmsK, so in the SI system it is , in the mmgsK and mm   N   s   K system it is .

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.

 symbol meaning m,kg,s,K mm,N,s,K cm,g,s,K E Young's Modulus Density 1 F Force Specific Heat Conductivity 1 h Film Coefficient Dynamic Viscosity

Typical values for air, water and steel at room temperature are:

• air
• JkgKmmsK
• mKNsK
• kgmsNsmm
• (specific gas constant) JkgKmmsK
• water
• kgmNsmm
• JkgKmmsK
• mKNsK
• PasNsmm
• steel
• NmNmm
• (Poisson coefficient)
• kgmNsmm
• JkgKmmsK
• WmKNsK