Body force potential edit

How do we find the body force potential? Before we proceed let us examine what conservative vector fields are.

Conservative vector fields edit

  • Work done in moving a particle from point A to point B in the field should be path independent.
  • The local potential at point P in the field is defined as the work done to move a particle from infinity to P.
  • For a vector field to be conservative



The field has to be irrotational.

Determining the body force potential edit

Suppose a body is rotating with an angular velocity   and an angular acceleration of  . Then,


Let us assume that the   coordinate system is oriented at an angle   to the   system. Then,








If the origin is accelerating with an acceleration   (for example, due to gravity), we have,


The body force field is given by


For this vector body force field to be conservative, we require that,


Hence, the field   is conservative only if the rotational acceleration is zero, i.e. = the rotational velocity is constant.=






Integrating equation (43),






Without loss of generality, we can set  . Then,




For a body loaded by gravity only, we can set  ,   and  , to get


For a body loaded by rotational inertia only, we can set  , and  , and get


We can see that an Airy stress function + a body force potential of the form shown in equation (49) can be used to solve two-dimensional elasticity problems of plane stress/plane strain.