Mechanics Of Solids: Rigid and deformable bodies

Rigid and deformable bodies

Rigid body motion theory is a fundamental and well-established part of physics. It is based on the approximation that for stiff materials, any force applied to a body produces a negligible deformation. Thus, the only change a force can produce is change in the center of mass motion and change in the rotational motion. This means that simulation of even complex bodies is relatively simple, and thus this method has become popular in the computer simulation field.

Given the forces acting on the body, the motion can be determined using ?? ??for translational motion, and a similar relation for rotational motion .

The rigid body motion model has traditionally been applied in range analysis in CAD and for computer animation where deformation is not required. If the deformation is not negligible, then the approximation does not hold, and we need to start over and come up with some other model. There exists many different models, but the two models which have emerged to become the most widely used in practice are: mass-spring models and statics models solved using the Finite Element Method (FEM).

Mass-spring models represent bodies as discrete mass-elements, and the forces between them are transmitted using explicit spring connections ('spring' is a historical term, and is not limited to pure Hooke interactions). Given the forces acting on an element, we can determine its motion using . The motion of the entire body is then implicitly described by the motion of its elements.

Mass-spring models have traditionally been applied mostly for cloth simulation. Statics models are based on equilibrium relations, and thus make the approximation that the effect of dynamics are negligible. Relations between the strain and stress fields of a body are set up, and through specifying known values of these fields, through for example specifying forces acting on the body, the unknown parts can be determined. These relations form differential equations, and the known values are boundary values. The FEM is an effective method for solving boundary value problems, and has thus given its name to these types of problems. Statics models have traditionally been applied in stress and displacement analysis systems in CAD.