Constraints like Fixed Constraint and
Prescribed Displacement will in general cause stresses near the constrained boundaries when the structure undergoes temperature changes. The same is true also for rigid objects like
Rigid Domain,
Rigid Connector, and
Attachment. By adding a
Thermal Expansion subnode to these features, you can allow the constrained boundaries to have a thermal expansion displacement.
Summation over double indexes is implied. The rigid body rotation term given in Ref. 3 is omitted, since it cannot be derived from the strain field. The reference point is chosen so that the displacement (caused by the strain field) is zero, so that the
ui(x0) term can be omitted. The integral is path independent when the compatibility is fulfilled. Because the constrained region is a virtual object, the integration path does not have to be inside a domain. For simplicity, a straight line from
X0 to
X is used for the integration. Let
p be the vector between the two points,
This integral can be computed using the built-in integrate() operator as long as the strain field is an explicit function of the material frame coordinates
X.
where εmni is the permutation tensor.