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 Material, 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.

The strain field must be converted into a displacement field u(X) such that

If the strain field fulfills the general compatibility relations, it is in principle possible to integrate the above relation. The procedure is outlined in Ref. 3, giving

Summation over double indices 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,

The distance along the integration path can then be parameterized by a parameter s running from 0 to 1as

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.

For the physics interfaces which have rotational degrees of freedom (Beam, Shell, and Plate), not only the displacement, but also the rotation of the constraint is needed. For a given displacement field u(X), the infinitesimal rotation vector Θ is given by

Applying the rotation operator to Equation 3-179 gives

where εmni is the permutation tensor.