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The reaction term enforcing a Prescribed Displacement on a solid model is a reaction force, similar to a Boundary Load boundary condition.
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The most consistent and general way to avoid spurious reaction terms affecting other physics is to start from the globally symmetric formulation and remove the terms entering equations belonging to other physics interfaces. This limits the reaction terms to affecting the current physics as if there were no other physics in the model, so the reaction terms preserve the symmetry. For Equation 3-1, this means that reaction terms are distributed over both u and v equations, in proportions 1:k(T).
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The other alternative is to apply the reaction terms only on certain individual variables. Another way to look at Equation 3-1 is to read it as prescribing a value for the x-displacement u, rather than prescribing a given relation between u and v. Accepting that view, it is reasonable to insert reaction terms only acting on u. Such reaction terms, in general, do not preserve symmetry even for a single physics interface.
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