  Residual stresses are self-equilibrated stresses on unloaded bodies. Owing to
their complex origins, it is useful to develop functions that can be linearly
combined to represent any sufficiently regular residual stress field. In this
work, we develop orthonormal sequences that span the set of all
square-integrable residual stress fields on a given three-dimensional region.
These sequences are obtained by extremizing the most general quadratic,
positive-definite functional of the stress gradient on the set of all
sufficiently regular residual stress fields subject to a prescribed
normalization condition; each such functional yields a sequence. For the
special case where the sixth-order coefficient tensor in the functional is
homogeneous and isotropic and the fourth-order coefficient tensor in the
normalization condition is proportional to the identity tensor, we obtain a
three-parameter subfamily of sequences. Upon a suitable parameter
normalization, we find that the viable parameter space corresponds to a
semi-infinite strip. For a further specialized spherically symmetric case, we
obtain analytical expressions for the sequences and the associated Lagrange
multipliers. Remarkably, these sequences change little across the entire
parameter strip. To illustrate the applicability of our theoretical findings,
we employ three such spherically symmetric sequences to accurately approximate
two standard residual stress fields. Our work opens avenues for future
exploration into the implications of different sequences, achieved by altering
both the spatial distribution and the material symmetry class of the
coefficient tensors, toward specific objectives.
