  Two triangular factorizations of the deformation gradient tensor are studied.
The first, termed the Lagrangian formulation, consists of an upper-triangular
stretch premultiplied by a rotation tensor. The second, termed the Eulerian
formulation, consists of a lower-triangular stretch postmultiplied by a
different rotation tensor. The corresponding stretch tensors are denoted as the
Lagrangian and Eulerian Laplace stretches, respectively. Kinematics (with
physical interpretations) and work conjugate stress measures are analyzed and
compared for each formulation. While the Lagrangian formulation has been used
in prior work for constitutive modeling of anisotropic and hyper\-elastic
materials, the Eulerian formulation, which may be advantageous for modeling
isotropic solids and fluids with no physically identifiable reference
configuration, does not seem to have been used elsewhere in a continuum
mechanical setting.
