On Dimensionality of Models
In the field of composites, a model consists of a system of equations governing some physical fields such as the displacement/stress/strain fields, which are functions of fundamental variables (usually coordinates). Hence there are three types of dimensionality that we are dealing with: dimensionality of coordinates, dimensionality of displacements, and dimensionality of stresses/strains.
Dimensionality of coordinates is the same as the number of coordinates used to describe the analysis domain. If the model is formulated in terms of one coordinate, then it is a 1D model, such as the beam model in terms of the beam reference axis. The governing differential equations are ordinary differential equations.
Dimensionality of displacements describes whether the model can predict deformation in one direction (1D), two directions (2D), or three directions (3D). Uniaxial stress/strain state could have up to three components including one normal component, and two shear components. Plane stress/strain state could have up to three components including two normal components and one shear components. 3D stress/strain state could have up to six components for a Cauchy continuum because stress/strain are second order symmetric tensors.
I prefer to describe the dimensionality of model as the dimensionality of the analysis domain (the number of coordinates used to describe the model). According to this definition, a beam theory is a 1D model, a plate/shell theory is a 2D model, and the elasticity theory is a 3D model. Note a beam model (1D) could predict 1D, 2D, or 3D displacements, Denoting the dimensionality of a model the same as that of the analysis domain also makes sense because it is also the same as the dimensionality of finite element meshes if the finite element analysis is used. Computational cost of a model is more significantly influenced by the dimensionality of analysis domain, not the dimensionality of field functions (displacement/stress/strain).