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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 stress/strain.

Dimensionality of coordinates is the same as the number of coordinates used to describe the analysis domain. Clearly dimensionality of coordinates is the same as the dimensionality of 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. If the model is formulated in terms of two coordinates, then it is a 2D model, such as the plate/shell model in terms of the coordinates describing the reference surfaces. The governing differential equations are 2D partial differential equations because the unknown field functions are functions of two coordinates. If the model is formulated in terms of three coordinates, then it is a 3D model. The governing differential equations are 3D partial differential equations because the unknown field functions are functions of three coordinates. 

 

Dimensionality of displacements describes whether the model can predict deformation in one direction (1D), two directions (2D), or three directions (3D). Dimensionality of stress/strain displacements describes whether the model can predict uniaxial stress/strain state (1D), plane stress/strain state (2D), or 3D stress/strain state. 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 because stress/strain are second order symmetric tensors. 

 

We frequently describe a mathematical model as 1D, 2D, or 3D without clearly describe what we mean by it, which could cause confusion.  I prefer to describe the dimensionality of model as the dimensionality of the analysis domain. 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, and 1D, 2D, 3D stress/strain states. Dimensionality of displacements are not necessarily corresponding to the dimensionality of stress/strain states. For example, the Timoshenko beam model, one can use it to predict 3D deformation including extension, bending in two directions, torsion and shearing in two directions.  But the stress/strain is only 1D for isotropic homogeneous beams. For composite beams, all six stress components could be significant unless you purposely assume them to vanish as most composite beam theory does. 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). 

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