Material and Spatial Coordinates

In a Cartesian coordinates system, the coordinates of a material point in a reference or initial configuration are denoted by $X$ . The coordinates of the same point in the deformed or final configuration are denoted by $x$ .

The motion or deformation of a body can thus be described by a function

ϕ (X, t)

where the material coordinates

X

and the time

t

are considered as independent variables:(1)

x = ϕ (X, t)

The function $ϕ$ gives the spatial positions of material points as a function of time.

The displacement of a material point is the difference between its original and final positions:(2)

u (X, t) = ϕ (X, t) - X

It is possible to consider displacements and, as a consequence final coordinates $x$ , as functions of initial coordinates $X$ . The initial configuration is assumed to be perfectly known and each coordinate $X$ identifies a specific material point. For this reason, the initial coordinates are called material coordinates.

On the other hand, the final coordinates $x$ identify a point of space which can be occupied by different material points according to the different analyzed configurations. For these reasons, the $x$ is called spatial coordinates.

In solid mechanics, material coordinates are usually called Lagrangian coordinates. In their general definition, they are given by the values of the integration constants of the differential equations of particle trajectories. A particular definition consists in using the coordinates $X$ of the particle in the initial configuration. This point of view corresponds to the definition of material coordinates in solid mechanics.

Use of material coordinates is well suited for solid mechanics as we seek to analyze the evolution of a set of points for which we search the final configuration and properties. Integration can be performed in the initial configuration for which geometric properties are usually simple.

In fluid mechanics however, the engineer is more interested in the evolution of a situation in a region defined by fixed boundaries in space. Boundaries are eventually crossed by fluid particles. It is the spatial configuration which is important while the set of particles may vary. This is the reason why fluid mechanics is usually developed using spatial or Eulerian coordinates.

In solid mechanics, the Eulerian formulation consists in considering displacements and initial coordinates as function of spatial coordinates $x$ . A problem for using Eulerian coordinates in solids mechanics is the difficulty of formulating constitutive equations, such as the relationship between stresses and strains that can take into account change of orientation. For this reason solid mechanics are principally developed using the Lagrangian point of view.

The reason for using the Lagrangian form for solids is primarily due to the need for accurate boundary modeling.