2.8.1 Material (Local) Description
The independent variables are the local (material) coordinates of a particle at time t,
regardless of where the point is in spatial coordinates.
2.8.2 Referential Description (Lagrangian Description)
This is usually the reference configuration description. The independent variables
are the x, y, z coordinates and time with respect to initial configuration and initial
state. The chosen reference configuration is the initial (unstressed) state at t ¼ 0.
2.8.3 Spatial Description (Eulerian Description)
This is the most common coordinate description used in fluid mechanics. The
coordinate system is defined with respect to current location of the object at time t.
As a result, the origin of the coordinate system is located at the new (deformed)
location of the object.
The origin of the material description and Eulerian description of coordinate
systems are usually the same, meaning that they use the same origin and the same
coordinate axes, although they could be different.
Of course transformation of coordinates between these descriptions can easily be
accomplished (Fig. 2.28). where r and s, are local coordinate system axes that move
with the point. O is the origin for referential (Lagrangian) coordinate system. O’ is
the origin for the material (local) and Eulerian coordinate system.
′
Fig. 2.28 Material (local),
spatial, and referential
coordinate system
description
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2 Stress and Strain in Continuum
The independent variables are the local (material) coordinates of a particle at time t,
regardless of where the point is in spatial coordinates.
2.8.2 Referential Description (Lagrangian Description)
This is usually the reference configuration description. The independent variables
are the x, y, z coordinates and time with respect to initial configuration and initial
state. The chosen reference configuration is the initial (unstressed) state at t ¼ 0.
2.8.3 Spatial Description (Eulerian Description)
This is the most common coordinate description used in fluid mechanics. The
coordinate system is defined with respect to current location of the object at time t.
As a result, the origin of the coordinate system is located at the new (deformed)
location of the object.
The origin of the material description and Eulerian description of coordinate
systems are usually the same, meaning that they use the same origin and the same
coordinate axes, although they could be different.
Of course transformation of coordinates between these descriptions can easily be
accomplished (Fig. 2.28). where r and s, are local coordinate system axes that move
with the point. O is the origin for referential (Lagrangian) coordinate system. O’ is
the origin for the material (local) and Eulerian coordinate system.
′
Fig. 2.28 Material (local),
spatial, and referential
coordinate system
description
42
2 Stress and Strain in Continuum