Embedded coordinate system

Note 2.5 (Embedded coordinates Jt>). In solids we can trace each material point X by attaching labels on its surface during deformation. Then, as shown in Fig. 2.4, it is easy to introduce a coordinate system in which the label values of the coordinates are not changed (x = X ) but the reference basis Gi (before deformation) is changed into the current basis gi (after deformation). This is referred to as the embedded coordinate system. 

In the embedded coordinate system, the transformation rule of the base vectors is given by ... 

Scheme C.l Geometries and coordinate systems of (a) a microspherical electrode, (b) a microdisc embedded in an insulating surface...

If we pick an arbitrary element we can see that it is represented by the xy-coordinates of the four nodal points, as depicted in Fig. 9.16. The figure also shows a -coordinate system embedded within the element. In the r/, or local, coordinate system, we have a perfectly square element of area 2x2, where the element spreads between —1 > < 1 and — 1 > rj < 1. This attribute allows us to easily allows us to use Gauss quadrature as a numerical integration scheme, where the limits vary between -1 and 1. The isoparametric element described in the //-coordinate system is presented in Fig. 9.17. 

Fig. 14.14 The melt exits the die at z = 0 the radius of the bubble R and the thickness 8 are a function of z. The coordinate system 7, is embedded into the inner surface of the bubble.

We first derive the kinematics of the deformation. The flow situation is shown in Fig. 14.14. Coordinate z is the vertical distance in the center of the axisymmetric bubble with the film emerging from the die at z = 0. The radius of the bubble R and its thickness 8 are a function of z. We chose a coordinate system C, embedded in the inner surface of the bubble. We discussed extensional flows in Section 3.1 where we defined the velocity field of extensional flows as... 

In order to get more experience with the newly proposed index ( ) we will consider the leading eigenvalue X of D/D matrices for several well-defined mathematical curves. We should emphasize that this approach is neither restricted to curves (chains) embedded on regular lattices, nor restricted to lattices on a plane. However, the examples that we will consider correspond to mathematical curves embedded on the simple square lattice associated with the Cartesian coordinates system in the plane, or a trigonal lattice. The selected curves show visibly distinct spatial properties. Some of the curves considered apparently are more and more folded as they grow. They illustrate the self-similarity that characterizes fractals. " A small portion of such curve has the appearance of the same curve in an earlier stage of the evolution. For illustration, we selected the Koch curve, the Hubert curve, the Sierpinski curve and a portion of another Sierpinski curve, and the Dragon curve. These are compared to a spiral, a double spiral, and a worm-curve. 

We consider the convected coordinate system embedded in a flowing fluid and deforming with it. The convected coordinates denoted by... 

A microdisc electrode is a micron-scale flat conducting disc of radius r. that is embedded in an insulating surface, with the disc surface flush with that of the insulator. It is assumed that electron transfer takes place only on the surface of the disc and that the supporting smlace is completely electroinactive under the conditions of the experiment. These electrodes are widely employed in electrochemical measurements since they offer the advantages of microelectrodes (reduced ohmic drop and capacitive effects, miniaturisation of electrochemical devices) and are easy to fabricate and clean for surface regeneration. In Chapter 2, we considered a disc-shaped electrode of size on the order of 1 mm. In that case we could approximate the system as one-dimensional because the electrode was large in comparison to the thickness of the diffusion layer, such that the current was essentially uniform across the entire electrode surface. Due to the small size of the microdisc, this approximation is no longer valid so we must work in terms of a three-dimensional coordinate system. While the microdisc can... 

Fig. 2.2. Supercritical (left) and subcriticcJ (right) Hopf bifurcations embedded in a three dimensional center manifold with a curved coordinate system (, i ) The arrows indicate the direction of the flow.

Now suppose we have a thin gap geometry - a mold, for example - in which flow occurs in both the x and z directions. The geometry is shown schematically in Figure 5.4, where the x-z coordinate system is embedded in the lower surface and the upper surface is located at a distance y = H(x,z). The lower surface moves relative to the upper with a velocity F F is a vector with components F and F in the x and z directions, respectively. Note that we could always align the coordinates in the x-z plane so that either Vx or F is zero but, as we saw with the screw extruder, this might not be the most physically meaningful choice. 

In describing the kinematics of a deformable body, instead of using a coordinate system fixed in space, it is convenient to use a coordinate system embedded in the moving object. This is frequently referred to as a convected coordinate system, and was first introduced by Oldroyd (1950). Any measure of deformation (strain) defined relative to such a coordinate system always refers to the same element of materials, and therefore should be independent of the local rate of translation or rotation. As will be shown in this section, if they are going to be useful, all kinematic variables defined in terms of the convected coordinates must be transformed to a fixed coordinate system as all physical measurements are made relative to the fixed coordinate system. 

Cylindrical polar coordinates (r, , z) are taken with z in the direction of flow (or axial direction). Also, symmetry around the z axis is assumed. Although the cylindrical system is space-fixed, for the calculation of stresses, a moving Cartesian coordinate system, f i, 2, ts, embedded in the inner surface of the bubble is used, as shown in Figure 9.21 (where both coordinate systems are shown). The 12 direction is normal to the film (i.e., thickness direction) the f 1 direction is in the direction of flow (i.e., machine direction) and the ts direction is perpendicular to 2 and i and tangent to the circumferential directions. 

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