If the strains are the final objective of the study it is necessary to perform additional transformations that require geometrical information of the body. 9.2 3-D.Determination of the strains Strains can be obtained directly from recorded holographic moiré by differentiation of the patterns in the frequency space [22]. This operation is straight forward when dealing with plane surfaces. If dealing with 3-D surfaces there is an important set of concepts that are a consequence of the description of Continuum Mechanics variables and the changes of these variables with the coordinates systems utilized to describe them [21], [23], [24]. In the case of displacements that are vectorial quantities the rules of transformation are simple and are contained in the vector algebra. The transformations of tensors are not so simple because when analyzing tensors on a 3-D surface the tensor must be contained in the tangent plane to the surface. This is well known in applications of 3-D elasticity solutions, (i.e. Theory of Shells). Figure 21 Fringe patterns of the displacements of a cylinder under internal pressure. In Experimental Mechanics a similar procedure must be followed and local strains in 3-D surfaces must be represented in 2D coordinate systems contained in the tangent plane to the surfaces. This operation of transformation of the strain tensor is very important in 3-D holographic interferometry of surfaces since displacements are obtained in a selected coordinate system, a global coordinate system that is utilized for displacement computation. Hence the derivatives of the displacements are obtained in the global system. The passage from the global coordinate system to the local involves the transformation of the strain tensor from the global coordinates system to the local coordinate systems, at each of the points of the surface. Figure 22 shows the local coordinate system 0-x1,x2, x3 and the local coordinate system contained in the tangent plane to the surface 0-x’1,x ’ 2, x’ 3. It involves a rotation of the coordinates system in the 3-D space. It involves the matrix transformation, [ ] [ ] ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ α α ε α α α α α α ε ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎣ ⎡ α α ε α α α α α α ε = 33 23 13 32 22 12 31 21 11 33 32 31 23 22 21 13 12 11 ' (25) 168
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