By Shabana A.A.
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Furthermore, one can show that the cross product between the two ÿ Á P vectors in Equation 181 can be written as u Â v ¼ 3i,j¼1 ui vj ii Â ij ¼ P3 i,j,k¼1 Cijk ui vj ik . In a similar manner to the third-order tensor, a Fourth-order tensor F can be defined as F ¼ ðu1 u2 u3 u4 Þ ¼ 3 X i,j,k,l¼1 f ijkl ii ij ik il ð1:82Þ where um , m ¼ 1, 2, 3, 4 is an arbitrary vector. As in the case of third-order tensors, one can write u4m ðu1 u2 u3 Þ ¼ Fim , where u4m is the mth component of the vector u4 .
Understanding the difference between these two transformations is crucial in understanding the definitions of the strain components that will be introduced in the next chapter. This subject is also important in the large deformation finite element formulation discussed in this book. Change of Parameters In order to explain the difference between the change of parameters and the coordinate transformation of vectors, we consider the vector r, which is expressed in terms of three coordinates x1 , x2 , and x3 .
This triad defines a coordinate system and the orthogonal matrix Q. 6 D’Alembert’s Principle 27 to the upper triangular matrix B defined as B ¼ QT A. The way the orthogonal matrix Q is defined here gives a physical interpretation for the QR decomposition of 3 Â 3 matrices. In the next chapter, it will be shown that the matrix of position vector gradients plays an important role in the formulation of the kinematic and strain equations. , 2006). Furthermore, the triangular matrix B, whose diagonal elements define the principal values, provides an alternate upper triangular form instead of the symmetric matrix U that results from the polar decomposition theorem.