tensor_analysis_basics - MY5400 MIDTERM REVIEW, FALL 2011,...

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Unformatted text preview: MY5400 MIDTERM REVIEW, FALL 2011, PATRICK K BOWEN In tensor notation, indices with only one occurrence are free indices, and those with two occurrences are dummy indices. More than two are not allowed. Free indices can assume any value from 13, and dummy indices must be summed over. The Kronecker Delta has a simple form: { The Kronecker Delta will switch the indices, in the term that comes after it ( will switch the existing index "i" with new index "j" that is: The product of two permutation indices can be written instead in terms of Kronecker Deltas: The determinant of a matrix is of the form (for column vector construction): And the determinant of a matrix product is: In this identity, the dummy index must be written first. The two permutation indices can be rewritten as long as the index sense is preserved, so if "i" is the dummy variable, for the same sense as above, or for the opposite sense For example, consider the dual cross product: ( ) . This can be written in terms of permutation indices and deltas: ( ) ( ) Also consider a mixed product ( ) (a scalar result), which is representative of a volume formed by the vectors: When a tensor is applied to a vector ), it is called a ( linear transformation. A vector can be transformed into new coordinates through (or, in indicial notation, ) where the Q matrix is defined as: [ ] The Kronecker Delta can be factored "out" of any vector or matrix, say to turn . The Permutation Symbol, , will assume a value of 1, -1, or 0 depending on the index order and value. This index will assume values: To transform a matrix to a new coordinate basis, In a simple case where an x-y coordinate system is rotated by with respect to the old coordinate axes (CCW being positive), [ ] { The even/odd permutation in determined by drawing a circle in which 1,2,3 are ordered clockwise. If the index is, say, 2,3,1, then the permutation will be even, because the indices follow the clockwise arrangement. If the index is 1,3,2, then the permutation will be odd because the indices run counter to the circle direction. If the index is 1,3,1, then the index will assume a value of zero. Remember that indices in equations are arbitrary, that is: different The dyadic product ( of and results in a second rank tensor instead of the cross product's first rank (vector) or dot product's zeroth rank tensor (scalar). The tensor takes the form: ( ) [ ] And the determinant of also evaluates to zero, to which there is a corresponding equation in which there are three invariant (I) terms which do not change with the reference coordinate system: To transform higher-order matrices, there must be more transformation matrices as well. For example, Recall that in eigenvalues and eigenvectors, a symmetric matrix can be described by: Recall that the directional cosine of a vector is the vector's components divided by the vector's length: || || Higher-order dyadic products are possible, and each one results in the addition of an extra rank. The dyadic product of , , and , for example, would result in a third rank (27-element) tensor. Each tensor () will have a symmetric ( and antisymmetric ( component such that , according to: ( ) [END OF DOCUMENT FOR Is] The eigenvectors comprise a set of mutually orthogonal vectors which, when used correctly, can yield the principal components of the matrix. The three vectors, n(i), can be arranged into a Q matrix with which the reference matrix can be transformed into its principal components. In the first equation, n(i) is expanded vertically to create a second rank tensor. The dot product is the amount of vector in the direction of vector , or: || |||| || ( ) The cross product is the vector ( ) orthogonal to the plane formed by and in a right-handed manner: ( ) The dual vector representation of is , where the three components come from the form of above. MY5400 MIDTERM REVIEW, FALL 2011, PATRICK K BOWEN [ index switching and the condition of symmetry: { Each pair has six independent combinations, which results in 36 overall combinations. However, the (ij)/(jk) matrix will be symmetric (C11 through C66 with 1=11, 2=22, 3=33, 4=23, 5=13, and 6=12), having 6 diagonal values and 15 symmetric values. Therefore, there are a maximum of 21 symmetric components by virtue of indicial symmetry. Returning to the case of monoclinic symmetry (symmetry about, for example, the (001) plane), we can eliminate any component in which the extended notation includes an odd number of threes. This results in 13 total components (eliminating 8). Rotation is carried out by multiplication of a vector/matrix by the matrix defined below (R) about for magnitude : { () [ The magnitude of A in a direction defined by vector is as follows in indicial notation: Invariants (because of length, are presented at the end, here): Also, some figures/tables that are useful: ...Or as follows in principal terms: The principal components can also be used to represent the matrix of interest (A) in a visual way through quadric representation. In this representation, when matrix A is represented on a coordinate system with bases xi { ...Or as follows in principal terms: This definition results in a radius vector on the quadric representation that is connected with the magnitude of the A property: For a reflection operation about a plane, the plane normal, , is used to calculate the transformation matrix applied to a vector in the following manner: In these definitions, "E" is an antisymmetric matrix which has a dual vector . Consider a simple example in which the vector (100) is rotated +90 about the (001) axis. [ ][ ] [ ] For example, consider the effect of symmetry about the (001) crystallographic plane on the stiffness matrix, Cijkl. [ ] [ ] When rotation occurs about the (001) axis, "1" and "2" become switchable. For (010), "1" and "3" become switchable. Neumann's principle states that properties of a crystal must have a minimum order of symmetry equal to the crystallographic symmetry. That is to say, a tetragonal crystal may not have triclinic symmetry in conductivity. It must have tetragonal or cubic symmetry. Gradient, divergence, and curl of a tensor have the effect to raise the order by 1, decrease the order by one, and not change the order, respectively. Consider C1113, which only sees one multiplication by the "-1" term: An even number of "threes" will result in no information about the stiffness properties, but an odd number will require that the C value is zero, as above. How many truly independent stiffness values are there? We can use index switching to find out...we can switch within pairs ( ) and switch pairs themselves ( ). The stiffness matrix is a 4th rank tensor with 81 total combinations, but that can be reduced by { () ...
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