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A scalar function that depends entirely on the principal invariants of a tensor is objective, i.e., independent of rotations of the coordinate system. This property is commonly used in formulating closed-form expressions for the strain energy density, or Helmholtz free energy, of a nonlinear material possessing isotropic symmetry.
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product [a] returns a pseudovector.
It can be shown that the stress tensor is a contravariant second order tensor, which is a statement of how it transforms under a change of the coordinate system. From an x i -system to an x i ' -system, the components σ ij in the initial system are transformed into the components σ ij ' in the new system according to the tensor transformation ...
Some useful relations in the algebra of vectors and second-order tensors in curvilinear coordinates are given in this section. The notation and contents are primarily from Ogden, [2] Naghdi, [3] Simmonds, [4] Green and Zerna, [1] Basar and Weichert, [5] and Ciarlet. [6]
A tensor whose components in an orthonormal basis are given by the Levi-Civita symbol (a tensor of covariant rank n) is sometimes called a permutation tensor. Under the ordinary transformation rules for tensors the Levi-Civita symbol is unchanged under pure rotations, consistent with that it is (by definition) the same in all coordinate systems ...
Now if we replace c by the 2nd rank tensor C: = [] Where c1 is the constant coefficient for the principal direction in x axis, and c2 is the constant coefficient for the secondary direction in y axis. In order to generate anisotropic effects, c1 and c2 must be different.
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors.A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics.
D and W are respectively the symmetric and antisymmetric part of the velocity gradient, while 1 represents the unit tensor. : {\displaystyle :} represents a contraction over two indices. Thus the Folgar Tucker is an differential equation for the second order tensor A, namely the orientation tensor.