By Anthony E. Armenàkas
CARTESIAN TENSORS Vectors Dyads Definition and principles of Operation of Tensors of the second one Rank Transformation of the Cartesian elements of a Tensor of the second one Rank upon Rotation of the method of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislation of Transformation of Its elements Symmetric Tensors of the second one Rank Invariants of the Cartesian elements of a Symmetric Tensor of the second one Rank desk bound Values of a functionality topic to a Constraining Relation desk bound Values of the Diagonal elements of a Symmetric Tensor of the Second. Read more...
summary: CARTESIAN TENSORS Vectors Dyads Definition and principles of Operation of Tensors of the second one Rank Transformation of the Cartesian parts of a Tensor of the second one Rank upon Rotation of the approach of Axes to Which they're Referred Definition of a Tensor of the second one Rank at the foundation of the legislations of Transformation of Its parts Symmetric Tensors of the second one Rank Invariants of the Cartesian parts of a Symmetric Tensor of the second one Rank desk bound Values of a functionality topic to a Constraining Relation desk bound Values of the Diagonal parts of a Symmetric Tensor of the second one
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Extra info for Advanced Mechanics of Materials and Applied Elasticity
9, we show that, with respect to the principal axes, the non-diagonal components of the tensor vanish. 77) In this case the diagonal components A1, A2 and A3 are called the principal components of the tensor. 7 Invariants of the Cartesian Components of a Symmetric Tensor ofthe Second Rank Consider a tensor of the second rank A and denote its components with respect to the Stationary Values of a Func tion Subject to a Constrai ning Relation 23 rectangular systems of axes x1, x2, x3 and xN, 1 xN, 2 xN3 by Aik (i, k = 1, 2, 3), Ajm (j, m = 1, 2, 3).
Moreover, the ordinates of points and of Mohr's circle represent the maximum and minimum values, respectively, of the non-diagonal components of the tensor with respect to any set of axes in the x1x2 plane. Notice that the x3 and x1 axes must be taken as shown in Fig. 11 Right-handed axes of reference used with Mohr's circle. A13 of the tensor and we proceed as previously. Moreover, notice that the x2 and x3 axes must be taken as shown in Fig. A32 of the tensor and we proceed as previously. The method of employing known elementary concepts in order to solve a problem which involves concepts which are more complicated and difficult to visualize is referred to in the literature as an analogy.
102) Thus, the Lagrange multipliers are equal to the stationary values An of the diagonal components of the symmetric tensor of the second rank [A]. 103) These three linear algebraic homogeneous equation, in other than the trivial and and have a solution , if the determinant of the coefficients of is zero. 80). 105) we see that the values An of a symmetric tensor of the second rank [A] are independent of the choice of the system of axes to which the components of the tensor are referred. 105) has three real roots A1, A2, A3 which are the three stationary values of the diagonal components of the tensor.
Advanced Mechanics of Materials and Applied Elasticity by Anthony E. Armenàkas