Advanced Mechanics of Materials and Applied Elasticity by Anthony E. Armenàkas

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 procedure 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 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 legislation 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 elements of a Symmetric Tensor of the second one

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126) 36 Cartesian Tensors A stationary value of the non-diagonal components of the tensor occurs in the , directions. 110) of a symmetric tensor of the second rank occur. If A11 = A22 and A12 = 0, then AN12 vanishes for any values of . 125) yields. 127) This equation has two solutions in the interval 0 < < B which differ by 90° ( and + B/2). 110) assume stationary values, are inclined 45° to the principal directions of this tensor in the x1 x2 plane (see Fig. 9). 116a). 116). 110) of a symmetric tensor of the second rank from one set of mutually perpendicular axes x1, x2, x3 to another obtained by rotating the set x1, x2, x3 about the x3 axis.

72). 72) is often used as the basis for the definition of a tensor of the second rank as an entity which possesses the following properties: 1. With respect to any set of rectangular axes it is specified by an array of nine numbers Aij (i,j = 1, 2, 3) — its nine cartesian components. 22 Cartesian Tensors 2. 72). 3 is not. 75) For instance, the tensor /A whose components with respect to a rectangular system of axes is given as is a symmetric tensor of the second rank. 9 we show that for any symmetric tensor of the second rank, there exists at least one system of rectangular axes x1, x2, x3, called principal, with respect to which the diagonal components of the tensor assume their stationary values.

98) where the constant 8 is the Lagrange multiplier. N). Each one of these sets of direction cosines and multiplier satisfies the following relations. 100) † A stationary value of a function could be a maximum, a minimum or a saddle point. 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.

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