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

By Anthony E. Armenàkas

CARTESIAN TENSORS Vectors Dyads Definition and ideas 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 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 parts Symmetric Tensors of the second one Rank Invariants of the Cartesian parts 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 process 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 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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Extra resources for Advanced Mechanics of Materials and Applied Elasticity

Sample text

7. 4). Ans. 315) (b) What are the components of a tensor A = ! 2i1i2 + i2i2 !

119) is satisfied for any value of . That is, any pair of two mutually perpendicular axes in the x1 x2 plane constitutes, with the x3 axis, a set of principal axes. 122) This transcendental equation has two solutions in the interval 0 # by 90o ( and # B which differ + B/2). That is, any quasi plane symmetric tensor of the second rank in the x1 x2 plane has two mutually perpendicular principal directions in this plane. 124) A1 is the algebraically largest and A2 is the algebraically smallest value of the diagonal components of the tensor referred to any system of axes in the x1 x2 plane.

It can Stationary Values of the Diagonal Components of a Symmetric Tensor 29 be shown that these sets of direction cosines specify three mutually perpendicular directions, referred to as the principal directions of the tensor. 106) In this case all directions are principal. Finally, it can be shown that the non-diagonal components referred to the principal axes vanish. Thus, with respect to its principal axes, a symmetric tensor of the second rank assumes a diagonal form. 107) Moreover, it can be shown that any direction along which the non-diagonal components of a symmetric tensor vanish, is a principal direction, that is, a direction along which the normal component of the tensor assumes a stationary value.

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