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By Edwin Bidwell Wilson

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7) (n) where tk are the components of t(n) ; and σik are the components of ti in the coordinate directions. 4. ) and tk is a tensor. Therefore, we conclude that σik is a tensor. 8) is that it relates the stress vector t(n) acting on a plane with unit normal n to the stress tensor σ in the material element. Knowing n and t(n) , σ may be found from the equation. 3 Stress tensor components. 11) We now discuss the meaning of the stress components σij . These are intensity of forces (force per unit area) along the coordinate directions.

2 Remark 2 — transformation of bases versus rotation of a tensor Note the difference in the transformation equations between a tensor T subjected to the transformation of bases and the tensor T itself undergoing rotation. 111) reduces to [T] obtained from the rotation of bases. Therefore, rotation of the tensor itself in the opposite direction has the same effect as the rotation of bases. 6. Consider the rotation of a vector v through an angle α into v. 2 Components of a vector. 92), vi = Qji∗ vj = Qij∗T vj (g) Hence, v1 v2 © 2005 by Chapman & Hall/CRC Press = cos α sin α − sin α cos α v1 v2 (h) 26 Continuum Mechanics and Plasticity Comparing (c) and (h), we conclude that rotation of a vector through an angle α has the same effect as rotation of axes through the same angle α in the opposite direction.

2 Continuum Mechanics and Plasticity Component Form of Rotation of a Tensor In this section, we derive the component form that describes the rotation of a tensor, and we point out that this transformation equation is different from the one which describes the change of bases. 93) for a vector vi , and by [Tij ] = [Qip ]T [Tpq ][Qqj ] for a tensor Tij . 94) Thus, vector v is rotated by Q into v. 96) shows that the components of v¯ in ei are the same as those of v in ei . This result is obvious, because both the vector and the coordinate system have been rotated by the same amount and their relative positions remain the same.

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A Statistical Discussion of Sets of Precise Astronomical Measurements Parallaxes by Edwin Bidwell Wilson


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