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By E. Poisson

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2. The symmetry does not depend on the choice of axes. 40 Elastic behavior 3. Any symmetric second-order tensor can be transformed by a rotation of axes to a form in which the tensor only contains the three diagonal components, the off-diagonal components being zero. For this state, the diagonal components are called the principal values and the particular set of axes are called the principal axes. 4. ; / / ^ ( r . r - r . 34) where Eijk is the called the permutation symbol. , the three principal values.

22. A unit square OABC is deformed into OPQR by the strains e n , e22and e12 (=e21 ). These strains will be referred to a new set of axes x\ and x'2, that are formed by rotating the original axes by an angle 0. The direction cosines of the transformed axes are an=a22= cos0 and an=—a2l = sin0. The shear strain component is then found from Eq. 21 Shear of diamond shape inscribed in a square undergoing equal and opposite extensions along the xx and x2 axes. 22. Two-dimensional strain in which a unit square OABC deforms to OPQR.

For the remainder of this chapter, we will return to the subject of linear elasticity and concentrate on the way it is formally described for macroscopic bodies. At first, uniform deformations in a body will be considered but then we will move to using continuum mechanics for the description of non-uniform deformations. 3 Engineering elastic constants For large bodies that are linear elastic, the version of Hooke's Law given as Eq. 2) is not very useful. Indeed, even the use offeree and displacement to describe the deformation becomes inappropriate.

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A Relativist's Toolkit - The Math of Black Hole Mechanics by E. Poisson


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