A note on maximum orthogonal shear stress and shear strain (Q804346)

The standard result concerning the determination of the maximum orthogonal shear stress or the maximum orthogonal infinitesimal shear strain is derived in a simple direct way. As in the previous note [(*) the author, ibid. 18, 279-282 (1987; Zbl 0619.73037)] concerning the determination of the maximum orthogonal finite shear, the derivation hinges on the two main properties of pairs of conjugate semi-diameters of ellipses, namely, for any ellipse (i) the sums of the squares of the lengths of pairs of conjugate semi-diameters is a constant and (ii) the areas of the parallelograms formed by pairs of conjugate semi-diameters are all equal to each other. Essentially the problem concerns the determination of the maximum value of \(S({\mathbf n};{\mathbf m})=S({\mathbf m};{\mathbf n})=\phi_{ij}n_ im_ j\), where \(\phi\) is a given real, symmetric, second order tensor and \({\mathbf n}, {\mathbf m}\) is any pair of orthogonal unit vectors. The previous note (*) was concerned with the determination of the maximum value of \(C_{KM}U_ KV_ M\), where \({\mathbf C}\) is a real positive definite, symmetric, second order tensor and \({\mathbf U}\), \({\mathbf V}\) are orthogonal vectors subject to \(C_{KM}U_ KU_ M=C_{KM}V_ KV_ M=1.\) If \(\phi\) is replaced by \({\mathbf t}\), the Cauchy stress tensor, then S is the shear stress associated with the pair of orthogonal directions \({\mathbf n}\) and \({\mathbf m}\), whilst if \(\phi\) is replaced by \(2{\mathbf e}\), where \({\mathbf e}\) is the infinitesimal strain tensor, then S is the orthogonal shear of the pair of directions \({\mathbf n}\) and \({\mathbf m}\).