By J. N. Reddy
This textbook on continuum mechanics displays the fashionable view that scientists and engineers might be proficient to imagine and paintings in multidisciplinary environments. The publication is perfect for complex undergraduate and starting graduate scholars. The e-book beneficial properties: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate platforms; various illustrative examples; chapter-end summaries; and workout difficulties to check and expand the knowledge of thoughts awarded.
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Additional resources for Cambridge Introduction To Continuum Mechanics
As a third example, we consider the stress tensor σ . In French, the word tension means stress and the word tensor originated from tension to denote a set of quantities that transforms as stress components under a coordinate transformation. 5. Components of stress in two dimensions. σ11 σ11 x1 θ O σ22 σ21 A x1 Fig. 5. 83) where we have assumed a unit thickness for the triangular lamina and we obtain the last two relations from the first two by setting θ → θ + π/2. 84) or σ = Qσ QT . 85) T = Qim Q jn σmn .
When we encounter an array of 3r elements, without examining how they transform from one coordinate system to another, we cannot conclude we have a tensor of rank r . We often encounter various arrays in spreadsheets (for example, homework grades); these are, obviously, not tensors! 1 Examples of Tensors In this subsection we examine three examples of tensors. 6 Tensors 15 We may construct second-rank tensors from vectors by using the idea of tensor products. We have seen the dot product and the cross product of two vectors.
The curvature tensor is obtained from the derivatives of the metric tensor. If R is zero, we have a flat space. By lowering the index m, we obtain the Riemann curvature as Ri jkl = gim Rm jkl . 70) and in three dimensions there are six, namely, R1212 , R1313 , R2323 , R1213 , R2123 , R3132 . 71) These curvatures are important when we consider the compatibility of strain components in Chapter 5. The Riemann curvature satisfies the Bianchi identity: Dm Ri jkl + Dk Ri jlm + Dl Ri jmk = 0. 74) and the Einstein curvature, which plays an important role in general relativity.
Cambridge Introduction To Continuum Mechanics by J. N. Reddy