Cambridge Introduction To Continuum Mechanics by J. N. Reddy

By J. N. Reddy

This textbook on continuum mechanics displays the fashionable view that scientists and engineers may be expert to imagine and paintings in multidisciplinary environments. The booklet is perfect for complex undergraduate and starting graduate scholars. The booklet gains: derivations of the elemental equations of mechanics in invariant (vector and tensor) shape and specializations of the governing equations to varied coordinate structures; a variety of illustrative examples; chapter-end summaries; and workout difficulties to check and expand the certainty of strategies provided.

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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.

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 . 86) Using index notation, we have Thus the stress components do transform in accordance with the transformation law for Cartesian tensors of rank 2. Using tensor notation, we have σ = σi j ei e j , σ = σi j ei e j , σ = Q · σ · QT . 6 Tensors 17 When the tensors involved are of the second rank or lower, we use matrix notation and write the preceding relation as σ = Qσ QT .

2. 8) S where the equation on the left has been multiplied by ei to get the vector form of the equation on the right. 3. ∂i A j dV = V ni A j dS S ∇ × AdV = or V where we have used multiplication by ei jkek. 10) S where ∗ stands for ·, ×, or ⊗, producing the dot product, cross product, or the tensor product, and A is a tensor of any rank. The Gauss theorem has also been attributed to Green and Ostrogradsky. We can also see that the preceding result applies to converting area integrals into contour integrals, instead of volume integrals into surface integrals.

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