By G. Fowles, G. Cassiday

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**Extra info for Analytical Mechanics [SOLUTIONS MANUAL]**

**Sample text**

16b) εxy = 1 ∂ux ∂uy + . 16c) These equations rely on the assumption that the displacements ux and uy are very small, so that the initial and ﬁnal positions of a given particle are practically the same. When the displacements are large, however, this is no longer the case and one must distinguish between initial and ﬁnal coordinates of particles. This is typically done by using capital letters X, Y for the initial positions and lower case x, y for the current coordinates. It would then be tempting to extend the use of the above equations to the nonlinear case by simply replacing derivatives with respect to x and y by their corresponding initial coordinates X, Y .

12b) Qji vj . 13b) where ⎡ e1 · e1 ⎢ [Q] = ⎣ e2 · e1 e3 · e1 e1 · e2 e1 · e3 ⎤ e2 · e2 ⎥ e2 · e3 ⎦ . 14) As a precursor to the discussion of second-order tensors, it is worth emphasizing the coordinate independent nature of vectors. For example, the vector equations w = u + v or s = u · v make sense without speciﬁc reference to the basis used to express the components of the vectors. Obviously, a vector will have different components when expressed in a different basis – but the vector remains unchanged.

7 Horizontal truss: tension stiffening. 3 NONLINEAR STRAIN MEASURES 9 Further insight into the nature of nonlinearity in the presence of large deformation can be revealed by this simple example if we consider the vertical stiffness of the truss member at joint B . This stiffness is the change in the equilibrium equation, R(x) = 0, due to a change in position x and is generally represented by K = dR/dx. 11b,c) together with the incompressibility condition a = V /l as K= = dT dx d σV x dx l2 ax dσ 2σax dl σa − 2 + l dl l dx l = dσ 2σ x2 σa − .