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By Gerhard A. Holzapfel

This article is the simplest one after i purchased a number of different continuum mechanics texts which made me pissed off repeatedly. this article is intensive, yet really easy to appreciate. the fashion the writer wrote the textual content is especially transparent and vintage. you could feel it really is an entertainment to check the textual content and make growth easily each day.

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Extra resources for Nonlinear Solid Mechanics: A Continuum Approach for Engineering

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26) (b) The conservative force field F is rotation-free, ∇× F (r) = 0. 27) (c) The line integral over the conservative force F is independent of the path C (dependent only upon the initial and final point of the path C), F (r) · dr independent of C. 28) C (d) The closed line integral over the conservative force F vanishes, F (r) · dr = 0 for all closed paths. 26) F · dr = − ∇V · dr = − dV C with the last integral depending only upon boundary points. Comment: Possibly, the condition (b) is not defined everywhere; for example, the force field of a point mass or of a point charge at their respective positions is not defined.

1 Definitions Definition 9. (for a single particle, characterized by the index i): mi ri pi = mi r˙ i l i = r i × pi Fi Ti = 12 mi r˙ 2i the mass of the particle; the coordinate (the position) of the particle; the momentum of the particle; the angular momentum of the particle with respect to the origin (at r = 0) the force acting on the particle; kinetic (translational22 ) energy23 of the particle. Definition 10. (for a system of N particles): The sums are understood as total mass ≡ N i=1 M= mi 1 M center of mass R= total momentum P = pi = total angular momentum L= li = E= 1 2mi internal total energy mi r i ˙ mi r˙ i = M R r i × pi p2i + 1 2 ij Vij (conservative systems) 22 23 Extended particles have also rotational energy in addition to the translational energy, see Chap.

261 with c = −1 and Δ = −4ϕ20 from the extremely voluminous tables in [11]. In order to perform one integration one needs one initial condition. 38 2 Newtonian Mechanics: First Applications K(z) p/2 0 0 z 1 Fig. 5. Graphical representation of K(z); see also [10], p. 592 with ω02 = l/g. From this one obtains by solving for ϕ ϕ = ϕ0 sin(ω0 t). 7) was to be demonstrated. 6 * Example: Plane Pendulum with Large Amplitude As an example, the plane pendulum shall be considered again, but now for displacements which are not necessarily small; the pendulum is assumed to oscillate, but not to rotate.

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