Section 1.1 Solid Mechanics Part II Kelly 31.1 The Equations of Motion In Part I, balance of forces and moments acting on any component was enforced in order to ensure that the component was in equilibrium. Here, allowance is made for stresses which vary continuously throughout a material, and force equilibrium of any portion of material is enforced. One-Dimensional Equation Consider a one-dimensional differential element of length xΔand cross sectional area A, Fig. 1.1.1. Let the averagebody force per unit volume acting on the element be band the averageacceleration and density of the element be aand ρ. Stresses σact on the element. Figure 1.1.1: a differential element under the action of surface and body forcesThe net surface force acting is AxAxx)()(σσ−Δ+. If the element is small, then the body force and velocity can be assumed to vary linearly over the element and the average will act at the centre of the element. Then the body force acting on the element is xAbΔand the inertial force is xaAΔρ. Applying Newton’s second law leads to abxxxxxAaxAbAxAxxρσσρσσ=+Δ−Δ+→Δ=Δ+−Δ+)()()()((1.1.1) so that, by the definition of the derivative, in the limit as 0→Δx, abdxdρσ=+1-d Equation of Motion(1.1.2) which is the one-dimensional equation of motion. Note that this equation was derived on the basis of a physical law and must therefore be satisfied for all materials, whatever they be composed of. The derivative dxd/σis the stress gradient– physically, it is a measure of how rapidly the stresses are changing. Example Consider a bar of length lwhich hangs from a ceiling, as shown in Fig. 1.1.2. AxΔ)(xσ)(xxΔ+σxxxΔ+ab,
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