1D_Elasticity_02_Elastodynamics

1D_Elasticity_02_Elastodynamics - Section 2.2 2.2...

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Section 2.2 Solid Mechanics Part II Kelly 23 2.2 One-dimensional Elastodynamics In rigid body dynamics, it is assumed that when a force is applied to one point of an object, every other point in the object is set in motion simultaneously. On the other hand, in static elasticity, it is assumed that the object is at rest and is in equilibrium under the action of the applied forces; the material may well have undergone considerable changes in deformation when first struck, but one is only concerned with the final static equilibrium state of the object. Elastostatics and rigid body dynamics are sufficiently accurate for many problems but when one is considering the effects of forces which are applied rapidly , or for very short periods of time, the effects must be considered in terms of the propagation of stress waves. 2.2.1 The Wave Equation Consider now the dynamic problem. In this case one considers the equation of motion: a b dx d ρ σ = + Equation of Motion (2.2.1a) dx du = ε Strain-Displacement Relation (2.2.1b) E = Constitutive Equation (2.2.1c) where a is the acceleration. Expressing the acceleration in terms of the displacement, one then obtains the dynamic version of Navier’s equation, 2 2 2 2 t u b x u E = + 1-D Navier’s Equation (2.2.2) In most situations, the body forces will be negligible, and so consider the partial differential equation 2 2 2 2 2 1 t u c x u = 1-D Wave Equation (2.2.3) where E c = (2.2.4) Equation 2.2.3 is the standard one-dimensional wave equation with wave speed c ; note from 2.2.4 that c has dimensions of velocity. The solution to 2.2.3 (see below) shows that a stress wave travels at speed c through the material from the point of disturbance, e.g. applied load. When the stress wave reaches a
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Section 2.2 Solid Mechanics Part II Kelly 24 given material particle, the particle vibrates about an equilibrium position, Fig. 2.2.1. Since the material is elastic, no energy is lost, and the solution predicts that the particles vibrate indefinitely, without damping or decay. Figure 2.2.1: stress wave travelling at speed c through an elastic rod This type of wave, where the disturbance (particle vibration) is in the same direction as the direction of wave propagation, is called a longitudinal wave . 2.2.2 Particle Velocities and Wave Speed Before examining the wave equation 2.2.3 directly, first re-express it as t v x = ρ σ (2.2.5) where v is the velocity. Consider an element of material which has just been reached by the stress wave, Fig. 2.2.2. The length of material passed by the stress wave in a time interval t Δ is t c Δ . During this time interval, the stressed material at the left-hand side of the element moves at (average) velocity v and so moves an amount t v Δ . The strain of the element is then the change in length divided by the original length: c v = ε (2.2.6) Under the small strain assumption, this implies that c v << 1 .
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 2 taught by Professor Staff during the Fall '11 term at Auckland.

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1D_Elasticity_02_Elastodynamics - Section 2.2 2.2...

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