Linear_Elasticity_01_The_Continuum

Linear_Elasticity_01_The_Continuum - Section 4.1 4.1 The...

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Section 4.1 Solid Mechanics Part I Kelly 93 4.1 The Continuum The Linear Elastic Model to be discussed in this Chapter is a continuum model . This type of model is discussed briefly in this section. 4.1.1 Stress and Scale In the definition of the traction vector, §3.3.1, it was assumed that the ratio S F Δ Δ / would indeed reach some definite limit as the area S Δ of the surface upon which the force F Δ acts was shrunk to zero. This issue can be explored further by considering Fig. 4.1.1. Assume first that the plane upon which the force acts is fairly large; it is then shrunk and the ratio S F / tracked. A schematic of this ratio is shown in Fig. 4.1.2. At first (to the right of Fig. 4.1.2) the ratio S F / undergoes change, assuming the stress to vary within the material, as it invariably will if a material is loaded in some complex way. Eventually the plane will be so small that the ratio changes very little, perhaps with some small variability ε . If the plane is allowed to get too small, however, down towards the atomic level, where one might encounter “intermolecular space”, there will be large changes in the ratio and the whole concept of a force acing on a single surface breaks down. Figure 4.1.1: A force acting on an internal surface In a continuum model, it is assumed that the ratio S F / follows the dotted path shown in Fig. 4.1.2. It should be kept in mind, then, that the traction in a real material should be evaluated through S F h S Δ Δ = Δ 2 *) ( lim t (4.1.1) point force plane force point plane
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Section 4.1 Solid Mechanics Part I Kelly 94 where * h is some minimum dimension below which there is no acceptable limit. It is
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 1 taught by Professor Staff during the Fall '11 term at Auckland.

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Linear_Elasticity_01_The_Continuum - Section 4.1 4.1 The...

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