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1DElasticity - CEE 220 1-D Elasticity Notes This handout...

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CEE 220 1-D Elasticity Notes This handout presents the basic concepts and analytical framework associated with engineering solid mechanics considered in the simple context of 1-Dimensional systems. Although this is not a particularly useful class of models from a practical point of view, it is very useful from a conceptual point of view because the big picture can be presented with a minimum of complicating factors. 1 Fundamental Quantities As discussed in class, the two fundamental quantities of interest in solid mechanics are stress and strain. In their 1-Dimensional manifestations we have the following conceptual definitions: Stress Intensity of force experienced by a material, i.e., amount of force carried normalized by the area of material available to carry it. For an axially-loaded (1-D) rod we have: σ = P A Strain Intensity of deformation, i.e., amount of displacement that must be accommodated normalized by the length of material loang which the displacement is distributed. For a 1-Dimensional rod with an initial length, L 0 , and an overall elongation, δ , we have = δ L 0 These conceptual descriptions both define the units of stress and strain, and provide a basic foundation we can build on. 2 Fundamental Relations To move from basic definitions to relations we can use to generate quantitative results, we need to cast our notions of stress and strain into a more mathematical context. To this end, we will develop mathematical relations governing the behavior of stresses/loads and strains/displacements in the 1-Dimensional case. x A b ( x ) Figure 1: A 1-D elastic rod. Figure 1 shows the basic set-up for our derivations. We have a 1-D rod of material with the location of each point given by a coordinate, x , and the displacement of each point from its original location denoted u ( x ). In general, we can have distributed applied axial loading, denoted here as b ( x ). 2.1 Kinematics: Strain-Displacement Relation To characterize the strain in this context, we need to determine the elongation intensity at each point in the body as function of the overall displacements. Applying the concept of strain as length change normalized 1
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CEE 220 2 x Δ x x u ( x ) u ( x + Δ x ) before deformation after deformation Figure 2: A 1-D elastic rod. by original length of a particular segment as shown in Figure 2, we can write = u ( x + Δ x ) - u ( x ) Δ x (1) To get the strain at the point x itself, we take the limit as Δ x 0, and this becomes our 1-D strain- displacement relation: ( x ) = lim Δ x 0 u ( x + Δ x ) -
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