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While the relation between the displacement field u(r) and the local strain tensor εij is rather elementary, it does not hurt to recall the decisive points. | |||||||||
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Let's take the simple example from the backbone and consider a rod that is uniformly elongated; i.e. u(r) = ux(x) = a · x; a is some constant. | |||||||||
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In other words, the vector u only has a component in x-direction, which only depends on x as variable. The geometry than looks like this: | |||||||||
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At any point in the rod a little cube will be deformed into a cuboid - the side in x-direction is somewhat longer than the others. | |||||||||
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What kind of strain do we have to put on a cube positioned a x, to produce the cuboid? | |||||||||
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Well, since there is only strain in x-direction, we simply write down the elementary formula for strain | |||||||||
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If we deform in all three directions, we get corresponding expressions for εyy and εzz. | |||||||||
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Since we also might have displacement components in x-direction that depend on y or z, e.g. ux(x, y, z) = a · y, we may, in general, also form mixed (partial) derivatives; e.g. ∂ux(x, y, z)/∂y. What do those derivatives signify? | |||||||||
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Shear stresses, of course. A little less easy to see, perhaps, but there can be no doubt about it. | |||||||||
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You may want to try to show that for yourself with the simple displacement field given above and the equations in the backbone as a guideline for what you are looking for. | |||||||||
5.2.1 Elasticity Theory, Energy and Forces
5.2.2 Stress Field of a Straight Dislocation
© H. Föll
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