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Lecture Notes 2.2

# Lecture Notes 2.2 - STRESS We have previously discussed how...

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1 STRESS We have previously discussed how we can characterize the deformation of a material by the strain in the material. What happens to the material when we strain it? Undeformed Deformed Strains are the same! But are the forces required to deform these 2 rods the same? Of course not!

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2 STRESS Consider the larger rod If the rod is in static eq, and we are applying force to it, then Newton’s 3 rd law says there must be a force to balance it Make a slice through the rod Anywhere you make a slice, there is still that force balance in the new FBD These are “internal” forces We normalize these internal forces to make a geometry- independent measure called stress Normalize by cross sectional area, A F F F F F F A A F = σ
3 If stress is “on-axis”, it is either tensile (positive) or compressive (negative…like pressure). A F = σ Tension Compression Shear A A A If shear stress, typically designate stress as τ

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4 Recall this example, where we determined that the block had non-zero shear in the x-y plane and non-zero tension in the y- direction We needed a more complex measure of strain (a tensor) The same can be said for stress
5 σ xx σ zz σ yy xy σ yx σ yz σ zx σ zy σ xz σ = = = y x yy yx xy xx zz zy zx yz yy yx xz xy xx σ τ τ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ In 3D (note tensor is symmetric). And in 2D.

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6 Why do we care about stress? 1. It turns out the the way materials fail most often depends on the level of stress in the material. This could be different for tensile, compressive, and/or shear stresses, and may be defined by some combination of them 2. The level of stress in the material can be related to the amount of strain. Steel Rubber F F Same x-sectional area, same force, same stress, but obviously different strains!