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zmnotes14_15_16 - M14 Transformation of Stress (continued)....

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Unformatted text preview: M14 Transformation of Stress (continued). Introduction to Strain Principal Stresses/Axes There is a set of axes (in 2 or 3-D) into which any state of stress can be resolved such that there are no shear stresses. These are known as the principal axes of stress. s I , s II , s III (By convention, s I is most tensile, s III is most compressive) Can also see from Mohr's circle that two of these are the largest (and smallest) or extreme values of stress. Side Note: if we thought about stress as a matrix: È s 11 s 12 s 13 ˘ Í ˙ dF = s dA s = s 21 Í s 22 s 23 ˙ Í Î s 21 s 32 s 33 ˚ ˙ dF = s dA The principal stresses are the eigenvalues of s and the principal directions the eigenvectors. Important: We are transforming the axis system, not the stress state. Consider a bar with two grid squares (rectangle) on the surface, one rotated through 45°: the bar now undergoes a tensile loading, and this generates a tensile strain x n rectangle is stretched/elongated (angles remain the same = 90°.) x ˜ m is sheared (i.e angles changed) But both grid squares (rectangles) are experiencing the same stress state (and strain state) Stress Invariants - actually the basis for Mohr's Circle Note that in 2-D, = cons tan t s 11 + s 22 = s I + s II In 3-D, s = cons tan t ( In var iant ) 11 + s 22 + s 33 i.e., the diameter Of the Mohr's circle (will meet other invariants elsewhere in solid mechanics). M15 Introduction to Strain We have examined stress, the continuum generalization of forces, now let's look at the continuum generalization of deformations: Definition of Strain Strain is the deformation of the continuum at a point. Or, the relative deformation of an infinitessimal element. Two ways that bodies deform: By elongation and shear: Elongation (extension, tensile strain) Tensile strain can be thought of as the change in length relative to the original length L deformed - L undeformed tensile strain = L deformed but body can also deform in shear – angles change Shear This produces an angle change in the body (with no rotation for pure shear) Consider infinitesimal element. Undeformed:...
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This note was uploaded on 01/28/2012 for the course AERO 16.01 taught by Professor Markdrela during the Fall '05 term at MIT.

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zmnotes14_15_16 - M14 Transformation of Stress (continued)....

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