MATERIALS_Lecture4

MATERIALS_Lecture4 - Tr ansfor mation of str ess components...

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Unformatted text preview: Tr ansfor mation of str ess components and of elastic constants Principal material coordinates are natural coordinates: Sign convention: I t is often necessary to know the stress-strain relationships in non- principal coordinates (off-axis) such as x and y. Therefore : How do we transform stress and strain? How do we transform the elastic constants? Tr ansfor mation of str ess components between coor dinate axes This is obtained by wr iting a for ce balance equation in a given dir ection. For example, in the x dir ection: cos sin 2 sin cos 12 2 2 2 1 = +-- = dA dA dA dA F x x cos sin 2 sin cos 12 2 2 2 1- + = x By repeating this, the complete set of stress transformations in xy coordinates can be obtained: [ ] = --- = - 12 2 1 1 12 2 1 2 2 2 2 2 2 2 2 T s c cs cs cs c s cs s c xy y x where c = cos and s = sin And in the 12 system we have: [ ] = xy y x T 12 2 1 [ ] --- = 2 2 2 2 2 2 2 2 s c cs cs cs c s cs s c T with (*) [ ] = 2 2 12 2 1 xy y x T Similarly, we have: Now, remember that for a 2-dimensional lamina in its principal coordinates we showed that: [ ] = =...
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This note was uploaded on 01/13/2010 for the course CVE cve202 taught by Professor Atilatma during the Fall '08 term at Acadia.

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MATERIALS_Lecture4 - Tr ansfor mation of str ess components...

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