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MATERIALS_Lecture4

# MATERIALS_Lecture4 - T r a nsfor ma ti on of str ess...

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Transformation of stress components and of elastic constants Principal material coordinates are ‘natural’ coordinates:

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Sign convention: It 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?
Transformation of stress components between coordinate axes This is obtained by writing a force balance equation in a given direction. For example, in the x direction: 0 cos sin 2 sin cos 12 2 2 2 1 = + - - = Σ θ θ τ θ σ θ σ σ dA dA dA dA F x x

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θ θ τ θ σ θ σ σ 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 τ σ σ τ σ σ τ σ σ T s c cs cs cs c s cs s c xy y x where c = cos θ and s = sin θ And in the 12 system we have: [ ] = xy y x T τ σ σ τ σ σ 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 γ ε ε γ ε ε Similarly, we have: Now, remember that for a 2-dimensional lamina in its principal coordinates we showed that: [ ] = = = 2 2 2 0 0 0 0 12 2 1 12 2

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MATERIALS_Lecture4 - T r a nsfor ma ti on of str ess...

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