Mechanics 1Q - 2.001 MECHANICS AND MATERIALS I Lecture Prof...

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2.001 - MECHANICS AND MATERIALS I Lecture # Prof. Carol Livermore Recall from last time: Normal strains, changes in length ±u ( x, y, z )= u x ( x, y, z ) ˆ i + u y ( x, y, z ) ˆ j + u z ( x, y, z ) k ˆ ² xx = ∂u x ( x, y, z ) ∂x ² yy = y ( x, y, z ) ∂y ² zz = z ( x, y, z ) ∂z Shear Strain δ γ xy = θ = L 1 14
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± ± ± ± ± ± δ 1 δ 2 γ xy = θ 1 + θ 2 =+ LL ∂u x y γ xy ∂y ∂x γ xy ± xy = ± yx = 2 1 ² x y ³ ± xy 2 1 ² y z ³ ± yz 2 ∂z 1 ² x z ³ ± xz 2 ± ± xx ± xy ± xz ± [ ± ]= ± ± ± yy ± ± . ± ± zx ± zy ± zz ± What kind of strain you see (normal vs. shear) and its magnitude depend on the relative orientation of deformation and coordinates. EXAMPLE: ± xy =0 2
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± xy =0 ± Relationship between σ and ± Recall uniaxial loading: ± = δ/L ± xx = σ = = σ xx = xx σ 0 = xx Look at thicker bar: δ ± xx =; ± yy = (something) L 3
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In this case: ± xx = ± axial ± yy = ± lateral ± lateral = ν± axial ,where ν is Poisson’s Ratio (unitless). Typically ν 0 . 3 Range 0 ν 0 . 5 Note: ν =0 . 5 incompressible Microstructure view of Poisson’s Ratio Recall Young’s Modulus
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This note was uploaded on 08/16/2009 for the course PESS A taught by Professor Prof during the Spring '09 term at Ohio University- Athens.

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Mechanics 1Q - 2.001 MECHANICS AND MATERIALS I Lecture Prof...

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