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# Lecture04 - ME 382 Lecture 04 MULTIAXIAL STRAIN STIFFNESS...

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ME 382 Lecture 04 12/ix/07 1 M ULTIAXIAL STRAIN & STIFFNESS Isotropic linear elasticity Normal stresses and strains are related by: i) Young’s modulus: E ii) Poisson’s ratio: ν ! 1 " # " 0.5 For most materials 0.2 ! " ! 0.5 Shear stresses and strains related by: " xx = # xx / E ! zz = " zz / E " yy = " zz = # \$" xx = # \$% xx / E " xx = " yy = # \$" zz = # \$% zz / E (no cross terms for shear) ! yy = " yy / E ! xy = " xy / G ! xx = ! zz = " #! yy = " #\$ yy / E ! xz = " xz / G ! yz = " yz / G

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ME 382 Lecture 04 12/ix/07 2 In general : 3-D Hooke’s Law ! xx = " xx # \$" yy # \$" zz ( ) / E ! yy = " #\$ xx + \$ yy " #\$ zz ( ) / E ! zz = " #\$ xx " #\$ yy + \$ zz ( ) / E ! xy = " xy / G ! xz = " xz / G ! yz = " yz / G Special cases: (i) Plane stress: One stress = 0; e.g. , σ zz = 0 (such as at free surface) (ii) Plane strain: One strain = 0; e.g. , ε zz = 0 (constraint in rigid die) Volumetric strain: Original volume: V o = l 3 New volume: V = l + u ( ) l + v ( ) l + w ( ) V = l 3 1 + u / l ( ) 1 + v / l ( ) 1 + w / l ( ) V = l 3 1 + " xx ( ) 1 + " yy ( ) 1 + " zz ( ) dV = V " V o # l 3 ( \$ xx + \$ yy + \$ zz ) Volumetric strain or dilation ! = dV / V o = " xx + " yy + " zz Elastic strain energy: In general: u = ! xx d " xx # + ! yy d " yy # + ! zz d " zz + \$ xy d % xy # + \$ yz d % yz # + \$ xz d % xz # # But, in linear elasticity: ! " # , \$ " %
ME 382 Lecture 04 12/ix/07 3 u = !

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