L07_Generalized Hookes Law-10

L07_Generalized Hookes Law-10 - EAS 209-Fall 2010...

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EAS 209-Fall 2010 Instructors: Christine Human 8/24/2010 1 07-Gen. Hooke’s Law Lecture 7 Generalized Hooke’s Law To date we have looked at the deformation caused by an axial load, and learned that the axial (normal) stress is related to the normal strain through ε σ = E (Hooke’s Law) Today’s Objectives: To look at a more general state of loading. Poisson’s ratio Multiaxial Loading o Bulk Modulus Shear Modulus o Shear Strain Generalized Hooke’s Law Today’s Homework EAS 209-Fall 2010 Instructors: Christine Human 8/24/2010 2 07-Gen. Hooke’s Law Poisson’s Ratio Consider a homogeneous, isotropic, prismatic bar with a uniaxial centric load. Stresses 0 = = = z y x A P Strains 0 , = z y x x E Elongation in axial direction is accompanied by shortening in the transverse direction. Poisson’s Ratio is defined as (minus) the ratio of the lateral strain to the axial strain lateral strain () axial strain y z x x nu ν =− ν is constant for a given material and is always positive since the axial and lateral strains have opposite signs.
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EAS 209-Fall 2010 Instructors: Christine Human 8/24/2010 3 07-Gen. Hooke’s Law Using Hooke’s Law E x x σ ε = Therefore E x z y νσ = = Typical Values For most metals ν =0.25 to 0.35 For concrete ν =0.1 to 0.2 Range of possible values 5 . 0 0 < < ν The lower limit is obvious – a negative v would imply that if we stretch a cylinder, the diameter would increase. The upper limit is explained later. No change in lateral imension Constant volume EAS 209-Fall 2010 Instructors: Christine Human 8/24/2010 4 07-Gen. Hooke’s Law Multiaxial Loading For multiaxial loading (no shear stresses) σ x , σ y and σ z are all non-zero E E E E E E E E E z y x z z y x y z y x x + = + = = Does the volume of the material change?
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L07_Generalized Hookes Law-10 - EAS 209-Fall 2010...

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