ME85-lecture14 - Lecture 14 Generalized Hookes Law Lecture...

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Lecture 14 Generalized Lecture 14 Generalized Hooke Hooke s s Law Law
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Poisson’s Ratio For a slender bar subjected to axial loading: 0 = = = z y x x E σσ σ ε The elongation in the x-direction is accompanied by a contraction in the other directions. Assuming that the material is isotropic (no directional dependence), 0 = z y Poisson’s ratio is defined as x z x y ν = = = strain axial strain lateral
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strain normal strain lateral = ν Poisson’s ratio
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Materials with negative Poisson’s ratio
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2. Negative Poisson’s ratio materials 1. Normal Material The original article. " Foam structures with a negative Poisson's ratio", Science , 235 1038-1040 (1987).
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Question: What is the stress-strain relation in multiple dimension ?
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) 2(1 E G (1 E : constants Lame G G G G G G ip relationsh inverse The xz xz zz yy yy xx zz yz yz yy yy yy xx yy xy xy xx yy yy xx xx ν νν λ εσ εε λσ + = + = = + + + = = + + + = = + + + = , ) 2 1 )( 2 , 2 ) ( 2 , 2 ) ( 2 , 2 ) ( ) 3 (
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Generalized Hooke’s Law For an element subjected to multi-axial loading, the normal strain components resulting from the stress components may be determined from the principle of superposition . This requires: 1) strain is linearly related to stress 2) deformations are small E E E E E E E E E z y x z z y x y z y x x σ νσ ε + = + = + = With these restrictions:
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Dilatation: Bulk Modulus Relative to the unstressed state, the change in volume is ( ) ( ) () [ ] [ ] () e) unit volum per in volume (change dilatation 2 1 1 1 1 1 1 1 = + + = + + = + + + = + + + = z y x z y x z y x z y x E e σσ σ ν εε ε For element subjected to uniform hydrostatic pressure, ( ) () modulus bulk 2 1 3 2 1 3 = = = = E k k p E p e Subjected to uniform pressure, dilatation must be negative, therefore 2 1 0 < <
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ME85-lecture14 - Lecture 14 Generalized Hookes Law Lecture...

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