lec7 - Elements of Continuum Elasticity David M. Parks...

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Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004
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Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic constitutive relations Geometry of Deformation –Position, 3 components of displacement, and [small] strain tensor –Cartesian subscript notation; vectors and tensors –Dilatation (volume change) and strain deviator –Special cases: homogeneous strain; plane strain Equilibrium of forces and moments: –Stress and ‘traction’ –Stress and equilibrium equations –Principal stress; transformation of [stress] tensor components between rotated coordinate frames –Special cases: homogeneous stress; plane stress Constitutive connections: isotropic linear elasticity –Isotropic linear elastic material properties: E, ν , G, and K –Stress/strain and strain/stress relations –Putting it all together: Navier equations of equilibrium in terms of displacements –Boundary conditions and boundary value problems
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Geometry of Deformation •Origin : 0 ; Cartesian basis vectors , e 1 , e 2 , & e 3 •Reference location of material point : x ; 1 , x 2 , x 3 •Displacement vector of material point: u(x) ; specified by displacement components, u 1 , u 2 , u 3 •Each function, u i (I=1,2,3), in general depends on position x functionally through its components: e.g., u 1 = u 1 (x 1 ,x 2 ,x 3 ); etc. •Deformed location of material point: y(x)=x+u(x) specified by its cartesian components, x
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Displacement of Nearby Points •Neighboring points: x and x + x u ( x ) and u ( x + x ) •Deformed: y ( x ) and y ( x + x ) •Displacements: u ( x ) and u ( x + x ) •Vector geometry: y = x + u , where u = u ( x + x ) - u ( x ) • Displacements:
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Displacement Gradient Tensor Taylor series expansions of functions u: i Thus, on returning to the expression on previous the slide, u is given, for i each component (i=1,. .3), by Components of the displacement gradient tensor can be put in matrix form:
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lec7 - Elements of Continuum Elasticity David M. Parks...

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