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Unformatted text preview: 1.050 Engineering Mechanics I Summary of variables/concepts Lecture 1626 1 Variable Definition Notes & comments Define undeformed position Deformed position Displacement vector X r Position vector, underformed configuration Note: Distinction between capital “X” and small “x” x r Position vector, deformed configuration ξ r X x r v r − = ξ Displacement vector j i ij e F e F r r ⊗ = Grad( ) 1 Grad( ) ξ r r + = = x F j i ij x x F ∂ ∂ = F dX dx v r ⋅ = Deformation gradient tensor Relates position vector of undeformed configuration with deformed configuration Lectures 16 and 17: Introduction to deformation and strain Key concepts: Undeformed and deformed configuration, displacement vector, the transformation between the undeformed and deformed configuration is described by the deformation gradient tensor Derivation first for general case of large deformation 2 Variable Definition Largedeformation theory d Ω J = d = det F d Ω r T nda = J ( ) F − 1 ⋅ N r dA 2 2 E = F T F − 1 L d − L = dX r ⋅ ( F T F − 1 )⋅ dX r = dX r ⋅ 2 E ⋅ dX r λ α = ∆ L α 2 E αα + 1 − 1 L 0, α 2 E αβ sin θ = α , β (1 + λ α )(1 + λ β ) r Grad ξ << 1 1 r r T ε = ( grad ξ + ( grad ξ ) ) ε 2 ε = 1 ⎜ ⎛ ∂ ξ i + ∂ ξ j ⎟ ⎞ ij 2 ⎜ ⎝ ∂ x j ∂ x i ⎟ ⎠ Notes & comments J = Jacobian volume change Surface change (area & normal) Definition of strain tensor Relative length variation in the αdirection Angle change between two vectors Small deformation strain tensor For Cartesian coordinate system Lecture 18: How to calculate change of geometry (angle, volume, length..) Small deformation theory: The small deformation theory is valid for small deformations only; for this case the equations simplify. These concepts are most important for the remainder of 1.050. 3 Variable Definition Notes & comments αβ αβ β α ε γ θ = ) = , ( 2 1 e e r r αα α ε λ ) = ( e r n n n r r r ⋅ ⋅ = ε λ n m m n r r r r ⋅ ⋅ = ε θ , 2 1 Smalldeformation theory Angle change Dilatation Volume change Surface change II ε n E n r r r ⋅ = ε ( ) (strain vector) “The” Mohr circle Mohr circle of strain tensor Lecture 18: Small deformation  Mohr circle for strain tensor. Any strain tensor can be represented in the Mohr plane; this way, one can display a 3D tensor quantity in a 2D projection. All concepts are the same as for the stress tensor Mohr plane. The quantities on the x/yaxes are dilatations and angle change (shear). 4 Variable Definition Notes & comments δ W Work done by external forces d ψ Free energy change W d δ ψ = Nondissipative deformation= elastic deformation All work done on system stored in free energy Defines thermodynamics of elastic deformation j j i i d dx x ξ ξ ψ ψ ∂ ∂ = ∂ ∂ j i d dx ∀ ξ ∀ , Solution approach 1D truss systems...
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 Fall '08
 MarkusBuehler

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