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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 r X x r v r = Displacement vector j i ij e F e F r r = Grad( ) 1 Grad( ) 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 J = d = det F d 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 = L 2 E + 1 1 L 0, 2 E sin = , (1 + )(1 + ) r Grad << 1 1 r r T = ( grad + ( grad ) ) 2 = 1 i + 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 direction 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 = ) = , ( 2 1 e e r r ) = ( e r n n n r r r = n m m n r r r r = , 2 1 Smalldeformation theory Angle change Dilatation Volume change Surface change II n E n r r r = ( ) (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 W Work done by external forces d Free energy change W d = 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 = j i d dx , Solution approach 1D truss systems...
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 Fall '08
 MarkusBuehler

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