variables2

variables2 - 1.050 Engineering Mechanics I Summary of...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 1.050 Engineering Mechanics I Summary of variables/concepts Lecture 16-26 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 Large-deformation 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 Small-deformation 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/y-axes are dilatations and angle change (shear). 4 Variable Definition Notes & comments W Work done by external forces d Free energy change W d = Non-dissipative 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...
View Full Document

Page1 / 15

variables2 - 1.050 Engineering Mechanics I Summary of...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online