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1.050 Engineering Mechanics I Summary of variables/concepts Lecture 16-26 1Variable 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 Xx rvr −=ξ Displacement vector jiij eF eF rr ⊗= Grad( )1Grad( ) ξ rr +== xF j i ij x xF ∂ ∂ = F dXdx vr ⋅= 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 2Variable Definition Large-deformation theory dΩJ = d = det F dΩ0 r Tnda = J ()F −1 ⋅ Nr dA 2 2E = FTF −1 Ld − L0 = dX r ⋅(FTF − 1)⋅ dX r = dX r ⋅ 2E ⋅ dX r λα =∆Lα 2Eαα +1 −1 L0,α 2Eαβsinθ= α ,β (1+ λα )(1+λβ ) r Grad ξ << 1 1 rr Tε= (gradξ +(gradξ) )ε 2 ε= 1 ⎜⎛ ∂ξi + ∂ξ j ⎟⎞ ij 2 ⎜⎝∂xj ∂xi ⎟⎠ Notes & comments J = Jacobian volume change Surface change (area & normal) Definition of strain tensor Relative length variation in the α-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. 3Variable Definition Notes & comments αβαββα εγθ =) =,(2 1 ee rr ααα ελ ) =(e r nnn rr r ⋅⋅= ελ nmm n rr rr ⋅⋅= εθ ,2 1Small-deformation theory Angle change Dilatation Volume change Surface change IIε nE n rrr ⋅= 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/y-axes are dilatations and angle change (shear). 4Variable Definition Notes & comments δW Work done by external forces dψ Free energy change Wd δψ = Non-dissipative deformation= elastic deformation All work done on system stored in free energy Defines thermodynamics of elastic deformation j j i i ddx x ξξ ψψ ∂ ∂ = ∂ ∂ ji ddx ∀ ξ∀ , Solution approach 1D truss systems klijklij c εσ = εσ = c : Link between stress and strain Also called “generalized Hooke’s law” Lectures 20 and 21: Elasticity, basic definitions. The most important concept of this lecture is that elastic deformation is a thermodynamic process under which no energy dissipation occurs. This concept can be generally applied to characterize any elasticity problem. We derived elasticity for 1D systems (including solution strategy), and then generalized it to 3D. This led to the link between stress and strain. 5Variable Definition Notes & comments Isotropic elasticity Elastic properties of material do NOT depend on direction Isotropic elasticity described uniquely by 2 parameters, K and G ε ( ) ∑∑== i j ij T 2 2 1:2 1 εε εε “Length” of a tensor tr(ε) () 0 0 332211:1tr Ω Ω − Ω =++== d dd dεεεεε “Volume change” of a tensor ),( dv εΨ ε 22 2 1 2 1 dv GK εε +Ψ = Free energy due to volume strain and shear strain (assumption, mathematical model to describe elastic behavior of isotropic solids) ( ) εεεεεεσ GGKGGK v 21 3 221 3 2 332211 +++⎟ ⎠ ⎞⎜ ⎝ ⎛ −=+⎟ ⎠ ⎞⎜ ⎝ ⎛ −= Linear isotropic elasticity Tensor notation Lecture 22: Isotropic elasticity, basic concepts. The most important equation on this slide is the one on the bottom, for linear isotropic elasticity. Note that isotropic elasticity is fully characterized by two constants, K and G. These two parameters have physical meaning; K describes how the free energy changes under volume changes, and G describes how the free energy changes under shear (shape) changes. 6Variable Definition σ11 = ⎜⎛ K − 2 G⎟⎞(ε11 +ε22 +ε33 )+ 2Gε11 ⎝ 3 ⎠ σ22 = ⎜⎛ K − 2 G ⎟⎞(ε11 +ε22 +ε33 )+ 2Gε22 ⎝ 3 ⎠ σ33 = ⎜⎛ K − 2 G ⎟⎞(ε11 +ε22 +ε33 )+ 2Gε33 ⎝ 3 ⎠ σ12 = 2Gε12 σ23 = 2Gε23 σ13 = 2Gε13 σ11 = ⎜⎛ K + 4 G ⎟⎞ε11 + ⎜⎛ K − 2 G ⎟⎞ε22 + ⎜⎛ K − 2 G ⎟⎞ε33 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ σ 22 = ⎜⎛ K − 2 G ⎟⎞ε11 + ⎜⎛ K + 4 G ⎟⎞ε22 + ⎜⎛ K − 2 G ⎟⎞ε33 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ σ 33 = ⎜⎛ K − 2 G ⎟⎞ε11 + ⎜⎛ K − 2 G ⎟⎞ε22 + ⎜⎛ K + 4 G ⎟⎞ε33 ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ σ12 = 2Gε12 σ 23 = 2Gε23 σ13 = 2Gε13 Notes & comments Linear isotropic elasticity Written out for individual stress tensor coefficients Linear isotropic elasticity Written out for individual stress tensor coefficients, collect terms that multiply strain tensor coefficients 4 c1111 = c2222 = c3333 = K + G3 2 c1122 = c1133 = c2233 = K − G 3 c1212 = c2323 = c1313 = 2G Lecture 22: Isotropic elasticity, equations that relate stress and strain. Here we summarize the equations in different forms. On the bottom, right, you see how to calculate the elasticity tensor coefficients from K and G. 7Overview: 3D linear elasticity 0div =+ g rρσ )(xrσStress tensor Basis: Physical laws (Newton’s laws) BCs on boundary of domain Ω nnT d T d rrr r ⋅=Ω∂σ)(: :Ω nnT rrr ⋅= σ)( jiij σσ = )(xrεStrain tensor Basis: Geometry BCs on boundary of domain Ω ξξξ rr r =Ω∂d d : Linear deformation theory 1Grad <<ξ r ( )( )Tξξε rr gradgrad2 1 +=Statically admissible (S.A.)Kinematically admissible (K.A.) Elasticity εσ :c= klijklij c εσ = εεσ GGK v 21 3 2 +⎟ ⎠ ⎞⎜ ⎝ ⎛ −= Basis: Thermodynamics Isotropic solid ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ + ∂ ∂ = i j j i ij xx ξξε 2 1 Summary, 3D linear elasticity. This page may be useful to keep an overview over the methods and approaches covered here. This summary is valid for any linear elasticity problem. 8Variable Definition • Step 1: Write down BCs (stress BCs and displacement


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