Introduction to 3D Elasticity - Introduction to Finite Elements - Lecture Slides, Slides of Mathematical Methods for Numerical Analysis and Optimization

Download Introduction to 3D Elasticity - Introduction to Finite Elements - Lecture Slides and more Slides Mathematical Methods for Numerical Analysis and Optimization in PDF only on Docsity! 1 Introduction to 3D Elasticity Reading assignment: Appendix C+ 6.1+ 9.1 + Lecture notes Summary: • 3D elasticity problem •Governing differential equation + boundary conditions •Strain-displacement relationship •Stress-strain relationship •Special cases 2D (plane stress, plane strain) Axisymmetric body with axisymmetric loading • Principle of minimum potential energy 1D Elasticity (axially loaded bar) x y x=0 x=L A(x) = cross section at x b(x) = body force distribution (force per unit length) E(x) = Young’s modulus u(x) = displacement of the bar at x x F 1. Strong formulation: Equilibrium equation + boundary conditions Lxb dx d  0;0  LxatF dx du EA xatu   00Boundary conditions Equilibrium equation 3. Stress-strain (constitutive) relation : ε(x) E(x)  E: Elastic (Young’s) modulus of bar 2. Strain-displacement relationship: dx du ε(x)  Docsity.com 2 Problem definition 3D Elasticity Surface (S) Volume (V) u v w V: Volume of body S: Total surface of the body The deformation at point x =[x,y,z]T is given by the 3 components of its    u x y z x displacement       w vu NOTE: u= u(x,y,z), i.e., each displacement component is a function of position 3D Elasticity: EXTERNAL FORCES ACTING ON THE BODY Two basic types of external forces act on a body 1. Body force (force per unit volume) e.g., weight, inertia, etc 2. Surface traction (force per unit surface area) e.g., friction BODY FORCE Surface (S) Volume (V) u v w Xa dV Xb dV Xc dV Volume element dV Body force: distributed force per unit volume (e.g., weight, inertia, etc)          b a X X X x y z x  cX NOTE: If the body is accelerating, then the inertia force may be considered as part of X            w        v u u u ~  XX z ST Volume (V) u v w Xa dV Xb dV Xc dV Volume element dV py pz px Traction: Distributed force per unit surface area       xp T SURFACE TRACTION x y x      z y p pS Docsity.com 5 Traction: Distributed force per unit area            z y x p p p T S n nx ny nz ST TS py px pz n If the unit outward normal to ST :         z y x n nn Then zzyzyxxz zyzyyxxy zxzyxyxx nnn nnn nnn       z y x p p p nx ny ST In 2D dy dx ds y n   TSpy px  dy dx dsx y xy xy C id h ilib i f h d ix x y n ds dy n ds dx     cos sin ons er t e equ r um o t e we ge n x-direction yxyxxx xyxx xyxx nnp ds dx ds dy p dxdydsp       Similarly yyxxyy nnp   3D elasticity problem is completely defined once we understand the following three concepts Strong formulation (governing differential equation + boundary conditions) Strain-displacement relationship Stress-strain relationship 2. Strain-displacement relationships: vu z w y v x u z y x              x w z u y w z v xy zx yz xy                    Docsity.com 6 Compactly; u                    y x 00 00 00    u         y x    (2)                               xz yz xy z 0 0 0       w vu             zx yz xy z     x y A B C A’ B’ C’ v udy dx dx x v   d u dy y u   dy y v v       1 2 In 2D x x u   x u x v tan βtan βββ)B'A'(C'angle 2 π y v dy dyvdy y v vdy AC ACC'A' x u dx dxudx x u udx AB ABB'A' 2121                                             xy y x    3D elasticity problem is completely defined once we understand the following three concepts Strong formulation (governing differential equation + boundary conditions) Strain-displacement relationship Stress-strain relationship 3. Stress-Strain relationship: Linear elastic material (Hooke’s Law)  D (3) Linear elastic isotropic material      0001                            2 21 00000 0 2 21 0000 00 2 21 000 0001 0001 )21)(1(       E D Docsity.com 7 Special cases: 1. 1D elastic bar (only 1 component of the stress (stress) is nonzero. All other stress (strain) components are zero) Recall the (1) equilibrium, (2) strain-displacement and (3) stress- strain laws 2. 2D elastic problems: 2 situations PLANE STRESS PLANE STRAIN 3. 3D elastic problem: special case-axisymmetric body with axisymmetric loading (we will skip this) PLANE STRESS: Only the in-plane stress components are nonzero Area element dA Nonzero stress components xyyx  ,, y x xy xy h D A i x y ssumpt ons: 1. h<<D 2. Top and bottom surfaces are free from traction 3. Xc=0 and pz=0 PLANE STRESS Examples: 1. Thin plate with a hole y x xy xy 2. Thin cantilever plate Nonzero strains: xyzyx  ,,, Isotropic linear elastic stress-strain law                    y x y x E       01 01 1 2  D  yxz   1 Nonzero stresses: xyyx  ,, PLANE STRESS         xyxy   2 1 00 Hence, the D matrix for the plane stress case is               2 1 00 01 01 1 2     E D  Docsity.com