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Unformatted text preview: Notes prepared by: (1) Senem Ayşe HASER (2) İmren UYAR COMPONENTS AND INDEX NOTATION Range Convention • • Indicies in Latin alphabet: i, j,k… range varies from 1, 2, 3 Indicies in Greek alphabet: , … range varies from 1, 2 Example: 1) xi=yi x1=y1 x2=y2 x3=y3 (3eqns) A11=B11, A12=B12, A13=B13, A21=B21, A22=B22, A23=B23, A31=B31, A32=B32, A33=B33 (9 eqns) 2) Aij =Bij 3) Aijk=Bijk A111=B111, A112=B112, A113=B113 A211=B211, A212=B212, A213=B213 (27 eqns) Summation Convention 1) A repeated index in a term implies summation over that index: Example i) ii) iii) xiyi = x1y1 + x2y2 + x3y3 (xαyα = x1y1 + x2y2) Aijxi =Ai1x1 Ai2xj +Ai3x3 i=1,2,3 Aij Aij = A11B11 + A12B12 + A13B13 +A21B21 + A22B22 + A23B23 +A31B31 + A32B32 + A33B33 (Aij)2=(Aij)2 2) An index cannot appear more than twice in a term . Non-example xiyizi (WRONG!) 3) A repeated index is called “dummy” index. The letter does not matter provided that coice does not violate other rules. Example Amm= Aii=Ajj Aijxi = Aimxm Aiixi WRONG! 4) An index which is not repeated is called “free” index. Note: The free index must appear on both sides of an equality. (But not the summed ones) Example yi =Aijxj = Aimxm • xj Force [Fi] = = vector (i’s have to match , you can change dummy indicies.) = Fi i=1, 2, 3 Fij =tensor=matrix Example A =b A11x1+A12x2+A13x3 = b1 A21x1+A22x2+A23x3 = b2 A31x1+A32x2+A33x3 = b23 A1JXJ = b1 A2JXJ = b2 A2JXJ = b3 Aijxj = bi 5) The summation convention may be suspended by writing “No Sum” to the right of the equation. Example xi + yi zi = λi (no sum on i) x1 + y1z1 = λ1 x2 + y2z2 = λ2 x3 + y3z3 = λ3 previous example Aijxj = bi (no sum on j) Indexed Variables in Equations 1) Any free index must agree in all terms. Exception: constants Examples: o o o ui + vi = wi Aijxj = zi +Bkkupvpxi Aijxj = 0 A A A x x A A A x A A A where i is a free index and j, p and k are dummy indices. 0 =0 0 A1jxj = 0 A2jxj = 0 A3jxj = 0 (1) A11x1 + A12x2 + A13x3 = 0 (2) A21x1 + A22x2 + A23x3 = 0 (3) A31x1 + A32x2 + A33x3 = 0 Non-examples: o o u i = vj zi + Apjwr = 0 2) You can multiply both sides of an equation by an indexed variable provides you do not violate any other rules. Example: xi + yi = zi xi xi vj xi Non-example: xixi + yryr =zizi yrxixi +yryryr =yrzizi 3) It is illegal to divide through by an indexed variable where index already appears in the equation. Example: xi xi =1 xi = 1/xi + yi + yi = zi = zi = vj zi ( cannot multiply by vi) + vj y i Example: Index notation is an extremely useful tool for performing vector algebra. In coordinate system, instead of using the typical axis labels x, y and z, x1, x2 and x3 are used. xi i= 1,2,3 The corresponding unit basis vectors are ê1, ê2 and ê3. êi i= 1,2,3 = ê1 + ê2 + ê3 Kronecker Delta 1) Kronecker Delta is represented by 9 numbers defined by δ δ δ δ 2) δ 1 if i 0 if i 1 δ 0 δ 0 δ δ δ j j 0 δ 1 δ 0 δ δ 1 δ 0 0 1 1 1 3 1 0 0 0 1 0 0 0 1 3) δ can be used to “sum on ” indices with a term it multiplies. Example: δu δu i=1 i=2 i=3 u δ u δu δu δu δu δu δu δu δu δu δu δu u u u 4) In an orthonormal basis ê1, ê2 , ê3 êi . êj = δ ê1 . ê1 = ê2 . ê2 = ê3 . ê3 = 1 ê1 . ê2 = ê2 . ê3 = ê1 . ê3 = 0 Permutation or Alternator Symbol εijk is represented by 27 numbers defined by • if (ijk) is an even (cyclic) permutation of (123), ijk = 1 • ijk = −1 if (ijk) is an odd (noncyclic) permutationof (123), • ijk = 0 if two or more subscripts are the same, Properties 1) Switching any 2 indices changes the sign. Examples 312 = - 321 ijk = - ikj kji = - ijk iij = 0 ijk ijk = 6 identities =δ δ δδ = 2δ = 2δ 6 k r 2)Cross product i.e. 123= 231 = 312 = 1 i.e. 213 = 321= 132 = −1 i.e. 111 = 112 = 313 = 0 etc. 3) Partial differentiation denoted by “comma” Examples: o o x, y, z x1, x2, x3 Stress equation
σ f, f, u , + + σ
, τ + + τ + Bx = 0 + B1 =0 σ σ σ + Bi =0 STRAIN Deformation Consider a body subjected to external loading that causes it to deform; Displacement due to 1) Deformation (strain) 2) Rigid body motion (translation and rotation) Displacement at any point within the body: u = u(x,y,z) v = v(x,y,z) w = w(x,y,z) Assumption: 1) Small deformations (h.o.t. neglected) 2) The principle of superposition can be used. Definition of Strain Consider an axially loaded member Normal strain, unit change in length, is defined ε lim∆
∆ ∆ If deformation is uniformly distributed over the original length, ε= =
δ L 2-D Plane Strain Case Normal strain ε ε ε ε Shear strain Small angles α Similarly, α tanα tanα Total angular change Shear strain 3-D PLANE STRAIN 1 u 2, i=j=1 i =1, j=2 u, Strain Tensor at a Point 1,2,3 form a symmetric matrix. They are called the components of The nine quantities , , are strain of the particle with respect to the system of axes x1, x2 and x3. The quantities called the normal components of strain of the particle in the directions of the axes x1, x2 and x3, respectively, while , are called shearing components of strain in the directions of the axes x1, x2 and x1, x3 and x2, x3, respectively. STRAIN TRANSFORMATION
′ ϵ ϵ 1 + cos 2 sin 2 ( where 12 2 12 ) The normal stress
′ 12 1 2 ′ 22 is 11 determined by replacing 12 /2 . ′12 22 2 sin 2 2 cos 2 REFERENCES
1) Ugural and Fenster, Advanced Strength and Applied Elasticity, 4th Ed., Prentice Hall, PTR, 2003 2) Armenakas A.E., Advanced Mechanics of Materials and Applied Elasticity, 2006, Taylor&Francis Group 3) Lecture notes ...
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- Spring '11