slide chương 1 cơ học vật chất rắn new mathematical preliminaries

.c om ng co an th cu u du o ng Chapter 1: Mathematical Preliminaries TDT  University  -­‐  2015 CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   co ng c om Elasticity theory is a mathematical model of material deformation Using principles of continuum mechanics, it is formulated in terms of many different types of field variables specified at spatial points in the body under study Some examples include: th an Scalars - Single magnitude mass density ρ , temperature T, modulus of elasticity E, du o ng Vectors – Three components in three dimensions e1 ,e ,e3 are unit basis vectors displacement vector u = ue1 + ve + we3 cu u Matrices – Nine components in three dimensions stress matrix ⎡σ x τ xy τ xz ⎤ ⎢ ⎥ σ = ⎢τ yx σ y τ yz ⎥ ⎢τ τ σ ⎥ ⎣ zx zy z ⎦ Other – Variables with more than nine components   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om With the wide variety of variables, elasticity formulation makes use of a tensor formalism using index notation This enables efficient representation of all variables and governing equations using a single standardized method th an co ng ⎡ a1 ⎤ ⎡ a11 a12 a13 ⎤ Index notation is a shorthand scheme whereby a whole set of numbers or components can be = ⎢ a2 ⎥ , aij = ⎢ a21 a22 a23 ⎥ ⎢ ⎥ ⎢ ⎥ represented by a single symbol with subscripts ⎢⎣ a3 ⎥⎦ ⎢⎣ a31 a32 a33 ⎥⎦ In general a symbol aij…k with N distinct indices represents 3N distinct numbers du o ng Addition, subtraction, multiplication and equality of index symbols are defined in the normal fashion; e.g cu u ⎡ a1 ± b1 ⎤ ⎡ a11 ± b11 a12 ± b12 ± bi = ⎢⎢ a2 ± b2 ⎥⎥ , aij ± bij = ⎢⎢ a21 ± b21 a22 ± b22 ⎢⎣ a3 ± b3 ⎥⎦ ⎢⎣ a31 ± b31 a32 ± b32 ⎡ λ a1 ⎤ ⎡ λ a11 λ a12 λ = ⎢⎢λ a2 ⎥⎥ , λ aij = ⎢⎢λ a21 λ a22 ⎢⎣ λ a3 ⎥⎦ ⎢⎣ λ a31 λ a32 CuuDuongThanCong.com λ a13 ⎤ λ a23 ⎥⎥ λ a33 ⎥⎦ a13 ± b13 ⎤ a23 ± b23 ⎥⎥ a33 ± b33 ⎥⎦ ⎡ a1b1 a1b2 b j = ⎢⎢ a2b1 a2b2 ⎢⎣ a3b1 a3b2 a1b3 ⎤ a2b3 ⎥⎥ a3b3 ⎥⎦ https://fb.com/tailieudientucntt   .c om h"p://incos.tdt.edu.vn   Summation Convention - if a subscript appears twice in the same term, then summation over that subscript from one to three is implied; for example ng aii = ∑ aii = a11 + a22 + a33 i =1 co aij b j = ∑ aij b j = ai1b1 + 2b2 + 3b3 an j =1 ng th A symbol j m n is said to be symmetric with respect to index pair mn if j m n = a n m j i cu u du o A symbol j m n is said to be antisymmetric with respect to index pair mn if j m n = − a n m j i If j m n is symmetric in mn while bp q m n is antisysmetric in mn, then the product is zero a b =0 i j m n p q m n 1 a = ( a + a ) + (aij − a ji ) = a(ij ) + a[ij ] Useful Identity ij ij ji 2 a(ij ) 1 = (aij + a ji ) symmetric a[ij ] = (aij − a ji ) antisymmetric 2 CuuDuongThanCong.com https://fb.com/tailieudientucntt   h"p://incos.tdt.edu.vn   Example 1-1: Index Notation Examples c om ⎡1 ⎤ ⎡2⎤ ⎢0 3⎥ , b = ⎢4⎥ a = ij Determine the following quantities: ⎢ ⎥ i ⎢ ⎥ aii , aij aij , aij a jk , aijb j , aijbb ⎢⎣ 2 ⎥⎦ ⎢⎣ ⎥⎦ i j , bb i i , bb i j , a(ij ) , a[ij ] Indicate whether they are a scalar, vector or matrix co ng The matrix aij and vector bi are specified by an Following the standard definitions given in section 1.2, th aii = a11 + a22 + a33 = (scalar) ng aij aij = a11a11 + a12 a12 + a13 a13 + a21a21 + a22 a22 + a23 a23 + a31a31 + a32 a32 + a33 a33 du o = + + + + 16 + + + + = 39 (scalar) cu u ⎡ ⎤ ⎡0 ⎤ ⎡0 0 ⎤ ⎡ 10 ⎤ aij a jk = 1a1k + a2 k + a3 k = ⎢0 0 ⎥ + ⎢0 16 12 ⎥ + ⎢6 ⎥ = ⎢6 19 18 ⎥ (matrix) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣ ⎥⎦ ⎢⎣0 ⎥⎦ ⎢⎣ 4 ⎥⎦ ⎢⎣6 10 ⎥⎦ ⎡ ⎤ ⎡ ⎤ ⎡0 ⎤ ⎡10 ⎤ aij b j = 1b1 + 2b2 + 3b3 = ⎢0 ⎥ + ⎢16 ⎥ + ⎢0 ⎥ = ⎢16 ⎥ (vector) ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣4 ⎥⎦ ⎢⎣ ⎥⎦ ⎢⎣0 ⎥⎦ ⎢⎣ ⎥⎦ aij bi b j = a11b1b1 + a12b1b2 + a13b1b3 + a21b2b1 +L = 84 (scalar) CuuDuongThanCong.com https://fb.com/tailieudientucntt   h"p://incos.tdt.edu.vn   Example 1-1: Index Notation Examples c om ⎡1 ⎤ ⎡2⎤ ⎢0 3⎥ , b = ⎢4⎥ a = ij Determine the following quantities: ⎢ ⎥ i ⎢ ⎥ ⎢⎣ 2 ⎥⎦ ⎢⎣ ⎥⎦ aii , aij aij , aij a jk , aijb j , aijbb i j , bb i i , bb i j , a(ij ) , a[ij ] Indicate whether they are a scalar, vector or matrix co ng The matrix aij and vector bi are specified by an Following the standard definitions given in section 1.2, u du o ng 0⎤ 16 ⎥ (matrix) ⎥ 0 ⎥⎦ 0⎤ ⎡ ⎤ ⎡1 ⎥ + ⎢ ⎥ = ⎢1 ⎥ 2⎢ ⎥ ⎢ ⎢⎣0 ⎥⎦ ⎢⎣1 ⎥⎦ ⎡1 ⎤ ⎡1 2⎤ ⎡ 1 = ( aij − a ji ) = ⎢⎢0 ⎥⎥ − ⎢⎢ ⎥⎥ = ⎢⎢ −1 2 ⎢⎣ 2 ⎥⎦ ⎢⎣0 ⎥⎦ ⎢⎣ cu ⎡2⎤ ⎡4 bi b j = ⎢4 ⎥ [ ] = ⎢8 ⎢ ⎥ ⎢ ⎢⎣0 ⎥⎦ ⎢⎣0 ⎡1 1 a( ij ) = ( aij + a ji ) = ⎢0 2⎢ ⎢⎣ th bi bi = b1b1 + b2b2 + b3b3 = + 16 + = 20 (scalar) a[ ij ] CuuDuongThanCong.com 1⎤ ⎥ (matrix) ⎥ 2 ⎥⎦ −1⎤ ⎥⎥ (matrix) −1 ⎥⎦ https://fb.com/tailieudientucntt   .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems 10   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   co   ng c om It is always possible to identify a right-handed Cartesian coordinate system such that each axes lie along principal directions of any given symmetric second order tensor Such axes are called the principal axes of the tensor, and the basis vectors are the principal directions {n(1), n(2) , n(3)} x3 ⎡λ1 0 ⎤ aij = ⎢ λ ⎥ ⎢ ⎥ ⎢⎣ 0 λ ⎥⎦ x¢3   cu x1 n(3)   n(2)   x¢2   x2 u du o ng th an ⎡ a11 a12 a13 ⎤ aij = ⎢ a21 a22 a23 ⎥ ⎢ ⎥ ⎢⎣ a31 a32 a33 ⎥⎦ Fig  3  Original  given  axes   n(1)   x¢1   Fig  4  Principle  axes   22   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Example 1-3 Principal Value Problem Determine the invariants, and principal values and directions of ⎡2 0 ⎤ + + = − 25 − = −25 −3 −3 ng th I a = aii = + − = , II a = = 2(−9 − 16 ) = −50 du o III a = co an First determine the principal invariants ng aij = ⎢0 ⎥ ⎢ ⎥ ⎢⎣0 −3 ⎥⎦ u −3 cu The characteristic equation then becomes det ⎡⎣ aij − λδij ⎤⎦ = −λ + 2λ + 25λ − 50 = ⇒ ( λ − ) ( λ − 25 ) = ∴ λ = , λ = , λ = −5 23   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Example 1-3 Principal Value Problem Determine the invariants, and principal values and directions of ⎡2 0 ⎤ co ng aij = ⎢0 ⎥ ⎢ ⎥ ⎢⎣0 −3 ⎥⎦ th an Thus  for  this  case  all  principal  values  are  dis=nct       For  the  λ1  =  5  root,  equa=on  (1.6.1)  gives  the  system   ng ( 1) which  gives  a  normalized  solu=on     n = ± (2e + e3 ) −3n1(1) = −2n2(1) + 4n3(1) = 4n2(1) − 8n3(1) = du o In  similar  fashion  the  other  two  principal  direc=ons  are  found  to  be     u n ( ) = ± e1 , n ( ) = ± (e − 2e3 ) cu It  is  easily  verified  that  these  direc=ons  are  mutually  orthogonal       Note  for  this  case,  the  transforma=on  matrix  Q        defined  by  (1.4.1)  becomes   ij ⎡0 / ⎢ Qij = ⎢ ⎢0 / ⎣ CuuDuongThanCong.com 1/ ⎤ ⎡5 0 ⎤ ⎥ ⎥ ⇒ aij′ = ⎢0 ⎥ ⎢ ⎥ ⎥ ⎢⎣0 −5 ⎥⎦ −2 / ⎦ 24   https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems 25   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om a ⋅ b = a1b1 + a2b2 + a3b3 = aibi Scalar or Dot Product e3 a × b = a1 a2 a3 = εijk a j bk ei b1 b2 b3 th aT A = {a} [ A ] = Aij = Aij AB = [ A ][ B] = Aij B jk T a3 b3 = εijk aib j ck c3 AB T = Aij Bkj Second Order Transformation Law aij′ = QipQ jq a pq A T B = Aji B jk ⇒ a′ = QaQ T du o u cu a2 b2 c2 Aa = [ A ]{a} = Aij a j = a j Aij ng Common Matrix Products a1 a.b × c = b1 c1 co e2 an e1 ng Vector or Cross Product tr ( AB ) = Aij B ji CuuDuongThanCong.com tr ( AB T ) = tr ( A T B ) = Aij Bij AijT = A ji ; tr A = Aii = A11 + A22 + A33 https://fb.com/tailieudientucntt 26   .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems 27   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om a = a ( x1 , x2 , x3 ) = a ( xi ) = a ( x ) Field concept for tensor components = ( x1 , x2 , x3 ) = ( xi ) = ( x ) co ng aij = aij ( x1 , x2 , x3 ) = aij ( xi ) = aij ( x ) ! a,i = ∂ ∂ ∂ a , , j = , aij ,k = aij , L ∂xi ∂x j ∂xk th an Comma notation for partial differentiation cu u du o ng If differentiation index is distinct, order of the tensor will be increased by one; e.g derivative operation on a vector produces a second order tensor or matrix CuuDuongThanCong.com ⎡ ∂a1 ⎢ ⎢ ∂x1 ⎢ ∂a , j = ⎢ ⎢ ∂x1 ⎢ ∂a3 ⎢ ⎣ ∂x1 ∂a1 ∂x2 ∂a2 ∂x2 ∂a3 ∂x2 ∂a1 ⎤ ⎥ ∂x3 ⎥ ∂a2 ⎥ ⎥ ∂x3 ⎥ ∂a3 ⎥ ⎥ ∂x3 ⎦ 28   https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   an co ng c om Directional Derivative of Scalar Field df = ∂ f dx + ∂ f dy + ∂ f dz = n ⋅∇f ds ∂x ds ∂ y ds ∂z ds dx dy dz n = unit normal vector in direction of s = e1 + e + e ds ds ds ∂ ∂ ∂ ∇ = vector differential operator = e1 + e + e ∂x ∂y ∂z th ∇f = grad f = gradient of scalar function f = e1 du o ng Common Differential Operations ∂f ∂f ∂f + e2 + e3 ∂x ∂y ∂z ∇ϕ = ϕ ,i e i Gradient of a Vector ∇u = ui, j e ie j cu u Gradient of a Scalar Laplacian of a Scalar ∇ 2ϕ = ∇ ⋅∇ϕ = ϕ ,ii Divergence of a Vector ∇ ⋅ u = ui,i Curlof a Vector ∇ × u = ε ijk uk , j e i Laplacian of a Vector ∇ u = ui,kk e i CuuDuongThanCong.com 29   https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om Example 1-4: Scalar/Vector Field Example 2 Scalar and vector field functions are given by φ = x − y , u = 2xe1 + yze + xye Calculate the following expressions, ∇φ, ∇ φ, ∇ ⋅ u, ∇u, ∇ × u ng Note vector field ∇φ is orthogonal to ϕ-contours, Gradient Vector Distribution a result true in general for all scalar fields co Using the basic relations 10 an ∇φ = 2xe1 − ye du o cu u ∇u = ui , j = ⎡ ⎤ 0 ⎢ ⎥ ⎢ 3z y ⎥ ⎢ ⎥ ⎢⎣ y x ⎥⎦ ∇×u = e1 e2 e3 ∂ / ∂x ∂ / ∂y ∂ / ∂z ng th ∇ 2φ = − = - (satisfies Laplace equation) ∇ ⋅ u = + 3z + = + 3z 2x yz xy = ( x − y )e1 − ye CuuDuongThanCong.com y x -2 -4 -6 -8 -10 -10 -5 Fig  5  Contours ϕ=constant and vector distributions of ∇φ https://fb.com/tailieudientucntt 10 30   Divergence Theorem c om h"p://incos.tdt.edu.vn   ∫∫ S u ⋅ n dS = ∫∫∫ ∇ ⋅ u dV ⇒ ∫∫ aij k nk dS = ∫∫∫ aij k ,k dV ∫ u ⋅ dr = ∫∫ (∇ × u) ⋅ n dS ⇒ ∫ aij dxt = ∫∫ ε rst aij k ,s nr dS S V ng V Green’s Theorem in the Plane an S C S th C co Stokes Theorem du o ng Apply Stoke theorem to a planar domain S with the vector field selected as u = f e1 + ge cu u ⎛ ∂g ∂f ⎞ ∂g ∂f − dxdy = ( fdx + gdy ) ⇒ dxdy = gn ds , ∫∫ S ⎜⎝ ∂x ∂y ⎟⎠ ∫C ∫∫ S ∂x ∫ C x ∫∫ S ∂y dxdy = ∫ C fny ds Zero-Value Theorem ∫∫∫ V fij k dV = ⇒ fij k = ∈ V 31   CuuDuongThanCong.com https://fb.com/tailieudientucntt .c om h"p://incos.tdt.edu.vn   1.1 Common Variable Types in Elasticity   co ng 1.2 Index/Tensor Notation an 1.3 Kronecker Delta & Alternating Symbol th 1.4 Coordinate Transformations du o ng 1.5 Cartesian Tensors General Transformation Laws 1.6 Principal Values and Directions for Symmetric Second Order Tensors cu u 1.7 Vector, Matrix and Tensor Algebra 1.8 Calculus of Cartesian Tensors 1.9 Orthogonal Curvilinear Coordinate Systems 32   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn       x3   z   c om x3   eˆz an x2   e1 th e2 r   du o x1   ng θ   co eˆr e3 e1 ng eˆθ cu u Fig  6  Cylindrical Coordinate System (r,θ,z) x1 = r cos θ , x2 = sin θ , x3 = z r = x12 + x22 , θ = tan −1 x2 , z = x3 x1 CuuDuongThanCong.com φ   e3 R   θ   eˆr eˆθ eˆφ e2 x2   x1   Fig  7  Spherical Coordinate System (R,ϕ,θ) x1 = R cos θ sin ϕ , x2 = R sin θ sin ϕ , x3 = R cos ϕ R = x12 + x22 + x32 x3 ϕ = cos −1 x12 + x22 + x32 x θ = tan −1 , 33   x1 https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om   Common Differential Forms x3   ξ3   ∇ = eˆ eˆ3 eˆ2 co an ξ1   th x2   e2   ng e1 ∇f = eˆ eˆ1 e3 du o x1   cu u Fig  8  Curvilinear coordinates m m ∂f ∂f ∂f ∂f ˆ ˆ ˆ + e + e = e ∑i i h ∂ξi h1 ∂ξ1 h2 ∂ξ h3 ∂ξ3 i ⎞ ∂ ⎛ h1h2 h3 u< i > ⎟ ∑ i ⎜ h1h2 h3 i ∂ξ ⎝ hi ⎠ ∇ ⋅u = ∂ ⎛ h1h2 h3 ∂ϕ ⎞ ⎜ ⎟ ∑ h1h2 h3 i ∂ξi ⎝ (hi ) ∂ξi ⎠ ε ∂ ∇ × u = ∑∑ ∑ ijk (u hk )eˆ i j i j k h j hk ∂ξ ∇ 2ϕ = ∇u = ∑∑ ξ = ξ ( x , x , x ) , x = x (ξ , ξ , ξ ) m ∂ ∂ ∂ ∂ + eˆ + eˆ = ∑ eˆ i h1 ∂ξ h2 ∂ξ h3 ∂ξ hi ∂ξi i ng ξ2   m (ds ) = (h1d ξ1 ) + (h2 d ξ ) + (h3 d ξ3 ) i j ∂eˆ j ⎞ eˆ i ⎛ ∂u< j > ˆ e + u j < j > ⎜ ⎟ hi ⎝ ∂ξi ∂ξi ⎠ ⎛ eˆ ∂ ⎞ ⎛ eˆ ∇ 2u = ⎜ ∑ i i ⎟ ⋅ ⎜ ∑∑ k ⎝ i hi ∂ξ ⎠ ⎝ j k hk ∂eˆ j ⎤ ⎞ ⎡ ∂u< j > ˆ e + u ⎢ ∂ξ k j < j > ∂ξ k ⎥ ⎟ ⎣ ⎦⎠ 34   CuuDuongThanCong.com https://fb.com/tailieudientucntt h"p://incos.tdt.edu.vn   c om From relations (1.9.5) or simply using the geometry shown in Figure eˆ r = cos θe1 + sin θe ∂eˆ ∂eˆ ∂eˆ r ∂eˆ ⇒ = eˆ θ , θ = −eˆ r , r = θ = eˆ θ = − sin θe1 + cos θe ∂θ ∂θ ∂r ∂r ng x2   co ∂ ∂ ∇ = eˆ r + eˆ θ ∂r r ∂θ ∂ϕ ∂ϕ ∇ϕ = eˆ r + eˆ θ ∂r r ∂θ ∂ ∂uθ ∇ ⋅u = (rur ) + r ∂r r ∂θ ∂ ⎛ ∂ϕ ⎞ ∂ ϕ ∇ ϕ= ⎜r ⎟+ r ∂r ⎝ ∂r ⎠ r ∂θ2   eˆr e2 r   ng du o The basic vector differential operations then follow to be th an eˆθ θ     x1   e1 Fig  9  Polar  coordinate system cu u ∂ur ⎞ ⎛1 ∂ ∇ ×u = ⎜ (ruθ ) − (ds ) = (dr ) + (rd θ) ⇒ h1 = , h2 = r ⎟ eˆ z r ∂θ ⎠ ⎝ r ∂r ∂u ∂u ⎛ ∂u ⎛ ∂u ⎞ ⎞ ∇u = r eˆ r eˆ r + θ eˆ r eˆ θ + ⎜ r − uθ ⎟ eˆ θeˆ r + ⎜ θ − ur ⎟ eˆ θeˆ θ ∂r ∂r r ⎝ ∂θ r ⎝ ∂θ ⎠ ⎠ ∂u u ⎞ ∂u u ⎞ ⎛ ⎛ ∇ 2u = ⎜ ∇ 2ur − θ − 2r ⎟ eˆ r + ⎜ ∇ 2uθ + r − θ2 ⎟ eˆ θ r ∂θ r ⎠ r ∂θ r ⎠ ⎝ ⎝ where u = ur eˆ r + uθeˆ θ , eˆ z = eˆ r × eˆ θ CuuDuongThanCong.com 35   https://fb.com/tailieudientucntt .c om ng co an th ng du o u cu CuuDuongThanCong.com https://fb.com/tailieudientucntt ... cu u ⎡ a1 ± b1 ⎤ ⎡ a 11 ± b 11 a12 ± b12 ± bi = ⎢⎢ a2 ± b2 ⎥⎥ , aij ± bij = ⎢⎢ a 21 ± b 21 a22 ± b22 ⎢⎣ a3 ± b3 ⎥⎦ ⎢⎣ a 31 ± b 31 a32 ± b32 ⎡ λ a1 ⎤ ⎡ λ a 11 λ a12 λ = ⎢⎢λ a2 ⎥⎥ , λ aij = ⎢⎢λ a 21 λ a22... definitions given in section 1. 2, th aii = a 11 + a22 + a33 = (scalar) ng aij aij = a11a 11 + a12 a12 + a13 a13 + a21a 21 + a22 a22 + a23 a23 + a31a 31 + a32 a32 + a33 a33 du o = + + + + 16 + + + + = 39 (scalar)... permutation of 1, 2,3 ⎪ ε ijk = ⎨ ? ?1 , if ijk is an odd permutation of 1, 2,3 ⎪ , otherwise ⎩ ? ?12 3 = ε 2 31 = ε 312 = ε 3 21 = ? ?13 2 = ε 213 = ? ?1 ? ?11 2 = ? ?13 1 = ε 222 = = CuuDuongThanCong.com 11   https://fb.com/tailieudientucntt