Mathematical Preliminary

Mathematical Preliminary

Outline Scalar, Vector and Matrix( 标量 矢量 矩阵 ) Indicial Notation and Summation Convention( 指标记法与求和约定 ) Kronecker Delta( 克罗内克 δ) Levi-Civita symbol( 轮换记号 ) Coordinate Transformation( 坐标变换 ) Tensor( 张量 ) Principal Values and Directions( 特征值与特征向量 ) Tensor Algebra( 张量代数 ) Tensor Calculus( 张量微分 ) Integral Theorems( 积分定理 ) Cylindrical and Spherical Coordinates( 柱与球坐标 )

Scalar Scalar: representing a single magnitude at each point in space Examples: Material density Young s modulus Poisson s ratio Shear modulus Temperature 3

Vector Vector: representing physical quantities that have both magnitude and direction Examples: Force Displacement of material points Rotation of material points 4

Matrix (Array) Matrix: a rectangular array of numbers Array: a data structure in which similar elements of data are arranged in a table 5

Indicial Notation Orthogonal unit vectors: Vector decomposition: a OP a i i a j j a k k 3 1 3 1 3 i1 a e a e a e a e i i j k i e 1 =i x =x 1 O z =x 3 Indicial notation: a shorthand scheme whereby a whole set of components is represented by a single symbol with subscripts k=e 3 a j=e x1 e1 a1 a11 a1 a13 xi x, ei e, ai a, aij a1 a a 3 x e a a a a 3 3 3 3 3 33 P y=x 6

Indicial Notation Addition and subtraction Scalar multiplication Outer multiplication 7

Indicial Notation Commutative, associative and distributive laws Equality of two symbols a1 b1 a11 b11 a1 b1 a13 b13 ai bi a b, aij b ij a1 b1 a b a3 b 3 a b a b a b a b Avoid 3 3 31 31 3 3 33 33 a b, a b i j ij kl 8

Summation Convention Summation convention: if a subscript appears twice in the same term, then summation over that subscript from one to three is implied a x x 3 3 a x x ij i j ij i j i1 j1 a a a ii jj kk In a single term, no index can appear more than twice. Dummy (repeated) indices vs. free (distinct) indices Among terms, index property must match. 9

Contraction and Symmetry Contraction: for example, a ii is obtained from a ij by contraction on i and j Outer multiplication contraction inner product a b contraction on i j i i a b contraction on ij a b ij kl ik kl jk a b Symmetric vs. antisymmetric (skewsymmetric) w.r.t. two indices, i.e. m and n Useful identity: 10

Kronecker Delta Kronecker delta: a useful special symbol commonly used in index notational schemes Symmetric Replacement property 11

Levi-Civita (Permutation) Symbol Antisymmetric w.r.t. any pair of its indices Cross product of two vectors a b a e b e a b e e a b e a b e i i j j i j i j i j ijk k ijk i j k Scalar triple product of three vectors abc a b e c e a b c e e a b c a b c ijk i j k m m ijk i j m k m ijk i j m km ijk i j k 1

Levi-Civita (Permutation) Symbol Determinant of a matrix (easily verifiable) a a a 11 1 13 det aij aij a1 a a3 ijka1 ia ja3k ijkai1a jak 3 a a a 3 3 33 By the definition of Levi-Civita and note the following a a a a a a a a a a a a ijk rst ir js kt ijk i1 j k 3 ijk i1 j3 k ijk i j3 k1 a a a a a a a a a ijk i j1 k 3 ijk i3 j1 k ijk i3 j k1 ijk rstaira jsakt 6det aij, ijkaila jmakn lmn det aij 13

Levi-Civita (Permutation) Symbol ε-δ property T A B A B AB i1 i i3 ijk i j k ip p jq q kr r ijk j1 j j3 ijk pqr T e e e e e e k1 k k 3 k1 k k 3 p3 q3 r3 km i1 i i3 p1 q1 r1 im pm im qm im rm j1 j j3 p q r jm pm jm qm jm rm pm km qm km rm ip iq ir ip iq ik ijk pqr jp jq jr ijk pqk jp jq jk ip jq iq jp kp kq kr kp kq kk 14

Sample Problem The matrix a ij and vector b i are specified by 1 0 aij 0 4 3, b i 4 1 0 Determine the following quantities, and indicate whether they are a scalar, vector or matrix. a, a a, a a, a b, a bb, bb, bb, a, a ii ij ij ij jk ij j ij i j i i i j Solution: a a a a 7 (scalar) ii ij ij 11 33 symm. a a a a a a a a a a a a a a a a a a a a anti. 11 11 1 1 13 13 1 1 3 3 31 31 3 3 33 33 1 4 0 0 16 9 4 1 4 39 (scalar) 15

Sample Problem - Solution 1 10 6 aija jk ai1a1 k ai ak ai3a3k 6 19 18 (matrix) 6 10 7 10 aijbj ai 1b1 ai b ai3b3 16 (vector) 8 a b b a b b a b b a b b a b b 84 (scalar) ij i j 11 1 1 1 1 13 1 3 1 1 b b b b b b b b 4 16 0 0 (scalar) i bb i i j 1 1 3 3 4 8 0 8 16 0 (matrix) 0 0 0 1 0 1 0 1 1 1 1 1 1 asymm. aij a ji 0 4 3 4 1 1 4 (matrix) 1 0 3 1 1 0 1 0 0 1 1 1 1 1 aanti. aij a ji 0 4 3 4 1 1 0 1 (matrix) 1 0 3 1 1 0 16

Coordinate Transformation Unit base vectors for each frame e e,e,e ; e e,e,e i 1 3 i 1 3 Transformation matrix Q cos( x, x ) ij i j e Q e Q e Q e 1 11 1 1 13 3 e Q e Q e Q 1 1 3 3 e Q e Q e Q e 3 31 1 3 33 3 Relations between base vectors e Q e ; e Q e i ij j i ji j e 17

Coordinate Transformation Transformation between vectors ν e e e e e e e e 1 1 3 3 i i 1 1 3 3 i i i Qij j; i Qji j Property of transformation matrix T i Q ji j Q jiq jk k Q jiq jk ik Q Q I T Q Q Q ; Q Q QQ i ij j ij kj k ij kj ik I det Q ij 1 For right-handed coordinates det Q ij 1 18

Sample Problem The components of a first and second order tensor in a particular coordinate frame are given by a i 1 1 0 3 4, aij 0 3 4 Determine the components of each tensor in a new coordinate system found through a rotation of 60 o about the x 3 -axis. Choose a counterclockwise rotation when viewing down the negative x 3 -axis. 19

Sample Problem - Solution Q ij cos60 cos30 cos90 1 3 0 cos150 cos 60 cos 90 3 1 0 cos 90 cos 90 cos 0 0 0 1 1 3 0 1 1 3 a i Qij a j 3 1 0 4 3 0 0 1 1 3 0 1 0 3 1 3 0 a ij QipQjqapq 3 1 0 0 3 1 0 0 0 1 3 4 0 0 1 7 4 3 4 3 3 3 4 5 4 13 3 3 3 13 3 4 0

Tensor - Definition Those matrices that satisfy the following rule are called second-order tensors. A Akl ekel Akl Qkie i Qlje j QkiQlj Akl eiej Aij eiej A Q Q A A Q Q A ij ki lj kl ij ik jl kl Exceptional matrices do exist, i.e. Spinors( 旋量 ) Fortunately, this is not our concern. 1

Tensor - Definition A tensor can be any order and must satisfy For general order A Q Q Q Q A ijkl im jn ko lp mnop A Q Q Q Q A ijkl mi nj ok pl mnop

Isotropic Tensors Defined as those tensors whose components remain the same under all transformations All scalars (zero order) are isotropic No non-trivia isotropic tensor of the first order Kronecker Delta is the most general second order isotropic tensor T ij ij T Q Q T Q Q Q Q T ij ik jl kl ik jl kl ik jk ij ij 3

Isotropic Tensors Levi-Civita Symbol is the most general third order isotropic tensor det a a a a T ijk det 1m n 3o mno m1 n o3 mno a ijk im jn ko mno ijk a a a a a a T Q Q Q T Q Q Q det Q T ijk im jn ko mno im jn ko mno ijk ijk The most general fourth order isotropic tensor T ijkl ij kl ik jl il jk T Q Q Q Q T Q Q Q Q ijkl im jn ko lp mnop im jn ko lp mn op mo np mp on Q Q Q Q Q Q Q Q Q Q Q Q T im jm ko lo im jn km ln im jn kn lm ijkl 4

Tensor Algebra The distinction between a tensor and its components Equality of two tensors: Addition/subtraction Scalar multiplication: A A i e Cross product of two vectors Triple product of three vectors A B Ae Be A B i i i i i i i 5

Tensor Algebra Tensor product (outer multiplication ) A B Ae B e A B e e A B e e i i j j i j i j i j i j Contraction on a pair of indices (Tensor product inner product) A B Ae B e A B e e A B i i j j i j i j i i Generalized inner product (matrix product) 6

Principal Values and Directions - D For the state of plane stress shown, determine the principal stresses and principal directions. Solution 1. Principal stresses. Principal directions 50 40 n1 0 40 10 n 0 50 40 0 50 10 1600 0 40 10 40 100 0 70, 30 1 50 70 40 n1 0 0 40 n1 0 n1 5 1 40 10 70 n n n 0 40 80 n 0 n 1 5 1 50 30 40 n1 0 80 40 n1 0 n1 5 n n1 40 10 30 n 0 40 0 n 0 n 5 7

Principal Values and Directions - 3D For the state of 3-D stress shown, determine the principal stresses and principal directions. 0 10 0 n 0 Solution 1. Principal stresses: z y 0MPa 0MPa I I I 1 3 10MPa 0MPa x 0 0 0 60 x y z x y y z z x xy yz zx 1 10 0 0 n 0 0 0 0 n 3 0 det ij ij 0 3 I1 I I3 0 30, 0, 10. 1 1 1 400 400 400 100 1100 x y z xy yz zx x yz y zx z xy 8000 000 6000. Principal directions: 0 10 0 n 0 1 10 0 0 n 0 0 0 0 n 3 0 8

Tensor Field and Comma Notation The field concept of a tensor component The Comma Notation for partial differentiation Gradient operator: Laplacian operator: Gradient of a scalar: Laplacian of a scalar: e e e e x x x x 1 3 1 3 ei e j xi x j xixi e e x i i, i i xx i i, ii i i 9

Tensor Calculus d dx1 dx dx3 Directional derivative: n e1 e e3 e1 e e3 ds x1 x x3 ds ds ds Gradient of a vector: Divergence of a vector: Gradient of a tensor: Divergence of a tensor: Curl of a vector: Laplacian of a vector: u ue e u e e u e e i i i j i j i, j i j x j xj u e j x u e u u j i i i, j ij i, i σ e e e e e e e e e ij ij i j k i j k ij, k i j k xk xk ij σ e e e e e e e xk k ij i j k i j ij, i j xk u k u e j u k x ek ijkei ijkuk, jei j x j u ukek uk, iiek xixi 30

Useful Identities All can be proved with Indicial and Comma Notations. 31

Sample Problem Scalar and vector field functions are given by n by x y and u xe1 3yze xye 3., Calculate u, u, the following u. expressions: ns are given by x y and u xe1 3yze xye 3 sions,,, u, u, u. s (1.8.4) Solution xe ye 0 u u 3z 0 3z u ii, 1 0 0 ui, jeiej 0 3z 3y y x 0. u e u e u e x i j j kij j, i k u u u u u u 1 3,,3 1,3 3,1 3,1 1, x 3y e y 1 i e e e e 3

Integral Theorems Gauss (Divergence) theorem S S u nds udv ; u n ds u dv. V k k k, k V Stokes theorem C C k k ijk k, j i S u dr u nds; S u dx u n ds. Green s theorem on a plane C u dx u dx u u dx dx. 1 1,1 1, 1 S 33

Cylindrical Coordinates 1 y x r cos, y r sin, r x y, tan x r sin r cos cos,, sin, x x r y y r r sin cos x r x x r r r cos sin y r y y r r er cos sin 0 ex excos eysin sin cos 0 e ey exsin eycos e z 0 0 1 ez ez er e e, er 34

Cylindrical Coordinates cos sin 0 x 1 c sin cos 0 Q er e ez y r r z 0 0 1 z 1 uc u e u e u e e e e r r z r r z z r z ur 1 ur ur erer u ere erez r r z u 1 u u ee r ur ee ee r r z uz 1 uz uz ezer eze ezez r r z z 35

Cylindrical Coordinates Double gradient of a scalar (symmetric as expected) 1 er e ez r r z 1 e rer ere erez r r r rz 1 1 1 1 eer r r r r r e e e e r z 1 ezer eze e ze z rz r z z z 36

Cylindrical Coordinates 1 c u er e e z urer u e uze z r r z ur ur 1 u uz r r r z 1 c u er e e z urer u e uze z r r z 1 uz u ur uz u 1 ur 1 er e e z u r z z r r r r 1 1 c er e ez er e e z r r z r r z 1 1 r r r r z 37

Cylindrical Coordinates rerer re re rzerez reer ee 1 σ r z z z zr z r z z e e e e e e e e e zezez r r z r r rz r ererer ereer ere zer eerer eeer r r r r r z zr z z ee zer e zerer e zeer e ze zer r r r r 1 r r r 1 r r 1 rz z erere ere e ere ze r r r r r r 1 r r 1 r r 1 z rz e ere e e e e e ze r r r r r r 1 zr z 1 z zr 1 z ezere e zee e ze ze r r r r r r r rz r ererez eree z ere ze z eere z eee z z z z z z z zr z z ee ze z e zere z e zee z e ze ze z z z z z 38

Cylindrical Coordinates Gradient of a tensor: Divergence of a tensor: σ e e e e e e e e e ij ij i j k i j k ij, k i j k xk xk ij σ e e e e e e e xk k ij i j k i j ij, i j xk σ contraction on the first and third index of σ r 1 r zr r σ er r r z r r 1 z r r e r r z r rz 1 z z rz e z r r z r 39

Cylindrical Coordinates er e er e e er e, er, e, r e ur ur ur urer ure er, urer er e u rer u u e e u u u e e u e e r u e 1 1 c r r r r z u u e u e u e r r z z u u u u u e u e u e r r r r r r c r r c c z z 40

Cylindrical Coordinates er e er e e, er, e, r e 1 1 r r r r r rz r z r r cσ r r r z r zee z zrezer z eze zezez e e e e e e e e e e r r r r r r r r r ee r r ee 1 e e r e e r e e r z z z z r r r r rz rz r z r 1 rz r r r r z r ee r 1 1 r e e r e e z zr zr zr z r z z z The Laplacian is symmetric if the tensor is. z ee z 41

Spherical Coordinates 1 r z R cos, r Rsin, R z r, tan z R sin R cos cos,, sin, z z R r r R R sin cos z R z z R R R cos sin r R r r R R er cos sin 0 ez siner cosez sin cos 0 e er coser sinez e 0 0 1 e e e e R e e R e e, er, sin e, cos e, siner cose e sine cose r R 4

Spherical Coordinates cos sin 0 z R 1 s c sin cos 0 Q r R 0 0 1 1 1 r Rsin 1 1 us urer u e u e er e e R R R sin u R 1 u u R 1 ur u erer ere ee R R R R R sin R u ur 1 u cotu 1 u e er e e e e R R R R R sin u 1 u u 1 cot u R u eer ee ee R R R Rsin R 43

Spherical Coordinates 1 1 s u er e e urer u e u e R R R sin u cot u R u R 1 u 1 u R R R R R sin 1 1 s u er e e urer u e u e R R R sin 1 u cot u 1 u e R R Rsin 1 ur u u 1 u u R u e e Rsin R R R R R 1 1 1 1 s er e e er e e R R R sin R R R sin cot 1 1 R R R R R R sin R 44

ReReR Re Re R ere ReeR ee 1 1 σ R R R e e e e e e e e e e e R R Rsin Spherical Coordinates R R R R R ererer ereer ereer eerer eeer eeer eerer R R R R R R R 1 R R R 1 R R 1 R eee R ee er erere eree eree R R R R R R R R 1 R R 1 R R 1 R 1 R eere eee eee eere R R R R R R R R 1 1 1 1 R cot R R R R R e e e e e e erere ere e R R R R sin R R sin R 1 R cot R 1 R cot R R eree eere Rsin R Rsin R cot 1 cot 1 R e e e e e e Rsin R Rsin R 1 cot 1 R cot R R R e ere e e e Rsin R R R sin R 1 R R cot Rsin R eee 45

Spherical Coordinates Gradient of a tensor: Divergence of a tensor: σ e e e e e e e e e ij ij i j k i j k ij, k i j k xk xk ij σ e e e e e e e xk k ij i j k i j ij, i j xk σ contraction on the first and third index of σ 1 cot R R 1 R R R σ er R R R sin R 1 1 R R cot R e R R R sin R 1 1 R R cot R R R R sin R e 46

Spherical Coordinates e e R e e R e e, er, sin e, cos e, sine R cose e e R e, R e e e R e sin e sin, cos, R cose sin e R cos e e R R R R R R sin cot 1 1 s u urer ue u e u cotu R u u sur e R R R R sin u u R cot u su e R R sin R sin u cot u ur su e R sin R sin R sin R 47

Outline Scalar, vector and matrix( 标量 矢量 矩阵 ) Indicial notation and summation convention( 指标记法与求和约定 ) Kronecker delta( 克罗内克 δ) Levi-Civita symbol( 轮换记号 ) Coordinate transformation( 坐标变换 ) Tensor( 张量 ) Principal values and directions( 特征值与特征向量 ) Tensor algebra( 张量代数 ) Tensor calculus( 张量微分 ) Integral theorems( 积分定理 ) Cylindrical and Spherical Coordinates( 柱与球坐标 ) 48