Vector and tensor calculus

Transcription

1 1 Vector and tensor calculus

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3 1.1 Examples Example 1.1 Consider three vectors a = 2 e 1 +5 e 2 b = 3 e1 +4 e 3 c = e 1 given with respect to an orthonormal Cartesian basis { e 1, e 2, e 3 }. a. Compute the length of vector a. a = = 29 b. Compute the unit vector in the direction of vector a. e = a/ a = (2 e 1 +5 e 2 )/ 29 = (2/ 29 e 1 +5/ 29 e 2 ) c. Compute the sum a+ b. a+ b = (2 e 1 +5 e 2 )+(3 e 1 +4 e 3 ) = 5 e 1 +5 e 2 +4 e 3 d. Compute the scalar product (inner product, dot product) a b. a b = (2 e 1 +5 e 2 ) (3 e 1 +4 e 3 ) = 2 e 1 3 e 1 +2 e 1 4 e 3 +5 e 2 3 e 1 +5 e 2 4 e 3 = = 6 e. Compute the dyadic product (tensor product, open product) a b. a b = (2 e 1 +5 e 2 )(3 e 1 +4 e 3 ) = 6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 f. Compute the cross product (vector product) a b. a b =(2 e 1 +5 e 2 ) (3 e 1 +4 e 3 ) = 2 e 1 3 e 1 +2 e 1 4 e 3 +5 e 2 3 e 1 +5 e 2 4 e 3 =0 8 e 2 15 e e 1 = 20 e 1 8 e 2 15 e 3 g. Compute the volume spanned by the three vectors a, b en c. V = ( a b) c = ((2 e 1 +5 e 2 ) (3 e 1 +4 e 3 )) e 1 = (20 e 1 8 e 2 15 e 3 ) e 1 = 20 Example 1.2 Consider a vector a, two second-order tensors A and B, and a fourth-order tensor 4 C a = 2 e 1 +5 e 2 A = 6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 B = 3 e 1 e 1 +4 e 3 e 1 4 C = 3 e 1 e 1 e 1 e 1 +4 e 1 e 3 e 1 e 1 +4 e 1 e 1 e 3 e 1 +8 e 1 e 3 e 3 e 1 given with respect to an orthonormal Cartesian basis { e 1, e 2, e 3 }. 3

4 a. Compute the square of tensor A, A 2. A 2 = A A =(6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) (6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) =36 e 1 e e 1 e e 2 e e 2 e 3 b. Compute the first, second and third invariants of tensor A. J 1 (A) = tr(a) = A 11 +A 22 +A 33 = = 6 J 2 (A) = 1 ( 2 tr 2 (A) tr(a 2 ) ) = 1 2 (62 36) = J 3 (A) = det(a) = det = c. Compute the deviatorc part of tensor A, A d. A d = A 1 3 tr(a)i =(6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) 1 3 6( e 1 e 1 + e 2 e 2 + e 3 e 3 ) d. Compute the transposed of tensor A, A T. A T = 6 e 1 e 1 +8 e 3 e e 1 e e 3 e 2 e. Compute the dot product A a. =4 e 1 e 1 2 e 2 e 2 2 e 3 e 3 +8 e 1 e e 2 e e 2 e 3 A a =(6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) (2 e 1 +5 e 2 ) =6 e 1 e 1 2 e 1 +6 e 1 e 1 5 e 2 +8 e 1 e 3 2 e 1 +8 e 1 e 3 5 e e 2 e 1 2 e e 2 e 1 5 e e 2 e 3 2 e e 2 e 3 5 e 2 = 12 e e 2 f. Compute the dot a A. a A =(2 e 1 +5 e 2 ) (6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) g. Compute A e 1. =2 e 1 6 e 1 e 1 +2 e 1 8 e 1 e 3 +2 e 1 15 e 2 e 1 +2 e 1 20 e 2 e 3 +5 e 2 6 e 1 e 1 +5 e 2 8 e 1 e 3 +5 e 2 15 e 2 e 1 +5 e 2 20 e 2 e 3 = 87 e e 3 A e 1 =(6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) e 1 h. Compute A : B. =6 e 1 e 1 e 1 +8 e 1 e 3 e e 2 e 1 e e 2 e 3 e 1 = 8 e 1 e e 2 e 2 A : B =(6 e 1 e 1 +8 e 1 e e 2 e e 2 e 3 ) : (3 e 1 e 1 +4 e 3 e 1 ) i. Compute 4 C : B. =6 e 1 e 1 : 3 e 1 e 1 +8 e 1 e 3 : 3 e 1 e e 2 e 1 : 3 e 1 e e 2 e 3 : 3 e 1 e 1 +6 e 1 e 1 : 4 e 3 e 1 +8 e 1 e 3 : 4 e 3 e e 2 e 1 : 4 e 3 e e 2 e 3 : 4 e 3 e 1 = = 50 4 C : B = (3 e 1 e 1 e 1 e 1 +4 e 1 e 3 e 1 e 1 +4 e 1 e 1 e 3 e 1 +8 e 1 e 3 e 3 e 1 ) : (3 e 1 e 1 +4 e 3 e 1 ) = 9 e 1 e e 1 e 3 Example 1.3 Show that vector n = 2 5 e e 3 is an eigenvector of the second-order tensor D = 6 e 1 e 1 + 4( e 1 e 3 + e 3 e 1 ). Compute the corresponding eigenvalue. 4

5 The eigenvalues λ and eigenvectors of a second-order tensor should satisfy: D n = λ n. For the given vector n and tensor D it holds 2 5 D n =(6 e 1 e 1 +4( e 1 e 3 + e 3 e 1 )) ( e ) e 3 = 12 e e e ( 2 =8 5 e ) e 3 = λ n 5 From here it can be concluded that, indeed, n is an eigenvector of D and the corresponding eigenvalue λ = 8. Example 1.4 Tensor E is given in spectral form as E = 25 n 1 n 1, with the eigenvector n 1 = 3 5 e e 2. Give the expression of the tensor E with respect to the Cartesian basis { e 1, e 2, e 3 }. E = 25 n 1 n 1 = 25 ( 3 5 e e 2)( 3 5 e e 2) = 9 e1 e e 2 e 2 +12( e 1 e 2 + e 2 e 1 ) Example 1.5 A vector field is given as u( x) = 3x 1 e 1 +4(x 2 +x 3 ) e 2 +2x 2 e 3, with (x 1,x 2,x 3 ) the coordinates with respect to the Cartesian basis { e 1, e 2, e 3 }. Compute the divergence u of this vector field. ( ) u = e 1 + e 2 + e 3 (3x 1 e 1 +4(x 2 +x 3 ) e 2 +2x 2 e 3 ) = 3+4 = 7 x 1 x 2 x 3 Example 1.6 Avector fieldisgivenwithrespecttoacylindrical basis{ e r (θ), e θ (θ), e z }as u = z e θ (θ). Compute the gradient u of this vector field. ( u = e r (θ) r + e θ(θ) 1 ) r θ + e z (z e θ (θ)) = z z r e e θ (θ) θ + e z e θ = z θ r e θ e r + e z e θ where it has been taken into account that e θ(θ) θ = e r (θ). Example 1.7 Are the following expressions correct for arbitrary differentiable vector fields φ en ψ? a. ( φ ψ) = ( φ) ψ + φ ( ψ) b. ( φ ψ) = ( φ) ψ +( ψ) φ c. ( φ ψ) = ( φ) ψ + φ ( ψ) In applying the product rule for the gradient operator acting on a product of vectors and tensors, it is important to respect all the contractions as well as the order of the different vectors composing each dyad (or quadrad) in each tensor. To assess the correctness of a product rule, it is therefore necessary to compare the sequence of all vector directions and contractions on the 5

6 left-hand side and compare them with those on the right-hand side. Product rules can be easily derived on this basis as well. The expression (a) preserves the order and type of multiplications correctly. Indeed, according to the left-hand side, the vector of the gradient operator should be multiplied using dot-product with vector φ. The direction of vector ψ should remain, multiplied by the scalar resulting from the dot-product. The expression (a) is therefore correct. Expression (b) is incorrect: in the second term on the right-hand side vectors φ and ψ are being multiplied with the dot product. Likewise, expression (c) is also incorrect: there the dot product between φ and ψ occurs in the first term on the right-hand side. 6

7 1.2 Exercises Exercise 1.1 The vector basis { e 1, e 2, e 3 } is right-handed and orthonormal. a. Determine e i for i = 1,2,3. b. Determine e i e j for i,j = 1,2,3. c. Determine e 1 e 2 e 3. d. Why is e 1 e 2 = e 3? Exercise 1.2 Consider the parallelepiped of Figure 1.1, spanned by the following vectors: a = e 1, b = 4 e2, c = 3 e 3 where { e 1, e 2, e 3 } is a right-handed orthonormal vector basis. c d e 3 a = e 1 e 2 b Figure 1.1: Parallelepiped a. Verify that the vector d is given by d = 4 e 2 +3 e 3 b. Verify that the vectors a and d are perpendicular, i.e. that the dot product of these vectors equals zero. c. Construct a vector m normal to the plane spanned by a and d. d. Determine the length m of m. e. Compute the unit vector in the direction of m. f. Compute the volume V of the parallelepiped spanned by a, b and c. Exercise 1.3 Consider a vector a = 2 e e 2 + e 3, where { e 1, e 2, e 3 } is a right-handed orthonormal vector basis. a. Determine the components a i of a with respect to this basis. Now consider a second basis { ε 1, ε 2, ε 3 }, given by: ε 1 = 1 2 ( e 1 + e 2 ), ε 2 = 1 2 ( e 2 e 1 ), ε 3 = e 3 b. Verify that these vectors also form a right-handed orthonormal basis. 7

8 c. With respect to { ε 1, ε 2, ε 3 }, the vector a can be written as a = α 1 ε 1 +α 2 ε 2 +α 3 ε 3. Determine the components α i. d. Give the column representations ã = [a 1 a 2 a 3 ] T and α = [α 1 α 2 α 3 ] T of a with respect to { e 1, e 2, e 3 } and { ε 1, ε 2, ε 3 } respectively. Does a column of components uniquely define a vector? Exercise 1.4 The basis { e 1, e 2, e 3 } is right-handed and orthonormal. The vectors a 1, a 2, a 3 and b are given by: a 1 = 4 e 1 +3 e 2, a 2 = 3 e 1 4 e 2, a 3 = a 1 a 2, b = 2 e1 +3 e 2 + e 3 a. Determine the expression for a 3 in terms of e 1, e 2 en e 3. b. Compute a i for i = 1,2,3. c. Determine the volume of the parallelepiped spanned by a 1, a 2 and a 3. d. Compute the angle between a 1 and a 2. e. Determine the vectors α i satisfying a i = a i α i (i = 1,2,3) Does { α 1, α 2, α 3 } form a right-handed, orthonormal vector basis? f. Determine the matrix representation of b with respect to each of the bases { e 1, e 2, e 3 }, { a 1, a 2, a 3 } and { α 1, α 2, α 3 }. g. Verify that a 1 a 2 b = a 1 a 2 b = a 2 b a 1. Exercise 1.5 Consider the orthonormal vector basis { e 1, e 2, e 3 } and three vectors a = 4 e 1 +3 e 2 e 3, b = 6 e2 e 3, c = 8 e 1 e 3 a. What is the condition for two vectors to be linearly independent? b. What is the condition for three vectors to be linearly independent? c. Are the vectors a, b and c independent? If not, what is the relation between them? Exercise 1.6 With respect to a right-handed orthonormal vector basis { e 1, e 2, e 3 } we define vectors a, b and p as: a = e 1 + e 2, b = e2 +2 e 3, p = e 1 The tensor A is defined as the dyad a b. a. Show that A is given in terms of the base vectors by A = e 1 e 2 +2 e 1 e 3 + e 2 e 2 +2 e 2 e 3 b. Determine the transpose A T of A. c. Compute the dot products A p and A T p; verify that the latter product is equal to p A 8

9 d. Determine the components A ij = e i A e j of A with respect to { e 1, e 2, e 3 }. e. Give the matrix representation Ā of A with respect to { e 1, e 2, e 3 }. f. Compute the invariants J 1 (A), J 2 (A) and J 3 (A) of A. g. Compute the deviatoric part A d. h. Can the inverse tensor A 1 be computed and why? Exercise 1.7 The basis { e 1, e 2, e 3 } is orthonormal and right-handed. The tensor A satisfies: A a = m( a n) with m = ( e 1 + e 2 ), n = e 1 + e 3 a. Show that A is given in terms of basis vectors dyads by A = e 1 e 1 + e 1 e 3 e 2 e 1 + e 2 e 3 b. Compute tr(a). c. Compute the second invariant J 2 (A). d. Compute the determinant of A. e. Determine A 2 = A A, A T A and A A T. f. Compute tr(a 2 ). g. Is tr(a 2 ) = tr 2 (A) always true? Exercise 1.8 Compute the invariants J 1 (A), J 2 (A), J 3 (A) and the deviator A d of the following tensors A: a. A = I b. A = e 1 e 1 + e 2 e 2 c. A = e 1 e 2 e 2 e 1 + e 3 e 3 d. A = e 1 e 1 + e 2 e 2 + e 3 e 3 +2( e 1 e 2 e 2 e 1 )+3( e 2 e 3 e 3 e 2 )+4( e 3 e 1 e 1 e 3 ) Exercise 1.9 Consider the symmetric tensor A defined as: A = 3 e 1 e 1 e 1 e 2 e 2 e 1 +3 e 2 e 2 +2 e 3 e 3 a. Give the matrix representation Ā of A with respect to { e 1, e 2, e 3 }. b. Compute the inverse tensor A 1. c. Compute the eigenvalues λ i of A. d. Determine theeigenvectors N i ; normalisethem such that N i = 1. Isthis set of eigenvectors unique? e. Give the spectral representation of tensor A. f. Give the spectral representation of tensor A 1. 9

10 Exercise 1.10 Consider the following scalar function a: a( x) = a 0 +a 1 x x where x represents the position vector and a 0 and a 1 are constants. a. Show that the infinitesimal difference da = a( x+d x) a( x) can be linearised as da = 2a 1 x d x. b. The gradient of a satisfies: da = d x a Show by comparison of this expression with the one obtained under a. that the gradient of a equals a = 2a 1 x c. Verify that the function a can be written with respect to a Cartesian basis { e x, e y, e z } as: a( x) = a 0 +a 1 (x 2 +y 2 +z 2 ) d. Compute the gradient a in this Cartesian basis and verify that the result is consistent with the expression obtained under b. e. Verify that the same result is obtained in a cylindrical basis { e r, e θ, e z }. Exercise 1.11 Consider a cylindrical base { e r, e θ, e z }. a. Show that the base vectors satisfy e r θ = e θ e θ θ = e r Write the cylindrical base vectors in terms of a fixed, Cartesian base for this purpose. b. Using the relations obtained under a., show that in cylindrical coordinates x = I. Does this relation also hold in other types of bases? Exercise 1.12 Consider the cylindrical basis { e r, e θ, e z } in which position is indicated by a position vector x = r e r (θ)+z e z. A vector u is defined in this basis as u = γrz e θ (θ) with γ a constant. a. Compute the gradient u of u 10

11 Exercise 1.13 Vector a, second-order tensor B and fourth-order tensor 4 C are given in a right-handed, orthonormal basis { e 1, e 2, e 3 } as according to a = e 1 + e 2 B = e 1 e 1 2 e 1 e 2 2 e 2 e 1 + e 2 e 2 4 C = e 1 e 1 e 1 e 1 4 e 1 e 1 e 2 e 2 4 e 2 e 2 e 1 e 1 + e 2 e 2 e 2 e 2 Compute the following products: a. 4 C a b. 4 C : B c. 4 C : 4 C Exercise 1.14 Demonstrate that for all v: v = ( v) : I = tr( v) Exercise 1.15 Assuming that v, w and B are given differentiable functions of the position vector x, show that: a. ( v w) = ( v) w + v w b. (B v) = B v + v B T c. (B v) = ( B) v +B : ( v) T Exercise 1.16 Consider two orthonormal bases { e 1, e 2, e 3 } and { ε 1, ε 2, ε 3 } which do not coincide. Let the relation between the base vectors be as follows: ε i = A e i, i = 1,2,3 a. Determine the dyadic products ε 1 e 1, ε 2 e 2 and ε 3 e 3 in terms of A and the base vectors e 1, e 2 and e 3. b. Show by adding the three results obtained above that the tensor A is given by A = ε 1 e 1 + ε 2 e 2 + ε 3 e 3 c. What is the effect of applying A to a arbitrary vector p, i.e. A p? And of A T p? Exercise 1.17 Given a right-handed orthonormal vector basis { e 1, e 2, e 3 }, what is the effect of the following tensors on a vector a = a 1 e 1 +a 2 e 2 +a 3 e 3? a. e 1 e 1 b. e 1 e 1 + e 2 e 2 c. e 1 e 1 + e 2 e 2 + e 3 e 3 11

12 d. e 1 e 2 e 2 e 1 + e 3 e 3 e. e 1 e 1 e 2 e 2 + e 3 e 3 Exercise 1.18 Thebasis{ e 1, e 2, e 3 }is orthonormal andright-handed. Thetensor Ais given by A = e 2 e 1 e 1 e 2. Show that for all v: A v = e 3 v What kind of tensor is A? 12

13 A Answers

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15 Chapter a. e i = 1 b. e i e j = δ ij c. e 1 e 2 e 3 = 1 d. because the basis is right-handed 1.2 c. m = a d = 4 e 3 3 e 2 (or another vector in this direction) d. m = 5 e. e m = 4 5 e e 2 f. V = a. a 1 = 2, a 2 = 3, a 3 = 1 b. ε 1 ε 1 = ε 2 ε 2 = ε 3 ε 3 = 1, ε 1 ε 2 = ε 2 ε 3 = ε 3 ε 1 = 0 ε 1 ε 2 = ε 3, ε 2 ε 3 = ε 1, ε 3 ε 1 = ε 2 c. α 1 = 5 2, α 2 = 1 2, α 3 = 1 d. ã = [ ] T, α = [ 5 2 ] T; no, the vector basis must be given. 1.4 a. a 3 = 25 e 3 b. a 1 = 5, a 2 = 5, a 3 = 25 c. V = 625 d. φ = 1 2 π e. α 1 = 4 5 e e 2, α 2 = 3 5 e e 2, α 3 = e 3 ; yes f. b e = [ ] T, b a = 1 [ ] T, b α = 1 [ ] T a. a b 0 b. a b c 0 c. no; 2 a b c = b. A T = e 2 e 1 + e 2 e 2 +2 e 3 e 1 +2 e 3 e 2 c. A p = 0, A T p = e 2 +2 e 3 d. A 12 = 1, A 13 = 2, A 22 = 1, A 23 = 2, A 11 = A 21 = A31 = A 32 = A 33 = e. Ā =

16 f. J 1 (A) = 1, J 2 (A) = 0, J 3 (A) = 0 g. A d = 1 3 ( e 1 e 1 + e 3 e 3 )+ e 1 e 2 +2 e 1 e e 2 e 2 +2 e 2 e 3 h. No, since A is a singular tensor. 1.7 b. tr(a) = 1 c. J 2 (A) = 0 d. det(a) = 0 e. A 2 = e 1 e 1 e 1 e 3 + e 2 e 1 e 2 e 3 A T A = 2 e 1 e 1 2( e 1 e 3 + e 3 e 1 )+2 e 3 e 3 A A T = 2 e 1 e 1 +2( e 1 e 2 + e 2 e 1 )+2 e 2 e 2 f. tr(a 2 ) = 1 g. no 1.8 a. J 1 (A) = 3, J 2 (A) = 3, J 3 (A) = 1, A d = 0 b. J 1 (A) = 2, J 2 (A) = 1, J 3 (A) = 0, A d = 1 3 ( e 1 e 1 + e 2 e 2 ) 2 3 e 3 e 3 c. J 1 (A) = 1, J 2 (A) = 1, J 3 (A) = 1, A d = 1 3 ( e 1 e 1 + e 2 e 2 )+ 2 3 e 3 e 3 + e 1 e 2 e 2 e 1 d. J 1 (A) = 3, J 2 (A) = 32, J 3 (A) = 30, A d = 2( e 1 e 2 e 2 e 1 )+3( e 2 e 3 e 3 e 2 )+4( e 3 e 1 e 1 e 3 ) a. Ā = b. A 1 = 3 8 e 1 e e 1 e e 2 e e 2 e e 3 e 3 c. λ 1 = 4, λ 2 = λ 3 = 2 d. N1 = 1 2 ( e 1 e 2 ), N2 = e 3, N3 = 1 2 ( e 1 + e 2 ); no e. A = 4 N 1 N1 +2 N 2 N2 +2 N 3 N3 f. A 1 = 1 4 N 1 N N 2 N N 3 N d. a = 2a1 x e x +2a 1 y e y +2a 1 z e z = 2a 1 x e. a = 2a1 r e r +2a 1 z e z = 2a 1 x 1.11 b. yes 1.12 a. u = γz( er e θ e θ e r )+γr e z e θ 1.13 a. 4 C a = e 1 e 1 e 1 4 e 1 e 1 e 2 4 e 2 e 2 e 1 + e 2 e 2 e 2 b. 4 C : B = 3 e 1 e 1 3 e 2 e 2 16

17 c. 4 C : 4 C = 17 e 1 e 1 e 1 e 1 8 e 1 e 1 e 2 e 2 8 e 2 e 2 e 1 e e 2 e 2 e 2 e a. ε 1 e 1 = A e 1 e 1, ε 2 e 2 = A e 2 e 2, ε 3 e 3 = A e 3 e 3 c. rotation; rotation in the opposite direction 1.17 a. projection on e 1 b. projection on the e 1 - e 2 plane c. no effect d. rotation around e 3 e. mirroring in the e 1 - e 3 plane 1.18 rotation and projection tensor 17