LEC-3 CL601 Tensor algebra and its application in continuum mechanics.pptx

Constitutive modelling of geomaterials;
vector and tensor calculus;
vector calculus;
vector operations;
The Kronecker Delta;
Order of tensor;
M tech Geotech Engineering;
Geotechnical Engineering

1. Constitutive Modelling of Geomaterials Prof. Samirsinh P Parmar Mail: [email protected] Asst. Prof. Department of Civil Engineering, Faculty of Technology, Dharmasinh Desai University, Nadiad , Gujarat, INDIA Lecture: 3 : Tensor algebra and its application in continuum mechanics

2. 2. Tensor algebra and its application in continuum mechanics Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 2

3. Vector: recapitulation Any quantity which encountered in analytical description of physical phenomena, have magnitude and direction and satisfy the parallelogram law of addition known as vector Where e ˆ 1 ,e ˆ 2 ,e ˆ 3 are the independent orthogonal unit vectors (base vectors) and ( A 1 , A 2 , A 3 ) are the scalar component of A relative to the base vectors unit vector along A Magnitude of A Cartesian basis A Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 3

4. Properties of Vector (Addition) A + B = B + A (commutative) . ( A + B ) + C = A + ( B + C ) (associative) . A + = A (existence of zero vector) . A + (− A ) = (existence of negative vector) . Properties of Vector (scalar multiplication) α ( β A ) = ( αβ ) A (associative) . ( α + β ) A = α A + β A (distributive scalar addition) . α ( A + B ) = α A + α B (distributive vector addition) . ( 4 ) 1  A = A  1 = A ,  A = . Properties of Vector (Linear Independence) β 1 A 1 + β 2 A 2 + …+ β n A n = ( A 1 , A 2 … A n ) are the linearly dependent β 1 A 1 + β 2 A 2 + …+ β n A n ≠ ( A 1 , A 2 … A n ) are the linearly independent If ( A 1 , A 2 ) are the linearly dependent, they called collinear If ( A 1 , A 2, A 3 ) are the linearly dependent, they called coplanar Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 4

5. Properties of Vector (scalar and vector products) A  B  D  AB cos  = Scalar product (Commutative and holds distributive law) A  B  C  AB sin  e ˆ c = Vector product/cross product (non-­‐Commutative and holds distributive law) Few important triple products A  B  C  A  B  C A  B  C  C  A  B  B  C  A A   B  C    A  C  B   A  B  C Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 5

6. Change of basis Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 6

7. Vector calculus Scalar field r  x 1 e ˆ 1  x 2 e ˆ 2  x 3 e ˆ 3 denote the position vector of a point in space. A scalar field is a scalar valued function of position in space. A scalar field is a function of the components of the position vector, and so may be expressed as φ ( x 1 , x 2 , x 3 ). The value of φ at a particular point in space must be independent of the choice of basis vectors . e.g . temperature distribution throughout space, the pressure distribution in a fluid Vector field A vector field is a vector valued function of position in space. A vector field is a function of the components of the position vector, and so may be expressed as v (x 1 , x 2 , x 3 ) v = v 1 ( x 1 , x 2 , x 3 ) ˆe 1 + v 2 ( x 1 , x 2 , x 3 ) ˆe 2 + v 3 ( x 1 , x 2 , x 3 ) ˆe 3 e.g . the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point. Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 7

8. Vector calculus 1     2   3 grad   e ˆ 1  x  e ˆ 2  x  e ˆ 3  x        e ˆ 1  x  e ˆ 2  x   e ˆ 3  x 1 2 3 Divergence of a vector field div v   v 1   v 2   v 3  x 1  x 2  x 3 Curl of a vector field slope of the tangent of the graph of the function magnitude of a vector field's source or sink at a given point infinitesimal rotation of a 3D vector field Gradient of scalar field Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 8

9. MATRIX Operations on Cartesian components of vectors and tensors may be expressed very efficiently and clearly using index notation. Lower case Latin subscripts ( i, j, k…) have the range (1,2,3) Components of the vector A are, A 1 , A 2 , A 3 ⎧ A 1 ⎫ ⎪ ⎪ A  A i  ⎨ A 2 ⎬ ⎪ A 3 ⎪ ⎩ ⎭ S 11 S 12 S 13 S 22 S 23 S 32 S 33 ⎡ ⎣ ⎢ S  S i j  ⎢ S 21 ⎢ S 31 ⎤ ⎥ ⎥ ⎥ ⎦ A coordinate-­‐free, or component-­‐free , treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system… Wikipedia Indicial notation Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 9

10. i  1 3 L   A i B i  A 1 B 1  A 2 B 2  A 3 B 3  A i B i Dummy and free index The repeated index is called a dummy index because it can be replaced by any other symbol that has not already been used in that expression L  A i B i  A j B j  A k B k Summation convention (Einstein convention) If an index is repeated in a product of vectors or tensors, summation is implied over the repeated index. i  1 3 S 11 x 1  S 21 x 2  S 31 x 3 ⎧ c j   S ij x i  ⎨ S 12 x 1  S 22 x 2  S 32 x 3 ⎪ ⎪ ⎩  S ij x i 3 3 L    S ij S ij  ? i  1 j  1 S 13 x 1  S 23 x 2  S 33 x 3 3 c ?   s ij t jk  ? j  1 c ik  s ij d jk  s im d mk  s in d nk The same index (subscript) may not appear more than twice in a product of two (or more) vectors or tensors i i s  ? Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 10

11. The Kronecker Delta ⎡ 1  i j  ⎢ ⎤ ⎥ ⎥ ⎦ ⎥ Definition  ij  ⎨ ⎧ 1 i  j ⎩ i  j  ii  ? 3 0 0 ⎢ 1 0 ⎣ ⎢ 0 0 1  ij s jk  ? s ik  ij s ij  ? s 11  s 22  s 33 Consider an orthogonal coordinate system defined by unit vectors e 1 , e 2 , e 3 , such that: e 1 e 1  e 2 e 2  e 3 e 3  1 e 1 e 2  e 2 e 3  e 3 e 1   ij  e i e j  ij  e ˆ i  e ˆ j Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 11

12. The Permutation ( alternating tensor ) Symbol 1 if i, j, k in cyclic order and not repeated  1 if i, j, k not in cyclic order and not repeated otherwise ⎧ ⎪ ⎩ e ijk  ⎨ ⎪ e ˆ i  e ˆ j  e ijk e ˆ k In an orthonormal basis, the scalar and vector products can be expressed in the index form using the Kronecker delta and the alternating symbols: A  B   A i e ˆ i    B j e ˆ j   A i B j  ij  A i B i A  B   A i e ˆ i    B j e ˆ j   A i B j e ijk e ˆ k e ijk e imn   jm  kn   jn  km ( e -­‐ δ identity) Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 12

13. Partial differentiation is denoted by a comma followed by the index of differentiation. Notation for partial differentiation  f f ,i   x i and x j,i   x j  x i Now if f ( x i ) f ,i   x  f  f  x 1  f  x 2 i 1 i 2 i 3  f  x 3     x  x  x  x  x  x i  f , j x j,i where i = free index and j = dummy index  x i  x j   i j  Q i j  Q k l    i k j l Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 13

14. T ensor A mathematical object analogous to but more general than a vector, represented by an array of components that are functions of the coordinates of a space… (Wikipedia) Tensors are a further extension of ideas we already use when defining quantities like scalars and vectors. stress = force Magnitude and direction area Orientation of plane Complete definition of Stress must include its type, either tensile/compressive or shear . Thus, specification of stress at a point requires two vectors , one perpendicular to the plane on which the force is acting and the other in the direction of the force . Such an object is known as a dyad , or what we shall call a second-­‐order tensor . Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 14

15. T ensor A linear operator that transforms a vector or a tensor to another vector or tensor. Change of basis in index notation  i  Q ij A j Q ij   i A j Q =    Q ij (transformation matrix) is a second order tensor Creation of a second order tensor from the product of two vectors also known as dyadic product Index notation Tensor notation Gradient of a vector field another example of dyadic product to create tensor grad v    v Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 15

16. The rank (or order) of a tensor is defined by the number of directions (and hence the dimensionality of the array) required to describe it. Order of tensor Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 16

17. References : Wikipedia: continuum mechanics/stress/strain… Prof. Samirsinh P Parmar/Dept. of Civil Engg./DDU Nadiad/ INDIA 17

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