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Dr. Wang Xingbo Fall ， 2005

Mathematical & Mechanical Method in Mechanical Engineering. Dr. Wang Xingbo Fall ， 2005. Mathematical & Mechanical Method in Mechanical Engineering. Introduction to Tensors. Concept of Tensors Tensor Algebra Tensor Calculus Application of Tensors. Mathematical & Mechanical

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Dr. Wang Xingbo Fall ， 2005

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  1. Mathematical & Mechanical Method in Mechanical Engineering Dr. Wang Xingbo Fall，2005

  2. Mathematical & Mechanical Method in Mechanical Engineering Introduction to Tensors • Concept of Tensors • Tensor Algebra • Tensor Calculus • Application of Tensors

  3. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors A coordinate transformation in an n-dimensional space

  4. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors T is a quantity with ns components represented by one of the following three forms

  5. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the first one and transformed by T is called a contravariant tensor of order s.

  6. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the second one and transformed by T is called a covariant tensor of order s.

  7. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Components of T are represented by the third one and transformed by T is called a mix-variant tensor of order s.

  8. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor product Let U, V be two vector spaces of dimension m, n, Tensor product of U and V is an mn–dimensional vector space W denoted by W=UV. Symbol  is used to denote a tensor product

  9. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Symbol  is used to denote a tensor product

  10. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Since UV is an mn-dimensional space, it has mn basis vectors. All pairs (i,j) produce exactly mn pairs of (ui,vj) It often uses symbol to denote the basis of W=UV. The elements of W=UV are

  11. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Covariant tensor basis are defined by tensor product of covariant vector basis

  12. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Contravariant tensor basis are defined by tensor product of contravariant vector basis

  13. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Tensor basis Mix-variant tensor basis are defined by tensor product of covariant and contravariant vector basis

  14. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors A vector is a first-order tensor Take s =1, the two forms of components The transformations This is what a contravariant vector or a covariant vector is!

  15. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor contravariant tensor T can be represented by The transformations

  16. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor covariant vector T can be represented by The transformations

  17. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor Mix covariant vector T can be represented by The transformations

  18. Mathematical & Mechanical Method in Mechanical Engineering Concept of Tensors Second-order tensor Matrix Form

  19. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Sample is a covariant tensor of order 2

  20. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Quantity to illustrate strain in an elastic material is a covariant tensor of order 2 Let A, B be two points in an elastic body and let .

  21. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor After deformation

  22. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor

  23. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor

  24. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor

  25. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor Let us change the Cartesian coordinate transformation O toAssume

  26. Mathematical & Mechanical Method in Mechanical Engineering Second order tensor

  27. Mathematical & Mechanical Method in Mechanical Engineering Strain tensor

  28. Mathematical & Mechanical Method in Mechanical Engineering Tensor algebra Addition and Subtract of Two Tensors Contraction of Tensors : Forcing one upper index equal to a lower index and invoking the summation convention A special operation on mix-variant tensors

  29. Mathematical & mechanical Method in Mechanical Engineering Geometric Meanings of Cross Product Contraction of Tensors

  30. Mathematical & mechanical Method in Mechanical Engineering Out Product of Two Tensors The product of two tensors is a tensor whose order is the sum of the orders of the two tensors, and whose components are products of a component of one tensor with any component of the other tensor.

  31. Mathematical & mechanical Method in Mechanical Engineering Out Product of Two Tensors • A=AikEik , B=BlmElm, • Ciklm= AikBlm

  32. Mathematical & mechanical Method in Mechanical Engineering Inner product of Two Tensors Multiplying two tensors and then contracting the product with respect to indices belonging to different factors

  33. Mathematical & mechanical Method in Mechanical Engineering Quotient Law Assume are two arbitrary tensors. IF Then A is a tensor

  34. Mathematical & mechanical Method in Mechanical Engineering Some Useful and Important Tensors • Metric tensors

  35. Mathematical & mechanical Method in Mechanical Engineering Metric tensors In 3-dimensional space

  36. Mathematical & mechanical Method in Mechanical Engineering The Alternating Tensor of Third Order εjkl= 1, if j, k, l cyclic permutation of 1, 2, 3 εjkl= -1, if j, k, l cyclic permutation of 2, 1, 3 εjkl= 0, otherwise.

  37. Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential be a vector field in covariant frame-vectors dA is called absolute differential or covariant differential of vector field A

  38. Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential Absolute differential of A is composed of two parts reflects the relationship of the contravariant components changing with spatial position

  39. Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential reflects that of frame-vectors components changing with spatial position

  40. Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential Let then absolute derivative represented by contravariant components

  41. Mathematical & mechanical Method in Mechanical Engineering Absolute Derivatives & Differential We also can derive absolute derivative represented by covariant components

  42. Mathematical & Mechanical Method in Mechanical Engineering Absolute Derivatives & Differential the absolute derivative of a vector field can be represented by either contravariant components or covariant components.

  43. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols are vectors , be linear combination of frame-vectors

  44. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols the first kind of Christoffel symbols the second kind of Christoffel symbols

  45. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols

  46. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols Partial Derivative Operator

  47. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols

  48. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols Relationships between and the metric tensor

  49. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols

  50. Mathematical & Mechanical Method in Mechanical Engineering Derivatives of Frame-vectors and Christoffel Symbols The Christoffel symbols are only symbols but not components of any tensor though they look like the form of tensor-components

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