1. Continuum Mechanics, part 1
cm_part1
Continuum Mechanics, part 1
Background, notation, tensors

2. The Concept of Continuum
The molecular nature of the structure of matter
is well established.
In many cases, however, the individual molecule
is of no concern.
Observed macroscopic behavior is based on
assumption that the material is continuously distributed
throughout its volume and completely fills the space.
The continuum concept of matter is the fundamen-
tal postulate of continuum mechanics.
Adoption of continuum concept means that field
quantities as stress and displacements are expressed
as piecewise continuous functions of space coordinates
and time.

4. cm0001b

5. cm0002a

6. cm0002b

7. cm0003a

8. Cm0003b

9. cm0004a

10. cm0004b

11. cm0005ab

12. cm0006

13. Tensors, vectors, scalars
General tensors .. transformation in curvilinear systems
Cartesian tensors .. transformation in Cartesian systems
Tensors are classified by the rank or order according to the particular form of the transformation law they obey.
In a three-dimensional space (n = 3) the number of components of a tensor is nN, where N is the rank (order) of that tensor.
Cm0007a

14. Cm0007b

15. Cm0008a

16. Cm0008b

17. cm0009a

18. cm0009b

19. Cm0010a

20. cm0010b

21. Summation rule
When an index appears twice in a term, that index is understood to take on all values of its range, and the resulting terms summed.
So the repeated indices are often referred to as dummy indices, since their replacement by any other letter, not appearing as a free index, does not change the meaning of the term in which they occur.

22. Vectors will be denoted in a following way Symbolic or Gibbs notation
Indicial notation; a component or all of them
Matrix algebra notation
Note
Tensor indicial notation does not distinguish between row and column vectors
cm0011b

23. cm0012a

24. Inverse transformation
Combining forward and inverse transformations for an arbitrary vector
cm0012b

25. The coefficient
gives the symbol or variable which is equal
either to one or to zero according to whether
the values i and k are the same or different.
This may be simply expressed by
i.e. by Kronecker delta or unit matrix

26. cm0013b

27. cm0014a

28. cm0014b

29. Second order tensor transformation
cm0015a

30. cm0015b

31. cm0016a

32. cm0016b

33. cm0016c

34. cm0017a

35. cm0017b

36. cm0018a

37. cm0018b

38. cm0019a

39. cm0019b

40. cm0020a

41. cm0020b

42. cm0021a

43. cm0021b

44. cm0022a

45. cm0022b

46. cm0023a

47. cm0023b

48. cm0024a

49. cm0024b

50. cm0025a

51. cm0025b