1. The Deformable Body

2. The flexible body
The elastic energy

3. Kinematics

4. The transplacement The transplacement: The deformation gradient:
Material line element
The transplacement:
The deformation gradient:

5. The deformaton gradient

6. The deformaton gradient

7. The length ratio
Length:
The length ratio:
The strain:

8. The length ratio

9. Locally length preserving transplacement
Local isometry
The rigid transplacement

10. The shear
The shear:

11. The shear

12. The volume ratio
The volume:
The volume ratio:

13. The volume ratio

14. The area ratio
The area:
The area ratio:

15. The normal vector

16. Polar factorization theorem

17. The displacement
The displacement gradient:

18. Strain tensors Deformation gradient: Left Cauchy-Green strain tensor:
Green-St. Venant strain tensor:
Infinitesimal strain tensor:

19. The Green-St. Venant strain tensor

20. Small displacement gradient

21. Principal directions of strain and principal stretches

22. Principal directions of strain and principal stretches

23. Representations of basic tensors

24. Strain, shear, volume and area ratios

25. The local deformation

26. Velocity and acceleration

27. Velocity gradient
Velocity gradient:
Stretching:
Spin:

28. Velocity gradient and divergence

29. The rigid transplacement

30. Summary

31. Mass
Mass density:

32. Conservation of Mass
Local balance of mass:

33. Equations of motion
Euler equations:
Contact force:
Body force:

34. Cauchy’s fundamental Lemma

35. Cauchy’s fundamental Theorem

36. Equations of motion, spatial description
Global equations of motion:
Local equations of motion:

37. Gauss theorem and the divergence

38. Referential description

39. The Piola-Kirchhoff stress tensor

40. Equations of motion, referential description
Global equations of motion:
Local equations of motion:

41. Gauss theorem and the divergence

42. Summary

43. Kinetic energy and the Power theorem
Net power:
Rigid body:

44. Power and Energy
External power:
Local equation of motion:

45. Power and Energy

46. The balance of mechanical energy
Specific internal energy:
Internal energy:
Total energy:
Heat supply (”heating”):
The balance of energy:

47. The first law of thermodynamics

48. The local balance of energy

49. The net power per unit volume
Stress tensor
Conjugated strain tensor
The second Piola-Kirchhoff stress tensor:
Relations between stress tensors:

50. Summary

51. with hyperelastic material
The elastic body
with hyperelastic material
Elastic potentials:
Constitutive equations:

52. Strain energy density
Strain energy density:

53. Elastic energy Total elastic energy: The internal energy:
Thermal energy neglected!

54. Balance of energy for an elastic body

55. The linear elastic body
Linear elastic material:
The elasticity tensor:
independent elasticities

56. The elasticity tensor
Elasticities:
36 independent elasticities

57. The elasticity tensor Symmetry: 21 independent elasticities
Positive definiteness:
Compliance tensor exists

58. The elastic energy
Green-St. Venant strain tensor:
Elastic energy:

59. Crystal systems There are in all 7 different Crystal systems:
- these have the following unit cells with associated number of elasicities
cubic (3), tetragonal (7,6), orthorombic (9),
triklinic (21), hexagonal (7,6,5), rhombohedral (9),
monoklinic (13),

61. Isotropic linear elastic material
Lame moduli:
Elastic potential:
Stress tensor:
Elastic energy:

62. Homogeneous isotropic linear elastic material
Lame moduli and independent of
Relation to Young’s modulus and Poisson’s ratio :

63. Example 7.2: The elastic bar
Present placement
Reference placement

64. Principal of virtual power
Euler equations (Eu1, Eu2):
Local equation of motion (Lem):
Internal forces zero system (Int):

65. Principal of virtual power
Virtual velocity field:

66. Virtual power Virtual power of external, internal and inertial forces:
are linear mappings:

67. The principal of virtual power
Note that:

68. Rigid virtual velocity field
Note that if the internal forces constitute a zero system then :

69. The principal of virtual power

70. The principal of virtual power

71. The principal of virtual power in continuum mechanics
(Lem 1)
(Lem 2)
Virtual velocity field:
Virtual powers:

72. The principal of virtual power in continuum mechanics

73. The principal of virtual power
Equivalences

74. The principal of virtual power in continuum mechanics

75. Exercise 2b:17

76. Exercise 2b:17 Solution
Disc: 𝒟
Shaft: 𝒮
Ground: G

77. Exercise 2b:17 Solution

78. Exercise 2b:17 Solution

79. Exercise 2b:17 Solution

80. Exercise 2b:17 Solution

81. Exercise 2b:17 Solution Angular velocity of shaft:
Angular velocity of disc:

82. Exercise 2b:17 Solution

83. Exercise 2b:17 Solution
Free body diagrams:

84. Exercise 2b:17 Solution
Equations of motion for Disc:

85. Exercise 2b:17 Solution

86. Moments of inertia for thin wheel

87. Exercise 2b:17 Solution
Equations of motion for Shaft:

88. Exercise 2b:17 Solution
Combining the equations by eliminating :

89. Exercise 2b:17 Solution
Neglecting inertia of the shaft:

90. Exercise 2b:17 Solution
Equations of motion in matrix representation:

91. Exercise 2b:17 Solution

92. Exercise 2b:17 Solution Equations of motion:
Five equations and five unknowns:

93. Exercise 2b:17 Solution

94. Exercise 2b:17 Solution

95. Exercise 3:3

96. Exercise 3:3

97. Exercise 3:9

98. Exercise 3:12

99. Exercise 3:15

100. Exercise 3:21

101. Exercise 3:21, continued