1. Lecture 6: Elastic Deformation of laminates
At the end of this lecture you will have an understanding of:
Basic knowledge of tensors and their link to the mechanical properties of materials
Concepts behind the elastic deformation of a lamina
Concepts behind the elastic deformation of laminates
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2. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Scalar: physical quantity not related to any directions (temperature for example)
Vector: physical quantity related to 1 or more directions and is expressed in relation to a set of axis
Tensor of second rank: physical quantity which relates 2 vectors via and array of 9 components, each of which is associated with a pair of axis of a given coordinate system
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3. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Consider vectors:
𝑃 = 𝑃 1 , 𝑃 2 , 𝑃 3
𝑄 = 𝑄 1 , 𝑄 2 , 𝑄 3
We can define a tensor T which will relate 𝑃 and 𝑄
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4. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Or in matrix form
With Tij a second rank tensor
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5. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of axis
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6. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of axis
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7. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of axis
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8. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of vector components X3
X’3 P P’
X’1 X1 X2
X’2
𝑃 = 𝑃 1 , 𝑃 2 , 𝑃 3
𝑃′ = 𝑃′ 1 , 𝑃′ 2 , 𝑃′ 3
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9. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of components of the second rank tensor
We want to find the relation between the new and old components of tensor Tij, after changing the coordinate set of axis X3
X’3
aij
X’1 X1 X2
X’2
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10. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of components of the second rank tensor
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11. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Transformation of components of the second rank tensor
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12. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Stress tensor
𝜎 𝑖𝑗 𝜎 11 𝜎 12 𝜎 13 𝜎 21 𝜎 22 𝜎 23 𝜎 31 𝜎 32 𝜎 33
Principal/shear
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13. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Strain tensor
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14. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Elasticity a 4th rank tensor
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15. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Hooke’s law
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16. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Compliance/stiffness tensors
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17. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Compliance/stiffness tensors: effect of symmetry
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18. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Compliance/stiffness tensors: effect of symmetry
With Cij=Cji
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19. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Form of the compliance/stiffness tensors
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20. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Form of the compliance/stiffness tensors
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21. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Isotropic material
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22. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Isotropic material
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23. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Isotropic material
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24. Lecture 6: Elastic Deformation of laminates
Basics of tensors and mechanical properties materials
Isotropic material
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25. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Isotropic material
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26. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina
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27. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina
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28. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina
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29. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina
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30. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina
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31. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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32. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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33. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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34. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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35. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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36. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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37. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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38. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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39. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Single lamina loaded off axis
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40. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Engineering constant of a lamina
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41. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Engineering constant of a lamina
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42. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Engineering constant of a lamina
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43. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Examples
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44. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Examples
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45. Lecture 6: Elastic Deformation of laminates
Elastic deformation of a lamina
Examples
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46. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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47. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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48. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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49. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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50. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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51. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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52. Lecture 6: Elastic Deformation of laminates
Loading of a stack of plies
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53. Lecture 6: Elastic Deformation of laminates
Balanced laminates
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54. Lecture 6: Elastic Deformation of laminates
Balanced laminates
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55. Lecture 6: Elastic Deformation of laminates
Balanced laminates
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56. Lecture 6: Elastic Deformation of laminates
Balanced laminates and internal stresses
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57. Lecture 6: Elastic Deformation of laminates
Balanced laminates
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