1. the strength of common materials is actually dictated not so much by bond strength but by something else: Defects

3. NANOSCALE Vs MICROSCALE Griffith’s experiments with glass fibers (1921) FIBER DIAMETER (micron) Strength of bulk glass: 170 MPa Extrapolates to 11 GPa 1 2 3 TENSILE STRENGTH (GPa) 0 20 40 60 80 100 120 0

5. INGLIS, 1913

6. A If x = a (point A), then  YYlocal = 3  0 If the local stress reaches the theoretical strength, then the applied stress (  0 ) at failure is  0 =  th /3 And assuming  th ≈ E/10, we get, at failure:  0  E/30 A more realistic situation is that of a sharper crack:  0  0

9. The strength of fibers is statistical

11. Probability of occurrence of a critical defect (F(V)= Probability of failure) against size for a given defect concentration At very small volume, low P of occurrence of a critical defect – Thus: strength tends to be very high At larger volume, F(V) climbs rapidly: A plateau is reached where size has no more effect.

13. Density function: As  increases, the distribution is more narrow, and  is proportional to the average of the distribution

22. By definition, the (Helmholtz) free energy of the body is  = E-TS Thus: So that for an isothermal deformation process (T = constant), we have: Therefore, we need to know the free energy per unit volume,  , as a function of  ik

23. This is easily calculated: since we have small deformations,  can be expanded in a Taylor series: where  0 is the free energy of the undeformed body, and the  ’s are given as follows:

24. By differentiating, we obtain: And we know that this is equal to  ik (for an isothermal process). If there is no deformation, there are no internal stresses in the body, thus  ik = 0 for  ik = o, from which we obtain  = 0. Thus, no linear term in the expansion of  in powers of  ik : by limiting the expansion to the second order:  ~  2

25. And we can therefore compute the stress tensor in terms of the strain tensor: or This very simple expression provides a linear dependence between stress and strain: it is the basic form of Hooke’s law ! Also, remember the connection between Young’s modulus and the potential from the previous class?

26. A common general form (valid for anisotropic bodies) of Hooke’s law is the following: Where  ij and  kl are 2d rank tensors and C ijkl is a 4 th rank tensor with 3x3x3x3 = 81 components [or 9 stress components x 9 strain components = 81]. The C ijkl are called the elastic constants. Historical parenthesis: Robert Hooke’s legacy

28. “ ut tensio, sic vis” (load ~ stretch) UNDER TENSION:

29. Similarly, under bending: load ~ deflection

30. Under shear… … and torsion: load ~ shear deformation load ~ angular deformation

31. Thus, in all cases, Hooke observed: The ratio applied force/distortion is a constant for the material. This is (almost) Hooke’s Law

32. Hooke’s Law This definition is valid whatever the mode of testing (tension, bending, torsion, shearing, hydrostatic compression, etc, and a specific modulus is then defined)

33. The stress-strain curve

34. A general stress-strain curve Elastic (fully reversible) Plastic (irreversible)

35. Comparing various stress-strain curves

38. 2 5 Isotropic 4 5 Transversely isotropic 4 5 Orthorombic [Orthotropic] 6 9 Monoclinic 6 9 Triclinic 2D 2 12 Isotropic 5 12 Transversely isotropic 9 12 Orthorombic [Orthotropic] 13 20 Monoclinic 21 36 Triclinic 3D Number of independent coefficients Number of nonzero coefficients Class of Material

39. (From J.F. Nye, ‘Physical Properties of Crystals’) Form of the C ijkl matrix

41. Notations:  ij = C ijkl  kl (i, j, k, l = 1,2,3)  ij = S ijkl  kl Historical paradox: The C ijkl are called the Stiffness components The S ijkl are called the Compliance components Contracted notations in the mechanics of composites:

42. Contracted notations in the mechanics of composites:

45. Transversely isotropic lamina (5 constants)

46. Isotropic lamina (2 constants)

49. Elementary experiments

50. Remember: an orthotropic lamina (9 constants)

52. All other elementary experiments provide similar links. Eventually we obtain what we wanted, the stress-strain relations in terms of engineering constants (E,  , G):

54. Finally, the connection between the stiffness constants C ij and the compliance constants S ij are as follows: