Chapter 2: Elasticity and Plasticity

1. Chapter 2:Elasticity and Plasticity

2. Tensile Strength Testing Machine

3. Elasticity Stress–strain curves in an elastic regime. (a) Typical curve for metals and ceramics. (b) Typical curve for rubber.

4. Strain Energy Density

5. Shear Stress and Strain (a) Specimen subjected to shear force. (b) Strain undergone by small cube in shearregion. (c) Specimen (cylinder)subjected to torsion by a torque T.

6. Poisson’s Ratio In an isotropic material, ε11 is equal to ε22. (a) Unit cube being extended in direction Ox3. (b) Unit cube in body subjected to tridimensional stress; only stresses on the three exposed faces of the cube are shown.

7. More Complex State of Stresses

8. Mohr Circle (a) Biaxial (or bidimensional) state of stress. (b) Mohr circle and construction of general orientation 0X1 X2 (c) Mohr circle and construction of principal stresses and maximum shear stresses (Method I).

9. Mohr Circle

10. Pure Shear

11. Anisotropic Effects

12. Elastic Modulus

13. Elastic Compliance and Stiffness Matrix

14. Anisotropy of Cubic System Zener’s anisotropy ratio Young’s modulus Shear modulus Bulk modulus Poisson’s ratio Lame constants

15. Young’s Modulus and Shear Modulus of Monocrystalline Cu

16. Young’s Modulus and Shear Modulus Monocrystalline Zirconia

17. Tridimensional Polar Plot for Zirconium

18. Polycrystal Voigt average: assume strain is same everywhere Reuss average: assume stress is same everywhere

19. Porosity on Young’s Modulus Watchman and Mackenzie:

20. Microcracks vs. Young’s Modulus

21. Microcracks vs. Young’s Modulus (cont’d) 1973: Salganik model 1974: O’connel & Budiansky model

22. Elastic Properties of Polymers

23. Viscoelasticity n=0: plastic n=1: linear viscous (Newtonian) n: power law Viscosity coefficient Fluidity:

24. Viscoelasticity (cont’d) Tensile storage modulus Tensile loss modulus

25. Rubber Elasticity From thermodynamics, one can derive: Extension ratio:

26. Elastic Properties of Biological Materials (a) Stress–strain response of human vena cava: circles-loading; squares-unloading. (Adapted from Y. C. Fung, Biomechanics (New York: Springer, 1993),p. 366.) (b) Representation of mechanical response in terms of tangent modulus (slope of stress–strain curve) vs. stress. (Adapted from Y. C. Fung. Biomechanics, New York: Springer,1993), p. 329.)

27. Blood Vessels Dimensions

28. Residual Stresses in Arteries

29. Cartilage Fiber Network

30. Mesostructure of Cartilage (a) Mesostructure of cartilage (consisting of four zones) showing differences in structure as a function of distance from surface; the bone attachment is at bottom. (From G. L. Lucas, F. W. Cooke, and E. A. Friis, A Primer on Biomechanics (New York: Springer, 1999), p. 273.) (b) Cross-section of human cartilage showing regions drawn schematically in (a). (Courtesy of K. D. Jadin and R. I. Shah.)

31. Mechanical Behavior of Superficial Zone of Cartilage Stress–strain curve for samples from the superficial zone of articular cartilage. Samples were cut parallel and perpendicular to collagen fiber orientation. (From G. E. Kempson, Mechanical Properties of Articular Cartilage. In Adult Articular Cartilage, ed. M. A. R. Freeman (London: Sir Isaac Pitman and Sons Ltd., 1973), pp. 171–228.)

32. Mechanical Properties of a DNA

33. Stretching Force vs. Relative Extension for a DNA Molecule

34. Stresses Acting on a Thin Film Effect of stresses acting on thin film on bending of substrate; (a) tensile stresses in thin film; (b) compressive stresses in thin film.

35. Elastic Constant and Bonding Two atoms with an imaginary spring between them; (a) equilibrium position; (b) stretched configuration under tensile force; (c) compressed configuration under compressive force.

36. Attraction and Repulsion Between Two Atoms (a) Interaction energies (attractive and repulsive terms) as a function of separation; (b) Force between two atoms as a function of separation; notice decrease in slope as separation increases.