Mechanical Properties Session 07-14

1. Mechanical PropertiesSession 07-14 Subject : S1014 / MECHANICS of MATERIALS Year : 2008

2. Mechanical Properties

3. What is Stress ? Much Work with limited time  High Stress

4. What is Stress ? Less Work with long time  Low Stress

5. What is Stress ? stress is according to strength and failure of solids. The stress field is the distribution of internal "tractions" that balance a given set of external tractions and body forces

6. Stress stress is according to strength and failure of solids. The stress field is the distribution of internal "tractions" that balance a given set of external tractions and body forces

7. Stress Look at the external traction T that represents the force per unit area acting at a given location on the body's surface.

8. Stress Traction T is a bound vector, which means T cannot slide along its line of action or translate to another location and keep the same meaning. In other words, a traction vector cannot be fully described unless both the force and the surface where the force acts on has been specified. Given both DF and Ds, the traction T can be defined as

9. Stress The internal traction within a solid, or stress, can be defined in a similar manner.

10. Stress Suppose an arbitrary slice is made across the solid shown in the above figure, leading to the free body diagram shown at right.

11. Stress Surface tractions would appear on the exposed surface, similar in form to the external tractions applied to the body's exterior surface.

12. Stress The stress at point P can be defined using the same equation as was used for T.

13. Stress Stress therefore can be interpreted as internal tractions that act on a defined internal datum plane. One cannot measure the stress without first specifying the datum plane.

14. Stress Surface tractions, or stresses acting on an internal datum plane, are typically decomposed into three mutually orthogonal components. One component is normal to the surface and represents direct stress. The other two components are tangential to the surface and represent shear stresses.

15. Stress What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses? Direct stresses tend to change the volume of the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus (which depends on the Young's modulus and Poisson ratio).

16. Stress What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses? Shear stresses tend to deform the material without changing its volume, and are resisted by the body's shear modulus.

17. Stress These nine components can be organized into the matrix:

18. Stress where shear stresses across the diagonal are identical (sxy = syx, syz = szy, and szx = sxz) as a result of static equilibrium (no net moment).

19. Stress This grouping of the nine stress components is known as the stress tensor (or stress matrix).

20. Stress The subscript notation used for the nine stress components have the following meaning:

21. What is Strain? A propotional dimensional change ( intensity or degree of distortion )

22. What is Strain measure? a total elongation per unit length of material due to some applied stress.

23. What are the types of strain ? • Elastic Strain • Plastic Deformation

24. Elastic Strain Transitory dimensional change that exists only while the initiating stress is applied and dissapears immediately upon removal of the stress.

25. Elastic Strain The applied stresses cause the atom are displaced the same amount and still maintain their relative geometic. When streesses are removed, all the atom return to their original positions and no permanent deformation occurs

26. Plastic Deformation a dimentional change that does not dissapear when the initiating stress is removed. It is usually accompanied by some elastic strain.

27. Elastic Strain & Plastic Deformation The phenomenon of elastic strain & plastic Deformation in a material are called elasticity & Plasticity respectively

28. Elastic Strain & Plastic Deformation Most of Metal material At room temperature they have some elasticity, which manifests itself as soon as the slightest stress is applied. Usually, they are also posses some plasticity , but this may not become apparent until the stress has been raised appreciablty.

29. Elastic Strain & Plastic Deformation Most of Metal material The magnitude of Plastic strain, when it does appear , is likely to be much greater than that of the elastic strain for a given stress increment

30. F Constitutive Solid material by force, F, at a point, as shown in the figure.

31. F d Constitutive Let the deformation at the the point be infinitesimal and be represented by vector d, as shown. The work done = F .d

32. F d y x z Constitutive For the general case: W = Fx dx i.e., only the force in the direction of the deformation does work.

33. F x Displacement Amount of Work done Constant Force If the Force is constant, the work is simply the product of the force and the displacement, W = Fx

34. Amount of Work done Linear Force: If the force is proportional to the displacement, the work is Fo F xo x Displacement

35. x F Strain Energy A simple spring system, subjected to a Force is proportional to displacement x; F=kx. Now determine the work done when F= Fo, from before: This energy (work) is stored in the spring and is released when the force is returned to zero

36. Hooke’s Law For systems that obey Hooke's law, the extension produced is directly proportional to the load: F=kx • where: • x = the distance that the spring has been stretched or compressed away from the equilibrium position, which is the position where the spring would naturally come to rest (usually in meters), • F = the restoring force exerted by the material (usually in newtons), and • K = force constant (or spring constant). The constant has units of force per unit length (usually in newtons per meter).

37. Hooke’s Law

38. Hooke’s Law

39. y a a a x Strain Energy Density Consider a cube of material acted upon by a force, Fx, creating stress sx=Fx/a2

40. Fx a x d Strain Energy Density y causing an elastic displacement, d in the x direction, and strain ex=d/a Where U is called the Strain Energy, and u is the Strain Energy Density.

41. u=1/2(300)(0.0015) N.mm/mm3 =0.225 N.mm/mm3 (a) For a linear elastic material

42. u=1/2(350)(0.0018) +350(0.0022) =1.085 N.mm/mm3 (b) Consider elastic-perfectly plastic

43. y a a a x y txy x gxy d = gxya Shear Strain Energy Consider a cube of material acted upon by a shearstress,txy causing an elastic shearstraingxy

44. Total Strain Energy for a Generalized State of Stress

45. D L A F Strain Energy for axially loaded bar F= Axial Force (Newtons, N) A = Cross-Sectional Area Perpendicular to “F” (mm2) E = Young’s Modulus of Material, MPa L = Original Length of Bar, mm

46. Da L A F Comparison of Energy Stored in Straight and Stepped bars (a)

47. L/2 L/2 nA A F Comparison of Energy Stored in Straight and Stepped bars Db (b)

48. What is Torsion ? an external torque is applied and an internal torque, shear stress, and deformation (twist) develops in response to the externally applied torque.

49. What is Torsion ? For solid and hollow circular shafts, in which assume the material is homogeneous and isotropic , that the stress which develop remain within the elastic limits, and that plane sections of the shaft remain plane under the applied torque.

50. What is isotropic ? is properties of the materials are the same in all directions in the material