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

2. Mechanical Properties
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3. What is Stress ? Much Work with limited time  High Stress
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4. What is Stress ?
Less Work with long time  Low Stress
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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
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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
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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.
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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
                                          
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9. Stress
The internal traction within a solid, or stress, can be defined in a similar manner.
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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.
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11. Stress
Surface tractions would appear on the exposed surface, similar in form to the external tractions applied to the body's exterior surface.
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12. Stress
The stress at point P can be defined using the same equation as was used for T.
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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.
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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.
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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).
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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.
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17. These nine components can be organized into the matrix:
Stress
These nine components can be organized into the matrix:
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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).
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19. Stress
This grouping of the nine stress components is known as the stress tensor (or stress matrix).
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20. Stress
The subscript notation used for the nine stress components have the following meaning:
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21. What is Strain?
A propotional dimensional change ( intensity or degree of distortion )
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22. What is Strain measure?
a total elongation per unit length of material due to some applied stress.
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23. What are the types of strain ?
Elastic Strain
Plastic Deformation
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24. Elastic Strain
Transitory dimensional change that exists only while the initiating stress is applied and dissapears immediately upon removal of the stress.
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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
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26. It is usually accompanied by some elastic strain.
Plastic Deformation
a dimentional change that does not dissapear when the initiating stress is removed.
It is usually accompanied by some elastic strain.
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27. Elastic Strain & Plastic Deformation
The phenomenon of elastic strain & plastic Deformation in a material are called elasticity & Plasticity respectively
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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.
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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
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30. Solid material by force, F, at a point, as shown in the figure.
Constitutive F
Solid material by force, F, at a point, as shown in the figure.
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31. The work done = 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
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32. W = Fx dx Constitutive For the general case:z y
For the general case:
W = Fx dx
i.e., only the force in the direction of the deformation does work.
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33. Constant Force W = Fx Amount of Work done
If the Force is constant, the work is simply the
product of the force and the displacement,
W = Fx F
Displacement x
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34. Amount of Work done
Linear Force:
If the force is proportional to the displacement, the work is Fo F xo x
Displacement
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35. Strain Energy F x
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
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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).
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37. Hooke’s Law
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38. Hooke’s Law
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39. Strain Energy Density
Consider a cube of material acted upon by a force, Fx, creating stress sx=Fx/a2 y x a
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40. Strain Energy Density y
causing an elastic displacement, d in the x direction, and strain ex=d/a x a Fx d
Where U is called the Strain Energy, and u is the Strain Energy Density.
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41. (a) For a linear elastic material
u=1/2(300)(0.0015) N.mm/mm3
=0.225 N.mm/mm3
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42. (b) Consider elastic-perfectly plastic
u=1/2(350)(0.0018)
+350(0.0022)
=1.085 N.mm/mm3
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43. Shear Strain Energy y x a
d = gxya
txy
gxy
Consider a cube of material acted upon by a shear stress,txy causing an elastic shear strain gxy
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44. Total Strain Energy for a Generalized State of Stress
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45. 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
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46. Comparison of Energy Stored in Straight and Stepped barsDa A L
(a)
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47. Comparison of Energy Stored in Straight and Stepped bars
L/2 nA
(b)
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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.
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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.
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50. What is isotropic ?
is properties of the materials are the same in all directions in the material
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51. Torsion of shafts
Shafts are members with length greater than the largest cross sectional dimension used in transmitting torque from one plane to another
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52. Internal Torque
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53. The axis of the shaft is denoted
Consider circular shaft AC subjected to equal and opposite torques T and T’. A cutting plane is passed through the shaft at B.
The FBD for section BC
must include the applied torque
and elementary shearing forces
dF. These forces are perpendicular to radius of the shaft and must
balance to maintain equilibrium.
The axis of the shaft is denoted
as r.
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54. dF is related to the shearing stress: dF = tdA
Taking moments about the Axis of the shaft results in
dF is related to the shearing stress: dF = tdA
So the applied torque can be related to the shearing stress as
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55. Equation is independent of material model as it represents static equivalency between shear stress and internal torque on a cross section
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56. Shear stress can’t exist on one plane only.
The applied torque produces a shear stress to the axis of the shaft. The equilibrium require equal stresses on the faces
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57. What is Torsion ?
To obtain a formula for the relative rotation f2-f1 in terms of the internal torque T.
To obtain a formula for the shear stress txq in terms of the internal torque T.
f - angle of twist
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58. Shearing Strain
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59. Assume: Material is linearly elastic and isotropic
Polar moment of inertia for the cross section
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60. Torsion formula
Circular hollow shaft with outer radius R, inner radius r
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61. Sign Convention
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62. Relative rotation f2-f1 in terms of the internal torque T.
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63. Shear stress txq in terms of the internal torque T.
Maximum occurs at shaft’s outer radius
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64. Direction of Shearing
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