Mechanical Properties of Materials Chapter 3 Mechanical Properties of Materials

3.6 Poisson’s Ratio When compressed, the bar is shortened longitudinally, but expands in lateral direction.

Under tension force, the bar increases its length by , and the radius reduces by . The strains in longitudinal and lateral directions are

S.D. Poisson found (1880s) that , within the elastic range and for a homogeneous and isotropic material , * Since long and lat have opposite signs,  is positive. *  is a non-dimensional quantity. * For most nonporous solids,  is in the range of 1/4~1/3. * It was proved theoretically  max =1/2. Thus 0    1/2

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3.7 The Shear Stress-Strain Diagram Tests can be conducted for pure shear case, and draw  -  curve.

* Hooke’s law for shear: * G – shear modulus modulus of rigidity * G, E, ,  - same unit force/area * Relationship:

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Chapter 5 Torsion

5.1 Torsional Deformation of a Circular Shaft Concept of Torsion * Torque – a moment tending to twist a member about its longitudinal axis. * Shaft or axis – transmit torque from motor to drive machine * Torsion – deformation due to torque

Observation * Every cross-section remains planar and circular. * Radial lines remain straight, and rotate through the same angle. * Longitudinal lines are twisted. * The length and radius remain unchanged.

Angle of Twist * Radial line on a cross-section rotate through an angle (x). * The angle of twist is a function of x.  = 0 at a fixed end (x = 0) * Since the angle is changed, there is shear strain. * Shear strain is the largest at outer surface.

5.2 The Torsion Formula Torsion Formula * Shear strain and shear stress are proportional to radius. * At the center, they are zero, and At outer surface they are maximum.

J – polar moment of inertia of the cross-section

Solid Shaft

Tabular Shaft

5.3 Power Transmission

Concepts of Work and Power Unit - SI system: J, FPS: lbft, IPS: lb in * Power – work done per unit time Unit - SI system: W, FPS: ftlb/s, IPS: inlb/s hp = 550 ftlb/s

Power Transmission  – angular velocity, circular frequency, radians/second f – f = 2, frequency, Hz, cycles/second Note: If the speed is given as n rpm (revolutions per minute), then

Shaft Design allow – from material, c, J – from geometry T – from working requirement * Given material (allow), to determine the size. * Give size, to determine the material.

5.4 Angle of Twist Angle of Twist

 - angle of twist of one end (x = L) with respect to the other end (x = 0), measured in radians T(x) – internal torques at location x J(x) – polar moment of inertia at x G – shear modulus of the shaft material

Constant Torque and Cross-Section Area G can be determined from pure shear tests.

Multiple Torques Sign Convention Use right-hand rule: Thumb is directed outward. Finger curl gives positive T and 

* Divide the shaft into several segments. * Cut each segment in middle, and find internal torque. * Draw torque diagram.