Lecture 7-Elasticity-chapter 12.pptx phy
- 2. Static Equilibrium •Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial reference frame. •Will deal now with the special case in which both of these velocities are equal to zero • This is called static equilibrium. •Static equilibrium is a common situation in engineering. •The principles involved are of particular interest to civil engineers, architects, and mechanical engineers.
- 3. Elasticity •We will disuss how objects deform under load conditions. •An elastic object returns to its original shape when the deforming forces are removed. •Various elastic constants will be defined, each corresponding to a different type of deformation. LECTURE GOALS
- 4. Center of Gravity •All the various gravitational forces acting on all the various mass elements are equivalent to a single gravitational force acting through a single point called the center of gravity (CG). •Each particle contributes a torque about an axis through the origin equal in magnitude to the particle’s weight multiplied by its moment arm. CG m x m x m x x m m m 1 1 2 2 3 3 1 2 3
- 5. Center of Gravity, cont •The torque due to the gravitational force on an object of mass M is the force Mg acting at the center of gravity of the object. •If g is uniform over the object, then the center of gravity of the object coincides with its center of mass. •If the object is homogeneous and symmetrical, the center of gravity coincides with its geometric center.
- 6. Definitions Associated With Deformation •Stress • Is proportional to the force causing the deformation • It is the external force acting on the object per unit cross- sectional area. •Strain • Is the result of a stress • Is a measure of the degree of deformation
- 7. Elasticity •So far we have assumed that objects remain rigid when external forces act on them. • Except springs •Actually, all objects are deformable to some extent. • It is possible to change the size and/or shape of the object by applying external forces. •Internal forces resist the deformation.
- 8. Elastic Modulus •The elastic modulus is the constant of proportionality between the stress and the strain. • For sufficiently small stresses, the stress is directly proportional to the strain. • It depends on the material being deformed. • It also depends on the nature of the deformation. •The elastic modulus, in general, relates what is done to a solid object to how that object responds. •Various types of deformation have unique elastic moduli. stress elastic ulus strain mod
- 9. ThreeTypes of Moduli •Young’s Modulus • Measures the resistance of a solid to a change in its length •Shear Modulus • Measures the resistance of motion of the planes within a solid parallel to each other •Bulk Modulus • Measures the resistance of solids or liquids to changes in their volume
- 10. Young’s Modulus •The bar is stretched by an amount DL under the action of the force F. •The tensile stress is the ratio of the magnitude of the external force to the cross-sectional areaA. •The tension strain is the ratio of the change in length to the original length. •Young’s modulus,Y, is the ratio of those two ratios: • Units are N / m2 i L L A F strain tensile stress tensile Y D
- 11. Stress vs. Strain Curve •Experiments show that for certain stresses, the stress is directly proportional to the strain. •This is the elastic behavior part of the curve. •The elastic limit is the maximum stress that can be applied to the substance before it becomes permanently deformed.
- 12. Stress vs. Strain Curve, cont’d. •When the stress exceeds the elastic limit, the substance will be permanently deformed. • The curve is no longer a straight line. •With additional stress, the material ultimately breaks.
- 13. Shear Modulus •Another type of deformation occurs when a force acts parallel to one of its faces while the opposite face is held fixed by another force. •This is called a shear stress. •For small deformations, no change in volume occurs with this deformation.
- 14. Shear Modulus, cont. •The shear strain is Dx / h. • Dx is the horizontal distance the sheared face moves. • h is the height of the object. •The shear stress is F / A. • F is the tangential force. • A is the area of the face being sheared. •The shear modulus is the ratio of the shear stress to the shear strain. •Units are N / m2 h x A F strain shear stress shear S D
- 15. Bulk Modulus •Another type of deformation occurs when a force of uniform magnitude is applied perpendicularly over the entire surface of the object. •The object will undergo a change in volume, but not in shape. •The volume stress is defined as the ratio of the magnitude of the total force, F, exerted on the surface to the area, A, of the surface. • This is also called the pressure. •The volume strain is the ratio of the change in volume to the original volume.
- 16. Bulk Modulus, cont. •The bulk modulus is the ratio of the volume stress to the volume strain. •The negative indicates that an increase in pressure will result in a decrease in volume. i i F volume stress P A B V V volume strain V V D D D D
- 17. Compressibility •The compressibility is the inverse of the bulk modulus. •It may be used instead of the bulk modulus.
- 18. Moduli andTypes of Materials •Both solids and liquids have a bulk modulus. •Liquids cannot sustain a shearing stress or a tensile stress. • If a shearing force or a tensile force is applied to a liquid, the liquid will flow in response.
- 19. Problem A 20-m long steel wire (cross-section 1.0 cm2,Young's modulus 2.0 x 1011 N/m2), is subjected to a load of 25 000 N. How much will the wire stretch under the load? Ans: 2.5 cm
- 20. Problem How large a force is necessary to stretch a 2.0-mm diameter copper wire (Y = 11 x 1010 N/m2) by 1.0%? Ans: 3.5 kN