Chapter 12
Objectives: After completion of this module, you should be able to: Demonstrate your understanding of elasticity, elastic limit, stress, strain, and ultimate strength. Write and apply formulas for calculating Young s modulus, shear modulus, and bulk modulus. Solve problems involving each of the parameters in the above objectives.
12-4 Elastic Properties of Solid An elastic body is one that returns to its original shape after a deformation. Golf Ball Rubber Band Soccer Ball
12-4 Elastic Properties of Solid An inelastic body is one that does not return to its original shape after a deformation. Dough or Bread Clay Inelastic Ball
An Elastic Spring A spring is an example of an elastic body that can be deformed by stretching. x F A restoring force, F, acts in the direction opposite the displacement of the oscillating body. F = -kx
12-4 Elastic Properties of Solid So far we have assumed that objects remain rigid when external forces act on them. (Except springs). Actually, objects are deformable It is possible to change the size and/or shape of the object by applying external forces. Internal forces resist the deformation. Atomic lattices can be approximated as sphere/spring repetitive arrangements
12-4 Elastic Properties of Solid Stress It is the external force acting on the object per unit area. Strain Is the result of a stress. Is a measure of the degree of deformation.
12-4 Elastic Properties of Solid Elastic Modulus The elastic modulus is the constant of proportionality between the stress and the strain. It relates what is done to a solid object to how the object responds. It depends on the material being deformed.
12-4 Elastic Properties of Solid Elastic Modulus
Tension and compression Strain is a dimensionless ratio fractional change in length of the specimen ΔL/L The modulus for tensile and compressive strength is called the Young s modulus F A E L L Thomas Young (1773 1829)
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.
Stress vs. Strain Curve 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.
Tensile testing
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 See the active figure to vary the values This is called a shear stress
The shear stress is F / A F is the tangential force A is the area of the face being sheared The shear strain is Δx / h Δx is the horizontal distance the sheared face moves h is the height of the object
Bulk Modulus: Elasticity of volume 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
Bulk Modulus: Elasticity of volume EThe negative indicates that an increase in pressure will result in a decrease in volume.
Modulus and Types 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
For the three parts of this Quick Quiz, choose from the following choices the correct answer for the elastic modulus that describes the relationship between stress and strain for the system of interest, which is in italics: (a) Young s modulus (b) shear modulus (c) bulk modulus (d) none of those choices (i) A block of iron is sliding across a horizontal floor. The friction force between the block and the floor causes the block to deform. (ii) A trapeze artist swings through a circular arc. At the bottom of the swing, the wires supporting the trapeze are longer than when the trapeze artist simply hangs from the trapeze due to the increased tension in them. (iii) A spacecraft carries a steel sphere to a planet on which atmospheric pressure is much higher than on the Earth. The higher pressure causes the radius of the sphere to decrease.
Stress and strain All deformations result from a stress deforming force per unit area Deformations are described by a strain unit deformation Coefficient of proportionality between stress and strain is called a modulus of elasticity stress = modulus * strain
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