Elastic Properties of Solids PDF

CHAPTER 2 Elastic Properties of Solid Slide 1  Elasticity We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is a quantity that is proportional to the force causing a deformation. More specifically, stress is the external force F actin...

CHAPTER 2 Elastic Properties of Solid Slide 1  Elasticity We shall discuss the deformation of solids in terms of the concepts of stress and strain. Stress is a quantity that is proportional to the force causing a deformation. More specifically, stress is the external force F acting on an object per cross-sectional area A. Stress = F/A The result of a stress is strain, which is a measure of the degree of deformation. Slide 2 Elasticity It is the ability of a substance to restore its original shape and size after deformation. Elastic materials Materials that restore its original shape and size after deformation are called elastic materials. Inelastic materials Materials that cannot restore its original shape and size after deformation are called inelastic materials. Slide 3 Slide 4 Fig. 12.15, p.374 It is found that, for sufficiently small stresses, strain is proportional to stress; the constant of proportionality depends on the material being deformed and on the nature of the deformation. We call this proportionality constant the elastic modulus. The elastic modulus is therefore defined as the ratio of the stress to the resulting strain: λ = Elastic modulus = stress/strain Slide 5  λ = Elastic modulus = stress/strain The elastic modulus in general relates what is done to a solid object (a force is applied) to how that object responds (it deforms to some extent). We consider three types of deformation and define an elastic modulus for each: Slide 6  1.Young’s modulus, which measures the resistance of a solid to a change in its length. 1.Shear modulus, which measures the resistance to motion of the planes within a solid parallel to each other. 2.Bulk modulus, which measures the resistance of solids or liquids to changes in their volume. Slide 7  Young’s Modulus: Elasticity in Length Y = tensile stress/ tensile : (UNIT) ? Stress (σ) =Force / Area=F/Ao (N/m2) Strain (ε) =elongation/ original length=ΔL/Lo Slide 8 Fig. 12.16a, p.374 Slide 9 Fig. 12.14, p.373 Shear Modulus: Elasticity of Shape Another type of deformation occurs when an object is subjected to a force parallel to one of its faces while the opposite face is held fixed by another force (Fig.). The stress in this case is called a shear stress. If the object is originally a rectangular block, a shear stress results in a shape whose cross section is a parallelogram. A book pushed sideways, as shown in Figure, is an example of an object subjected to a shear stress. To a first approximation (for small distortions), no change in volume occurs with this deformation. Slide 10 Slide 11 Slide 12 Fig. 12.16b, p.374  Bulk Modulus: Volume Elasticity Bulk modulus characterizes the response of an object to changes in a force of uniform magnitude applied perpendicularly over the entire surface of the object, as shown in Figure (We assume here that the object is made of a single substance.) Slide 13 Slide 14 Slide 15 Fig. 12.17, p.375 Slide 16 Table 12.1, p.374  Poisson's Ratio σ We shall now return to the stretch or longitudinal strain which is controlled by Young's modulus. The longitudinal strain was seen to be ΔL/L. There exists, however, in addition to the longitudinal strain a lateral strain responsible for the decrease in cross-section of the rod or wire, and is represented by : (Δr /r ), The ratio of lateral to longitudinal strain is called Poisson's ratio: σ = (Δr/r) / (ΔL/L) Slide 17 Pre-stressed Concrete If the stress on a solid object exceeds a certain value, the object fractures. The maximum stress that can be applied before fracture occurs depends on the nature of the material and on the type of applied stress. Slide 18  For example, concrete has: a tensile and shear strength of about 2x106 N/m2, a compressive strength of 20x106 N/m2 If the applied stress exceeds these values, the concrete fractures. It is common practice to use large safety factors to prevent failure in concrete structures. Slide 19  (a)A concrete slab with no reinforcement tends to crack under a heavy load. (b) The strength of the concrete is increased by using steel reinforcement rods. (c) The concrete is further strengthened by pre-stressing it with steel rods under tension Slide 20 Fig. 12.18, p.376 Concrete is normally very brittle when it is cast in thin sections. Thus, concrete slabs tend to sag and crack at unsupported areas, as shown in Figure a. The slab can be strengthened by the use of steel rods to reinforce the concrete, as illustrated in Figure b. Because concrete is much stronger under compression (squeezing) than under tension (stretching) or shear, vertical columns of concrete can support very heavy loads, whereas horizontal beams of concrete tend to sag and crack. Slide 21  However, a significant increase in shear strength is achieved if the reinforced concrete is pre- stressed, as shown in Figure c. As the concrete is being poured, the steel rods are held under tension by external forces. The external forces are released after the concrete cures; this results in a permanent tension in the steel and hence a compressive stress on the concrete. This enables the concrete slab to support a much heavier load. Slide 22  Ductile and Brittle Materials Ductility is another important mechanical property. It is a measure of the degree of plastic deformation that has been sustained at fracture. In this case, if large plastic deformation takes place between the elastic limit and the fracture point, the material is said to be Ductile. Ductility may be expressed quantitatively as either percent elongation or percent area reduction. The percent elongation (% EL) is the percentage of plastic strain fracture, Slide 23 %EL = (ΔL/Lo) x 100  A material that experiences very little or no plastic deformation upon fracture is termed Brittle. Slide 24 Slide 25 Fig. 12.15, p.374