get bulk modulus of elasticity formula in fluid mechanics example problems with solutions ?
The ability of a body to regain its original shape and size when the deforming force is withdrawn, is known as elasticity. A body which completely regains its original shape and size after the removal of the deforming force is said to be perfectly elastic. A body which retains its deformed shape and shows no tendency at all to regain its original shape, after the removal of the external force, is said to be perfectly plastic. In fact, no material body can be perfectly elastic or perfectly plastic.
When a deforming force is applied to a body, it reacts to the applied force by developing a reaction (or restoring) force which, from Newton’s third law, is equal in magnitude and opposite in direction to the applied force. The reaction force per unit area of the body which is called into play due to the action of the applied force is called stress. The mangnitude of the stress is equal to the magnitude of the applied force divided by the area of the body, but the direction of the stress is opposite to that of the applied force. Stress is measured in units of force per unit area, i.e. Nm–2. Thus
Stress = F /A
where F is the applied force and A is the area over which it acts
When a deforming force is applied to a body, it may suffer a change is size or shape. Strain is defined as the ratio of the change in size or shape to the original size or shape of the body. Strain is a number; it has no units or dimensions. The ratio of the change in length to the original length is called longitudinal strain. The ratio of the change in volume to the original volume is called volume strain. The strain resulting from a change in shape is called shearing strain.
Hookes’ law states that, within the elastic limit, the stress developed in a body is proportional to the strain produced in it. Thus, the ratio of stress to strain is a constant. This constant is called the modulus of elasticity. Thus
Modulus of elasticity = stress /strain
Since strain has no unit, the unit of the modulus of elasticity is the same as that of stress, namely, Nm–2
Suppose that a rod of length L and a uniform cross-sectional area A is subjected to a longitudinal pull. In other words, two equal and opposite forces are applied at its ends. As a result of applying the deforming forces, there is an extension in length which we denote by △L. The strain is given by.
Strain = △L /L
Since the deformation involves a change in length, the strain is called longitudinal strain or linear strain. Since the length increases (△L > 0) it is also called extensional strain or tensile strain. The stress is given by
Stress = F/A
The stress in the present case is called linear stress, tensile stress, or extensional stress. If the direction of the forces is reversed so that △L is negative, we speak of compressional strain and compressional stress. If the elastic limit is not exceeded, then from Hooke’s law
where Y, the constant of proportionality, is called the Young’s modulus of the material of the rod and may be defined as the ratio of the linear stress to linear strain, provided the elastic limit is not exceeded. Since strain has no unit, the unit of Y is Nm–2
Solids, liquids and gases can be deformed by subjecting them to a uniform normal pressure P in all directions. Stress and pressure have the same dimension (force per unit area), but pressure is not the same thing as stress. Pressure is the force per unit area acting on the surface of a system, the force being everywhere perpendicular to the surface so that, for a uniform pressure, the force per unit area is the same. Pressure is a particular kind of stress which changes only the volume of the substance and not its shape. The substance may be a solid, liquid or gas. A small increase in pressure DP applied to a substance decreases its volume
from, say, V to V – △V so that △V is the small decrease in volume. The volume strain is given by
Volume strain = – △V /V
The bulk modulus is defined as the ratio of the excess pressure and the corresponding volume strain, i.e
If △P is positive, △V will be negative and vice versa. The negative sign in our definition of bulk modulus B ensures that B is always positive. The SI unit of B is Nm–2. The reciprocal of B is known as compressibility. The bulk modulus of a gas depends on the pressure. Under isothermal conditions (i.e. when the temperature is kept constant), the bulk modulus of a gas is equal to its pressure P. Under adiabatic conditions (i.e. when heat is not allowed to leave or enter the system), the bulk modulus is equal to Isothermal bulk modulus = P Adiabatic bulk modulus = gP.yP, where y (= Cp/Cv) is the ratio of the molar heat capacities of the gas at constant pressure and constant volume. Thus Isothermal bulk modulus = P
Adiabatic bulk modulus = yP.
Shear Modulus or Modulus of Rigidity
Shear is a particular kind of stress which only solids can withstand. The solid is deformed by changing its shape without changing its size. The body does not move or rotate as a whole: there is a relative displacement of its contiguous layers. Consider a solid in the form of a rectangular cube as in Fig. below
Suppose the lower face PQRS is held fixed and a force F is applied parallel to the upper face MNUV. If we do this to a liquid, it will begin to flow; it will not remain in equilibrium. In other words, liquids cannot withstand this kind of shear force. But when it is applied to a solid, it will remain in equilibrium because no net force and no net torque act on it. There is a couple produced by this
force and an equal and opposite force coming into play on the lower face. As a result of this, the lines joining the two faces turn through an angle θ. We say that the face MNRS is sheared through an angle θ (measured in radians). The angle θ is called the shear strain or the angle of shear and is a measure of the degree of deformation. If A is the area of the face MNUV, the ratio F/A is the shearing stress. It is found that for small deformation, the shearing stress is proportional to the shear strain, i.e.
The quantity η is called the shear modulus or the modulus of rigidity. Referring to Fig. below, if θ is small,
If a rod or wire of length l and radius r is fixed at one end and a torque τ is aplied at the other end, the rod or wire is twisted about its axis. If θ is the angle of twist, the torque is related to modulus of rigidity (η) by the relation
When a wire is stretched with a force, apart from an increase in its length, there is a slight decrease in its diameter, i.e. both shape and volume change under longitudinal stress. The ratio of the decrease △D in diameter to the original diameter D is called lateral strain, i.e. strain at right angles to the deforming force. Thus
It is found experimentally that within the elastic limit, the lateral strain is proportional to the longitudinal strain. The ratio of the two is called Poisson’s ratio and is denoted by σ. Hence,
Since it is a ratio between two types of strain, σ is dimensionless. Theoretically, one can show that it must be less than 0.5. For most solids it lies between 1/4 and 1/3, and for rubber it is very nearly 0.5.
Energy Stored in a Strained Wire: Strain Energy
If a wire is stretched, the potential energy stored per unit volume is given by
Where S = strees, ε = strain and Y = Young’s modulus of the material of the wire.
If a metal rod fixed rigidly at its ends is heated or cooled, then due to expansion or contraction, tensile or compressive stress is set up in the rod. These stresses are called thermal stresses. If a rod of length L is free to expand or contract and its temperature is changed by △T, the change in its length is given by
where α is the coefficient of linear expansion of the rod. Now
Similarly, if a fluid is contained in a vessel such that its volume cannot change, then a change in temperature results in a change in pressure. The thermal stress is then given by
where B is the bulk modulus of the fluid and y is its coefficient of volume expansion or contraction.
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