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What is elastic fatigue? What is its importance?
Elastic fatigue is a condition in which a material loses its elastic properties slowly over time as it experiences repeated or cyclic loading and unloading. Continuous cycles of stress cause the material to lose its elastic recovery properties, thus failing to return to its original shape. Elastic fRead more
Elastic fatigue is a condition in which a material loses its elastic properties slowly over time as it experiences repeated or cyclic loading and unloading. Continuous cycles of stress cause the material to lose its elastic recovery properties, thus failing to return to its original shape. Elastic fatigue is very important in engineering and in material selection, especially when it comes to components in bridges, vehicles, and machinery that are subject to fluctuating loads.
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Define Poisson’s ratio.
Poisson's ratio (ν) is defined as the ratio of the transverse strain to the axial strain when a material is subjected to uniaxial stress. It denotes how much a material deforms in the lateral direction when it is stretched or compressed along its length. Mathematically it can be represented as folloRead more
Poisson’s ratio (ν) is defined as the ratio of the transverse strain to the axial strain when a material is subjected to uniaxial stress. It denotes how much a material deforms in the lateral direction when it is stretched or compressed along its length. Mathematically it can be represented as follows:
u = – (transverse strain)/(axial strain) = – (Δd/d)/(ΔL/L)
Where:
– Δd is the change in diameter (transverse deformation),
– d is the original diameter,
– ΔL is the change in length (axial deformation),
– L is the original length.
Important Points
Poisson’s ratio is a dimensionless quantity.
It generally lies between 0 and 0.5 for most materials with values near to 0.5 showing that the material is almost incompressible.
A value of 0 means that there is no transverse deformation when the material is stretched or compressed.
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A given quantity of an ideal gas is at pressure P and absolute temperature T. The isothermal bulk modulus of the gas is
To find the isothermal bulk modulus K of an ideal gas we can use the formula: K = - V (∂P/∂V)_T For an ideal gas at constant temperature (isothermal) the relation between pressure P and volume V is given by Boyle's Law: PV = nRT Differentiating this equation while keeping the temperature constant giRead more
To find the isothermal bulk modulus K of an ideal gas we can use the formula:
K = – V (∂P/∂V)_T
For an ideal gas at constant temperature (isothermal) the relation between pressure P and volume V is given by Boyle’s Law:
PV = nRT
Differentiating this equation while keeping the temperature constant gives us:
∂P/∂V = -nRT/V²
Thus the isothermal bulk modulus becomes:
K = -V (-nRT/V²) = nRT/V
Since nRT = PV we can substitute that in the equation too;
K = PV/V = P
Thus the isothermal bulk modulus of the gas is:
K = P
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A wire of length L, and cross-sectional area A is made of a material of Young’s modulus Y. If the wire is stretched by an amount x, the work done is
To calculate the work done W by stretching a wire of length L and cross-sectional area A through an amount x, we could use the following relationship between stress strain and Young's modulus: The stress in the wire, σ, can be found by the relation as follows: σ = F/A where F represents the appliedRead more
To calculate the work done W by stretching a wire of length L and cross-sectional area A through an amount x, we could use the following relationship between stress strain and Young’s modulus:
The stress in the wire, σ, can be found by the relation as follows:
σ = F/A
where F represents the applied force. The strain ε is described by:
ε = x/L
As related by Young’s modulus Y,
Y = σ/ε = (F/A)/(x/L)
From this we can write the force F:
F = (YAx)/L
The work done W when the wire is stretched by an amount x is given by the area under the stress-strain curve which is the integral of force over displacement:
W = ∫ F dx = ∫ (YAx/L) dx
Evaluating the integral we get:
W = (Y A/L) ∫ x dx = (Y A/L) * [x²/2] from 0 to x = (Y A/L) * (x²/2)
Work done is hence,
W = (Y A x²)/(2 L)
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Define the term strain. Why it has no units and dimensions? What are different types of strain?
Definition of Strain: Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as: ε = ΔL / L₀ Here, ΔLRead more
Definition of Strain:
Strain is a measure of deformation, which gives the displacement between the particles in a material body. Strain is defined as the ratio of change in dimension to the original dimension of the material. In mathematical terms, strain (ε) is expressed as:
ε = ΔL / L₀
Here, ΔL represents the change in length and L₀ represents the original length.
Why Strain Has No Units and Dimensions:
Strain is a dimensionless quantity because it is a ratio of two lengths (change in length to original length), which means that both numerator and denominator have the same units. Hence, the units cancel out, and one gets a unitless value. Since it is dimensionless, therefore it has no dimensions in the context of physical measurement.
Different Types of Strain:
1. Tensile Strain: It takes place when a material is stretched. It is defined as the change in length divided by the original length (elongation).
2. Compressive Strain: It takes place when a material is compressed. It is defined as the change in length divided by the original length (shortening).
3. Shear Strain: It is that strain which arises due to shear forces applied on a material. It is defined as the change in angle between two lines originally at right angles, expressed as the ratio of lateral displacement to the original length.
4. Volumetric Strain: It is the change in volume per unit original volume of a material, occurring under uniform pressure.
These types of strain are important for understanding how materials respond to various forces and deformations.
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