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Solids and Young's Modulus

Young's Modulus

1. Definitions

At a specified temperature, substances can exist in four different forms: solid, liquid, gas, and plasma.

Exerting forces on substances, solid tends to keep its volume and its shape, liquid tends to keep its volume not its shape, and gas and plamsa tend to keep neither their shape nor their volume. Generally, the separation between the two phases is not always clear, at 0o C, water can be solid or liquid. A phase of substance is a matter of temperature. In the normal conditions (pressure p = 1 atm, and temperature = room temperature t = 20o C), we can know the phase of many substances such as steel, water, and air.

Solids are not easy to deform, that is changing their shape or size. Depending on the force exerted on a solid, it can change slightly. Deformation on solids depends also on their nature that is anisotropic or isotropic. We will focus here on isotropic solids.
To deform a solid, force alone is not sufficient to consider, the area where this force is applied is important as well. We define the stress on the solid as the force exerted on the solid per unit area.

The force inward F is applied on a surface of area S, It can be the sum of two components: normal force Fn and a parallel force Fp. In vector notation, we can write F = Fn + Fp.
The stress σ is then:
σ = Fn/S + Fp/S = σn + σp.
σn is called the normal stress, and σp is called the shear stress

To maintain the solid in the equilirium, The same force equal in magnitude and in the opposite direction must be applied. This force is decomposed into its parallel and normal components that will balance the same force components of the first force applied.

If the force is directed away from the solid, in this case the normal stress is called tensile stress. If the same perpendicular force is applied inward on all the four surfaces of the solid, the stress is called pressure
The dimention of stress is N/m2 that is Pascal Pa ( lb/in2 or psi: pounds per square inch in British system. 1 lb/in2 = 6.9 kPa)

2. Young's modulus

The change in length is defined as:
εtc = Δx/d
called a tensile strain and a compressive strain for a compression and elongation respectively.
The shear strain is defined as:
εs = φ = Δx/d

When the force applied on solid becomes important; that is in the fracture case, deformation (then strain) can be proportional to the steress. This proportionality is called the Young's mudulus. We can write then:
Y = σ/ε = (Fn/S)/(Δx/d)
Its dimension is a force per unit area; its SI unit is the pascal(Pa).
Similarly, the shear mudulus is defined as M = σss = (Fp/S)/(Δx/d)
The bulk mudulus is defined as:
B = - V dP/dV = - ΔP/(ΔV/V)
That measures the required pressure P to compress a substance of volume V by the fraction ΔV/V.

Some examples:

Substance Y M B
Steel 200 160 84
Glass 14 3 8
water 0 0 2.2
Most gases 0 0 1 x 10- 4


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