Derive the bulk modulus (volume modulus of elasticity) for a bar subjected under pressure p.
The bulk modulus (K) is defined as the ratio of the pressure change (Δp) applied to a material to the resulting relative volume change (ΔV/V) that it undergoes.
Mathematically, K is given by the equation:
K = -V(dP/dV)
where V is the initial volume of the material, P is the pressure, and dP/dV represents the derivative of P with respect to V.
To derive this equation, we start with a bar of initial volume V0 and apply a pressure P to it, resulting in a change in volume ΔV. The relationship between pressure and volume change is given by:
(dP/dV) = -K/V
where K/V represents the bulk modulus of the material.
Rearranging the equation, we have:
(dP/dV) * V = -K
Integrating both sides with respect to V, we get:
∫(dP/dV) * V dV = -∫K dV
Simplifying, we have:
∫P dV = -K * ∫dV
Integrating, we obtain:
PV = -KV + C
where C is the constant of integration.
Rearranging the equation, we have:
KV + PV = C
Since P is the external pressure and ΔV is the change in volume, we can write the equation as:
K(V0 + ΔV) + P(V0 + ΔV) = C
Expanding and rearranging the equation, we get:
KV0 + KΔV + PV0 + PΔV = C
Since C is a constant, KV0 + PV0 = C.
Substituting this value back into the equation, we have:
KΔV + PΔV = 0
Factoring out ΔV, we obtain:
(ΔV)(K + P) = 0
Since ΔV cannot be zero, we must have:
K + P = 0
Thus, we find that the bulk modulus (K) is given by:
K = -P
Therefore, the bulk modulus (volume modulus of elasticity) for a bar subjected to pressure P is equal to the negative of the applied pressure.
To derive the bulk modulus (volume modulus of elasticity) for a bar subjected to pressure, we can use the concept of stress and strain.
The stress (σ) on the bar is defined as the force (F) applied per unit area (A). In this case, the stress is the pressure (p) applied to the bar:
σ = p
The strain (ε) on the bar is defined as the change in volume (ΔV) divided by the original volume (V).
ε = ΔV / V
The bulk modulus (K) is defined as the ratio of stress to strain:
K = σ / ε
To find the bulk modulus, we need to relate the change in volume (ΔV) to the pressure (p).
Consider a bar with initial length L, cross-sectional area A, and volume V = AL.
When a pressure p is applied, the bar compresses in the direction of the force, resulting in a decrease in the length by ΔL.
The change in volume (ΔV) is equal to the initial volume multiplied by the change in length:
ΔV = (A * ΔL)
Substituting this into the equation for strain, we get:
ε = (ΔV) / V
= (A * ΔL) / (AL)
= ΔL / L
Substituting the expressions for stress (σ) and strain (ε) into the equation for the bulk modulus (K), we get:
K = σ / ε
= p / (ΔL / L)
= p * (L / ΔL)
Therefore, the bulk modulus (K) for a bar subjected to pressure (p) is given by:
K = p * (L / ΔL)
To derive the bulk modulus (K) for a bar subjected to pressure (p), we need to understand the concept of volume strain (ε) and the stress-strain relationship.
The volume strain (ε) is defined as the change in volume (∆V) divided by the original volume (V) of the bar:
ε = (∆V / V)
Now, let's consider a bar with an initial volume V and subjected to pressure p. As a result of the pressure, the bar will experience a change in volume (∆V) and a corresponding change in length (∆L).
From the stress-strain relationship, we know that the stress (σ) is equal to the pressure (p) applied to the bar:
σ = p
The strain (ε) can be expressed as the change in length (∆L) divided by the original length (L) of the bar:
ε = (∆L / L)
Since we are interested in the volume strain (ε), which is related to both ∆V and V, we can express ∆L in terms of ∆V using the equation for volume change:
∆V = V - V0
where V0 is the original volume.
Now, let's relate the strain (ε) to the stress (σ) using Hooke's Law for isotropic materials. Hooke's Law states that the stress is proportional to the strain:
σ = E * ε
where E is the modulus of elasticity.
Substituting the expressions for stress (σ) and strain (ε), we get:
p = E * (∆L / L)
Rearranging the equation, we can express ∆L in terms of ∆V:
∆L = (∆V * L) / V
Substituting this expression into the previous equation, we have:
p = E * (∆V * L) / (V * L)
p = E * (∆V / V)
Now, comparing this equation with the definition of volume strain (ε) as ∆V / V, we can see that:
ε = p / E
Finally, the bulk modulus (K) is defined as the ratio of pressure (p) to the volume strain (ε):
K = p / ε = p / (p / E) = E
Hence, the bulk modulus (K) is equal to the modulus of elasticity (E) for a bar subjected to pressure.