>>30267841Simple example: Quartz square prism
Geometry: 1x1x10 centimeters.
W = 1 cm
H = 1 cm
L = 10 cm
Square step impulse of 1000 N over 100 microseconds
Impulse duration: T = 100 ms
Chuubanite concentration: Cc = 1 Mh/m^3
Chuubanite concentration outside of prism: Cc_out = 0 Mh/cm^3
Young's Modulus: E = 80 GPa
Vitubium detente: Vd = 0.1 ms
Activation via activator factor: Ac_activator_factor = 1000
Impulse travels through material as a longitudinal wave, a stress wave
Surface area of impact: A = W * H = 1 cm * 1 cm = 1 cm^2
Volume of prism: V = A * L = 10 cm^3
Force during impulse: F = 1000 N
Pressure on impact during impulse: P = F/A = 1N / 1 cm^2 = 10 MPa
VFE = V * Cc * P * Ac_activator_factor= 10 cm^3 * 1 Mh/m^3 * 10 MPa * 1000 = 100000 Ami (Note delP = P since the prism begins at a state of 0 stress)
Ratio = (Cc - Cc_out) / (Cc + Cc_out) = 1 Mh/m^3 / 1 Mh/m^3 = 1
E_mag = Ratio * VFE * Vd = 1 * 100000 Ami * 0.1 ms = 10 mJ
Stress and Strain from material compression
Base Stress: S_base = P = 10 MPa (Note, this is only true if the Impulse duration is longer than the travel time of the stress wave through the material).
Base Strain: del = S_base / E = 10 MPa / 80 GPa = 1.25E-4
Base Strain Energy: U_base = V * S_base * del / 2 = 6.25 mJ
The proportion of energy converted to strain energy vs heat and light varies. These proportions are set by the factors K_strain, K_heat and K_light, which always add up to 1.
Assume K_strain = 1
Strain energy: U = E_mag * K_strain + U_base = 16.25 mJ
Back to strain: del_total = sqrt(2 * U / V / E) = 2.016E-4
Calculate total stress as well: S_total = del_total * E = 16.12 MPa
If you have an object sitting against the end of the prism, assuming ideal (impossible) conditions, all this energy is transferred to the object and converted into kinetic energy.
For example, 1*1*1 cm cube of steel.
mass: m = 1 cm^3 * 8 g/cm^3 = 8 g
Kinetic energy: K = m*v^2/2
Velocity of cube after transfer of energy: v = sqrt(2*K/m) = 2.016 m/s
Practical example: Circular Kronie air gun equivalent
Geometry: Cylinder, 10 centimeters long, R6xL100 millimeters
R = 6 mm
L = 10 cm
Square step impulse of 1000 N over 100 microseconds
Impulse duration: T = 100 ms
Chuubanite concentration: Cc = 20000 Mh/m^3
Chuubanite concentration outside of prism: Cc_out = 0.0001 Mh/m^3
Young's Modulus: E = 80 GPa
Vitubium detente: Vd = 0.1 ms
Activation via activator factor: Ac_activator_factor = 1000
Impulse travels through material as a longitudinal wave, a stress wave
Surface area of impact: A = pi * R^2 = 1 cm * 1 cm = 1.13 cm^2
Volume of prism: V = A * L = 15.8 cm^3
Force during impulse: F = 1000 N
Pressure on impact during impulse: P = F/A = 1N / 1 cm^2 = 8.84 MPa
VFE = V * Cc * P * Ac_activator_factor = 15.8 cm^3 * 20000 Mh/m^3 * 8.84 MPa * 1000 = 2.8 GAmi (Note delP = P since the prism begins at a state of 0 stress)
Ratio = (Cc - Cc_out) / (Cc + Cc_out) = 0.9999... = ~ 1
E_mag = Ratio * VFE * Vd = 1 * 2.8 GAmi * 0.1 ms = 280 J
Stress and Strain from material compression
Base Stress: S_base = P = 8.84 MPa (Note, this is only true if the Impulse duration is longer than the travel time of the stress wave through the material).
Base Strain: del = S_base / E = 8.84 MPa / 80 GPa = 1.11E-4
Base Strain Energy: U_base = V * S_base * del / 2 = 7.74 mJ
The proportion of energy converted to strain energy vs heat and light varies. These proportions are set by the factors K_strain, K_heat and K_light, which always add up to 1.
Assume K_strain = 0.4
Strain energy: U = E_mag * K_strain + U_base = 112 J
Back to strain: del_total = sqrt(2 * U / V / E) = 1.33E-2
Calculate total stress as well: S_total = del_total * E = 1.064 GPa (Note, this is almost the ultimate compression strength of the quartz. The end of the rod undergoes stresses higher than this and is chipped away with each firing, shortening the life of the firing rod.)
For example, 9.48 g bullet of same radius
Kinetic energy: K = m*v^2/2 = U
Velocity of cube after transfer of energy: v = sqrt(2*K/m) = 153.7 m/s (Same as Girardoni Air Rifle)
I don't really expect anyone to actually be autistic enough to read this, this is deep deep I Just Want That Crystal Math territory. But if someone does, it's still a WIP and open for feedback.