A syntactic foam is made by mixing thin-walled hollow glass spheres (radius, r, and wall thickness, t=0.05r) into an epoxy resin, which is then cured and hardened. The glass from which the spheres are made is twice as dense as the hardened resin.
If the spheres are identical and randomly packed with a volume fraction of 0.45, what is the density of the syntactic foam, in terms of the resin density, ρr? Note that the volume of a sphere is Vs=43πr3 and that, for a thin-walled sphere, the volume of the spherical shell is Vshell=4πr2t .
To find the density of the syntactic foam, we need to consider the volume fraction of the spheres and the density of both the glass spheres and the hardened resin.
Given information:
- Volume fraction of the spheres (ε): ε = 0.45
- Density of the glass spheres (ρg): ρg = 2 times the density of the hardened resin (2ρr)
To start, let's assume a unit volume of the syntactic foam. Within this unit volume, a fraction ε is occupied by the glass spheres, and the remainder (1 - ε) is the volume of the cured and hardened resin.
Now, let's calculate the ratio of the two volumes.
Volume of the glass spheres:
Vg = ε * Vs,
where Vs is the volume of a single glass sphere.
Volume of the hardened resin:
Vr = (1 - ε),
since (1 - ε) portion of the unit volume is occupied by the resin.
The volume of a thin-walled hollow glass sphere can be written as:
Vs = Vshell = 4πr^2t,
where t represents the wall thickness.
Substituting Vs into the equation for Vg:
Vg = ε * 4πr^2t.
Knowing that Vr = (1 - ε), we need to calculate the total volume of the syntactic foam, Vf.
Vf = Vg + Vr.
Substituting the expressions for Vg and Vr:
Vf = ε * 4πr^2t + (1 - ε).
The density of the syntactic foam (ρf) can be expressed as the total mass divided by the total volume. The total mass consists of the mass of the glass spheres (mg) and the mass of the resin (mr).
ρf = (mg + mr) / Vf.
To calculate the density of the syntactic foam, we need to express the mass of the glass spheres (mg) and the resin (mr) in terms of their volumes. Since the density (ρ) is equal to mass/volume, we can write:
mg = ρg * Vg,
mr = ρr * Vr.
Substituting the expressions for Vg and Vr:
mg = ρg * ε * 4πr^2t,
mr = ρr * (1 - ε).
Substituting these values back into the expression for ρf:
ρf = [(ρg * ε * 4πr^2t) + (ρr * (1 - ε))] / [ε * 4πr^2t + (1 - ε)].
Now you can calculate the density of the syntactic foam by plugging in the given values for ε, ρg, and ρr, and performing the necessary calculations.