Rubber Apparent Compression Modulus
Computes the apparent compression modulus of a rubber block squeezed between two bonded plates, Ec = E₀ × (1 + 2 × k × S²), where E₀ is the Young's modulus of the compound, k is a numerical constant tabulated by hardness (running from about 0.93 for soft 30 IRHD rubber to 0.53 for hard 75 IRHD) and S is the shape factor, which for a circular block equals the diameter divided by four times the thickness. Rubber is essentially incompressible in volume, so what resists the load is not compression of the material but friction on the bonded faces, which stops the sides from bulging; that is why the apparent modulus can sit many times above the compound's own Young's modulus. The result, in megapascals, is what you use to get the vertical stiffness of the mount (stiffness = Ec × area ÷ thickness) and from there the natural frequency of the isolated system. Because the shape factor is S = D ÷ 4t and the dominant term goes with S squared, diameter and thickness are levers of equal weight and opposite sign: halving the thickness and doubling the diameter do exactly the same thing, and in the example both take the apparent modulus from 11.3 to 35.3 MPa — a thin mount is a stiff mount, and that ruins vibration isolation. Enter the compound's Young's modulus, the constant k, the block diameter and the thickness.
Result
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Why a thin rubber mount turns out stiff
Anyone sizing an engine mount, a bridge bearing or a machine pad meets the same surprise early on: the compound carries a Young's modulus of a little over 3 MPa, yet the bonded block behaves as if it had 11. Measured stiffness lands far above the figure calculated from E₀, natural frequency climbs with it, and a mount meant to isolate ends up amplifying inside the machine's running range. The apparent compression modulus supplies the missing correction, and it leans on block shape more than on the rubber.
Ec = E₀ × (1 + 2k S²). E₀ is the compound's Young's modulus, roughly 1 to 9 MPa across common hardnesses. The constant k comes from a hardness table, running from about 0.93 for soft 30 IRHD rubber to a little over 0.5 above 70 IRHD, since soft rubber bulges more and gains more from bonded faces. S is the shape factor: loaded area divided by the area free to bulge, which for a bonded disc reduces to diameter ÷ (4 × thickness). With the defaults — 3.25 MPa, k = 0.64, a 100 mm disc 18 mm thick — S works out at 1.39 and Ec at 11.275 MPa, nearly three and a half times E₀.
The expression holds for small strain, up to some 10% to 15% compression, and for faces that genuinely stay bonded; let the block slip on the plate and bulging returns, taking stiffness down with it. It also ignores volumetric compressibility, which takes over once S climbs past somewhere between 8 and 10 — beyond that the bulk modulus near 2,000 MPa sets the ceiling and Ec stops tracking S². Keep both lengths in one unit: the shape factor carries no dimension, so diameter in millimetres against thickness in centimetres inflates S tenfold and the modulus a hundredfold.
Frequently asked questions
Where do the 11.275 MPa from the defaults come from?
Which k applies when all I have is Shore A hardness?
Does this work for a rectangular block or only for a disc?
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