Modulus of Rupture, Four-Point Bending

Nearly every strength figure on a ceramic datasheet came from this test. A rectangular bar, ground to size, laid on two supports, loaded by two inner rollers at the quarter points, and broken. The peak load and the geometry go into beam theory and out comes a stress at the tensile face — the modulus of rupture, or flexural strength. It is used because a proper tensile test on a ceramic is genuinely difficult: gripping a brittle specimen without breaking it in the grips, and aligning it well enough that the bending stress is negligible, takes expensive fixturing that a bend test does not.

Read the form of the equation before the answer. Depth is squared and width is not. A 10% error in measuring $d$ costs 20% in the reported strength, while the same error in $b$ costs 10%. Measure the cross-section of every bar with a micrometer and use its own dimensions, never the nominal ones — the difference between a 2.98 mm bar and a 3.00 mm bar is 1.3% of the answer, and it is free to get right.

The 3/4 is the four-point geometry, and it is the easiest mistake in the whole test. This is the quarter-point configuration, where the inner span is exactly half the outer. The three-point formula is $\sigma = 3PL/(2bd^2)$ — a factor of two different. Substituting a four-point result into the three-point formula, or the reverse, doubles or halves the answer, and the result still looks like a plausible ceramic strength. Check which fixture produced the load before touching the arithmetic.

Four-point is preferred for a reason worth understanding. In three-point bending the peak stress exists on one line only, directly under the single roller; in four-point the entire inner span sits at the same peak stress. That means the test interrogates a real volume of material rather than a line, and the odds of the peak-stress region containing a serious flaw go up accordingly. The measured strength therefore comes out lower — and more honest. A three-point figure and a four-point figure for the same material are different numbers and always will be; the effective-volume page turns the difference into arithmetic.

The same reasoning explains why flexural strength always exceeds tensile strength on the same ceramic, typically by 20 to 50%. Only a thin layer near the tensile face is highly stressed, so the test samples a small effective volume and is flattered by it. Taking a datasheet's flexural strength and using it as a tensile allowable is a real and common design error, and the size-scaling page is where the correction lives.

And one specimen is not a strength. A ceramic has a distribution; a single bar tells you almost nothing, ten tell you roughly where the middle is, and thirty start to constrain the shape. The whole first half of this category is about what to do with those thirty numbers, and the median-rank page is where to start.

Two more things the formula assumes. It assumes linear elasticity right up to failure, which for a ceramic at room temperature is true and at high temperature may not be — creep redistributes the stress, the linear distribution the formula depends on no longer holds, and the reported strength comes out too high. And it assumes the bar is loaded in pure bending with negligible shear, which needs a span-to-depth ratio comfortably above about 4; standard fixtures use far more. Below that the bar can fail in shear and the number means nothing at all.

ASTM C1161 is the authority: it fixes the standard specimen sizes, the fixture spans, the chamfers on the tensile-face edges that stop the specimen failing from a corner, the articulating rollers that let the fixture accommodate a bar that is not perfectly parallel, and the loading rate. It is not reproduced here and this page is not a substitute for it — but every one of those details exists because it was found to change the answer, which is itself the most useful thing to know about ceramic strength testing.