Does the square cube law always hold?

Question

Water has surface tension but surface tension doesn't follow the square cube law. To reduce the size of its meniscus by half, you require 4 times the gravitational field strength.

According to the Wikipedia article Fracture toughness, the units of fracture toughness are not the same as the units of material strength which has the same units as pressure. From the units and the Wikipedia article Fracture mechanics, I'm guessing that when a brittle material has a crack, the tension is magnified at the tip of the crack by a number proportional to the square root of the length of the crack and when it gets magnified to more than the theoretical strength of the material, it starts to grow making it get magnified even more easily and the crack propagates at the speed of sound in the material although the rapid crack growth probably prevents the tension from getting magnified to more than the theoretical strength at the tip when it's longer, i.e. no part can be under more tension than the theoretical strength so rapid crack growth must occur to prevent the tension from getting magnified to more than the theoretical strength of the material. From the way crack growth from magnified tension works, that means that if you have a rod with a crack of a given length, than a rod of twice the thickness and twice the length of a crack requires less force of tension per area to break so the square cube law does not apply to an object with a crack.

A believe that even for a sphere of a given stable perfect crystal with only point dislocations and no cracks that's smooth, the square cube law very closely holds doesn't exactly hold. You may be thinking that it must hold because once the tension exceeds a certain amount, it will break. I think it doesn't exactly work that way. I believe that according to a simplified quantum mechanical theory where there are only 3 spatial dimensions and electrons and nuclei are point charges that are infinitely stable with no nuclear chemistry and the gravitational constant is zero, when a sphere of an infinitely stable substance, corundum, is under tension, there can be homogeneous nucleation of crack growth from the interior and the higher the tension, the higher the rate of homogeneous nucleation of crack growth. Let's define the theoretical strength of a corundum sphere to be the amount of tension it must be under to have a 50% chance of fracturing within the number of seconds equal to the number of meters in its diameter. Then for any corundum sphere, the theoretical strength of a sphere twice its size is the tension that gives a rate of homogeneous nucleation of crack growth 16 times smaller than the homogeneous nucleation rate for tension equal to the theoretical strength of that sphere. I believe that for sufficiently large size, the theoretical strength of a corundum sphere that size varies approximately as the reciprocal of the log of its size. If that's the case, than the fraction of the theoretical strength lost with a doubling of the size also varies approximately as the reciprocal of the log of the size. Thus, the deviation from the square cube law that an observer observes would vary as the reciprocal of the log of how many times larger than an atom that observer is.

Answer 1

The square-cube law states that when an object's dimensions are scaled by some multiplier $k$, the volume is multiplied by $k^3$ and the surface area is multiplied by $k^2$. This means that any quantity that is proportional to the volume also is multiplied by $k^3$, and any quantity that is proportional to the surface area is also multiplied by $k^2$. Most of the time, what people refer to as the "square-cube law" is actually a corollary of the above: any quantity proportional to the surface-area to volume ratio is multiplied by $k$ when all dimensions are multiplied by $k$.

As such, I'm not quite sure why you would expect any of your examples to follow the square-cube law, as none of them depend on the surface-area to volume ratio. The "size" (which I am assuming means height) of the meniscus depends on several things, none of which are the volume of the water (see this post: How far can water rise above the edge of a glass?). Surface tension is, as its name implies, a surface effect, depending only on the surface area, not the volume. And the fracture toughness is dependent on crack length and ultimate strength, as well as the particular crack geometry; hence, it also does not depend on the surface area or volume.