Tag Archives: cracking

Thermal stress failure

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Take a glass, preferably a cheap glass, and set it in a bowl of ice-cold water so that the water goes only half-way up the glass. Now pour boiling hot water into the glass. In a few seconds the glass will crack from thermal stress, the force caused by heat going from the inside of the glass outside to the bowl of cold water. This sort of failure is not mentioned in any of the engineering material books that I had in college, or had available for teaching engineering materials. To the extent that it is mentioned mentioned on the internet, e.g. here at wikipedia, the metric presented is not derived and (I think) wrong. Given this, I’d like to present a Buxbaum- derived metric for thermal stress-resistance and thermal stress failure. A key aspect: using a thinner glass does not help.

Before gong on to the general case of thermal stress failure, lets consider the glass, and try to compute the magnitude of the thermal stress. The glass is being torn apart and that suggests that quite a lot of stress is being generated by a ∆T of 100°C temeprarture gradient.

To calcule the thermal stress, consider the thermal expansivity of the material, α. Glass — normal cheap glass — has a thermal expansivity α = 8.5 x10-6 meters/meter °C (or 8.5 x10-6 foot/foot °C). For every degree Centigrade a meter of glass is heated, it will expand 8.5×10-6 meters, and for every degree it is cooled, it will shrink 8.5 x10-6 meters. If you consider the circumference of the glass to be L (measured in meters), then
∆L/L = α ∆T.

where ∆L is the change in length due to heating, and ∆L/L is sometimes called the “strain.”. Now, lets call the amount of stress caused by this expansion σ, sigma, measured in psi or GPa. It is proportional to the strain, ∆L/L, and to the elasticity constant, E (also called Young’s elastic constant).

σ = E ∆L/L.

For glass, Young’s elasticity constant, E = 75 GPa. Since strain was equal to α ∆T, we find that

σ =Eα ∆T 

Thus, for glass and a ∆T of 100 °C, σ =100°C x 75 GPa x 8.5 x10-6 /°C  = 0.064  GPa = 64MPa. This is about 640 atm, or 9500 psi.

As it happens, the ultimate tensile strength of ordinary glass is only about 40 MPa =  σu. This, the maximum force per area you can put on glass before it breaks, is less than the thermal stress. You can expect a break here, and wherever σu < Eα∆T. I thus create a characteristic temperature difference for thermal stress failure:

The Buxbaum failure temperature, ß = σu/Eα

If ∆T of more than ß is applied to any material, you can expect a thermal stress failure.

The Wikipedia article referenced above provides a ratio for thermal resistance. The usits are perhaps heat load per unit area and time. How you would use this ratio I don’t quite know, it includes k, the thermal conductivity and ν, the Poisson ratio. Including the thermal conductivity here only makes sense, to me, if you think you’ll have a defined thermal load, a defined amount of heat transfer per unit area and time. I don’t think this is a normal way to look at things.  As for including the Poisson ratio, this too seems misunderstanding. The assumption is that a high Poisson ratio decreases the effect of thermal stress. The thought behind this, as I understand it, is that heating one side of a curved (the inside for example) will decrease the thickness of that side, reducing the effective stress. This is a mistake, I think; heating never decreases the thickness of any part being heated, but only increases the thickness. The heated part will expand in all directions. Thus, I think my ratio is the correct one. Please find following a list of failure temperatures for various common materials. 

Stress strain properties of engineering materials including thermal expansion, ultimate stress, MPa, and Youngs elastic modulus, GPa.

You will notice that most materials are a lot more resistant to thermal stress than glass is and some are quite a lot less resistant. Based on the above, we can expect that ice will fracture at a temperature difference as small as 1°C. Similarly, cast iron will crack with relatively little effort, while steel is a lot more durable (I hope that so-called cast iron skillets are really steel skillets). Pyrex is a form of glass that is more resistant to thermal breakage; that’s mainly because for pyrex, α is a lot smaller than for ordinary, cheap glass. I find it interesting that diamond is the material most resistant to thermal failure, followed by invar, a low -expansion steel, and ordinary rubber.

Robert E. Buxbaum, July 3, 2019. I should note that, for several of these materials, those with very high thermal conductivities, you’d want to use a very thick sample of materials to produce a temperature difference of 100*C.

Why concrete cracks and why sealing is worthwhile

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The oil tanker Palo Alto is one of several major ships made with concrete hulls.

The oil tanker Palo Alto is one of several major ships made with concrete hulls.

Modern concrete is a wonderful construction material. Major buildings are constructed of it, and major dams, and even some ships. But under the wrong circumstances, concrete has a surprising tendency to crack and fail. I thought I’d explain why that happens and what you can do about it. Concrete does not have to crack easily; ancient concrete didn’t and military or ship concrete doesn’t today. A lot of the fault lies in the use of cheap concrete — concrete with lots of filler — and with the cheap way that concrete is laid. First off, the major components of modern concrete are pretty uniform: sand and rock, Portland cement powder (made from cooked limestone, mostly), water, air, and sometimes ash. The cement component is what holds it all together — cements it together as it were — but it is not the majority of even the strongest concretes. The formula of cement has changed too, but the cement is not generally the problem. It doesn’t necessarily stick well to the rock or sand component of concrete (It sticks far better to itself) but it sticks well enough that spoliation, isn’t usually a problem by itself.

What causes problem is that the strength of concrete is strongly affected (decreased) by having lots of sand, aggregate and water. The concrete used in sidewalks is as cheap as possible, with lots of sand and aggregate. Highway and wall concrete has less sand and aggregate, and is stronger. Military and ship concrete has little sand, and is quite a lot stronger. The lowest grade, used in sidewalks, is M5, a term that refers to its compressive strength: 5 Mega Pascals. Pascals are European (Standard International) units of pressure and of strength. One Pascal is one Newton per square meter (Here’ a joke about Pascal units). In US (English) units, 5 MPa is 50 atm or 750 psi.

Ratios for concrete mixes of different strength.

Ratios for concrete mixes of different strength; the numbers I use are double these because these numbers don’t include water; that’s my “1”.

The ratio of dry ingredients in various concretes is shown at right. For M5, and including water, the ratio is 1 2 10 20. That is to say there is one part water, two parts cement, 10 parts sand, and 20 parts stone-aggregate (all these by weight). Added to this is 2-3% air, by volume, or nearly as much air as water. At least these are the target ratios; it sometimes happens that extra air and water are added to a concrete mix by greedy or rushed contractors. It’s sometimes done to save money, but more often because the job ran late. The more the mixer turns the more air gets added. If it turns too long there is extra air. It the job runs late, workers will have to add extra water too because the concrete starts hardening. I you see workers hosing down wet concrete as it comes from the truck, this is why. As you might expect, extra air and water decrease the strength of the product. M-10 and M-20 concrete have less sand, stone, and water as a proportion to cement. The result is 10 MPa or 20 MPa strength respectively.

A good on-site inspector is needed to keep the crew from adding too much water. Some water is needed for the polymerization (setting) of the concrete. The rest is excess, and when it evaporates, it leaves voids that are similar to the voids created by having air mix in. It is not uncommon to find 6% voids, in commercial concrete. This is to say that, after the water evaporates, the concrete contains about as much void as cement by volume. To get a sense of how much void space is in the normal concrete outside your house, go outside to a piece of old concrete (10 years old at least) on a hot, dry day, and pour out a cup of water. You will hear a hiss as the water absorbs, and you will see bubbles come out as the water goes in. It used to be common for cities to send inspectors to measuring the void content of the wet (and dry) concrete by a technique called “pycnometry” (that’s Greek for density measurement). I’ve not seen a local city do this in years, but don’t know why. An industrial pycnometer is shown below.

Pyncnometer used for concrete. I don't see these in use much any more.

Pycnometer used for concrete. I don’t see these in use much any more.

One of the main reason that concrete fails has to do with differential expansion, thermal stress, a concept I dealt with some years ago when figuring out how cold it had to be to freeze the balls off of a brass monkey. As an example of the temperature change to destroy M5, consider that the thermal expansion of cement is roughly 1 x 10-5/ °F or 1.8 x10-5/°C. This is to say that a 1 meter slab of cement that is heated or cooled by 100°F will expand or shrink by 10-3 m respectively; 100 x 1×10-5 = 10-3. This is a fairly large thermal expansion coefficient, as these things go. It would not cause stress-failure except that sand and rock have a smaller thermal expansion coefficients, about 0.6×10-5 — barely more than half the value for cement. Consider now what happens to concrete that s poured in the summer when it is 80°F out, and where the concrete heats up 100°F on setting (cement setting releases heat). Now lets come back in winter when it’s 0°F. This is a total of 100°F of temperature change. The differential expansion is 0.4 x 10-5/°F x 100°F =  4 x10-4 meter/meter = 4 x10-4 inch/inch.

The force created by this differential expansion is the elastic modulus of the cement times the relative change in expansion. The elastic modulus for typical cement is 20 GPa or, in English units, 3 million psi. This is to say that, if you had a column of cement (not concrete), one psi of force would compress it by 1/3,000,000. The differential expansion we calculated, cement vs sand and stone is 4×10-4 ; this much expansion times the elastic modulus, 3,000,000 = 1200 psi. Now look at the strength of the M-5 cement; it’s only 750 psi. When M-5 concrete is exposed to these conditions it will not survive. M-10 will fail on its own, from the temperature change, without any help needed from heavy traffic. You’d really like to see cities check the concrete, but I’ve seen little evidence that they do.

Water makes things worse, and not only because it creates voids when it evaporates. Water also messes up the polymerization reaction of the cement. Basic, fast setting cement is mostly Ca3SiO5

2Ca3SiO5 + 6 H2O –> 3Ca0SiO2•H2O +3Ca(OH)2•H2O.

The former of these, 3Ca0SiO2•H2O, forms something of a polymer. Monomer units of SiO4 are linked directly or by partially hydrated CaO linkages. Add too much water and the polymeric linkages are weakened or do not form at all. Over time the Ca(OH)2 can drain away or react with  CO2 in the air to form chalk.

concrete strength versus-curing time. Slow curing of damp concrete helps; fast dry hurts. Carbonate formation adds little or no strength. Jehan Elsamni 2011.

Portland limestone cement strength versus curing time. Slow curing and damp helps; fast dry hurts. Carbonate formation adds little or no strength. Jehan Elsamni 2011.

Ca(OH)2 + CO2 → CaCO3 + H2O

Sorry to say, the chalk adds little or no strength, as the graph at right shows. Concrete made with too much water isn’t very strong at all, and it gets no stronger when dried in air. Hardening goes on for some weeks after pouring, and this is the reason you don’t drive on 1 too 2 day old concrete. Driving on weak concrete can cause cracks that would not form if you waited.

You might think to make better concrete by pouring concrete in the cold, but pouring in the cold makes things worse. Cold poured cement will expand the summer and the cement will detach from the sand and stone. Ideally, pouring should be in spring or fall, when the temperature is moderate, 40-60°F. Any crack that develops grows by a mechanism called Rayleigh crack growth, described here. Basically, once a crack starts, it concentrates the fracture forces, and any wiggling of the concrete makes the crack grow faster.

Based on the above, I’ve come to suspect that putting on a surface coat can (could) help strengthen old concrete, even long after it’s hardened. Mostly this would happen by filling in voids and cracks, but also by extending the polymer chains. I imagine it would be especially helpful to apply the surface coat somewhat watery on a dry day in the summer. In that case, I imagine that Ca3SiO5 and Ca(OH)2 from the surface coat will penetrate and fill the pores of the concrete below — the sales pores that hiss when you pour water on them. I imagine this would fill cracks and voids, and extend existing CaOSiO2•H2O chains. The coat should add strength, and should be attractive as well. At least that was my thought.

I should note that, while Portland cement is mostly Ca3SiO5, there is also a fair amount (25%) of Ca2SiO4. This component reacts with water to form the same calcium-silicate polymer as above, but does so at a slower rate using less water per gram. My hope was that this component would be the main one to diffuse into deep pores of the concrete, reacting there to strengthen the concrete long after surface drying had occurred.

Trump tower: 664', concrete and glass. What grade of concrete would you use?

Trump tower: 664′, concrete and glass. What grade of concrete would you use?

As it happened, I had a chance to test my ideas this summer and also about 3 years ago. The city inspector came by to say the concrete flags outside my house were rough, and thus needed replacing, and that I was to pay or do it myself. Not that I understand the need for smooth concrete, quite, but that’s our fair city. I applied for a building permit to apply a surface coat, and applied it watery. I used “Quickrete” brand concrete patch, and so far it’s sticking OK. Pock-holes in the old concrete have been filled in, and so far surface is smooth. We’ll have to see if my patch lasts 10-20 years like fresh cement. Otherwise, no matter how strong the concrete becomes underneath, the city will be upset, and I’ll have to fix it. I’ve noticed that there is already some crumbling at the sides of flags, something I attribute to the extra water. It’s not a problem yet, but hope this is not the beginning of something worse. If I’m wrong here, and the whole seal-coat flakes off, I’ll be stuck replacing the flags, or continuing to re-coat just to preserve my reputation. But that’s the cost of experimentation. I tried something new, and am blogging about it in the hope that you and I benefit. “Education is what you get when you don’t get what you want.” (It’s one of my wise sayings). At the worst, I’ll have spent 90 lb of patching cement to get an education. And, I’m happy to say that some of the relatively new concrete flags that the city put in are already cracked. I attribute this to: too much sand, air, water or air (they don’t look like they have much rock): Poor oversight.

Dr. Robert E. Buxbaum. March 5, 2019. As an aside, the 664 foot Trump Tower, NY is virtually the only skyscraper in the city to be built of concrete and glass. The others are mostly steel and glass. Concrete and glass is supposed to be stiffer and quieter. The engineer overseeing the project was Barbara Res, the first woman to oversee a major, NY building project. Thought question: if you built the Trump Tower, which quality of concrete would you use, and why.