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Strength of Materials

May 11, 2020

At a very early age, children discover than some things, like eggs, are very fragile, while things such as rocks are more likely to break things than to be broken themselves. Knowledge of the strength of materials becomes useful in life when you're doing such things as deciding which objects you can stand on safely to replace a light bulb. Some people find that the topic of material strength is so interesting that they become mechanical engineers and materials scientists.

I pondered the strength of materials in my youth when my bicycle hand brake (which I now surmise as having been die cast from zinc) fractured; but, I was more interested in physics, and that's what I studied as an undergraduate. Lack of physics funding for graduate students in the early 1970s forced me to explore options other than physics. At the suggestion of one of my physics professors, I pivoted into a materials science Ph.D. program. As I like to joke, a physicist is the type of scientist who tries to find steel in the periodic table. Some of the course topics were quite foreign to me, but I began to acclimate. Eventually, there were quite a few things that captured my interest, such as chemical thermodynamics.

After some post-doctoral research, I accepted a position in industrial research under the directorship of one of the country's most prominent materials scientists, J. J. (Jack) Gilman (1925-2009). Gilman started as a mechanical engineer, but received a Ph.D. in physical metallurgy from Columbia University (New York, New York). After spending time in various industrial research organizations, he became a professor of engineering at Brown University (Providence, Rhode Island) in 1960, and then a professor of physics and metallurgy at the University of Illinois in 1963.[1]

Gilman became director of the Allied Chemical Materials Research Center in 1968, and he remained at Allied for somewhat more than a decade.[1] At Allied, he promoted research and development of the solid state laser material, Alexandrite (chromium-doped Chrysoberyl), and metallic glasses for use as core materials for power transformers, and as magnetic shielding. In his later career, Gilman served as director of the Center for Advanced Materials at the Lawrence Berkeley Laboratory and a member of the Department of Materials Science and Engineering at UCLA.[1]

J.J. (Jack) Gilman

J. J. (Jack) Gilman (right), and his office at the Allied Chemical Materials Research Center (left), circa 1975. Note the ashtrays on the desk, since many people were still smokers in the 1970s. It was in this office in 1977 that Jack and I had a gentleman scientists' disagreement about the existence of short-range order in metallic glasses.

Gilman published four books, more than 250 scientific papers, and 73 articles on industrial management.[1] His publications included a theory of ball lightning.[2] He was a fellow of the American Physical Society and the American Society for Metals in 1971, and a member of the National Academy of Engineering.[1]

(Photographs from Allied Chemical promotional materials. Click for larger image.)

Material strength is related to scratch resistance. Testing a diamond by scratching glass is a plot point in many old detective novels and B-movies, the idea being that glass jewelry won't scratch glass, but diamond will. This scratch test is useless, for more reason than the present complications of cubic zirconia (yttria-stabilized zirconia, YSZ) being used in jewelry that simulates diamonds, and the presumed inferiority of synthetic diamonds over natural diamonds. Many glasses, however, have always been able to scratch other glasses, and there are many other materials that will scratch glass. Window glass has a hardness of about 5.5, but there are other glasses with hardness up to 6.6, YSZ has a hardness of 8.0, and YAG (yttrium aluminum garnet) has a hardness of 8.5.

The idea that harder materials will scratch softer materials was mentioned in antiquity by Theophrastus in his treatise, On Stones, and Pliny the Elder in his Naturalis Historia (77 AD), both of which have been mentioned in this blog. The scratch test of mineral hardness was placed on a scientific basis in 1812 by German mineralogist, Friedrich Mohs (1773-1839), who created what's now called the Mohs scale of mineral hardness. The Mohs scale is an ordinal scale of minerals in which members of higher rank will scratch members of lower rank (see figure).

Figure caption

Relating hardness scales. The Vickers hardness test produces a material hardness number by pressing a pyrimidal indenter into the surface and measuring the dimensions of the indentation.

This graph relates Mohs hardness and Vickers hardness for the ten Mohs minerals, Talc, Mg3Si4O10(OH)2; Gypsum, CaSO4·2H2O; Calcite, CaCO3; Fluorite, CaF2; Apatite, Ca5(PO4)3(OH-,Cl-,F-); Orthoclase, KAlSi3O8; Quartz, SiO2; Topaz, Al2SiO4(OH-,F-)2; Corundum, Al2O3; and Diamond, C.

(Data from Gilman,[3] graphed using Inkscape. Click for larger image.)

At the most fundamental level, a material's strength arises from the bonding energy that hold its atoms together. In 1918, shortly after the lattice structure of crystals was confirmed, German physicist, Erwin Madelung (1881-1972), calculated the electrostatic forces that hold ionic crystals together. The calculation of this lattice energy, which is -786 kJ/mol for NaCl, gives a theoretical yield strength that's much larger than that observed. The theoretical yield strength, generally given in terms of shear strength G (see figure), is G/30, where G is the shear modulus. The experimental yield stress of NaCl crystals can be as low as 1 MPa, while the theoretical value is 1000 MPa.[4]

tension and shear

Pull and tug - Visualization of tension (left) and shear (right) stress acting on a body. (Created using Inkscape)

The fact that actual materials are not ideal crystals is the reason behind this diminished strength. Crystals are imperfect at the atomic level, and this imperfection is manifested as dislocations that allow easy glide across crystal planes. Polycrystalline materials and glass will have surface cracks that promote fracture, and these might be as small as a scratch. In 1921, aeronautical engineer, Alan Arnold Griffith (1893-1963), conducted experiments on freshly-drawn glass fibers in which he added a controlled flaw, a notch at the surface of the fibers.[5]

Griffith's data led to what's now called Griffith's criterion, that for brittle materials the product of the stress at fracture and the square root of the notch length is nearly a constant; that is,
σf√a ≈ C
where σf is the stress at fracture, a is the notch length, and C is a constant that depends on the energy required to create two new surfaces at each side of the fracture.

While Diamond is at the top of the hardness scales, with a Mohs hardness of 10 and a Vickers hardness up to 150 GPa, materials scientists are on a quest for other superhard materials that might be useful for cutting tools and wear-resistant coatings. Some limitations of diamond, aside from its high cost, is that it oxidizes at temperatures above 800 °C and it's inefficient in cutting ferrous alloys since diamond dissolves in iron to form iron carbides at high temperatures. Since the strength of diamond arises from its short, directional covalent bonds between carbon atoms, there's hope that similarly bonded compounds of low atomic number elements like boron, carbon, nitrogen, and oxygen will yield high hardness.

C3N4 and B-C-N ternary compounds were the first of these materials to be investigated, followed by some transition metal borides, such as ReB2, OsB2, and WB4. In theory, beta-carbon nitride (β-C3N4) may be harder, but it has not been synthesized. Cubic boron nitride (β-BN) is nearly as hard as diamond; in fact, it will scratch diamond. rhenium diboride (ReB2) is another material that scratches diamond.

Rhenium diboride is an excellent candidate for a superhard material, since the electronegativities of rhenium (1.9) and boron (2.04) are quite close, so their bonds have a mostly covalent character. The superhard properties of this material, which is synthesized at ambient pressures, were discovered in 2007.[6] Microindentation hardness tests showed an average hardness of 48 GPa, and X-ray diffraction analysis gave a bulk modulus of 360 GPa, which is not that much lower than that of diamond (445 GPa).[6-7] However, there are reasons to believe that diamond will all reign as the king of the superhards.[7]

While superhards will find application as cutting tools, most useful high strength components are made from metal alloys, not carbides and borides. In an open access paper in a recent issue of Physical Review letters, Michael Chandross and Nicolas Argibay of Sandia National Laboratories (Albuquerque, New Mexico), present a simple theory that allows prediction of the ultimate strength of pure metals and alloys.[8-10] Unlike most theories of this type, their theory does not need fitting parameters.[10] It's based on thermodynamics and the concept of amorphization, and it closely predicts the ultimate strength of nearly 20 different metals (see graph).[8-10]

Chandross and Argibay, PRL. 2020, fig_3

Sandia National Laboratories material shear strength model. (From fig. 3 of ref. 8, licensed under the Creative Commons Attribution 4.0 International license.[8])

Metals are polycrystalline, being made from small crystal grains that are bonded together at grain boundaries that have an influence on their strength. The peak strength of a metals generally occurs for a grain size between 10-20 nm.[9] Large grains lead to softer metals, since dislocations can more more easily. The Hall-Petch law predicts that metal strength will increase as grain size is reduced,[11] but grains smaller than about 10 nm are associated with lower strength.[9] At about 10 nm, the grain boundaries become more important, and the conclusion is that the less ordered nature of the grain boundary material is responsible, since amorphous materials are softer.[9]

Chandross and Argibay developed a model based on the activation energy needed to cause deformation by amorphization that predicts the peak strength of polycrystalline metals.[8] The model is based on material properties alone.[8] The model predicts the strength of the three principal crystal types, body-centered cubic, hexagonal close packed, and face-centered cubicfcc, and also one alloy.[8] Chandross and Argibay make the assumption that the energy needed to completely disorder a grain boundary is equal to the energy needed to melt it, and this would be the activation energy for grain boundary sliding.[9] The result is an equation for the strength of a metal in terms of its melting point, heat of fusion, and the volume fraction of grain boundaries.[9]

Crystal to amorphous transition of a lattice

Transformation of a crystalline metal (left) to an amorphous material (right), with an energy difference related to the heat of fusion. (Image from an American Physical Society Press Release, reformatted for clarity.[10])

Their model shows that the crystalline grains of a polycrystalline metal and its grain boundaries have equal strength near a grain size of about 10 nm, consistent with experiments.[9] These results might be extended to prediction of the glass transition in polymers.[9] since the input parameters of the model are usually known, it should be possible to rapidly screen alloy compositions to determine the optimum grain size for maximum strength.[9]


  1. John D. Mackenzie, "John J. Gilman 1925–2009," Memorial Tributes: Volume 14 (National Academies Press, 2011), pp.108ff..
  2. John Gilman, "Ball Lightning and Plasma Cohesion," arXiv, February 19, 2003.
  3. J. J. Gilman, "Hardness of pure alkali halides," Journal of Applied Physics, vol. 44, no. 3 (March, 1973), pp. 982ff., https://doi.org/10.1063/1.1662382.
  4. John J. Gilman, "Micromechanics of Shear Banding," Lawrence Berkeley Laboratory Report (August 1992), and the Proceedings of the Twenty-Ninth Annual Technical Meeting of the Society of Engineering Science (San Diego, CA, September 14, 1992), PDF file.
  5. A.A. Griffith, A. A. (1921), "The phenomena of rupture and flow in solids," Philosophical Transactions of the Royal Society of London, vol. A221 (1921), pp. 163–198. Available here, also.
  6. Hsiu-Ying Chung, Michelle B. Weinberger, Jonathan B. Levine, Abby Kavner, Jenn-Ming Yang, Sarah H. Tolbert, and Richard B. Kaner, "Synthesis of Ultra-Incompressible Superhard Rhenium Diboride at Ambient Pressure," Science, vol. 316. no. 5823 (20 April 2007), pp. 436-439.
  7. Vadim V. Brazhkin1,a) and Vladimir L. Solozhenko, "Myths about new ultrahard phases: Why materials that are significantly superior to diamond in elastic moduli and hardness are impossible," Journal of Applied Physics, vol. 125, no. 13 (April 3, 2019, Article no. 130901, https://doi.org/10.1063/1.5082739. Also at arXiv
  8. Michael Chandross and Nicolas Argibay, "Ultimate Strength of Metals," Phys. Rev. Lett., vol. 124, no. 12 (March 27, 2020), Article no. 125501, DOI:https://doi.org/10.1103/PhysRevLett.124.125501.
  9. Christopher A. Schuh, "Viewpoint: Deadlocked Order and Disorder in the Strongest Metals," Physics, vol 13, no. 4 (March 25, 2020)
  10. Ultimate strength of metals, American Physical Society Press Release, March 25, 2020.
  11. E O Hall, "The Deformation and Ageing of Mild Steel: III Discussion of Results," Proceedings of the Physical Society, Section B, vol. 64, no. 9 (September 1, 1951), pp. 747ff., https://doi.org/10.1088/0370-1301/64/9/303.

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