Solid solution strengthening

Last updated May 09, 2024

In metallurgy, solid solution strengthening is a type of alloying that can be used to improve the strength of a pure metal.^[1] The technique works by adding atoms of one element (the alloying element) to the crystalline lattice of another element (the base metal), forming a solid solution. The local nonuniformity in the lattice due to the alloying element makes plastic deformation more difficult by impeding dislocation motion through stress fields. In contrast, alloying beyond the solubility limit can form a second phase, leading to strengthening via other mechanisms (e.g. the precipitation of intermetallic compounds).

Types

Depending on the size of the alloying element, a substitutional solid solution or an interstitial solid solution can form.^[2] In both cases, atoms are visualised as rigid spheres where the overall crystal structure is essentially unchanged. The rationale of crystal geometry to atom solubility prediction is summarized in the Hume-Rothery rules and Pauling's rules.

Substitutional solid solution strengthening occurs when the solute atom is large enough that it can replace solvent atoms in their lattice positions. Some alloying elements are only soluble in small amounts, whereas some solvent and solute pairs form a solution over the whole range of binary compositions. Generally, higher solubility is seen when solvent and solute atoms are similar in atomic size (15% according to the Hume-Rothery rules) and adopt the same crystal structure in their pure form. Examples of completely miscible binary systems are Cu-Ni and the Ag-Au face-centered cubic (FCC) binary systems, and the Mo-W body-centered cubic (BCC) binary system.

Interstitial solid solutions form when the solute atom is small enough (radii up to 57% the radii of the parent atoms)^[2] to fit at interstitial sites between the solvent atoms. The atoms crowd into the interstitial sites, causing the bonds of the solvent atoms to compress and thus deform (this rationale can be explained with Pauling's rules). Elements commonly used to form interstitial solid solutions include H, Li, Na, N, C, and O. Carbon in iron (steel) is one example of interstitial solid solution.

Mechanism

The strength of a material is dependent on how easily dislocations in its crystal lattice can be propagated. These dislocations create stress fields within the material depending on their character. When solute atoms are introduced, local stress fields are formed that interact with those of the dislocations, impeding their motion and causing an increase in the yield stress of the material, which means an increase in strength of the material. This gain is a result of both lattice distortion and the modulus effect.

When solute and solvent atoms differ in size, local stress fields are created that can attract or repel dislocations in their vicinity. This is known as the size effect. By relieving tensile or compressive strain in the lattice, the solute size mismatch can put the dislocation in a lower energy state. In substitutional solid solutions, these stress fields are spherically symmetric, meaning they have no shear stress component. As such, substitutional solute atoms do not interact with the shear stress fields characteristic of screw dislocations. Conversely, in interstitial solid solutions, solute atoms cause a tetragonal distortion, generating a shear field that can interact with edge, screw, and mixed dislocations. The attraction or repulsion of the dislocation to the solute atom depends on whether the atom sits above or below the slip plane. For example, consider an edge dislocation encountering a smaller solute atom above its slip plane. In this case, the interaction energy is negative, resulting in attraction of the dislocation to the solute. This is due to the reduced dislocation energy by the compressed volume lying above the dislocation core. If the solute atom were positioned below the slip plane, the dislocation would be repelled by the solute. However, the overall interaction energy between an edge dislocation and a smaller solute is negative because the dislocation spends more time at sites with attractive energy. This is also true for solute atom with size greater than the solvent atom. Thus, the interaction energy dictated by the size effect is generally negative.^[3]

The elastic modulus of the solute atom can also determine the extent of strengthening. For a “soft” solute with elastic modulus lower than that of the solvent, the interaction energy due to modulus mismatch (U_modulus) is negative, which reinforce the size interaction energy (U_size). In contrast, U_modulus is positive for a “hard” solute, which results in lower total interaction energy than a soft atom. Even though the interaction force is negative (attractive) in both cases when the dislocation is approaching the solute. The maximum force (F_max) necessary to tear dislocation away from the lowest energy state (i.e. the solute atom) is greater for the soft solute than the hard one. As a result, a soft solute will strengthen a crystal more than a hard solute due to the synergistic strengthening by combining both size and modulus effects.^[3]

The elastic interaction effects (i.e. size and modulus effects) dominate solid-solution strengthening for most crystalline materials. However, other effects, including charge and stacking fault effects, may also play a role. For ionic solids where electrostatic interaction dictates bond strength, charge effect is also important. For example, addition of divalent ion to a monovalent material may strengthen the electrostatic interaction between the solute and the charged matrix atoms that comprise a dislocation. However, this strengthening is to a less extent than the elastic strengthening effects. For materials containing a higher density of stacking faults, solute atoms may interact with the stacking faults either attractively or repulsively. This lowers the stacking fault energy, leading to repulsion of the partial dislocations, which thus makes the material stronger.^[3]

Surface carburizing, or case hardening, is one example of solid solution strengthening in which the density of solute carbon atoms is increased close to the surface of the steel, resulting in a gradient of carbon atoms throughout the material. This provides superior mechanical properties to the surface of the steel without having to use a higher-cost material for the component.^[4]

Governing equations

Solid solution strengthening increases yield strength of the material by increasing the shear stress, $\tau$ , to move dislocations:^[1]^[2]

$\Delta \tau =Gb\epsilon ^{\tfrac {3}{2}}{\sqrt {c}}$

where c is the concentration of the solute atoms, G is the shear modulus, b is the magnitude of the Burger's vector, and $\epsilon$ is the lattice strain due to the solute. This is composed of two terms, one describing lattice distortion and the other local modulus change.

$\epsilon =|\epsilon _{G}-\beta \epsilon _{a}|$ Here, $\epsilon _{G}$ the term that captures the local modulus change, $\beta$ a constant dependent on the solute atoms and $\epsilon _{a}$ is the lattice distortion term.

The lattice distortion term can be described as:

$\epsilon _{a}={\dfrac {\Delta a}{a\Delta c}}$ , where a is the lattice parameter of the material.

Meanwhile, the local modulus change is captured in the following expression:

$\epsilon _{G}={\dfrac {\Delta G}{G\Delta c}}$ , where G is shear modulus of the solute material.

Implications

In order to achieve noticeable material strengthening via solution strengthening, one should alloy with solutes of higher shear modulus, hence increasing the local shear modulus in the material. In addition, one should alloy with elements of different equilibrium lattice constants. The greater the difference in lattice parameter, the higher the local stress fields introduced by alloying. Alloying with elements of higher shear modulus or of very different lattice parameters will increase the stiffness and introduce local stress fields respectively. In either case, the dislocation propagation will be hindered at these sites, impeding plasticity and increasing yield strength proportionally with solute concentration.

Solid solution strengthening depends on:

Concentration of solute atoms
Shear modulus of solute atoms
Size of solute atoms
Valency of solute atoms (for ionic materials)

For many common alloys, rough experimental fits can be found for the addition in strengthening provided in the form of:^[2]

$\Delta \sigma _{s}=k_{s}{\sqrt {c}}$

where $k_{s}$ is a solid solution strengthening coefficient and $c$ is the concentration of solute in atomic fractions.

Nevertheless, one should not add so much solute as to precipitate a new phase. This occurs if the concentration of the solute reaches a certain critical point given by the binary system phase diagram. This critical concentration therefore puts a limit to the amount of solid solution strengthening that can be achieved with a given material.

Examples

Aluminum alloys

An example of aluminum alloys where solid solution strengthening happens by adding magnesium and manganese into the aluminum matrix. Commercially Mn can be added to the AA3xxx series and Mg can be added to the AA5xxx series.^[5] Mn addition to the Aluminum alloys assists in the recrystallization and recovery of the alloy which influences the grain size as well.^[5] Both of these systems are used in low to medium-strength applications, with appreciable formability and corrosion resistance.^[6]

Nickel-based superalloys

Many nickel-based superalloys depend on solid solution as a strengthening mechanism. The most popular example is the Inconel family, where many of these alloys contain chromium and iron and some other additions of cobalt, molybdenum, niobium, and titanium.^[7] The nickel-based superalloys are well known for their intensive use in the industrial field especially the aeronautical and the aerospace industry due to their superior mechanical and corrosion properties at high temperatures.^[8]

An example of the use of the nickel-based superalloys in the industrial field would be turbine blades. In practice, this alloy is known as MAR—M200 and is solid solution strengthened by chromium, tungsten and cobalt in the matrix and is also precipitation hardened by carbide and boride precipitates at the grain boundaries^[9]^[10]. The key impacting factor for these turbine blades lies in the grain size which an increase in grain size can lead to a significant reduction in the strain rate. An example of this reduced strain rate in MAR--M200 can be seen in the figures to the right where the figure on the bottom has a grain size of 100um and the figure on the top has a grain size of 10mm^[11].

This reduced strain rate is extremely important for turbine blade operation because they undergo significant mechanical stress and high temperatures which can lead to the onset of creep deformation. Therefore, the precise control of grain size in nickel-based superalloys is key to creep resistance and mechanical reliability and longevity. Some ways to control the grain size lie in the manufacturing techniques like directional solidification and single crystal casting^[12].

Stainless steel

Stainless steel is one of the most commonly used metals in many industries. Solid solution strengthening of steel is one of the mechanisms used to enhance the properties of the alloy. Austenitic steels mainly contain chromium, nickel, molybdenum, and manganese.^[13] It is being used mostly for cookware, kitchen equipment, and in marine applications for its good corrosion properties in saline environments.

Titanium alloys

Titanium and titanium alloys have been wide usage in aerospace, medical, and maritime applications. The most known titanium alloy that adopts solid solution strengthening is Ti-6Al-4V. Also, the addition of oxygen to pure Ti alloy adopts a solid solution strengthening as a mechanism to the material, while adding it to Ti-6Al-4V alloy doesn’t have the same influence.^[14]

Copper alloys

Bronze and brass are both copper alloys that are solid solution strengthened. Bronze is the result of adding about 12% tin to copper while brass is the result of adding about 34% zinc to copper. Both of these alloys are being utilized in coins production, ship hardware, and art.

Related Research Articles

In materials science, creep is the tendency of a solid material to undergo slow deformation while subject to persistent mechanical stresses. It can occur as a result of long-term exposure to high levels of stress that are still below the yield strength of the material. Creep is more severe in materials that are subjected to heat for long periods and generally increase as they near their melting point.

In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to slide over each other at low stress levels and is known as glide or slip. The crystalline order is restored on either side of a glide dislocation but the atoms on one side have moved by one position. The crystalline order is not fully restored with a partial dislocation. A dislocation defines the boundary between slipped and unslipped regions of material and as a result, must either form a complete loop, intersect other dislocations or defects, or extend to the edges of the crystal. A dislocation can be characterised by the distance and direction of movement it causes to atoms which is defined by the Burgers vector. Plastic deformation of a material occurs by the creation and movement of many dislocations. The number and arrangement of dislocations influences many of the properties of materials.

In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.

Precipitation hardening, also called age hardening or particle hardening, is a heat treatment technique used to increase the yield strength of malleable materials, including most structural alloys of aluminium, magnesium, nickel, titanium, and some steels, stainless steels, and duplex stainless steel. In superalloys, it is known to cause yield strength anomaly providing excellent high-temperature strength.

A superalloy, or high-performance alloy, is an alloy with the ability to operate at a high fraction of its melting point. Key characteristics of a superalloy include mechanical strength, thermal creep deformation resistance, surface stability, and corrosion and oxidation resistance.

<span class="mw-page-title-main">Cottrell atmosphere</span> Concept in materials science

In materials science, the concept of the Cottrell atmosphere was introduced by A. H. Cottrell and B. A. Bilby in 1949 to explain how dislocations are pinned in some metals by boron, carbon, or nitrogen interstitials.

Hardening is a metallurgical metalworking process used to increase the hardness of a metal. The hardness of a metal is directly proportional to the uniaxial yield stress at the location of the imposed strain. A harder metal will have a higher resistance to plastic deformation than a less hard metal.

A solid solution, a term popularly used for metals, is a homogeneous mixture of two different kinds of atoms in solid state and having a single crystal structure. Many examples can be found in metallurgy, geology, and solid-state chemistry. The word "solution" is used to describe the intimate mixing of components at the atomic level and distinguishes these homogeneous materials from physical mixtures of components. Two terms are mainly associated with solid solutions – solvents and solutes, depending on the relative abundance of the atomic species.

In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and will return to its original shape when the applied stress is removed. Once the yield point is passed, some fraction of the deformation will be permanent and non-reversible and is known as plastic deformation.

In materials science, hardness is a measure of the resistance to localized plastic deformation, such as an indentation or a scratch (linear), induced mechanically either by pressing or abrasion. In general, different materials differ in their hardness; for example hard metals such as titanium and beryllium are harder than soft metals such as sodium and metallic tin, or wood and common plastics. Macroscopic hardness is generally characterized by strong intermolecular bonds, but the behavior of solid materials under force is complex; therefore, hardness can be measured in different ways, such as scratch hardness, indentation hardness, and rebound hardness. Hardness is dependent on ductility, elastic stiffness, plasticity, strain, strength, toughness, viscoelasticity, and viscosity. Common examples of hard matter are ceramics, concrete, certain metals, and superhard materials, which can be contrasted with soft matter.

<span class="mw-page-title-main">Nanocrystalline material</span>

A nanocrystalline (NC) material is a polycrystalline material with a crystallite size of only a few nanometers. These materials fill the gap between amorphous materials without any long range order and conventional coarse-grained materials. Definitions vary, but nanocrystalline material is commonly defined as a crystallite (grain) size below 100 nm. Grain sizes from 100 to 500 nm are typically considered "ultrafine" grains.

Methods have been devised to modify the yield strength, ductility, and toughness of both crystalline and amorphous materials. These strengthening mechanisms give engineers the ability to tailor the mechanical properties of materials to suit a variety of different applications. For example, the favorable properties of steel result from interstitial incorporation of carbon into the iron lattice. Brass, a binary alloy of copper and zinc, has superior mechanical properties compared to its constituent metals due to solution strengthening. Work hardening has also been used for centuries by blacksmiths to introduce dislocations into materials, increasing their yield strengths.

<span class="mw-page-title-main">Grain boundary strengthening</span> Method of strengthening materials by changing grain size

In materials science, grain-boundary strengthening is a method of strengthening materials by changing their average crystallite (grain) size. It is based on the observation that grain boundaries are insurmountable borders for dislocations and that the number of dislocations within a grain has an effect on how stress builds up in the adjacent grain, which will eventually activate dislocation sources and thus enabling deformation in the neighbouring grain as well. By changing grain size, one can influence the number of dislocations piled up at the grain boundary and yield strength. For example, heat treatment after plastic deformation and changing the rate of solidification are ways to alter grain size.

In materials science, segregation is the enrichment of atoms, ions, or molecules at a microscopic region in a materials system. While the terms segregation and adsorption are essentially synonymous, in practice, segregation is often used to describe the partitioning of molecular constituents to defects from solid solutions, whereas adsorption is generally used to describe such partitioning from liquids and gases to surfaces. The molecular-level segregation discussed in this article is distinct from other types of materials phenomena that are often called segregation, such as particle segregation in granular materials, and phase separation or precipitation, wherein molecules are segregated in to macroscopic regions of different compositions. Segregation has many practical consequences, ranging from the formation of soap bubbles, to microstructural engineering in materials science, to the stabilization of colloidal suspensions.

Oxide dispersion strengthened alloys (ODS) are alloys that consist of a metal matrix with small oxide particles dispersed within it. They have high heat resistance, strength, and ductility. Alloys of nickel are the most common but includes iron aluminum alloys.

Dislocation creep is a deformation mechanism in crystalline materials. Dislocation creep involves the movement of dislocations through the crystal lattice of the material, in contrast to diffusion creep, in which diffusion is the dominant creep mechanism. It causes plastic deformation of the individual crystals, and thus the material itself.

An antiphase domain (APD) is a type of planar crystallographic defect in which the atoms within a region of a crystal are configured in the opposite order to those in the perfect lattice system. Throughout the entire APD, atoms sit on the sites typically occupied by atoms of a different species. For example, in an ordered AB alloy, if an A atom occupies the site usually occupied by a B atom, a type of crystallographic point defect called an antisite defect is formed. If an entire region of the crystal is translated such that every atom in a region of the plane of atoms sits on its antisite, an antiphase domain is formed. In other words, an APD is a region formed from antisite defects of a parent lattice. On either side of this domain, the lattice is still perfect, and the boundaries of the domain are referred to as antiphase boundaries. Crucially, crystals on either side of an antiphase boundary are related by a translation, rather than a reflection or an inversion.

Dynamic strain aging (DSA) for materials science is an instability in plastic flow of materials, associated with interaction between moving dislocations and diffusing solutes. Although sometimes dynamic strain aging is used interchangeably with the Portevin–Le Chatelier effect (or serrated yielding), dynamic strain aging refers specifically to the microscopic mechanism that induces the Portevin–Le Chatelier effect. This strengthening mechanism is related to solid-solution strengthening and has been observed in a variety of fcc and bcc substitutional and interstitial alloys, metalloids like silicon, and ordered intermetallics within specific ranges of temperature and strain rate.

Nabarro–Herring creep is a mode of deformation of crystalline materials that occurs at low stresses and held at elevated temperatures in fine-grained materials. In Nabarro–Herring creep, atoms diffuse through the crystals, and the creep rate varies inversely with the square of the grain size so fine-grained materials creep faster than coarser-grained ones. NH creep is solely controlled by diffusional mass transport. This type of creep results from the diffusion of vacancies from regions of high chemical potential at grain boundaries subjected to normal tensile stresses to regions of lower chemical potential where the average tensile stresses across the grain boundaries are zero. Self-diffusion within the grains of a polycrystalline solid can cause the solid to yield to an applied shearing stress, the yielding being caused by a diffusional flow of matter within each crystal grain away from boundaries where there is a normal pressure and toward those where there is a normal tension. Atoms migrating in the opposite direction account for the creep strain. The creep strain rate is derived in the next section. NH creep is more important in ceramics than metals as dislocation motion is more difficult to effect in ceramics.

Anelasticity is a property of materials that describes their behaviour when undergoing deformation. Its formal definition does not include the physical or atomistic mechanisms but still interprets the anelastic behaviour as a manifestation of internal relaxation processes. It is a behaviour differing from elastic behaviour.

References

1 2 Pelleg, Joshua (2013). Mechanical Properties of Materials. New York: Springer. pp. 236–239. ISBN 978-94-007-4341-0.
1 2 3 4 Soboyejo, Wole O. (2003). "8.3 Solid Solution Strengthening". Mechanical properties of engineered materials. Marcel Dekker. ISBN 0-8247-8900-8. OCLC 300921090.
1 2 3 Courtney, Thomas H. (2005). Mechanical Behavior of Materials. Illinois: Waveland Press, Inc. pp. 186–195. ISBN 978-1-57766-425-3.
↑ Li, Donglong; Zhang, Mengqi; Xie, Lechun; Wang, Zhanjiang; Zhou, Zhongrong; Zhao, Ning; Palmer, David; Jane Wang, Q. (2020-07-15). "Contact Yield Initiation and Its Influence on Rolling Contact Fatigue of Case-Hardened Steels". Journal of Tribology. 142 (12). doi:10.1115/1.4047581. ISSN 0742-4787.
1 2 Ryen, Øyvind; Holmedal, Bjørn; Nijs, Oscar; Nes, Erik; Sjölander, Emma; Ekström, Hans-Erik (2006-06-01). "Strengthening mechanisms in solid solution aluminum alloys". Metallurgical and Materials Transactions A. 37 (6): 1999–2006. doi:10.1007/s11661-006-0142-7. ISSN 1543-1940.
↑ Zhao, Qinglong; Holmedal, Bjørn (2013-02-15). "The effect of silicon on the strengthening and work hardening of aluminum at room temperature". Materials Science and Engineering: A. 563: 147–151. doi:10.1016/j.msea.2012.11.062. hdl: 11250/2469312 . ISSN 0921-5093.
↑ Hodge, F. Galen (2006-09-01). "The history of solid-solution-strengthened Ni alloys for aqueous corrosion service". JOM. 58 (9): 28–31. Bibcode:2006JOM....58i..28H. doi:10.1007/s11837-006-0078-9. ISSN 1543-1851.
↑ Akca, Enes; Gürsel, Ali (2015-06-26). "A review on superalloys and IN718 nickel-based INCONEL superalloy". Periodicals of Engineering and Natural Sciences. 3 (1). doi: 10.21533/pen.v3i1.43 . ISSN 2303-4521.
↑ Gu, Shuning; Gao, Hangshan; Wen, Zhixun; Pei, Haiqing; Li, Zhenwei; Zhao, Yanchao; Yue, Zhufeng (December 2021). "Creep characteristics of directionally solidified turbine blades based on the difference in original casting characteristics". Journal of Alloys and Compounds. 884: 161055. doi:10.1016/j.jallcom.2021.161055. ISSN 0925-8388.
↑ Zhang, Chun-Yi; Wei, Jing-Shan; Wang, Ze; Yuan, Zhe-Shan; Fei, Cheng-Wei; Lu, Cheng (2019-10-29). "Creep-Based Reliability Evaluation of Turbine Blade-Tip Clearance with Novel Neural Network Regression". Materials. 12 (21): 3552. Bibcode:2019Mate...12.3552Z. doi: 10.3390/ma12213552 . ISSN 1996-1944. PMC 6861887 . PMID 31671898.
↑ "CHAPTER 19". defmech.engineering.dartmouth.edu. Retrieved 2024-05-08.
↑ Coudon, F.; Gourdin, S.; Boucicaud, A.; Rose, T.; Cailletaud, G. (February 2020). "A stochastic approach applied to directionally solidified turbine blades". International Journal of Solids and Structures. 184: 193–201. doi:10.1016/j.ijsolstr.2019.04.007. ISSN 0020-7683.
↑ Sieurin, Henrik; Zander, Johan; Sandström, Rolf (2006-01-15). "Modelling solid solution hardening in stainless steels". Materials Science and Engineering: A. 415 (1): 66–71. doi:10.1016/j.msea.2005.09.031. ISSN 0921-5093.
↑ Oh, J. -M.; Lee, B. -G.; Cho, S. -W.; Lee, S. -W.; Choi, G. -S.; Lim, J. -W. (2011-10-01). "Oxygen effects on the mechanical properties and lattice strain of Ti and Ti-6Al-4V". Metals and Materials International. 17 (5): 733–736. Bibcode:2011MMI....17..733O. doi:10.1007/s12540-011-1006-2. ISSN 2005-4149.

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[:0-1] 1 2 Pelleg, Joshua (2013). Mechanical Properties of Materials. New York: Springer. pp. 236–239. ISBN 978-94-007-4341-0.

[:2-2] 1 2 3 4 Soboyejo, Wole O. (2003). "8.3 Solid Solution Strengthening". Mechanical properties of engineered materials. Marcel Dekker. ISBN 0-8247-8900-8. OCLC 300921090.

[:1-3] 1 2 3 Courtney, Thomas H. (2005). Mechanical Behavior of Materials. Illinois: Waveland Press, Inc. pp. 186–195. ISBN 978-1-57766-425-3.

[4] Li, Donglong; Zhang, Mengqi; Xie, Lechun; Wang, Zhanjiang; Zhou, Zhongrong; Zhao, Ning; Palmer, David; Jane Wang, Q. (2020-07-15). "Contact Yield Initiation and Its Influence on Rolling Contact Fatigue of Case-Hardened Steels". Journal of Tribology. 142 (12). doi:10.1115/1.4047581. ISSN 0742-4787.

[:3-5] 1 2 Ryen, Øyvind; Holmedal, Bjørn; Nijs, Oscar; Nes, Erik; Sjölander, Emma; Ekström, Hans-Erik (2006-06-01). "Strengthening mechanisms in solid solution aluminum alloys". Metallurgical and Materials Transactions A. 37 (6): 1999–2006. doi:10.1007/s11661-006-0142-7. ISSN 1543-1940.

[6] Zhao, Qinglong; Holmedal, Bjørn (2013-02-15). "The effect of silicon on the strengthening and work hardening of aluminum at room temperature". Materials Science and Engineering: A. 563: 147–151. doi:10.1016/j.msea.2012.11.062. hdl: 11250/2469312 . ISSN 0921-5093.

[7] Hodge, F. Galen (2006-09-01). "The history of solid-solution-strengthened Ni alloys for aqueous corrosion service". JOM. 58 (9): 28–31. Bibcode:2006JOM....58i..28H. doi:10.1007/s11837-006-0078-9. ISSN 1543-1851.

[8] Akca, Enes; Gürsel, Ali (2015-06-26). "A review on superalloys and IN718 nickel-based INCONEL superalloy". Periodicals of Engineering and Natural Sciences. 3 (1). doi: 10.21533/pen.v3i1.43 . ISSN 2303-4521.

[9] Gu, Shuning; Gao, Hangshan; Wen, Zhixun; Pei, Haiqing; Li, Zhenwei; Zhao, Yanchao; Yue, Zhufeng (December 2021). "Creep characteristics of directionally solidified turbine blades based on the difference in original casting characteristics". Journal of Alloys and Compounds. 884: 161055. doi:10.1016/j.jallcom.2021.161055. ISSN 0925-8388.

[10] Zhang, Chun-Yi; Wei, Jing-Shan; Wang, Ze; Yuan, Zhe-Shan; Fei, Cheng-Wei; Lu, Cheng (2019-10-29). "Creep-Based Reliability Evaluation of Turbine Blade-Tip Clearance with Novel Neural Network Regression". Materials. 12 (21): 3552. Bibcode:2019Mate...12.3552Z. doi: 10.3390/ma12213552 . ISSN 1996-1944. PMC 6861887 . PMID 31671898.

[11] "CHAPTER 19". defmech.engineering.dartmouth.edu. Retrieved 2024-05-08.

[12] Coudon, F.; Gourdin, S.; Boucicaud, A.; Rose, T.; Cailletaud, G. (February 2020). "A stochastic approach applied to directionally solidified turbine blades". International Journal of Solids and Structures. 184: 193–201. doi:10.1016/j.ijsolstr.2019.04.007. ISSN 0020-7683.

[13] Sieurin, Henrik; Zander, Johan; Sandström, Rolf (2006-01-15). "Modelling solid solution hardening in stainless steels". Materials Science and Engineering: A. 415 (1): 66–71. doi:10.1016/j.msea.2005.09.031. ISSN 0921-5093.

[14] Oh, J. -M.; Lee, B. -G.; Cho, S. -W.; Lee, S. -W.; Choi, G. -S.; Lim, J. -W. (2011-10-01). "Oxygen effects on the mechanical properties and lattice strain of Ti and Ti-6Al-4V". Metals and Materials International. 17 (5): 733–736. Bibcode:2011MMI....17..733O. doi:10.1007/s12540-011-1006-2. ISSN 2005-4149.

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