Theoretical strength of a solid

When a solid is in tension, its atomic bonds stretch, elastically. Once a critical strain is reached, all the atomic bonds on the fracture plane rupture and the material fails mechanically. The stress at which the solid fractures is the theoretical strength, often denoted as $\backslash sigma\_\{th\}$. After fracture, the stretched atomic bonds return to their initial state, except that two surfaces have formed.

The theoretical strength is often approximated as: {{Cite book|title=Mechanical behavior of materials|last=H.|first=Courtney, Thomas|date=2005|publisher=Waveland Press|isbn=978-1577664253|oclc=894800884}}{{Cite book|last1=Jin|first1=Z.|url=https://www.worldcat.org/oclc/770668002|title=Fracture mechanics|last2=Sun|first2=C.|date=2011|publisher=Academic Press|isbn=978-0-12-385001-0|location=Waltham, MA|pages=11–14|oclc=770668002}}

:$\backslash sigma\_\{th\}\; \backslash cong\; \backslash frac\{E\}\{10\}$

where

• $\backslash sigma\_\{th\}$ is the maximum theoretical stress the solid can withstand.
• E is the Young's Modulus of the solid.

The stress-displacement, or $\backslash sigma$ vs x, relationship during fracture can be approximated by a sine curve, $\backslash sigma\; =\; \backslash sigma\_\{th\}\; sin(2\backslash pi\; x/\backslash lambda)$, up to $\backslash lambda$/4. The initial slope of the $\backslash sigma$ vs x curve can be related to Young's modulus through the following relationship:

:$\backslash left\; (\backslash frac\{d\backslash sigma\}\{dx\}\; \backslash right\; )\_\{x=0\}\; =\; \backslash left\; (\backslash frac\{d\backslash sigma\}\{d\backslash epsilon\}\; \backslash right\; )\_\{x=0\}\; \backslash left\; (\backslash frac\{d\backslash epsilon\}\{dx\}\; \backslash right\; )\_\{x=0\}\; =\; E\backslash left\; (\backslash frac\{d\backslash epsilon\}\{dx\}\; \backslash right\; )\_\{x=0\}$

where

• $\backslash sigma$ is the stress applied.
• E is the Young's Modulus of the solid.
• $\backslash epsilon$ is the strain experienced by the solid.
• x is the displacement.

The strain $\backslash epsilon$ can be related to the displacement x by $\backslash epsilon\; =\; x/a\_0$, and $a\_0$ is the equilibrium inter-atomic spacing. The strain derivative is therefore given by

$\backslash left\; (\backslash frac\{d\backslash epsilon\}\{dx\}\; \backslash right)\_\{x=0\}\; =\; 1/a\_0$

The relationship of initial slope of the $\backslash sigma$ vs x curve with Young's modulus thus becomes

:$\backslash left\; (\backslash frac\{d\backslash sigma\}\{dx\}\; \backslash right\; )\_\{x=0\}\; =\; E/a\_0$

The sinusoidal relationship of stress and displacement gives a derivative:

:$\backslash left\; (\backslash frac\{d\backslash sigma\}\{dx\}\; \backslash right\; )\; =\; \backslash left\; (\backslash frac\{2\backslash pi\}\{\backslash lambda\}\; \backslash right\; )\; \backslash sigma\_\{th\}\; cos\backslash left\; (\backslash frac\{2\backslash pi\; x\}\{\backslash lambda\}\; \backslash right\; )\; =\; \backslash left\; (\backslash frac\{2\backslash pi\; \backslash sigma\}\{\backslash lambda\}\; \backslash right\; )\_\{x\backslash rightarrow0\}$

By setting the two $d\backslash sigma/dx$ together, the theoretical strength becomes:

:$\backslash sigma\_\{th\}\; =\; \backslash frac\{\backslash lambda\; E\}\{2\backslash pi\; a\_0\}\; \backslash cong\; \backslash frac\{E\}\{2\backslash pi\}\; \backslash cong\; \backslash frac\{E\}\{10\}$

The theoretical strength can also be approximated using the fracture work per unit area, which result in slightly different numbers. However, the above derivation and final approximation is a commonly used metric for evaluating the advantages of a material's mechanical properties.{{Cite journal|last1=Wu|first1=Ge|last2=Chan|first2=Ka-Cheung|last3=Zhu|first3=Linli|last4=Sun|first4=Ligang|last5=Lu|first5=Jian|title=Dual-phase nanostructuring as a route to high-strength magnesium alloys|journal=Nature|volume=545|issue=7652|pages=80–83|doi=10.1038/nature21691|year=2017|pmid=28379942|bibcode=2017Natur.545...80W |s2cid=4463565 }}

• Strength of materials
• Fracture mechanics
• Solid mechanics
• Stress (mechanics)
• Ultimate tensile strength
• Fracture
• Creep (deformation)
• Material Failure Theory

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Category:Solid mechanics

Category:Materials science