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 ruptures and the material mechanically fails. The stress at which the solid fractures is the theoretical strength, often denoted as ${\displaystyle \sigma _{th}}$. After fracture, the stretched atomic bonds return to its initial state, except that two surfaces has formed.

The theoretical strength is often approximated as: [1][2]

${\displaystyle \sigma _{th}\cong {\frac {E}{10}}}$

where

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

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

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

where

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

The strain ${\displaystyle \epsilon }$ can be related to the displacement x by ${\displaystyle \epsilon =x/a_{0}}$, and ${\displaystyle a_{0}}$ is the equilibrium inter-atomic spacing. The strain derivative is therefore given by ${\displaystyle \left({\frac {d\epsilon }{dx}}\right)_{x=0}=1/a_{0}}$

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

${\displaystyle \left({\frac {d\sigma }{dx}}\right)_{x=0}=E/a_{0}}$

The sinusoidal relationship of stress and displacement gives a derivative:

${\displaystyle \left({\frac {d\sigma }{dx}}\right)=\left({\frac {2\pi }{\lambda }}\right)\sigma _{th}cos\left({\frac {2\pi x}{\lambda }}\right)=\left({\frac {2\pi \sigma }{\lambda }}\right)_{x\rightarrow 0}}$

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

${\displaystyle \sigma _{th}={\frac {\lambda E}{2\pi a_{0}}}\cong {\frac {E}{2\pi }}\cong {\frac {E}{10}}}$

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.[3]

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