The theoretical strength of a solid is the maximum possible stress a perfect solid can withstand. It is often much higher than what current real materials can achieve. The lowered fracture stress is due to defects, such as interior or surface cracks. One of the goals for the study of mechanical properties of materials is to design and fabricate materials exhibiting strength close to the theoretical limit.
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
\sigmath
The theoretical strength is often approximated as: [1] [2]
\sigmath\cong
E | |
10 |
\sigmath
The stress-displacement, or
\sigma
\sigma=\sigmathsin(2\pix/λ)
λ
\sigma
\left(
d\sigma | |
dx |
\right)x=0=\left(
d\sigma | |
d\epsilon |
\right)x=0\left(
d\epsilon | |
dx |
\right)x=0=E\left(
d\epsilon | |
dx |
\right)x=0
\sigma
\epsilon
The strain
\epsilon
\epsilon=x/a0
a0
\left(
d\epsilon | |
dx |
\right)x=0=1/a0
The relationship of initial slope of the
\sigma
\left(
d\sigma | |
dx |
\right)x=0=E/a0
The sinusoidal relationship of stress and displacement gives a derivative:
\left(
d\sigma | |
dx |
\right)=\left(
2\pi | |
λ |
\right)\sigmathcos\left(
2\pix | |
λ |
\right)=\left(
2\pi\sigma | |
λ |
\right)x → 0
By setting the two
d\sigma/dx
\sigmath=
λE | |
2\pia0 |
\cong
E | |
2\pi |
\cong
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]