Page 55 - Mechanical Behavior of Materials

54 Chapter 2 Structure and Deformation in Materials

Figure 2.18 Basis of estimates of theoretical shear strength, where it is assumed that entire
planes of atoms shift simultaneously, relative to one another.

changes direction beyond this as the atoms try to snap into a second stable configuration at x = b.
A reasonable estimate is a sinusoidal variation
2πx
τ = τ b sin (2.3)
b
where τ b is the maximum value as τ varies with x; hence, it is the theoretical shear strength.
The initial slope of the stress–strain relationship must be the shear modulus, G, in a manner
analogous to E for the tension case previously discussed. Noting that the shear strain for small
values of displacement is γ = x/h,wehave

dτ dτ
G = = h (2.4)
dγ x=0 dx x=0
Obtaining dτ/dx from Eq. 2.3 and substituting its value at x = 0gives τ b :

Gb
τ b = (2.5)
2πh
The ratio b/h varies with the crystal structure and is generally around 0.5 to 1, so this estimate is
on the order of G/10.
In a tension test, the maximum shear stress occurs on a plane 45 to the direction of uniaxial
◦
stress and is half as large. Thus, a theoretical estimate of shear failure in a tension test is
Gb
σ b = 2τ b = (2.6)
πh
Since G is in the range E/2to E/3, this estimate gives a value similar to the previously mentioned
σ b = E/10 estimate based on the tensile breaking of bonds. Estimates of theoretical strength are
discussed in more detail in the first chapter of Kelly (1986).