The question asks for the relationship between yield strength in shear ($\tau_y$) and yield strength in tension ($\sigma_t$) according to the Maximum Shear Stress Theory of failure. This theory is also known as the Tresca criterion.

Understanding Maximum Shear Stress Theory

The Maximum Shear Stress Theory is a failure theory applied to ductile materials. It states that yielding begins when the maximum shear stress in any element under any combination of stresses becomes equal to the maximum shear stress developed at the point of yielding in a simple tensile test of the same material.

Applying the Theory to Standard Tests

1. Simple Tensile Test

Consider a simple tensile test specimen loaded uniaxially in tension. At the point of yielding, the stress is the yield strength in tension, denoted as $\sigma_t$.

In this uniaxial stress state, the principal stresses are:

• $\sigma_1 = \sigma_t$
• $\sigma_2 = 0$
• $\sigma_3 = 0$

The maximum shear stress ($\tau_{max}$) in this stress state is given by half the difference between the largest and smallest principal stresses:

\(\tau_{max, tension\_yield} = \frac{|\sigma_1 - \sigma_3|}{2} = \frac{|\sigma_t - 0|}{2} = \frac{\sigma_t}{2}\)

According to the Maximum Shear Stress Theory, yielding in tension occurs when the applied tensile stress reaches $\sigma_t$, and at this point, the maximum shear stress is $\frac{\sigma_t}{2}$. This value, $\frac{\sigma_t}{2}$, represents the critical maximum shear stress that causes yielding.

2. Pure Shear Test

Consider a pure shear stress state. The yield strength in pure shear is denoted as $\tau_y$.

In a state of pure shear ($\tau$), the principal stresses are:

• $\sigma_1 = +\tau$
• $\sigma_2 = 0$
• $\sigma_3 = -\tau$

At the point of yielding in pure shear, the applied shear stress is $\tau_y$. So, the principal stresses are:

• $\sigma_1 = +\tau_y$
• $\sigma_2 = 0$
• $\sigma_3 = -\tau_y$

The maximum shear stress ($\tau_{max}$) in this pure shear yield state is:

\(\tau_{max, shear\_yield} = \frac{|\sigma_1 - \sigma_3|}{2} = \frac{|\tau_y - (-\tau_y)|}{2} = \frac{|2\tau_y|}{2} = \tau_y\)

Deriving the Relationship

The Maximum Shear Stress Theory states that yielding occurs when the maximum shear stress in any stress state equals the maximum shear stress at yielding in a simple tensile test.

So, for yielding to occur in pure shear, the maximum shear stress in pure shear yield ($\tau_{max, shear\_yield}$) must be equal to the maximum shear stress in tensile yield ($\tau_{max, tension\_yield}$).

Equating the two:

\(\tau_{max, shear\_yield} = \tau_{max, tension\_yield}\)

\(\tau_y = \frac{\sigma_t}{2}\)

This relationship can also be written as:

\(\tau_y = 0.5 \sigma_t\)

This equation relates the yield strength in shear ($\tau_y$) to the yield strength in tension ($\sigma_t$) based on the Maximum Shear Stress Theory (Tresca criterion).

Conclusion

Based on the Maximum Shear Stress Theory, the relation between yield strength in shear ($\tau_y$) and yield strength in tension ($\sigma_t$) is $\tau_y = 0.5 \sigma_t$. This means the yield strength in shear is predicted to be half of the yield strength in tension for ductile materials.