Distortion Energy Theory gives the higher value shear yield strength.

Explanation:
In the field of mechanics, there are different theories that are used to predict the failure of materials under shear stress. Two such theories are the Maximum Shear Stress Theory (also known as Tresca's Theory) and the Distortion Energy Theory (also known as von Mises' Theory).

Maximum Shear Stress Theory:
According to the Maximum Shear Stress Theory, failure occurs when the maximum shear stress in a material reaches a certain critical value. This theory assumes that failure is governed by the principal shear stress, which is the difference between the maximum and minimum principal stresses. The maximum shear stress is given by the formula:

τ_max = (σ_max - σ_min) / 2

where τ_max is the maximum shear stress, σ_max is the maximum principal stress, and σ_min is the minimum principal stress.

Distortion Energy Theory:
On the other hand, the Distortion Energy Theory considers failure to occur when the distortion energy (also known as the von Mises stress) in a material reaches a critical value. The distortion energy is given by the formula:

σ_von = √((σ_x - σ_y)^2 + (σ_y - σ_z)^2 + (σ_z - σ_x)^2 + 6(τ_xy^2 + τ_yz^2 + τ_zx^2)) / √2

where σ_von is the von Mises stress, σ_x, σ_y, and σ_z are the normal stresses in the x, y, and z directions respectively, and τ_xy, τ_yz, and τ_zx are the shear stresses on the corresponding planes.

Difference in Shear Yield Strength:
The shear yield strength is the shear stress at which a material begins to deform plastically. The Distortion Energy Theory predicts a higher shear yield strength compared to the Maximum Shear Stress Theory. This is because the Distortion Energy Theory takes into account the effect of all the normal and shear stresses on the failure of the material, whereas the Maximum Shear Stress Theory only considers the maximum shear stress.

Conclusion:
In summary, the Distortion Energy Theory gives a higher value for the shear yield strength compared to the Maximum Shear Stress Theory. This is because the Distortion Energy Theory considers the effect of all the stresses on failure, while the Maximum Shear Stress Theory only considers the maximum shear stress. It is important to note that the shear yield strength can vary from material to material, and the choice of theory depends on the specific characteristics of the material being analyzed.