(I) Maximum Shearing Stress Criterion:
The Maximum Shearing Stress Criterion, also known as the Tresca criterion, is a yield criterion used to predict the yield of ductile materials under complex stress states. According to this criterion, yielding occurs when the maximum shearing stress in a material reaches a critical value.
The maximum shearing stress (
�
max
σ
max
) is given by the following equation:
�
max
=
�
1
−
�
3
2
σ
max
=
2
σ
1
−σ
3
where
�
1
σ
1
and
�
3
σ
3
are the principal stresses acting on the material.
The yield condition based on the Maximum Shearing Stress Criterion is that
�
max
σ
max
should not exceed the yield strength (
�
yield
σ
yield
) of the material. Therefore, the yield criterion can be written as:
�
max
≤
�
yield
σ
max
≤σ
yield
This criterion is applicable to ductile materials like metals, where yielding occurs due to the onset of plastic deformation and the movement of dislocations.
(II) Maximum Distortion Energy Criterion:
The Maximum Distortion Energy Criterion, also known as the von Mises criterion, is another yield criterion used to predict the yield of ductile materials. According to this criterion, yielding occurs when the distortion energy in a material reaches a critical value.
The distortion energy (
�
dist
σ
dist
) is given by the following equation:
�
dist
=
1
2
[
(
�
1
−
�
2
)
2
+
(
�
2
−
�
3
)
2
+
(
�
3
−
�
1
)
2
]
σ
dist
=
2
1
[(σ
1
−σ
2
)
2
+(σ
2
−σ
3
)
2
+(σ
3
−σ
1
)
2
]
where
�
1
σ
1
,
�
2
σ
2
, and
�
3
σ
3
are the principal stresses acting on the material.
The yield condition based on the Maximum Distortion Energy Criterion is that the distortion energy (
�
dist
σ
dist
) should not exceed the yield strength (
�
yield
σ
yield
) of the material. Therefore, the yield criterion can be written as:
�
dist
≤
�
yield
σ
dist
≤σ
yield
This criterion is also applicable to ductile materials and is often preferred over the Maximum Shearing Stress Criterion for its ability to accurately predict yielding under multiaxial stress states.
To summarize, the Maximum Shearing Stress Criterion is based on the maximum shearing stress in the material, while the Maximum Distortion Energy Criterion is based on the distortion energy. Both criteria provide guidelines to predict the yield of ductile materials, with the choice of criterion depending on the specific material and the stress state under consideration.