The document discusses yield criteria, which is the relationship between stresses that predicts material yielding. It describes two common yield criteria: the von Mises yield criterion and Tresca criterion. The von Mises criterion states that yielding occurs when the second invariant of the deviatoric stress tensor exceeds a critical value. The Tresca criterion states that yielding occurs when the difference between the maximum and minimum principal stresses exceeds a critical value. The document provides equations and examples to explain these criteria. It concludes that the Tresca criterion guarantees no yield, so it is commonly used for safe design, while von Mises is widely used for plasticity analysis since it guarantees yielding.

1. Yield Criteria
Course Instructor:
Dr.Venkata Girish Kotnur
Assistant Professor
School of Engineering Sciences & Technology
University of Hyderabad
Presented By:
Mahfooz Alam
17ETMM10

2. States of Stress
• When a body is subjected to a stress below the yield strength, it will deform
elastically. The moment the stress is removed, the body comes to initial position.
• In contrast, when the body is stressed beyond the yield point, it will undergo
permanent deformation. If it is a ductile material, it will plastically deform
continuously with increase in load applied.
• If a certain object is subjected to uniaxial tensile load, it will start yielding –
deforming plastically – when the stress reaches the uniaxial yield stress.
• However, when the state of stress is triaxial, a single shear stress can not be used to
predict yielding.
• It is the combination of the three stress states which alone can predict yielding.

3. Yield Criterion:
The relationship among the stresses which predict the yielding of a
material is called yield criterion.
Theinherent assumptions involved in defining the yielding are:
1) The material is isotropic & incompressible
2) Poisson’s ratio equals 0.5
3) Hydrostatic or mean stress does not causeyielding of the
material
4) Porous materials like powder metallurgy alloys can be
assumed compressible
5) They have Poisson’s ratio lessthan 0.5
Assumptions involved:
Commonly, for ductile materials, there are two important yield criteria.
Von Mises yield criterion – also called distortion energy criterion and
Tresca criterion also called Maximum shear stress theory.

4. States Of Stress on a Plane:
From the above figures, we could understand that the given state of biaxial stress
can be replaced by a sum of hydrostatic and deviatoric stresses. Hydrostatic
stress, though does not influence the yielding, it does increase ductility of a
material, when it is applied.

5. Yield Criteria:
Commencement ofplastic deformation in materials is predicted by yield
criteria.
Commonly, for ductile materials, there are two important yield criteria.
Von Mises yield criterion – also called distortion energy criterion and
Tresca criterion also called Maximum shear stress theory.

6. Stress Tensor:
Stress is a second order symmetric tensor

7. Von Mises Yield Criteria:
S3-J1S2+J2S-J3=0
When the second invariant of deviatoric stress tensor exceeds a critical
value yielding occurs.
Yielding occurs when J2≥K2 where K2 is a constant
J2=1/6[(σx-σy)2+(σy-σz)2+(σz-σx)2]+6(τxy
2+τyz
2+τzx
2)]=K2 Equation 1
J2< K2 : No Yielding
J2=K2: Yield point
J2>K2 : Yielded
J2=1/6[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]=K2 Equation 2
The constant K2 can be found out by substituting the known
uniaxial condition σ1=σ=σ0,σ2=0,σ3=0
1/6[(σ0-0)2+(0-0)2+(0-σ0)2]=K2
1/6[σ0
2+ σ0
2]= K2
σ0
2/3 = K2
K=σ0/√3
Substituting K2 in Equation 2, we get
1/√2[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]1/2=σ0
Substituting K2 in Equation 1, we get
1/√2[(σx-σy)2+(σy-σz)2+(σz-σx)2]+6(τxy
2+τyz
2+τzx
2)] 1/2= σ0

8. L.H.S<R.H.S No Yielding
L.H.S=R.H.SYield Point
L.H.S>R.H.SYielded (Material is already in plastic state)
NOTE:
It is Same as distortion energy theory or octahedral shear stress theory of
Theories of failure

9. Tresca Criteria:
(4J2
3-27J3
3-36K2J2
2+96K4J2-64K6)
On Simplification, σmax-σmin=K
When the difference between maximum and minimum stress exceeds a
critical value there is yielding
When σmax-σmin≥K, where ‘K’ is a constant
So, yield condition is σmax-σmin=K
Substituting uniaxial yield condition, σ1=σ0,σ2=0,σ3=0 [σmax=σ0,σmin=0]
σ0-0=K K=σ0
Substituting the value of ‘K’ in the yield condition gives
σmax-σmin= σ0

10. Shear Yield Stress according to Tresca:
σx= 0,σy= 0,τxy=τ
Principal Stresses:σ1,2=σx+σy/2±√{((σx-σy)/2)2+τxy
2}
σ1,2= ±τ, σ1=τ,σ2=-τ,σ3=0
For pure shear σ1=τ,σ2=-τ,σ3=0
According to Tresca Yielding Occurs When
σmax-σmin= σ0
So, τ-(-τ)= σ0
2τ = σ0
τ = σ0/2
Yield stress in Shear τ0=σ0/2

11. Shear Yield Stress according to VonMises:
Substituting the condition for pure shear in VonMises
Condition:
1/√2[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]1/2=σ0
1/√2[(τ-(-τ))2+(-τ-0)2+(0-τ)2]1/2=σ0
1/√2[(2τ)2+(-τ)2+(-τ)2]1/2= σ0
1/√2[4τ2+τ2+τ2]1/2= σ0
1/√2[6τ2]1/2 = σ0
1/√2[√6*τ] = σ0
√3τ = σ0
τ= σ0/√3
τ0= σ0/√3

12. Problem:
The principal components of stress at a point are 300MPa and 200MPa tensile and
50MPa Compressive. Find whether yielding takes place according to VonMises if
yield strength of the material is 325MPa.
VonMises:
L.H.S1/√2[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]1/2=σ0 R.H.S
σ0=325MPa(given)
Solving L.H.S,1/√2[(300-200)2+(200-(-50))2+(-50-300)2]1/2
1/√2[10000+62500+122500]1/2 = 312.24MPa(L.H.S)<<325(R.H.S)
[L.H.S<R.H.SNo Yielding]
Tresca:
σmax-σmin= σ0
300-(-50)=325
350>325
L.H.S>R.H.SAlready Yielded

13. [(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]=2σ0
2 VonMises Equation
(x-y)2+(y-z)2+(z-x)2 = a2 Equation of a Cylinder
σ1
σ2
σ3

14. 15% deviation

15. Conclusion:
Tresca Criteria guarentees no yield condition so,the designers(structural
elements design) uses Tresca criteria mostly for safe design.
Yielding is guarenteed by VonMises. So, vonmises is widely used for
plasticity analysis.

16. Reference:
1) Prof. Amitabha Ghosh, formerly of IIT Kanpur and Director IIT
Kharagpur is now the Platinum Jubilee Senior Scientist of The National
Academy of Sciences India and Hon. Distinguished Professor IIEST
Shibpur, will be visiting Dept. of Mechanical Engineering IIT Delhi.
2) Ashok Kumar Mallik is a professor in the Department of Mechanical
Engineering, IIT, Kanpur.
A Text book Of Manufacturing Science by:

17. Thank You