1. ME 612 Metal Forming and Theory of Plasticity
10. Plastic Instability
Assoc.Prof.Dr. Ahmet Zafer Şenalp
Mechanical Engineering Department
Gebze Technical University

2. Mechanical Engineering Department, GTU
10. Plastic Instability
General instability classification:
Elastic instability
Plastic instability
The instability behavior of columns under compression is an example of elastic instability (determination of load that instability starts).
Plastic instability anaylsis searches for the load or pressure that will cause rupture or crack in plastic deformation zone.
In this section the plastic instabillity anaylsis for:
Simple tension test
Thin walled cylinder
Thin walled pipe
will be presented.
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

3. 10.1. Tensile Plastic Instability
Figure Load elongation curve for tensile test
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

4. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
Stress and strain states in tension test:
(10.1)
(10.2)
(10.3)
(10.4)
(10.5)
(10.6)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

5. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
Levy-Mises equations: If related equalities are placed in Levy-Mises equations:
(10.7)
(10.8)
(10.9)
(10.10)
(10.11)
(10.12)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

6. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
From here: is obtained. The equalities (10.2) and (10.3) are placed in the equivalent stress equation: As a result: is obtained.
(10.13)
(10.14)
(10.15)
(10.18)
(10.19)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

7. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
To find equivalent strain equalities (10.13) and (10.14) are placed into the equivalent strain equation: As a result: is obtained. Eq (10.1) is written as: If ln is applied to both sides of the equality:
(10.20)
(10.21)
(10.22)
(10.23)
(10.24)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

8. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
At maximum load ( F=Fmax) : At instability point work hardening rate is equal to the area reduction rate. From constancy of volume: As volume is constant the term defining volume change:
(10.25)
(10.26)
(10.27)
(10.28)
(10.29)
(10.30)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

9. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
If Eq (10.26) and (10.31) are equated: instability equation is obtained. This equation can be written in terms of equivalent stress and equivalent strain:
(10.31)
(10.32)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

10. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
It is assumed that material obeys Swift equation: is obtained. Eq (10.32) and (10.34) are equated: term is obtained, simplifying this: strain instability equation is obtained. In this equation; n : Work hardening power B : Prestrain coefficient
(10.33)
(10.34)
(10.35)
(10.36)
If n is high work hardenening is high,
if n is low work hardenening is less.
If B is high small deformation
If B is small large deformation
occurs
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

11. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
Strain instability equation given in Eq (10.36)is placed into Swift equation: is obtained. At the same time: If Eq (10.38) is placed into Eq (10.39): is obtained.
(10.37)
(10.38)
(10.39)
(10.40)
(10.41)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

12. Mechanical Engineering Department, GTU
10.1. Tensile Plastic Instability
10. Plastic Instability
Eq (10.41) is placed into Eq (10.40): Term is obtained. This term is force instability term. Figure shows generelized instability strain. Here z is defined as: For simple tension test the above obtained term is placed into z equation z=1 is obtained.
(10.42)
(10.43)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

13. 10.1. Tensile Plastic Instability
Figure Generalized instability strain
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

14. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
Stress and strain states of a thin walled sphere is:
(10.44)
(Plane stress problem)
(10.45)
(10.46)
(10.47)
Figure Free body diagram of a spherical shell subjected to internal pressure
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

15. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
Instability analysis will be conducted on maximum pressure criteria. Maximum P means maximum Levy-Mises equations: If Eq (10.44) and (10.45) are placed into Eq (10.48) and (10.50):
(10.48)
(10.49)
(10.50)
(10.51)
(10.52)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

16. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
From Eq (10.51) and (10.52) From Eq (10.44) is obtained. For maximum pressure criteria at instability dP=0 If Eq (10.46) and (10.47) are used in Eq (10.55) is obtained.
(10.53)
(10.54)
(10.55)
(10.56)
(10.57)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

17. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
If Eq (10.53) is placed into the above Eq: Equivalent stress: If Eq (10.44) and (10.45) are placed into Eq (10.59) : Equivalent strain:
(10.58)
(10.59)
(10.60)
(10.61)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

18. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
If Eq (10.47) and (10.53) are placed into Eq (10.61) If Eq (10.60) and (10.62) are placed into Eq (10.58) It is assumed that material obeys Swift’s Law:
(10.62)
(10.63)
(10.64)
(10.65)
(10.66)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

19. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
From Eq (10.64) and (10.65) By simplifying: If Eq (10.68) is placed into Swift equation: is obtained. Using Eq (10.60) and (10.69):
(10.67)
(10.68)
(10.69)
(10.79)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

20. Mechanical Engineering Department, GTU
10.2. Plastic Instability Analysis
for Thin Walled Sphere
10. Plastic Instability
and using Eq (10.44) and (10.79) is obtained. Using Eq (10.62), (10.68) and (10.46), (10.47) is obtained. Here the value obtained in Eq (19.80) is the critical pressure value according to maximum pressure criteria obeying Swift’s law. After this pressure value a crack or rupture or explosion in the material should be expected.
(10.80)
(10.81)
(10.82)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

21. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
Plastic instability analysis will be performed according to maximum pressure criteria. For a thin walled pipe stess, strain states are: Hoop stress: Longitudinal stress: Max. P Max
(10.83)
(10.84)
(Plane stress problem)
(10.85)
(10.86)
(10.87)
(10.88)
(Plane strain problem)
(10.89)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

22. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
This problem can be accepted as both plane stress and plane strain problem. Levy-Mises equations: If Eq (10.85) and (10.86) are placed into (10.90) and (10.91)
(10.90)
(10.91)
(10.92)
(10.93)
(10.94)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

23. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
is obtained. From Eq (10.93) and (10.94) From Eq (10.83) For maximum pressure criteria at instability: dP=0 Eq (10.87),(10.89) and (10.98) are replaced into Eq (10.97): is obtained. If Eq (10.95) is placed into Eq (10.99):
(10.95)
(10.96)
(10.97)
(10.98)
(10.99)
(10.100)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

24. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
Equivalent stress is: Eq (10.85) and (10.86) are placed into Eq (10.101) : Equivalent strain is: If Eq (10.88) and (10.95) are placed into Eq (10.103) : Placing Eq (10.102) and (10.104) into Eq (10.100):
(10.101)
(10.102)
(10.103)
(10.104)
(10.105)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

25. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
From Eq (10.105) is obtained. From Eq (10.106) and (10.107) is obtained. From here: is obtained. If Eq (10.110) is placed into Eq (10.107) :
(10.106)
(10.107)
(10.108)
(10.109)
(10.110)
(10.111)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU

26. Mechanical Engineering Department, GTU
10.3. Plastic Instability Analysis
for Thin Walled Pipe
10. Plastic Instability
Using Eq (10.102) and (10.111) : is obtained. Using Eq (10.83) and (10.112) term is obtained. Using Eq (10.104), (10.110) and (10.87), (10.89) Equations are obtained. Here the value obtained in Eq (10.113) is the critical pressure value. After this pressure value a crack or rupture or explosion in the material should be expected. Eq (10.114) and (10.115) give the critical radius and critical thickness values respectively.
(10.112)
(10.113)
(10.114)
(10.115)
Dr. Ahmet Zafer Şenalp ME 612
Mechanical Engineering Department, GTU