10. Plastic Instability e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: [email protected]@gmail.com Mechanical Engineering Department Gebze

10. Plastic Instability10. Plastic Instability

Assoc.Prof.Dr. Ahmet Zafer Şenalpe-mail: e-mail: [email protected]@gmail.com

Mechanical Engineering DepartmentGebze Technical University

ME 612ME 612 Metal Forming and Theory of Plasticity Metal Forming and Theory of Plasticity

mailto:[email protected]

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

2Mechanical Engineering Department, GTU


Figure 10.1. Load elongation curve for tensile test




10.1.10.1. Tensile Plastic Instability Tensile Plastic Instability

Stress and strain states in tension test:





A

F1

02 03

01 ln

02 w

wln

03 t

tln

(10.1)

(10.2)

(10.3)

(10.4)

(10.5)

(10.6)

Levy-Mises equations:

If related equalities are placed in Levy-Mises equations:





(10.7)

(10.8)

(10.9)

(10.12)

(10.10)

(10.11)

3211 2

1d

3

2d

3122 2

1d

3

2d

2133 2

1d

3

2d

11 d3

2d

12 2

1d

3

2d

13 2

1d

3

2d

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)

2d

d

2

1

2d

d

3

1

32

21

231

232

221

2

1

1

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)

21

231

232

221 dddddd

3

2d

1dd

AF 1

AlnlnFln 1

A

dAd

F

dF

1

1

(10.24)

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)

0F

dF

A

dAd

A A V 00

lnAlnVln

d

A

dA

V

dV

0V

dV (10.30)

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:




10.1. Tensile Plastic Instability10.1. Tensile Plastic Instability

(10.31)

(10.32)

d

A

dA

11

1 ddd

11

1

d

d

d

d

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 powerB : Prestrain coefficient





(10.33)

(10.35)

n)B(C

)B(

n

)B(

)B(nC

d

d n

(10.34)

)B(

n

Bn (10.36)

If n is high work hardenening is high,if n is low work hardenening is less.If B is high small deformationIf B is small large deformation occurs

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)

(10.39)

(10.38)

(10.40)

nnn Cn)BnB(C)B(C

n1 Cn

AF 1

ACnF n

A

Alnln 0

01

1e

AA 0

(10.41)

Eq (10.41) is placed into Eq (10.40):

Term is obtained. This term is force instability term. Figure 10.2. 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.




10.1. Tensile Plastic Instability10.1. Tensile Plastic Instability

(10.42)

(10.43)

1e

ACnF 0n

n

)B(z

Bn

Figure 10.2. Generalized instability strain





Stress and strain states of a thin walled sphere is:



10. Plastic Instability10. Plastic Instability10.2. Plastic Instability Analysis10.2. Plastic Instability Analysis

for Thin Walled Spherefor Thin Walled Sphere

(10.44)

(10.45)t2

Pr21

03 (Plane stress problem)

021 r

rln

03 t

tln

(10.46)

(10.47)

Figure 10.3. Free body diagram of a spherical shell subjected to internal pressure

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):





(10.51)

(10.48)

(10.49)

(10.50)

1

3211 2

1d

3

2d

3122 2

1d

3

2d

2133 2

1d

3

2d

11 2

1d

3

2d

13 d3

2d (10.52)

From Eq (10.51) and (10.52)

From Eq (10.44)

is obtained. For maximum pressure criteria at instabilitydP=0If Eq (10.46) and (10.47) are used in Eq (10.55)

is obtained.





(10.56)

(10.53)

(10.54)

(10.55)

(10.57)

312

2lntlnrlnPlnln 1

0t

dt

r

dr

P

dPd

1

1

311

1 ddd

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:





(10.60)

(10.58)

(10.59)

(10.61)

11

1 d3d

21

231

232

221

2

1

1

21

231

232

221 dddddd

3

2d

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:





(10.64)

(10.62)

(10.63)

(10.65)

dd2 1

d2

3d

2

3

d

d

n)B(C

B

n

B

)B(nC

d

dn

(10.66)

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):





(10.69)

(10.67)

(10.68)

(10.79)

B

n

2

3

B3

n2

n

3

n2C

n

1 3

n2C

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.





(10.82)

(10.80)

(10.81)

r

t2

3

n2CP

n

2

B

3

n

0err

B

3

n2

0ett

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




for Thin Walled Pipefor Thin Walled Pipe

(10.85)

(10.83)

(10.84)t

Pr1

t2

Pr2

03

21

2

(Plane stress problem)

1(10.86)

01 r

rln

0l

lln

02

03 t

tln

(Plane strain problem)

(10.87)

(10.88)

(10.89)

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)





(10.91)

(10.90)

(10.92)

(10.93)

(10.94)

3211 2

1d

3

2d

3122 2

1d

3

2d

2133 2

1d

3

2d

11 4

3d

3

2d

13 4

3d

3

2d

is obtained. From Eq (10.93) and (10.94)

From Eq (10.83)

For maximum pressure criteria at instability:dP=0Eq (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):





(10.96)

(10.95)

(10.97)

(10.99)

(10.98)

31

tlnrlnPlnln 1

t

dt

r

dr

P

dPd

1

1

311

1 ddd

11

1 d2d

(10.100)

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)

(10.104)

(10.103)

(10.105)

21

231

232

221

2

1

3

21

21

231

232

221 dddddd

3

2d

d2

3d 1

d3d

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)

(10.109)

(10.108)

(10.110)

3d

d

n)B(C

)B(

n

)B(

)B(nC

d

d n

)B(

n3

B3

n

n

3

nC

(10.111)

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)

(10.114)

(10.113)

(10.115)

n

13

nC

3

2

r

t

3

nC

3

2P

n

B

3

n

2

3

0err

B

3

n

2

3

0ett

Documents

10. Plastic Instability e-mail: Assoc.Prof.Dr. Ahmet Zafer Şenalp e-mail: [email protected]@gmail.com Mechanical Engineering Department Gebze