ME 612 Metal Forming and Theory of Plasticity

ME 612 Metal Forming and Theory of Plasticity 10. Plastic Instability Assoc. Prof. Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail. com Mechanical Engineering Department Gebze Technical University

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 2

10. 1. Tensile Plastic Instability 10. Plastic Instability Figure 10. 1. Load elongation curve for tensile test Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 3

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 4

10. Plastic Instability 10. 1. Tensile Plastic Instability Levy-Mises equations: (10. 7) (10. 8) (10. 9) If related equalities are placed in Levy-Mises equations: (10. 10) (10. 11) (10. 12) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 5

10. Plastic Instability 10. 1. Tensile Plastic Instability From here: (10. 13) (10. 14) (10. 15) is obtained. The equalities (10. 2) and (10. 3) are placed in the equivalent stress equation: (10. 18) As a result: (10. 19) is obtained. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 6

10. Plastic Instability 10. 1. Tensile Plastic Instability To find equivalent strain equalities (10. 13) and (10. 14) are placed into the equivalent strain equation: (10. 20) As a result: is obtained. Eq (10. 1) is written as: (10. 21) (10. 22) If ln is applied to both sides of the equality: (10. 23) (10. 24) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 7

10. Plastic Instability 10. 1. Tensile Plastic Instability At maximum load ( F=Fmax) : (10. 25) (10. 26) At instability point work hardening rate is equal to the area reduction rate. From constancy of volume: (10. 27) (10. 28) (10. 29) As volume is constant the term defining volume change: (10. 30) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 8

10. Plastic Instability 10. 1. Tensile Plastic Instability (10. 31) 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. 32) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 9

10. Plastic Instability 10. 1. Tensile Plastic Instability It is assumed that material obeys Swift equation: (10. 33) (10. 34) is obtained. Eq (10. 32) and (10. 34) are equated: (10. 35) term is obtained, simplifying this: (10. 36) strain instability equation is obtained. In this equation; n : Work hardening power B : Prestrain coefficient Dr. Ahmet Zafer Şenalp ME 612 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 Mechanical Engineering Department, GTU 10

10. Plastic Instability 10. 1. Tensile Plastic Instability Strain instability equation given in Eq (10. 36)is placed into Swift equation: (10. 37) is obtained. At the same time: (10. 38) (10. 39) If Eq (10. 38) is placed into Eq (10. 39): (10. 40) (10. 41) is obtained. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 11

10. Plastic Instability 10. 1. Tensile Plastic Instability Eq (10. 41) is placed into Eq (10. 40): (10. 42) Term is obtained. This term is force instability term. Figure 10. 2. shows generelized instability strain. Here z is defined as: (10. 43) For simple tension test the above obtained is obtained. Dr. Ahmet Zafer Şenalp ME 612 term is placed into z equation z=1 Mechanical Engineering Department, GTU 12

10. 1. Tensile Plastic Instability 10. Plastic Instability Figure 10. 2. Generalized instability strain Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 13

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere Stress and strain states of a thin walled sphere is: (10. 44) (Plane stress problem) (10. 45) (10. 46) (10. 47) Figure 10. 3. Free body diagram of a spherical shell subjected to internal pressure Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 14

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere Instability analysis will be conducted on maximum pressure criteria. Maximum P means maximum Levy-Mises equations: (10. 48) (10. 49) (10. 50) If Eq (10. 44) and (10. 45) are placed into Eq (10. 48) and (10. 50): (10. 51) (10. 52) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 15

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere From Eq (10. 51) and (10. 52) From Eq (10. 44) (10. 53) (10. 54) (10. 55) is obtained. For maximum pressure criteria at instability d. P=0 (10. 56) If Eq (10. 46) and (10. 47) are used in Eq (10. 55) (10. 57) is obtained. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 16

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere If Eq (10. 53) is placed into the above Eq: (10. 58) Equivalent stress: (10. 59) If Eq (10. 44) and (10. 45) are placed into Eq (10. 59) : (10. 60) Equivalent strain: (10. 61) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 17

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere If Eq (10. 47) and (10. 53) are placed into Eq (10. 61) (10. 62) If Eq (10. 60) and (10. 62) are placed into Eq (10. 58) (10. 63) (10. 64) It is assumed that material obeys Swift’s Law: (10. 65) (10. 66) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 18

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere From Eq (10. 64) and (10. 65) (10. 67) By simplifying: (10. 68) If Eq (10. 68) is placed into Swift equation: (10. 69) is obtained. Using Eq (10. 60) and (10. 69): (10. 79) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 19

10. Plastic Instability 10. 2. Plastic Instability Analysis for Thin Walled Sphere and using Eq (10. 44) and (10. 79) (10. 80) is obtained. Using Eq (10. 62), (10. 68) and (10. 46), (10. 47) (10. 81) (10. 82) 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. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 20

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe Plastic instability analysis will be performed according to maximum pressure criteria. For a thin walled pipe stess, strain states are: Hoop stress: (10. 83) Longitudinal stress: (10. 84) (Plane stress problem) Max. P (10. 85) (10. 86) Max (10. 87) (Plane strain problem) (10. 88) (10. 89) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 21

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe This problem can be accepted as both plane stress and plane strain problem. Levy-Mises equations: (10. 90) (10. 91) (10. 92) If Eq (10. 85) and (10. 86) are placed into (10. 90) and (10. 91) (10. 93) (10. 94) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 22

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe is obtained. From Eq (10. 93) and (10. 94) (10. 95) From Eq (10. 83) (10. 96) (10. 97) For maximum pressure criteria at instability: (10. 98) d. P=0 Eq (10. 87), (10. 89) and (10. 98) are replaced into Eq (10. 97): (10. 99) is obtained. If Eq (10. 95) is placed into Eq (10. 99): (10. 100) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 23

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe Equivalent stress is: (10. 101) Eq (10. 85) and (10. 86) are placed into Eq (10. 101) : (10. 102) Equivalent strain is: (10. 103) If Eq (10. 88) and (10. 95) are placed into Eq (10. 103) : (10. 104) Placing Eq (10. 102) and (10. 104) into Eq (10. 100): (10. 105) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 24

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe From Eq (10. 105) (10. 106) (10. 107) (10. 108) is obtained. From Eq (10. 106) and (10. 107) is obtained. From here: (10. 109) (10. 110) is obtained. If Eq (10. 110) is placed into Eq (10. 107) : (10. 111) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 25

10. Plastic Instability 10. 3. Plastic Instability Analysis for Thin Walled Pipe Using Eq (10. 102) and (10. 111) : (10. 112) is obtained. Using Eq (10. 83) and (10. 112) (10. 113) term is obtained. Using Eq (10. 104), (10. 110) and (10. 87), (10. 89) (10. 114) (10. 115) 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. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 26