ME 612 Metal Forming and Theory of Plasticity

ME 612 Metal Forming and Theory of Plasticity 8. Stress-Strain Relations Assoc. Prof. Dr. Ahmet Zafer Şenalp e-mail: azsenalp@gmail. com Mechanical Engineering Department Gebze Technical University

8. Stress-Strain Relations Experiments have shown that in uniaxial loading strain corresponding to certain stress is composed of two parts: § Recoverable elastic strain § Irrecoverable plastic strain Experiments have shown that elastic strain can be related to stress by linear elastic equations. The equations valid for isotropic solid materials are: (8. 1) (8. 2) (8. 3) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 2

8. Stress-Strain Relations (8. 4) (8. 5) (8. 6) : Poisson’s ratio E : Young elasticity modulus G : Shear elasticity modulus The above ex, ey and ez equations can be rearranged to express in terms of hydrostatic and deviatoric stresses: (8. 7) (8. 8) (8. 9) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 3

8. Stress-Strain Relations Here sm: hydrostatic stress: (8. 10) is deviatoric stress: (8. 11) In terms of indicial notation: (8. 12) (8. 13) =1 if i=j =0 if i j Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 4

8. Stress-Strain Relations Figure 8. 1. Elastic and plastic strains Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 5

8. Stress-Strain Relations Theory of plasticity involves with irrecoverable plastic strain. In multiaxial loading general strain term can be decomposed into elastic and plastic parts: (8. 14) : Total strain : Elastic strain component : Plastic strain component In differential form; Dr. Ahmet Zafer Şenalp ME 612 (8. 15) Mechanical Engineering Department, GTU 6

8. Stress-Strain Relations 8. 1. Prandl-Reuss Equations Reuss assumed that the plastic strain increment is at any instant proportional to the instantaneous stress deviation and shear stresses, thus: (8. 16) In terms of indicial notation: (8. 17) : is an instantaneous non-negative constant of proportionality : deviatoric stress The above equation can be expressed in terms of principal stress directions: (8. 18) These equations give only ratio but does not give information about quantity. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 7

8. Stress-Strain Relations 8. 1. Prandl-Reuss Equations (8. 19) (8. 20) These equations are called Prandl-Reuss equations and can be written this form: (8. 21) (8. 22) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 8

8. Stress-Strain Relations 8. 2. Levy-Mises Equations Levy-Mises equations can be defined as a special case of Prandl-Reuss equations. These are (8. 23) In terms of total strains (8. 24) (8. 25) (8. 26) Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 9

8. Stress-Strain Relations 8. 2. Levy-Mises Equations (8. 27) (8. 28) (8. 29) As seen Levy-Mises equations discard elastic behavior. Hence when elastic deformation is important Prandl-Reuss equations should be used. Dr. Ahmet Zafer Şenalp ME 612 Mechanical Engineering Department, GTU 10