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8. Stress-Strain Relations Assoc.Prof.Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical University ME 612 Metal Forming and Theory of Plasticity

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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: Dr. Ahmet Zafer Şenalp ME 612 2Mechanical Engineering Department, GTU 8. Stress-Strain Relations (8.1) (8.2) (8.3)

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: Poisson’s ratio E : Young elasticity modulus G : Shear elasticity modulus The above e x,e y and e z equations can be rearranged to express in terms of hydrostatic and deviatoric stresses: Dr. Ahmet Zafer Şenalp ME 612 3Mechanical Engineering Department, GTU 8. Stress-Strain Relations (8.4) (8.5) (8.6) (8.7) (8.8) (8.9)

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Here m : hydrostatic stress: is deviatoric stress: In terms of indicial notation: Dr. Ahmet Zafer Şenalp ME 612 4Mechanical Engineering Department, GTU 8. Stress-Strain Relations (8.10) (8.11) (8.12) (8.13) =1 if i=j =0 if i j

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Dr. Ahmet Zafer Şenalp ME 612 5Mechanical Engineering Department, GTU 8. Stress-Strain Relations Figure 8.1. Elastic and plastic strains

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Theory of plasticity involves with irrecoverable plastic strain. In multiaxial loading general strain term can be decomposed into elastic and plastic parts: : Total strain : Elastic strain component : Plastic strain component In differential form; Dr. Ahmet Zafer Şenalp ME 612 6Mechanical Engineering Department, GTU 8. Stress-Strain Relations (8.14) (8.15)

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Reuss assumed that the plastic strain increment is at any instant proportional to the instantaneous stress deviation and shear stresses, thus: In terms of indicial notation: : is an instantaneous non-negative constant of proportionality : deviatoric stress The above equation can be expressed in terms of principal stress directions: These equations give only ratio but does not give information about quantity. Dr. Ahmet Zafer Şenalp ME 612 7Mechanical Engineering Department, GTU 8. Stress-Strain Relations 8.1. Prandl-Reuss Equations (8.16) (8.17) (8.18)

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These equations are called Prandl-Reuss equations and can be written this form: Dr. Ahmet Zafer Şenalp ME 612 8Mechanical Engineering Department, GTU 8. Stress-Strain Relations 8.1. Prandl-Reuss Equations (8.20) (8.19) (8.21) (8.22)

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Levy-Mises equations can be defined as a special case of Prandl-Reuss equations. These are In terms of total strains Dr. Ahmet Zafer Şenalp ME 612 9Mechanical Engineering Department, GTU 8. Stress-Strain Relations 8.2. Levy-Mises Equations (8.23) (8.24) (8.25) (8.26)

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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 Mechanical Engineering Department, GTU 8. Stress-Strain Relations 8.2. Levy-Mises Equations (8.27) (8.28) (8.29)

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