A complete Plasticity Model

Yield criterion-stress state @ yielding occurs Flow rule-increment on plastic strain after

yielding Hardening law-Evolution of flow stress with

increased plastic deformation Loading-unloading condition-determines how

stress path moves w.r.t yield surface.

J2 flow theory(Von-mises criterion) Only 2nd invariant of stress deviator tensor used for plasticity analyses.

Hydrostatic stress has no effect on plastic flow.

Flow stress independent of 3rd invariant of stress deviator.

2 stress state with equal distortion energy have equal von-mises stress.

I1-J2-J3 model(Gao et al.)

To distinguish 2 stress states with same energy concept of lode angle is proposed.

Since hydrostatic stress(I1)-significant role in accelerating failure & negligible effect on the stress-plastic strain relationship,

while the Lode angle(J2,J3) -considerable effect in modifying the stressstrain curves & does not significantly affect the failure strains.

Overview of I1-J2-J3 model

Yield function(F) and flow potential(G) are assumed to be 1st order homogenous function of stress.

Yield condition of following form is proposed.

Where is the hardening parameter. Flow rule:

I1-J2-J3 model

Enforcing the equivalence of plastic work:

Stress Update:

I1-J2-J3 model

Hardening law and evolution equation are used to update plastic strain:

Consistent tangent moduli:

Other plasticity models

The Mohr-Coulomb yield (failure) criterion Drucker proposed a yield function that depends on the J2-

J3, by which the yield surface lies between the von Mises yield surface and theTresca yield surface

Brunig (1999) presented an I1J2 yield criterion, which is similar to the DruckerPrager yield condition in soil mechanics.

Brunig et al., the authors described an I1-J2-J3 flow theory in their numerical simulation of the deformation and localization behavior of hydrostatic-stress-sensitive metals

Other plasticity models

Subramanya et al employed an extended Drucker-Prager yield model to study the roles of pressure sensitivity, plastic dilatancy and yield locus shape on theinteraction between the notch and a nearby void.

Methods for analysing fracture mechanism

1.Modeling individual voids explicitly using finite element mesh.-Void growth performance can be accurately simulated.-Require a large number of elements to model the voids in a structural component.

2.Porous continuum method.-Benzerga and Leblond.-McClintock and Rice and Tracy.-Gurson model-GTN model-Rousselier-Goludanu, Leblond and Devaux model(GLD model)

3.Continuum Damage mechanics model-Lemaitre-Rice and Johnson

GTN model

Gurson model

Load carrying capacity loss when f tends to 1

GTN model

-f* is used to model rapid loss of load carrying capacity after void coalescence

-Voids are spherical and remain spherical throughout growth process.

GLD model

Since the void shape may change to probate or oblate shape after deformation depending on the applied stress state.

In GLD model both void volume fraction and void shape evolve with deformation.

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