09_constitutive-models-21

nullConstitutive modelsConstitutive modelsPart 2 ElastoplasticElastoplastic material modelsElastoplastic material modelsElastoplastic materials are assumed to behave elastically up to a certain stress limit after which combined elastic and plastic behaviour occurs. Plasticity is path dependent – the changes in the material structure are irreversible nullStress-strain curve of a hypothetical material Idealized results of one-dimensional tension test Engineering stressEngineering strainYield point Yield stressJohnson’s limit … 50% of Young modulus valueReal life 1D tensile test, cyclic loadingReal life 1D tensile test, cyclic loadingHysteresis loops move to the right - rachetingWhere is the yield point?Conventional yield pointLin. elast. limitMild carbon steel before and after heat treatmentMild carbon steel before and after heat treatmentConventional yield point … 0.2%The plasticity theory covers the following fundamental pointsThe plasticity theory covers the following fundamental pointsYield criteria to define specific stress combinations that will initiate the non-elastic response – to define initial yield surface Flow rule to relate the plastic strain increments to the current stress level and stress increments Hardening rule to define the evolution of the yield surface. This depends on stress, strain and other parameters Yield surface, functionYield surface, functionYield surface, defined in stress space separates stress states that give rise to elastic and plastic (irrecoverable) states For initially isotropic materials yield function depends on the yield stress limit and on invariant combinations of stress components As a simple example Von Mises … Yield function, say F, is designed in such a way that Three kinematic conditions are to be distinguishedThree kinematic conditions are to be distinguishedSmall displacements, small strains material nonlinearity only (MNO) Large displacements and rotations, small strains TL formulation, MNO analysis 2PK stress and GL strain substituted for engineering stress and strain Large displacements and rotations, large strains TL or UL formulation Complicated constitutive models nullRheology models for plasticityIdeal or perfect plasticity, no hardeningnullLoading, unloading, reloading and cyclic loading in 1DIsotropic hardening in principal stress spaceIsotropic hardening in principal stress spacenullLoading, unloading, reloading and cyclic loading in 1DnullKinematic hardening in principal stress spaceVon Mises yield condition, four hardening modelsVon Mises yield condition, four hardening models1. Perfect plasticity – no hardening2. Isotropic hardening3. Kinematic hardening4. Isotropic-kinematicDifferent types of yield functionsDifferent types of yield functionsPlasticity models – physical relevancePlasticity models – physical relevance Von Mises - no need to analyze the state of stress - a smooth yield sufrace - good agreement with experiments Tresca - simple relations for decisions (advantage for hand calculations) - yield surface is not smooth (disadvantage for programming, the normal to yield surface at corners is not uniquely defined) Drucker Prager a more general model1D example, bilinear characteristics1D example, bilinear characteristicsStrain hardening parameter… means total or elastoplastic… elastic modulus… tangent modulusStrain hardening parameter againStrain hardening parameter againElastic strains removedInitial yieldUpon unloading and reloading the effective stress must exceedGeometrical meaning of the strain hardening parameter is the slope of the stress vs. plastic strain plotHow to remove elastic partHow to remove elastic part1D example, bar (rod) element elastic and tangent stiffness1D example, bar (rod) element elastic and tangent stiffnessElastic stiffnessTangent stiffnessResults of 1D experiments must be correlated to theories capable to describe full 3D behaviour of materialsResults of 1D experiments must be correlated to theories capable to describe full 3D behaviour of materials Incremental theories relate stress increments to strain increments Deformation theories relate total stress to total strainRelations for incremental theories isotropic hardening example 1/9Relations for incremental theories isotropic hardening example 1/9Parameter onlyRelations for incremental theories isotropic hardening example 2/9Relations for incremental theories isotropic hardening example 2/9Eq. (i) … increment of plastic deformation has a direction normal to F while its magnitude (length of vector) is not yet knowndefines outer normal to F in six dimensional stress spaceRelations for incremental theories isotropic hardening example 3/9Relations for incremental theories isotropic hardening example 3/9elastictotalplastic deformationsmatrix of elastic moduliRelations for incremental theories isotropic hardening example 4/9Relations for incremental theories isotropic hardening example 4/9Dot product and quadratic form … scalarRow vectorColumn vectorLambda is the scalar quantity determining the magnitude of plastic strain increment in the flow ruleStill to be determinedRelations for incremental theories isotropic hardening example 5/9Relations for incremental theories isotropic hardening example 5/9equal to zero for perfect plasticity diadic productRelations for incremental theories isotropic hardening example 6/9Relations for incremental theories isotropic hardening example 6/9A new constant definedAt time tRelations for incremental theories isotropic hardening example 7/9Relations for incremental theories isotropic hardening example 7/9WRelations for incremental theories isotropic hardening example 8/9Relations for incremental theories isotropic hardening example 8/9J2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatmentJ2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatmentJ2 theory, numerical treatment …2/6J2 theory, numerical treatment …2/6J2 theory, numerical treatment …3/6J2 theory, numerical treatment …3/6Six nonlinear differential equations + one algebraic constraint (inequality) There is exact analytical solution to this. In practice we proceed numericallyJ2 theory, numerical treatment …4/6J2 theory, numerical treatment …4/6System of six nonlinear differential equations to be integratedJ2 theory, numerical treatment …5/6 predictor-corrector method, first part: predictorJ2 theory, numerical treatment …5/6 predictor-corrector method, first part: predictor1. known stress2. test stress (elastic shot)3a. elastic part of increment 3b. plastic part of incrementJ2 theory, numerical treatment …6/6 predictor-corrector method, second part: correctorJ2 theory, numerical treatment …6/6 predictor-corrector method, second part: correctornullSecant stiffness method and the method of radial return