Presentation on theme: "Constitutive models Part 2 Elastoplastic"— Presentation transcript:
1 Constitutive models Part 2 Elastoplastic Version telemachos c:\education_ppt_doc_jpg\constitutive models 21.ppt
2 Elastoplastic material models Elastoplastic 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
3 Stress-strain curve of a hypothetical material Idealized results of one-dimensional tension testJohnson’s limit … 50% of Young modulus valueEngineering stressYield pointYield stressHigh quality alloyed steels do not have a pronounced separation between elastic and plastic curvesThey are tough, ductile, able to survive 10^5 loading cycles to failureStandard (mild) carbon steels have a more pronounced plateau after yielding occursThis is what we find in textbooksISO norm defines the yield point differentlyEngineering strain
4 Real life 1D tensile test, cyclic loading Conventional yield pointWhere is the yield point?Lin. elast. limitTensile test on a notched rod. Alloyed steel, cyclic loading.Where is the yield point?By ISO norm the the conventional yield stress corresponds to the occurrence of 0.2% of permanent deformationThe linear elasticity limit (where the straight line ends) is defined as that causing 0.005% of permanent deformationAbout 100 cycles to failure. Racheting is also called cyclic creepStrachota, A.: Experimental determination of elastoplastic strains in the vicinity of a stress concentrator.Research Report , SVUSS, Bechovice, 1989Hysteresis loops moveto the right - racheting
5 Mild carbon steel before and after heat treatment Conventional yield point … 0.2%In this respect the behaviour of heat treated steels is similar to that of alloyed steelsYield stress went up, the strength was increased as wellAgain, conventional yield point corresponds to 0.2% of permanent deformationStrachota, A.: Experimental determination of elastoplastic strains in the vicinity of a stress concentrator.Research Report , SVUSS, Bechovice, 1989The existence of ‘jump’ can be explained by the stick-slip phenomenon caused by the devil machine during the experiment
6 The plasticity theory covers the following fundamental points Yield criteria to define specific stress combinations that will initiate the non-elastic response – to define initial yield surfaceFlow rule to relate the plastic strain increments to the current stress level and stress incrementsHardening rule to define the evolution of the yield surface. This depends on stress, strain and other parametersThe theory of plasticity is of phenomenological natureActual changes of material structure (dislocations, etc) are attempted to be described by continuum theory termsDefine dislocationsAlso the data from 1D experiment tests are often used to describe a fully 3D behaviour.
7 Yield surface, function Yield surface, defined in stress space separates stress states that give rise to elastic and plastic (irrecoverable) statesFor initially isotropic materials yield function depends on the yield stress limit and on invariant combinations of stress componentsAs a simple example Von Mises …Yield function, say F, is designed in such a way thatExperiments indicate that yielding is unaffected my moderate hydrostatic stress so it is possible to present yield conditions as functions of the second and third stress invariants (the first invariant does not enter into considerations since it represents hydrostatic stress (pressure))Yield surface, defined in principal stress space, separates stress states that give rise to elastic and plastic (irrecoverable) states.In cyclic loading the surface may move in space … kinematic hardeningYield function depends on the yield stress limit and on invariant combinations of stress components, on temperature, history of loading, etc.The yield condition is a mathematical relationship among stress components an each point that must be satisfied for the onset of plastic behaviour at the point
8 Three kinematic conditions are to be distinguished Small displacements, small strainsmaterial nonlinearity only (MNO)Large displacements and rotations, small strainsTL formulation, MNO analysis2PK stress and GL strain substituted for engineering stress and strainLarge displacements and rotations, large strainsTL or UL formulationComplicated constitutive models
9 Rheology models for plasticity Ideal or perfect plasticity, no hardening
10 Loading, unloading, reloading and cyclic loading in 1D Good model for monotonic loading onlyIsotropic hardening dictates the same shape of stress strain curve regardless of whether the stress is tensile or compressive or multiaxial.This way we ignore the material anisotropy generated by plastic strain and Bauschinger effect.Bauschinger effectExceed (or go beyond) the yield stress value in tension causes the decreasing (in absolute value) the yield stress value in compression.The kinematic hardening model describes this phenomenon.
11 Isotropic hardening in principal stress space The assumption of isotropic hardening under loading conditions postulates that the yield surface simply increases in size while maintaining its original shape. Yield surface is defined in six-dimensional space of stresses, or could also be considered in the three-dimensional space of principal stresses or in the deviatoric spacePi-plane is defined in such a way that each principal stress axis makes the same angle (arccos(1/sqrt(3)) with pi-plane.This is a so called isometric projectionThe intersection of the yield surface with pi-plane is called yield curveVon Mises – distortion energy theory, expressed by effective stress. Also called J2 theory – second invariant of deviatoric stressTresca – maximum shear theory
12 Loading, unloading, reloading and cyclic loading in 1D Good model for cyclic loading after saturation is attained and richeting vanished
13 Kinematic hardening in principal stress space In incremental hardening the initial yield surface is translated to a new location in stress space without changing its shape
14 Von Mises yield condition, four hardening models 1. Perfect plasticity – no hardening2. Isotropic hardeningModels 1,2 … good for monotonic loading, the directions of principal stresses should not change muchModels 3,4 … good for changing and/or cyclic loadings, agree well with experimental results, cover 1,2 as well4. Isotropic-kinematic3. Kinematic hardening
15 Different types of yield functions Dislocations represent localized damage of crystalic structure of material.The damage usually has a linear shapeThere are either vacancies in crystalic structure or ‘foreign’ atomsAt yield, the dislocations start to move, material flows.Yield function could in addition depend on strain rate – rate dependent case, then the time becomes a ‘real’ variable.Also on temperature, in the simple case it is a parameter dependence.Also on heat exchange – then the temperature becomes ‘real’ variable.
16 Plasticity models – physical relevance Von Mises- no need to analyze the state of stress- a smooth yield sufrace- good agreement with experimentsTresca- 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 Pragera more general modelSituations where these models are usefulDifferences between von Mises and Tresca are up 20%
17 1D example, bilinear characteristics … tangent modulusStrain hardening parameterIncrement of strain (total, sometimes elastoplastic) is composed of elastic and plastic partsElastic part is recoverable, plastic one is not … permanent strain, damage of material, material is hardenedTotal response means elastoplasticStrain hardening parameter is a proportionality coefficient between the stress increment and the plastic strain incrementThis is crucial for understanding the plastic deformation… elastic modulus… means total or elastoplastic
18 Strain hardening parameter again Upon unloading and reloading the effective stress must exceedInitial yieldInitial yielding starts at sigma0The total deformation is composed of elastic and plastic partsGeometrical meaning of the strain hardening parameter Hprime is the slope of the stress vs. plastic strain plotElastic strains removedGeometrical meaning of the strain hardening parameter isthe slope of the stress vs. plastic strain plot
19 How to remove elastic part Telemachos c:\education_ppt_doc_jpg\constitutive_models\elast_plast_increments.bmp
20 1D example, bar (rod) element elastic and tangent stiffness Elastic stiffnessTangent stiffness
21 Results of 1D experiments must be correlated to theories capable to describe full 3D behaviour of materialsIncremental theories relate stress increments to strain incrementsDeformation theories relate total stress to total strainDeformation theories are suitable only for the proportional loading which means that all stress components are being changed proportionally to the same parameter. Attributed to Drucker. Not frequently used today.
22 Relations for incremental theories isotropic hardening example 1/9 Parameter onlyIncrement and limit of time derivative, time is a parameter onlyStress increments and stress rates could be used alternatively in this caseF>0, no physical meaning, analogy to a friction force that cannot be greater than (normal force) x (coefficient of friction)It is however a powerful tool to find equilibrium conditionsWe will come to this later.Rate formulation is used here. It is, however, a way of useful notation.Actually, the process is considered rate independent.
23 Relations for incremental theories isotropic hardening example 2/9 Eq. (i) … increment of plastic deformation has a directionnormal to F while its magnitude (length of vector) is not yet knowndefines outer normal to Fin six dimensional stress spaceDuring plastic deformations F = 0 and dF = 0If the same function F appears in the flow rule expression than we say that we have an associate flow plasticity model
24 Relations for incremental theories isotropic hardening example 3/9 elastictotalplastic deformationsmatrix of elastic moduli
25 Relations for incremental theories isotropic hardening example 4/9 Row vectorColumn vectorLambda is the scalar quantity determining the plastic strain increment in flow ruleE is a known matrix of elastic moduliVector q is known provided the yield function F is knownVector p is still to be determinedStill to be determinedDot product and quadratic form … scalarLambda is the scalar quantity determining the magnitudeof plastic strain increment in the flow rule
26 Relations for incremental theories isotropic hardening example 5/9 diadic productequal to zero for perfect plasticity
27 Relations for incremental theories isotropic hardening example 6/9 At time tA new constant defined
28 Relations for incremental theories isotropic hardening example 7/9 WAccepting bilinear characteristic
29 Relations for incremental theories isotropic hardening example 8/9 sigma_m … average or medium stresss … deviatoric stress, pressure part of the stress removed since it does not play significant role in plasticityA … strain hardening parameterE_ep … elastoplastic modulusIncrement of stress is obtained as a function of total strain incrementThere is no 9/9 slide so far
30 J2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatment Preliminaries, different notationAs an example for numerical treatmentJ2 theory actually means von Mises’ plasticity modelNotation and procedure shown for treating ideal plasticity (no hardening)
31 J2 theory, numerical treatment …2/6 End of preliminaries and the yield criterion
32 J2 theory, numerical treatment …3/6 Flow rule and resulting equations + inequality conditionSix nonlinear differential equations + one algebraic constraint (inequality)There is exact analytical solution to this. In practice we proceed numerically
33 J2 theory, numerical treatment …4/6 Typical for incremental approachIncrement of stress is expressed as a function of total strain incrementThis time for ideal plasticityTo get stress the differential equations have to be integratedTo do that numerically we have to go into the forbidden space beyond the yield surface where F>0and find a way how to come back to the surface satisfying the equilibrium conditions at the same time.System of six nonlineardifferential equationsto be integrated
34 3b. plastic part of increment 1. known stress J2 theory, numerical treatment …5/6 predictor-corrector method, first part: predictor3b. plastic part of increment1. known stress2. test stress (elastic shot)Predictor-corrector method (tangent stiffness-radial corrector method)3a. elastic part of increment