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**Constitutive models Part 2 Elastoplastic**

Version telemachos c:\education_ppt_doc_jpg\constitutive models 21.ppt

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**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

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**Stress-strain curve of a hypothetical material **

Idealized results of one-dimensional tension test Johnson’s limit … 50% of Young modulus value Engineering stress Yield point Yield stress High quality alloyed steels do not have a pronounced separation between elastic and plastic curves They are tough, ductile, able to survive 10^5 loading cycles to failure Standard (mild) carbon steels have a more pronounced plateau after yielding occurs This is what we find in textbooks ISO norm defines the yield point differently Engineering strain

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**Real life 1D tensile test, cyclic loading**

Conventional yield point Where is the yield point? Lin. elast. limit Tensile 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 deformation The linear elasticity limit (where the straight line ends) is defined as that causing 0.005% of permanent deformation About 100 cycles to failure. Racheting is also called cyclic creep Strachota, A.: Experimental determination of elastoplastic strains in the vicinity of a stress concentrator. Research Report , SVUSS, Bechovice, 1989 Hysteresis loops move to the right - racheting

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**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 steels Yield stress went up, the strength was increased as well Again, conventional yield point corresponds to 0.2% of permanent deformation Strachota, A.: Experimental determination of elastoplastic strains in the vicinity of a stress concentrator. Research Report , SVUSS, Bechovice, 1989 The existence of ‘jump’ can be explained by the stick-slip phenomenon caused by the devil machine during the experiment

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**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 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 The theory of plasticity is of phenomenological nature Actual changes of material structure (dislocations, etc) are attempted to be described by continuum theory terms Define dislocations Also the data from 1D experiment tests are often used to describe a fully 3D behaviour.

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**Yield surface, function**

Yield 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 Experiments 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 hardening Yield 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

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**Three kinematic conditions are to be distinguished**

Small 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

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**Rheology models for plasticity**

Ideal or perfect plasticity, no hardening

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**Loading, unloading, reloading and cyclic loading in 1D**

Good model for monotonic loading only Isotropic 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 effect Exceed (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.

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**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 space Pi-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 projection The intersection of the yield surface with pi-plane is called yield curve Von Mises – distortion energy theory, expressed by effective stress. Also called J2 theory – second invariant of deviatoric stress Tresca – maximum shear theory

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**Loading, unloading, reloading and cyclic loading in 1D**

Good model for cyclic loading after saturation is attained and richeting vanished

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**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

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**Von Mises yield condition, four hardening models**

1. Perfect plasticity – no hardening 2. Isotropic hardening Models 1,2 … good for monotonic loading, the directions of principal stresses should not change much Models 3,4 … good for changing and/or cyclic loadings, agree well with experimental results, cover 1,2 as well 4. Isotropic-kinematic 3. Kinematic hardening

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**Different types of yield functions**

Dislocations represent localized damage of crystalic structure of material. The damage usually has a linear shape There are either vacancies in crystalic structure or ‘foreign’ atoms At 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.

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**Plasticity 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 model Situations where these models are useful Differences between von Mises and Tresca are up 20%

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**1D example, bilinear characteristics**

… tangent modulus Strain hardening parameter Increment of strain (total, sometimes elastoplastic) is composed of elastic and plastic parts Elastic part is recoverable, plastic one is not … permanent strain, damage of material, material is hardened Total response means elastoplastic Strain hardening parameter is a proportionality coefficient between the stress increment and the plastic strain increment This is crucial for understanding the plastic deformation … elastic modulus … means total or elastoplastic

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**Strain hardening parameter again**

Upon unloading and reloading the effective stress must exceed Initial yield Initial yielding starts at sigma0 The total deformation is composed of elastic and plastic parts Geometrical meaning of the strain hardening parameter Hprime is the slope of the stress vs. plastic strain plot Elastic strains removed Geometrical meaning of the strain hardening parameter is the slope of the stress vs. plastic strain plot

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**How to remove elastic part**

Telemachos c:\education_ppt_doc_jpg\constitutive_models\elast_plast_increments.bmp

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**1D example, bar (rod) element elastic and tangent stiffness**

Elastic stiffness Tangent stiffness

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Results 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 strain Deformation 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.

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**Relations for incremental theories isotropic hardening example 1/9**

Parameter only Increment and limit of time derivative, time is a parameter only Stress increments and stress rates could be used alternatively in this case F>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 conditions We 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.

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**Relations for incremental theories isotropic hardening example 2/9**

Eq. (i) … increment of plastic deformation has a direction normal to F while its magnitude (length of vector) is not yet known defines outer normal to F in six dimensional stress space During plastic deformations F = 0 and dF = 0 If the same function F appears in the flow rule expression than we say that we have an associate flow plasticity model

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**Relations for incremental theories isotropic hardening example 3/9**

elastic total plastic deformations matrix of elastic moduli

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**Relations for incremental theories isotropic hardening example 4/9**

Row vector Column vector Lambda is the scalar quantity determining the plastic strain increment in flow rule E is a known matrix of elastic moduli Vector q is known provided the yield function F is known Vector p is still to be determined Still to be determined Dot product and quadratic form … scalar Lambda is the scalar quantity determining the magnitude of plastic strain increment in the flow rule

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**Relations for incremental theories isotropic hardening example 5/9**

diadic product equal to zero for perfect plasticity

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**Relations for incremental theories isotropic hardening example 6/9**

At time t A new constant defined

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**Relations for incremental theories isotropic hardening example 7/9**

W Accepting bilinear characteristic

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**Relations for incremental theories isotropic hardening example 8/9**

sigma_m … average or medium stress s … deviatoric stress, pressure part of the stress removed since it does not play significant role in plasticity A … strain hardening parameter E_ep … elastoplastic modulus Increment of stress is obtained as a function of total strain increment There is no 9/9 slide so far

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**J2 theory, perfect plasticity 1/6 alternative notation … example of numerical treatment**

Preliminaries, different notation As an example for numerical treatment J2 theory actually means von Mises’ plasticity model Notation and procedure shown for treating ideal plasticity (no hardening)

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**J2 theory, numerical treatment …2/6**

End of preliminaries and the yield criterion

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**J2 theory, numerical treatment …3/6**

Flow rule and resulting equations + inequality condition Six nonlinear differential equations + one algebraic constraint (inequality) There is exact analytical solution to this. In practice we proceed numerically

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**J2 theory, numerical treatment …4/6**

Typical for incremental approach Increment of stress is expressed as a function of total strain increment This time for ideal plasticity To get stress the differential equations have to be integrated To do that numerically we have to go into the forbidden space beyond the yield surface where F>0 and find a way how to come back to the surface satisfying the equilibrium conditions at the same time. System of six nonlinear differential equations to be integrated

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**3b. plastic part of increment 1. known stress**

J2 theory, numerical treatment …5/6 predictor-corrector method, first part: predictor 3b. plastic part of increment 1. known stress 2. test stress (elastic shot) Predictor-corrector method (tangent stiffness-radial corrector method) 3a. elastic part of increment

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**J2 theory, numerical treatment …6/6 predictor-corrector method, second part: corrector**

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**Secant stiffness method and the method of radial return**

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