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 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

4. 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

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 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

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 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.

7. 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

8. 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

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 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.

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 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

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 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

15. 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.

16. 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%

17. 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

18. 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

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 stiffness
Tangent stiffness

21. 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.

22. 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.

23. 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

24. Relations for incremental theories isotropic hardening example 3/9
elastic
total
plastic deformations
matrix of elastic moduli

25. 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

26. Relations for incremental theories isotropic hardening example 5/9
diadic product
equal to zero for perfect plasticity

27. Relations for incremental theories isotropic hardening example 6/9
At time t
A new constant defined

28. Relations for incremental theories isotropic hardening example 7/9W
Accepting bilinear characteristic

29. 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

30. 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)

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 condition
Six 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 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

34. 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

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

36. Secant stiffness method and the method of radial return