Materials like mild steel have defined yield point hence it is easy to distinguish between the elastic region and plastic region of deformation. But for materials that do not have specified yield point, it is hard to distinguish between elastic and plastic deformation region. In that case may be plastic deformation starts from beginning of the application of the load. For elastic region, stress and strain are in linear relationship with each other hence Hook’s law valid true. But for plastic region, the relation between stress and strain is nonlinear and complicated. So need for continuum plasticity model arises. The main aim of continuum plasticity model is to formulate mathematical model based on experimental results that can predict the plastic deformation of material under varying loading conditions and at different elevated temperature.

1. A review of constitutive models for
plastic deformation
MM 694 Seminar Presentation
by
More Samir Dhanaji
173110042
Under the guidance of
Prof. Anirban Patra
Department of Metallurgical Engineering and Material Science
1

2. Outline
• Need for Continuum Plasticity Models
• Yield Criterion
• Yield Surface
• Isotropic and Kinematic Hardening
• Theory of Plasticity
• Summary
2

3. Need for continuum plasticity models
Fig 1 Stress-strain curve under uniaxial loading for mild
steel [1]
• Yield point – distinguish elastic and plastic deformation
• In elastic region, stress-strain relationship is linear.
• In plastic region, stress-strain relationship is non-linear
and complicated.
• Main goal of continuum plasticity models:
To obtain stress-strain and load deflection relationship.
• The stress-strain response will depends on Temperature,
composition, strain rate, heat treatment, state of stress,
history, etc.
3

4. Yield Criterion
• In uniaxial loading, plastic flow begins when σ = 𝜎 𝑦, the tensile yield stress
• Pure hydrostatic pressure or mean stress tensor, p , doesn’t cause yielding in metals.
• Only the deviatoric stress, 𝑺𝒊𝒋, which represents the shear stresses causes plastic flow.
• Yielding surface separate the elastic and plastic domains.
• Yield criterion- determine the onset of the plastic deformation
• Yield criterion must be some function of the invariants ( J and J’ represents invariant of stress tensor, 𝝈𝒊𝒋
and deviatoric stress, 𝑺𝒊𝒋).
• In plasticity theory,
𝝈𝒊𝒋 = p 𝜹𝒊𝒋 + 𝑺𝒊𝒋 (1)
where 𝜹𝒊𝒋 represents kronecker delta.
4

5. Von Mises’Yield Criterion
• “Yielding would occurs when the second invariant J2
’ of the deviatoric stress tensor S reaches a critical
value k2”
J2
’ - k2 = 0 for yielding or plastic deformation (2)
J2
’ < k2 for elastic deformation
• For uniaxial tensile test, σ1 = σy and σ2 = σ3 = 0,
k =
𝝈 𝒚
3
(3)
• In pure shear stress (τy), we have 𝝈 𝟏 = - 𝝈 𝟑 = τy and 𝝈 𝟐 = 0,
k = 𝝉 𝒚 (4)
• From eq. (3) and (4) we get
k =
𝝈 𝒚
3
= 𝝉 𝒚 (5)
• For a von Mises material, the yield strength in uniaxial tension is 3 times the yield strength in pure shear.
5

6. Tresca Criterion
• “Yielding would occurs when the maximum shear stress (𝝉 𝒎𝒂𝒙) reaches the critical value k of the a
material.”
𝝉 𝒎𝒂𝒙 = 𝑘 for yielding or plastic deformation (6)
𝝉 𝒎𝒂𝒙 < 𝑘 for elastic deformation
• For uniaxial tensile test, σ1 = σy and σ2 = σ3 = 0,
k =
𝝈 𝒚
2
(7)
• In pure shear stress (τy), we have 𝝈 𝟏 = - 𝝈 𝟑 = τy and 𝝈 𝟐 = 0,
k = 𝝉 𝒚 (8)
• From eq. (7) and (8) we get
k = 𝝉 𝒚 =
𝝈 𝒚
2
(9)
• Thus, for Tresca’s materials, the shear strength in pure shear is half of that in uniaxial tension.
6

7. Von Mises and Tresca criterion for plane
stress condition
Fig. 2 von Mises and Tresca criterion
for plane stress conditions [2]
Von Mises represents ellipse whose major semi-axis is 2𝜎 𝑦 and minor
semi-axis is
2
3
𝜎 𝑦.
7

8. Yield surface
Fig. 3 Schematic representation of
Yield surface [3]
• The yield criteria can be represented geometrically by a cylinder oriented at
equal angles to the 𝜎1, 𝜎2 and 𝜎3 axes.
• A state stress which gives a point inside the cylinder represents the elastic
behavior.
• Yielding begins when the state of stress reaches the surface of cylinder.
• MN, the cylinder radius is the deviatoric stress.
• The cylinder axis, OM, which makes equal angles with the principal axes
represents the hydrostatic component of the stress tensor.
• The generator of the yield surface is the line parallel to OM. If stress state
characterized by (𝜎1, 𝜎2, 𝜎3) lies on the yield surface, so does (𝜎1 + 𝐻, 𝜎2 +
𝐻, 𝜎3 +𝐻)
8

9. Normality
Fig. 4 Yield locus [3]
• Drucker (1951): “The total plastic strain vector, must be normal
to the yield surface.”
• Net work has to be expended during the plastic deformation of
body. So the rate of energy dissipation is nonnegative:
𝝈𝒊𝒋 𝒅𝜺𝒊𝒋
𝒑
≥ 0
• The 𝒅𝜺𝒊𝒋
𝒑
is the incremental plastic strain vector and must be
normal to the yield surface.
• Because of normality rule, the yield locus is always convex.
9

10. Isotropic Hardening
Fig. 5 Schematic diagram of the deviatoric plane and
stress vs plastic strain response [4]
• The yield surface expands uniformly, but with a fixed
shape and center.
• Area inside yield surface represents elastic region. The
circumference of yield surface denotes elasto-plastic
region.
• The yield function (F) for isotropic hardening of
pressure-insensitive materials represented as:
F = f (𝐽2
′
, 𝐽3
′
) – k (𝜺 𝒆
𝒑
) = 0 (10)
Where 𝐽2
′
and 𝐽3
′
are the second and third invariants of the
deviatoric stress tensor S, k is function of equivalent or
effective plastic strain, 𝜺 𝒆
𝒑
.
10

11. Bauschinger Effect
Fig. 6 Bauschinger effect [5]
• When we change nature of stress from tension to compression yield
strength of the metal decreases. This effect is called as Bauschinger
effect.
• Because of this effect modelling of plastic deformation becomes
complicated.
• This effect is a generally seen in most polycrystalline metals as well
as in single crystals.
11

12. Kinematic Hardening
Fig. 7 Schematics of the deviatoric plane: stress vs plastic strain
response for kinematic hardening model []
• The yield surface does not change its shape and size, but
simply translates in the stress space in the direction of its
normal.
• It is used to represents Bauschinger effect.
• The initial yield surface is described by
F = f (σ) – k = 0 (11)
• After kinematic hardening, subsequent yield surface takes
the form
f (σ - 𝜶) – k = 0 (12)
Where F is a yield function, 𝜶 is a tensorial hardening
parameter, generally called as back stress is center of the
yield surface in the stress space, k represents size of the
yield surface.
12

13. Prager Model (1955)
• By observing behavior of kinematic hardening Prager proposed the following linear constitutive equation
for the back stress ‘𝜶’ [3]
dα = c dεp (12)
Where c is a material constant.
• According to this model, yield surface moves in the direction of the normal to the yield surface at the
loading point, due to the normality condition for the plastic strain rate or strain increments.
Drawbacks
• It does not give consistent results for 3D and 2D cases.
• Transverse hardening or softening effect.
13

14. Ziegler Model (1959)
• After overcoming the drawbacks of Prager’s model, Ziegler proposed the following constitutive
equation [3]:
dα = (σ - α) dµ (13)
where dµ is a proportionality scalar constant determined by yield criterion, α is back stress.
Fig. 8 Prager’s and Ziegler’s Kinematic hardening
model [3]
• In Prager’s model, yield surface moves along the normal
direction where as in Ziegler’s model, the movement takes
place in the radial direction determined by the vector σ – α.
14

15. Theory of Plasticity
Fig. 9 Uniaxial stress-strain curve [3]
• The plastic deformation is history or path dependent
process because of its dissipation feature.
• According to the history dependent nature of the total εp,it
should be expressed as
εp = ʃdεp = ʃ 𝜺 dt (14)
Two general categories of plastic stress-strain relationships.
• Incremental or flow theories relate stresses to plastic strain
increments.
• Deformation or total strain theories relate the stresses to
total plastic strains.
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16. Levy-Mises Equations
• Ideal plastic solids where elastic strains are negligible.
• Consider yielding under uniaxial tension:
𝝈 𝟏 ≠ 0, 𝝈 𝟐 = 𝝈 𝟑 = 0, and 𝝈 𝒎 =
𝝈 𝟏
3
• Since only deviatoric stresses cause yielding
𝑺 𝟏 = 𝝈 𝟏 - 𝝈 𝒎 =
2𝝈1
3
; 𝑺 𝟐 = 𝑺 𝟑 =
−𝝈 𝟏
3
(15)
• Constant volume condition: d𝜺 𝟏= -2d𝜺 𝟐 = -2 d𝜺 𝟑
d 𝜺 𝟏
d 𝜺 𝟐
= -2 =
𝑺 𝟏
𝑺 𝟐
(16)
• Generalization:
d 𝜺 𝟏
𝑺 𝟏
=
d 𝜺 𝟐
𝑺 𝟐
=
d 𝜺 𝟑
𝑺 𝟑
= d 𝜆 (17)
• At any instant of deformation, the ratio of the plastic strain increments to the current deviatoric stresses is
constant.
• In terms of actual stresses,
d𝜺 𝟏 =
2
3
d 𝜆 𝝈 𝟏 −
1
2
𝝈 𝟐 + 𝝈 𝟑 (18)
16

17. Levy-Mises Equations
• The principal axes of stress 𝝈 are same as deviatoric stress S, thus 𝜺 or d𝜺 is coaxial with S. Levy-Mises
equation can be expressed as
𝛆 = 𝜆 S or 𝛆ij = 𝜆Sij (19)
Where 𝜆 is a proportionality factor and determined by yield criterion.
• Von Mises criterion can be expressed as
𝑺𝒊𝒋 𝑺𝒊𝒋 =
2
3
𝝈 𝒚
𝟐 (20)
where 𝜎 𝑦 is yield stress. From eq. (19) and (20) we can get
𝜆 =
𝟑 𝜺 𝒊𝒋 𝜺 𝒊𝒋
𝟐𝝈 𝒚
𝟐 =
3
2
𝜺 𝒆
𝝈 𝒚
(21)
Where 𝜺 𝒆 is effective incremental strain rate, 𝜺 𝒆=
2
3
𝜺 𝒊𝒋 𝜺 𝒊𝒋
2
• Drawbacks: Only plastic strains are considered.
17

18. Prandtl-Reuss Equations
• Elastic strain considered along with plastic strain. Prandtl (1927) and Reuss (1930) proposed equation for
the plastic strain rate. Elastic deformation causes volume and shape change [3].
• For shape change, the rate of deviatoric elastic strain rate 𝜺′𝒆 can be represented as
𝜺′𝒆 =
1
2𝑮
𝑺 or 𝜺𝒊𝒋
′𝒆
=
1
2𝑮
𝑺𝒊𝒋 (22)
• For volume change, the rate of deviatoric elastic strain,
𝜺kk
e =
𝑷
𝑲
=
𝝈 𝒌𝒌
9𝑲
𝜹ij where 𝑷 = (σ1 + σ2 +σ3 )/3 (23)
• Rate of plastic strain rate 𝜺p can be expressed by assuming to be coaxial with the deviatoric stress S as
𝜺p = 𝜆 S or 𝜺p
ij = 𝜆 𝑺𝒊𝒋 (24)
• Total strain rate:
𝜺ij =
𝝈 𝒌𝒌
9𝑲
𝜹ij +
1
2𝐺
𝑺ij + 𝜆 𝑺𝒊𝒋 (on the yield surface)
𝜺ij =
𝝈 𝒌𝒌
9𝑲
𝜹ij +
1
2𝑮
𝑆𝑖𝑗 (inside the yield surface) (25)
where K and G are bulk modulus of rigidity and modulus of rigidity respectively.
18

19. Summary
General Theory of Plasticity requires the following
• A yield criterion, which specifies the onset of plastic deformation for different combination of applied load.
e.g. von Mises, Tresca.
• A hardening rule, which prescribes the work hardening of the material and the change in yield condition
with the progression of plastic deformation.
e.g. Isotropic or Kinematic Hardening
• A flow rule which relates increments of plastic deformation to the stress components.
e.g. Levi-Mises or Prandtl-Reuss
19

20. References
1. https://nptel.ac.in/courses/112107146/lects%20&%20picts/image/lect11/lecture11.htm
2. http://www.failurecriteria.com/misescriteriontr.html (Von Mises and Tresca criterion fpr plane stress
condition graph)
3. Continuum Theory of Plasticity by A. S. Khan, Sujian Huang
4. A review of some plasticity and viscoplasticity constitutive theories
J.L. Chaboche 2008 International Journal of Plasticity.
5. https://www.researchgate.net/figure/Schematic-loading-reverse-loading-curves-with-without-
permanent-softening_fig2_225776824
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