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TO STUDY THE MATERIAL BEHAVIOUR OF SA333-Mn
STEEL UNDER PROPORTIONAL AND NON
PROPORTIONAL LOADING
Pruthviraj B. Parmar1, Chaitanya K. Desai2
P. G. Student, CKPT,Surat,Gujarat Technological University, India 1
Associate Professor, CKPT,Surat,Gujarat Technological University,Ahmadabad, India
2
Abstract: In a most of non proportional cyclic loading, the investigated material show
additional hardening in comparison to the hardening observed under proportional loading
also opposite effect is observed. In the present paper the above mentioned phenomena are
simulationally investigated under non proportional cyclic loading in the form of circular
path and proportional cyclic loading for the commercial material: SA333 C-Mn steel
according to the polish standards. Use the mechanical properties and chaboche model to
the cyclic loading condition. The simulation study was carried out on the strain cyclic
characteristics and ratcheting of the material subjected to proportional loading and from
that shows loading depend on the strain amplitude but slightly on the history and
ratcheting changes with the different condition.
Keywords: Cyclic plastic zone, Chaboche kinematic hardening model, EPFM,
Ratcheting.
I. INTRODUCTION
Mechanical behaviour of structural materials under non-proportional loading was
investigated in many research centres [H.S. Lamba, O.M. Sidebottom(1978) ,E. Tanaka,
(1985)]. Therefore, it is well-known that the cyclic loading of metals along nonproportional paths is much more significant than that for proportional ones. In
majority of the cases of non-proportional cyclic loading the investigated materials
showed additional hardening in comparison to the hardening observed under
proportional loading path [S. Murakami, M. Kawai (1989)]. It is usually assumed that
such effect is due to a higher number of slip systems activated by the complex nonproportional loading [S.H. Doong, D.F. Socie (1991)]. From previous experimental
works it is also known that for certain class of materials the softening effect can be
observed under the non-proportional cyclic loadings [L. Dietrich, G. Socha, Z.L.
Kowalewski (2000)]. Such difference between material behaviour under cyclic
loading leads to the essential difficulties in the constitutive modeling. Therefore, for the
rational formulation of multiaxial cyclic constitutive equation, it is necessary to study a
series of representative non-proportional cyclic behavior, and to identify the property of
the multiaxial cyclic hardening or softening mechanisms.
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Proportionality and additional hardening are two common terms in multiaxial
fatigue that increase the hardening of the materials: Proportional loading cases
increase the hardening of the material by fatigue damage mechanisms and the
additional hardening is related with alteration of the dislocation substructure caused by
the cyclic plastic strain along multiple slip mechanisms of the material structure.
Nonproportional load can be defined mechanically in terms of the rotation of the
principal strain planes. From the fatigue viewpoint a strain path that results in a fixed
orientation of the principal axis associated with the alternation of strain components is
proportional to the strain history and is nonproportional if the principal axis rotates
with time.
II. MATERIAL MODEL DETAIL
In order to determine the behaviour of the material various linear and non linear material
models are used and there for there are four basic aspects that are used to derive the response
of the material like: (i) The yield surface, (ii) The flow rule derived from the normality
condition of plastic strain increments, (iii) The hardening rule and (iv) consistency condition
(Bari and Hassan, 2000). In chaboche model Von-mises yield function, associated flow rule
and kinematic hardening model is used to capture the cyclic response.
Kinematic hardening model proposed by Chaboche (Chaboche, 1991) is a superposition
of three Armstrong and Frederick hardening models.
The model can be written in the form
(1)
(2)
Where,
(3)
Where total back stress α is a summation of the three decomposed back stresses (α = α 1 +
α2 + α3); Ci is the kinematic hardening coefficient; Di is the kinematic hardening exponent;
and
are the plastic strain increment vector and equivalent plastic strain.
Loading part of the stress strain curve can be represent as
(4)
Where, σ and σ0 are the stresses at any point and cyclic yield stress respectively.
Saturated value of this back stress is in the model is given as a summation of all three
individual saturated value of decomposed back stress, as expressed by
(5)
Where αs is the saturated value of the back-stress.
Following the suggestion by chaboche to determine the material parameter from strain
controlled stable hysteresis loop and all the parameters are determine from the 1.6% stable
Low cyclic Fatigue (LCF) hysteresis loop. The value of C1 is determined from the slope of
stress-plastic strain curve of loading branch of stable hysteresis loop at cyclic yield point.
Corresponding D1 value should be large enough such that α1 saturate immediately. The value
of C3 is determined from the slope of stress-plastic strain curve of unloading branch of stable
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hysteresis loop at cyclic yield point. C2 and D2 are estimated by trial and error so that
following Equation (8) is satisfied, C1 > C2 > C3 and D1 > D2 > D3 = 0
(6)
Where,
is the plastic strain at the cyclic yield point in loading branch of hysteresis
loop. Material parameter of chaboche Kinematic hardening model is shown in table 1. [Paul
et al., 2010].
TABLE 1:- MATERIAL PARAMETER OF SA333 C-MN STEEL
Parameter E
ν
σ0
C1
C2
C3
D1
D2
D3
Value
200.0 0.3
225.0 140.0 25.0
1950.0 1750.0 238.0 0.0
GPa
MPa
GPa
GPa
MPa
III. FINITE ELEMENT MODEL OF SPECIMEN
A standard dogbone specimen simulation as shown in Fig. 1. To determining the stressplastic strain diagram, ratcheting cycles for the different loading magnitudes and tensile
stress-shear stress under proportional and non proportional loading cyclic loading
conventional software ABAQUS is used.
Figure 1: Dogbone specimen
For the Analysis purpose C3D6 and CD38R is used for producing the very fine mesh as
shown in Fig. 2.
Figure 2: A dogbone Specimen with Meshing
Exactly similar to experimental boundary conditions were applied in the simulation work;
bottom part is fixed and upper part give the tensile loading and torsion fir the non
proportional loading in all directions. For the proportional give the different magnitude of the
loading as tensile loading in the upward direction and fix at the bottom part.
IV. SIMULATION RESULT OF NON PROPORTIONAL LOADING CONDITION
In order to describe the loading paths, stress and strain states are characterised by the
Huber–Mises effective stress, and the Huber–Mises effective total strain. Thus, the biaxial
stress and strain states realised on a dogbone specimens can be described using the following
expressions:
(7)
where σ and τ are, respectively, axial and shear stresses. The stress amplitudes and the
strain amplitudes are defined as follows:
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(8)
Figure 3: Axial stress-Shear stess (Non proportional loading)
V. SIMULATION RESULT OF PROPORTIONAL LOADING CONDITION
Proportional loading analysis is performing on dogbone specimen for 20 cycles for
determining the stress-strain behaviour. For determining the stress-strain behaviour of the
specimen at three different loading conditions. Stress and Plastic strain response along the
loading direction with number of cycles is observed and Stress-strain hysteresis loop are
shown in Fig. 4 for all four position with Cycle number 1, 5, 10, 15 and 20. Out of all the
loading condition, for the first loading condition the cycles nearer so the hardening in the
material shown and another two loading conditions the softening shown because the cycle is
going the far with each other. It is well established that strain evolution taking place during
stress controlled loading and progressive accumulation of strain in the mean stress direction
is observed which is known as „ratcheting‟ shown in Fig.5.
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Figure 5: Ratcheting evolution with number of cycles
VI. CONCLUSION



Elastic plastic fracture mechanics based on the proportional loading and non
proportional cyclic loading analysis is performed on dogbone specimen using a
Chaboche kinematic hardening material model.For the non proportional loading
analysis the circular behaviour of the plastic stress-plastic starin.
Proportional loading analysis the three different loading condition says that first
loading condition strain-stress cycles are nearer so there is lower plastic strain and
loading is increase there is higher plastic strain for another two condition.
Progressive ratcheting behavior has been also observed in the cyclic for the low
loading condition the ratcheting is lower then the higher loading conditon.
ACKNOWLEDGMENT
I wish to thank all the members of Computational Mechanics Group of C.K.Pithawalla
College of engineering and technology for their co-operation, guidance and support during
the work. I further would like to whole-heartedly express my gratitude to all my family
members.
REFERENCES
[01] Chaboche, J.L. International Journal of Plasticity, 1991.
[02] E. Tanaka, S. Murakami and M. Ooka, “Effects of Strain Path Shapes on
Nonproportional Cyclic Plasticity”, I. Mech.Phys.Solids, 1984.
[03] H. S. Lamba and O. M. Sidebottom, “Cyclic Plasticity for Nonproportional Paths”,
ASME Journal of Engineering Materials and Technology, 1978.
[04] L. Dietrich, G. Socha and Z.L. Kowalewski, in: Proceedings of the 19th Symposium on
Exp. Mech. Sol., Jachranka, ZGPW, Warsaw, 2000.
[05] S. Bari and T. Hassan, “Anatomy of coupled constitutive models for ratcheting
simulation”. International Journal of Plasticity 16, 381-409, 2000.
[06] S. Paul and S. Dhar, “Simulation of cyclic plastic deformation response in SA333 C–Mn
steel by a kinematic hardening model.”, Computational Materials Science 48, 2010.
[07] S. H. Doong and D. F. Socie, “Constitutive Modeling of Metals under Nonproportional
Cyclic Loading”, ASME Journal of Materials and Technology, 1991.
[08] S. Murakami, M. Kawai, K. Aoki and Y. Ohmi, ASME J. Eng. Mater.Technol.1989.
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