Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model
A representative volume element is developed based on the Voronoi tessellation to reveal the mechanism of shape instability behavior. In the model, a damage-coupled crystal plastic model is established to describe the shape instability behavior. The heterogeneity of materials is introduced into the...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2017
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| Zitieren: | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model / G.C. Wu, Y.F. Li, X.D. Pan, G.L. Wang // Проблемы прочности. — 2017. — № 1. — С. 132-139. — Бібліогр.: 20 назв. — англ. |
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| citation_txt | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model / G.C. Wu, Y.F. Li, X.D. Pan, G.L. Wang // Проблемы прочности. — 2017. — № 1. — С. 132-139. — Бібліогр.: 20 назв. — англ. |
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| container_title | Проблемы прочности |
| description | A representative volume element is developed based on the Voronoi tessellation to reveal the mechanism of shape instability behavior. In the model, a damage-coupled crystal plastic model is established to describe the shape instability behavior. The heterogeneity of materials is introduced into the model with the aim of simulating the microstructure of materials. The experimental and simulation results show that the fatigue damage in the elastic deformation stage with high cyclic stress level is the initial motivation of shape instability behavior. The cyclic softening and ratcheting properties of materials accelerate the plastic strain accumulated rate.
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| first_indexed | 2025-12-07T18:03:14Z |
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UDC 539.4
Numerical Simulation of Fatigue Damage and Shape Instability Behavior of
Steel 40Cr by the Damage-Coupled Crystal Plastic Model
G . C. W u , Y. F. L i, X . D. P an , an d G. L . W ang
School of Mechatronics Engineering, Harbin Institute of Technology, Harbin, China
A representative volume element is developed based on the Voronoi tessellation to reveal the
mechanism o f shape instability behavior. In the model, a damage-coupled crystal plastic model is
established to describe the shape instability behavior. The heterogeneity o f materials is introduced
into the model with the aim o f simulating the microstructure o f materials. The experimental and
simulation results show that the fatigue damage in the elastic deformation stage with high cyclic
stress level is the initial motivation o f shape instability behavior. The cyclic softening and ratcheting
properties o f materials accelerate the plastic strain accumulated rate.
K eyw ords: shape instability failure m echanism , plastic deformation, finite elem ent analysis,
disposable m echanical elements.
In tro d u c tio n . A s the m ain failure m ode for disposable m echanical elements that
usually undertake critical cyclic loads, shape instability behavior is defined to describe the
process from the total elastic deform ation to the elastic-plastic deform ation o f m aterials [1].
Figure 1 shows the stress-strain hysteresis cycle curves recorded in stress-controlled
low-cycle fatigue tests. The shape instability failure is defined w hen the accum ulated total
plastic strain, £ t , in the specim en reaches 0.2% in the fatigue process, £p and £ r are
cyclic plastic strain and ratcheting strain, respectively.
Q 4 t
Omax
•Tmin
Fig. 1
The shape instability life data are vital for the reliability design o f disposable
m echanical elements. However, as shape instability failure that occurs during the transition
from elastic deform ation to plastic deform ation concerns only the initial stage o f plastic
deform ation, the shape instability life tests are fairly com plex and inefficient. Therefore, in
order to m inim ize the num ber o f life tests and predict shape instability life, the mechanism
o f shape instability behavior should be firstly investigated to find out the m ain control
factors. Shape instability behavior is sim ilar to com m on ratcheting one. The latter
phenom enon investigations are generally focused on the plastic evolution in the total
fatigue process, and such phenom ena are w ell understood [2-6]. Cyclic softening and
© G. C. WU, Y. F. LI, X. D. PAN, G. L. WANG, 2017
132 ISSN 0556-I7IX. Проблемы прочности, 2017, N2 1
Numerical Simulation o f Fatigue Damage
fatigue dam age can cause a sim ilar behavior w ith shape instability [7-9]. However, as
shape instability behavior is observed in the initial fatigue process, it is fairly difficult to
search and observe inconspicuous m icrostructure evolution features in macrospecimens.
Therefore, in order to reveal the m echanism o f shape instability, the num erical m ethod has
to be used in investigations, although not as stand-alone application. In this study, the
num erical m ethod is independently used based on the sim ilarity betw een the shape
instability behavior and the ratcheting behavior. The proved results on the ratcheting
behavior can be im proved to establish a proper num erical m odel to simulate the shape
instability behavior. Firstly, a three-dim ensional cyclic crystal plastic m odel w ith
inhom ogeneity o f m aterial properties based on the cyclic plastic theory is developed to
sim ulate the shape instability behavior. Then, the m odel param eters are m odified by
com paring the features o f shape instability behavior described by the m odel and the
experim ental data. I f the m acrobehavior described by the num erical m odel is coincident
w ith the one tested by the experim ent data, it can be assum ed that the num erical model
represents the details o f the shape instability process and the m ain factors controlling the
shape instability behavior can be identified, w hich is instrum ental for revealing the
m echanism o f shape instability behavior.
1. M a te ria ls an d Specim ens. In this study, steel 40Cr specim ens w ere prepared for
the param eter identification and shape instability experiments. The chem ical com position o f
steel 40Cr is as follows (wt.%): 0.45 C, 0.5 Mn, 0.37 Si, 1.0 Cr, 0.025 Ni, 0.03 P, 0.025 S,
and 0.03 Cu. The hardness o f steel 40Cr bar w as HRC 28-30 by quenching and tem pering
treatm ent. Specimens w ith a gauge diam eter o f 12 m m and a gauge length o f 25 m m were
m achined from steel bars w ith diam eter o f 22 m m from the same batch o f hot-rolled steel
40Cr. Turning and polishing processes were successively used to decrease and eliminate
scratches at the gauge position and transition arc o f the specimens. The roughness o f the
specim en gauges w ere m easured on a surface profiler and the roughness value w as no more
than 0.4 ,Mm. The cyclic stress-accum ulated plastic strain curves w ere tested to determine
the backstress and all other cyclic hardening parameters. Full ten sion - com pression tests
w ere conducted to obtain the fatigue life data and to determ ine the fatigue dam age
parameters.
2. C onstitu tive M odel. In order to sim ulate the shape instability behavior, a proper
cyclic crystal plasticity m odel should conclude both cyclic softening and fatigue dam age
parameters. It m eans that the constitutive m odel should be established based on a fatigue
dam age-coupled cyclic plasticity model. Therefore, a m odel based on the F rederick-
A rm strong m odel is used to describe the cyclic plastic deform ation behavior o f traditional
m etallic m aterials [10]. The evolution law o f Ohno and Wang is com bined w ith the
fram ework o f the Frederick-A rm strong m odel to adjust the ratcheting strain rate to
im prove the prediction accuracy o f strain [11, 12]. The elastic-plastic fatigue damage
m odel provided by Lem aitre and Chaboche is involved to take the fatigue dam age into
consideration [13]. The m ain equations in the established m odel are presented as followed:
A ccording to the small am ounts o f deform ation theory, the total strain, £ ̂ , is defined
as
(1)
where £ j is the elastic and £ P is the plastic strains.
Value o f £ j is expressed as
(2)
ISSN 0556-171X. npoôëeubi 2017, N2 1 133
G. C. Wu, Y. F. Li, X. D. Pan, and G. L. Wang
where v is the Poisson’s ratio, E is the elastic modulus, o y is the Cauchy stress tensor,
D is dam age parameter, o is the m ain diagonal com ponent o f o y , and (5y is
K ronecker delta.
The evolution law o f plastic strain, pp , is defined as
e p
e ij
- dF
do ij
(3)
where X is a non-negative scalar multiplier, F is the M ises y ield function for damage,
w hich is as follows:
F =
. , . , , 1/2
d" - x W - r ^ - d l - Q,2 1 1 - D 1 - D (4)
where o dev is the deviatoric stress tensor and % is the backstress tensor. V alue o f Q is the
radius o f the y ield surface and defined as [14]:
Q = P (Q™- Q )bv. (5)
where Qx and bi are m aterial constants. Value o f p is defined as
P = ( 1(~ P P )f = - > ■ (6)
The backstress tensor % in the Eq. (5) is expressed as
E n = 1 %(f} (i = 1 ,2 , . . . , n), (7)
where n is a backstress com ponent, and %( ̂ is the respective total num ber o f backstress
com ponents, defined as follows [11, 12]:
(1- D )c ( ' ) 1 r (i} e p - fix (i T
m
!e p „ % !'3 r (i )
̂ ) \ lk (!)l /
x (8)
(i ) (9)
where c (1 ̂ and r (1 ̂ are m aterial constants and m is an enhancem ent rate exponential.
Value o f D is calculated as follows [13]:
— = [1 - ( 1 - D )l+ß f
dN
A II
M 0 (1 3b2r H , mean )(1 D )
(10)
where a , /3, M o , and are m aterial constants and are determ ined based on the w ork of
Bogard et al. [15], o H , mean is the m ean value o f the hydrostatic stress, w hich is zero, and
A u is the param eter determ ined by the Sines yield criterion [16].
134 ISSN 0556-171X. npoôëeMbi npounocmu, 2017, № 1
Numerical Simulation o f Fatigue Damage
3. M odel D evelopm ent. The dam aged-coupled cyclic crystal plastic m odel was
program ed by using the UMAT function o f the software ABAQUS. The cyclic plastic strain
and ratcheting strain in the initial fatigue process could be accurately predicted by using
appreciate m aterial property parameters. Voronoi tessellation is used to build a
representative volum e element. The features o f m aterial m icrostructures are approxim ately
duplicated in the software ABAQUS.
Figure 2a, obtained via electron backscatter diffraction (EBSD) measurem ents, shows
that the average grain size is — 5 [im. As the steel 40Cr was hot rolled, the m easured
specim en w as cut o ff from the small cylindrical bar along the radial direction. The grain
size was then m easured along the transverse direction. Because the grains in the m aterial
are generally equiaxed, each o f the grain-size ratios, RD/ND and RD/TD, is assum ed to be
close to unity. The crystalline phase images confirm that the grains in the m icrostructure are
equiaxed. A ccording to the grain size, the grain density in the finite elem ent m odel is
approxim ately 8 -106 polyhedrons/m m 3 in the sim ulation model. Obviously, the proper
dim ension for the m odel should be determ ined to reduce the analysis time. Therefore, the
sim ulated zone is lim ited to a cube w ith an area o f 50 ,«m3 and 1000 polyhedrons. As
shown in Fig. 2b, the polyhedrons in the cube are generated conform ing to the rules o f
Voronoi tessellation. In Voronoi tessellation, the angles at the triple junctions m ay
sometim es differ from 120°. In order to reduce the influence o f the artificial stress, sharp
angles at the junctions could be avoided by assigning seeds based on a uniform distribution.
O nly some o f the junctions are triple junctions, and m ost o f the angles at the junctions are
not very sharp. In addition, the artificial concentration by angular differences at the triple
junctions is insignificant as com pared w ith that resulting from differences in the properties
o f the grains. This low concentration occurs because the total plastic strain is only 0.2%.
Grain diameter [ M. m]
a
Fig. 2. The basic information used to simulate the micromechanical properties of grains: (a) grain size
distribution in the region of interest from the EBSD measurement; (b) representative volume element
of steel 40Cr; (c) nanoindentation test conducted at 50 random points.
ISSN 0556-171X. npoôëeuu npouHocmu, 2017, № 1 135
G. C. Wu, Y. F. Li, X. D. Pan, and G. L. Wang
Therefore, the error associated w ith the sim ulation is acceptable. By applying the periodic
boundary condition on the cube, a representative volum e elem ent m odel is established. The
grain m echanics properties are generated based on the stochastic distribution, w hich is
determ ined according to nanoindentation test data. Figure 2c shows a part o f 50 random
positions on the specim en used to conduct the tests. The elastic m odule o f the grains could
thus be obtained directly. The yield strength o f the grains could be experim entally
determ ined from stress-strain curve o f nanoindentation tests w ith relative ease based on the
w ork o f L iu and Chen [17]. These m aterial data are then used to fit a G aussian distribution.
For the steel 40Cr in this study, the expected value and standard deviation o f y ield strength
are 790 and 92 M Pa, respectively. The effect o f phases has been involved in the
determ ination o f distribution param eters and finally presented in the sim ulations o f
m echanical behaviors.
The cyclic crystal plasticity m odel is program m ed and introduced into a representative
volum e elem ent (RVE) m odel established in the software ABAQUS. As the stress range
w hich is around the elastic lim it o f m aterials is fairly narrow, inhom ogeneity o f m aterial
properties is introduced into the RVE m odel to describe shape instability behavior. The
RVE m odel w ith m ore realistic m icrostructure has the capacity to reflect the detail in the
shape instability process. Figure 3 shows the m ethod to incorporate the m icrostructural
features and grain m echanics properties into the sim ulation m odel in ABAQUS. Figure 3b
shows the RVE m odel established in ABAQUS. Figure 3b shows the procedure to assign
random groups o f m aterial property data into the RVE model. A group o f grain m echanics
properties is random ly generated by using the G aussian distribution function o f ABAQUS.
A fter that, the generated data are assigned to the grains in the RVE model. The established
sim ulation m odel represents sim ilar geom etric features and m echanical properties as the
real specim ens on a micro level. By assigning the developed m aterial constitutive model
into the elements, the established RVE m odel is able to sim ulate the shape instability
behavior.
Random material data
Fig. 3. Shape instability behavior simulation in ABAQUS: (a) representative volume element; (b) the
method to introduce inhomogeneity of material properties into RVE.
4. S im ula tion an d E x p e rim e n ta l R esults. Figure 4 shows the com parison results
betw een experim ental and sim ulated results. Generally, m ean stress and stress am plitude
are m ore frequently used in fatigue investigations. However, they are not intuitive
param eters in the m echanical design. M axim um load and fatigue feature, corresponding to
the m axim um stress and stress ratio, are usually given as design indexes. Therefore, the
experim ent in this paper w as conducted under stress ratios —1 and —2/3, and the constant
m axim um stress, w hich is equal to 95% o f the yield strength. It is noted that the yield
136 ISSN 0556-171X. npoôneMb npounocmu, 2017, N 1
Numerical Simulation o f Fatigue Damage
strength is usually set as an input param eter in norm al finite elem ent models. However, the
random generation and assignm ent strategies o f m aterial properties in the grains o f the
m odel w ill all influence the m acroperform ance o f the sim ulated yield strength, although the
expected value o f the m aterial properties is the tested yield strength. Therefore, the
sim ulated yield strength should be firstly sim ulated for the RVE m odel, but not directly
specified as the test yield strength. The stress-strain curve w ith 0.2% total plastic strain is
picked up from the w hole fatigue process, both experim entally and via the simulations.
S tress-strain hysteresis loops w ith sim ilar features are picked out for both the experiments
and the sim ulations by adjusting the m aterial param eter bi from Eq. (6) and the damage
param eter D. It m eans that these two param eters are the m ain factors influencing shape
instability behavior.
Fig. 4. Comparison results of steel 40Cr between experimental data and simulated results.
5. D iscussion. Param eter bi represents the cyclic softening and ratcheting properties
o f materials. Damage param eter D reflects the fatigue perform ance o f materials. These two
properties play different roles in the w hole shape instability process. A ccording to the
plasticity deform ation features, shape instability behavior could be divided into two stages
including the plastic strain generation stage and the plastic strain increasing stage.
In the previous stage, according to the sim ulation m odel, the plastic strain is generated
due to the fatigue dam age o f materials. In Eq. (6), the cyclic softening property o f m aterials
w ill cause plastic strain accum ulation, only w hen the m axim um stress is higher than the
elastic limit, that is, the RVE m odel should be loaded in the plastic deform ation zone (the
constitutive m odels have been validated in m any studies [18-20]). As the m axim um stress
in the sim ulation is the 95% o f the elastic limit, it could be considered that the cyclic
softening properties are not the trigger o f shape instability behavior. In the micro level, the
plastic strain occurs in the RVE m odel w hen the w eakest grain exceeds its ow n elastic limit,
although at this tim e the m acro elastic lim it o f the RVE m odel is higher than the value o f
the w eakest grain. It m eans that in the m icro level the plastic strain in the RVE m odel is
also irrelevant w ith the cyclic softening properties o f m aterials. For the specim ens in the
experiment, flaws and defects are the w eakest parts. The plastic strain w ill occur around
these stress-concentration locations. W ithout fatigue damage, the accum ulated plastic strain
w ill tend to form stable hysteresis loops after a certain num ber o f the sym m etrical or
unsym m etrical cyclic loading. Obviously, it is fairly difficult for m aterials arriving at 0.2%
total plastic deform ation only depending on the plastic strain from the stress-concentration.
Therefore, it could be concluded that fatigue dam age is prim ary in driving shape instability
behavior at the initial stage.
In the plastic strain increasing stage, after the plastic strain is generated, the cyclic
softening and ratcheting properties o f m aterials w ill accelerate the accum ulation rate o f the
plastic strain. The rapid increasing o f plastic strain w ill lead to fatigue fracture m uch faster
ISSN 0556-171X. npoôëeubi 2017, N2 1 137
G. C. Wu, Y. F. Li, X. D. Pan, and G. L. Wang
than expected. In this stage, neglecting cyclic softening and ratcheting properties will
overestim ate the fracture life and lead to fatal risks.
As the lim itation o f experim ent observation methods, the conclusion is not obtained
based on the experim ent result but from the num erical simulation. However, the conclusion
is credible, because the num erical m odel is established based on the previous proved
theories according to the sim ilarity betw een the shape instability behavior and the
ratcheting behavior. Since the m echanics behavior in m acrolevel could be described by
using the num erical model, it is reasonable to assume that the detail in the num erical model
is sim ilar w ith the m aterial m icrostructures in the m icrolevel, therefore, the num erical
sim ulation results are reliable.
C onclusions. The investigation shows that the fatigue dam age in the elastic
deform ation stage w ith high stress level is the initial m otivation o f shape instability
behavior. The cyclic softening and ratcheting properties o f materials accelerate the plastic
strain accum ulated rate. Shape instability behavior is not the essential property o f cyclic
softening materials. It explains that some cyclic softening materials, such as T i-6A l-4V , do
not show shape instability behavior. I f cyclic softening m aterials have excellent fatigue
perform ance in the elastic deform ation stage, shape instability behavior could be ignored. It
m eans that it is not necessary to test the shape instability life o f these cyclic softening
materials.
Acknow ledgm ents. This study is supported by the National Natural Science Foundation
o f China (51405101), the research and innovation fund o f H arbin Institute o f Technology
(Grant N um ber HIT.NSRIF.2015 053), the China Postdoctoral Science Foundation (Grant
N um bers 2014M 561340 and 2016T90277) and Heilongjiang Postdoctoral Fund (Grant
N um ber LBH-Z14100).
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Received 30. 08. 2016
ISSN 0556-171X. npoôëeMbi npounocmu, 2017, N 1 139
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| id | nasplib_isofts_kiev_ua-123456789-173591 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T18:03:14Z |
| publishDate | 2017 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Wu, G.C. Li Y., F. Pan, X.D. Wang, G.L. 2020-12-12T14:40:01Z 2020-12-12T14:40:01Z 2017 Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model / G.C. Wu, Y.F. Li, X.D. Pan, G.L. Wang // Проблемы прочности. — 2017. — № 1. — С. 132-139. — Бібліогр.: 20 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173591 539.4 A representative volume element is developed based on the Voronoi tessellation to reveal the mechanism of shape instability behavior. In the model, a damage-coupled crystal plastic model is established to describe the shape instability behavior. The heterogeneity of materials is introduced into the model with the aim of simulating the microstructure of materials. The experimental and simulation results show that the fatigue damage in the elastic deformation stage with high cyclic stress level is the initial motivation of shape instability behavior. The cyclic softening and ratcheting properties of materials accelerate the plastic strain accumulated rate. This study is supported by the National Natural Science Foundation of China (51405101), the research and innovation fund of Harbin Institute of Technology (Grant Number HIT.NSRIF.2015 053), the China Postdoctoral Science Foundation (Grant Numbers 2014M561340 and 2016T90277) and Heilongjiang Postdoctoral Fund (Grant Number LBH-Z14100). en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model Численное моделирование формоизменения стали 40Сг в условиях циклического нагружения в упругопластической постановке Article published earlier |
| spellingShingle | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model Wu, G.C. Li Y., F. Pan, X.D. Wang, G.L. Научно-технический раздел |
| title | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model |
| title_alt | Численное моделирование формоизменения стали 40Сг в условиях циклического нагружения в упругопластической постановке |
| title_full | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model |
| title_fullStr | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model |
| title_full_unstemmed | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model |
| title_short | Numerical Simulation of Fatigue Damage and Shape Instability Behavior of Steel 40Cr by the Damage-Coupled Crystal Plastic Model |
| title_sort | numerical simulation of fatigue damage and shape instability behavior of steel 40cr by the damage-coupled crystal plastic model |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173591 |
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