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Numerical Simulation of Elastic-Plastic Non-Conforming Contact
271
Plastic strains induce residual stresses, namely elastic stresses that would persist after elastic
unloading. These stresses superimpose the ones induced by contact pressure. The resulting
state generates further plastic strain if stress intensity exceeds yield strength. Consequently,
an accurate estimation of stress field in the elastic-plastic body is essential to plastic strain
increment prediction.
Figures 6 and 7 depict distributions of equivalent von Mises contact stress (stress induced by
contact pressure) and total stress in the elastic-plastic half-space. Residual stress intensity,
Fig. 8, is one order of magnitude smaller than equivalent contact stress. Comparison of
distributions depicted in Figs. 6 and 7, using the same scale, suggests that residual stress
reduces peaks in contact stress intensity, thus making the resulting field more uniform. This
behavior is also suggested by the curves traced in Fig. 9. Maximum intensity of contact
stress increase more rapidly than the maximum of the total field, due to contribution of
residual stress. Consequently, residual stresses, which represent material response to plastic
flow, act to impede further plastic yielding.
Fig. 6. Von Mises stress induced by contact pressure
Fig. 7. Maximum intensities of stress fields versus loading level
Numerical Simulations - Applications, Examples and Theory
272
Fig. 8. Von Mises residual stress
Fig. 9. Total (contact and residual) Von Mises stress in the elastic-plastic body
Profiles of residual prints corresponding to the same six loading levels are depicted in Fig.
10. These profiles show that residual displacement increase contact conformity in
investigated non-conforming contact, leading to a more uniform distribution of contact
pressure.
The variation of residual print maximum depth with the loading level is presented in Fig.
11. This curve was also obtained experimentally by El Ghazal, (El Ghazal, 1999), numerically
by Jacq et al., (Jacq et al., 2002), and using FEA by Benchea and Cretu, (Benchea & Cretu,
2008).
Numerical Simulation of Elastic-Plastic Non-Conforming Contact
273
Fig. 10. Residual print profiles in elastic-plastic spherical contact
Fig. 11. Residual print depth versus loading level
6.2 E-EP and EP-EP Contact
Normal residual displacement enters interference equation, by superimposing the
deflections induced by contact pressure. When only one of the contacting bodies, let it be
body (2), is elastic-plastic and the other one, let it be body (1), is elastic, the following
interference equation can be written by superimposing the residual part of
displacement
2
3
()r
u
, related to development of plastic zone in the elastic-plastic body (2), in
elastic contact interference relation, Eq. (7):
12
2
12
33
ω
+
+
=
++−
()
()
()
(, ) (, ) (, ) (, ) .
pr
r
hi j hi i j u i j u i j
(35)
Numerical Simulations - Applications, Examples and Theory
274
On the other hand, when contacting bodies are both elastic-plastic, Eq. (35) encloses residual
displacements of both surfaces, namely
12
3
+()
(, )
r
uij. If the hardening behavior or contacting
bodies is dissimilar, residual displacement should be computed for every body separately.
The model is simplified considerably if the bodies follow the same hardening law and have
the same initial contact geometry, because, due to symmetry of the problem about the
common plane of contact,
12
33
=
() ( )rr
uu. Consequently, Eq. (35) becomes:
12
2
12
33
2
ω
+
+
=
++−
()
()
()
(, ) (, ) (, ) (, ) .
pr
r
hi j hi i j u i j u i j (36)
To validate Eq. (36), the contact between two spheres of radius
0 015= .Rm is simulated
numerically, for two different material behaviors: elastic, and elastic-plastic following
Swift's law, with the following parameters: 945
=
BMPa, 20
=
C , 0 121
=
.n .
The contact is loaded up to a level of 11 179
=
,WN, corresponding to a hertzian pressure
8=
H
p
GPa and to a Hertz contact radius 817
μ
=
H
am.
Pressure distributions obtained using Eqs. (35) and (36) respectively, depicted in Fig. 12,
agree well with already published results, (Boucly et al., 2007). As expected, in the EP-EP
contact, pressure appears more flattened compared to the E-EP case, due to a more
pronounced increasing in contact conformity related to doubling of the residual term.
Fig. 12. Pressure profiles for various material behaviors
Variations of maximum effective plastic strain with loading level, in the E-EP and in the EP-
EP contact respectively, are depicted in Fig. 13. Intensity of plastic strains in the E-EP contact
is up to 40% higher than the one corresponding to the EP-EP scenario.
Variations of maximum pressure with the loading level in the E-E, the E-EP and the EP-EP
contact, are depicted in Fig. 14. The curves presented in Figs. 13 and 14 also match well the
results of Boucly, Nélias, and Green, (Boucly et al., 2007).
Numerical Simulation of Elastic-Plastic Non-Conforming Contact
275
Fig. 13. Maximum effective accumulated plastic strain versus loading level
Fig. 14. Maximum pressure versus loading level
6.3 R-EPP contact and experimental validation
As Contact Mechanics uses simplifying assumptions in order to circumvent the
mathematical complexity of the arising equations, experimental validation is needed to
verify model viability. An extended program of experimental research was conducted in
the Contact Mechanics Laboratory of the University of Suceava, aiming to assess residual
print parameters in rough elastic-plastic non-conforming contacts. The stand used for the
loading experiments was originally designed by Nestor et al., (Nestor et al., 1996).
Microtopography of deformed surface was scanned with a laser profilometer UBM14.
Contact between a steel ball, assumed as a rigid indenter, and a lead specimen, simulating
the elastic-plastic half-space, was loaded up to an equivalent hertzian pressure
Numerical Simulations - Applications, Examples and Theory
276
094= .
H
p
GPa . The contact was also simulated using the numerical formulation. As lead is
best described as an EPP material, a linear hardening law with a very small slope was
considered in the numerical model. As stated in (Jacq, 2001), the plastic strain increment is
undefined when assuming a purely EPP material behavior.
Residual prints at a hertzian pressure of
094. GPa is depicted in Fig. 15.
Fig. 15. Experimental residual print in R-EPP contact,
094= .
H
p
GPa
Fig. 16. Residual print depth versus loading level
Variation of print depth with loading level is presented in Fig. 16. The agreement between
the values predicted numerically and those obtained experimentally is considered
satisfactory, giving the complexity of the phenomena involved.
Numerical Simulation of Elastic-Plastic Non-Conforming Contact
277
6. Conclusions
A numerical approach for simulating the elastic-plastic contact with isotropic hardening,
based on Betti’s reciprocal theorem, is overviewed in this paper. Problem decomposition, as
originally suggested by Mayeur and later by Jacq, is employed to assess pressure and plastic
strain distribution, on three nested iterative levels.
The newly proposed algorithm has two major advantages over other existing methods.
Firstly, the plastic strain increment is determined in a fast convergent Newton-Raphson
procedure which iterates a scalar, namely the effective accumulated plastic strain. The
method, originally suggested by Fotiu and Nemat-Nasser, employs an elastic predictor,
which places the trial state outside yield surface, and a plastic corrector, used to derive the
return path to the yield locus. The algorithm is fast, stable, and accurate even for large
loading increments.
An additional advantage arises from moving residual stress computation, which is very
computationally intensive, to an upper iterative level.
Secondly, the use of three-dimensional spectral methods for solving the intrinsically three-
dimensional inclusion problem improves dramatically the overall algorithm efficiency.
Solution is obtained by problem decomposition, following a method originally suggested by
Chiu. Subproblem of stresses due to eigenstrains in infinite space is solved using influence
coefficients also derived by Chiu. Traction-free surface condition is imposed with the aid of
Boussinesq fundamental solutions, in a simplified formulation, well adapted to elastic-
plastic contact modeling.
With the newly advanced three-dimensional convolution and convolution-correlation
hybrid algorithm, based on the DCFFT technique, the computational effort is reduced
dramatically, allowing for finer grids in problem discretization.
The newly proposed algorithm was used to simulate, with a high resolution of 120 120 80××
elementary cells, the spherical contact between bodies with various behaviors: R-EP, E-EP,
EP-EP and R-EPP.
Elastic-plastic pressure appears flattened compared to the elastic case, due to changes in
hardening state of the EP material, and in contact conformity due to superposition of
residual displacement in interference equation.
Plastic zone, initially occupying a hemispherical region located at hertzian depths, advances
toward half-space boundary with increased loading, enveloping an elastic core. This
development is consistent with existing models for the elastic-plastic process, marking the
passing from elastic-plastic range to fully plastic.
Residual stress intensity is one order of magnitude smaller than equivalent stresses induced
by contact pressure. They contribute to total elastic field by decreasing the peaks in contact
stress intensity, thus impeding further plastic flow.
A modified interference equation is used for solving the EP-EP contact with similar
hardening behavior and symmetry about the common plane of contact.
Furthermore, residual displacement predicted numerically for the R-EPP contact match well
print depths obtained experimentally in indentation of a lead specimen, assumed as an EPP
half-space, with a steel ball assumed as a rigid indenter.
Numerical Simulations - Applications, Examples and Theory
278
7. Acknowledgement
This paper was supported by the project “Progress and development through post-doctoral
research and innovation in engineering and applied sciences – PRiDE - Contract no.
POSDRU/89/1.5/S/57083”, project co-funded from European Social Fund through
Sectorial Operational Program Human Resources 2007-2013.
Grant CNCSIS 757/2007-2008, entitled “Research upon the Effects of Initial Stresses in
Dental Biocontacts” provided partial support for this work.
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