(16)
where ξ is the smoothing parameter to switch viscosity between
loading and unloading.
Now, in the subsequent paragraphs, the procedure for determining
the viscosity constants
(Al, Au, q and n) will be discussed followed by the smoothing
parameter (ξ).
Bearing
/Pier
MPa MPa
Table 3. Rate-dependent viscosity parameters of the HDRB
Using the strain histories of the SR loading tests at different
strain levels (Figure 8), the
overstress-dashpot strain rates relationships are determined
(Figures 27 (a) to (c)), which
correspond to Eq.16 for both loading and unloading conditions. A
standard method of
nonlinear regression analysis is employed in Eq. 16 to identify the
viscosity constants. As
motivated by the relationships of the overstress-dashpot strain
rates obtained in the SR/MSR
test results, the value of n is kept the same in loading and
unloading conditions. The
nonlinear viscosity parameters obtained in this way are presented
in Table 3. Figures 27(a)
to (c) present the overstress-dashpot strain rates relationships
obtained using the proposed
model and the SR test results; the solid lines show the model
results and the points do for
the experimental data.
A sinusoidal loading history is utilized to determine the smoothing
parameter of the model
(Eq.16). The sinusoidal loading history corresponds to a horizontal
shear displacement
history applied at the top of the bearing at a frequency of 0.5 Hz
with amplitude of 1.75. An
optimization method based on Gauss-Newton algorithm [45] is
employed to determine the
smoothing parameter. The optimization problem is mathematically
defined as minimizing
the error function presented as
2
, , 1
Mechanical Characterization of Laminated Rubber Bearings and Their
Modeling Approach 329
where N represents the number of data points of interest, τexp,n
and τm,n imply the shear stress
responses at time tn obtained from the experiment and the model,
respectively, and ξ stands
for the parameter to be identified. Using the Gauss-Newton
algorithm, following condition
is satisfied for obtaining the minimum error function as
2 1, , 0i i i iE t E t , (18)
where refers the gradient operator. During the iteration process,
the updated parameter in
each iteration is determined using
1 2
, (19)
where δ is the numerical coefficient between 0 and 1 to satisfy the
Wolfe conditions at each
step of the iteration. The values of ξ determined in this way for
the bearings are presented in
Table 3.
Figure 26. Overstress-dashpot strain rate relation obtained from
the MSR tests at different strain levels
in loading and unloading regimes of (a) HDR2, (b) RB2, and (c)
LRB2; the values in the legend stand for
the total strain in respective relaxation processes, and 50,100,
150% correspond to relaxation processes
after loading and unloading.
-0.2 -0.1 0 0.1 0.2
O v e r s t r e s s ( M P a )
Dashpot strain rate(1/sec)
strain:1.5 strain:1.0 strain:0.5
-0.1 -0.05 0 0.05 0.1 0.15
O v e r s t r e s s ( M P a )
Dashpot strain rate(1/sec)
MSR test[RB2]
100% 150% 50%
-0.2 -0.1 0 0.1 0.2
O v e r s t r e s s ( M P a )
Dashpot strain rate(1/sec)
strain:0.5 strain:1.0 strain:1.5
Earthquake Engineering 330
Figure 27. Identification of viscosity parameters of (a) HDR2, (b)
RB2, and (c) LRB2; the model results
represented by solid lines are obtained by τ = τoe with parameters
given in Table 3 and the relations
between oe d as calculated from SR test data are shown by points.
The values in the legend stand for
the total strain in respective relaxation processes, and 100,150,
175% correspond to relaxation processes
after loading, and 0% after unloading.
4. Thermodynamic consistency of the rheology model
The Clausisus-Duhem inequality is a way of expressing the second
law of
thermodynamics used in continuum mechanics. This inequality is
particularly useful in
determining whether the given constitutive relations of
material/solid are
thermodynamically compatible [46]. This inequality is a statement
concerning the
irreversibility of natural resources, especially when energy
dissipation is involved. The
compatibility with the Clausisus-Duhem inequality is also known as
the thermodynamic
consistency of solid. This consistency implies that constitutive
relations of solids are
formulated so that the rate of the specific entropy production is
non-negative for arbitrary
temperature and deformation processes.
-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5
O v e r s t r e s s ( M P a )
Dashpot strain rate (1/sec)
strain:1.75 strain:1.5 strain:1.0
-0.4 -0.2 0 0.2 0.4 0.6
O v e r s t r e s s ( M P a )
Dashpot strain rate (1/sec)
strain:1.75 strain:1.50 strain:1.00
-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
O v e r s t r e s s ( M P a )
Dashpot strain rate (1/sec)
strain:1.75 strain:1.50 strain:1.00
In this context, the Clausius-Duhem inequality reads
0ρψ τγ , (21)
where is the mass density, is the Helmholtz free energy per unit
mass, and τγ is the
stress power per unit volume. It states that the supplied stress
power has to be equal or
greater than the time rate of the Helmholtz free energy. For the
rheology model, the
Helmholtz free energy is the mechanical energy stored in the three
springs shown in Figure
17 can be represented as
12 2 23 1 2 4
1 1 1
a c a c
C ρψ γ,γ ,γ C γ C γ γ C γ
m
1 2 3 4
m
a c a a c cρψ γ,γ ,γ C γ γ C γγ C γ γ C γ γ (23)
The stress power of the model is
ee ep oe ee ep a s oe c dτγ τ τ τ γ τ γ τ γ γ τ γ γ (24)
Inserting the stress power (Eq.24) and the time-rate of the
Helmholtz free energy (Eq.23) into
the 2nd law of thermodynamics (Eq. 21) and rearranging the terms
leads to the following
expression (Eq.25)
1 2 3 4 0 m
ep a a ee oe c c ep s oe dτ C γ γ τ C γ C γ γ τ C γ γ τ γ τ γ
(25)
In order to satisfy this inequality for arbitrary values of the
strain-rates of the variables in
the free energy, the following equations for the three stress
components which correspond
to Eqs.(5), (9) ,and (11), respectively, yield
1ep aτ C γ , 2 3 sgn m
eeτ C γ C γ γ , and 4oe cτ C γ (26)
The residual inequality to be satisfied is
0ep s oe dτ γ τ γ (27)
It states that the inelastic stress-powers belonging to the two
dissipative elements (Element S
and Element D) have to be non-negative for arbitrary deformation
processes. Assuming that
the time-derivatives of the inelastic deformations are the same
sign as the corresponding
stress quantities, each of the product terms of Eq. (27) is ensured
to be non-negative with
parameters given in Table 3. The non-negative dissipation energy of
the bearings is ensured
only when all the parameters responsible for expressing the
elasto-plastic stress ( ep ) and
Earthquake Engineering 332
the viscosity induced overstress ( oe ) are non-negative. The
parameters of Table 3 have
confirmed this condition.
This chapter discusses an experimental scheme to characterize the
mechanical behavior of
three types of bearings and subsequently demonstrates the modeling
approaches of the
stress responses identified from the experiments. The mechanical
tests conducted under
horizontal shear displacement along with a constant vertical
compressive load
demonstrated the existence of Mullins’ softening effect in all the
bearing specimens.
However, with the passage of time a recovery of the softening
effect was observed. A
preloading sequence had been applied before actual tests were
carried out to remove the
Mullins’ effect from other inelastic phenomena. Cyclic shear tests
carried out at different
strain rates gave an image of the significant strain-rate dependent
hysteresis property. The
strain-rate dependent property in the loading paths was appeared to
be reasonably stronger
than in the unloading paths. The simple and multi-step relaxation
tests at different strain
levels were carried out to investigate the viscosity property in
the loading and unloading
paths of the bearings. Moreover, in order to identify the
equilibrium hysteresis, the multi-
step relaxation tests were carried out with different maximum
strain levels. The dependence
of the equilibrium hysteresis on the experienced maximum strain and
the current strain
levels was clearly demonstrated in the test results.
The mechanical tests indicated the presence of strain-rate
dependent hysteresis with high
strain hardening features at high strain levels in the HDRB. In the
other bearing specimens,
strain-rate dependent phenomenon was seen less prominent; however,
the strain hardening
features at high strain levels in the RB showed more significant
than any other bearings. In
this context, an elasto-plastic model was proposed for describing
strain hardening features
along with equilibrium hysteresis of the RB, LRB and HDRB. The
performance of the
proposed model in representing the strain rate-independent
responses of the bearings was
evaluated. In order to model the strain-rate dependent hysteresis
observed in the
experiments, an evolution equation based on viscosity induced
overstress was proposed for
the bearings. In doing so, the Maxwell’s dashpot-spring model was
employed in which a
nonlinear viscosity law is incorporated. The nonlinear viscosity
law of the bearings was
deduced from the experimental results of MSR and SR loading tests.
The performance of the
proposed evolution equation in representing the rate-dependent
responses of the bearings
was evaluated using the relaxation loading tests. On the basis of
the physical interpretation
of the strain-rate dependent hysteresis along with other inelastic
properties observed in the
bearings, a chronological method comprising of experimentation and
computation was
proposed to identify the constitutive parameters of the model. The
strain-rate independent
equilibrium response of the bearing was identified using the
multi-step relaxation tests.
After identifying this response, the elasto-plastic model (the top
two branches of the
rheology model) was used to find out the parameters for the
elasto-plastic response. A series
Mechanical Characterization of Laminated Rubber Bearings and Their
Modeling Approach 333
of cyclic shear tests were utilized to estimate the strain-rate
independent instantaneous
response of the bearings. A linear elastic spring element along
with the elasto-plastic
rheology model (the rheology model without the dashpot element) was
used to determine
the parameters of the instantaneous responses. After determining
the elasto-plastic
parameters of the bearings, the proposed evolution equation based
on the viscosity induced
overstress was used to find out the viscosity parameters by
comparing the simple relaxation
test data. Moreover, a mathematical equation involving smoothing
function was proposed
to establish loading and unloading conditions of the overstress;
and sinusoidal loading data
was then used to estimate the smoothing parameter of the
overstress. Finally, the
thermodynamic compatibility was confirmed by expressing the
rheology model by using
the Clausius-Duhem inequality equation.
Chittagong, Bangladesh
Acknowledgement
The experimental works were conducted by utilizing the laboratory
facilities and bearings-
specimens provided by Rubber Bearing Association, Japan. The
authors indeed gratefully
acknowledge the kind cooperation extended by them. The authors also
sincerely
acknowledge the funding provided by the Japanese Ministry of
Education, Science, Sports
and Culture (MEXT) as Grant-in-Aid for scientific research to carry
out this research work.
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