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Ucak-HY80CyclicResponse-SCC2010.pdf
Modeling the Cyclic Response of HY-80 Steel Under
Dynamic Loading
Alper UCAK1, David R. HUFNER2, and Ken ARPIN2
1)
Department of Civil Engineering, The Catholic University of America,
Washington, DC 20064
2)
General Dynamics Electric Boat, Groton, CT 06340
Abstract: A time-independent constitutive model for HY-80 (High Yield) steels
with a yield plateau is outlined. The model couples the non-linear kinematic
hardening concept with a memory surface in the plastic strain space, to account
for progressive cyclic hardening/softening, and a pseudo memory surface in the
deviatoric stress space, to correctly describe the plateau response. The model is
incorporated into Abaqus/Standard through a UMAT. The performance of the
model is validated against experimental data.
Keywords: Constitutive Model, Dynamics, Experimental Verification, Plasticity
1. Introduction
As the engineering community moves towards design concepts based on the
performance of a component or structure, finite element methods gain popularity
and become an indispensable tool, especially in applications involving cyclic
loading in the inelastic range. In finite element applications dealing with large
elastic–plastic cyclic deformations, accurate description of the material response
is essential in order to arrive at an accurate and reliable prediction of the member
or structural response. Inelastic cyclic characteristics of engineering materials are
quantified with cyclic plasticity models, that is, mathematical models based
continuum mechanics.
Finite element simulations presented by Hodge and Minicucci (1997), Arpin
(1999), and Ucak and Tsopelas (2008) show that later time dynamic, and the
quasi-static cyclic hysteretic response of structural components made of structural
steel with a yield plateau cannot be accurately captured, unless the macroscopic
response of the base material is correctly modeled. This can be attributed to the
fact that for structural steels with a yield plateau, the stress-strain response
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recorded under monotonic and cyclic load paths are different. Generally an
amplitude-dependent cyclic hardening/softening phenomenon is observed, where
the amount of hardening/softening depends on the curvature of the load path.
In this paper, a constitutive model for HY-80 (High Yield) steel, used in naval
applications, is developed and validated. The material model is coupled with an
efficient integration algorithm and integrated into the general-purpose finite
element code ABAQUS via user material subroutines (UMAT). A brief
description of the model, which is capable of capturing the response of the
material for monotonic and cyclic loading conditions, is presented. The
performance of the model is validated against experimental data.
2. Observed Material Behavior of HY-80 Steel
All experimental coupon data presented in this paper for HY-80 steel were
reported by Arpin and Trimble (1997), and Hodge et al. (2003). The experiments
were conducted on round specimens under displacement controlled loading
protocol.
The monotonic stress versus plastic strain curve of a HY-80 specimen is depicted
in Figure 1. During monotonic loading, after the homogeneous elastic
deformation, HY-80 steel shows a rather sharp yield point, followed by a yield
plateau. The plastic deformation along the yield plateau is caused by Luders band
propagation. During this process the plastic deformation along the gage is
inhomogeneous. Once the Luders bands cover the whole gage, the plastic
deformation becomes homogeneous, and the specimen starts hardening. The
hardening curve of the material is non-linear with respect to loading amplitude.
From a macroscopic point of view, the yield plateau region is treated as plastic
material without hardening.
Material response under cyclic loading is much more complex as compared with
monotonic loading. Generally the monotonic hardening curve will not be
representative of the cyclic characteristics of the material. Figure 2 depicts the
hysteresis curves of HY-80 steel under constant amplitude cyclic loading, for
different loading amplitudes. As evident from Figure 2, for fully reversed loading
amplitudes smaller than approximately 1%, the observed material response is
cyclic softening, while for larger amplitudes the material will undergo cyclic
hardening.
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Figure 1. Monotonic true plastic strain vs. true stress curve of HY-80
steel (experimental data after Hodge et al. 2003).
Figure 2. Hysteretic curves of HY-80 steel under constant amplitude
cyclic loading (experimental data after Hodge et al. 2003).
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3. Description of the Constitutive Model
Detailed description of the constitutive model used to describe the cyclic behavior
of structural steels with a yield plateau was outlined in Ucak and Tsopelas (2008).
In this section only the constitutive relations that apply to HY-80 steel is
presented.
The constitute model used to describe cyclic response of HY-80 steel is based on
the basic principles of time-independent theory of plasticity with a von Mises
yield surface and an associative flow rule
,
(1)
,
,
,
(2)
where, bold letters indicate second order tensors, s is the stress deviator, x is the
back stress, is the size of the yield surface, k is the initial size of the yield
surface, R is the isotropic hardening/softening variable, is the plastic multiplier,
and p is the accumulated (or equivalent) plastic strain.
A memory surface is incorporated into the plastic strain space, to measure the
loading amplitude
,
(3)
(4)
,
,
(5)
where, and are the center and radius of the memory surface respectively,
is the Heaviside function, i.e.
if
and
if
,
denotes the Macauley brackets, i.e.
if
and
if
,
and
are the unit normal vectors to the yield surface and memory surface
respectively, and
is a material dependent parameter.
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Consistent with the observed material behavior, existence of a plateau region and
hardening region is assumed. In the plateau region, the (equivalent) stress cannot
exceed the initial yield stress of the virgin material. For ideal von Mises behavior,
this implies that,
. The transition from the plateau region to the
hardening region depends on the loading amplitude and the accumulated plastic
strain, such that
if
if
and
(7)
→ hardening region.
In the plateau region the material dependent parameter
hardening region
.
3.1
(6)
→ plateau region,
, whereas in the
Initial Plateau Response
A bounding surface is incorporated into the deviatoric stress-space, to simulate
the plateau behavior during monotonic loading. The bounding surface has the
same shape and size as the yield surface of the virgin material, and is not allowed
to translate nor change size. During initial plastic loading, the yield surface
softens and translates at the same time, and is always in contact with the bounding
surface at the loading point. The bounding surface is expressed as
.
(8)
Softening of the yield surface is explicitly defined with the following differential
equation
,
where, R∞ is the asymptotic (fully saturated) value of the isotropic softening
function (–k<R∞<0), and b is a sufficiently large softening coefficient to cause
almost instantaneous softening.
During initial monotonic loading, on the yield plateau, the center of the yield
surface is evaluated using the consistency condition that the outward normal of
the yield surface and the bounding surface are identical
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(9)
(10)
.
The bounding surface, which is used to simulate the plateau response during
initial loading will vanish: (a) when (7) holds during monotonic loading, or (b) if
an unloading-plastic reloading occurs while (6) is holding. At this instant, the
bounding surface is deactivated and the kinematic hardening variable x is
decomposed into 2 short-term ( ) and 2 long-term ( ) components
(11)
.
3.2
Unloading From the Plateau
Upon first unloading in the plateau region, while (6) is holding, the long-range
and short-range kinematic hardening variables are defined respectively from
(12)
,
(for
),
(13)
where,
are material parameters, and are material dependent functions. In
the plateau region, the material dependent functions are assumed in the form
, (for
),
(14)
where,
are material parameters. For the two short-range kinematic hardening
functions the following relation has to hold
.
3.3
(15)
Hardening Behavior
At the instant, when (7) is satisfied, the existing memory of the material is erased
and re-set with the following initial conditions
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,
(16)
,
where,
is the plastic strain tensor at which (7) is satisfied.
The evolution law for the long-term kinematic hardening variable is assumed to
be in the form
, (for
(17)
).
Cyclic hardening observed under proportional loading paths is incorporated into
the model by modifying the first short range kinematic hardening variable as
F=0 and
F<0 or
with the initial value
. In (18), w and
parameters such that
,
(18)
are material dependent
.
4. Numerical Simulations
Trimble and Krech (1997) documented a series of experiments, in which
cantilever beam specimens made of HY-80 grade steel were subjected to transient
dynamic loads. The test program was designed to study the nonlinear dynamic
response of simple structures, and consequently to investigate the ability of
elastic-plastic analysis techniques to predict dynamic inelastic response of
components made of HY-80 steel. Some of the tested specimens in the abovecited study were used as benchmark verification examples to demonstrate the
accuracy of the constitutive model proposed.
The schematic presentation of the experimental setup documented by Trimble and
Krech (1997) is shown in Figure 3. The assemblies used in the experiments
consisted of horizontal [Figure 3(a)], and vertical [Figure 3(b)] cantilever beams,
made of HY-80 grade steel, attached to drop tables. In the experiments, the
horizontal cantilever beams were subjected to short-duration pulses, where as the
vertical cantilever beams were subjected to long duration pulses. The specimens
were designed to produce surface strains in the range of 2~3%. During the
experiments the acceleration and the deformation of the cantilever tip and the
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surface strain at different locations were measured. In this study, one horizontal
(referred to as CS-2) and two vertical cantilever beams (referred to as CL-1 and
CL-3) are considered.
(a)
(b)
Figure 3. Schematic presentation of the experimental setup
documented by Trimble and Krech (1997); (a) for the horizontal
cantilever beams; (b) for the vertical cantilever beams.
Figure 4 depicts a photograph of the instrumented horizontal cantilever beam
assembly used in the experiments, and the corresponding finite element (FE)
model used to represent the assembly. Figure 5 depicts the FE model used for the
vertical cantilever beam assembly.
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Figure 4. Photograph of the instrumented horizontal cantilever beam
assembly and the corresponding finite element model.
Figure 5. Finite element model for the vertical cantilever beam
assembly.
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In the FE models the plates, mass blocks, and support plates are mathematically
represented by 2nd order quadratic reduced integration shell elements (Abaqus
element type S8R5). The models are constructed using the nominal construction
dimensions of the specimens, and excited at the cantilever base with the actual
transient accelerations.
The material dependent properties used for the proposed model are calibrated
using the experimental data reported by Arpin and Trimble (1997), and Hodge et
al. (2003). The simulated material response under uni-axial tension and constant
amplitude proportional cyclic loading are presented in Figure 1 and Figure 2
respectively. A comparison of the experimentally obtained results with the
simulations shows that the model can capture the essential characteristics of the
material. The plateau response, curvature of the monotonic loading curve, the
shape of the hysteresis curves, and the cyclic characteristics of the material are
correctly simulated.
Figure 6 through Figure 8 depict the correlation between the recorded and
predicted acceleration, relative displacement, and strain time histories for vertical
and horizontal cantilever specimens. The acceleration and relative displacement
time histories were recorded at the tip of the specimens, where as the strain time
histories were recorded close to the cantilever base support. The strain gage
locations are given in Figure 6 through Figure 8. Here it is important to note that
due to the test configuration, the relative displacement of the mass could not be
physically measured for specimen CS-2, since there was no place to attach a
displacement transducer. In all figures the calculated Russell error factor
(Russell, 1997) is also shown for reference. The Russell error factor is an
unbiased error measure used for comparing transient signals. Separate error
measures are calculated for both magnitude and phase error. These are then
combined in a single comprehensive error measure, which accounts for both
sources of error. In general, a comprehensive error of less than 0.15 is considered
excellent correlation.
A comparison of the time histories presented in Figure 6 through Figure 8 shows
that the predicted results are in good agreement with the experimental ones. The
peak acceleration, relative displacement, and strain values are very well captured.
Excellent correlation between the recorded and simulated time histories is
observed, and the phase of the response is correctly captured. Here it is important
to note that the peak strain values the specimens experienced is well above the
yield strain, and the response is highly inelastic.
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Figure 6. Specimen CL-1: recorded and predicted acceleration,
relative deformation and strain time histories
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Figure 7. Specimen CL-3: recorded and predicted acceleration,
relative displacement and strain time histories
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Figure 8: Specimen CS-2: recorded and predicted acceleration and
strain time histories
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Figure 8: Specimen CS-2: continued.
For all the specimens, the Russell comprehensive error factors calculated is below
0.15. For specimen CL-1 and CL-3 the error factors range from 0.069 to 0.13,
and for specimen CS-2 the observed error factors are between 0.10 and 0.15.
Generally, for the strain time histories, the error factor increases, as the gage gets
closer to the fixed support. This can be attributed to the welds at the support, and
initial stresses caused by the welds, which are not explicitly modeled. Never the
less, for all specimens the observed Russell comprehensive error factor is below
0.15, which indicates the correlation is excellent.
Conclusions
A constitutive model is outlined for HY-80 steel, used in naval applications. The
model can correctly capture both the yield plateau and the progressive cyclic
hardening observed in the experiments. The proposed model is validated against
published experimental data. It is shown that the constitutive model can
accurately replicate the inelastic dynamic response of simple structures under
short and long duration impulse loading.
5. References
Arpin, K. R., and Trimble, T. F., “Material Properties Test to Determine
Ultimate Strain and True Stress-True Strain Curves for High Yield Steels,”
General Dynamics Electric Boat, Report No TDA-19194, Groton, CT, 1997.
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1.
2.
3.
4.
5.
6.
7.
Arpin, K. R., “ Prediction of Inelastic Shock Response: Evaluation of
ABAQUS Metal Plasticity Models,” Electric Boat Corporation Report, No
AD-20366, Groton, CT, 1999.
Hodge, S. C., and Minicucci, J. M., “Elastic-Plastic Finite Element Analysis –
To-Test Correlation for Structures Subjected to Dynamic Loading,” KAPL
Laboratory, Report No KAPL-P-000193, Schenectady, NY, 1997.
Hodge, S. C., Minicucci, J. M., and Trimble, T. F., “Cyclic Material
Properties Test to Determine Hardening/Softening Characteristics of HY-80
Steel,” General Dynamics Electric Boat, Report No TDA-19195, Groton, CT,
2003.
Russell, D. M., “Error Measures for Comparing Transient Data; Part I:
Development of a Comprehensive Error Measure; Part II: Error Measures
Case Study,” in Proceedings of the 68th Shock and Vibration Symposium,
Hunt Valley, MD, 1997.
Trimble, T. F., and Krech, G. R., “Simple Structures Test for Elastic-Plastic
Strain Acceptance Criterion Validation,” KAPL Laboratory, Report No
KAPL-P-000206, Schenectady, NY, 1997.
Ucak, A., and Tsopelas, P., “Realistic Modeling of Structural Steels with
Yield Plateau Using Abaqus/Standard,” in Proceedings of the 2008 Abaqus
Users’ Conference, Newport, RI, 2008.
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