unified mechanics theory -finite element modeling on
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Unified Mechanics Theory - Finite Element Modeling
on Mechanical Damage
for Lead-Rubber Bearings
by
Claudia Patricia Murillo Melo
May 17th, 2019
A thesis submitted to the
Faculty of the Graduate School of the
University at Buffalo, The State University of New York
in partial fulfillment of the requirements for the
degree of
Master of Science
Department of Civil, Structural and Environmental Engineering
Copyright by
Claudia Patricia Murillo Melo
2019
To start a great project, you need courage… to finish it, takes perseverance... and
determination.
Anonymous
First, to GOD, to MY MOM, to HERBERT,
to MY LOVE MIKE and to the AMAZING
FULBRIGHT PROGRAM for
AfroColombian Leaders
DAMAGE MECHANICS
ACKNOWLEDGEMENTS
First, I want to thank God for the “astral alignment” that brought me through this amazing
Fulbright experience, the great people I met, the safe trips, the great and bad moments.
My sincere gratitude to Professor Cemal Basaran for giving me the opportunity to gain
knowledge and critical thinking about the Damage Mechanics Theory and how to
implement it in structural control systems like Seismic isolation, specifically Lead Rubber
Bearings. To professor Jongmin Shim, for being part of the committee member. To
Professor Michael Constantinou for the recommendations and suggested solutions
during this research, and for the borrowed books.
To Herbert Giraldo Gomez who has change my life, giving me support and
encouragement since day one. Thanks for being such great “godfather” to the
afrocolombian young students crossing your way at the National University of Colombia.
I also want to thank professor Diego Duenas for all the advices, recommendations and
support in the long distance.
To my classmates, who lend me a hand through all the process involving the advance of
this thesis. Also to my friends, Lisbeth Vallecilla, Leydi Vidal, Ramla Qureshi, Constanza
Flores, brave women who gave me strength, and boosting laughs. Also my sincere
gratitude to my aunty Teresa Melo. To Michael Degnan, thanks for your support, warm
love, great care, smart advices and beautiful smile! God bless your family for making me
feel part of it, specially your father.
The Fulbright program for afro -Colombian leaders is an open window when all the doors
are close for my community in my country. Thank you for giving me the opportunity to
make my dream come true, and to Laspau for the remarkable administrative work.
My heart is full of gratitude to my mom Maria Ines Melo, whose hard work, dedication and
support gave me the courage to reach this point. During the several weak moments, you
were always there keeping me from falling. I love you beyond any measurement system…
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TABLE OF CONTENTS
ABSTRACT .....................................................................................................................X
RESUMEN .....................................................................................................................XI
Chapter 1 INTRODUCTION ......................................................................................... 1
1.1 OBJETIVES (scope of research)........................................................................ 4
1.1.1 General........................................................................................................ 4
1.1.2 Specific ........................................................................................................ 4
1.2 METHODOLOGY............................................................................................... 5
Chapter 2 LITERATURE REVIEW ............................................................................... 6
2.1 SEISMIC ISOLATION ........................................................................................ 6
2.2 LEAD RUBBER BEARINGS-LRB ...................................................................... 9
2.2.1 General definition ........................................................................................ 9
2.2.2 Mechanical Properties of the LRB ............................................................. 10
2.2.3 Design and Construction Requirements .................................................... 12
2.2.4 Test Evaluation Requirements................................................................... 13
Chapter 3 UNIFIED MECHANICS THEORY.............................................................. 13
3.1 NEWTONIAN MECHANICS............................................................................. 14
3.2 THERMODINAMYC LAWS.............................................................................. 15
3.2.1 Conservation of energy, mass and momentum ......................................... 15
3.3 ENTROPY BALANCE ...................................................................................... 16
3.4 FULLY COUPLED THERMO MECHANICAL EQUATION ............................... 17
3.5 DAMAGE EVOLUTION FUNCTION ................................................................ 18
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3.6 ENTROPY PRODUCTION EQUATION ........................................................... 19
3.7 CONSTITUTIVE RELATIONSHIP FUNCTION ................................................ 19
3.8 CONSTITUTIVE MODEL EQUATIONS ........................................................... 20
3.8.1 Yield surface.............................................................................................. 20
3.8.2 Isotropic hardening .................................................................................... 21
3.8.3 Nonlinear kinematic hardening rule ........................................................... 22
3.8.4 Flow rule and Consistency Parameter ....................................................... 22
3.8.5 Viscoplastic creep law ............................................................................... 23
3.8.6 Damage coupled model............................................................................. 24
3.8.7 Return mapping algorithm ......................................................................... 24
Chapter 4 FINITE ELEMENT MODEL AND ANALYSIS............................................. 26
4.1 GEOMETRIC DEFINITION OF THE LRB ........................................................ 26
4.2 MECHANICAL PROPERTIES OF THE MATERIALS ...................................... 27
4.2.1 Rubber....................................................................................................... 27
4.2.2 Steel .......................................................................................................... 28
4.2.3 Lead........................................................................................................... 29
4.3 FINITE ELEMENT MODEL .............................................................................. 32
4.3.1 Model Assembly ........................................................................................ 32
4.3.2 Mesh definition .......................................................................................... 33
4.3.3 Boundary Conditions ................................................................................. 34
4.4 FINITE ELEMENT MODEL VERIFICATION .................................................... 36
4.4.1 Frequencies............................................................................................... 36
4.4.2 Elastic Behavior......................................................................................... 37
4.4.3 Elasto Plastic Behavior .............................................................................. 39
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4.5 UMAT SUBROUTINE ALGORITHM ................................................................ 41
Chapter 5 EVALUATION AND RESULTS.................................................................. 42
5.1 LABORATORY TEST FOR CALIBRATION AND VALIDATION OF RESULTS46
Chapter 6 CONCLUSIONS ........................................................................................ 48
Chapter 7 RECOMMENDATIONS ............................................................................. 49
Chapter 8 APPENDICES ........................................................................................... 50
8.1 Appendix A UMAT SUBROUTINE ................................................................... 50
8.2 Appendix B....................................................................................................... 64
- Create link Abaqus - compiler in Windows (researchgate.net/, n.d.) ................... 64
- Create link Abaqus - compiler in Linux (redhat) (Imechanica.org, n.d.) ............... 66
8.3 Appendix C Variables for UMAT Subroutine (Dassault Systems, n.d.) ............ 67
Chapter 9 REFERENCES .......................................................................................... 73
LIST OF TABLES
Table 4.1 Thickness of the LRB layers.......................................................................... 26
Table 4.2 Diameter of the LRB layers ........................................................................... 26
Table 4.3 Rubber properties (left: original values, right: adjusted for Abaqus) .............. 28
Table 4.4 Steel properties ............................................................................................. 28
Table 4.5 Lead properties ............................................................................................. 31
Table 5.1 laboratory test results for calibration and validation of results ....................... 46
LIST OF FIGURES
Figure 1.1 NQS with 6th Roundabout Bridge, Colombia (Jcortes, 2015) ....................... 2
Figure 1.2 Corredor Honda-Manizales Bridge, Colombia (Eriksen et al., 2018).............. 2
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........................................................................................................................................ 6
Figure 2.1 Effects by the addition of flexible isolators (Dynamic Isolation Systems, 2007)
Figure 2.2 Graphic description of base isolation (Dynamic Isolation Systems, 2007) ..... 7
Figure 2.3 Friction Pendulum
Figure 4.20 Force- Displacement Loops for the Elasto-Plastic material / Model for
.......................................................................................... 8
Figure 2.4 Elastomeric bearing ....................................................................................... 8
Figure 2.5 Lead Rubber Bearing (Dynamic Isolation Systems, 2007) ........................... 10
Figure 2.6 hysteresis loop (Dynamic Isolation Systems, 2007) ..................................... 11
Figure 3.1 Schematics of isotropic hardening (Chaboche, 1989).................................. 21
Figure 3.2 Schematics of the linear kinematic hardening (Chaboche, 1989) ................ 22
Figure 4.1 Geometric definition of the LRB ................................................................... 27
Figure 4.2 Temperature distribution of the led core (Buckle et al., 2006) ...................... 30
Figure 4.3 effect of temperature on UTS (Guruswamy, 1999)....................................... 30
Figure 4.4 part instances and solid revolutions ............................................................. 32
Figure 4.5 Sections defined........................................................................................... 33
Figure 4.6 Element type definition for the mesh ............................................................ 33
Figure 4.7 Amplitude definition for boundary conditions................................................ 34
Figure 4.8 Boundary condition- Lateral displacement ................................................... 34
Figure 4.9 Boundary condition- Fixed base................................................................... 35
Figure 4.10 Amplitude definition for predefined field ..................................................... 35
Figure 4.11 Boundary condition- Temperature increase ............................................... 36
Figure 4.12 Modes and frequencies calculated in Abaqus ............................................ 37
Figure 4.13 Elastic properties for lead........................................................................... 38
Figure 4.14 Time – Displacement / Elastic material ...................................................... 38
Figure 4.15 Time – Force / Elastic material................................................................... 39
Figure 4.16 Force- Displacement elastic behavior of the material lead ......................... 39
Figure 4.17 Elasto-Plastic properties for lead................................................................ 40
Figure 4.18 Time – Displacement / Elastoplastic material............................................. 40
Figure 4.19 Time – Force / Elastoplastic material ......................................................... 41
validation ....................................................................................................................... 41
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Figure 5.1 Maximum displacement of 114mm............................................................... 42
Figure 5.7 Force Displacement Loops of LRB test studied herein for calibration purposes
“Example No 10” Displacement amplitude = 114mm and frequency = 0.35 Hz (Peak
Figure 5.8 Energy dissipated per cycle “Example 10” (Kalpakidis & Constatinou, 2008)
Figure 5.2 Time – Displacement / UMAT Subroutine .................................................... 43
Figure 5.3 Time – Force / UMAT Subroutine................................................................. 43
Figure 5.4 Time – Force / UMAT Subroutine (adjusted)................................................ 44
Figure 5.5 Force- Displacement / UMAT Subroutine (adjusted).................................... 45
Figure 5.6 Energy dissipated by cycle (adjusted) .......................................................... 45
velocity = 250mm/s) (Kalpakidis & Constatinou, 2008) ................................................. 47
...................................................................................................................................... 47
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ABSTRACT
Several studies have been developed to estimate the reduction of the characteristic
strength of the Lead Rubber Bearing (LRB) respect to the increment of the cycles. This
phenomenon has been observed in several experimental studies involving cyclic motion,
but no means to predict this behavior using analytical analysis, and involving temperature
changes of the lead has been reached yet.
LRBs are used worldwide as devices for seismic isolation of buildings and bridges, for
new structures and retrofit projects. This document presents the first attempt for the
implementation of the Unified Mechanics Theory to simulate entropy generation (or
damage evolution) of the mechanical properties of high purity lead core of a Base isolator
type LRB. A finite element model was made in ABAQUS using 3D elements.
Thermodynamic based constitutive models comprise the UMAT subroutine to update the
Jacobian matrix of the material properties of the lead core, for each time increment with
the respective temperature increment associated. These constitutive models involve,
viscoplasticity defined by a yield surface, kinematic hardening and finally damage in the
material. The results are compared to experimental test done at State University of New
York at Buffalo (UB) by (Kalpakidis & Constatinou, 2008), specifically the example No 10
was taken as the validation test, which dimensions are described as “typical” for an LRB
in base isolation in Latin America. After several computationally expensive versions of the
model, convergence was reached and degradation of the lead core was observed, the
shear force obtained was bigger than expected. However, the energy dissipated by cycle
has similar tendency to the actual laboratory test, with adjusted values.
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RESUMEN
Se han desarrollado varios estudios para estimar la reducción de la resistencia
característica del aislador de caucho con núcleo de plomo (LRB) con respecto al
incremento de los ciclos. Este fenómeno se ha observado en varios estudios
experimentales que involucran desplazamientos cíclicos, pero aún no se ha llegado a
ningún medio para predecir este comportamiento mediante análisis analítico e
involucrando los cambios de temperatura en el plomo.
Los LRB se utilizan en todo el mundo como dispositivos para el aislamiento sísmico de
edificios y puentes, para nuevas estructuras y proyectos de reforzamiento. Este
documento presenta el primer intento de implementación de la Teoría de la Mecánica
Unificada para simular la generación de entropía (o la evolución del daño) de las
propiedades mecánicas del núcleo de plomo de alta pureza de un aislador sísmico tipo
LRB. Se realizó un modelo en elementos finitos en ABAQUS utilizando elementos 3D.
Los modelos constitutivos basados en la termodinámica comprenden la subrutina UMAT
para actualizar la matriz jacobiana con las propiedades del material del núcleo principal
del aislador, para cada incremento de tiempo con la correspondiente temperatura
asociada. Estos modelos constitutivos implican, la viscoplasticidad definida por una
superficie de cedencia, el endurecimiento cinemático y finalmente el daño en el material.
Los resultados se comparan con la prueba experimental realizada en la Universidad
Estatal de Nueva York en Buffalo (UB) por (Kalpakidis y Constatinou, 2008),
específicamente el ejemplo No 10 se tomó como el ejemplo de validación, cuyas
dimensiones se describen como "típicas" para un LRB en aislamiento sísmico en América
Latina. Después de varias versiones computacionalmente demandantes del modelo, se
alcanzó la convergencia y se observó una degradación del núcleo de plomo, el valor para
la fuerza de corte obtenida fue mayor de lo esperado. Sin embargo, la energía disipada
por ciclo tiene una tendencia similar a la prueba de laboratorio real, una vez se han
ajustado los valores a lo esperado.
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Chapter 1 INTRODUCTION
Seismic isolation is used to create a flexible interface between the support of the structure
and the structure itself, the objective of this flexible interface is to reduce the seismic
effects that the structural system would transfer to the substructure, seismic isolation is a
type of structural control system to localize the ductility demand to an specific
element(Buckle, Constantinou, Diceli, & Ghasemi, 2006). In the area of highway
infrastructure is a matter of to guarantee completely operational bridges through the
highway system during and after an earthquake; this characteristic makes the use of base
isolators the most popular way for seismic hazard mitigation (Eriksen, Mohammed, &
Coria, 2018). Indeed, during the structural design phase of a new bridge involves high
cost savings regarding the foundation, due to the reduction in the forces transferred to
the soil; on existing bridges also benefits from cost savings since the reinforcements of
the existing piers and abutments could be avoided by the implementation of base
isolators; also, a seismic isolator is easy to replace in case of complete damage in the
device.
Many earthquake prone countries located in Latin America are decidedly moving towards
the use of seismic isolators, since their location on a high seismic hazard zone given by
the Andes Mountains, part of the Pacific Ring of Fire. Despite the interest to increase the
infrastructure resilience, the implementation of LRBs has slowly increase during the last
10 years in Colombia, in some cases as a result of the lack of knowledge and in other
cases due to the initial high budget considerations given to the implementation of the
devices (Eriksen et al., 2018). One of the most recent infrastructure projects, including
LRBs in Colombia is the corridor Honda-Manizales bridge (Figure 1.2), which is part of
the highway that connects the center of the country to the east and the west of the
National territory.
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Figure 1.1 NQS with 6th Roundabout Bridge, Colombia (Jcortes, 2015)
Figure 1.2 Corredor Honda-Manizales Bridge, Colombia (Eriksen et al., 2018)
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In the context of Civil Engineering, predicting the response of the materials, including the
failure modes, or the ultimate load associated to the failure mechanism, or even safety
factors, is essential for the development of structural engineering. Physical and structural
characteristics of materials change with time, some of this changes are admissible (i.e.
the change in color due to UV exposure) due to the lack of influence in the mechanical
response of the material; others may be inadmissible like concrete cracking or the alkali-
aggregate reaction in a containment structure, (Murillo Melo, 2014), these inadmissible
changes in the mechanical properties could be associated with inelastic irreversible
processes or damage in the material. There is very limited research in the literature about
damage in the materials, several studies assume “damage” as the loss of section, not
taking into account the mechanical properties changes due to temperature variations (i.e.
combustion process), or the environment influence (i.e. marine environments).
The possibility to count on numerical models to evaluate and predict the macroscopic
degradation through the material, taking into account the susceptibility to diverse
parameters controlling the process, while being influenced by effects of time or
temperature, is a requirement in progress towards substantial improvements in the quality
of the final product (Sánchez, 2006). Previous research has involved the effect on the
characteristic strength of the bearing due to heating of the lead core, without taking into
account the strain rate effects on the strength, post-elastic stiffness or strain effects in the
stiffness (Kalpakidis, Constantinou, & Whittaker, 2010). Earlier attempts to apply the
Unified Mechanics Theory to model the mechanical behavior of LRB, were not conclusive
in the results. Nevertheless, it was a good starting point to advance this investigation.
Hence, this is the first attempt to model the degradation of the mechanical properties of
high purity lead core in a Lead Rubber Bearing (LRB), by means of the Unified Mechanics
Theory as the mathematical base for the subroutine called UMAT, to be use in a finite
element model developed in ABAQUS.
The Unified Mechanics Theory was first proposed on solder joints by (C. Basaran & Yan,
1998), it was later proposed on solids by (Cemal Basaran & Nie, 2004). This theory brings
together the Newtonian Mechanics Theory and Thermodynamics laws to study the
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damage in the material, based on the entropy generation in the system as means to
describe the degradation mechanism. The Unified Mechanics Theory was recently proved
mathematically by (Sosnovskiy & Sherbakov, 2016) to analyze damage state of complex
systems.
Newtonian Mechanics does not take into account the life time degradation of the material,
but gives the response of any material to different external loads applied to one body. On
the other hand, Thermodynamics provides information on how such material goes
through irreversible processes to total entropy. Nowadays, empirical curve fitting is used
to introduce thermodynamics into Newtonian mechanics to reproduce phenomenological
degradation functions. This document intends to implement the Unified Mechanics Theory
to reproduce the hysteretic curves, which represents the degradation of the element, of
one Lead Rubber Bearing using the mechanical properties for high purity lead. One first
attempt for this research was made by (Hernandez, 2018), a good approximation to the
validation test was obtained even though the properties of the material were not for pure
lead.
1.1 OBJETIVES (scope of research)
1.1.1 General
Predict the mechanical response of Lead Rubber Bearings (LRB) without empirical
degradation, fatigue or fracture, using a Finite Element Model in Abaqus where a UMAT
subroutine is used to implement the Unified Mechanics Theory.
1.1.2 Specific
- Gain knowledge about Lead Rubber Bearings as a control system for bridges.
- Study the behavior of the LRBs under lateral load.
- Learn about the Continuum damage mechanics model
- Study the mechanical damage in materials, specifically High purity lead.
- Gain knowledge about the Unified Mechanics Theory
- Develop a Finite Element Model to simulate the damage in the lead core of an
LRB, due to lateral displacements emulating earthquake forces.
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1.2 METHODOLOGY
Chapter 2 includes the literature review on seismic isolation, and the specific information
regarding Lead Rubber Bearings to understand the nature of the forces that create
damage in the LRB, the mechanical properties and current normativity is also included.
Chapter 3 describes the Unified Mechanics Theory, base for the UMAT subroutine.
Chapter 4 shows the LRB dimensions selected based on a typical isolator used in Latin
America, and the description of the initial properties of the materials involved in this study,
also the model assembly and the UMAT subroutine will be defined. Chapter 5 present the
information from test data and the results obtained with finite element analysis: a) for a
model where the materials are defined as elastic, b) for a model where the rubber and
the steel have elastic properties and the lead is defined with elasto-plastic behavior, c)
damage in the lead will be obtained from the third model and compared to actual
experiments done in the past to validate the results. It should be noted that the design of
the LRB is not included. Finally, appendices including the complete UMAT subroutine,
the description of the variables used in the subroutine, and the ways to create the link in
Windows and Linux to compile the subroutine in Abaqus using Fortran.
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Chapter 2 LITERATURE REVIEW
2.1 SEISMIC ISOLATION
Seismic isolation is defined to reduce the effects of earthquakes on several structures
including bridges; in other words, reduces the forces created in the superstructure during
the ground motion and transferred to the substructure. Basically, the fundamental period
of the structure shift by the addition of flexible isolators (Figure 2.1), allowing the structure
to reduce the lateral displacements. This effect leads to an improved performance and an
increase in the service life in both cases, new or existing bridges (Buckle et al., 2006).
The use of devices like seismic isolators to reach a performance superior to the current
national regulations, covers also purposes of protect the investment or to protect the
operability of the community, including bridges that are complex structures, critical to the
national transportation system of each country.
Increasing
Period shift damping
Figure 2.1 Effects by the addition of flexible isolators (Dynamic Isolation Systems, 2007)
Several types of devices are used for seismic isolation worldwide, but they all are
designed to accomplish the same goal: uncouple the displacements of the bridge
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superstructure to the bearing support to reduce seismic effects on the substructure, and
likewise in the ground. The same concept applies when talking about new projects or
retrofitting existing structures (Buckle et al., 2006).
Figure 2.2 Graphic description of base isolation (Dynamic Isolation Systems, 2007)
Seismic isolation of buildings, bridges and other structures is obtained with the used of
base isolators, a brief explanation for the most commonly used devices is shown below:
- Friction Pendulum Systems, which consist of a concave lubricated surface base
in stainless steel, a sliding element and a cover plate. During the ground motion
the sliding element travels along the concave lubricated surface, absorbing
earthquake energy in each cycle. See Figure 2.3 (Claudio Nitsche M., 2018)
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Figure 2.3 Friction Pendulum
- Elastomeric bearings, the best known and most commonly used base isolator,
could be made of single hard rubber, or comprised of elastomer layers intercalated
with steel plates, vulcanized together, fabricated under standard regulations like
AASHTO. This device can have a rectangular or round shape, according to the
structure solicitation. See Figure 2.4 (Claudio Nitsche M., 2018)
Figure 2.4 Elastomeric bearing
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The behavior of base isolators is based on the maximum amplitude of the displacement,
the frequency and the damage of the material. The type of base isolator object of this
study is the LRB, which is a type of elastomeric bearing including a lead core; this device
is further described below.
2.2 LEAD RUBBER BEARINGS-LRB
2.2.1 General definition
Seismic isolator like the LRBs are flexible devices under lateral loads, and very rigid under
vertical loads. Its configuration consists of two fixing plates made of steel and located at
the top and bottom of the device, layers of rubber intercalated with shims of steel between
the steel plates, and a lead core in the center of this configuration which provides most of
the energy dissipation capacity of the LRB. The lead plug is usually longer than the total
height of rubber to guarantee compression in the core once the device is assembled, the
diameter of the lead plug will define the energy dissipation capacity of the isolator
(Kalpakidis & Constatinou, 2008) see Figure 2.5. These devices, whose purpose is to
assume most of the deformation the superstructure generates during earthquakes, are
installed as an interface between the superstructure and the bearings, the remaining
forces and displacements will be assumed by the substructure giving as a result a
structure with reduced damage.
The layers of rubber provide the devices with enough elasticity to assume horizontal
deformations, the steel shims together with the top and bottom plates improve the vertical
resistance of the device keeping it from bulging, as mentioned before the confined lead
core provides the energy dissipation through a process of plastic deformation while the
lead core yields releasing heat during the lateral deformations (Claudio Nitsche M., 2018).
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Bottom fixing plate
Top fixing plate
Figure 2.5 Lead Rubber Bearing (Dynamic Isolation Systems, 2007)
Once the seismic event is finished, the lead core hardens back to its original configuration
while the rubber layers return to its original position; in theory, the initial properties remain
intact (Claudio Nitsche M., 2018).
2.2.2 Mechanical Properties of the LRB
The implementation of base isolation in a project should be based on the Design
Earthquake, an upper and lower bound are determined to stablish the nominal properties
of the device, the estimated range of values must be confirmed with laboratory tests made
for previous projects with similar devices for isolation, or for actual isolators to be used in
the project under review. The range for the nominal mechanical properties can also be
obtained from manufacturers of isolators, since they are supposed to test each batch of
fabricated isolators (Kalpakidis & Constatinou, 2008).
The idealized behavior of the Lead Rubber Bearing under lateral forces is described as a
bilinear hysteretic loop, obtained during shear- displacement test where the shear strain
should not exceed the 250% (shear strain is defined as the displacement divided by the
height of the isolator). In structural design, the shear forces are obtained with the Design
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Basis Earthquake (DBE), where the ground motion has 10% probability of being
exceeded in 50 years, and with the Maximum Credible Earthquake (MCE) where the
ground motion has a 2% probability of being exceeded in 50 years (Dynamic Isolation
Systems, 2007). The main parameters that could be obtained from the first hysteresis
loop is shown in the Figure 2.6.
Figure 2.6 hysteresis loop (Dynamic Isolation Systems, 2007)
Ke: Elastic stiffnes, controlled by the lead core size, is the initial stiffness of the isolator.
Kd: Yielded stiffness, controlled by the shear modulus of the rubber,G, total rubber
thickness, Tr, and the bonded rubber area A. Is the secondary stiffness of the isolator
calculated as:
𝐺𝐴 𝐾𝑑 =
𝑇𝑟
Keff: Effective stiffness, the isolator maximum force divided by the maximum
displacement.
Qd: Characteristic strength, the point where the hysteresis loop crosses the force axis.
Defines the isolator response to service loads, and depends on the area of the lead core
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AL, and the effective yield strength of the lead, which varies from cycle to cycle according
the core heating, σYL for the first loop. Qd is calculated as: Qd= AL * σYL
Fy: Yield force, corresponds to the yield displacement, Y, parameter used to determine
the effective damping of the isolator. Thus, is the point in the hysteresis loop where Ke
changes to Kd.
EDC: Energy dissipated per cycle, is the area within the hysteresis loop. This value is a
measure of the damping of the isolator. The material absorbs a certain amount of energy
in each load cycle, the reduction in the total energy may be characterized as fatigue,
which means that the accumulated fatigue increases an amount corresponding to the
hysteresis energy dissipated, until the material reaches the critical value of fatigue where
the produces the failure. The total energy dissipated until failure leads to determine the
fatigue life prediction of the material (Nie, 2005).
The estimated lower bound values for Qd and Kd, are assumed as the average of the first
three loops, the corresponding upper bound values are taken from the first loop, then
adjusted with the modification factors in AASHTO (Buckle et al., 2006).
2.2.3 Design and Construction Requirements
Earthquake ground motions for the analysis, determine the superstructure displacement
based on the requirements of the AASHTO and ASCE-7 codes, this step determine the
spectral demands for the bridge, which is used in the design of the isolation system to
meet the required performance, manufacture and installation. Seismic isolation should be
employed when improved seismic performance is required (Buckle et al., 2006).
The design review previous the manufacturing of the devices, involves analysis of the
criteria followed during the project design, including 1) Site-specific spectra and ground
motion histories, 2) Preliminary design regarding definition total maximum and design
displacement and the corresponding devices, plus the lateral force level; 3) Definition of
the appropriate property modification factors for the device selected, 4)Prototype testing
program, 5) Final design of the structural system with the respective analysis and
modelling of the isolators, 6) Isolator production testing program (ASCE-7, 2016).
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2.2.4 Test Evaluation Requirements
Testing of lead rubber bearings involve a set of characterization tests that must be done
by the LRB manufacturer, to stablish generic databases regarding the effects of velocity,
pressure, temperature changes, etc. Also, a set of prototype tests to determine the
mechanical properties of the device, to then compare with the properties assume by the
structural designer prior the production of the final LRBs, at least two of each kind of
bearing designed should be tested for thermal response (thermal movement of the
bridge), wind and brake forces, seismic response and stability. Finally, a set of
compression and combined compression with shear tests are required to verify the
mechanical properties of the bearings under production as quality control (Buckle et al.,
2006). One of the purposes of this research is to calibrate a Finite Element Model in
ABAQUS to reduce the amount of physical laboratory test to determine the future
response of the device under design and review, by means of complex formulations to
simulate damage through calculations for the Thermodynamical State Index using
increments of time and temperature as explained in the next chapter of this document.
Chapter 3 UNIFIED MECHANICS THEORY
The main objective of the Unified Mechanics Theory is to determine the laws of nature to
help predict the long term structural response of organic or inorganic materials without
the use of empirical phenomenological curve fitting models, which involves the fatigue,
degradation or fracture of the solid in study through testing (Cemal Basaran & Nie, 2004).
Accurate understanding about the damage process of the mechanical properties of the
materials is required to attempt to determine constitutive equations to model the
mentioned process as an approach to real laboratory test. The mechanism of fatigue
failure is by plastic deformation, which is due to shear in metals when sliding of atomic
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planes over one another at molecular levels, causes plastic deformation at the continuum
scale (C. Basaran & Yan, 1998).
The Unified Mechanics Theory is the mean to unify the classical Newtonian Mechanics
with the thermodynamics laws to describe the evolution damage parameter, which is a
non-negative value dependent on the entropy generation of the system, “entropy can
always be created, never destroyed” (C. Basaran & Yan, 1998). The theory was probed
mathematically by (Sosnovskiy & Sherbakov, 2016)
The theory to predict a physical fatigue evolution model has its foundation in the laws of
thermodynamics, which are used to determine the internal entropy generation of the
system. Thus, the lifetime predictor is shown as the damage evolution parameter
(Temfack & Basaran, 2015).
3.1 NEWTONIAN MECHANICS
As it is well known, the Newtonian mechanics provides the solution for any structure
deformed by any external load, neglecting the internal changes in the solid related to time
evolution. This is an elastic constitutive relationship theory where only the displacement
“u” or the mass “m” are the nodal unknowns, while the stiffness “k” or the acceleration “a”
remain constant (Cemal Basaran & Nie, 2004).
F = ma & F = ku
Involving the Unified Mechanics theory: besides the displacement “u”, the entropy
generation rate is also a nodal unknown in the coupled thermomechanical equation. In
this case, the stiffness or the acceleration change continuously with time evolution:
F = ma (1- Ф(ṡ)) and F = ku (1- Ф(ṡ))
In this equation, Ф is the Thermodynamic State Index, which defines the degradation
evolution function based on the entropy generation rate (ṡ).
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3.2 THERMODINAMYC LAWS
To gather information about the changes in a solid over time, is required the use of the
thermodynamic laws. The first law of thermodynamics defines the conservation of energy.
The second law of thermodynamics describes the natural tendency of any body, organic
or inorganic to degenerate to a disorder state. This phenomenon is called entropy, which
is a function of the probability of disorder (Cemal Basaran & Nie, 2004).
3.2.1 Conservation of energy, mass and momentum
Conservation of mass: The total mass remains unaltered since there is no flow of matter
in or out of the volume element. The following balance equation shows that the local
change of density ρ is equal to the negative divergence of the flow of mass, where 𝑣 is
the velocity (Cemal Basaran & Nie, 2004).
𝜕𝜌 = −𝑑𝑖𝑣𝜌𝑣
𝜕𝑡
Momentum principle: The sum of the kinetic energy (1/2) 𝜌𝑣2 and the potential energy
ρψ, provides the rate of change trough the equation:
1𝜕𝜌 ((2) 𝜐2 + 𝜓)
1 = −𝑑𝑖𝑣 {𝜌 ( 𝜐2 + 𝜓) ∙ 𝜐 − 𝜎 ∙ 𝜐} − 𝜎: 𝐷
𝜕𝑡 2
Where σ is the symmetric stress tensor, ψ is the potential independent of time and D is
the symmetric rate of deformation tensor (De Groot & Mazur, 1962)
Conservation of energy: The total energy of the system is given by varying percentages
of the internal energy, the potential energy and the kinetic energy. Despite the changes
in these percentages, the total energy of the system remains unaltered (De Groot &
Mazur, 1962):
1 𝑒 = 𝑢 + 𝜓 + 𝜐2
2
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Assuming that the only transfer of energy to the system is through mechanical work, by
means of surface tractions, body forces, and heat exchange, the total energy could
change only if there is flow into or out of the volume considered through its boundary (De
Groot & Mazur, 1962):
𝜕𝜌𝑒 = −𝑑𝑖𝑣 𝐽𝑒 + 𝜌𝑟
𝜕𝑡
Where r is the distributed internal heat source per unit mass and Je is the total energy flux
per unit surface and unit time, has a negative sign since the energy flux normal to the
surface is outwards. Consequently, a negative value of energy flux means a positive input
of energy to the system (Temfack & Basaran, 2015):
𝐽𝑒 = 𝜌𝑒𝜐 − 𝜎 ∙ 𝜐 + 𝐽𝑞
Where 𝜎 ∙ 𝜐 is the energy flux due to mechanical work performed on the system, and Jq is
the energy flux (De Groot & Mazur, 1962).
The balance equation for the specific internal energy u, is defined by:
𝑑𝑢 𝜌 = −𝑑𝑖𝑣𝐽𝑞 + 𝜎: 𝐷 + 𝜌𝑟
𝑑𝑡
3.3 ENTROPY BALANCE
Entropy is defined as the damage evolution in the system, entropy flows into the volume
leading the natural tendency of the bodies to change into a more disordered state, known
as degeneration process. There is an entropy source due to an irreversible phenomenon
inside the volume, which related to the variation in temperature. When the irreversible
entropy generation rate reaches the value of zero, the system fails or “die”. The variation
of entropy dS depends on only two terms, dSe the entropy regarding the transfer of heat,
across the boundary of the system, due to external sources, and dSi regarding the internal
entropy (Cemal Basaran & Nie, 2004):
dS= dSe + dSi
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where dSi =0 for reversible transformation and dSi ≥ 0 for irreversible transformation of
the system, according to the second law of thermodynamic states (Sosnovskiy &
Sherbakov, 2016). The differential version of the same equation is given by:
𝑑𝑠 𝜌 = −𝑑𝑖𝑣𝐽𝑠 + 𝛾
𝑑𝑡
Where s is the entropy per unit mass, Js is the entropy flux and γ is the entropy production
per unit volume and unit time, defined by:
1 𝜌 𝑑𝑤𝑎𝑣𝑎 𝑘 𝜌𝑟 𝛾 = 𝜎: 𝐷 − − |𝑔𝑟𝑎𝑑 𝑇|2 +
𝑇 𝑇 𝑑𝑡 𝑇2 𝑇
where 𝑤𝑎𝑣𝑎 is the work stored in the system during the process or available work, T is
the absolute temperature, and k is the thermal conductivity of the solid (Cemal Basaran
& Nie, 2004).
3.4 FULLY COUPLED THERMO MECHANICAL EQUATION
The Unified Mechanics Theory involves two state variables: the temperature T and the
total strain ε for elasticity. Then the fully coupled thermodynamic equation simulates the
evolution of the temperature in the solid due to external mechanical work, under adequate
boundary conditions:
𝜕𝜎 𝜕𝐴𝑘 𝑘∇2𝑇 = 𝜌𝐶�̇� − 𝜎: 𝜀𝑝 + 𝐴𝑘𝑉𝑘 − 𝜌𝑟 − 𝑇 ( : 𝜀𝑒 + 𝑉�̇�)
𝜕𝑇 𝜕𝑇
Where Ak represent the thermodynamic forces associated with the in the internal
variables Vk, 𝜀𝑝 denotes the elastic strain, C the specific heat, r is the distributed internal
heat source per unit mass (Lemaitre & Chaboche, 1990).
𝐴𝑘𝑉𝑘 represents the nonrecoverable energy stored in the materials associated to a
different dissipation phenomena, which is often negligible, 𝐴𝑘𝑉𝑘 ≈ 0. Then the fully
coupled elastoplastic thermomechanical equation is:
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𝜕𝜎 𝑘∇2𝑇 = 𝜌𝐶�̇� − 𝜎: 𝜀𝑝 − 𝜌𝑟 − 𝑇 ( : 𝜀𝑒)
𝜕𝑇
Which allows to calculate the heat flux Jq generated in a solid body due to elastic or
inelastic work, the equation is also used to simulate the thermal effects on the material
behavior.
The damage coupled viscoplasticity characterize the interaction between the macro-level
mechanical properties and the damage in the material due to microstructure degradation,
where the fatigue failure can be associated to the number and velocity of cycles when
talking about cyclic loading (Nie, 2005).
3.5 DAMAGE EVOLUTION FUNCTION
As mentioned before, damage is the progressive deterioration of the material prior to
failure. An analysis on the cumulative damage, which is an irreversible process, leads to
prediction of life service for any material. (C. Basaran & Yan, 1998) proposed for the first
time, the relation between the entropy per unit mass and the disorder parameter
𝑅 𝑠 = 𝑙𝑛𝑊
𝑚𝑠
Where s is the entropy per unit mass, ms is the specific mass and R is the gas constant.
Accordingly, the disorder function is
𝑠𝑚𝑠 𝑊 = 𝑒 𝑅
The Thermodynamic State Index (TSI) is the term to measure the damage evolution (Ф)
related to the entropy (ṡ) in the system at any arbitrary time with respect to an initial
reference state, entropy flows into the volume, there is an entropy source due to
irreversible phenomena inside the volume (Cemal Basaran & Nie, 2004). Then, the
damage evolution function is given by
𝑊 − 𝑊0] = [1 − 𝑒−(𝑠−𝑠0)𝑚𝑠/(𝑁0𝑘)]Φ = 𝑓 [
𝑊
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Where W is the probability of disorder, W0 is the disorder associated to the initial state
with entropy s0, No is the Avogadro number, k is the Bolztmann’s constant and ms is the
specific mass of the material.
3.6 ENTROPY PRODUCTION EQUATION
Entropy generation rate related to the variation in entropy Δs (s-s0) can be calculated by
Where the first term relates to the internal heat generation, the second term integrates
the diffusion mechanisms such as electromigration, stress gradient, thermomigration and
chemical concentration gradient. The third term is associated to the internal mechanical
work, and is the one that will be taken in this study to determine Δs.
Since Δs is a non-negative quantity, Ф is always greater or equal to zero. Likewise, Ф=0
if Δs=0 and Ф=1 when Δs→∞.
3.7 CONSTITUTIVE RELATIONSHIP FUNCTION
Damage mechanics provides the frame to develop damage evolution models to be able
to simulate damage behavior of solid materials at small strains. According to the Hooke’s
law and the strain equivalent principle, the elasticity constitutive relationship is defined as
𝑑𝜎 = (1 − Φ)𝐶0(𝑑𝜀 − 𝑑𝜀𝑝 − 𝑑𝜀𝑇)
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Where 𝑑𝜎 is the total stress increment vector, C0 s the original stiffness matrix, 𝑑𝜀𝑝 is the
inelastic strain increment, 𝑑𝜀𝑇 is the incremental thermal strain.
Then, the fully coupled elastoplastic thermomechanical equation, the damage evolution
function, the entropy production equation, and the constitutive relationship function,
together, characterize the progressive damage behavior of any solid material.
3.8 CONSTITUTIVE MODEL EQUATIONS
Lead exhibits time, stress and temperature dependent deformations, as it will be shown
in the next chapter, Similar behavior was study in (Gomez & Basaran, 2006a) for eutectic
Pb/Sn solder joints. Based on this similarities, the UMAT subroutine created by Gomez
will be employed to simulate the degradation of the mechanical properties of the lead core
in the LRB, the subroutine was created to incorporate the constitutive model equations
for small strains, this is an attempt to calibrate the model in order to obtain similar
hysteretic loops corresponding to the laboratory tests.
ABAQUS demands an integration algorithm to update the stress tensor, the material
Jacobian comprising stress to strain plus the variables involved in one integration point.
Gomez created an algorithm to integrate and develop each constitutive model equation
to update the stresses and the material Jacobian matrix for each time increment with the
corresponding temperature increment, damage was coupled into the constitutive model
by means of the effective stress concept and the strain equivalence principle (Gomez &
Basaran, 2006a).
3.8.1 Yield surface
If the Von Mises yield surface type with isotropic and kinematic hardening comprise the
constitutive model, then:
2 𝐹(𝜎, 𝛼) = ‖𝑆 − 𝑋‖ − √ 𝐾(𝛼) ≡ ‖𝑆 − 𝑋‖ − 𝑅(𝛼)
3
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Where 𝐹(𝜎, 𝛼) is a yield surface that divides the elastic and the inelastic domain, α is the
hardening parameter that defines the evolution of the radius of the yield surface, σ is the
second order stress tensor, X is the deviatoric component of the back stress tensor that
indicates the position of the center of the yield surface in stress space, R(α) is the radius
of the yield surface in stress space and S is the deviatoric component of the stress tensor
(Gomez & Basaran, 2006a).
1 𝑆 = 𝜎 − �̂�𝐼
3
Where p is the hydrostatic pressure and I is the second order identity tensor.
3.8.2 Isotropic hardening
The isotropic hardening described the uniform yield expansion, which means the
evolution of the radius of the yield surface.
2 𝑌0 + 𝑅∞(1 − 𝑒−𝑐𝛼)𝐾 (𝛼) = √
3
Where α in this case is the plastic hardening parameter or plastic strain trajectory evolving
2 as �̇� = √ 𝛾, Y0 is the initial yield stress and R∞ is an isotropic hardening saturation value,
3
c is the isotropic hardening rate (Chaboche, 1989). See Figure 3.1- Left: in the deviatoric
plane; right: the stress vs plastic strain response.
Figure 3.1 Schematics of isotropic hardening (Chaboche, 1989)
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3.8.3 Nonlinear kinematic hardening rule
Describes the shift of yield surface without change in size or shape, in other words, the
evolution of the center of the yield surface in the stress space. The equation to account
for the transient hardening effects in each stress- strain loop is
�̇� = 𝑐1𝜀̇𝑃 − 𝑐2𝑋�̇�
Where c1 and c2 are material parameters, the first term relates the linear kinematic
hardening rule, the second term represents the dynamic recovery term which introduces
the nonlinearity between the back stress X and the actual plastic strain (Gomez &
Basaran, 2006a). See Figure 3.2 - Left: in the deviatoric plane; right: the stress vs plastic
strain response.
Figure 3.2 Schematics of the linear kinematic hardening (Chaboche, 1989)
3.8.4 Flow rule and Consistency Parameter
The flow rule to describe the evolution of the plastic strain, involving �̂� as a vector normal
to the yield surface in the stress space, 𝜀̇𝑃 as the plastic strain rate (Gomez & Basaran,
2006a).
𝜕𝐹 𝑃 𝜀̇ = �̇� = �̇��̂�
𝜕𝜎
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The nonnegative parameter for the amount of plastic flow �̇�, is also defined as the
consistency parameter, which obeys specific properties for rate dependent or
independent material. For rate dependent material the constitutive equation is
⟨𝜙(𝐹)⟩ �̇� =
𝜂
Where 𝜙(𝐹) is a specified function describing the character of the viscoplastic flow, ⟨ ⟩
are Macauley brackets and 𝜂 represents a viscosity material parameter, when 𝜂 → 0 the
constitutive model becomes rate independent. The magnitude of the viscoplastic flow is
proportional to the distance of the stress state to the surface defined by F(σ,α)=0 for the
case of rate dependent material (Gomez & Basaran, 2006a).
3.8.5 Viscoplastic creep law
According to literature, creep is defined as the permanent deformation of the materials,
due to constant load or stress over long periods of time, this effect could limit the service
lifetime of any structure.
The evolution of the viscoplastic strain is described by the equation for creep law
proposed by Kashyap and Murty (1981), including the multiaxial case by (C. Basaran &
Yan, 1998)
𝑛 𝑝 𝑣𝑃 𝐴𝐷0𝐸𝑏 ⟨𝐹⟩ 𝑏 𝑄 𝜕𝐹
𝜀̇ = ( ) ( ) 𝑒−𝑅𝑇
𝑘𝑇 𝐸 𝑑 𝜕𝜎
Where 𝐴 is a dimensionless material parameter which is temperature and rate dependent,
𝐷0 is the frequency factor used to calculate the diffusion coeficient, 𝐸(𝑇) is a temperature
dependent Young’s modulus, 𝑏 is the magnitude of Burger’s vector for the material, 𝑘 is
the Boltzmann constant, 𝑇 is the absolute temperature in Kelvin, 𝑛 is a stress exponent
for plastic deformation rate (where 1/𝑛 defines the strain sensitivity), 𝑑 is the average
grain size, 𝑝 is a grain size exponent, 𝑄 is the creep activation energy and 𝑅 is the
universal gas constant (Gomez & Basaran, 2006b). From the flow rule and the
consistency parameter equations, the viscosity material parameter is redefined as
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𝑝 𝑘𝑇 𝑑 𝑄 𝜂 = ( ) 𝑒𝑅𝑇
𝐴𝐷0𝐸1−𝑛𝑏 𝑏
3.8.6 Damage coupled model
According to the strain equivalent principle, the damage couple model can be written as
𝑣𝑃 − 𝜀𝑇)𝜎 = (1 − Φ)𝐶: (𝜀̇ − 𝜀̇ ̇
2 𝐹 = ‖𝑆 − 𝑋Φ‖ − (1 − Φ)√ 𝐾(𝛼) ≡ ‖𝑆 − 𝑋Φ‖ − (1 − Φ)𝑅(𝛼)
3
3.8.7 Return mapping algorithm
The return mapping algorithm adopted as the UMAT subroutine was initially developed
by (Gomez & Basaran, 2006b) for thin layer eutectic Pb/Sn solder joints, for small strains.
In further research this algorithm needs to be calibrated for large strains, suitable for the
physics of the nature of the device under study.
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Chapter 4 FINITE ELEMENT MODEL AND ANALYSIS
4.1 GEOMETRIC DEFINITION OF THE LRB
The geometry selected for the analysis is the one describe in the example number
ten of the technical report MCEER-08-27 “Effects of heating and Load history on the
Behavior of Lead Rubber Bearings” (Kalpakidis & Constatinou, 2008).
The element under study is compound of 15 steel shims, 3/16” thick; 16 rubber
layers, 3/8” thick; and four steel plates each of 1” thickness. Table 4.1 and Table 4.2
show the summary of the dimensions used to create the model in Abaqus for the
LRB. The Figure 4.1 shows the sketch taken from the original report.
Table 4.1 Thickness of the LRB layers
Table 4.2 Diameter of the LRB layers
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Figure 4.1 Geometric definition of the LRB
4.2 MECHANICAL PROPERTIES OF THE MATERIALS
4.2.1 Rubber
Rubber is defined as a Neo-hookean material since a single modulus of elasticity is not
enough to define its stress-strain behavior. One of the most important properties of this
material is the ability to recover, almost completely, its original characteristics after large
deformations (Shahzad, Kamran, Siddiqui, & Farhan, 2015). However, due to the
softening occurring during first stages of deformation of the rubber, known as Mullins
effect and considering the damage parameter, these structural properties are assumed
to change significantly (Diani, Fayolle, & Gilormini, 2009).
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The strain energy function W, used to define rubber is:
1𝑊 = 𝐶10(𝐼1 − 3) +
𝐷1 (𝐽𝑒𝑙 − 1)2
Where C10 and D1 are material constants controlling the shear behavior and bulk
compressibility respectively (Shahzad et al., 2015), defined for this study as shown in
Table 4.3. The Poisson ratio of rubber is almost 0.5, any material created in Abaqus with
a Poisson ratio bigger than 0.495 has potential convergence problems. The bulk modulus
was modified accordingly to cope with this restriction of the FEM program. In
consequence, the final coefficient D1 used for the strain energy function is slightly smaller,
thus it is assumed that this change has no significant influence in the final result.
Table 4.3 Rubber properties (left: original values, right: adjusted for Abaqus)
C10 = G/2 (G: shear modulus)
D1 = 2/K (K: bulk modulus)
4.2.2 Steel
Commercial steel is modeled as an elastic material at the top and bottom plates, as well
as in the steel shims. Properties for this material are shown in Table 4.4
Table 4.4 Steel properties
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It should be noted that the shims are cutted by laser during the fabrication process for
better precision. In fact, all the fabrication process is computer controlled to combine
agility in the production with precision (Dynamic Isolation Systems, 2007).
4.2.3 Lead
This metal is a material to be found in the nature, where is extracted and processed to be
commercialized. The most important properties of this material is the malleability,
optimum resistance to corrosion while in air, and for purposes of this research, its low
strength and high ductility. Thus, the tensile strength of lead is around 12-17 MPa, less
than other common metals, but can be increased considerably with additions of other
metals to create alloys, like tin or copper (Thornton, Rautiu, & Brush, 2001).
During fabrication of the LRB, lead should be tightly confined, with a diameter within the
range B/6<dL<B/3, where B is the bonded diameter of the circular bearing, the rubber
layer thickness should be less than 3/8in (9.5mm), and the extreme surfaces of the core
should be sealed to guarantee protection against damage and confinement of the lead
(Buckle et al., 2006).
Pure lead is one of the few metals to reach recrystallization at room temperature, which
means that the material regains the original shape after deformations due by shear or
extrusion. In other words, lead maintain its ductility under very low temperatures.
However, this property could be lost by even the slightest impurity in very small
concentrations (Buckle et al., 2006).
The mechanical properties for lead depend on several factors, some of them are the
speed of deformation, grain size, dislocation density, degree of plastic deformation (strain
hardening effects), temperature, etc. In other words, properties like the yield point
depends on the duration of the test and the strain rate, in the same way, ultimate tensile
strength (UTS) decreases with higher temperature as observed in Figure 4.3
(Guruswamy, 1999).
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Figure 4.2 Temperature distribution of the led core (Buckle et al., 2006)
Figure 4.3 effect of temperature on UTS (Guruswamy, 1999)
It can also be noted that exposure to lead can have serious consequences for health,
ranging from metal poisoning to cancer and affecting body organs like the brain and
kidneys. In consequence, the exposure to lead in industry has been reduced
significantly… (Thornton et al., 2001), also this study will rely on data collected from
literature, and experimental data obtained in previous studies.
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The mechanical properties used in the Abaqus model for lead are defined as follows: the
properties 1 to 8,18 and 25 are taken from (Guruswamy, 1999); the properties related to
kinematic hardening are complex to obtain without laboratory test results for an specific
material, thus the properties 10, 13 to 15 and 19 to 21 were taken from (Gomez &
Basaran, 2006b), the property 9 was taken from (Hernandez, 2018), since it could be
assumed that the deviatoric planes will be similar for this study. The values for isotropic
hardening (values 11 and 12) were calculated using the values in the “TABLE 4-18” from
(Kalpakidis & Constatinou, 2008) according to (Chaboche, 1989) constitutive theories.
The properties 16 and 17 are taken from (Edalati & Horita, 2011) and finally, values 22 to
24 are universal constants. The summary of these values is shown below in Table 4.5.
Table 4.5 Lead properties
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4.3 FINITE ELEMENT MODEL
4.3.1 Model Assembly
The finite element model in ABAQUS was be defined as an Axisymmetric model, created
with 3D elements. The assembly is comprised of 72 parts, each defining one respective
instance. Each section of the LRB was defined as a solid revolve, connected together
using constraints type Tie. To define each constraint, it is required to define a master and
a slave surface. For this model, the lead and the steel surface where defined as master,
and the rubber surface was defined as a slave.
Instance defined by a section revolved
Figure 4.4 part instances and solid revolutions
Seven different kind of sections were defined, according to the material and geometry
associated. As shown in Figure 4.5. The base of the entire model is set as fixed in
boundary conditions and the lateral with the vertical displacements are defined as input
boundary conditions.
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Upper plate
Bottom plate
Internal bottom plate
Internal Upper plate
Lead core
Rubber Layers
Steel Shims
Figure 4.5 Sections defined
4.3.2 Mesh definition
A very important aspect when doing a model in Abaqus is the mesh definition, and the
respective element type definition, which should be defined according to the properties of
the material itself. The mesh of the model is defined as follows:
- Steel: Quadratic tetrahedral elements of type C3D10
- Rubber: Quadratic tetrahedral elements of type C3D10
- Lead: Quadratic tetrahedral elements of type C3D10
Figure 4.6 Element type definition for the mesh
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4.3.3 Boundary Conditions
The model is subject to lateral displacements, vertical displacement, amplitudes were
defined specifically for cyclic displacement as well as for vertical displacement, and
temperature increase as a predetermined field for the surface of the lead core in contact
with the steel and the rubber.
Figure 4.7 Amplitude definition for boundary conditions
Figure 4.8 Boundary condition- Lateral displacement
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Figure 4.9 Boundary condition- Fixed base
Figure 4.10 Amplitude definition for predefined field
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Figure 4.11 Boundary condition- Temperature increase
4.4 FINITE ELEMENT MODEL VERIFICATION
4.4.1 Frequencies
It is required to determine the mode frequencies to verify the stability of the model. All the
modes and frequencies must present positive values, to make sure that the model is
stable. See Figure 4.12.
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Figure 4.12 Modes and frequencies calculated in Abaqus
4.4.2 Elastic Behavior
To verify that the materials in the model are adequately defined, a model was made where
the mechanical properties of the lead were defined as elastic and isotropic, the
mechanical response and respective curve force - displacement was as expected. Since
the model is purely elastic, there is no energy dissipated per cycle.
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Figure 4.13 Elastic properties for lead
Figure 4.14 Time – Displacement / Elastic material
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Figure 4.15 Time – Force / Elastic material
Figure 4.16 Force- Displacement elastic behavior of the material lead
4.4.3 Elasto Plastic Behavior
Another model was developed to verify the elastoplastic response of the material to the
assigned lateral displacements. The results were as well as expected, since the material
is defined as elasto-plastic and no degradation of the material is involved yet, the energy
dissipated per cycle (EDC) remains with an estimated constant value of 100 kN-m respect
to the number of cycles.
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Figure 4.17 Elasto-Plastic properties for lead
Figure 4.18 Time – Displacement / Elastoplastic material
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Figure 4.19 Time – Force / Elastoplastic material
Figure 4.20 Force- Displacement Loops for the Elasto-Plastic material / Model for validation
4.5 UMAT SUBROUTINE ALGORITHM
The UMAT subroutine is usually used when any of the predefined materials included in
the ABAQUS material library accurately represents the behavior of the material to be
modeled. UMAT subroutine define complex, constitutive models for materials, this
subroutine requires a code in FORTRAN for the integration algorithm where the Jacobian
matrix and the respective explicit definition of stresses are modified according to the time
and temperature evolution. In other words, the algorithm updates the stresses in the
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material with the corresponding Jacobian matrix, at each time increment each value is
calculated through iteration until the solution for each state converges. The UMAT
subroutine involves the kinematic hardening effects described by (Lubarda & Benson,
2002), and follows the procedure stablished by (Gomez & Basaran, 2006a) for Pb/Sn
solder joints on small strains with small deformations. The FORTRAN code for this
research is shown in Appendix A UMAT SUBROUTINE.
Chapter 5 EVALUATION AND RESULTS
Finite element modeling of Lead Rubber Bearings could lead to a reduction in the budget
of any related projects, since the amount of devices to test could be reduced in at least
one of the three test stages: characterization, prototype and quality control. With the use
of model prediction of the mechanical properties of a base isolator, including thermal
effects, it would be possible to determine, at a preliminary phase, the characteristic
strength, yield force, yield displacement, elastic stiffness and post-elastic stiffness, which
are the minimum values for the preliminary design. The results obtained from the finite
element model using the UMAT subroutine are shown below.
Figure 5.1 Maximum displacement of 114mm
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Figure 5.2 Time – Displacement / UMAT Subroutine
The Figure 5.3 clearly shows a reduction on the force as response for the same
amplitude of motion, which means that effectively the mechanical properties of the lead
are reduced by the temperature increment for the respective time increment.
Figure 5.3 Time – Force / UMAT Subroutine
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The shear force obtained is three times bigger than the validation model, which means
that the flow rule parameters need to be adjusted to obtain a maximum value as reaction
force around 350 kN as observed in the graph force displacement for validation.
If the values for the reaction forces are adjusted by calibrating the values for the plastic
rule, the expected graphic for the Time-Force and the Force – Displacement loops would
be as shown in Figure 5.4 and Figure 5.5 respectively.
Figure 5.4 Time – Force / UMAT Subroutine (adjusted)
The results obtained by the finite element model are not as close as expected respect to
the validation laboratory test. However, it is a good step towards further investigation to
get close to a finite element model that allows predictions of the fatigue life of the LRBs,
based on the degradation of the mechanical properties of high purity lead core, due to
temperature increase by cycle.
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Figure 5.5 Force- Displacement / UMAT Subroutine (adjusted)
The respective energy dissipated by cycle is shown in Figure 5.6
Figure 5.6 Energy dissipated by cycle (adjusted)
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5.1 LABORATORY TEST FOR CALIBRATION AND VALIDATION OF
RESULTS
The data used for the input was taken from (Kalpakidis & Constatinou, 2008) as
mentioned earlier in this document as shown in Table 5.1, Figure 5.7 and Figure 5.8.
Table 5.1 laboratory test results for calibration and validation of results
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Figure 5.7 Force Displacement Loops of LRB test studied herein for calibration purposes “Example No 10” Displacement amplitude = 114mm and frequency = 0.35 Hz (Peak velocity =
250mm/s) (Kalpakidis & Constatinou, 2008)
Figure 5.8 Energy dissipated per cycle “Example 10” (Kalpakidis & Constatinou, 2008)
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Chapter 6 CONCLUSIONS
One of the main objectives was to simulate the fatigue mechanism that the isolator type
Lead Rubber Bearing (LRB) goes through while being in laboratory test for the
development of hysteretic loops under lateral loads based on design earthquakes, taking
into account the effects of temperature for each increment of time. The results obtained
are satisfactory since it was observed that the properties of the lead core decrease with
each time increment and the respective temperature increase. However, the flow rule
needs to keep being adjusted to obtained reaction forces closer to the ones observed in
the actual laboratory test.
The UMAT subroutine, developed in previous research at the University at Buffalo for
acrylic composites and alloys solders, based on the Unified Mechanics Theory to
determine the constitutive model, was used to recreate the degradation of the mechanical
properties of the high purity lead core, using the commercial finite element package
ABAQUS. The material parameters involved in the subroutine were obtained from the
literature.
The results of the simulation were compared with actual laboratory test developed in
previous research at the University at Buffalo for verification purposes. The outcomes
obtained by the finite element model are not as close as expected respect to the validation
laboratory test. However, it is a good step towards further investigation to get close to a
finite element model that allows predictions of the fatigue life of the LRBs, based on the
degradation of the mechanical properties of high purity lead core, due to temperature
increase by cycle.
The damage in terms of hysteresis energy drop was also measure, as an internal state
variable corresponding to the material degradation under fatigue loading. The
Thermodynamic Index State, which is a measure of the irreversibility of the
thermodynamic system, is used as the units of damage. The damage coupled constitutive
model recreates the interaction between the mechanical response of the material under
cyclic loading, fatigue and damage.
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The results obtained using simulation on a finite element model, have good correlation
with the test data used for validation. The results obtained are optimal, being aware that
this is among the initial attempts to simulate damage using the Unified Mechanics Theory,
on the lead core of a complete device for isolation systems, composed of materials of
different nature mechanically.
Chapter 7 RECOMMENDATIONS
The mechanical properties for the high purity lead core were taken from the literature.
However, there were found some differences in the values regarding the parameters used
in the constitutive model, some authors describe one value, some others showed ranges
of values especially for the flow rule of the material. The model presented in this thesis
obtained a good approximation, but still requires tuning to define the appropriate values
according to the size of the element and the kind of study.
The UMAT subroutine requires modifications to updated the constitutive equations to
create matrix for large strains. As can be seen in results there is a wide range of research
to keep advancing in the topic. Also, further studies could be developed to understand
the effect of higher or lower frequencies for the cyclic displacements in the material.
The model is computationally very expensive. To obtain 15 seconds of data, the model
takes around 10 complete days. Due to limitations in time, it was not possible to advance
further in the calibration of the parameters for the flow rule, despite of the great
achievement obtaining a model that converges. Further develop in the research is
recommended to obtain a model fully tune for LRBs.
One of the appendices describes how to create the link between ABAQUS and
FORTRAN in Windows and in Linux. This was a major limitation at the beginning of the
investigation. The use of this procedures are strongly recommended to successfully
compile the subroutine “inside” ABAQUS.
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Chapter 8 APPENDICES
8.1 Appendix A UMAT SUBROUTINE
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8.2 Appendix B
- Create link Abaqus - compiler in Windows (researchgate.net/, n.d.)
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- Create link Abaqus - compiler in Linux (redhat) (Imechanica.org,
n.d.)
Linking Abaqus/Fortran for running subroutine in UBUNTU (linux)-Abaqus GUI
Graphical issue(Transparent-/transluscent)
Instruction on how to link Abaqus and Fortran for running subroutines (UMAT or VUMAT) (In
Linux-redHat) including: intel c++ /Fortran and Abaqus
1) These are the requirements:
a) Abaqus version 2018.
b) Intel fortran and C++ ifort version 16.0.2
c) In linux, the abaqus website says it is only compatible with Suse and red-hat.
2) Linking the Fortran with Abaqus and running the UMAT
a) First of all in linux machine the fortran files are not known as .fort or .For which were used in
windows. Linux knows them as .f . so we need to change them to .f
b) For running abaqus with subroutine we should run "abaqus job=myjobname
user=myfortranfilename int".
If we run it now, an error will occure: Some error like: "ifort is not available in the PATH". We
need to add the "ifort" path to the environment variable "PATH". it can be done temporarily
(Should be repeated in each terminal individually) by running
"export PATH=/path to ifort.var/:$PATH" (Example in my case: "export
PATH=/opt/intel/Compiler/11.1/080/bin/intel64:$PATH") before running the "abaqus
job=myjobname user=myfortranfilename int" .
If you want to make it a permanent change in PATH (Which is preferable since we do not need to
run it manually each time and if you have administrator permission), you should do "gksudo gedit
~/.bashrc" and add the line " export PATH=/opt/intel/Compiler/11.1/080/bin/intel64:$PATH" to
the end of the file.
by doing that you should be able to run the abaqus job=myjobname user=myfortranfilename int
and the error about the "ifort PATH" should not appear again.
c) Another error might happen " about the missed shared libraries "libiomp5.so" which is located
in intel libraries (you have previously installed the intel C++ and Fortan softwares). It should be
in "/opt/intel/Compiler/11.1/080/lib/intel64".
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8.3 Appendix C Variables for UMAT Subroutine (Dassault Systems, n.d.)
In all situations
DDSDDE(NTENS,NTENS)
Jacobian matrix of the constitutive model, ∂Δσ/∂Δε, where Δσ are the stress increments and Δε are
the strain increments. DDSDDE(I,J) defines the change in the Ith stress component at the end of
the time increment caused by an infinitesimal perturbation of the Jth component of the strain
increment array. Unless you invoke the unsymmetric equation solution capability for the user-
defined material, Abaqus/Standard will use only the symmetric part of DDSDDE. The symmetric
part of the matrix is calculated by taking one half the sum of the matrix and its transpose.
For viscoelastic behavior in the frequency domain, the Jacobian matrix must be dimensioned as
DDSDDE(NTENS,NTENS,2). The stiffness contribution (storage modulus) must be provided in
DDSDDE(NTENS,NTENS,1), while the damping contribution (loss modulus) must be provided
in DDSDDE(NTENS,NTENS,2).
STRESS(NTENS)
This array is passed in as the stress tensor at the beginning of the increment and must be updated
in this routine to be the stress tensor at the end of the increment. If you specified initial stresses
(Initial conditions in Abaqus/Standard and Abaqus/Explicit), this array will contain the initial
stresses at the start of the analysis. The size of this array depends on the value of NTENS as defined
below. In finite-strain problems the stress tensor has already been rotated to account for rigid body
motion in the increment before UMAT is called, so that only the corotational part of the stress
integration should be done in UMAT. The measure of stress used is “true” (Cauchy) stress.
If the UMAT utilizes a hybrid formulation that is total (as opposed to the default incremental
behavior), the stress array is extended beyond NTENS. The first NTENS entries of the array
contain the stresses, as described above. The additional quantities are as follows:
STRESS(NTENS+1) Read only: ˆJ,
STRESS(NTENS+2) Write only: ˆK=J∂2U∂ˆJ2, and
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STRESS(NTENS+3) Write only: ∂ˆK∂ˆJ=J∂3U∂ˆJ3, where U is the volumetric part of the
strain energy density potential.
STATEV(NSTATV)
An array containing the solution-dependent state variables. These are passed in as the values at the
beginning of the increment unless they are updated in user subroutines USDFLD or UEXPAN, in
which case the updated values are passed in. In all cases STATEV must be returned as the values
at the end of the increment. The size of the array is defined as described in Allocating space.
In finite-strain problems any vector-valued or tensor-valued state variables must be rotated to
account for rigid body motion of the material, in addition to any update in the values associated
with constitutive behavior. The rotation increment matrix, DROT, is provided for this purpose.
SSE, SPD, SCD
Specific elastic strain energy, plastic dissipation, and “creep” dissipation, respectively. These are
passed in as the values at the start of the increment and should be updated to the corresponding
specific energy values at the end of the increment. They have no effect on the solution, except that
they are used for energy output.
Only in a fully coupled thermal-stress or a coupled thermal-electrical-structural analysis
RPL Volumetric heat generation per unit time at the end of the increment caused by
mechanical working of the material.
DDSDDT(NTENS) Variation of the stress increments with respect to the temperature.
DRPLDE(NTENS) Variation of RPL with respect to the strain increments.
DRPLDT Variation of RPL with respect to the temperature.
Only in a geostatic stress procedure or a coupled pore fluid diffusion/stress analysis for pore
pressure cohesive elements
RPL
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RPL is used to indicate whether or not a cohesive element is open to the tangential flow of pore
fluid. Set RPL equal to 0 if there is no tangential flow; otherwise, assign a nonzero value to RPL
if an element is open. Once opened, a cohesive element will remain open to the fluid flow.
Variables that can be updated
PNEWDT
Ratio of suggested new time increment to the time increment being used (DTIME, see discussion
later in this section). This variable allows you to provide input to the automatic time incrementation
algorithms in Abaqus/Standard (if automatic time incrementation is chosen). For a quasi-static
procedure, the automatic time stepping that Abaqus/Standard uses, which is based on techniques
for integrating standard creep laws (see Quasi-static analysis), cannot be controlled from within
the UMAT subroutine.
PNEWDT is set to a large value before each call to UMAT.
If PNEWDT is redefined to be less than 1.0, Abaqus/Standard must abandon the time increment
and attempt it again with a smaller time increment. The suggested new time increment provided
to the automatic time integration algorithms is PNEWDT × DTIME, where the PNEWDT used is
the minimum value for all calls to user subroutines that allow redefinition of PNEWDT for this
iteration.
If PNEWDT is given a value that is greater than 1.0 for all calls to user subroutines for this iteration
and the increment converges in this iteration, Abaqus/Standard may increase the time increment.
The suggested new time increment provided to the automatic time integration algorithms is
PNEWDT × DTIME, where the PNEWDT used is the minimum value for all calls to user
subroutines for this iteration.
If automatic time incrementation is not selected in the analysis procedure, values of PNEWDT that
are greater than 1.0 will be ignored and values of PNEWDT that are less than 1.0 will cause the
job to terminate.
Variables passed in for information
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STRAN(NTENS)
An array containing the total strains at the beginning of the increment. If thermal expansion is
included in the same material definition, the strains passed into UMAT are the mechanical strains
only (that is, the thermal strains computed based upon the thermal expansion coefficient have been
subtracted from the total strains). These strains are available for output as the “elastic” strains.
In finite-strain problems the strain components have been rotated to account for rigid body motion
in the increment before UMAT is called and are approximations to logarithmic strain.
DSTRAN(NTENS)
Array of strain increments. If thermal expansion is included in the same material definition, these
are the mechanical strain increments (the total strain increments minus the thermal strain
increments).
TIME(1) Value of step time at the beginning of the current increment or frequency.
TIME(2) Value of total time at the beginning of the current increment.
DTIME Time increment.
TEMP Temperature at the start of the increment.
DTEMP Increment of temperature.
PREDEF
Array of interpolated values of predefined field variables at this point at the start of the increment,
based on the values read in at the nodes.
DPRED Array of increments of predefined field variables.
CMNAME
User-defined material name, left justified. Some internal material models are given names starting
with the “ABQ_” character string. To avoid conflict, you should not use “ABQ_” as the leading
string for CMNAME.
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NDI Number of direct stress components at this point.
NSHR Number of engineering shear stress components at this point.
NTENS Size of the stress or strain component array (NDI + NSHR).
NSTATV
Number of solution-dependent state variables that are associated with this material type (defined
as described in Allocating space).
PROPS(NPROPS) User-specified array of material constants associated with this user material.
NPROPS User-defined number of material constants associated with this user material.
COORDS
An array containing the coordinates of this point. These are the current coordinates if geometric
nonlinearity is accounted for during the step (see Defining an analysis); otherwise, the array
contains the original coordinates of the point.
DROT(3,3)
Rotation increment matrix. This matrix represents the increment of rigid body rotation of the basis
system in which the components of stress (STRESS) and strain (STRAN) are stored. It is provided
so that vector- or tensor-valued state variables can be rotated appropriately in this subroutine: stress
and strain components are already rotated by this amount before UMAT is called. This matrix is
passed in as a unit matrix for small-displacement analysis and for large-displacement analysis if
the basis system for the material point rotates with the material (as in a shell element or when a
local orientation is used).
CELENT
Characteristic element length, which is a typical length of a line across an element for a first-order
element; it is half of the same typical length for a second-order element. For beams and trusses it
is a characteristic length along the element axis. For membranes and shells it is a characteristic
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length in the reference surface. For axisymmetric elements it is a characteristic length in the (r,z)
plane only. For cohesive elements it is equal to the constitutive thickness.
DFGRD0(3,3)
Array containing the deformation gradient at the beginning of the increment. If a local orientation
is defined at the material point, the deformation gradient components are expressed in the local
coordinate system defined by the orientation at the beginning of the increment. For a discussion
regarding the availability of the deformation gradient for various element types, see Deformation
gradient.
DFGRD1(3,3)
Array containing the deformation gradient at the end of the increment. If a local orientation is
defined at the material point, the deformation gradient components are expressed in the local
coordinate system defined by the orientation. This array is set to the identity matrix if nonlinear
geometric effects are not included in the step definition associated with this increment. For a
discussion regarding the availability of the deformation gradient for various element types, see
Deformation gradient.
NOEL Element number.
NPT Integration point number.
LAYER Layer number (for composite shells and layered solids).
KSPT Section point number within the current layer.
JSTEP(1) Step number.
JSTEP(2) Procedure type key (see Results file output format).
JSTEP(3) 1 if NLGEOM=YES for the current step; 0 otherwise.
JSTEP(4) 1 if current step is a linear perturbation procedure; 0 otherwise.
KINC Increment number.
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