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\journal{Comput. Methods Appl. Mech. Engrg.}
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\begin{document}
\begin{frontmatter}
\title{Energy Dissipation Analysis of Elastic-Plastic Materials }
%\title{Energy dissipation analysis of elastic-plastic materials\tnoteref{mytitlenote}}
%\tnotetext[mytitlenote]{This research is supported by the US-DOE}
%% or include affiliations in footnotes:
\author[mymainaddress]{Han Yang}
%\ead{hhhyang@ucdavis.edu}
\author[mymainaddress]{Sumeet Kumar Sinha}
\author[mymainaddress]{Yuan Feng}
\author[mysecondaddress]{David B McCallen}
\author[mymainaddress,mysecondaddress]{Boris Jeremi{\'c}\corref{mycorrespondingauthor}}
\ead{jeremic@ucdavis.edu}
\cortext[mycorrespondingauthor]{Corresponding author}
\address[mymainaddress]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA}
\address[mysecondaddress]{Earth Science Devision, Lawrence Berkeley National Laboratory, Berkeley, CA, USA}
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\begin{abstract}
Presented is an energy dissipation analysis framework for granular
material that is based on thermodynamics.
%
%Through many papers on seismic energy analysis of soil-structure-interaction
%(SSI) systems, the confusion between plastic work and plastic dissipation caused
%apparent violation of the second law of thermodynamics. A seismic energy
%analysis framework based on thermodynamics and computational geomechanics is
%presented.
%
Theoretical formulations are derived from the second law of
thermodynamics, in conjunction with a few plausible assumptions on energy
transformation and dissipation.
%
The role of plastic free energy is emphasized by a conceptual experiment
showing its physical nature.
%
Theoretical formulation is adapted in order to be applied in elastic-plastic
finite element method (FEM) simulations.
%
Developed methodology is verified through comparison of input work, stored
energy, and energy dissipation of the system.
%
Separation of plastic work into plastic free energy and energy dissipation
removes a common mistake, made in a number of publications, where energy
dissipation can attain negative values (energy production) which is impossible.
% %
% Examples presented show that, in most cases, more that 50\% of the input energy
% (coming from, for example, seismic excitations) is dissipated due to plasticity
% on material level.
% %For materials with linear kinematic hardening,
% %linear increase of plastic dissipation is observed. For nonlinear
% %hardening materials, energy is dissipated nonlinearly. Large-scale simulation
% %results on realistic SSI systems is being processed and will presented in future
% %papers. The proposed method is capable of energy analysis for large-scale
% %realistic SSI systems and hence can be used to improve structure and foundation
%designs on the aspect of proper seismic energy dissipation.
\end{abstract}
\begin{keyword}
Seismic energy dissipation \sep FEM \sep Computational geomechanics \sep
Thermodynamics \sep Elastic-plastic materials \sep Plastic free energy
\end{keyword}
\end{frontmatter}
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\section{Introduction}
Energy dissipation in elastic plastic solids and structures is the result of an
irreversible dissipative process in which energy is transformed from one form to
another and entropy is produced.
%
The transformation and dissipation of energy is related to permanent deformation
and damage within an elastic-plastic material.
%
Of particular interest here is the dissipation of mechanical energy that is input into
elastic-plastic solids by static or dynamic excitations.
%
% Upon deformation, elastic-plastic materials will absorb energy.
% %
% Absorbed energy is composed of the elastic strain energy, plastic free
% energy and plastic dissipation.
Early work on plastic dissipation was done by \citet{Farren1925} and \citet{Taylor1934}.
They performed experiments on
metals and proved that a large part, but not all, of the input mechanical energy
is converted into heat. The remaining part of the non-recoverable plastic work
is known as the stored energy of cold work. The ratio of plastic work converted
into heating (Quinney--Taylor coefficient), usually denoted as $\beta$, has been
used in most later work on this topic. Based on large number of experiments,
this ratio was determined to be a constant between 0.6 to 1.0
\citep{Clifton1984,Belytschko1991,Zhou1996,Dolinski2010,Ren2010,Osovski2013}.
%It has been realized that this assumption is not valid in all cases, but it's
%simply too complicated to involve the evolution of Quinney--Taylor coefficient
%in thermomechanical constitutive models.
More recently Rittel
\citep{Rittel2000a,Rittel2000b,Rittel2003} published several insightful papers
on the energy dissipation (heat generation) of polymers during cyclic loading,
presenting both experimental and theoretical works.
%
Rosakis et al. \cite{Rosakis2000} presented
a constitutive model for metals based on thermoplasticity that is able to
calculate the evolution of energy dissipation.
%
Follow up papers
\citep{Hodowany2000,Ravichandran2002} present assumptions to
simply the problem.
%
%One widely used assumption is the adiabatic condition, since
%air conducts heat much slower than metal. This assumption is reasonable in rapid
%monotonic or cyclic loading (impact, vibration, earthquake).
%
One direct application of
plastic dissipation to geotechnical engineering is presented by
Veveakis et al. \citep{Veveakis2007,Veveakis2012}, using thermoporomechanics to model
the heating and pore pressure increase in large landslides, like the 1963 Vajont
slide in Italy.
%OVDE
In the past few decades, extensive studies have been conducted on energy
dissipation in structures and foundations.
%
Work by \citet{Uang1990} has been considered a source and a reference for
many recent publications dealing with energy as a measure of structural demand.
% by many
%researchers
%\citep{Leger1992,Cosenza1993,Kalkan2007a,Kalkan2008,Symans2008,Gajan2011,
%Moustafa2011,Moustafa2014,Mezgebo2017,Deniz2017}.
%
\citet{Uang1990} developed an energy analysis
methodology based
on absolute input energy (or energy demand).
Numerical analysis results were compared with experiments on a multi-story
building.
%
In their work, Uang and Bertero \cite{Uang1990},
calculated hysteretic energy indirectly by taking
the difference of absorbed energy and elastic strain energy.
%
The term
{absorbed energy} of each time step is simply defined as restoring force times
incremental displacement.
%
It is also stated that {hysteretic energy} is irrecoverable, which
indicates that this parameter was considered the same as {hysteretic
dissipation} or {plastic dissipation}.
%
An equation for
energy balance, is given by (\citet{Uang1990}) as:
%
\begin{equation}
E_i = E_k + E_{\xi} + E_a = E_k + E_{\xi} + E_s + E_h
\label{Eq1}
\end{equation}
%
where $E_i$ is the (absolute) input energy, $E_k$ is the (absolute) kinetic
energy, $E_{\xi}$ is the viscous damping energy, $E_a$ is the absorbed energy,
which is composed of elastic strain energy $E_s$ and hysteretic energy $E_h$.
The problem with this approach is the absence of plastic free energy, which is
necessary to correctly evaluate energy dissipation of elastic-plastic materials
and to uphold the second law of thermodynamics. While there is no direct plot of
plastic dissipation (hysteretic energy) in \cite{Uang1990}, since it was not defined
directly, there are plots of other energy components. Plastic dissipation
can be easily calculated from these plots.
%
After doing this,
indications of negative incremental energy dissipation, which violates the basic
principles of thermodynamics, were found in various sections of the paper.
This misconception could be clarified by renaming hysteretic energy as
{plastic work}, a sum of plastic dissipation and
plastic free energy. Both plastic work and plastic free energy can be
incrementally negative, but plastic dissipation (defined as the difference of
plastic work and plastic free energy) must be incrementally non-negative during
any time period. Unfortunately, this misconception has been inherited (if not
magnified) by many following studies on energy analysis of earthquake
soils and structures (hundreds of papers).
% % %
% % In the past few decades, extensive
% % studies have been conducted on energy dissipation in structures and foundations
% % \cite{Uang1990}, \cite{Leger1992}, \cite{Symans1998},
% % \cite{Soong2002}, \cite{Symans2008}, \cite{Wong2008},
% % \cite{Nehdi2010}.
% %
% A numbe
% Despite different methodology used, the calculations of energy dissipation due to
% hysteretic damping (material plasticity) are all performed using stress-strain and/or
% force-displacement response curves.
% %
% Traditionally in elastic-plastic computations,
% it is assumed that all plastic deformation will lead to energy dissipation.
% %
% In other words, plastic energy dissipation is calculated by integrating the
% product of generalized plastic strain and generalized stress.
% %
% In some cases, researchers simply take the area of hysteresis loop as an
% evaluation of energy dissipation.
% %
% However, close examination of results (in a number of plots) from the previously
% mentioned papers reveals that accumulated (total) energy dissipation was
% observed to decrease during dynamic loading.
% %
% This is a clear violation of the second law of thermodynamics.
% %
% It also means that the approach used to compute energy dissipation is
% inappropriate.
%
%OVDE
The basic principles of
thermodynamics are frequently used to derive new constitutive models, for example by
\citet{Dafalias1975}, \citet{Ziegler1987},
\citet{Collins97},
\citet{Houlsby2000}, \citet{Collins2002a},
\citet{Collins2002b}, \citet{Collins2003b} and
\citet{Feigenbaum2007}.
%
%In these studies, the definition of plastic energy dissipation is clarified.
%
The concept of plastic free energy is introduced to enforce the second law of
thermodynamics for developed constitutive models.
%
It is important to distinguish between energy dissipation due to plasticity
and plastic work, which is often a source of a confusion.
%
The physical nature of plastic free energy is illustrated later in this
paper through a conceptual example that is analyzed on particle scale.
%
Essentially, development of plastic free energy is caused by particle
rearrangement in granular assembly under external loading.
Specific formulation of free energy depends on whether the elastic and plastic behavior
of the material is coupled.
%
According to Collins et al. \cite{Collins97},
\cite{Collins2002a},
\cite{Collins2003b}, material coupling
behavior can be divided into {modulus coupling}, where the instantaneous
elastic stiffness (or compliance) moduli depend on the plastic strain, and
{dissipative coupling}, where the rate of dissipation function depends
not only on the plastic strains and their rates of change but also on the
stresses (or equivalently the elastic strains).
%
The modulus coupling describes the degradation of
stiffness as in for rock and concrete, and is usually modeled by employing a coupled
elastic-plastic constitutive model or by introducing damage variables.
%
The dissipative coupling is considered to be one of the main reasons for
non-associative behavior in geomaterials
\cite{Collins97}, \cite{Ziegler1981}.
%
%In this paper, various elastic-plastic material
%models are applied and tested, involving both coupled stiffness tensor and
%non-associative plasticity.
%
A number of stability postulates are commonly used to prevent
violation of principles of thermodynamics.
%
Stability postulates include Drucker's
stability condition
\cite{Drucker1956}, \cite{Drucker1957}, Hill's stability condition
\cite{Bishop1951}, \cite{Hill1958}, and Il'Iushin's stability postulate
\cite{Il1961}, \cite{local-63}.
%
As summarized in a paper by Lade
\cite{Lade2002},
theoretical considerations by
\citet{Nemat-Nasser83} and \citet{local-25} have suggested that they are
sufficient but not necessary conditions for stability.
%
These {stability
postulates} can indeed ensure the admissibility of the constitutive models by
assuming certain restrictions on incremental plastic work. As demonstrated by
\citet{Collins2002a}, if the plastic strain rate is replaced by
the irreversible stain rate in Drucker's postulate, then all the standard
interpretations of the classical theory still apply for coupled materials.
\citet{Dafalias77b} also modified Il'Iushin's postulate in a
similar way and applied it to both coupled and uncoupled materials.
%Other than theoretical considerations, deficiency in the current practice of
%seismic energy analysis also exists on numerical modeling. A number of numerical
%studies on SSI systems have been conducted with significant results and
%inspirations \cite{Chen1977},
%\cite{Makris1994}, \cite{McKenna97}, \cite{Bielak2001},
%\cite{Sett2005a}, \cite{Sett2005b},
%\cite{Fenves1998}, \cite{Elgamal2008}, \cite{Jeremic2008d},
%\cite{Jeremic2013}, \cite{Jeremic2004g},
%\cite{Boulanger1999}, \cite{Yang2003}.
%
% HAN:
%
% What are you trying to say with the above references?
%
% Han's comment:
% This part is not useful, should be deleted...
%
%
%
%However, among all
%the papers on SSI problems, most of them have documented over-simplified or even
%no energy analysis in their cases.
%
It is important to note that development of inelastic deformation in geomaterials
involves large changes in entropy, and significant energy dissipation.
%
It is thus useful to perform energy dissipation (balance) analysis for all
models with inelastic deformation.
%
%Thus, it is necessary to perform energy dissipation analysis in all numerical
%studies of SSI systems. Unfortunately, little effort has been done regarding to
%this topic.
In this paper we focus on energy dissipation
on material level.
%More results of large-scale modeling is being
%processed and will be discussed in our following papers.
%
Focus is on proper modeling that follows thermodynamics.
%
Comparison is made between accumulated plastic dissipation and accumulated
plastic work, since these quantities can
be quite different in most cases.
%Depending on the material constitutive model
%used in simulation, the energy dissipation accumulates with unique patterns.
As
a way of verification, the input work, which is introduced by applying external
forces, is compared with the stored energy and dissipation in the entire system.
Finally, conclusions on plastic energy dissipation are drawn from the verified
results.
% Han's comment:
% I saw many papers have a paragraph like this to explain the contents of the paper
%
%
% In this study, a seismic energy analysis framework is established based on
% thermodynamics and computational geomechanics. Different mechanisms of seismic
% energy transformation and dissipation are identified and analyzed. Derived from
% the second law of thermodynamics and reasonable assumptions, formulations of
% energy parameters in multiple forms are presented. In order to apply them to
% numerical simulations, these theoretical equations are modified to satisfy
% computational requirements of FEM. To show the effectiveness of our newly
% developed methodology, a series of numerical simulations under different
% boundary conditions are conducted with emphasize on energy dissipation analysis.
% In particular, values of accumulated plastic dissipation are compared with
% accumulated plastic work. It can be observed later that the two quantities can
% be very different in most cases. Depending on the material constitutive model
% used in simulation, the energy dissipation accumulates with unique patterns. As
% a way of verification, the input work, which is introduced by applying external
% forces, is compared with the stored energy and dissipation of the entire system.
% Finally, conclusions on plastic energy dissipation are drawn from the verified
% results.
%
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\section{Theoretical and Computational Formulations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\su
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\su
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\subsection{Thermo-Mechanical Theory}
For energy analysis of elastic-plastic materials undergoing {isothermal
process}, it is beneficial to start from the statement of the first and second
laws of thermodynamics:
%
\begin{equation}
\hat{W} = \dot{\Psi} + \Phi
%, \quad \mathrm{where} \; \Phi \ge 0
\end{equation}
%
where $\Phi \ge 0$ and $\hat{W} \equiv \sigma : \dot{\epsilon} = \sigma_{ij}
\dot{\epsilon}_{ij}$ is the rate of work per unit volume.
%
The function $\Psi$ is the {Helmholtz free
energy}, and $\Phi$ is the rate of dissipation; both defined per volume. The
free energy $\Psi$ is a function of the {state variables} (also known as
{internal variables}), but $\Phi$ and $\hat{W}$ are not the time
derivatives of the state functions.
%
The choice of state variables depends on the
complexity of constitutive model that is being used, as cyclic loading with
certain hardening behaviors usually requires more state variables. This will be
elaborated in the following sections as we discuss specific elastic-plastic
material models. Note that in this paper all stresses are defined as effective
stresses. In order to avoid confusion, the common notation ($\sigma_{ij}^{'}$) will not be used.
Standard definition of stress from mechanics of materials, i.e. positive in tension, is used.
For general elastic-plastic materials, the free energy depends on both the
elastic and plastic strains. In most material models, it can be assumed
that the free energy $\Psi$ can be decomposed into elastic and plastic parts:
%
%
% HAN: is this Phi or Psi in the equation below, I am skipping editing
%
% Han's comment:
% Should be Psi, this was a typo. Corrected.
%
\begin{equation}
\Psi = \Psi_{el} + \Psi_{pl}
\label{Eq2}
\end{equation}
The total rate of work associated with the effective stress can be written as
the sum of an elastic and plastic component:
%
% HAN: here you use effective stress, while it should be just stress, right?
%
% Han's comment:
% Plastic energy dissipation comes from inter-particle frictions, which should be associated
% with effective stresses. So, as stated in line 370, all stresses are defined as effective
% stresses. In order to avoid confusion, the common notation ($\sigma_{ij}^{'}$) will not be used.
%
\begin{equation}
\hat{W}^{el} \equiv \sigma_{ij} \dot{\epsilon}^{el}_{ij} = \dot\Psi_{el}
\end{equation}
%
and
%
\begin{equation}
\hat{W}^{pl} \equiv \sigma_{ij} \dot{\epsilon}^{pl}_{ij} = \dot\Psi_{pl} + \Phi
\label{Eq4}
\end{equation}
%
%
%
In the case of a {decoupled material}, the elastic free energy $\Psi_{el}$ depends only on the
elastic strains, and the plastic free energy $\Psi_{pl}$ depends only on the plastic strains, as shown
by \citet{Collins97}:
%
\begin{equation}
\Psi = \Psi_{el}(\epsilon^{el}_{ij}) + \Psi_{pl}(\epsilon^{pl}_{ij})
\label{Eq5}
\end{equation}
The effective stress can also be decomposed into two parts:
\begin{equation}
\sigma_{ij} = \alpha_{ij} + \chi_{ij}, \quad \text{where} \; \alpha_{ij} \equiv \frac{\partial \Psi_{pl}}{\partial \epsilon^{pl}_{ij}} \quad \text{and} \quad \chi_{ij} \equiv \frac{\partial \Phi}{\partial \dot{\epsilon}^{pl}_{ij}}
\label{Eq6}
\end{equation}
The stress tensors $\alpha_{ij}$ and $\chi_{ij}$ are termed the shift (drag,
back or quasi-conservative) stress and dissipative stress respectively.
Ziegler's orthogonal postulate \cite{Ziegler1987} ensures the validity
of Equation~\ref{Eq6}. It is equivalent to the maximum entropy production
criterion, which is necessary to obtain unique formulation. Also, this is a weak
assumption so that all the major continuum models of thermo-mechanics are
included. Equation~\ref{Eq4} of plastic work rate can hence be
rewritten as:
%
\begin{equation}
\hat{W}^{pl} \equiv \sigma_{ij} \dot\epsilon^{pl}_{ij} = \dot\Psi_{pl} + \Phi = \alpha_{ij} \dot\epsilon^{pl}_{ij} + \chi_{ij} \dot\epsilon^{pl}_{ij}
\end{equation}
%
% CONTINUING editing from here, until we correct Psi or Phi above
%
The {plastic work} is the product of the {true stress} with
the plastic strain rate, while the {dissipation rate} is the product of
the {dissipative stress} with the plastic strain rate. They are only
equal if the back stress is zero, or equivalently, if the free energy depends
only on the elastic strains.
In kinematic hardening models, where the back stress describes the translation (or rotation)
of the yield surface, the decomposition of the true stress (sum of back
stress and dissipative stress) is a default assumption. Although such a shift
stress is important for anisotropic material models,
\citet{Collins2002b} have pointed out that it is also necessary in
isotropic models of geomaterials with different strength in tension and
compression.
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\subsection{Plastic Free Energy}
A popular conceptual model, which focused on particulate materials and demonstrated the
physical occurrence of shift stresses, was described by
\citet{Besseling1994} and \citet{Collins2002b}. On macro (continuum) scale, every
point in a given element is at yield state and deforms plastically. But on
meso-scale, only part of this element is undergoing plastic deformations, the
remaining part is still within yield surface and respond elastically. The
elastic strain energy stored in the elastic part of a plastically deformed
macro-continuum element is considered to be locked into the macro-deformation,
giving rise to the plastic free energy function $\Psi_{pl}$ and its associated
back stress $\alpha_{ij}$. This energy can be released only when the plastic
strains are reversed.
For better explanation, the nature of plastic free energy in particulate materials
is illustrated through a finite element simulation combined with
considerations of particle rearrangement on mesoscopic scale.
Figure~\ref{example_DP_AF_cyclic_response} shows stress-strain response of
Drucker-Prager with nonlinear Armstrong-Frederick kinematic hardening, a typical
elastic-plastic model for metals and geomaterials.
% As shown in
% Figure~\ref{example_DP_AF_cyclic_response}, the stress-strain curves match the
% typical
% response of an elastic-plastic material under cyclic loading. Numerically, the
% constitutive model is Drucker-Prager yield function with Armstrong-Frederick
% nonlinear kinematic hardening.
% which is usually used to simulate behavior of
% geotechnical materials.
Six states during shear are chosen to represent
evolution of micro fabric of the numerical sample.
%
Correspondingly, Figure~\ref{2D_assembly_cyclic} shows the process of particle
rearrangement of the 2D granular assembly under cyclic shearing from microscopic
level. The square window can be roughly considered as a representative volume (a constitutive level or a finite element)
in FEM.
%Note
%that this example only shows one arbitrary result of infinite particle
%arrangements, but it is representative nonetheless.
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Theo_Stress-Strain.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Theo_Stress-Plastic_Strain.pdf}}
\caption{
\label{example_DP_AF_cyclic_response}
Elastic-plastic material modeled with Drucker-Prager yield function and
Armstrong-Frederick kinematic hardening under cyclic shear loading: (a) Stress-strain curve;
(b) stress and plastic strain versus time.}
\end{figure}
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_1.pdf}}
\quad
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_2.pdf}}
\\
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_3.pdf}}
\quad
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_4.pdf}}
\\
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_5.pdf}}
\quad
\subfloat[]{\includegraphics[width=0.4\columnwidth]{Figures/Plastic_Free_Energy_6.pdf}}
\caption{
\label{2D_assembly_cyclic}
Particle rearrangement of a 2D granular assembly under cyclic shearing:
(a) Initial state; (b) Loading (accumulating plastic free energy); (c) End of
loading (maximum plastic free energy); (d) Unloading (plastic free energy
unchanged); (e) Reverse loading (releasing plastic free energy); (f) End of
reverse loading (plastic free energy released).}
\end{figure}
By discussing movement and energy of particle A in
Figure~\ref{2D_assembly_cyclic}, the physical nature of plastic free energy is
illustrated. At state (a), which is the beginning of deformation, particle A does not
bear any load other than its self weight. State (b) is in middle of
loading, when particle B pushes downwards to particle A until it makes contact
with particle D and E. Load reaches peak at state (c), and there's no
space for particle A to move. Then the sample is unloaded to state (d).
Particle A is now {stuck} between particles C, D,
and F, which means that certain amount of elastic energy is stored due to particle elastic
deformation. Compared with state (a), this part of elastic energy is not
released when the sample is unloaded, which indicates that it's not classic
strain energy. {This part of elastic energy on particle level which can't
be released by unloading is defined as the plastic free energy in granular
materials.} Reverse loading starts at state (e), where particle D pushes
particle A upwards, making it {squeeze through} particle C and F.
Elastic energy on particle level, which is now defined as plastic free energy, is
released during reverse loading.
By analyzing this example, an explanation on particle scale is provided for the origin of
plastic free energy in granular materials. It is important
to note that the concept of plastic free energy also exists in metals and
other materials, as studied by \citet{Dafalias2002} and
\citet{Feigenbaum2007}. The physical nature of plastic free energy
in these materials can be different and probably should be studied on
molecular and/or crystalline level.
Collins \cite{Collins2002b}, \cite{Collins2003b} suggested
that in the case of granular materials, the particle-level plastic energy
dissipation during normal compaction, arises from the plastic deformations
occurring at the inter-granular contacts on the strong force chains, that are
bearing the bulk of the applied loads. Collins also suggested that the {locked-in} elastic
energy is produced in the weak force networks, where the local stresses are not
large enough to produce plastic deformation at the grain contacts. The plastic
strains can be associated with the irreversible rearrangement of the particles,
whilst the elastic energy arises from the elastic compression of the particle
contacts. Part of this elastic strain energy will be released during unloading,
however other part of this energy will be trapped as a result of the irreversible
changes in the particle configuration.
% This idea is best illustrated using
% Discrete Element Method (DEM), where granular assemblies are modeled
% with discrete elements and contacts with elastic-plastic contact models.
%More detailed explanations of DEM and its applications can be found in the
%papers by \cite{Cundall1979}, \cite{Radjai1996},
%\cite{Thornton2000}, \cite{Yang2015}, and \cite{Yang2016}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Plastic Dissipation}
As pointed out, plastic work and energy dissipation are not the same physical
quantity. The confusion of these two
concepts often leads to incorrect results and conclusions, especially in seismic
energy dissipation analysis. Of major concern in this paper is the computation
of plastic dissipation, as elaborated in this section.
With the decoupling assumption (Equation~\ref{Eq5}), the second law of
thermodynamics (positive entropy production) directly leads to the dissipation
inequality, which states that the energy dissipated due to the difference of the
plastic work rate and the rate of the plastic part of the free energy must be
non-negative:
%
\begin{equation}
\Phi =
\sigma_{ij} \dot{\epsilon}^{pl}_{ij} - \dot{\Psi}_{pl} =
\sigma_{ij} \dot{\epsilon}^{pl}_{ij} - \rho \dot{\psi}_{pl} \ge 0
\label{Eq8}
\end{equation}
%
where $\dot{\psi}_{pl}$ is the rate of plastic free energy, per unit volume, and
$\rho$ is the density. In addition, $\psi_{pl}$ denotes plastic free energy density,
which is generally not constant at different locations in a body. This expression is closer
to physics and makes it convenient for further derivations.
Now we proceed to consider how to calculate plastic free energy, which can then
be used to calculate dissipation. According to
\citet{Feigenbaum2007}, plastic free energy density $\psi_{pl}$ is
assumed to be additively decomposed into parts which correspond to the
isotropic, kinematic and distortional hardening mechanisms as follows:
%
\begin{equation}
\psi_{pl} =
\psi_{pl}^{iso} +
\psi_{pl}^{ani};
\quad \psi_{pl}^{ani} =
\psi_{pl}^{kin} - \psi_{pl}^{dis}
\end{equation}
%
where $\psi_{pl}^{iso}$, $\psi_{pl}^{ani}$, $\psi_{pl}^{kin}$, and
$\psi_{pl}^{dis}$ are the isotropic, anisotropic, kinematic, and distortional
parts of the plastic free energy, respectively. The anisotropic part is assumed to decompose
into kinematic and distortional parts, which correspond to different hardening
models. The subtraction, instead of addition, of $\psi_{pl}^{dis}$ from
$\psi_{pl}^{kin}$, to obtain the overall anisotropic part $\psi_{pl}^{ani}$ of the
plastic free energy, is a new concept proposed by
\citet{Feigenbaum2007}. This expression can better fit experimental
data, as well as satisfy the plausible expectations for a limitation of
anisotropy development.
As pointed out by \citet{Dafalias2002}, the thermodynamic
conjugates to each of the internal variables exist and each part of the
plastic free energy can be assumed to be only a function of these
conjugates. The explicit expressions for the isotropic and kinematic components
of the plastic free energy are:
%
\begin{equation}
\psi_{pl}^{iso} =
\psi_{pl}^{iso}(\bar{k}) =
\frac{\kappa_1}{2 \rho} \bar{k^2};
\quad \psi_{pl}^{kin} =
\psi_{pl}^{kin}(\bar{\alpha}_{ij}) =
\frac{a_1}{2 \rho} \bar{\alpha}_{ij} \bar{\alpha}_{ij}
\label{Eq10}
\end{equation}
%
where $\bar{k}$ and $\bar{\alpha}_{ij}$ are the
thermodynamic conjugates to $k$ (size of the yield surface) and $\alpha_{ij}$
(deviatoric back stress tensor representing the center of the yield surface), respectively.
Material constants $\kappa_1$ and $a_1$ are non-negative material constants whose
values depend on the choice of elastic-plastic material models.
According to definition, the thermodynamic conjugates are related to the
corresponding internal variables by:
%
\begin{equation}
k =
\rho\frac{\partial \psi_{pl}^{iso}}{\partial \bar{k}} =
\kappa_1 \bar{k};
%
\quad \alpha_{ij} =
\rho\frac{\partial \psi_{pl}^{kin}}{\partial \bar{\alpha}_{ij}} =
a_1 \bar{\alpha}_{ij}
\label{Eq11}
\end{equation}
%
By substituting Equation~\ref{Eq11} back into Equation~\ref{Eq10}, the plastic free
energy can be expressed in terms of the internal variables:
%
\begin{equation}
\psi_{pl}^{iso} =
\frac{1}{2 \rho \kappa_1} k^2;
%
\quad \psi_{pl}^{kin} =
\frac{1}{2 \rho a_1} \alpha_{ij} \alpha_{ij}
\label{Eq12}
\end{equation}
With Equation~\ref{Eq12}, the components of plastic free energy can be computed,
as long as the internal variables are provided.
Combining Equation~\ref{Eq8} with \ref{Eq12}, the plastic dissipation in a given
elastic-plastic material can be accurately obtained at any location, at any time.
This approach allows engineers and designers to correctly identify energy
dissipation in time and space and make appropriate conclusions on material behavior.
% In the above presented methodology, the important role
% of plastic free energy is recognized and applied to numerical modeling studies.
% %Compared to earlier energy analysis approaches, one major advantage is that the
% Using this approach, plastic dissipation of an SSI model can be obtained at any
% location, at any time during simulation.
%
%This allows us to identify components with high energy
%dissipation, which indicates significant deformation or damage, from a large
%numerical model. Based on this information, structures can be better designed
%against seismic or other types of dynamic loading.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Energy Computation in Finite Elements}
Formulations from the previous section are applied to FEM analysis in order
to follow energy dissipation.
%
%Since the intent is to analyze the location of energy dissipation in large
%Earthquake Soil Structure Interaction (ESSI) models,
Energy density is chosen as the physical parameter for energy analysis.
%
Energy density in this study is defined as the amount of energy stored in a
given region of space per unit volume.
For FEM simulations, both external forces and displacements can be prescribed.
The finite element program accepts either (or both) forces and/or displacements as input and
solves for the other.
Either way, the {rate of input work} can be calculated by simply
multiplying force and displacement within a time step. Therefor {input
work} of a finite element model is:
%
\begin{equation}
W_{Input}(t) =
\int_{0}^{t} \dot W_{Input}(T) dT =
\int_{0}^{t} \sum_{i} F_{i}^{ex}(\boldsymbol{x},T) \dot{u}_{i}(\boldsymbol{x},T) dT
\label{Eq_input_work}
\end{equation}
%
where $F_i^{ex}$ is the external force and $u_i$ is the displacement computed at
the location of the applied load, at given time step, for a load controlled analysis.
The external load can have many forms, including nodal loads, surface loads,
and body loads. All of them are ultimately transformed into nodal forces. As shown
in Equation~\ref{Eq_input_work}, input
work is computed incrementally at each time step, in order to obtain the
evolution of total input work at certain time.
As shown in Figure~\ref{energy_classification}, when loads and/or
displacements are introduced
into a finite element model, the input energy will be converted in a number of
different forms as it propagates
through the system.
%
Input energy will be converted into {kinetic
energy}, {free energy}, and {dissipation}. As mentioned before,
free energy can be further separated into elastic part, which is traditionally
defined as strain energy, and plastic part, which is defined as the plastic free
energy. Kinetic energy and strain energy can be considered as the recoverable
portion of the total energy since they are transforming from one to
another. Plastic free energy is more complicated in
the sense that it is conditionally recoverable during reverse loading, as has
been discussed in detail in previous sections. Other than kinetic energy and
free energy, the rest of the input energy is dissipated, transformed
into heat or other forms of energy that are irrecoverable.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.7\columnwidth]{Figures/Energy_Transformation.pdf}
\caption{\label{energy_classification}
% FIGURE TO BE CHANGED SO THAT IS A REGULAR FINITE ELEMENT, WITH INPUT LOADS
% AND/OR DISPLACEMENTS, AND THEN THE REST (DISSIPATION) IS THE SAME...
%
Different forms of energy in a dynamic soil-structure system.}
\end{figure}
Calculation of kinetic energy and strain energy is rather straight
forward:
%
\begin{equation}
U_K(\boldsymbol{x},t) =
\frac{1}{2} \rho \dot{u}_{ij}(\boldsymbol{x},t) \dot{u}_{ij}(\boldsymbol{x},t)
\label{Eq_kinetic_energy_density}
\end{equation}
\begin{equation}
U_S(\boldsymbol{x},t) =
\int_{0}^{t} \dot {U_S}(\boldsymbol{x},T) dT =
\int_{0}^{t} \sigma_{ij}(\boldsymbol{x},T) \dot \epsilon_{ij}^{el}(\boldsymbol{x},T) dT
\label{Eq_strain_energy_density}
\end{equation}
%
where $U_K$ and $U_S$ are the {kinetic energy density} and {strain energy
density}, respectively.
%
%
Similar to the input energy, strain energy density and plastic free energy are
also computed incrementally. Integrating energy density over the entire model,
corresponding energy quantities are expressed as:
%
\begin{eqnarray}
E_K(t) = \int_V U_K(\boldsymbol{x},t) dV
\label{EK_kinetic_energy}
\end{eqnarray}
\begin{eqnarray}
E_S(t) = \int_V U_S(\boldsymbol{x},t) dV
\label{ES_strain_energy}
\end{eqnarray}
\begin{eqnarray}
E_{P}(t) = \int_V \Psi_{pl}(\boldsymbol{x}, t) dV
\label{EP_plastic_free_energy}
\end{eqnarray}
where $E_K$, $E_S$, and $E_P$ are the kinetic energy, strain energy, and plastic
free energy of the entire model, respectively. Energy densities, defined in
Equations~\ref{Eq_kinetic_energy_density} and \ref{Eq_strain_energy_density} are functions of
both time and space, while energy components, defined in the above equations
(Equation~\ref{EK_kinetic_energy}, \ref{ES_strain_energy}, and \ref{EP_plastic_free_energy}) are only functions of time, since
they are integrated over the whole model.
%
% It is more intuitive to compare
% results between different models by the total energy, since it is one scalar
% quantity. However, in order to understand the distribution of energy inside a
% model, it is necessary to analyze energy density.
Although the plastic free energy is conditionally recoverable, it is still
considered to be stored in the system, rather than dissipated. Summing up all
the stored energy $E_{Stored}$, one obtains:
%
\begin{equation}
E_{Stored} = E_K + E_S + E_P
\end{equation}
Rate of plastic dissipation, given by Equation~\ref{Eq8}, can be integrated over
time and space:
%
\begin{equation}
D_P(t) = \int_V \int_{0}^{t} \Phi(\boldsymbol{x},T) dT dV
\end{equation}
%
where $D_P$ is the dissipation due to plasticity of the entire model at certain
time.
Finally the energy balance of a finite element model is given by:
%
\begin{equation}
W_{Input} = E_{Stored} + D_P = E_K + E_S + E_P + D_P
\end{equation}
%Another important issue is how to choose the location for energy density calculation, which is originated from FEM. As shown in Figure~\ref{energy_density_calculation}, in this case we choose to use the Gauss points. Two aspects are considered in this decision. Firstly, we have the solution for stress and strain on the Gauss points, which makes it straight forward to calculate the strain energy density of this location. Secondly, in our 1D beam case with 8-node-brick elements, displacements at Gauss points can be linearly interpolated from nodal displacements, which we already calculated and stored in output files. Then we just need to take its derivative to get velocity, thus the kinetic energy density.
%\begin{figure}
% \centering
% \includegraphics[width=0.5\columnwidth]{energy_density_calculation.pdf}
% \caption{\label{energy_density_calculation}
% Position to calculate energy density components in 1D case with 8-node-brick elements}
%\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Studies}
Numerical simulation results presented in this paper are performed using
the Real ESSI (Real Earthquake Soil Structure Interaction)
Simulator \cite{Real_ESSI_Simulator}.
% Real ESSI Simulator is a software,
% hardware and documentation system for high fidelity, high performance, time
% domain, nonlinear, 3D, finite element modeling and simulation of
% earthquake-soil/rock-structure interaction of Infrastructure Objects (Nuclear
% Facilities, Nuclear Power Plants, Dams, Bridges, Buildings). Real ESSI contains
% options for various materials, elements, constrains, loads, and computing
% algorithms, especially newly developed constitutive models for geotechnical and
% structural components. These features make Real ESSI highly effective on seismic
% energy dissipation studies, which is important yet rare to exist in many classic
% seismic numerical simulation frameworks. The proposed methodology is implemented
% and verified in the Real ESSI Simulator.
%
Examples in this paper focus on constitutive behavior of elastic-plastic
material from the perspective of energy dissipation. All cases are modeled with
solid brick elements, using static, load control analysis.
% Static algorithm is employed
%to eliminate the influence of inertial forces and dynamic reactions, which will
%be the focus of our following papers.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Elastic Material}
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.3\columnwidth]{Figures/Model_Single_Block.pdf}}
\subfloat[]{\includegraphics[width=0.6\columnwidth]{Figures/Model_Cantilever.pdf}}
\caption{\label{figure_models}
Numerical models used in this paper: (a) Single brick element; (b) Cantilever with 10 brick elements.}
\end{figure}
Initial investigation of energy dissipation is focused on linear elastic material.
%
It is noted that linear elastic material does not dissipate energy.
%
However, use of linear elastic material model is suitable for preliminary
verification of the newly developed energy analysis methodology. In this section,
energy balance in a single brick element and a cantilever beam is studied,
as shown in Figure~\ref{figure_models}.
The simplest case is a single element model under uniform shear load. The model is
constrained appropriately to simulate simple shear test. In order to show the
influence of different material parameters and loads, a set of simulations are
performed and the results are presented in
Table~\ref{table_linear_elastic_single} and Figure~\ref{figure_linear_elastic}.
\begin{table}[htb]
\centering
\caption{Energy analysis results for linear elastic materials (single element).}
\label{table_linear_elastic_single}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|cc|cc|cccc|c}
\hline
\multirow{2}{*}{Case} & \multicolumn{2}{c|}{Material Property} & \multicolumn{7}{c}{Simulation Results} \\ \cline{2-10}
& $E$ (GPa) & $\nu$ & $u$ (m) & \boldmath$W_{Input}$ (J) & $E_{K}$ (J) & $E_{S}$ (J) & $E_{P}$ (J) & \boldmath$E_{Stored}$ (J) & \boldmath$D_P$ (J) \\
\hline
1a & 100 & 0.30 & 2.60E-5 & 13.00 & 0.00 & 13.00 & 0.00 & 13.00 & 0.00 \\
1b & 150 & 0.30 & 1.73E-5 & 8.67 & 0.00 & 8.67 & 0.00 & 8.67 & 0.00 \\
1 & 200 & 0.30 & 1.30E-5 & 6.50 & 0.00 & 6.50 & 0.00 & 6.50 & 0.00 \\
1c & 250 & 0.30 & 1.04E-5 & 5.20 & 0.00 & 5.20 & 0.00 & 5.20 & 0.00 \\
1d & 300 & 0.30 & 8.67E-6 & 4.33 & 0.00 & 4.33 & 0.00 & 4.33 & 0.00 \\
\hline
1e & 200 & 0.20 & 1.20E-5 & 6.00 & 0.00 & 6.00 & 0.00 & 6.00 & 0.00 \\
1f & 200 & 0.25 & 1.25E-5 & 6.25 & 0.00 & 6.25 & 0.00 & 6.25 & 0.00 \\
1 & 200 & 0.30 & 1.30E-5 & 6.50 & 0.00 & 6.50 & 0.00 & 6.50 & 0.00 \\
1g & 200 & 0.35 & 1.35E-5 & 6.75 & 0.00 & 6.75 & 0.00 & 6.75 & 0.00 \\
1h & 200 & 0.40 & 1.40E-5 & 7.00 & 0.00 & 7.00 & 0.00 & 7.00 & 0.00 \\
\hline
\end{tabular}
}
\end{table}
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.45\columnwidth]{Figures/Elastic_E.pdf}}
\subfloat[]{\includegraphics[width=0.45\columnwidth]{Figures/Elastic_Poisson.pdf}}
\caption{\label{figure_linear_elastic}
Relationships between energy storage and different simulation parameters
(single element model): (a) Young's modulus; (b) Poisson's ratio.}
\end{figure}
Since linear elastic material is used with static algorithm,
energy components related to dynamics (kinetic energy) and plasticity (plastic
free energy and plastic dissipation) are equal to zero. This means that all input
work is stored in the system, as observed in all cases.
%The match
%between input work and the summation of stored energy and dissipation, which is
%zero in this case, verifies the reliability of the presented energy analysis
%methodology in single element cases.
% HAN: is is inversely proportional to Poisson or directly proportional (as per
% equation
%
% Han's comment:
% Should be proportional to Poisson's ratio. Corrected.
Figure~\ref{figure_linear_elastic} shows that energy stored in the system is
inversely proportional to Young's moduli and proportional to Poisson's ratio. This is
expected because of the following equations for strain energy under
static shear loading:
%
\begin{equation}
E_S = \frac{1}{2} \tau \gamma = \frac{1}{2G} \tau^2 = \frac{1+\nu}{E} \tau^2
\label{equation_20}
\end{equation}
Note that these relationships are only valid at constitutive level. For models with
more finite elements, stress and strain are generally not uniform.
The computation of energy depends on the distribution of energy density, and nonuniform stress/strain
distribution will result in nonuniform energy density distribution.
% HAN: need to make a sketch/drawing of this model and of the simple block of
% material from the previous example, just so that readers can see what
% is going on
% Han's comment:
% Added two figures at the beginning of the section
In order to study the influence of simulation parameters in larger models,
another set of simulations with cantilever model (Figure~\ref{figure_models}b) are performed.
Vertical loads
are applied to the nodes of the free end. In this case, both shearing
and bending occurs, which means that in general a full 3D state of
stress and strain is present. The results are
presented in
Table~\ref{table_linear_elastic_cantilever} and Figure~\ref{figure_linear_elastic}. As
expected, energy behavior of cantilever is different than the
single-element/constitutive example.
\begin{table}[htb]
\centering
\caption{Energy analysis results for linear elastic materials (cantilever model).}
\label{table_linear_elastic_cantilever}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|cc|cc|cccc|c}
\hline
\multirow{2}{*}{Case} & \multicolumn{2}{c|}{Material Property} & \multicolumn{7}{c}{Simulation Results}
\\
\cline{2-10}
& $E$ (GPa) & $\nu$ & $u$ (m) & \boldmath$W_{Input}$ (J) & $E_{K}$ (J) & $E_{S}$ (J) & $E_{P}$ (J) & \boldmath$E_{Stored}$ (J) & \boldmath$D_P$ (J) \\
\hline
2a & 100 & 0.30 & 2.33E-3 & 116.57 & 0.00 & 116.57 & 0.00 & 116.57 & 0.00 \\
2b & 150 & 0.30 & 1.55E-3 & 77.71 & 0.00 & 77.71 & 0.00 & 77.71 & 0.00 \\
2 & 200 & 0.30 & 1.17E-3 & 58.28 & 0.00 & 58.28 & 0.00 & 58.28 & 0.00 \\
2c & 250 & 0.30 & 9.33E-4 & 46.63 & 0.00 & 46.63 & 0.00 & 46.63 & 0.00 \\
2d & 300 & 0.30 & 7.77E-4 & 38.86 & 0.00 & 38.86 & 0.00 & 38.86 & 0.00 \\
\hline
2e & 200 & 0.20 & 1.20E-5 & 65.89 & 0.00 & 65.89 & 0.00 & 65.89 & 0.00 \\
2f & 200 & 0.25 & 1.26E-3 & 62.97 & 0.00 & 62.97 & 0.00 & 62.97 & 0.00 \\
2 & 200 & 0.30 & 1.17E-3 & 58.28 & 0.00 & 58.28 & 0.00 & 58.28 & 0.00 \\
2g & 200 & 0.35 & 1.02E-3 & 51.17 & 0.00 & 51.17 & 0.00 & 51.17 & 0.00 \\
2h & 200 & 0.40 & 8.12E-4 & 40.60 & 0.00 & 40.60 & 0.00 & 40.60 & 0.00 \\
\hline
\end{tabular}
}
\end{table}
For all cases, the energy balance between input and storage is maintained, which
gives us confidence on the energy calculation methodology for elastic material.
According to results in Figure~\ref{figure_linear_elastic},
energy stored in the system is still inversely proportional to Young's modulus.
This is because the general equation for elastic strain energy density is:
%
\begin{equation}
E_S =
\frac{1}{2E}
\left( \sigma^2_{xx} + \sigma^2_{yy} + \sigma^2_{zz} +
2(1+\nu)(\sigma^2_{xy} + \sigma^2_{yz} + \sigma^2_{zx}) \right)
\label{equation_21}
\end{equation}
\noindent
So as long as all the elements have the same Young's modulus, the relationship
between stored energy and Young's modulus will remain valid.
%
% Due to the
% existence of bending (normal stresses), the curve of stored energy against
% Poisson's ratio is generally nonlinear for realistic models. Even more, the
% increase of Poisson's ratio can lead to a rise or fall of energy storage in
% different cases. The influence of loading is more complicated in models with
% multiple elements. Large amount of tests have shown that in a cantilever-like
% model, the plot of elastic strain energy stored in the system against vertical
% force on the free end is usually close to a quadratic curve.
%
%
%
%
% From two shown examples, is is apparent that even for
% linear elastic material, calculated energy can vary non-linearly.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Elastic-Perfectly Plastic Material}
% By performing energy analysis for linear elastic materials, the input work and
% storage computations are verified. The next step is to apply the analysis
% framework to elastic-plastic materials in order to analyze energy dissipation due
% to plasticity.
In this section, elastic-perfectly plastic material is used.
Equations~\ref{Eq8} and \ref{Eq12} indicate that in the case of no hardening the
rate of plastic free energy is zero. Then
the incremental plastic work is equal to incremental plastic dissipation. Note
that this is one of the rare cases where plastic dissipation equals to plastic work.
Figure~\ref{figure_elastic_perfectly_plastic} shows
stress--strain curve (left) and energy calculated for elastic-perfectly plastic
constitutive model (right) used here.
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Perfectly_Plastic_Stress-Strain.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Perfectly_Plastic_Energy.pdf}}
\caption{\label{figure_elastic_perfectly_plastic}
Energy analysis of elastic perfectly plastic material: (a) Stress--strain curve;
(b) Input work, plastic dissipation, strain energy and plastic work.}
\end{figure}
In this case, the plastic dissipation is equal to the
plastic work. This means that the plastic free energy does not
develop at all during loading and unloading. Zero plastic free energy
points out the absence of fabric evolution
of a particulate, elastic-plastic material, as all the input work is dissipated
through particle to particle friction.
%
Since there is no plastic free energy $E_P$ in this case, the stored energy
equals to
mechanical energy, which is the combination of
%
% HAN: always put variable names before their symbol!
%
% NAME THAM ALL!
%
strain energy $E_S$ and kinetic energy $E_K$. Total stored energy $E_{Stored}$
develops nonlinearly and always has the same value at the beginning of every
loop after the first one. Plastic dissipation $D_P$ increases linearly when the material yields.
This can be explained by rewriting
Equation~\ref{Eq8} with $\Psi_{pl}=0$:
\begin{equation}
\Phi = \sigma_{ij} \dot\epsilon^{pl}_{ij}
\end{equation}
%
where stress $\sigma_{ij}$ is constant after elastic perfectly plastic material yields,
and rate plastic deformation $\dot\epsilon^{pl}_{ij}$ is also constant. Then the rate
of plastic dissipation is constant which makes the plastic dissipation $D_P$ increase linearly.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Elastic-Plastic Material with Isotropic Hardening}
Next material model used is elastic-plastic with
linear isotropic hardening. First used to model monotonic behavior of elastic-plastic
materials, isotropic hardening assumes that the yield surface maintains shape, while isotropically
(proportionally) changing its size.
%
% Failed to
% model the Bauschinger effect by itself, isotropic hardening is usually used to
% derive more sophisticated elastic-plastic materials or applied in combination
% with other hardening rules. Either way, it is valuable to be analyzed from the
% perspective of energy dissipation.
%
%
Figure~\ref{figure_isotropic_hardening} illustrates the stress-strain
response as well as energy balance for elastic-plastic material with
isotropic hardening.
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/IH_Stress-Strain.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/IH_Energy.pdf}}
\caption{\label{figure_isotropic_hardening}
Energy analysis of elastic-plastic material with isotropic hardening: (a)
Stress-strain curve; (b) Input work, plastic dissipation, strain energy, and
plastic work.}
\end{figure}
As can be observed from Figure~\ref{figure_isotropic_hardening}, plastic free
energy is equal to the plastic work, which means that the plastic dissipation is
zero during cycles of loading. Even though this might sound surprising, it can
be explained using basic thermodynamics. Linear isotropic hardening,
used in this case, can be described through a rate of the internal variable
(size of the yield surface) $\dot{k}$ as:
%
\begin{equation}
\dot k = \kappa_1 | \dot \epsilon^{pl}_{ij} |
\end{equation}
%
where $| \dot \epsilon^{pl}_{ij} |$ is the magnitude of the rate of plastic
strain while $\kappa_{1}$ is a hardening constant. Substituting previous
equation into Equation~\ref{Eq12} yields:
%
% HAN: put names for variables, name \psi_{pl) and so on...
%
%
\begin{equation}
\psi_{pl}
=
\psi_{pl}^{iso}
=
\frac{\kappa_1}{2 \rho} \epsilon^{pl}_{ij} \epsilon^{pl}_{ij}
\end{equation}
Take the time derivative of the above equation:
%
\begin{equation}
\dot{\psi}_{pl}
=
\frac{\kappa_1}{\rho} \epsilon^{pl}_{ij} \dot{\epsilon}^{pl}_{ij}
\end{equation}
Then the rate of dissipation due to plasticity can be expressed as:
%
\begin{equation}
\Phi
=
\sigma_{ij} \epsilon^{pl}_{ij} - \rho \dot{\psi}_{pl}
=
(\sigma_{ij} - \kappa_1 \epsilon^{pl}_{ij}) \dot \epsilon^{pl}_{ij}
=
(\sigma_{ij} - k m_{ij}) \dot \epsilon^{pl}_{ij}
\end{equation}
%
where $m_{ij}$ is the plastic flow direction. The plastic flow direction
defines the direction of incremental plastic strain, which can be different from the
direction of total plastic strain. But in the case of von Mises type elastic-plastic material
with only isotropic hardening, whose yield surface is always a circle with center at the origin,
the plastic flow direction $m_{ij}$ is the same as the direction of the total plastic strain
$\epsilon^{pl}_{ij}$. Thus we have $\kappa_1 \epsilon^{pl}_{ij} = k m_{ij}$ in the above equation.
If we assume, for simplicity
sake, that plastic flow direction is associated with the yield function, that is
there is only deviatoric plastic flow, as yield function is of von Mises type,
the gradient of the yield surface $n_{ij} (=\partial F/ \partial \sigma_{ij})$
is equal to the
plastic flow direction $m_{ij} (=n_{ij})$. Noting that $\sigma_{ij} \dot
\epsilon^{pl}_{ij} = s_{ij} \dot \epsilon^{pl}_{ij}$, where $s_{ij} (=\sigma_{ij} - 1/3 \sigma_{kk} )$ is the
deviatoric part of the stress tensor, the rate of plastic dissipation can be
rewritten as:
%
\begin{equation}
\Phi = (s_{ij} - k n_{ij}) \dot \epsilon^{pl}_{ij} = \alpha_{ij} \dot \epsilon^{pl}_{ij}
\end{equation}
%
Realizing that the back stress $\alpha_{ij}$ is always zero since we assume no
kinematic hardening, then the rate of plastic dissipation becomes zero, which
means there is no energy dissipation during cycles of loading for isotropically hardening material.
Obviously, the observed response is not physical from the perspective of energy dissipation.
%
%
% Also, this happens not only during cyclic but monotonic loading as well, which
% is commonly considered suitable for isotropic hardening. Therefore, one should
% pay extra attention to energy computation when applying isotropic hardening
% rules.
%
% Therefore, isotropically hardening material models, can mimic/replicate material
% stress-strain behavior for monotonic loading.
%
% It is known that
% isotropic hardening material models cannot be used to properly model cyclic
% loading response, as they are unable to model
%
Therefore, isotropic hardening material models cannot properly model energy
dissipation, even for monotonic loading.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Elastic-Plastic Material with Kinematic Hardening}
Compared with isotropic hardening, kinematic hardening can better describe the
constitutive, stress-strain behavior of elastic-plastic materials, particularly
for cyclic loading.
%In order to model the Bauschinger effect, kinematic
%hardening assumes that the yield surface remains the same size and shape but
%translates or rotates in stress space.
Elastic-plastic material that relies on kinematic
hardening is used to analyze energy dissipation.
Both linear and nonlinear kinematic hardening rules are
investigated in relation to energy dissipation.
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\subsubsection{Prager Linear Kinematic Hardening}
Prager's linear kinematic hardening rule is given as:
%
\begin{equation}
\dot \alpha_{ij} = a_1 \dot{\epsilon}^{pl}_{ij}
\label{Eq16}
\end{equation}
%
where $a_1$ is a hardening constant.
%
If only linear kinematic hardening (Equation~\ref{Eq16}) is assumed, the back
stress $\alpha_{ij}$ is expressed explicitly, and can be substituted into
Equation~\ref{Eq12} yielding:
%
\begin{equation}
\psi_{pl} = \psi_{pl}^{kin} = \frac{a_1}{2 \rho} \epsilon^{pl}_{ij} \epsilon^{pl}_{ij}
\end{equation}
Take the time derivative of the above equation:
\begin{equation}
\dot{\psi}_{pl} = \frac{a_1}{\rho} \epsilon^{pl}_{ij} \dot{\epsilon}^{pl}_{ij}
\label{Eq32}
\end{equation}
If we again assume that the gradient of the yield surface $n_{ij}$
is equal to the plastic flow direction $m_{ij}$, as in the case of linear isotropic hardening,
then the rate of dissipation due to plasticity can be rewritten as:
%
\begin{equation}
\Phi
=
\sigma_{ij} \dot{\epsilon}^{pl}_{ij} - \rho \dot{\psi}_{pl}
=
(s_{ij} - \alpha_{ij}) \dot\epsilon^{pl}_{ij}
=
k m_{ij} \dot\epsilon^{pl}_{ij}
\label{Eq33}
\end{equation}
%
Notice that the term $m_{ij} \dot\epsilon^{pl}_{ij}$ denotes the
magnitude of the rate of plastic strain. Since only linear kinematic hardening is assumed,
the internal variable $k$ will remain constant.
%
% HAN, what do you mean if load are applied in a certain way
%
% It was bad expression, I see you changed this already... This is what I intended to say...
%
So if loads are applied in such a way
that the rate of plastic strain is constant, then the
rate of dissipation will also remain constant. In other words, the accumulated
dissipation will be linearly increasing under the assumption of linear kinematic
hardening.
Figure~\ref{figure_linear_kinematic_hardening} shows stress--strain response (left) and energy computation
results (right) of an elastic-plastic material with linear kinematic hardening.
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Linear_KH_Stress-Strain.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/Linear_KH_Energy.pdf}}
\caption{\label{figure_linear_kinematic_hardening}
Energy analysis of elastic-plastic material with linear kinematic
hardening: (a) Stress--strain curve; (b) Input work, plastic dissipation
strain energy and plastic work.}
\end{figure}
%
% HAN: below, this 50% is somewhat ad hoc, that is there is nothing significant
% about 50%, or any other number, as they are influenced by material parameters...
% Agree. Changing 50% into other expressions
%
As expected, the plastic dissipation increases linearly once the material
yields. In contrast to the isotropic hardening case, a significant amount of the input
work is dissipated due to material plasticity. The ratio of dissipated energy to input work is largely influenced by the material parameters. However, in general, energy dissipation will be
observed if kinematic hardening model is used.
% From the perspective of
% thermodynamics, this is physically reasonable, which provides support to the
% point that kinematic hardening is more realistic than isotropic hardening.
Another important observation is that the plastic work decreases during certain
phases of reverse loading, while the actual rate of energy dissipation is
always nonnegative. It is important not to confuses the definitions of plastic work to plastic
dissipation, as plastic work can increase and decrease which can lead to a
(impossible) conclusion, violating the second law of thermodynamics.
% And more importantly, all the energy analysis will be invalid in such
%circumstances. Unfortunately, this is a fairly common oversight among many
%papers on seismic energy dissipation.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Armstrong-Frederick Kinematic Hardening}
Armstrong-Frederick kinematic hardening model \cite{Armstrong66} is often used
to simulate elastic-plastic material behavior under cyclic loading.
%Its nonlinear feature makes the
%constitutive models better describe the behaviors of realistic materials.
%The
%material parameters are usually defined differently among various models.
Material parameters of the Armstrong Frederic kinematic hardening rule can be
derived from basic thermodynamics.
%model To
%clarify these parameters, it is beneficial to derive the Armstrong-Frederick
%kinematic hardening rule from thermodynamics, which has been the base for all
%formulations in this paper.
\citet{Feigenbaum2007} gave the sufficient (but not necessary)
conditions of the inequality in Equation~\ref{Eq8}. These conditions are more
restrictive but satisfy the frameworks of most classical elastic-plastic models.
One of the sufficient conditions corresponds to kinematic hardening is
expressed as:
%
%
% HAN: perhaps expand a bit this part as that equation (AF) pops out of blue skies!
% In fact, you need to show von Mises yield surface F, all of then, isotropic and kinematic hardening...).
% this one will become F=(s_ij-alpha_ij)(s_ij-alpha_ij) - k ...
% I rewrote this part. The previous discussion focuses on how to derive hardening
% equations from thermodynamics, but that's not our purpose. Now I just cite the equation
% from Feigenbaum and Dafalias (2007) to give the expression of AF hardening. Then
% add a little further derivation to point out that the energy dissipation in
% this case is non-negative and nonlinear.
%
\begin{equation}
\alpha_{ij} (\dot{\epsilon}^{pl}_{ij} - \frac{1}{a_1} \dot{\alpha}_{ij}) \ge 0
\label{Eq14}
\end{equation}
%
%
The following relation is a sufficient condition to satisfy Equation~\ref{Eq14}:
%
%
\begin{equation}
\dot{\alpha}_{ij} = a_1 \dot{\epsilon}^{pl}_{ij} - a_2 \dot{\lambda} \alpha_{ij}
\label{Eq15}
\end{equation}
%
where $\dot{\lambda}$ is a non-negative scalar plastic multiplier and
$a_2$ is a non-negative material hardening constant. It can be proven that
$a_1/a_2$ is related to the limit of back stress magnitude $\left| \alpha_{ij
}\right|$.
%Note that during the derivation of Equation~\ref{Eq14}, it is assumed
%to be associated plasticity.
Equation~\ref{Eq15} should be recognized as the
classical Armstrong-Frederick nonlinear kinematic hardening.
Taking the time derivative of the kinematic part of plastic free energy
(Equation~\ref{Eq32}), and substituting the expression of back stress $\alpha_{ij}$
(Equation~\ref{Eq15}) gives:
%
\begin{equation}
\dot{\psi}_{pl}^{kin}
=
\frac{1}{\rho a_1} \alpha_{ij} \dot{\alpha}_{ij}
=
\frac{1}{\rho} \alpha_{ij} (\dot{\epsilon}^{pl}_{ij} - \frac{a_2}{a_1} \dot{\lambda} \alpha_{ij})
\end{equation}
If the gradient of the yield surface $n_{ij}$ is assumed to be
equal to the plastic flow direction $m_{ij}$, as was done in previous sections, then the rate of plastic energy dissipation of an Armstrong-Frederick kinematic hardening
elastic-plastic material is given by:
%
\begin{equation}
\Phi
=
\sigma_{ij} \dot{\epsilon}^{pl}_{ij} - \rho \dot{\psi}_{pl}
=
s_{ij} \dot\epsilon^{pl}_{ij} - \alpha_{ij} \dot\epsilon^{pl}_{ij} + \frac{a_2}{a_1} \dot{\lambda} \alpha_{ij} \alpha_{ij}
=
k m_{ij} \dot\epsilon^{pl}_{ij} + \frac{a_2}{a_1} \dot{\lambda} \alpha_{ij} \alpha_{ij}
\end{equation}
Compared with Equation~\ref{Eq33}, the above expression has an additional term
which makes the rate of plastic dissipation non-constant even if the rate of
plastic strain is constant. As the back stress $\alpha_{ij}$ becomes larger when
load increases, the rate of plastic dissipation also increases. This indicates a
nonlinear result of total plastic dissipation, which is exactly what we have
observed in our computations.
% In previously described general developments, until Equation~\ref{Eq12}, no plastic flow
% direction were assumed.
%so the above
%expression are considered general.
% Assuming associated elastic-plasticity, plastic flow rule can be expressed as:
% plasticity, plastic will be
%%considered, which is typical with metal. The plastic flow rule going to be used
%is expressed as:
%
% \begin{equation}
% \dot{\epsilon}^{pl}_{ij} = \dot{\lambda} \frac{\partial f}{\partial \sigma_{ij}} = \dot{\lambda} m_{ij}
% \label{Eq13}
% \end{equation}
%
% where $\dot{\lambda}$ is a non-negative scalar plastic multiplier and $m_{ij}$
% denotes plastic flow direction.
% Substituting Equation~\ref{Eq12} and \ref{Eq13} into \ref{Eq8}, one rewrites the
% inequality in Equation~\ref{Eq8} with decoupled components correspond to
% different hardening mechanisms.
% of the evanescence
%memory type.
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/AF_KH_Stress-Strain.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Figures/AF_KH_Energy.pdf}}
\caption{\label{figure_AF_kinematic_hardening}
Energy analysis of elastic-plastic material with Armstrong-Frederick kinematic
hardening: (a) Stress-strain diagram; (b) Input work, plastic dissipation,
strain energy, and plastic work.}
\end{figure}
Figure~\ref{figure_AF_kinematic_hardening} shows the energy computation results
of an elastic-plastic material with Armstrong-Frederick kinematic hardening.
Compared to all previous cases, the material response of this model is more
sophisticated and more realistic.
%Up to 60\% of the input energy is
%dissipated due to plasticity in each cycle.
Decrease of plastic work is
observed, again, while the plastic dissipation is always nonnegative during the entire
simulation. For both linear and nonlinear kinematic hardening cases, the plastic
free energy is relatively small compared to the plastic dissipation.
%It can be
%easily ignored in SSI simulations under seismic loading, since the number of
%load cycles is large and the response is wrong. Nevertheless, the
%plastic free energy should never be neglected, in the sense of satisfying the
%basic laws of thermodynamics.
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\section{Conclusions}
%OVDE
Presented was a methodology for (correct) computation of energy dissipation in
elastic-plastic materials based on the second law of thermodynamics.
%
A very important role of plastic free energy was analyzed, with highlights on
its physical nature and theoretical formulations. The proposed methodology has
been illustrated using a number of elasto-plastic material models.
An analysis of a common misconception that equates plastic work and dissipation,
which leads to the
violation of the basic principles of thermodynamics, was addressed.
%The
%introduction of plastic free energy has been proved to be effective on
%eliminating such violation.
%
A conceptual example, for granular materials, was used to explain the
physical meaning of plastic free energy. It was also shown that plastic free
energy is responsible for the evolution of internal variables.
It was shown that energy balance is ensured by taking into consideration all
energy components, including kinetic and strain energy. Input work was
balanced with the stored and dissipated energy, expressed as the summation of
all possible components.
Presented approach was illustrated and tested using several
elastic-plastic constitutive models with various hardening rules.
%
Elastic materials showed no energy dissipation (as expected), leading to the
input work being equal to the stored energy.
%
Elastic-perfectly plastic materials had no change in
plastic free energy, which led to the equality of plastic work and plastic
dissipation and indicated no evolution of particle arrangements. The
plastic dissipation, in that case, was observed to be increasing linearly.
Isotropic hardening materials experienced zero dissipation even after
yielding. This observation was surprising, but verified by further derivation
of energy equations. This observation also serves as a reminder that the
isotropic hardening rules can be used, but only with observed lack of energy
dissipation. Prager's linear and Armstrong-Frederick nonlinear kinematic
hardening materials both gave significant dissipations, with large fluctuation
of plastic free energy as well. In the case with linear kinematic hardening,
linear increase of dissipation was derived and observed, while energy was
dissipated nonlinearly in the case of nonlinear kinematic hardening. Although
the plastic free energy was
not significant for some materials, it is noted that it should always be
recognized and
considered during energy analysis, so that the basic principles of
thermodynamics are maintained.
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\section{Acknowledgments}
This work was supported by the US-DOE. We would like to thank Professor
Yannis Dafalias (UCD and NTUA) for inspiring discussions.
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\bibliography{refmech}
\bibliographystyle{abbrvnat}
\end{document}