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\begin{document}

\title{Energy Dissipation in Solids due to Material Inelasticity, Viscous Coupling, and Algorithmic Damping}

\author[1]{Han Yang}
\author[2]{Hexiang Wang}
\author[3]{Yuan Feng}
\author[4]{Fangbo Wang, Ph.D.}
% \author[2]{David B McCallen, Ph.D.}
\author[5]{Boris Jeremi{\'c}, Ph.D.}

\affil[1]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
\affil[2]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
\affil[3]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
\affil[4]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
\affil[5]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
%\affil[]{Department of Civil and Environmental Engineering, University of California, Davis, CA, USA.}
%\affil[2]{Earth Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, USA.}
%\affil[ ]{Email: jeremic@ucdavis.edu}

\maketitle


\begin{abstract}
Presented  is  a study on energy dissipation in dynamic inelastic systems due to
material inelasticity, viscous damping, and algorithmic damping.
%
Formulation    for    plastic    dissipation  is based  on  thermodynamics, with
consideration of plastic free energy.
%
Computation  of  viscous energy dissipation of the Rayleigh type is developed and discussed.
%
Energy dissipation due to algorithmic damping is discussed as well, and compared
with previous two, physical energy dissipation mechanisms.
%
Energy  dissipation due to all three dissipation  mechanisms  are  illustrated  and 
discussed  in  relation  to  single  element  tests and dynamic wave propagation
problems.
\end{abstract}


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% \tableofcontents

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% \newpage
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\section{Introduction}

Numerical simulation of inelastic dynamic solid systems has three mechanisms
of  energy  dissipation:
%
\begin{itemize}
\setlength\itemsep{0pt}
\item Material  inelasticity, hysteretic damping,
%
\item Viscous coupling between solid and fluid (external or internal),
%
\item Algorithmic, numerical damping.
%
\end{itemize}
%
The   first two energy dissipation  mechanisms have    physical interpretations,
while  the  last  mechanism is  primarily  due to numerical integration of dynamic
equations of motion.



Energy  dissipation  from inelasticity and viscous effects is physically correct
if properly modeled and calibrated.
%
Energy  dissipation  can  also  be  used  to  improve  performance  and optimize
engineering design.
%
Proper  modeling of different energy dissipation mechanics provides insight into
realistic response of engineering solids.
%
Substituting  one  mechanism of energy dissipation with another one  might  not 
provide proper insight into realistic response of engineering solids.

The  mechanical energy that irreversibly transforms into heat in inelastic 
materials is defined as the energy dissipation due to material inelasticity,
or simply referred to as plastic energy dissipation.
%
This    energy dissipation mechanism is dominant for any significant static
or dynamic loading, that leads to yielding and material damage.
%
Plastic energy dissipation is displacement proportional \cite{local-87}.



The  proper modeling of plastic energy dissipation in inelastic solids was
presented earlier \cite{Yang2017a}.
%
The  difference  between  plastic energy dissipation and plastic work, which was
first  described by \citeN{Farren1925}, was addressed
with  focus  on  the  accurate  computation  of  plastic free energy and plastic
dissipation in different inelastic material models.
%
Plastic energy  dissipation modeling and experiments are discussed in a number of recent
papers   \cite{Mason1994,Collins97,Rittel2000a,Rittel2000b,Rosakis2000,Collins2002b,Feigenbaum2007,Veveakis2012}.


The   dissipative  interaction  between  solid  and  viscous  fluid, internal in
pores  of porous solid, or external to solids and structures, can also dissipate
significant amount of mechanical energy.
%
This type of dissipation is referred to as viscous energy dissipation or viscous
damping.
%
Viscous energy dissipation is velocity  proportional \cite{local-87}.
%
Modeling  of  viscous damping without explicitly taking into account interacting
solid/structure and fluid is usually done using Caughey damping \cite{Caughey1960}.
%
Special  case  of  Caughey damping is Rayleigh damping, where damping matrix is
obtained by a linear combination of the mass and stiffness matrices.
%
It  was  pointed  out  by  \citeN{Hall2006} that  the  damping  forces obtained
from Rayleigh damping can be unrealistically high.
%
This makes the analysis result nonconservative, and require careful calibration of 
damping parameters.



Sometimes high  level  of Rayleigh damping is used to mimic inelastic material behavior.
%
In  other  words,  material  nonlinear/inelastic behavior is modeled using linear
elastic material and non-physical, high level of Rayleigh damping.
%
It  will be shown that energy dissipation using plastic dissipation and Rayleigh
viscous  damping  lead  to  differences  in  response,  particularly  when 
significant material nonlinear/inelastic behavior is present.


Algorithmic  damping, or numerical damping, is introduced into the system during
time  integration  of  equations  of motion, by most commonly used time marching
algorithms, for example the Newmark family algorithms \cite{Newmark1959,Hilber1977,Chung1993}.
%
These    time    integration   methods can be made energy conserving with proper
choice of integration parameters, for example by choosing $\gamma=0.500$ and $\beta=0.250$ for Newmark algorithm.


However, recent  studies  have  shown that the Newmark family of algorithm might
lose its energy conserving
properties   in   some   cases,   for  example,  geometric  nonlinear  problems,
impact/contact  problems,  and  large  time  step  simulations  \cite{Krenk2006,Krenk2014}.
%
Efforts  have  been  made  to  develop  various types of energy conserving time
integration algorithms \cite{Simo1991,Bathe2007,Gonzalez2000,Krenk2014}.
%
In  this  study,  the classic Newmark time integration algorithm is used, and no
energy  conserving  issue  mentioned earlier is observed in any of the presented
cases.
%
System  energy  conservation is in fact maintained in all cases when the Newmark
parameter $\gamma=0.500$ and $\beta=0.250$ .
%
For  any  choice  of  $\gamma > 0.500$, Newmark family algorithms will dissipate, or
produce     energy     in     the     elastic    or    inelastic    system
\cite{local-87,local-86}.
%
This algorithmic energy dissipation or production can have significant effect on
results of modeling and simulation of a dynamic system.



It  is  noted  that energy dissipation due to plastic dissipation and/or viscous
damping  and/or  algorithmic  damping  is  inherent  to loading of solids and
structures.
%
Plastic        dissipation     is   always present for yielding material, and is
calculated  on  the  constitutive  level  for  any type of loading, monotonic or
cyclic, static or transient.
%
Viscous    damping   is present for transient loading only, monotonic or cyclic,
and is calculated on a single finite element level.
%
Algorithmic  damping is present for transient loading only, monotonic or cyclic,
and  can  be  calculated for  a single finite element or finite element system
level.
%
Presented  framework allows for detailed energy dissipation calculations for all
of the above mechanisms.





The  next  section  presents the theoretical formulations used to compute energy
dissipation  due to material inelasticity, Rayleigh damping, and algorithmic
damping.
%
Next,  a  number of numerical examples are used to illustrate the differences in
displacement  and  energy responses when different energy dissipation mechanisms
are used.
%
Paper is concluded with a number of suggestions  on  the  choices  of  energy
dissipation mechanisms for dynamic modeling of inelastic solids.




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\section{Energy Dissipation Mechanisms}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Plastic Energy Dissipation}


Energy dissipation due to material inelasticity is a phenomenon that can be
observed  in most  structural  and  geotechnical  materials during deformation.
%
Plastic  energy  dissipation  can be used to estimate material damage.
%
According  to  the  second  law  of  thermodynamics, the energy dissipation of a
system should always increase or remain constant as deformation occurs.
%
Ability    to    follow  energy dissipation, a scalar, during static and dynamic
deformation  can  provide useful information about the performance of solids and
structures, and can be used to improve design and/or retrofit.


Unfortunately,     there  exists a misconception where plastic work is used as a
measure of plastic energy dissipation.
%
This  misconception originates from ignoring plastic free energy, which together
with plastic dissipation makes up plastic work.
%
Plastic  free  energy  can  make up a significant portion of plastic work, which
renders energy dissipation results incorrect.
%
The  correct  modeling  of  plastic energy dissipation follows the principles of
thermodynamics,  which ensures energy balance and nonnegative incremental energy
dissipation.
%
Based on the formulations derived from thermodynamics, a framework for computing
plastic  energy  dissipation  was  established  by \citeN{Yang2017a},
which is applied to the numerical examples presented in this paper.


Following  the first and second laws of thermodynamics, the equation for plastic
energy  dissipation  in  decoupled  material models was presented by \citeN{Yang2017a}:
%
\begin{equation}
\Phi =  \sigma_{ij} \dot{\epsilon}_{ij} - \sigma_{ij} \dot{\epsilon}_{ij}^{el} - \rho \dot{\psi}_{pl} \ge 0
\label{equation_plastic_dissipation_final}
\end{equation}
%
where  $\Phi$  is  the  rate  of  plastic  energy  dissipation  per unit volume,
$\sigma_{ij}$  is  the stress  tensor,  $\epsilon_{ij}$  is  the strain tensor,
$\epsilon_{ij}^{el}$  is  the  elastic  part of the strain tensor, $\rho$ is the
mass  density of the material, ${\psi}_{pl}$ is the plastic free energy per unit
volume.  
%
All  stresses are defined as the effective stresses.
%
Sign
convention for the  stress  is  consistent  with  the  standard definition in
mechanics    of    materials,    i.e.    positive    in   tension.   
%
Note     that Equation~\ref{equation_plastic_dissipation_final}  holds under the
assumption  of isothermal process, which is a reasonable assumption for analysis
of most civil engineering problems \cite{Collins97}.



For  most inelastic material models, the stress $\sigma_{ij}$, total strain
$\epsilon_{ij}$  and  elastic  strain  $\epsilon_{ij}^{el}$  are  parts  of  the
standard output for the material response.
%
A  challenge  is  to  evaluate  plastic  free  energy ${\psi}_{pl}$, that is not
defined explicitly, if at all, for many classic material models.
%
The  physical  interpretation  of plastic free energy is related to the state of
the material's micro-structure.
%
Physical  interpretation of plastic free energy was discussed and illustrated in
earlier publications \cite{Besseling1994,Collins2002b,Yang2017a,Yang2017b}.


For   pressure-independent   inelastic  material  models  (e.g.  von  Mises
plasticity)  with  isotropic and/or kinematic hardening, the plastic free energy
can be calculated from
%
\begin{equation}
\psi_{pl} = \psi_{pl}^{iso} + \psi_{pl}^{kin} = 
\frac{1}{2 \rho \kappa_1} k^2 + \frac{1}{2 \rho a_1} \alpha_{ij} \alpha_{ij}
\label{equation_plastic_free_energy_von_mises}
\end{equation}
%
where  the plastic free energy $\psi_{pl}$ is decomposed into two parts that are
related   to  isotropic  hardening  $\psi_{pl}^{iso}$  and  kinematic  hardening
$\psi_{pl}^{kin}$.
%
Here  scalar  $k$ is the size  of  the  yield  surface and tensor $\alpha_{ij}$ 
is  the  back  stress tensor representing the center of the yield surface, while
$\kappa$ and $a_1$ are nonnegative material constants.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Rayleigh Damping}
\label{Rayleigh_Damping} 

To  compute  the  energy  dissipation  due to viscous damping, we start with the
general form of the equation of motion:
%
\begin{equation}
M_{ij} \ddot{u}_j(t) + C_{ij} \dot{u}_j(t) + K_{ij}^{elpl}(t) u_j(t) = f_i(t)
\label{equation_of_motion}
\end{equation}
%
where  $u_j(t)$ is the vector of generalized displacements, $M_{ij}$ is the mass
matrix,   $C_{ij}$   is   the   damping   matrix,   $K_{ij}^{elpl}(t)$   is  the
inelastic  stiffness  matrix that generally evolves with time, $f_i(t)$ is
the external load vector.
%
For linear viscous damping of the Rayleigh type, the damping matrix is
expressed as
%
\begin{equation}
C_{ij} = a_M M_{ij} + a_{K} K_{ij}^{el}
\label{equation_damping_matrix}
\end{equation}
%
where  $a_M$  and  $a_{K}$  are  damping constants with units of s$^{-1}$ and s,
respectively.
%
The  stiffness matrix used to construct the damping matrix is usually the initial tangent
stiffness  matrix,  which  is  also  the  elastic  stiffness  $K_{ij}^{el}$  for
inelastic materials.
%
Equation~\ref{equation_damping_matrix}   indicates   that   the  damping  matrix
$C_{ij}$ is constant through the entire simulation.


The expression used to compute the damping constants for a desired damping ratio
$\xi$ in a specified frequency range from $\hat{\omega}$ to $R \hat{\omega}$ was
given by \citeN{Hall2006} as
%
\begin{equation}
a_M = 2 \xi \hat{\omega} \frac{2 R}{1 + R + 2 \sqrt{R}}
\quad , \quad
a_K = 2 \xi \frac{1}{\hat{\omega}} \frac{2}{1 + R + 2 \sqrt{R}}
\label{equation_damping_constants}
\end{equation}


It   can  be  seen  in  Equation~\ref{equation_damping_matrix}  that  there  are
mass-proportional and stiffness-proportional parts in Rayleigh damping.
%
The    combination  of  mass-proportional and stiffness-proportional damping can
provide a desired control over modal damping ratios.
%
However,  as  pointed out by \citeN{Hall2006}, classic Rayleigh damping must
be used with appropriate damping coefficients, which should give a near-constant
value of damping for all modes with frequencies that are of interest.
%
For  the  modes  outside  the prescribed frequency range, the damping ratios are
unrealistically high.


%
The incremental form of energy balance for a dynamic system with viscous damping can be expressed as
%
\begin{equation}
\Delta {W}_{Input} = \Delta {E}_K + \Delta {D}_V + \Delta {W}_M
\label{equation_energy_balance}
\end{equation}
%
The left hand side of Equation~\ref{equation_energy_balance}  is the
increment of external input work
%
\begin{equation}
\Delta {W}_{Input}=  f_i \Delta {u}_i
\label{equation_energy_balance_W_Input}
\end{equation}
%
%
%%
The three terms on the right hand side of Equation~\ref{equation_energy_balance} are
the  increment of kinetic energy $\Delta {E}_K$, the increment of viscous energy 
dissipation $\Delta {D}_V$, and the increment of material work of the system $\Delta {W}_M$
%
\begin{equation}
\begin{aligned}
\Delta {E}_K &= M_{ij} \ddot{u}_j \Delta u_i \\
\Delta {D}_V &=  C_{ij} \dot{u}_j \Delta {u}_i \\
\Delta {W}_M &= K_{ij}^{elpl} {u}_j \Delta {u}_i = \Delta {E}_S + \Delta {E}_P + \Delta {D}_P
\end{aligned}
\label{equation_energy_balance_components}
\end{equation}
%
Note  that  the term of material work $W_M$ can be separated into an elastic part
and  a  plastic  part.
These two components are known as  the elastic strain energy $E_S$ and the plastic
work of the system, respectively.
%
Then,  as  mentioned  in  the  previous  section,  plastic  work can be further
decomposed into plastic free energy $E_P$ and plastic energy dissipation $D_P$.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Algorithmic Energy Dissipation}


Newmark time integration method \cite{Newmark1959} is used for all cases in this study.
%
The  forward  displacements $^{n+1}u_{i}$ and velocities $^{n+1}\dot{u}_{i}$ are
expressed in terms of their current values and the forward and current values of
the acceleration
%
\begin{equation}
\begin{aligned}
^{n+1}\dot{u}_{i} &= \; ^{n}\dot{u}_{i} + (1 - \gamma) h \; ^{n}\ddot{u}_{i}  + \gamma h \; ^{n+1}\ddot{u}_{i} \\
^{n+1}u_{i} &= \; ^{n}u_{i} + h \; ^{n}\dot{u}_{i} + (\frac{1}{2} - \beta) h^{2} \; ^{n}\ddot{u}_{i} + \beta h^{2} \; ^{n+1}\ddot{u}_{i}
\end{aligned}
\end{equation}
%
where  $\Delta  t$ is the length of each time step, $\gamma$ and $\beta$ are the
Newmark  integration parameters that controls the amount of algorithmic damping in
the system.


\citeN{Krenk2006} gave the incremental form of the energy balance equation
\ref{equation_energy_balance} over increment $t_n$ to $t_{n+1}$, for Newmark algorithm
%
\begin{equation}
\begin{aligned}
& \left. \left[ 
\frac{1}{2} M_{ij}^{\star} \dot{u}_{i} \dot{u}_{j} 
+ 
\frac{1}{2} K_{ij} {u}_i {u}_j 
+ 
\left( \beta - \frac{1}{2} \gamma \right) \frac{1}{2} h^{2} M_{ij}^{\star} \ddot{u}_{i} \ddot{u}_{j} 
\right] \right|^{t_{n+1}}_{t_n} 
= \\
& \quad \quad + \Delta u_{i} \left[ \frac{1}{2} (f_{i}^{n+1} + f_{i}^{n}) + \left( \gamma - \frac{1}{2} \right) \Delta f_{i} \right] \\
& \quad \quad  - \left( \gamma - \frac{1}{2} \right) \left[ K_{ij} \Delta {u}_i \Delta {u}_j  + \left( \beta - \frac{1}{2} \gamma \right) h^{2} M_{ij}^{\star} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \right] \\
& \quad \quad  - \frac{1}{2} h \left[ h^{-2} C_{ij} \Delta {u}_i \Delta {u}_j + \frac{1}{4} C_{ij} (\dot{u}_{i}^{n+1} + \dot{u}_{i}^{n}) (\dot{u}_{j}^{n+1} + \dot{u}_{j}^{n}) \right] \\
& \quad \quad  + \frac{1}{2} \left( \beta - \frac{1}{2} \gamma \right)^{2} h^{3} C_{ij} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} 
\end{aligned}
\label{equation_energy_balance_increment_krenk}
\end{equation}
%
where the equivalent mass matrix $M_{ij}^{\star}$ is defined as
%
\begin{equation}
M_{ij}^{\star} = M_{ij} + \left( \gamma - \frac{1}{2} \right) h C_{ij}
\end{equation}
%



Rearranging   Equation~\ref{equation_energy_balance_increment_krenk}  gives  the
explicit expression for the amount of algorithmic energy dissipation over an
increment
%
\begin{equation}
\begin{aligned}
& \left[ {E}_K + {D}_V + {W}_M - {W}_{Input} \right]^{t_{n+1}}_{t_n} = \\
& \quad \quad + \left( \gamma - \frac{1}{2} \right) \Delta f_{i} \Delta u_{i} + \frac{1}{2} \left( \beta - \frac{1}{2} \gamma \right)^{2} h^{3} C_{ij} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \\
& \quad \quad  - \left( \gamma - \frac{1}{2} \right) \left[ K_{ij} \Delta {u}_i \Delta {u}_j  + \left( \beta - \frac{1}{2} \gamma \right) h^{2} M_{ij}^{\star} \Delta \ddot{u}_{i} \Delta \ddot{u}_{j} \right] \\
& \quad \quad  - \left. \left[ \frac{1}{2} \left( \gamma - \frac{1}{2} \right) h C_{ij} \dot{u}_{i} \dot{u}_{j} + \left( \beta - \frac{1}{2} \gamma \right) \frac{1}{2} h^{2} M_{ij}^{\star} \ddot{u}_{i} \ddot{u}_{j} \right] \right|^{t_{n+1}}_{t_n} \\
\end{aligned}
\label{equation_energy_balance_increment_krenk_new}
\end{equation}
%
When  $\gamma  =  0.500$ and $\beta=0.250$, all the terms on the right hand side of
Equation   \ref{equation_energy_balance_increment_krenk_new}   vanish,   thus   no
algorithmic energy dissipation exists.
%
For  other  values  of  $\gamma$ and $\beta$, algorithmic energy dissipation, or
even energy production, is observed in the system.









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Numerical Examples}

Numerical  simulations  presented  in this paper are performed using the MS ESSI
Simulator system \cite{Real_ESSI_Simulator}, and all the examples presented here
are available at \url{http://ms-essi.info/}.
%
Energy dissipation due to material inelasticity and
Rayleigh  damping is investigated through a series of single-element tests with
different material parameters and loading conditions.
%
Then,    elastic and inelastic materials  are  used  in  a  stack of solid brick
elements  model  to  further  illustrate  energy dissipation in wave propagation
problems.
%
Finally,   cases   with   plastic  energy  dissipation,  Rayleigh  damping,  and
algorithmic    damping    are   analyzed to demonstrate the importance of energy
dissipation mechanisms in dynamic finite element modeling.


The   inelastic  material  model  used in this study is associated von Mises plasticity
with nonlinear Armstrong-Frederick kinematic hardening.
%
The yield function of von Mises plasticity is
%
\begin{equation}
f = \sqrt{(s_{ij} - \alpha_{ij})(s_{ij} - \alpha_{ij})} - \sqrt{\frac{2}{3}} k
\label{equation_vm_yield_function}
\end{equation}
%
The general expression for Armstrong-Frederick kinematic hardening rule is
%
\begin{equation}
d{\alpha}_{ij} = \frac{2}{3} h_a d{\epsilon}^{pl}_{ij} - \sqrt{\frac{2}{3}} c_r d{\lambda} \alpha_{ij}
\label{equation_af_hardening}
\end{equation}
%
where  $d{\lambda}$    is a nonnegative scalar plastic multiplier and $h_a$ and
$c_r$  are  nonnegative  material  hardening  constants.  
% 
The  parameter  $h_a$  controls  the  initial  stiffness  of  the material after
yielding, while ratio $h_a/c_r$ controls the limit of stress magnitude, or shear strength.



For all dynamic problems analyzed, Newmark integration algorithm is used.
%
In    most  cases  with  elasticity  and  inelasticity and Rayleigh damping, the
numerical  integration  parameters  are  chosen  in a way so that no algorithmic
damping exists in the system, i.e. Newmark parameters $\gamma = 0.500$ and $\beta = 0.250$.
%
Only  for  the  last  example, algorithmic damping was used, by choosing Newmark
parameters $\gamma > 0.500$ and $\beta = 0.250(\gamma+0.500)^2$.
%


External  loads  are  applied  incrementally using load- or displacement-control
scheme.
%
Nonlinear system of equations is solved using Newton-Raphson algorithm with line
searching to help convergence.
%
Standard  8-node-brick elements are used in all examples.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Single Element Tests}

Energy  dissipation in single 8-node-brick element is studied in this section.
%
The  individual  and  combined effects of material inelasticity and Rayleigh
damping are investigated through a series of numerical examples.



The single element model, with boundary conditions and loads, is shown
in Fig.\ref{Model_Single_Element}.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.4\columnwidth]{Fig01.pdf}
\caption{\label{Model_Single_Element}
Single element model under shear loading.}
\end{figure}
%
During the shearing stage, the bottom of the element is fixed and imposed motion
is applied on the top.
%
The    main    reason  for using  displacement-control scheme is to eliminate any
resonance  effects that might occur during faster dynamic tests, if load-control
is used.
%
The time increment $\Delta t$ for the Newmark algorithm is set to $\Delta t = 0.01$s.




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Elastic Material without Rayleigh Damping}

The first example uses linear elastic material with no Rayleigh damping.
%
No energy dissipation in any form is expected in this case.
%
The equivalence between external work and mechanical energy stored in the system
(energy balance) in single-element model is verified through this example.


The    material    properties    used    for this   case  are  summarized   in
Table~\ref{table_elastic_material}.
%
\begin{table}[!htb]
\centering
\caption{Model parameters of the linear elastic material.}
\label{table_elastic_material}
\small
\begin{tabular}{lcc}
\hline\hline
\multicolumn{1}{c}{Parameter} & \multicolumn{1}{c}{Unit} & \multicolumn{1}{c}{Material} \\ \hline
\texttt{mass\_density} ($\rho$)                  & $kg/m^3$   & 2650                      \\
\texttt{poisson\_ratio}   ($\nu$)               &                          & 0.3                       \\
\texttt{shear\_wave\_velocity}  ($v_s$)    & $m/s$                      & 800                       \\
\texttt{elastic\_modulus} ($E$)              & $MPa$                     & 4409.6                      \\
\hline\hline
\end{tabular}
\end{table}




Fig.\ref{figure_single_elastic}  shows  the  displacement, stress-strain, and
energy  responses  of  the  case  using  linear elastic material without Rayleigh
damping.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig02a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig02b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig02c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig02d.pdf}}
\caption{\label{figure_single_elastic}
Energy computation of single brick element modeled using linear elastic material and no Rayleigh damping: (a) Displacement time series; (b) Stress-strain response; (c) Energy results; (d) External work.}
\end{figure}
%
The  stress-strain response indicates that the material is linear elastic, which
is expected.
%
Only  kinetic  energy and strain energy appear in this example since there is no
energy dissipation.
%
Notice  that  the  kinetic  energy is not zero at  the beginning because of the
initial velocity that comes with the sine-wave imposed motion.



The  comparison  between  external  work,  which  is  computed using the imposed
displacement  time  series and nodal force response, and total mechanical energy
stored  in the system, calculated at each Gauss point, shows that energy balance
is achieved.
%
Note        that    the    frequency    of   oscillation   of   energy   curves, shown in
Fig.\ref{figure_single_elastic}(c)(d), is twice of that of the displacement
time  series, Fig.\ref{figure_single_elastic}(a).
%
This is expected for linear elastic systems under displacement control loading.










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Inelastic  Material without Rayleigh Damping}
\label{section_single_inelastic}

The   next   example   focuses   on  the  energy  dissipation  due  to  material
inelasticity only.
%
A  more  systematic  study of plastic energy dissipation in solids can  be found
in \citeN{Yang2017a}, where the energy dissipation behavior of various inelastic
material models was investigated.
%
In   this   section,   a  material  model  defined with  von Mises   yield 
function  and  Armstrong-Frederick  nonlinear kinematic hardening is used.
%
No Rayleigh damping is added to any model in this section.



Table~\ref{table_inelastic_material} summarizes material parameters for these studies.
%
\begin{table}[!htb]
\centering
\caption{Model parameters of the inelastic materials.}
\label{table_inelastic_material}
\small
\begin{tabular}{lcccc}
\hline\hline
\multirow{2}{*}{Parameter} & \multirow{2}{*}{Unit} & \multicolumn{3}{c}{Material} \\ \cline{3-5} 
                     &                   & Soft & Medium & Hard \\ \hline
\texttt{mass\_density} ($\rho$) & $kg/m^3$ & 1800  & 1800  & 1800 \\
\texttt{elastic\_modulus} ($E$) & $MPa$ & 748.8  & 748.8  & 748.8   \\
\texttt{poisson\_ratio} ($\nu$) & & 0.3  & 0.3  & 0.3   \\
\texttt{von\_mises\_radius} & $MPa$ & 1.0  & 1.0  & 1.0   \\
\texttt{armstrong\_frederick\_ha} ($h_a$) & $MPa$  & 374.4 & 748.8 & 1497.6 \\
\texttt{armstrong\_frederick\_cr} ($c_r$) &  & 50  & 100  & 200 \\ 
\hline\hline
\end{tabular}
\end{table}
%
Three materials with the same shear strength, controlled by ratio $h_a/c_r$, but different post-yield stiffness, $h_a$, are modeled.
%
These three materials are used in single element examples and wave propagation problems.
%
They will be referred to as soft, medium, and hard inelastic materials in the remaining parts of this paper.
%
Fig.\ref{figure_single_inelastic} shows the
stress-strain  response  and  energy  computation  results  for the cases using
inelastic material and no Rayleigh damping.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig03a.pdf}}
\\
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig03b.pdf}}
\\
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig03c.pdf}}
\caption{\label{figure_single_inelastic}
Energy computation for a single  brick  element modeled using inelastic material and
no Rayleigh damping: (a) Soft material; (b) Medium material; (c) Hard material.}
\end{figure}



According    to   Fig.\ref{figure_single_inelastic}, by keeping shear strength constant  and  increase post-yield stiffness, the
amount of plastic energy dissipation increases.
%
This  is consistent with the stress-strain response which shows that the degree
of plastification increases as the parameter $h_a$ increases.
%
It  is  also noted that changes in strain energy during cyclic shearing, is very
small,  due to a small elastic region of the material model.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Elastic Material with Rayleigh Damping}

The next example focuses on the energy dissipation due to viscous damping.
%
Viscous damping is modeled using Rayleigh damping.
%
Linear    elastic   material is used in all cases in this section so that energy
dissipation can only be caused by Rayleigh damping.


Energy dissipation by Rayleigh damping can be analyzed using equations presented
in section~\ref{Rayleigh_Damping}.
%
By substituting      the      expression      of      Rayleigh     damping     matrix
(Equation~\ref{equation_damping_matrix}   and  \ref{equation_damping_constants})
into      the      term      of      viscous      energy      dissipation     in
Equation~\ref{equation_energy_balance_components}, one obtains
%
\begin{equation}
\Delta {D}_V =  C_{ij} \dot{u}_j \Delta {u}_i =
\frac{4 \xi}{1 + R + 2 \sqrt{R}} \left( \hat{\omega} R M_{ij} + \frac{K_{ij}^{el}}{\hat{\omega}} \right) \dot{u}_j \Delta {u}_i
\label{equation_rayleigh_dissipation}
\end{equation}



Table~\ref{table_single_rayleigh} shows the parameters used in this section.
%
\begin{table}[!htb]
\centering
\caption{Rayleigh  damping  parameters  for  the  single  element examples using
linear elastic material with Rayleigh damping.}
\label{table_single_rayleigh}
\small
\begin{tabular}{cccccc}
\hline\hline
\multicolumn{1}{c}{Parameter}   & \multicolumn{1}{c}{Unit} & Case 1 & Case 2 & Case 3 & Case 4 \\ \hline
$R$   &  & 5.0 & 5.0 & 5.0 & 5.0 \\
$T_1$ & $s$ & 1.0 & 1.0 & 1.0 & 1.0 \\
$\xi$ &     & 0.02 & 0.05 & 0.10 & 0.20 \\ 
\hline\hline
\end{tabular}
\end{table}
%
The effect of damping ratio $\xi$ to the amount of viscous energy dissipation is
investigated.
%
Typical values of damping ratio $\xi$ for soils and structures are selected.
%
The lower limit of damping range $\hat{\omega}$ can be computed from the model's
fundamental frequency $\omega_1$.
%
It  is  common  practice  to  set  $\hat{\omega} = (2/{3}) \omega_1 =
(4 \pi)/(3 T_1)$ due to nonlinear response of the system \cite{Hall2006}.
%
In order to  cover  the  frequency  range from $({2}/{3}) \omega_1$ to $3
\omega_1$, a second-mode  frequency  of  a linear shear beam, the
parameter $R$ should be set to 4.5.
%
For    the    examples   in  this  section, $R$ is chosen to be slightly larger,
$R=5.0$, to cover a larger range.





The        energy       computation       results       are       shown       in
Fig.\ref{figure_single_rayleigh_xi}.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.48\columnwidth]{Fig04a.pdf}}
\subfloat[]{\includegraphics[width=0.48\columnwidth]{Fig04b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.48\columnwidth]{Fig04c.pdf}}
\subfloat[]{\includegraphics[width=0.48\columnwidth]{Fig04d.pdf}}
\caption{\label{figure_single_rayleigh_xi}
Energy  computation  of single brick element modeled using elastic material with
Rayleigh damping: (a) $\xi$ = 0.02; (b) $\xi$ = 0.05; (c) $\xi$ = 0.10; (d) $\xi$ = 0.20.}
\end{figure}
%
It    can    be    observed that the viscous energy dissipation increases as the
damping ratio increases.
%
The values indicate a linear relationship between the viscous energy dissipation
and        damping       ratio,   that is       consistent       with
Equation~\ref{equation_rayleigh_dissipation}.
%
All other forms of energy are the same in these cases.







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Inelastic Material with Rayleigh Damping}

In  the  previous examples, the separate effects of material inelasticity
and Rayleigh damping have been studied.
%
In  this  example,  both  energy  dissipation  mechanisms,  inelasticity and
Rayleigh damping, are used.
%
In  particular,  it will be shown that the relative amount of energy dissipation
due  to  these  two mechanisms is very sensitive to the choice of material and
damping parameters.




Fig.\ref{figure_single_inelastic_rayleigh}  shows  the stress-strain response and
energy  computation  results  for the three cases using inelastic material model
with Rayleigh damping.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig05a.pdf}}
\\
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig05b.pdf}}
\\
\subfloat[]{
\includegraphics[width=\columnwidth]{Fig05c.pdf}}
\caption{\label{figure_single_inelastic_rayleigh}
Energy computation of single brick element modeled using inelastic material with Rayleigh damping: (a) Soft material, $\xi = 0.1$; (b) Medium material, $\xi = 0.06$; (c) Hard material, $\xi = 0.02$.}
\end{figure}
%
%
As  can be observed, energy dissipation results are very different in these three cases.
%
Plastic    energy    dissipation    can   be much larger or much smaller or very
similar  to  the viscous  energy dissipation, depending on the choices of
simulation parameters.
%
More  discussion  on  the  combined  effect  of  plastic  energy dissipation and
Rayleigh  damping  is  shown in section~\ref{Wave_Propagation} where a wave
propagation problem is studied.










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Wave Propagation}
\label{Wave_Propagation}

The  numerical  examples for a single-element present  a fundamental understanding of
the  energy  behavior for a fully  controlled  system,  with fixed and
prescribed displacements.
%
In  this  section,  the  energy dissipation due to material inelasticity and
viscous damping in a wave propagation problem is investigated.
%
Compared  to the  single-element  cases where energy transformation happens in
one  element,  the   wave propagation problem can illustrate the evolution of
energy storage and dissipation in time and space.


It  is  important to select appropriate grid-spacing and time-step size to ensure
numerical stability and accuracy.
%
A  detailed study on the discretization effects in the finite element simulation
of  seismic  waves  in  elastic  and  inelastic  media  was  conducted  by
\citeN{Watanabe2015}.
%
According    to    \citeN{Jeremic2008d},    the   suitable  maximum grid spacing
$\Delta    x$    for shear wave propagation for a model using eight-node-brick
finite  elements  can  be determined by considering shear wave velocity $V_s$,
and the highest frequency of the input signal $f_{max}$:
%
%
\begin{equation}
\Delta x \le \frac{V_{s}}{10 f_{max}}
\label{equation_grid_spacing}
\end{equation}
%


The  time-step  size $\Delta t$ in wave propagation problem is usually chosen on the basis
that a given wave front does not reach two consecutive nodes at the same time.
%
The  following  condition,  given  by  \citeN{Watanabe2015},  is  used to ensure
numerical stability when using Newmark time integration algorithm:
%
\begin{equation}
\Delta t \le \frac{\Delta x}{V_{s}}
\label{equation_time_step}
\end{equation}



The  model used in this section is a stack of 8-node-brick elements, as shown in
Fig.\ref{figure_model_wave}.
%
\begin{figure}[!htbp]
\centering
\includegraphics[width=\columnwidth]{Fig06.pdf}
\caption{\label{figure_model_wave}
Numerical model of the wave propagation problem used in this paper.}
\end{figure}
%
%
It  is  noted that for inelastic material models, where shear stiffness is
reduced    due    to    inelasticity,    as    is   the    case    here,   
Equation~\ref{equation_grid_spacing} should still hold.
%
This  means  that  the  element size has to be small enough so that the model is
capable of propagating waves through inelastic material.
%
However, there is a limit of how small the mesh size can become, and what frequencies can
be accurately propagated.
%
For a  given example, assuming reduction of shear stiffness, shear wave velocity,
from   $V_s=400$m/s   (elastic)   to   $V_s=40$m/s   (inelastic).
Then,  based  on
Equation~\ref{equation_grid_spacing}, the length of each element is chosen to be
$\Delta x= 1$m.
%
Meanwhile, according to Equation~\ref{equation_time_step}, the time increment $\Delta
t$ for the Newmark algorithm is set to $\Delta t = 0.001$s.


Model  features  a total of 4000 eight-node-brick elements, with boundary conditions
that allow only shear waves to propagate.
%
The  left  3900 elements are used to model a layer of material that is assumed to
remain elastic during wave propagation.
%
The  material  model  properties of the elastic layer was given in Table~\ref{table_elastic_material}.
%
The right 100 elements represent a layer that can be elastic or inelastic.
%
In    some    examples,  this layer remains elastic, in order to investigate the
effect of Rayleigh damping.
%
In  other  examples,  this  layer  is  inelastic,  and dissipates energy through
material inelasticity and/or Rayleigh damping.




%
An    Ormsby    wavelet~\cite{Ryan1994},  with  peak value controlled by $A$ and
frequency  contents starting from $f_1$ to $f_4$, with a constant amplitude from
$f_2$ to $f_3$, is imposed to the fixed end of the model.
%
The function of Ormsby  wavelet is
%
\begin{equation}
\begin{aligned}
f(t) = &A \left[ \left( \frac{\pi {f_4}^2}{f_4 - f_3} {sinc(\pi f_4 (t-t_s))}^2 - \frac{\pi {f_3}^2}{f_4 - f_3} {sinc(\pi f_3 (t-t_s))}^2 \right) \right.\\
       &- \left. \left( \frac{\pi {f_2}^2}{f_2 - f_1} {sinc(\pi f_2 (t-t_s))}^2 - \frac{\pi {f_1}^2}{f_2 - f_1} {sinc(\pi f_1 (t-t_s))}^2 \right) \right]
\label{Ormsby_eq}
\end{aligned}
\end{equation}
%
where  $t_s$   is   the   time  when  maximum   amplitude   is  occurring,  and
sine cardinal is defined as $sinc(x)=sin(x)/x$. 
%
Please    note    that  from  Equation~\ref{Ormsby_eq}, peak value $f^{peak}$ is
defined at $t=t_s$ as
%
\begin{equation}
\begin{aligned}
 f^{peak} = A   \pi \left( {f_4 + f_3} - {f_2 - f_1}  \right) 
\label{Ormsby_eq_02}
\end{aligned}
\end{equation}
%
%
Fig.\ref{figure_imposed_displacement} shows the imposed displacement, in time domain and frequency domain, created
using  Ormsby  wavelet  with  $A=0.01$m$\cdot$s,  $f_1=1.0$Hz,
$f_2=2.0$Hz, $f_3=4.0$Hz, and $f_4=5.0$Hz.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig07a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig07b.pdf}}
\caption{\label{figure_imposed_displacement}
Imposed  displacement  of  the  wave  propagation  problem  created using Ormsby
wavelet with $A=0.01$m$\cdot$s, $f_1=1.0$Hz,  $f_2=2.0$Hz,  $f_3=4.0$Hz, and $f_4=5.0$Hz:  (a)  Time series; (b)
Frequency domain.}
\end{figure}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Elastic Material without Rayleigh Damping}

In   the  first  case, material     parameters     for     elastic  material are
used for the entire model.
%
This    simple    setup  illustrates the energy transformation process as  shear
waves propagate and reflect within a uniform media.



Fig.\ref{figure_wave_all_rock}  shows the displacement response at the free end of
the   model   in  time  and  frequency  domain,  as  well  as  the  energy  and
stress-strain  results,  for  the  case  of  wave propagation within an uniform
media.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig08a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig08b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig08c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig08d.pdf}}
\caption{\label{figure_wave_all_rock}
Energy computation of wave propagation within an uniform elastic media: (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
As  expected,  the  displacement at the free end is twice in
magnitude and same in frequency contents.
%
The  stress-strain  response  of  the  material is a straight line since linear
elastic material is used in this case.


Because  of  the  absence  of energy dissipation mechanism, the total mechanical
energy remains constant after the wave is input into the system.
%
The transformation between kinetic energy and strain energy is observed when the
shear wave hits the free end of the model and gets reflected back.
%
During  the  wave  propagation  process,  before  and  after the reflection, the
amounts  of  kinetic  energy  and  strain  energy are exactly the same, which is
expected in a wave propagation problem.










%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Elastic and Inelastic Material without Rayleigh Damping}

In this section, the energy dissipation due to material inelasticity during wave propagation is investigated.
%
The inelastic material model uses von Mises yield function in combination with Armstrong-Frederick kinematic hardening.
%
Soft,     medium,     and     hard     inelastic     materials,     shown     in
Table~\ref{table_inelastic_material}                                         on
page~\pageref{table_inelastic_material},  are  used to investigate the influence
of post-yield stiffness on plastic dissipation.
%


Fig.\ref{figure_wave_vm_1}, \ref{figure_wave_vm_2}, and \ref{figure_wave_vm_3} show the displacement response at the free end of the model in time and frequency domain, as well as the stress-strain and energy results, for the cases with material inelasticity and no Rayleigh damping.
%
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig09a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig09b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig09c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig09d.pdf}}
\caption{\label{figure_wave_vm_1}
Energy computation of wave propagation using soft inelastic material and no Rayleigh damping: (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig10a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig10b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig10c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig10d.pdf}}
\caption{\label{figure_wave_vm_2}
Energy computation of wave propagation using medium inelastic material and no Rayleigh damping: (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig11a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig11b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig11c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig11d.pdf}}
\caption{\label{figure_wave_vm_3}
Energy computation of wave propagation using hard inelastic material and no Rayleigh damping: (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
Peak displacement response at the free end of the model is larger when the inelastic material is stiffer after yield.
%
Due to the use of inelastic material, permanent deformation is observed in the displacement time series. 
%
This leads to the low frequency contents in the displacement spectrum.
%
% 

%
According to the energy plots, more energy is dissipated due to material inelasticity when the inelastic material has a larger post-yield stiffness.
%
This observation is consistent with the findings from single element examples (section~\ref{section_single_inelastic}) that harder inelastic material has more plastic dissipation.
%
When the inelastic layer is much softer than the elastic layer, the majority of the incoming wave is reflected back at the material interface instead of entering the inelastic layer.
%
As a result, energy dissipation and displacement response are less significant in the case of a softer inelastic material.








%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Elastic Material with Rayleigh Damping}


The next section focuses on the effect of Rayleigh damping in wave propagation problems.
%
Linear elastic material, shown in Table~\ref{table_elastic_material}, is used
in order to only have viscous dissipation mechanism.
%
Due  to  the  use  of Rayleigh damping, it is expected to observe viscous energy
dissipation  and  displacement reduction throughout the model.
%
Parametric study on the damping coefficient $\xi$ is performed.
%
The damping parameters used in this section are summarized in Table~\ref{table_wave_rayleigh}.
% %
%
\begin{table}[!htb]
\centering
\caption{Rayleigh damping parameters for the wave propagation examples using linear elastic material with Rayleigh damping.}
\label{table_wave_rayleigh}
\small
\begin{tabular}{ccccc}
\hline\hline
\multicolumn{1}{c}{Parameter}   & \multicolumn{1}{c}{Unit} & Case 1 & Case 2 & Case 3\\ \hline
$R$   &  & 5.0 & 5.0 & 5.0 \\
$T_1$ & $s$ & 1.0  & 1.0 & 1.0 \\
$\xi$ &     & 0.02 & 0.05 & 0.10 \\ 
\hline\hline
\end{tabular}
\end{table}
%



Fig.\ref{figure_wave_rayleigh_xi_1},   \ref{figure_wave_rayleigh_xi_2},   and
\ref{figure_wave_rayleigh_xi_3}  show  the  displacement  response  in  time and
frequency  domain,  as  well  as  the  energy  results, when different values of
damping ratio $\xi$ are used.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig12a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig12b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig12c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig12d.pdf}}
\caption{\label{figure_wave_rayleigh_xi_1}
Energy computation of wave propagation using linear elastic material and Rayleigh damping ($\xi$ = 0.02): (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig13a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig13b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig13c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig13d.pdf}}
\caption{\label{figure_wave_rayleigh_xi_2}
Energy computation of wave propagation using linear elastic material and Rayleigh damping ($\xi$ = 0.05): (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig14a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig14b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig14c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig14d.pdf}}
\caption{\label{figure_wave_rayleigh_xi_3}
Energy computation of wave propagation using linear elastic material and Rayleigh damping ($\xi$ = 0.10): (a) Displacement time series; (b) Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
As expected, the amount of surface displacement reduction and energy dissipation increases when larger $\xi$ is used.
%
Despite changes in the peak value, the frequency contents of the displacement response at the free end remain the same as those of the input motion.






For structural materials like steel and concrete, the Rayleigh damping ratio is usually chosen to be between 0.02 to 0.05.
%
As seen in Fig.\ref{figure_wave_rayleigh_xi_2}, about half of the input energy is dissipated due to Rayleigh damping when $\xi$ is 0.05.
%
It is noted that sometimes large  Rayleigh damping ratio are used, as  high  as 
0.20. 
%
These high Rayleigh damping ratios lead to a significant amount of energy
dissipation through viscous energy dissipation mechanism.
%
Such  high  level  of  energy  dissipation  may not be physical and will lead to
underestimation of the system response.
%





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Comparison between Plastic Dissipation and Viscous Damping}

As  previously  noted, sometimes Rayleigh damping of the system is set to a high
level to mimic inelastic material behavior.
%
By comparing the example using soft inelastic material without Rayleigh damping,
shown  in Fig.\ref{figure_wave_vm_1}, with the one using linear elastic material
and Rayleigh damping $\xi = 0.05$, shown in Fig.\ref{figure_wave_rayleigh_xi_2},
it  appears  that  the  amounts  of  energy dissipation are similar in these two
examples.
%
However,  this  correspondence is very sensitive to material properties, damping
parameters, loading conditions and frequency of loading.
%
In  this  section, the two examples with different energy dissipation mechanisms
are   reanalyzed  using  an  input  motion  with  different  frequency  content,
$f_1=0.5$Hz,   $f_2=1.0$Hz,   $f_3=3.0$Hz,   and   $f_4=3.5$Hz,   as   shown  in
Fig.\ref{figure_change_imposed_displacement}.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig15a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig15b.pdf}}
\caption{\label{figure_change_imposed_displacement}
Imposed  displacement  of  the  wave  propagation  problem  created using Ormsby
wavelet with $A=0.01$m$\cdot$s, $f_1=0.5$Hz,  $f_2=1.0$Hz,  $f_3=3.0$Hz, and $f_4=3.5$Hz:  (a)  Time series; (b)
Frequency domain.}
\end{figure}


Fig.\ref{figure_wave_change_motion} shows the displacement responses and energy computation results for the two examples using different energy dissipation mechanisms.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=\columnwidth]{Fig16a.pdf}}
\\
\subfloat[]{\includegraphics[width=\columnwidth]{Fig16b.pdf}}
\caption{\label{figure_wave_change_motion}
Energy computation of wave propagation using a different input motion: (a) Soft inelastic material without Rayleigh damping; (b) Linear elastic material with Rayleigh damping $\xi = 0.05$.}
\end{figure}
%
The     first     case     uses    soft    inelastic    material,    shown    in
Table~\ref{table_inelastic_material} on page~\pageref{table_inelastic_material},
with  no  Rayleigh  damping,  while the second case uses linear elastic material
with Rayleigh damping $\xi = 0.05$.
%
From  Fig.\ref{figure_wave_change_motion},  it  can  be seen that the amounts of
energy  dissipation  are  very  different  in these two cases with the new input
motion.
%
Thus,  material inelasticity and viscous damping may give equal, similar amounts
of  energy  dissipation  for a specific set of simulation parameters and loading
conditions,  but  this  similarity  will  not  hold if the model is subject to a
different loading.
%
In  other  words,  plastic  dissipation  and  viscous energy dissipation are not
interchangeable  energy  dissipation mechanisms, and should always be considered
independently.
%
Moreover,  viscous damping assumes that each node of the finite element model is
connected to a viscous damper, while in reality for civil engineering solids and
structures  most  energy  is  not  dissipated  in  the  form  of  linear viscous
damping~\cite{Ostadan2004,Hall2006}.






It  should also be pointed out that the response,  especially in
the  frequency  domain,  are  very  different  when comparing the examples using
inelastic material with the ones using only Rayleigh damping.
%
Permanent  deformation  and  change of stiffness, which can only be modeled with
inelastic materials, are not observed in the cases using linear elastic material
with Rayleigh damping.
%
For  the  cases  using  inelastic  materials, the displacement response contains
frequencies that are both higher and lower than those of the input motion.
%
For practical engineering designs, if only elastic material is used in numerical
simulations,  these  important  high frequency motions and permanent deformations
could be missed completely.
% 









%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Combined Effects of Plastic Dissipation and Viscous Damping}


In this section, the combined effect of energy dissipation through material inelasticity and Rayleigh damping is presented.
%
Three cases with different combinations of inelastic materials (Table~\ref{table_inelastic_material}) and damping coefficients ($\xi$ = 0.02, 0.04, or 0.10) are simulated and analyzed.
%
Fig.\ref{figure_wave_vm_rayleigh_1}, \ref{figure_wave_vm_rayleigh_2}, and \ref{figure_wave_vm_rayleigh_3} show the displacement, stress-strain, and energy results of these three cases with both material inelasticity and Rayleigh damping.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig17a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig17b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig17c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig17d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_1}
Energy  computation  of  wave  propagation  using  soft  inelastic  material and
Rayleigh   damping  with  $\xi$  =  0.01:  (a)  Displacement  time  series;  (b)
Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig18a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig18b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig18c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig18d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_2}
Energy computation of wave propagation using medium inelastic material model and
Rayleigh   damping  with  $\xi$  =  0.04:  (a)  Displacement  time  series;  (b)
Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig19a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig19b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig19c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig19d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_3}
Energy  computation  of wave propagation using hard inelastic material model and
Rayleigh   damping  with  $\xi$  =  0.10:  (a)  Displacement  time  series;  (b)
Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
Permanent deformation and low frequency response at the free end is observed.
%
Note that the high frequency content of motions is significantly reduced because
of the Rayleigh damping in the system.






Depending  on  the choices of simulation parameters, it is shown that the amount
of  energy  dissipation  due to material inelasticity can be larger, smaller, or
comparable to that caused by Rayleigh damping.
%
For  problems  where  significant  solid-fluid  interaction is expected, e.g.
underground  structure  in saturated soil, the viscous damping of the soil structure interaction (SSI) system
is likely to have significant contribution to energy dissipation.
%
For    other    problems    with    little  solid-fluid  interaction,  but large
material  damage,  e.g.  aboveground  structure suffering large seismic loading,
more energy is likely to be be dissipated due to material inelasticity.
%
Thus it  is important to model not only the correct amount of energy dissipation of a
system but the proper energy dissipation mechanisms as well.
%
The  system response can be very different, as shown, when different combinations
of energy dissipation mechanisms are used.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Influence of Algorithmic Damping}



Algorithmic  damping  was  excluded in all previously presented examples by setting
the  Newmark  parameter to  $\gamma=0.500$ and $\beta=0.250$ \cite{local-86}.
%
In practice, Newmark parameters $\gamma$ and $\beta$ are usually set to 
$\gamma > 0.500$ and $\beta = 0.250(\gamma+0.500)^2$ in order to introduce
algorithmic damping into the system. 
% 
To    illustrate  the  influence  of Newmark parameters  $\gamma$ and $\beta$ on
energy dissipation, two sets of examples with algorithmic damping are analyzed.



Fig.\ref{figure_wave_numerical_1} and \ref{figure_wave_numerical_2} show the
displacement,  stress-strain,  and  energy  results  of  two  cases where linear
elastic  material  and  no  Rayleigh  damping  is  used,  which  means that only
algorithmic damping dissipates system energy.
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig20a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig20b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig20c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig20d.pdf}}
\caption{\label{figure_wave_numerical_1}
Energy  computation  of  wave  propagation  using  linear  elastic  material and
algorithmic  damping  ($\gamma  =  0.505$, $\beta=0.253$): (a) Displacement time
series;  (b)  Displacement  spectrum;  (c)  Energy  results;  (d)  Stress-strain
response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig21a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig21b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig21c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig21d.pdf}}
\caption{\label{figure_wave_numerical_2}
Energy  computation  of wave propagation using linear elastic material model and
algorithmic  damping  ($\gamma =  0.700$, $\beta=0.360$):  (a)  Displacement  time  series;  (b)
Displacement spectrum; (c) Energy results; (d) Stress-strain response.}
\end{figure}
%
There  is a noticeable amount of algorithmic damping when the Newmark parameters
are increased to $\gamma=0.505$ and $\beta=0.253$.
%
As  Newmark  parameters are increased to $\gamma=0.700$ and $\beta=0.360$, it is
observed  that  the  algorithmic  damping  in the system becomes unrealistically
high.
%
More  importantly,  most  of  the  mechanical  energy  propagated by the wave is
numerically dissipated even before the wave reaches the free end of the column.
%












%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Combined Effects of Plastic Dissipation, Viscous Damping, and Algorithmic Damping}


For  realistic  modeling  and  simulation,  usually all three energy dissipation
mechanisms are used.
%
Fig.\ref{figure_wave_vm_rayleigh_numerical_1},
\ref{figure_wave_vm_rayleigh_numerical_2},                                   and
\ref{figure_wave_vm_rayleigh_numerical_3}  show the displacement, stress-strain,
and  energy  results  of  the  three  cases  where plastic dissipation, Rayleigh
damping, and algorithmic damping are all present.
%
The  medium  inelastic  material defined in Table~\ref{table_inelastic_material}
and Rayleigh damping coefficient $\xi = 0.04$ are used for all three cases.
%
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig22a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig22b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig22c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig22d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_numerical_1}
Energy    computation    of    wave  propagation using medium inelastic material
model,    Rayleigh    damping    ($\xi$    =    0.04),  and  algorithmic damping
($\gamma=0.505$,  $\beta=0.253$): (a) Displacement time series; (b) Displacement
spectrum; (c) Energy
results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig23a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig23b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig23c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig23d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_numerical_2}
Energy    computation    of    wave  propagation using medium inelastic material
model,    Rayleigh    damping    ($\xi$    =    0.04),  and  algorithmic damping
($\gamma=0.510$,  $\beta=0.255$): (a) Displacement time series; (b) Displacement
spectrum; (c) Energy
results; (d) Stress-strain response.}
\end{figure}
%
\begin{figure}[!htbp]
\centering
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig24a.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig24b.pdf}}
\\
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig24c.pdf}}
\subfloat[]{\includegraphics[width=0.49\columnwidth]{Fig24d.pdf}}
\caption{\label{figure_wave_vm_rayleigh_numerical_3}
Energy  computation  of  wave propagation using medium inelastic material model,
Rayleigh    damping  ($\xi$  =  0.04), and algorithmic damping ($\gamma= 0.550$,
$\beta=0.276$): (a)
Displacement  time  series;  (b)  Displacement spectrum; (c) Energy results; (d)
Stress-strain response.}
\end{figure}
%
It is observed that for a relatively small Newmark parameters $\gamma=0.505$ and
$\beta=0.253$,  the  amount  of energy dissipation due to algorithmic damping is
comparable  but less than energy dissipation caused by material inelasticity and
viscous damping.
%
As  Newmark  parameter $\gamma$ increases to $\gamma=0.510$ and $\beta=0.253$, algorithmic damping
becomes the comparable energy dissipation mechanism in the system.
%
Then,  when  Newmark  parameter $\gamma$ increases even higher to $\gamma=0.550$
and $\beta=0.276$, the motions in the system are almost completely damped out by
algorithmic  damping,  and material inelasticity and viscous damping do not play
any significant role.



As  noted  by  \citeN{local-87},  high frequencies can be introduced into finite
element models, during discretization process.
%
It  is  thus  important  to  be  able  to  try  to damp those unrealistic higher
frequencies by applying a reasonable amount of algorithmic damping.
%
It  is  noted that introduction of algorithmic damping has to be done carefully,
as  in  some  cases,  as  shown  here, algorithmic damping can surpass all other
physical forms of energy dissipation.
%
It  is  thus    suggested   that  a  sensitivity  study  should be conducted for
dynamic  finite  element  models,  so  that  influence of algorithmic damping on
results is better understood.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}


Presented  was  a  study  on  energy  dissipation  due to material inelasticity,
Rayleigh damping, and algorithmic damping for dynamic loading of solids.
%
The  importance  of  proper  modeling    of energy dissipation due to inelastic
material behavior, due to Rayleigh damping, and due to algorithmic damping
in dynamic nonlinear systems was investigated and discussed.
%
The differences in system response with these three energy dissipation mechanisms
were  illustrated through a series of numerical examples.



Thermomechanics formulation that correctly  computes  plastic energy
dissipation in inelastic materials was presented.
%
The  important  role  of plastic free energy was emphasized.
%
In  addition,  Rayleigh  damping  and  corresponding  viscous energy dissipation
formulation was presented as well.
%
Use       of    significant, potentially  unrealistic    Rayleigh   damping  was
illustrated and discussed.
%
This  high  Rayleigh  damping  might  lead  to  unrealistically    high  energy 
dissipation and underestimation of the system response.
%
Considering  both plastic energy dissipation and viscous damping of the Rayleigh
type, the rate form of energy balance for a dynamic system was obtained.



Presented  analysis approach was illustrated using several numerical examples
with  energy  dissipation due to material inelasticity, and/or Rayleigh damping, and/or
algorithmic damping.
%
Permanent  deformation  was  observed  in the cases using inelastic material (as
expected), leading to the low frequency contents in the displacement response at
the ground surface.
%
Rayleigh  damping can  significantly reduce  high frequency motions, which might have 
 important implications in evaluating  the  safety  of structures, systems and components.



The  influence  of  algorithmic  damping  was illustrated and discussed as well.
%
Algorithmic damping mainly reduces higher frequency  motions, and it 
exists throughout simulation process.
%
Unrealistically  high  values  of  algorithmic  damping  can  be  obtained,  and
sensitivity  studies  are  needed  for  proper  choice  of  algorithmic  damping
parameters.



Although    the   amounts  of  dissipated energy  were comparable in some cases,
the   observed   system  responses  were  very  different  when  different  energy
dissipation mechanisms were used.
%
All  three  energy  dissipation  mechanisms,  material  inelasticity,   viscous 
damping,  and  algorithmic damping, model  fundamentally different physical
or mathematical phenomena in finite element simulation.
%
Therefore, it is important to use appropriate energy dissipation mechanisms  by 
following  proper  physics and mathematics, allowing analysts to gain confidence
in obtained results, and use those results for design, assessment and retrofits.






%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Acknowledgments}

This work was supported in part by the US-DOE.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\newpage

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%
%\bibliography{refmech}
%


\end{document}
