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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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\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}