text:
\section{Introduction}\label{SecIntro}

Understanding the mechanisms of acoustic nonlinearity in heterogeneous materials is an object of intensive studies \cite{Guyer99,Ostrovsky01,Guyer09,Lebedev14}. Experimental evidence has shown that media such as rocks and concrete possess an anomalously strong acoustic nonlinearity, which is of great importance for the description of ultrasonic phenomena including damage diagnostics. Besides the widely-studied nonlinear and hysteretic stress-strain relation \cite{Kadish96}, a long-time relaxation is also reported by most of the authors \cite{TenCate96,TenCate00}. This slow dynamics is typically observed in experiments of softening / hardening \cite{Renaud12,Riviere13}, where a bar is forced by a monochromatic excitation on a time interval, before the source is switched-off. During the experiment, the elastic modulus is measured by Dynamic Acousto-Elastic Testing methods. It can be observed that the elastic modulus decreases gradually (softening), and then it recovers progressively its initial value after the extinction of the source (hardening). The time scales of each stage is much longer than the time scale of the forcing, which justifies the term "slow dynamics".

The modelling of this slow dynamic effect has been investigated by many authors. An essentially phenomenological model is widely used for this purpose: the Preisach-Mayergoyz model (P-M model) based on the integral action of hysteretic elements connecting stress and strain \cite{Scalerandi06,Scalerandi10,Lebedev14}. This model initially arose from the theory of magnetism, where the "hysteron" has a clear physical significance. In elasticity, such a physical interpretation is not available. To overcome this limitation and to develop a rigorous theory, various authors have proposed alternative models based on clear mechanical concepts. To our knowledge, the first physical model of slow dynamics was described in \cite{TenCate00}, where the relaxation was related to the recovery of microscopic contact impeded by a smooth spectrum of energy barriers. This theory was extended in \cite{Aleshin07a,Aleshin07b}, and recently improved based on the analysis of inter-grain contacts and the resulting surface force potential with a barrier \cite{Lebedev14}. Another approach was followed in \cite{Pecorari04}, where the author shows that two rough surfaces interacting via adhesion forces yield dynamics similar to that of the fictitious elements of the Preisach-Mayergoyz space \cite{Pecorari04}.

Here, we present an alternative mechanical description of slow dynamics based on the works of Vakhnenko and coauthors \cite{Vakhnenko04,Vakhnenko05}, where the following scenario is proposed:
\begin{itemize}
\item the Young's modulus $E$ varies with time. One can write $E(g)$, where $g$ is a time-dependent concentration of defects. It is closely related to the notion of damage in solids mechanics. But contrary to what happens in this irreversible case, where $g$ strictly increases with time, the evolution of $g$ is reversible. Waiting a sufficiently long time, the initial material properties are recovered;
\item at equilibrium, stress $\sigma$ yields a concentration of defects $g_\sigma$. The dependence of $g_\sigma$ with respect to $\sigma$ is monotonic;
\item out of equilibrium, relaxation times are required for $g$ to reach $g_\sigma$. Whether $g<g_\sigma$ (increase in the number of defects) or $g>g_\sigma$ (decrease in the number of defects), Vakhnenko et al state that the time scales differ. The argument is given in section III of \cite{Vakhnenko05}: "there are various ways for an already existing crack in equilibrium to be further expanded when surplus tensile load is applied. However, under compressive load a crack, once formed, has only one spatial way to be annhilated or contracted". In both cases, these relaxation times are much longer than the time scale of the excitation, which explains the slow dynamics.
\end{itemize}
Comparisons with experimental data are given in section V of \cite{Vakhnenko05}, where the authors reproduced experiments done on Berea sandstone \cite{TenCate96}. One current weakness is that no micro-mechanical description of the involved defects has been proposed so far. A possible analogy may be found with populations of open / closed cracks filled with air, equivalent to a population of bubbles that relax towards an equilibrium state, depending on the applied stress \cite{Emelianov04,Gavrilyuk04}. In counterpart, one attractive feature of Vakhnenko's model is that it combines hyperbolic equations and relaxation terms, which constitutes a sound basis of physical phenomena \cite{Godunov03}.

The present paper is a contribution to the theoretical analysis of this model and to its practical implementation to describe wave motion in damaged media. First, we point out that no mechanisms prevents the concentration of defects from exceeding 1, which is physically unrealistic. We fix this problem by proposing another expression for the equilibrium concentration. Second, the Stokes model describing viscoelasticity behaviour in \cite{Vakhnenko05} poorly describes the attenuation in real media, and it is badly suited to time-domain simulations of wave propagation. Instead, we propose a new nonlinear version of the Zener model. This viscoelastic model degenerates correctly towards a pure nonlinear elasticity model when attenuation effects vanish. Moreover, the usual Zener model in the linear regime is recovered \cite{Carcione07}. In practice, this model only requires one physical parameter under the assumption of constant quality factor. Third, hyperbolicity is analyzed. Depending on the chosen model of nonlinear elasticity, a real sound speed may obtained only on a finite interval of strains; this is true in  particular with the widely-used Landau's model.

The main effort of Vakhnenko et al was devoted to the construction of a model of slow dynamics. The resolution of the involved equations was quite rudimentary and not satisfying. Indeed, the equilibrium concentration of defects $g_\sigma$ was assumed to be known and was imposed (eq (17) in \cite{Vakhnenko05}), while it depends on $\sigma$. But treating the full coupled nonlinear equations is out of reach of a semi-analytical approach, which explains the strategy of these authors. On the contrary, we propose here a numerical method to integrate the full system of equations, involving the nonlinear elasticity, the hysteretic terms of viscoelasticity, and the slow dynamics. Due to the existence of different time scales, a splitting strategy is followed, ensuring the optimal time step for integration. The full system is split into a propagative hyperbolic part (resolved by a standard scheme for conservation laws) and into a relaxed part (resolved exactly).

Our numerical model is very modular. The various bricks (nonlinear elasticity, viscoelasticity, slow dynamics) can be incorporated easily. Numerical tests validate each part separately. When all the whole bricks are put together, typical features of wave motion in damaged media are observed. The softening / hardening experiments are qualitatively reproduced.

\section{Physical modeling}\label{SecPhys}

In this section, we write the basic components describing the wave motion in a 1D material with damage. The fundations rely on linear elastodynamics, whose equations are recalled in section \ref{SecPhysLin}. Then, the soft-ratchet model of Vakhnenko and coauthors is introduced and enhanced in section \ref{SecPhysSoft}. The fast dynamics is described in section \ref{SecPhysElasto}, where various known models of nonlinear elasticity are presented, and a nonlinear model of viscoelasticity is proposed. This latter degenerates correctly in the limit cases of linear elasticity or null attenuation.

\subsection{Linear elastodynamics}\label{SecPhysLin}

In the case of small deformations, the propagation of 1D elastic waves can be described by the following system \cite{Achenbach73}:
\begin{subnumcases}{\label{ElastoLin}}
\frac{\partial v}{\partial t}-\frac{1}{\rho}\frac{\partial \sigma}{\partial x}=\gamma,\label{ElastoLin1}\\
[6pt]
\frac{\partial \varepsilon}{\partial t}-\frac{\partial v}{\partial x}=0,\label{ElastoLin2}
\end{subnumcases}
where $t$ is the time, $x$ is the spatial coordinate, $\gamma$ is a forcing term, $u$ is the displacement, $v=\frac{\partial u}{\partial t}$ is the velocity, $\varepsilon=\frac{\partial u}{\partial x}$ is the strain, and $\sigma$ is the stress. The latter is a function of strain: $\sigma=\sigma(\varepsilon)$.

In the linear case, Hooke's law writes $\sigma=E\,\varepsilon$, where $E$ is the Young's modulus, which is assumed to be constant over time. In the particular case where $\gamma$ is a Dirac source at $x_s$ with time evolution ${\cal G}(t)$, then the exact solution of (\ref{ElastoLin}) is straightforward
\begin{equation}
\varepsilon=-\frac{\mbox{sgn}(x-x_s)}{2\,c^2}\,{\cal G}\left(t-\frac{|x-x_s|}{c}\right),
\label{ElastoExact}
\end{equation}
where sgn is the sign distribution, and $c=\sqrt{\frac{1}{\rho}\frac{\partial \sigma}{\partial \varepsilon}}\equiv\sqrt{E/\rho}$ is the speed of sound.

The goal of the forthcoming sections is to extend the model (\ref{ElastoLin}) in three ways:
\begin{itemize}
\item time variations of $E$ due to the stress;
\item nonlinear Hooke's law;
\item hereditary effects (viscoelasticity).
\end{itemize}
The time scales for the first effect (variation of $E$) are much greater than for the second and third effect. This is consequently referred to as {\it slow dynamics}.

\subsection{Slow dynamics: soft-ratchet model}\label{SecPhysSoft}

Here we follow the approach taken from \cite{Vakhnenko04,Vakhnenko05} with some modifications. The slow dynamics of the medium is assumed to rely on the concentration of activated defects $g$, which varies with $\sigma$. In the lowest approximation, the Young's modulus is written:
\begin{equation}
E=\left(1-\frac{\textstyle g}{\textstyle g_{cr}}\right)\,E^+,
\label{Young}
\end{equation}
where $g_{cr}$ and $E^+$ are the critical concentration of defects and the maximum possible value of Young's modulus, respectively (figure \ref{FigYoung}-(a)). The following constraints hold:
\begin{equation}
0 \leq g\leq g_{cr}\leq 1.
\label{HypVakh}
\end{equation}

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm}
\includegraphics[scale=0.42]{Young2.eps} &
\hspace{-0.4cm}
\includegraphics[scale=0.42]{Gcinetique2.eps}
\end{tabular}
\caption{\label{FigYoung} parameters of the slow dynamics. (a): Young's modulus $E$ in terms of the concentration of defects $g$ (\ref{Young}), for $E^+=14.28$ GPa; the vertical dotted line denotes the initial concentration of defects $g_0=0.3$ and the corresponding Young's modulus $E_0=E(g_0)=10$ GPa. (b): time evolution of the concentration of defects $g$ given an equilibrium stress $\sigma$ and two initial values $g_0$; the horizontal dotted line denotes $g_\sigma$.}
\end{center}
\end{figure}

The concentration $g$ is assumed to evolve to its stress-dependent equilibrium value $g_\sigma$ at a rate $f_r$ if $g>g_\sigma$ (restoration), or $f_d$ if $g<g_\sigma$ (destruction). This mechanism can be modeled by the ordinary differential equation
\begin{equation}
\frac{\textstyle dg}{\textstyle dt}=-\left(f_r\,H(g-g_\sigma)+f_d\,H(g_\sigma-g)\right)\,(g-g_\sigma),
\label{Kinetic}
\end{equation}
where $H$ is the Heaviside step distribution. The frequencies $f_r$ and $f_d$ differ substantially:
\begin{equation}
f_r \ll f_d \ll f_c,
\label{TimeScales}
\end{equation}
where $f_c$ is a typical frequency of the excitation. Figure \ref{FigYoung}-(b) represents the time evolution of $g$, given a constant equilibrium concentration $g_\sigma=0.3$ denoted by a horizontal dotted line. The restoration and rupturation frequencies are $f_r=25$ Hz and $f_d=250$ Hz, respectively. Two initial value of the concentration of defects are considered: $g_0=0.2$ and $g_0=0.4$. In both cases, $g$ tends towards $g_\sigma$ with different rates: destruction is much faster than restoration.

It remains to define the evolution of $g_\sigma$ with $\sigma$. In \cite{Vakhnenko04,Vakhnenko05}, the authors propose the expression
\begin{equation}
g_\sigma=g_0\,\exp(\sigma/\overline{\sigma}), \hspace{1cm}\overline{\sigma}=\frac{\textstyle kT}{\textstyle \upsilon},
\label{GsigVakh}
\end{equation}
where $g_0$ is the unstrained equilibrium concentration of defects, $k$ is the Boltzmann constant, $T$ is the temperature, and $\upsilon$ is a typical volume accounting for a single defect. If $ \sigma>\overline{\sigma}\ln g_{cr}/g_0$, then $g_\sigma>g_{cr}$; in this case, the concentration may evolve to $g>g_{cr}$ due to equation (\ref{Kinetic}), which contradicts the second assumption in (\ref{HypVakh}). To remove this drawback and to build a physically realistic expression of $g_\sigma$, we enforce (\ref{HypVakh}) together with the following requirements:
\begin{subnumcases}{\label{Hyp}}
0\leq g_\sigma<g_{cr},\label{Hyp1}\\
g_\sigma(0)=g_0,\label{Hyp3}\\
\lim_{\sigma\rightarrow -\infty}g_\sigma=0,\label{Hyp2}\\
\lim_{\sigma\rightarrow +\infty}g_\sigma=g_{cr},\label{Hyp4}\\
\frac{\textstyle \partial g_\sigma}{\textstyle \partial \sigma}>0.\label{Hyp5}
\end{subnumcases}
The simplest smooth function satisfying (\ref{Hyp}) is
\begin{equation}
g_\sigma=\frac{\textstyle g_{cr}}{\textstyle 2}\left(1+\tanh\left(\frac{\textstyle \sigma-\sigma_c}{\textstyle \overline{\sigma}}\right)\right),
\label{GsigFLP}
\end{equation}
where the central stress is
\begin{equation}
\sigma_c=\overline{\sigma}\,\tanh^{-1}\left(1-2\frac{\textstyle g_0}{\textstyle g_{cr}}\right).
\label{Sig0}
\end{equation}
Figure \ref{FigSRsigbar}-(a) illustrates the two expressions of the stress-dependent equilibrium value $g_\sigma$: the "exponential model" (\ref{GsigVakh}), and the "tanh model" (\ref{GsigFLP})-(\ref{Sig0}). The numerical values are $g_0=0.3$ and $\overline{\sigma}=10^5$ Pa. The two expressions are the same at null stress. But for tractions greater than 230 kPa, the value of $g_\sigma$ deduced from (\ref{GsigVakh}) exceeds 1, leading to non-physical negative Young's modulus. Figure \ref{FigSRsigbar}-(b) illustrates the influence of $\overline{\sigma}$ in (\ref{GsigFLP}). As $\overline{\sigma}$ decreases, $g_\sigma$ may evolve more easily towards the extreme values 0 and $g_{cr}$, and hence the damage may increase thanks to (\ref{Kinetic}).

\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm}
\includegraphics[scale=0.33]{Gsig3.eps} &
\hspace{-0.5cm}
\includegraphics[scale=0.33]{SigBar3.eps}
\end{tabular}
\caption{\label{FigSRsigbar} equilibrium concentration of defects $g_\sigma$ in terms of the applied stress $\sigma$. (a): "exponential model" (\ref{GsigVakh}) and "tanh model" (\ref{GsigFLP}). (b): "tanh model" (\ref{GsigFLP}) with various values of $\overline{\sigma}$. The horizontal dotted line denotes the critical concentration of defects $g_{cr}$; the vertical dotted line denotes the central stress $\sigma_c$. }
\end{center}
\end{figure}

\subsection{Fast dynamics: nonlinear viscoelasticity}\label{SecPhysElasto}

\paragraph{Nonlinear elasticity} ~\\
The stress-strain relation is given by a smooth function
\begin{equation}
s\equiv s(\epsilon,\,K,\,{\bf p}),
\label{ElastoNL}
\end{equation}
where $s$ is the stress, $\epsilon$ is the strain, $K$ is a stiffness, and ${\bf p}$ is a set of parameters governing the nonlinearity. No pre-stress is considered; $K$ is the slope of $s$ at the origin; lastly, $s$ is homogeneous of degree 1 in $K$. In other words, $s$ satisfies the following properties:
\begin{equation}
s(0,\,K,\,{\bf p})=0,\hspace{0.5cm} \frac{\partial s}{\partial \epsilon}(0,\,K,\,{\bf p})=K,\hspace{0.5cm} s(\epsilon,\,\alpha K,{\bf p})=\alpha\,s(\epsilon,\,K,\,{\bf p}).
\label{PropElastoNL}
\end{equation}
Three models of nonlinear elasticity (\ref{ElastoNL}) satisfying (\ref{PropElastoNL}) are now given and illustrated in figure \ref{FigElastoNL}.

\noindent
\underline{Model 1}. This model is from \cite{Vakhnenko05} and mimics the Lennard-Jones potential describing the interaction between a pair of neutral atoms:
\begin{equation}
s(\epsilon,\,K,\,{\bf p})=K \frac{\textstyle d}{\textstyle r-a}\left(\frac{1}{\left(\displaystyle 1+\frac{\epsilon}{d}\right)^{a+1}}-\frac{1}{\left(\displaystyle 1+\frac{\epsilon}{d}\right)^{r+1}}\right),\hspace{0.5cm} {\bf p}=(r,\,a,\,d)^T.
\label{Model1}
\end{equation}
The nonlinear parameters are the repulsion and attraction coefficients $r$ and $a$ ($0<a<r$). The strain is bounded below by the maximal allowable closure $d$. The function (\ref{Model1}) has an extremal point $\epsilon_c>0$, and then it decreases asymptotically towards 0 when $\epsilon>\epsilon_c$ (figure \ref{FigElastoNL}-(a)).

\noindent
\underline{Model 2}. A third-order Taylor expansion of the model 1 (\ref{Model1}) yields
\begin{equation}
s(\epsilon,\,K,\,{\bf p})=K\,\epsilon\left(1-\frac{1}{2}\left(r+a+3\right)\frac{\textstyle \epsilon}{\textstyle d}+\frac{\textstyle 1}{\textstyle 6}\left(r^2+ra+a^2+6r+6a+11\right)\left(\frac{\textstyle \epsilon}{\textstyle d}\right)^2\right),\hspace{0.5cm}{\bf p}=(r,\,a,\,d)^T.
\label{Model2}
\end{equation}
The nonlinear parameters are the same than in model 1. But contrary to what happened in model 1, the function (\ref{Model2}) is a strictly monotonically increasing function without extremal point (figure \ref{FigElastoNL}-(a)). Moreover, the strain is not bounded below.

\noindent
\underline{Model 3}. The most widely used law in ultrasonic NonDestructive Testing is the so-called Landau's model \cite{Landau70}
\begin{equation}
s(\epsilon,\,K,\,{\bf p})=K\,\epsilon\left(1-\beta\,\epsilon-\delta\,\epsilon^2\right),\hspace{0.5cm} {\bf p}=(\beta,\,\delta)^T.
\label{Model3}
\end{equation}
The parameters governing the nonlinear behavior are $\beta$ and $\delta$; in practice, $\beta \ll \delta$. Like what happens with model 1, the function (\ref{Model3}) has extremal points, but it is not bounded below (figure \ref{FigElastoNL}-(b)).

\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm}
\includegraphics[scale=0.35]{ElastoNL12.eps} &
\hspace{-0.9cm}
\includegraphics[scale=0.35]{ElastoNL3.eps}
\end{tabular}
\caption{\label{FigElastoNL} Stress-strain relations for the three models (\ref{ElastoNL}). In (a), the dotted lines denote the coordinates of the inflexion point for model 1. The physical parameters are: $E=10$ GPa, $d=4.3\,10^{-4}$ m, $a=2$, $r=4$ (models 1 and 2), $\beta=100$, $\delta=10^8$ (model 3).}
\end{center}
\end{figure}

\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.30]{Zener.eps}
\end{tabular}
\caption{\label{FigZener} Rheological model of a generalized Zener material.}
\end{center}
\end{figure}

\paragraph{Viscoelasticity} ~\\
To incorporate attenuation, the following criteria are used as a guideline:
\begin{description}
\item[${\cal C}_1$:] when the viscous effects are null, the nonlinear elasticity must be recovered (\ref{ElastoNL});
\item[${\cal C}_2$:] when a linear stress-strain relation holds, it is necessary to recover the standard linear solid model (or generalized Zener model), which accurately represents the behavior of usual solids \cite{Carcione07}.
\end{description}
For this purpose, a system with $N$ Zener elements connected in parallel is considered (figure \ref{FigZener}). The total stress acting on the system is
\begin{equation}
\sigma=\sum_{\ell=1}^N\sigma_{1\ell}=\sum_{\ell=1}^N(\sigma_{2\ell}+\sigma_{3\ell}),
\label{MailleSigma}
\end{equation}
where the index 1 refers to the springs in series, and indices 2-3 refer to the springs and dashpots in parallel. The strain $\varepsilon$ is
\begin{equation}
\varepsilon=\varepsilon_{1\ell}+\varepsilon_{2\ell},\hspace{1cm}\ell=1,\cdots,N.
\label{MailleEps}
\end{equation}
The index 1 springs satisfy nonlinear stress-strain relations (\ref{ElastoNL}) with stiffnesses $K_{1\ell}$. The parameters ${\bf p}$ governing the nonlinearity (for instance $\beta$ and $\delta$ in model 3 (\ref{Model3})) are assumed to be constant and identical for each element. The index 2 springs satisfy linear stress-strain relations with stiffnesses $K_{2\ell}$. Lastly, the dashpots satisfy linear Maxwell laws with coefficients of viscosity $\eta_\ell$. These laws are summed up as follows:
\begin{subnumcases}{\label{Rheo}}
\displaystyle
\sigma_{1\ell}(\varepsilon_{1\ell})=s(\varepsilon_{1\ell},\,K_{1\ell},\,{\bf p}),\label{Rheo1}\\
[6pt]
\displaystyle
\sigma_{2\ell}(\varepsilon_{2\ell})=s(\varepsilon_{2\ell},\,K_{2\ell},\,{\bf 0}),\label{Rheo2}\\
\displaystyle
\sigma_{3\ell}(\varepsilon_{2\ell})=\eta_\ell\frac{\textstyle \partial \varepsilon_{2\ell}}{\textstyle \partial t}\label{Rheo3}.
\end{subnumcases}
To determine the parameters $K_{1\ell}$, $K_{2\ell}$ and $\eta_\ell$, the relaxation times $\tau_{\sigma\ell}$, $\tau_{\varepsilon\ell}$ and the relaxed modulus $E_R$ are introduced:
\begin{equation}
\tau_{\sigma\ell}=\frac{\eta_\ell}{K_{1\ell}+K_{2\ell}},\hspace{0.5cm}
\tau_{\varepsilon\ell}=\frac{\eta_\ell}{K_{2\ell}},\hspace{0.5cm}
\frac{E_R}{N}=\frac{K_{1\ell}\,K_{2\ell}}{K_{1\ell}+K_{2\ell}}.
\label{TauEta}
\end{equation}
On the one hand, a procedure is given in \ref{SecViscoCoeff} to compute the relaxation times in terms of the quality factor $Q$. On the other hand, $E_R$ is related to the unrelaxed Young's modulus $E$ (\ref{Young}) and to the relaxation times previsously determined (see \cite{Carcione07}):
\begin{equation}
E_R=\frac{N}{\displaystyle \sum_{\ell=1}^N\frac{\tau_{\varepsilon\ell}}{\tau_{\sigma\ell}}}\,E.
\label{ErVsE}
\end{equation}
Once $\tau_{\sigma\ell}$, $\tau_{\varepsilon\ell}$ and $E_R$ are determined, inverting (\ref{TauEta}) provides the values of the viscoelastic model in terms of relaxed modulus and relaxation times ($\ell=1,\cdots,N$):
\begin{equation}
K_{1\ell}=\frac{\tau_{\varepsilon\ell}}{\tau_{\sigma\ell}}\,\frac{E_R}{N},\hspace{0.5cm}
K_{2\ell}=\frac{\tau_{\varepsilon\ell}}{\tau_{\varepsilon\ell}-\tau_{\sigma\ell}}\,\frac{E_R}{N},\hspace{0.5cm}
\eta_\ell=\frac{\tau^2_{\varepsilon\ell}}{\tau_{\varepsilon\ell}-\tau_{\sigma\ell}}\,\frac{E_R}{N}.
\label{E1E2Eta}
\end{equation}
From (\ref{ErVsE}) and (\ref{E1E2Eta}), it follows that the viscoelastic parameters depend indirectly on the Young's modulus $E$, and thus depend on $g$. In other words, the proposed model of viscoelasticity evolves with the concentration of defects and thus with the applied stress.

In the inviscid case, the stress-strain relation deduced from (\ref{MailleSigma})-(\ref{Rheo}) makes it possible to recover the nonlinear elasticity (\ref{ElastoNL}), whatever the number $N$ of relaxation mechanisms:
\begin{equation}
\sigma=s(\varepsilon,E,{\bf p}).
\label{Consistance}
\end{equation}
This property is proven in \ref{ProofHyp}.

\section{Mathematical modeling}\label{SecMath}

In this section, the basic components describing wave motion in damaged media are put together and analysed. Section \ref{SecMathEdp} collects the various mechanisms (nonlinear elastodynamics, slow dynamics, hysteresis) into a single system of first-order equations. Two important properties of this system are addressed in section \ref{SecMathProp}: hyperbolicity (finite sound velocity) and decrease in energy.

\subsection{First-order system}\label{SecMathEdp}

The conservation of momentum (\ref{ElastoLin1}) writes
\begin{equation}
\frac{\partial v}{\partial t}=\frac{1}{\rho}\frac{\partial \sigma}{\partial x}+\gamma,
\label{FO1}
\end{equation}
where $\gamma$ is a forcing term, and $\sigma$ is given by (\ref{MailleSigma}). The hypothesis of small deformations (\ref{ElastoLin2}) gives
\begin{equation}
\frac{\partial \varepsilon}{\partial t}=\frac{\partial v}{\partial x}.
\label{FO2}
\end{equation}
Lastly, manipulations on (\ref{MailleSigma}), (\ref{MailleEps}) and (\ref{Rheo3}) yield
\begin{equation}
\frac{\partial \varepsilon_{1\ell}}{\partial t}=\frac{\partial v}{\partial x}+\frac{\sigma_{2\ell}(\varepsilon-\varepsilon_{1\ell})-\sigma_{1\ell}(\varepsilon_{1\ell})}{\eta_\ell},\hspace{1cm}\ell=1,\cdots,N.
\label{FO3}
\end{equation}
In (\ref{FO3}), $\varepsilon_{1\ell}$ takes the place of the memory variables proposed in \cite{Lombard11} and is better suited to nonlinear elasticity. Putting together (\ref{FO1})-(\ref{FO3}) and the relaxation equation (\ref{Kinetic}) leads to the first-order system of $N+3$ evolution equations
\begin{subnumcases}{\label{EDP}}
\frac{\partial v}{\partial t}-\frac{1}{\rho}\frac{\partial \sigma}{\partial x}=\gamma,\label{EDP1}\\
[4pt]
\frac{\partial \varepsilon}{\partial t}-\frac{\partial v}{\partial x}=0,\label{EDP2}\\
[4pt]
\frac{\partial \varepsilon_{1\ell}}{\partial t}-\frac{\partial v}{\partial x}=\frac{\sigma_{2\ell}(\varepsilon-\varepsilon_{1\ell})-\sigma_{1\ell}(\varepsilon_{1\ell})}{\eta_\ell}, \hspace{1cm}\ell=1,\cdots,\,N,\label{EDP3}\\
[4pt]
\frac{\textstyle dg}{\textstyle dt}=-\left(f_r\,H(g-g_\sigma)+f_d\,H(g_\sigma-g)\right)\,(g-g_\sigma).\label{EDP4}
\end{subnumcases}
To close the system (\ref{EDP}), the following equations are recalled:
\begin{itemize}
\item The total stress $\sigma$ in (\ref{EDP1}) depends on $\varepsilon_{1\ell}$ via (\ref{MailleSigma}), (\ref{Rheo1}), and a nonlinear law (\ref{ElastoNL}):
\begin{equation}
\sigma=\sum_{\ell=1}^N s(\varepsilon_{1\ell},\,K_{1\ell},\,{\bf p}).
\label{EDPsigma}
\end{equation}
\item The stress components $\sigma_{1\ell}$ and $\sigma_{2\ell}$ in (\ref{EDP3}) depend on the stifnesses $K_{1\ell}$ and $K_{2\ell}$ (\ref{Rheo1}) and (\ref{Rheo2}). The latter, as well as the viscosity coefficients $\eta_\ell$, depend on the Young modulus $E$ via (\ref{ErVsE})-(\ref{E1E2Eta}), and thus on $g$:
\begin{equation}
E=\left(1-\frac{\textstyle g}{\textstyle g_{cr}}\right)\,E^+.
\label{EDPYoung}
\end{equation}
\item The equilibrium value of the defect concentration $g_\sigma$ in (\ref{EDP4}) satisfies (\ref{GsigFLP}) and (\ref{Sig0}):
\begin{equation}
g_\sigma=\frac{\textstyle g_{cr}}{\textstyle 2}\left(1+\tanh\left(\frac{\textstyle \sigma-\sigma_c}{\textstyle \overline{\sigma}}\right)\right).
\label{EDPgsig}
\end{equation}
\end{itemize}
The system (\ref{EDP}), together with equations (\ref{EDPsigma})-(\ref{EDPgsig}), generalizes the standard equations of linear elastodynamics (\ref{ElastoLin}). It accounts for softening / recovering of Young's modulus, nonlinearity and viscoelasticity.

For the sake of clarity, the vector of $N+3$ variables is introduced
\begin{equation}
{\bf U}=\left(v,\,\varepsilon,\,\varepsilon_{11},\cdots,\,\varepsilon_{1N},\,g\right)^T.
\label{VecU}
\end{equation}
Then the system (\ref{EDP}) can be put in the form
\begin{equation}
\frac{\partial}{\partial t}{\bf U}+\frac{\partial}{\partial x}{\bf F}({\bf U})={\bf R}({\bf U})+{\bf \Gamma}.
\label{SysCons}
\end{equation}
The flux function ${\bf F}$, the relaxation term ${\bf R}$, and the forcing ${\bf \Gamma}$ are
\begin{equation}
\begin{array}{l}
\displaystyle
{\bf F}({\bf U})=\left(-\frac{\sigma}{\rho},\,-v,\,-v,\cdots,\,-v,0\right)^T,\\
[10pt]
\displaystyle
{\bf R}({\bf U})=\left(0,\,0,\,\Delta_1,\cdots,\,\Delta_N,-\left(f_r\,H(g-g_\sigma)+f_d\,H(g_\sigma-g)\right)\,(g-g_\sigma)
\right)^T,\\
[10pt]
\displaystyle
{\bf \Gamma}=\left(\gamma,\,0,\cdots,\,0,\,0\right)^T,
\label{FluxFunc}
\end{array}
\end{equation}
where
\begin{equation}
\Delta_\ell=\frac{\sigma_{2\ell}(\varepsilon-\varepsilon_{1\ell})-\sigma_{1\ell}(\varepsilon_{1\ell})}{\eta_\ell}.
\label{DeltaL}
\end{equation}
To conclude, let us consider the limit-case where the viscoelastic attenuation is neglected. In this case, equation (\ref{Consistance}) states that the stress-strain relations degenerate rigorously towards pure nonlinear elasticity, whatever $ N$.

\subsection{Properties}\label{SecMathProp}

Hyperbolicity is a crucial issue in wave problems - physically, mathematically, and numerically. It amounts to saying that there exists a real and finite sound velocity $c$. This property was analysed in \cite{Ndanou14} for a particular nonlinear stress-strain relation in 3D. In 1D, it reduces to a simpler case detailed as follows.
Let us define the sound speed $c$ by
\begin{equation}
c^2=\sum_{\ell=1}^Nc_\ell^2=\frac{1}{\rho}\sum_{\ell=1}^N\frac{\partial \sigma_{1\ell}}{\partial \varepsilon_{1\ell}}.
\label{C2}
\end{equation}
The system (\ref{SysCons}) is hyperbolic if and only if $c^2>0$ in (\ref{C2}).
The proof, as well as sufficient conditions on the strain to ensure hyperbolicity, is given in \ref{ProofHyp}. From (\ref{C2}), the local elastic modulus $M$ can be deduced:
\begin{equation}
M=\rho\,c^2=\sum_{\ell=1}^N\frac{\partial \sigma_{1\ell}}{\partial \varepsilon_{1\ell}}.
\label{ModuleC}
\end{equation}
Note that the Stokes viscoelastic model used in \cite{Vakhnenko05} introduces a term $\frac{\partial^2 v}{\partial x^2}$ in the right-hand side of (\ref{EDP3}). This Laplacian term destroys the hyperbolic character of the system (\ref{SysCons}). The viscoelastic model used here has therefore better mathematical properties.

Now let us examine the spectrum of the relaxation function in (\ref{SysCons}).
Let us consider linear stress-strain relations. The parameters $K_{1\ell}$, $K_{2\ell}$ and $\eta_{\ell}$ are "freezed" in (\ref{ErVsE})-(\ref{E1E2Eta}), so that they do not depend on $g$ via $E$ (\ref{Young}). Then, the eigenvalues of the Jacobian matrix ${\bf J}=\frac{\partial {\bf R}}{\partial {\bf U}}$ are
\begin{equation}
\mathrm{Sp}({\bf J})=\left\{0^2,\,-f_\xi,\,-\frac{K_{1\ell}+K_{2\ell}}{\eta_\ell}\right\}=\left\{0^2,\,-f_\xi,\,-\frac{1}{\tau_{\sigma\ell}}\right\},\hspace{0.5cm}\ell=1,\,\cdots,\,N,
\label{SpectreJ}
\end{equation}
(see (\ref{TauEta})), with $f_\xi=f_r$ if $g>g_\sigma$, $f_\xi=f_d$ if $g<g_\sigma$, $f_\xi=0$ else.
The proof is detailed in \ref{ProofRelax}. Two observations can be made:
\begin{itemize}
\item ${\bf J}$ is definite-negative if the relaxation frequencies $\tau_{\sigma\ell}$ are positive. The latter parameters are deduced from an optimization process based on the quality factor (\ref{SecViscoCoeff}). To ensure the energy decrease, it is therefore crucial to perform nonlinear optimization with constraint of positivity.
\item The optimization procedure detailed in \ref{SecViscoCoeff} is performed on the frequency range $[f_{\min},\,f_{\max}]$ surrounding the excitation frequency $f_c$. These frequencies satisfy
\begin{equation}
f_{\min}\approx \frac{1}{\max \tau_{\sigma\ell}}<f_c<f_{\max}\approx \frac{1}{\min \tau_{\sigma\ell}}.
\label{SpectreRange}
\end{equation}
In (\ref{SpectreRange}), $\approx$ are replaced by equalities if a linear optimisation is used \cite{Lombard11}. From (\ref{TimeScales}), it follows the spectral radius of ${\bf J}$
\begin{equation}
\varrho({\bf J})=\frac{1}{\min \tau_{\sigma\ell}} \gg f_\xi,
\label{Rspectre}
\end{equation}
so that the system (\ref{SysCons}) is stiff.
\end{itemize}

\section{Numerical modeling}\label{SecNum}

In this section, a numerical strategy is proposed to integrate the first-order equations (\ref{SysCons}). For the sake of efficiency, a splitting approach is followed in section \ref{SecNumSplit}. The original equations are splitted into two parts, solved successively: a propagative part (section \ref{SecNumHyp}) and a relaxation part (section \ref{SecNumRelax}).

\subsection{Splitting}\label{SecNumSplit}

To integrate (\ref{SysCons}), a uniform spatial mesh $\Delta x$ and a variable time step $\Delta t^{(n)}\equiv \Delta t$ are introduced. An approximation ${\bf U}_i^n$ of the exact solution ${\bf U}(x_i=i\,\Delta x,\,t_n=t_{n-1}+\Delta t)$ is sought. A first strategy is to discretize explicitly the non-homogeneous system (\ref{SysCons}). But numerical stability implies a bound of the form
\begin{equation}
\Delta t\leq \min\left(\frac{\Delta x}{c_{\max}},\,\frac{2}{\varrho({\bf J})}\right),
\label{CFLdirect}
\end{equation}
where $c_{\max}=\max c_i^n$ is the maximal sound velocity at time $t_n$, and $\varrho({\bf J})$ is the spectral radius of the Jacobian of the relaxation term. As deduced from (\ref{Rspectre}), the second bound in (\ref{CFLdirect}) is penalizing compared with the standard CFL condition $\Delta t\leq \Delta x / c_{\max}$.

Here we follow another strategy: equation (\ref{SysCons}) is split into a hyperbolic step
\begin{equation}
\frac{\partial}{\partial t}{\bf U}+\frac{\partial}{\partial x}{\bf F}({\bf U})={\bf 0}
\label{SplittingHyp}
\end{equation}
and a relaxation step
\begin{equation}
\frac{\partial}{\partial t}{\bf U}={\bf R}({\bf U})+{\bf \Gamma}.
\label{SplittingRelax}
\end{equation}
The discrete operators associated with the discretization of (\ref{SplittingHyp}) and (\ref{SplittingRelax}) are denoted ${\bf H}_h$ and ${\bf H}_r$, respectively. The second-order Strang splitting is used, solving successively (\ref{SplittingHyp}) and (\ref{SplittingRelax}) with adequate time increments:
\begin{subnumcases}{\label{Splitting}}
\displaystyle
{\bf U}_i^{(1)}={\bf H}_r\left(\frac{\Delta t}{2}\right)\,{\bf U}_i^{n},\label{Splitting1}\\
[6pt]
\displaystyle
{\bf U}_i^{(2)}={\bf H}_h\left(\Delta t\right)\,{\bf U}_i^{(1)},\label{Splitting2}\\
[6pt]
\displaystyle
{\bf U}_i^{n+1}={\bf H}_r\left(\frac{\Delta t}{2}\right)\,{\bf U}_i^{(2)}.\label{Splitting3}
\end{subnumcases}
Provided that ${\bf H}_h$ and ${\bf H}_r$ are second-order accurate and stable operators, the time-marching (\ref{Splitting}) gives a second-order accurate approximation of the original equation (\ref{SysCons}) \cite{Leveque02}.

\subsection{Hyperbolic step}\label{SecNumHyp}

The homogeneous equation (\ref{SplittingHyp}) is solved by a conservative scheme for hyperbolic systems \cite{Leveque02}
\begin{equation}
{\bf U}_i^{n+1}={\bf U}_i^n-\frac{\textstyle \Delta t}{\textstyle \Delta x}\left({\bf F}_{i+1/2}-{\bf F}_{i-1/2}\right).
\label{Conservatif}
\end{equation}
Many sophisticated schemes can be used for this purpose \cite{LeVeque03}. For the sake of simplicity and robustness, the Godunov scheme is used here. The numerical flux function ${\bf F}_{i+1/2}$ is computed using the Rusanov method \cite{Toro99}
\begin{equation}
{\bf F}_{i+1/2}=\frac{\textstyle 1}{\textstyle 2}\left({\bf F}({\bf U}_{i+1}^n)+{\bf F}({\bf U}_{i}^n)-\lambda_{i+1/2}^n({\bf U}_{i+1}^n-{\bf U}_i^n)\right),
\label{Rusanov}
\end{equation}
where ${\bf F}$ is the flux function (\ref{FluxFunc}), and the diffusion parameter $\lambda_{i+1/2}^n$ is given by the Davis approximation \cite{Davis88}
\begin{equation}
\lambda_{i+1/2}^n=\max\left(c_i^n,c_{i+1}^n\right).
\label{Davis}
\end{equation}
The Godunov scheme is first-order accurate and stable under the usual Courant-Friedrichs-Lewy (CFL) condition
\begin{equation}
\Delta t= \frac{\alpha\,\Delta x}{c_{\max}},\mbox{  with } \alpha \leq 1.
\label{CFL}
\end{equation}

\subsection{Relaxation step}\label{SecNumRelax}

Let us denote $\overline{{\bf U}}=(\varepsilon,\,\varepsilon_{11},\cdots,\,\varepsilon_{1N})$ and $\overline{{\bf R}}$ the restriction of ${\bf R}({\bf U})$ to the strain components (\ref{FluxFunc})-(\ref{DeltaL}). The ordinary differential equation (\ref{SplittingRelax}) can then be written
\begin{subnumcases}{\label{ODE}}
\frac{\partial v}{\partial t}=\gamma,\label{ODE1}\\
[4pt]
\frac{\partial}{\partial t}\overline{{\bf U}}=\overline{{\bf R}}(\overline{{\bf U}}),\label{ODE2}\\
[4pt]
\frac{\textstyle dg}{\textstyle dt}=-\left(f_r\,H(g-g_\sigma)+f_d\,H(g_\sigma-g)\right)\,(g-g_\sigma),\label{ODE3}
\end{subnumcases}
The viscoelastic parameters in the relaxation function $\overline{{\bf R}}$ depend implicitly on $g$ (see section \ref{SecPhysElasto}), which complicates the resolution of (\ref{ODE1}). However, one can take advantage of the scaling (\ref{TimeScales}). Indeed, $\varepsilon$ and $\varepsilon_{1\ell}$ evolve much faster than $g$, so that the viscoelastic parameters $K_{1\ell}$, $K_{2\ell}$, $\eta_\ell$ are almost constant on a time step. Consequently, they are "freezed" and the three equations in (\ref{ODE}) can be solved separately.

The half-time step in the relaxation steps (\ref{Splitting1})-(\ref{Splitting3}) is denoted by $\tau=\frac{\Delta t}{2}$. One details the time-stepping from $t_n$ to the first intermediate step (\ref{Splitting1}); adaptation to the third intermediate step (\ref{Splitting3}) is straightforward.

The first equation (\ref{ODE1}) is integrated using the Euler method:
\begin{equation}
v_i^{n+1}=v_i^{(1)}+\Delta t\,\gamma(i,\,t_n).
\label{EulerODE2}
\end{equation}
To integrate the second equation (\ref{ODE2}), a first-order Taylor expansion of $\overline{{\bf R}}(\overline{{\bf U}})$ is performed
\begin{equation}
\frac{\partial}{\partial t}\overline{{\bf U}}\approx\overline{{\bf R}}(\overline{{\bf 0}})+\frac{\partial \overline{{\bf R}}}{\partial \overline{{\bf U}}}({\bf 0})\,\overline{{\bf U}}={\bf \overline{J}}\,\overline{{\bf U}},
\label{SemiLinear}
\end{equation}
where ${\bf \overline{J}}$ is the Jacobian matrix (\ref{MatJ}); the nullity of stress at zero strain has been used (\ref{Rheo1}). Then (\ref{SemiLinear}) is solved exactly, leading to the relaxation operator
\begin{equation}
\overline{{\bf U}}^{(1)}_i=e^{{\bf \overline{J}}\,\tau}\,\overline{{\bf U}}^{n}_i
\label{ExpoS1}
\end{equation}
with the matrix exponential
\begin{equation}
e^{{\bf \overline{J}}\,\tau}=
\left(
\begin{array}{cccc}
1   & 0 & \cdots & 0\\
[8pt]
\displaystyle \frac{E_{21}}{E_{11}+E_{21}}\left(1-e^{-\frac{E_{11}+E_{21}}{\eta_1}\,\tau}\right) & \displaystyle e^{-\frac{E_{11}+E_{21}}{\eta_1}\,\tau} &  &  \\
\vdots & & \ddots & \\
\displaystyle \frac{E_{2N}}{E_{1N}+E_{2N}}\left(1-e^{-\frac{E_{1N}+E_{2N}}{\eta_N}\,\tau}\right) & & & e^{-\frac{E_{1N}+E_{2N}}{\eta_N}\,\tau}
\end{array}
\right).
\label{ExpoS2}
\end{equation}
Lastly, the third equation (\ref{ODE3}) is solved exactly. The grid value $g_{\sigma i}$ is evaluated thanks to (\ref{GsigFLP}). Setting
\begin{equation}
f_\xi=
\left\{
\begin{array}{l}
f_r \mbox{  if } g_i^n\geq g_{\sigma i}^n,\\
[8pt]
f_d \mbox{  if } g_i^n< g_{\sigma i}^n,
\end{array}
\right.
\label{MuNu}
\end{equation}
leads to
\begin{equation}
g_i^{(1)}=g_{\sigma i}^n+\left(g_i^n-g_{\sigma i}^n\right)\,e^{-f_\xi\,\tau}.
\label{SolveODE3}
\end{equation}
The integrations (\ref{ExpoS1}), (\ref{EulerODE2}) and (\ref{SolveODE3}) are unconditionally stable. As a consequence, the splitting (\ref{Splitting}) is stable under the CFL condition (\ref{CFL}).

\subsection{Summary of the algorithm}\label{SecNumDiffu}

The numerical method can be divided in two parts:
\begin{enumerate}
\item initialisation
\begin{itemize}
\item bulk modulus $\rho$, Young's modulus $E=E_0=\rho\,c_\infty^2$;
\item soft-ratchet coefficients $g_{cr}=1$, $g=g_0$, $f_r$, $f_d$, $\overline{\sigma}$;
\item maximum Young's modulus $E^+$ (\ref{Young})
\item nonlinear coefficients (e.g. $\beta$ and $\delta$ in (\ref{Model3});
\item quality factor $Q$, frequency range of optimization $[f_{\min},\,f_{\max}]$, number of relaxation mechanisms $N$;
\item optimization of the viscoelastic coefficients (\ref{SecViscoCoeff});
\end{itemize}
\item time-marching $t_n \rightarrow t_{n+1}$, $x_i=i\,\Delta x$ ($n=0,\cdots,\,N_t$, $i=1,\cdots,\,N_x$)
\begin{itemize}
\item physical and numerical parameters
\begin{description}
\item[-] Young's modulus $E$ (\ref{Young}), viscoelastic parameters $E_R$ (\ref{ErVsE}), $K_{1\ell}$, $K_{2\ell}$ and $\eta_\ell$ (\ref{E1E2Eta});
\item[-] partial stresses $\sigma_{1\ell}$ (\ref{Rheo1}) and total stress $\sigma$ (\ref{MailleSigma});
\item[-] sound velocity $c$ (\ref{C2}) and (\ref{C2Modeles}), maximal velocity $c_{\max}$;
\item[-] time step $\Delta t$ (\ref{CFL});
\end{description}
\item relaxation step ${\bf H}_r$ (\ref{Splitting1})
\begin{description}
\item[-] strains (\ref{ExpoS1}) and (\ref{ExpoS2});
\item[-] velocity $v$ (\ref{EulerODE2});
\item[-] concentration of defects at equilibrium $g_\sigma$ (\ref{GsigFLP}) and out of equilibrium $g$ (\ref{SolveODE3});
\end{description}
\item hyperbolic step ${\bf H}_h$ (\ref{Splitting2})
\begin{description}
\item[-] coefficient $\lambda_{i+1/2}$ of Davis (\ref{Davis});
\item[-] computation of the flux ${\bf F}$ (\ref{FluxFunc}), e.g., by the Rusanov flux ${\bf F}_{i+1/2}$ (\ref{Rusanov});
\item[-] time-marching of the conservative scheme (\ref{Conservatif});
\end{description}
\item relaxation step ${\bf H}_r$ (\ref{Splitting3}).
\end{itemize}
\end{enumerate}

\section{Numerical experiments}\label{SecExp}

\subsection{Configuration}\label{SecExpConfig}

\begin{table}[htbp]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$\rho$ (kg/m$^{3}$) & $E_0$ (GPa) & $g_0$ & $f_r$ (Hz) & $f_d$ (Hz) & $\overline{\sigma}$ (GPa) & $\beta$ & $\delta$ & $Q$ \\
\hline
2054 & 2.21 & 0.1 & 25 & 250 & 0.1 & 40 & $3.5\,10^6$ & 20 \\
\hline
\end{tabular}
\end{small}
\end{center}
\vspace{-0.5cm}
\caption{\label{TabParam} Physical parameters.}
\end{table}

The physical parameters are detailed in table \ref{TabParam}. Depending on the test, some of these parameters are modified. In the limit-case of linear elasticity, the sound velocity is $c=\sqrt{E/\rho}=3280$ m/s. The maximal CFL number is $\alpha=0.95$ in (\ref{CFL}). The mesh size is $\Delta x=4\,10^{-3}$ m. Depending on the test, two lengths of domain are considered. For each test, a receiver put at $x_r=0.2$ m stores the numerical solution at each time step.

The wave fields are excited by a punctual source at $x_s=10^{-2}$ m, with a central frequency $f_c=10$ kHz. Depending on the expression of the forcing $\gamma$ in (\ref{EDP3}), it is possible to deduce the magnitude of the maximal strain $\varepsilon_{\max}$ emitted by the source in the limit-case of linear elasticity (\ref{ElastoExact}):
\begin{equation}
\varepsilon_{\max}=\frac{1}{2\,c^2}\max{\cal G}(t).
\label{EpsMax}
\end{equation}

The Landau model for nonlinear elasticity is used (\ref{Model3}). The coefficient $\beta$ is much smaller than $\delta$. The critical value of strain that ensures hyperbolicity (\ref{EpsC}) is $\varepsilon_c=3.08\,10^{-4}$. The viscoelastic effects are described by $N=4$ relaxation mechanisms. The relaxation times $\tau_{\sigma\ell}$ and $\tau_{\varepsilon\ell}$ (\ref{TauEta}) are computed by optimization on the frequency range $[f_{\min}=f_c/10,\,f_{\max}=f_c \times 10]$ (see \ref{SecViscoCoeff}); they are given in table \ref{TabRelax}.

\begin{table}[h!]
\begin{center}
\begin{small}
\begin{tabular}{|c|c|c|c|c|}
\hline
& $\ell=1$ & $\ell=2$ & $\ell=3$ & $\ell=4$\\
\hline
$\tau_{\sigma\ell}$ (s) & $1.16 \,10^{-3}$ & $2.05\,10^{-4}$ & $4.49\,10^{-5}$ & $7.75\,10^{-6}$ \\
\hline
$\tau_{\varepsilon\ell}$ (s) & $1.53 \,10^{-3}$ & $2.49\,10^{-4}$ & $5.50\,10^{-5}$ & $1.06\,10^{-5}$ \\
\hline
\end{tabular}
\end{small}
\end{center}
\vspace{-0.5cm}
\caption{\label{TabRelax} Relaxation times for a quality factor $Q=20$. Optimization with $N=4$ relaxation mechanisms on the frequency range [1 kHz, 100 kHz].}
\end{table}

\subsection{Test 1: nonlinear elastodynamics}\label{SecExpTest1}

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
$\varepsilon_{\max}=10^{-5}$ & $\varepsilon_{\max}=2.0\,10^{-4}$\\
\hspace{-0.9cm} \includegraphics[scale=0.35]{EpsVsX-1m5.eps} &
\hspace{-0.9cm} \includegraphics[scale=0.35]{EpsVsX-2m4.eps} \\
\end{tabular}
\caption{\label{FigTest1X} test 1. Snapshot of the strain after 400 time steps, for two amplitudes of the excitation. The vertical dotted line denotes the location $x_r$ of the receiver.}
\end{center}
\end{figure}

In the first test, the viscoelasticity is neglected, and the activation / restoration of defects is annihilated: $f_r=f_d=0$ Hz. This test corresponds to the example 12 of \cite{Whitham74}. Our goal is to show typical features of wave propagation in purely nonlinear elastic media. The source is a monochromatic excitation:
\begin{equation}
{\cal G}(t)=A\,\sin(\omega_c t)\,H(t),
\label{SourceSinus}
\end{equation}
where $A$ is the magnitude of the forcing, and $\omega_c=2\,\pi\,f_c$. From (\ref{EpsMax}) and (\ref{SourceSinus}), it is possible to estimate the maximal strain $\varepsilon_{\max}$ emitted by the source in the linear elastic case. The domain of propagation is $L_x=2$ m long and is discretized onto 400 grid nodes.

Figure \ref{FigTest1X} displays the spatial evolution of $\varepsilon$ after 400 time steps. For $\varepsilon_{\max}=10^{-5}$, almost no distorsion of the wave is seen. On the contrary, $\varepsilon_{\max}=2.0\,10^{-4}$ yields a high distorsion as the wave propagates. Shocks, as well as the attenuation due to the intersection of characteristic curves \cite{Leveque02}, are observed.

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm} \includegraphics[scale=0.35]{EpsVsT-2m4.eps} &
\hspace{-0.9cm} \includegraphics[scale=0.35]{Fourier-2m4.eps} \\
\end{tabular}
\caption{\label{FigTest1T} test 1. Time history of the strain at the receiver at $x_r$ (a), normalized Fourier coefficients (b). The amplitude of the excitation is $\varepsilon_{\max}=2.0\,10^{-4}$.}
\end{center}
\end{figure}

Figure \ref{FigTest1T} displays the time evolution of the strain recorded at the receiver (vertical dotted line in Figure \ref{FigTest1X}) for $\varepsilon_{\max}=2.0\,10^{-4}$. The normalized amplitudes of the Fourier series decomposition show a typical feature of cubic nonlinear elasticity: the spectrum involves mainly odd harmonics \cite{Hamilton98}.

\subsection{Test 2: linear viscoelasticity}\label{SecExpTest2}

The goal of the second test is to validate the numerical modeling of attenuation. For this purpose, a linear stress-strain relation is chosen ($\beta=\delta=0$), and the activation / restoration of defects is still annihilated ($f_r=f_d=0$ Hz). Consequently, the system (\ref{EDP}) simplifies into
\begin{subnumcases}{\label{EDPVisco}}
\frac{\partial v}{\partial t}-\frac{1}{\rho}\frac{\partial \sigma}{\partial x}=\gamma,\label{EDPVisco1}\\
[4pt]
\frac{\partial \varepsilon}{\partial t}-\frac{\partial v}{\partial x}=0,\label{EDPVisco2}\\
[4pt]
\frac{\partial \varepsilon_{1\ell}}{\partial t}-\frac{\partial v}{\partial x}=\frac{K_{2\ell}}{\eta_\ell}(\varepsilon-\varepsilon_{1\ell})-\frac{K_{1\ell}}{\eta_\ell}\varepsilon_{1\ell}.\label{EDPVisco3}
\end{subnumcases}
The domain of propagation is $L_x=2$ m long and is discretized onto 400 grid nodes. The time evolution of the source is a truncated combination of sinusoids with $C^6$ smoothness:
\begin{equation}
{\cal G}(t) =
\left\{
\begin{array}{l}
\displaystyle
\sum_{m=1}^4 a_m \sin\,(b_m\,\omega_c\,t)  \mbox{  if }\;0\leq t\leq \frac{1}{f_c},\\
[6pt]
\displaystyle
0 \qquad \mbox{otherwise},
\end{array}
\right.
\label{JKPS_C6}
\end{equation}
with parameters $b_m=2^{m-1}$, $a_1=1$, $a_2=-21/32$, $a_3=63/768$ and $a_4=-1/512$. Five receivers are put at abscissae $x_r=0.5+0.3\,(j-1)$, with $j=1,\,\cdots 5$.

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm} \includegraphics[scale=0.35]{SismoGodunov.eps} &
\hspace{-0.9cm} \includegraphics[scale=0.35]{AnalytiqueGodunov.eps}
\end{tabular}
\caption{\label{FigTest2} test 2. Wave propagation in a viscoelastic medium. (a): time evolution of $v$ at a set of receivers; (b): snapshot of $v$ at $t=0.46$ ms, and comparison between the numerical and the semi-analytical solution.}
\end{center}
\end{figure}

Figure \ref{FigTest2}-(a) shows a seismogram of the velocity recorded at the receivers. Attenuation and dispersion of the waves is clearly observed. Figure \ref{FigTest2}-(b) compares the numerical solution with the semi-analytical solution after 400 time steps. The computation of the semi-analytical solution is described in \ref{SecSolExact}; it is numerically evaluated with $N_f=512$ Fourier modes, with a frequency step $\Delta f=200$ Hz. Good agreement is observed between numerical and exact values. The attenuation is slightly overestimated by the scheme, due to the numerical diffusion of the Godunov scheme. This numerical artifact can be fixed by choosing a higher-order scheme \cite{Toro99}.

\subsection{Test 3: softening / recovering}\label{SecExpTest3}

The goal of the third test is to illustrate the softening / recovering of the elastic modulus, and to validate the numerical modeling of this phenomenon. For this purpose, linear elasticity is assumed and the viscoelasticity is neglected ($\beta=\delta=0$, $Q=+\infty$). Even if a linear stress-strain relation is used, the evolution problem (\ref{EDP}) is nonlinear by virtue of (\ref{EDP4}), (\ref{EDPYoung}) and (\ref{EDPgsig}). Like in test 1, the source is monochromatic; but is is switched off after a time $t^*$:
\begin{equation}
{\cal G}(t)=A\,\sin(\omega_c t)\,\left(H(t)-H(t^*)\right).
\label{SourceSwitch}
\end{equation}
As long as the source is switched on ($0<t<t^*$), the equilibrium concentration of defects increases from the initial value $g_0$ up to $g^*=g(t^*)$. At the same time, the Young's modulus decreases from $E_0$ to $E^*$ via (\ref{Young}).

\begin{figure}[htbp]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm}
\includegraphics[scale=0.42]{MlinSbar2.eps} &
\hspace{-0.3cm}
\includegraphics[scale=0.42]{MlinFr2.eps}
\end{tabular}
\caption{\label{FigTest3} test 3. Time evolution of the elastic modulus $M$ (\ref{ModuleC}) at $x_r$. (a): influence of the central stress $\overline{\sigma}=10^8$ Pa and $10^7$ Pa. (b): influence of the frequency of restoration $f_r=2.5$ Hz and 100 Hz. The vertical dotted line denotes the time $t^*$ where the source is switched off.}
\end{center}
\end{figure}

For $t>t^*$, the waves go out of the domain, and the elastodynamic fields vanish. From (\ref{EDPgsig}) and (\ref{Sig0}), $\sigma=0$ implies that the equilibrium concentration of defects becomes $g_\sigma=g_0$. As a consequence, the ordinary differential equation (ODE) (\ref{EDP4}) describing the evolution of defects simplifies into
\begin{equation}
\left\{
\begin{array}{l}
\displaystyle
\frac{\textstyle dg}{\textstyle dt}=-f_r\,(g-g_0),\\
[8pt]
\displaystyle
g(t^*)=g^*.
\end{array}
\right.
\label{ODEgstar}
\end{equation}
The solution of (\ref{ODEgstar}) is
\begin{equation}
g(t)=g_0+\left(g^*-g_0\right)\,e^{-f_r (t-t^*)}.
\label{gstar}
\end{equation}
Equation (\ref{gstar}) is injected into (\ref{Young}), which gives the time evolution of the Young's modulus during the recovering process ($t\geq t^*$):
\begin{equation}
E(t)=E_0-\frac{1}{g_{cr}}\left(g^*-g_0\right)\,e^{-f_r (t-t^*)}\,E^+.
\label{Estar}
\end{equation}
The domain of propagation is $L_x=0.4$ m long and is discretized onto 100 grid nodes. The maximal strain is $\varepsilon_{\max}=10^{-5}$. Time integration is performed up to $t=460$ ms. Figure \ref{FigTest3} shows the time evolution of the elastic modulus $M\equiv E$ (\ref{ModuleC}); this equality occurs only because a linear stress-strain relation is assumed. The numerical values of $M$ are shown from the beginning of the simulation, whereas the exact values of $E$ (\ref{Estar}) are shown from $t^*$. For the sake of clarity, the values are shown only each 5000 time steps. Logically, the elastic modulus decreases as long as the source is switched on (softening), and then increases up to its initial value (recovering).

Figure \ref{FigTest3}-(a) illustrates the influence of the central stress on the evolution of $M$: $\overline{\sigma}=10^8$ Pa or $10^7$ Pa (the other parameters are those of table \ref{TabParam}). According to the Vakhnenko's expression (\ref{GsigVakh}), these values correspond to spherical defects of radius $2.13\,10^{-10}$ m and $4.59\,10^{-10}$ m, respectively. In both cases, equilibrium has been reached at $t^*$. The lower value of $\overline{\sigma}$ yields a greater variation of the elastic modulus. This property follows from (\ref{GsigFLP}): as $\overline{\sigma}$ decreases, the curve $g\rightarrow g_\sigma$ stiffens and tend towards a Heaviside step function. Consequently, greater values of $g_\sigma$ are obtained when $\overline{\sigma}$ is smaller. This implies a greater evolution of $g$ (\ref{Kinetic}), and hence of $E$ (\ref{Young}).

Figure \ref{FigTest3}-(b) illustrates the influence of the frequency of restoration on the evolution of $M$: $f_r=2.5$ Hz or 100 Hz (the other parameters are those of table \ref{TabParam}). The lowest value of $f_r$ yields a greater variation of the elastic modulus. This is a consequence of the competition between restoration (with frequency $f_r$) and destruction (with frequency $f_d$). When $f_r$ is too low compared with $f_d$, restoration has almost no time to occur during one period $T=1/f_c$, and destruction plays a preponderant role.

\subsection{Test 4: full model}\label{SecExpTest4}

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.9cm}
\includegraphics[scale=0.44]{SigEps2.eps} &
\hspace{-0.9cm}
\includegraphics[scale=0.35]{Mnonlin.eps}
\end{tabular}
\caption{\label{FigSigvsEps} test 4. (a): stress-strain curves at $x_r$ for different quality factor $Q$ and a forcing amplitude $\varepsilon_{\max}=2.0\,10^{-4}$. (b): time evolution of the elastic modulus; the vertical dotted line denotes the time $t^*$ when the source is switched off.}
\end{center}
\end{figure}

The fourth and last test incorporates all the physical mechanisms of the model: nonlinear stress-strain law, viscoelasticity, activation / restoration of defects. The domain is $L_x=0.4$ m long and is discretized onto 100 grid nodes. The source is a monochromatic excitation (\ref{SourceSinus}). Time integration is performed during $5\,10^4$ time steps. The fields are recorded at $x_r$.

Figure \ref{FigSigvsEps}-(a) illustrates the influence of viscoelasticity on the stress-strain law. When viscous effects are neglected ($Q=+\infty$, where $Q$ is the quality factor), the behavior induced by the Landau law (\ref{Model3}) is observed. Moreover, the scaling (\ref{TimeScales}) induces that the evolution of defects on one cycle is insufficient to provide a measurable hysteretic effect. On the contrary, hysteresis is obtained when viscoelasticity is accounted for ($Q=20$). Figure \ref{FigSigvsEps}-(b) mimics the simulation of test 3, where the source a switched-on and off. But contrary to test 3, a nonlinear stress-strain relation is used. Large oscillations up to $t^*$ can be observed, contrary to what can be seen in figure \ref{FigTest3}.

\begin{figure}[h!]
\begin{center}
\begin{tabular}{cc}
(a) & (b)\\
\hspace{-0.5cm}
\includegraphics[scale=0.32]{Erel-ElastoNL-M6bis.eps} &
\hspace{-0.9cm}
\includegraphics[scale=0.32]{Erel-ElastoNL-M5bis.eps} \\
(c) & (d)\\
\hspace{-0.5cm}
\includegraphics[scale=0.32]{Erel-ViscoNL-M6bis.eps} &
\hspace{-0.9cm}
\includegraphics[scale=0.32]{Erel-ViscoNL-M5bis.eps}
\end{tabular}
\caption{\label{FigMvsEps} test 4. Relative variations in the elastic modulus $M$ for various amplitudes of forcing $\varepsilon_{\max}$, from $10^{-6}$ to $7.5\,10^{-5}$. Top (a-b): without viscoelasticity; bottom (c-d): with viscoelasticity.}
\end{center}
\end{figure}

Figure \ref{FigMvsEps} displays the relative variation of the elastic modulus $\Delta M=(M-M_0)/M_0$ in terms of the strain, for various amplitudes of the forcing. Three observations can be made. First, nonlinear curves are obtained, which is a signature of the nonlinear stress-strain relation. Second, $\Delta M$ increases with $\varepsilon_{\max}$: softening increases monotonically with the forcing. Third and last, loops are obtained if and only if viscoelasticity is incorporated (c-d). These three features are qualitatively similar to those obtained experimentally  \cite{Renaud12,Riviere13}.

\subsection{Conclusion}\label{SecConclu}

We have proposed a one-dimensional model that captures the behavior of real media under longitudinal bar excitation, including the following features: nonlinear elasticity; softening / recovering of the elastic modulus; hysteretic evolution of the elastic modulus with the strain. The proposed model is very modular. It involves three different bricks which can be used also independently: see for instance the numerical experiments in section \ref{SecExp}, in which are considered various combinations of elasticity, attenuation and slow dynamics. Experimentally, the parameters corresponding to each mechanisms can be identified separately:
\begin{itemize}
\item the measure of nonlinear elastic parameters is described in many books \cite{Guyer99,Hamilton98};
\item the measure of the quality factor must be performed in the linear regime. See the reference book \cite{Bourbie87} for a description of an experimental protocol;
\item lastly, measuring the parameters of the slow dynamics is detailed in many papers cited in the bibliography. The current challenge is to link the physical observations to the parameters of Vakhnenko's model. Our ambition, with the present paper, is to provide experimenters with a tool for testing various sets of parameters, and hence testing the validity of Vakhnenko's model.
\end{itemize}
A major interest of the numerical approach is the possibility to tackle with variable coefficients in space, which is representative of localized defects \cite{Pecorari14}. In particular, a random initial distribution of defects $g_0(x)$ can be considered  straightforwardly.

Many improvements can be investigated, to mention but a few. More sophisticated models can be built quite naturally, considering for instance relaxation of the nonlinear coefficients ${\bf p}$ in (\ref{Rheo1}), or a nonlinear law in (\ref{Rheo2}). Concerning the numerical simulations, higher-order schemes (such as WENO schemes \cite{Leveque02}) can easily be adapted to the proposed formulation. Lastly, theoretical analyses should be done to prove rigorously the well-posedness of the model and its thermodynamic properties.

Work is currently proceeding along two directions. First, numerical simulations are being done to recover quantitatively the experimental results of the litterature \cite{Renaud12,Riviere13}. Second, the extension of this model to 2D and 3D geometries is under progress.

redpajama_set_name: RedPajamaArXiv


How much does a Admissions Gate Attendant make in the United States?
The average admissions gate attendant salary in United States is $76,731 or an equivalent hourly rate of $37. In addition, they earn an average bonus of $13,044. Salary estimates based on salary survey data collected directly from employers and anonymous employees in United States.
An entry level admissions gate attendant (1-3 years of experience) earns an average salary of $54,642. On the other end, a senior level admissions gate attendant (8+ years of experience) earns an average salary of $94,921.
Based on our compensation data, the estimated salary potential for Admissions Gate Attendant will increase 16 % over 5 years.


text:
I was under the impression that "20 something" meant someone in his/her early 20s. Would a 29 year old be still considered "20 something"? When did this expression start?
20-something means you are from age 20-29 as 300something means 30-39
it would include people that fall inbetween of 20 -29|
passerby Nov-16-2004
There was a television programme called 'thirty something'. All the main characters in it were between 20 and 30.
M_Stevenson Apr-11-2004
It could depend on the perspective. If I'm talking about someone else and am asked what that person's age is, and I don't know for sure, I could say, "20-something." If I am that person and don't want to admit being 29, I would say, "20-something."
With the old TV show, I thought it meant in their 30s, but they prefer not to be more specific.
mike7 May-06-2003
Joe and Nina raise good points. My impression of the "thirty something" phrase was originally that it described a relatively amorphous group of around the same age (or decade of age) that acted and thought and consumed in similar ways. Thirty something was basically synonymous with yuppies, as I remember it being used. I think now it has devolved through common usage to its more mundane meaning, i.e., as Nina or Joe suggest, anyone 20-29.
owl Mar-19-2003
Naturally, I would assume that the companion expression "30 something" came before the TV show of the same name, but I believe that in that case, "30 something" would be skewed toward the latter end of the age group, since they don't want to say they are 36 or so, and would rather people think they might be 31. (whatever).
Anyway, intuitively, "20 something" means anything in 21-29 (since 20 has no "something" after it)... but since I don't know the origin of the phrase either, the original coiner could have intended the early 20's.
Joe4 Mar-18-2003
today, as I turn 25, still will use 20-something. I suppose a 29 year old could technically use 20-something.
nospam.chick Mar-18-2003

redpajama_set_name: RedPajamaCommonCrawl


NothingButTech88's Samsung Galaxy S7 Edge Camera Samples!
Jacklyn is a technology journalist and filmmaker, who brings a fresh and unique understanding of the youth, female, and professional market to all of her reviews. She specializes in consumer electronics and high quality tech products from lesser known brands. She is an editor, writer, and social media manager at TheTech.Ninja, where she writes articles on current technology launches, trends, news and reviews. You can also find her on YouTube at YouTube.com/NothingButTech88 .
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text:
Q: Compare between 2 tables in 2 different databases I have 2 access databases one called revision and the other called revision2. Each database has a table called batch.
I have DataGridView that shows the data in the first database.
I need to compare between the 2 tables from the different databases and what matches get a green color in the DataGridView.
I used that code but it only colors the first match not all,
private void dataGridView1_CellFormatting(object sender, DataGridViewCellFormattingEventArgs e)
{
    OleDbConnection con = new OleDbConnection(@"Provider=Microsoft.ACE.OLEDB.12.0;Data Source=|DataDirectory|\revision.accdb");
    OleDbConnection con2 = new OleDbConnection(@"Provider=Microsoft.ACE.OLEDB.12.0;Data Source=|DataDirectory|\revision2.accdb");
    OleDbCommand cmd = new OleDbCommand("select * From batch", con);
    OleDbCommand cmd2 = new OleDbCommand("select * From batches", con2);
    con.Open();
    con2.Open();
    OleDbDataReader dr = cmd.ExecuteReader();
    OleDbDataReader dr2 = cmd2.ExecuteReader();
    dr.Read();
    dr2.Read();
    DataTable dt = new DataTable();
    dt.Columns.Add("batch_no");
    if (dr["batch_no"].ToString() == dr2["batch_no"].ToString())
    {
        string num = dr["batch_no"].ToString();
        foreach (DataGridViewRow myrow in dataGridView1.Rows)
        {
            if (Convert.ToString(myrow.Cells[0].Value) == num)
            {
                myrow.DefaultCellStyle.BackColor = Color.Green;
            }
            else
            {

            }
        }
        con.Close();
        con2.Close();
    }
}

Any solution to get what all what matches and color it? Or any better ideas?

A: You did not loop through the reader(dr), the one you are supposed to find the matching values with your current grid.
Anyway, what's the DataTable and the other reader (dr2) for ?
A quick modification, not yet tested :
    if (dr["batch_no"].ToString() == dr2["batch_no"].ToString())
    {
        while (dr.Read())
        {
            string num = dr.GetString(dr.GetOrdinal("batch_no")); //dr["batch_no"].ToString();
            foreach (DataGridViewRow myrow in dataGridView1.Rows)
            {
                if (Convert.ToString(myrow.Cells[0].Value) == num)
                {
                    myrow.DefaultCellStyle.BackColor = Color.Green;
                }
                else
                {

                }
            }
        }

        con.Close();
        con2.Close();
    }

A: u can use "dataview" to check your condition
in this sample "tbl2.DefaultView.FindRows" can be helpful
    OleDbConnection con = new OleDbConnection(@"Provider=Microsoft.ACE.OLEDB.12.0;Data Source=|DataDirectory|\revision.accdb");
    OleDbConnection con2 = new OleDbConnection(@"Provider=Microsoft.ACE.OLEDB.12.0;Data Source=|DataDirectory|\revision2.accdb");
    OleDbCommand cmd = new OleDbCommand("select * From batch", con);
    OleDbCommand cmd2 = new OleDbCommand("select * From batches", con2);
    con.Open();
    con2.Open();
    OleDbDataReader dr = cmd.ExecuteReader();
    OleDbDataReader dr2 = cmd2.ExecuteReader();

    DataTable tbl1 = new DataTable();
    tbl1.Load(dr);

    DataTable tbl2 = new DataTable();
    tbl1.Load(dr2);

    dataGridView1.DataKeyNames = new string[] { "batch_no" };
    dataGridView1.DataSource = tbl1;
    dataGridView1.DataBind();

    for (int i = 0; i < dataGridView1.Rows.Count; i++)
    {
         if (tbl2.DefaultView.FindRows(dataGridView1.DataKeys[i].Value).Length > 0)
            dataGridView1.Rows[i].BackColor = Color.Green;
    }

redpajama_set_name: RedPajamaStackExchange


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text:
Home » Lifestyle » Candis Cayne Lists English Tudor Hillside Cottage in Glendale's Adams Hill 'Hood
Candis Cayne Lists English Tudor Hillside Cottage in Glendale's Adams Hill 'Hood
Groundbreaking actress and entertainer Candis Cayne has her home in the bohemian, up-and-coming Adams Hill neighborhood in the southeastern foothills of Glendale, Calif., up for sale at $1.285 million. Cayne, a legendary drag performer before she blazed showbiz trails as the first transgender actress to portray a recurring transgender character on primetime TV in ABC's "Dirty Sexy Money" and who more recently popped up on "Elementary" and portrayed the Fairy Queen on the third season of the fantasy drama series "The Magicians," hopes to just about double her money on the property that last traded in late 2012 for $650,000. Perched remarkably high on a precipitous slope, it's a long, thigh-burning and fanny-toning three flights up from the street level two-car garage to the front door of the imposing and ever-so-slightly spooky 1920s English Tudor Revival cottage that contains three bedrooms and 2.5 bathrooms plus a self-contained one-bedroom and one-bathroom guesthouse, all in 2,627 square-feet.
A postage stamp-sized porch and a proper if puny foyer leads to a voluminous living room under a steeply pitched and vaulted ceiling with a towering arched casement window that frames a northeasterly, cross-canyon view over Glendale. A glitzy, gold-painted fireplace is emblazoned with a regal crest and on the back wall behind the sofa hangs a risqué portrait drawing of the home's bombshell resident. Dark-stained oak floorboards continue from the living room into the adjoining dining room that features original diamond paned casement windows and an eclectic mélange of modern and antique furnishings that include a fully mirrored and gilt-trimmed dining table. A sunny, cozily proportioned breakfast nook has French doors to an arched loggia and mismatched archways in the dining room and breakfast nook lead into the kitchen, an extra-wide galley-style arrangement that was stylishly redone by San Francisco-based gamble + design's Jeni Gamble, a life-long friend of Cayne. Custom cabinets are painted a watery shade of washed denim, counter tops are marble and the Moroccan-inspired herringbone pattern black and white tile back splashes extend decadently all the way to the ceiling. The main floor also offers a powder room and a small office/den lined with simple plank wood shelving.
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An enormous stained glass window sends colored shards of light through the switchback stairway that winds up the top floor where a pair of modest guest bedrooms share a hall bathroom sheathed in dated, burgundy ceramic tiles and the master bedroom offers an open view through a picture window along with a walk-in closet, a small attached bathroom that could use a cosmetic overhaul and a second arched loggia that provides a panoramic, over the telephone wires vista.
Discreetly tucked under the main house with both interior and exterior entrances, the attached one-bedroom and one-bathroom guesthouse is unexpectedly spacious and completely independent with a parquet-floored living room anchored by a massive fireplace, a full kitchen with adjoining dining nook, a uniquely bi-level bedroom and a bathroom with claw-footed bathtub and, like the bathrooms in the main house, decoratively dated tile work.
Reminiscent of an elegant, Haussmann-era Parisian apartment, salvaged steel-trimmed French doors in the kitchen lead out to a verdantly planted backyard that includes flagstone terrace bordered by rock walls, a chunky outdoor fireplace, and a dining patio under a romantic canopy of vines that cling to a steel framework. A stone stairway climbs up to the rear of the up-sloped property that privacy loving potential buyers will take note backs up to a dead quiet cemetery.
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This dimmer unit is designed for controlling any 12V or 24V single colour 2-wire product, such as our LED strip light. Change the brightness up and down using the top arrow keys or by using the touch wheel dial. It features a mode with a turn off 30 second delay, so that you don't have to exit the room in the dark. It can also be programmed to remember your favourite setting as a 'scene', so that it can be easily set back to this after it has been changed.
It has a clean modern look with its glass touch panel, and it emits a pleasant sound to indicate when a touch has registered - however this can be turned on/off (muted) simply by pressing and holding the power button until it sounds. A blue indicator light shows just above the power button when the unit is powered off so that it is easy to locate in the dark.
Run up to 3 lengths of strip/12V products, each up to 4A load each - the maximum output of this controller is 4A per channel (12A max total). It dims from 5 to 100% brightness. Each length/load is controlled at the same time, so that strip installed in different areas dim at the same time (this controller does not have the ability to control the lengths separately).
Use this controller with our flexible strip light.


Arizona Center for Laser Dentistry is an internationally recognized treatment facility and a leader (TMJ Dentist ) in treating successfully TMJ conditions quickly and painlessly. Dr. DiVito uses medical grade imaging including 3D CT scans, magnetic resonance, CAD/CAM (computer assisted design and milling) and advanced laser technology to deliver low level light therapy applications allowing for reduced pain and quicker healing for damaged jaw joint tissues. Our Scottsdale, AZ office proudly serving patients in Phoenix area for over 35 years with our team consisting of a top rated best in his class in laser dentistry who specifically specialize in TMJ treatments.
Dr. Enrico DiVito has advanced training for treatment of TMJ. He can treat the underlying cause of your TMJ/TMD, thereby relieving your painful symptoms. You will be happy to know that TMJ/TMD can often be stopped and reversed with the proper care.
Dr. Enrico DiVito, clinical director for Arizona Center for Laser Dentistry, is a renowned dentist in the field of TMD/TMJ. Dr. DiVito has successfully treated several thousand patients using his non-surgical, noninvasive therapeutic treatment approach. Part of his commitment to serving his patients includes providing them with information that will help them to make a better informed decision about TMJ treatment. Dr. Enrico E. DiVito is a dentist with advanced training for the non-surgical management of TMJ disorders. He is a native of Phoenix and has practiced in Scottsdale since 1980. His staff is highly trained and experienced in helping manage sick and injured patients. A complete dental examination with appropriate x-rays is usually done to help determine if a victim of head and neck trauma and has sustained a TMJ injury. Sometimes an MRI study will also be needed to examine the soft structures that are damaged in these injuries. A complete TMJ evaluation can put the victim's mind at ease about the reason for face, jaw and headache pain.
After the initial evaluation has been completed, a pain-free, non-surgical plan is developed for each patient.
If your in Phoenix, and you have TMJ disorder and are looking for a specialist in the field, or have a question, please contact us. TMJ Symptoms: Below are a few symptoms that can help determine whether or not you suffer from TMJ: Headaches: sometimes of migraine proportions. Ear pain symptoms: ringing or stuffiness in the ears, hearing loss, "plugged" ears, ear pain, a feeling of "under water" Clicking: There is a pad or disc which ideally sits between the temporal bone and mandible in the TM Joint. This cushion, like a magic carpet, can slip out of place causing clicking when opening and closing. Clicking can lead to locking and grating or crepitus. Locking: When a disc is chronically displaced, it does not always click back into place. This leads to locking of the jaw joint with deflection of the jaw and limited range of motion. Popping: Some clicking can be staccato or popping by nature. Jaw Pain: Although it can originate from many sources, the most common reason for jaw pain is dental, then musculoskeletal. Most dentists can rule out dental decay, periodontal disease or root canal as the offending dental pathology. Muscles, tendons, ligaments and the temporomandibular joint can all cause jaw pain. Jaw pain can radiate from the neck.
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Qualified and capable managers is what the world needs as of today, therefore it is really important that you get a quality education. The global community is growing more interconnected today than at any other time in human civilization, and this means that new business strategies are needed to push organisations through this period. Western ideology and the principles of capitalism have been sweeping through the world, leading many students in consideration of studying abroad. Are you one of those students that would like to study business and management in New Zealand, then you should be aware of certain requirements as you begin the application process.
Choosing to study abroad is a major decision that should not be taken lightly. In fact, the planning and preparation process for this begins long before you ever board a plane for another country. You will want to ensure that you make the most out of your secondary school education in order to have your eventual application looked upon more favorably when the time comes. This includes earning a General Certificate of Education at the Advanced Level. In doing so, you will need to complete a minimum of three A-Level subjects with at least a 'C' grade. You can elevate your place in the applicant pool by earning even better grades and passing more than the required number of A-Level subjects. There are other options for secondary education as well, including the completion of an Australian Senior Secondary Certificate of Education Qualification with a minimum score of 69.6.
Remember that the medium of instruction will be English, so if you want to study business and management in New Zealand then you will need to demonstrate a certain level of language proficiency. While there are generally several accepted ways of demonstrating, most universities in New Zealand will be looking at your IELTS score. This is the most recognized international English language exam in Australia and New Zealand; so make sure that you begin preparing yourself early. You will want to achieve a minimum score of 6.0 on each strand of the test, but aim for 6.5 and your application will be more appealing.
Finally, in order to study business and management in New Zealand you will need to plan for the financial aspect of this decision. While there are certainly scholarships and grants that you can apply for, you can also make this move by paying out of your pocket. The general annual tuition cost of studying for an undergraduate degree in New Zealand is $25,150 NZD. Remember to also plan for miscellaneous course fees, travel expenses, and costs associated with your housing and food.


text:
Home›Production Function›Deakin University modernizes its software delivery – Strategy – Software
Deakin University modernizes its software delivery – Strategy – Software
Deakin University has spent the past 18 months modernizing its approach to software delivery, adopting standard CI / CD tools and removing manual efforts to get code into production faster.
In this week's CXO Challenge on the iTnews podcast, Director of Software Engineering Aaron Whitehand shares the story of the university's modernization so far, covering its adoption of DevOps practices, automation, and standard tools.
Software Engineering as a named function is "relatively new" and is found in the central IT division of the university, known as eSolutions.
"The team is made up of approximately 45 people, who cover the responsibilities of application architecture, software development, testing, software operations and DevOps practices," said Whitehand.
"Even though they're under eSolutions, they're usually scattered across a dozen or more different delivery teams.
"We work in a matrix organizational structure here at Deakin, so the majority of the team staff could work with all or part of the company at any time in cross-functional delivery teams. "
Whitehand said the "last few years" have seen a greater focus on software engineering as an internal discipline.
"Deakin has embarked on a relatively aggressive modernization of practices," Whitehand said.
"We've been looking to modernize the way we approach things there, and a key part of that is automation, which is obviously part of any DevOps transformation and focuses on removing all manual steps from the process. "
Software development practices within the university were already reasonably "contemporary"; Whitehand noted that iterative development and agile methodologies began to replace waterfall delivery in some (but not all) projects about seven years ago.
The adoption of DevOps practices and tools was more recent and occurred when the university hired a partner on an independent IT project that used DevOps, exposing internal software engineering teams to this way of working.
"We were working with a partner on an implementation, and through that we were exposed to new ways of working that really resonated with some people, including myself, in the organization," said Whitehand said.
"It was after this engagement that we started to think more seriously about DevOps.
"We started to strategize on how we were going to transform or start adopting DevOps practices, and it grew from there. "
DevOps – and the ability to go faster – was personally appealing to Whitehand as he revised his own work practices.
"I did manual tasks on a daily basis and they took a lot of time," he said.
"These were things like code reviews, working with peers, handoffs to security or to testers for other manual efforts, and obviously these workflows are code base by code base consideration.
"So for me the biggest eye-opener was the fact that I could start automating some of these low-value activities and start delivering more value to our customers faster. "
While the university had some IT process automation tools, these tended to be purchased by small teams or for specific projects.
"We've had duplicate tools and varying levels of maturity over time," Whitehand said.
"Some teams don't even use any form of IT process automation, and that manual effort was pretty big in those teams.
"With the duplication of tools and the need to couple less mature tools with other open source or commercial offerings to provide certain functionality, we were duplicating the expense or having to educate a large number of people on a large number of platforms. "
This duplication of licensing and effort has become unpleasant over the past year, especially as budgets have tightened.
"Certainly in the current environment, we have to be mindful of our spending," Whitehand said.
"Streamlining came to the fore and we started looking at how to streamline tools, but at the same time, how we can ensure consistency and empowerment of teams at scale.
"So we started to promote not only a consistent tool to use, but we also created a platform engineering team – a core team that could actually allow other teams to have a standard consistent approach. "
The consistent tool that was ultimately selected was GitLab, following proof of concept with three potential options.
The university "started with a very small 100-seat license" for GitLab, which Whitehand said was a deliberate move for a "proof of adoption phase."
"As we looked to increase capacity, we wanted to ensure success through a phased and phased approach," he said.
"And as we also modernized our behind-the-scenes practices, there was a lot to consider in how we would implement a tool like this in this phased approach to ensure our developers understand the concepts and not just the technology.
"Where we've gone is an increase in our license to what's called the Unite license, which effectively gives us unlimited users, and we can use it now to accelerate adoption of the platform. "
This accelerated adoption likely means that GitLab is being used outside of software engineering – by university research operations but also potentially in student study programs.
"From an education and research perspective, we are working with these areas of the business to understand how they can leverage the tool and how it would benefit them," Whitehand said.
"There is no question that DevOps is standard practice in the industry today, and I have no doubts that organizations that do not adopt it will not achieve the same level of success as their peers.
"This kind of exposure and success is also true for the new generation of learners that we have thanks to Deakin, and providing this tool and this industry best practice experience improves their level of employability.
"So as part of our plan now, we have started to engage with some key stakeholders in these areas on the possibilities of integrating the services into different learning outcomes and research outcomes.
"We [software engineering]can also use these best practice offerings to give student interns hands-on experience of the tools in a corporate context as well.
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We spend a fair amount of time at Rustici Software looking at problems people encounter in our SCORM Test Track and our other LMS implementations. We want every piece of content to work the first time, and so we dig around for the real source of problems. In an effort to improve interoperability in general, we're going to start posting some of these conversations on the blog so that others can link to them and avoid similar issues.
The part about the cmi.completion_status element is true, the LMS basically shouldn't mess with it….and we don't. If you take a close look at the debug logs when you launch the course for a second time, you'll see that the cmi.completion_status value is still unknown. You can also see this by drilling down into the detailed report at the top of the SCORM Test Track screen.
The cmi.completion_status for the SCO however is not what provides the overall determination of a course's completion status. This might be a bit confusing, but in SCORM 2004, there are two data models. The first is the simple CMI runtime data model that SCO's read from and write to via the SCORM API. Every SCO has a set of CMI runtime data associated with it. The second data model is the sequencing data model. The sequencing data model is closely related to, but separate from the runtime data model. Every item in a SCORM 2004 course has a set of sequencing data associated with it (in other words, SCOs and aggregations both have sequencing data in contrast to CMI data which is only available on SCOs). It is the sequencing data of the root aggregation of the course that determines the course's overall status. In this case, there is a single SCO wrapped in a single aggregation (every SCORM course always has an aggregation at the top level). The status you see in Test Track is the completion status of the root aggregation's sequencing data….which is closely related to, but not exactly the same as the single SCO's runtime data.
Now, why does the course show completion? For a detailed explanation, see section 3.13.2 of the SCORM 2004 3rd Edition Sequencing and Navigation spec. Basically, there is a sequencing rule that indicates whether or not the content is going to communicate any completion information via the SCORM API ("completion set by content"). If this setting is false (the default), then if the LMS doesn't receive any completion status information from the content, it is supposed to automatically mark the content as completed. Since this course isn't reporting any completion status information when it exits early on, the LMS is behaving correctly by automatically setting the sequencing completion status to completed.
There are two best practices that content vendors implement to avoid this behavior. The first is to override the default value in the manifest and set "completion set by content" to true. The second is to immediately set cmi.completion_status to incomplete upon launch of the content. This call tells the LMS that it should expect completion information to be explicitly recorded and should not auto-complete. This best practice dates back to SCORM 1.2.


One of the loudest claims the communists used to make was that the unity of proletarian internationalism would render obsolete any national or ethnic prejudice. It was supposed that the brotherhood of workers would consign all ethnic strife to the proverbial "dustbin of history".
Bulgaria under communist rule provided a stark example of the resilience of bigotry under Marxism.
In 1984, the Bulgarian government under Todor Zhivkov began a bizarre and brutal campaign to forcibly assimilate Bulgaria's Turks, who comprised about 10% of the population. During a period of two months, the entire Turkish population was forced to take on Bulgarian names. Villages were cordoned off and people were assigned new names. Often, brothers from the same family found that they now had different last names. The Turkish Weekly describes the brutality and chaos of this period as follows.
"The Bulgarian totalitarian state started an assimilation act towards the Turkish minority at the end of 1984 and the beginning of 1985. More than 1 million Turks were renamed by force within two months. The Bulgarian totalitarian state literally started a war against these undefended and poor people. The totalitarian regime mobilized all of its force: army, police troops, special soldiers, armed volunteers, even an organization of children. The authorities of the totalitarian state began persecuting the Turkish population in all of the regions [under its control]. Many people were detained in the police station for days and beaten to death. During this act of the totalitarian state, 517 people were arrested and sent to the concentration camp on Belene Island, and nearly 400 people were exiled or imprisoned by fake courts. Before being sent to the concentration camp, exiled or imprisoned we were subjected to unbearable cruelties and brutalities. We were made to live in conditions which can't be described. Our families were persecuted and put under psychological pressure again and again. In 1989 the Bulgarian totalitarian state launched [a campaign of] ethnic cleansing. Within 72 days 350,000 Turks were deported from the country."
In addition to those who were forcibly deported, there were those who found the pressure of living hated and hunted in Bulgaria to be unbearable. To the lasting credit, the Turkish republic opened its borders and its heart to its Turkish brothers in Bulgaria. It offered them the right of return and the opportunity to take Turkish citizenship. For thousands of Turks who had literally had their very names stolen, the kindness of the Turkish government and people was priceless. Many sold their property at great loss to save their lives. In Turkey, many were bewildered by a capitalist system that was alien to them as well as a form of spoken Turkish quite different from what they knew in Bulgaria.
Many people know of ethnic Germans driven from their homes at the end of World War Two. Many remember the images of streams of ethnic Germans trudging hundreds of miles on foot to an uncertain future in a new country that was in a state of physical and economic ruin.
Today these Germans have been absorbed into German society. Their grandchildren preserve the memory of the injustice done to their grandparents, but do so from established and comfortable homes and not refugee camps.
Bulgaria's Turks were as forlorn and desperate in the 80s as the Germans were in the 40s. The idea of jus sanguinis, the right to citizenship by virtue of ancestry was critical to minimising loss of life. As noble as the idea is of a nation of laws such as we enjoy in the US, states such as Germany, Turkey and Greece as well as Israel with their legal principal of jus sanguinis occupy a critical and essential place in the family of nations.
Turkey has had dark chapters in its history. Kurds and Armenians as well as Assyrians are all too well acquainted with this fact. It badly needs to develop legal and constitutional safeguards for its own national minorities. But it deserves respect and recognition for its humanitarian assistance to its Turkish compatriots in Bulgaria.
How does a nation preserve its language, its culture and its way of life and at the same time accommodate its national minorities? Turkey has taught the world valuable lessons, both in the failures of its history and in its successes. It is a nation to be studied by a world that sees new crises unfolding every day .
In the post communist era, the situation of Bulgaria's Turks has improved. With difficulty, names taken away have been returned, if not the property that was lost. The history of Bulgaria's Turks stands as stark testimony to the moral bankruptcy of communism and the fragility of human decency. I wish Bulgaria's Turks peace and tranquility, wherever they may choose to live.


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The Latest: Utilities president blasts PG&E over outages
SAN FRANCISCO — The Latest on a massive power shut-off last week in Northern California (all times local):
California's top utilities regulator says she's astounded by the lack of basic preparation Pacific Gas & Electric officials took in readying for a pre-emptive power shut-off that affected more than 2 million people.
California Public Utilities Commission President Marybel Batjer said at an emergency meeting Friday that the utility failed on so many levels on simple matters.
PG&E shut off power to 700,000 customer accounts as a wildfire prevention method. Customers complained of overloaded call centers and a website that crashed throughout the Oct. 9 event.
Local officials said they were left in the dark as well.
The chief executive of California's largest utility says it will take about a decade for Pacific Gas and Electric Co. to get to the point where widespread safety outages are not necessary when fire danger is high.
PG&E Corp. CEO Bill Johnson told state regulators Friday he expects the utility to get better with each new pre-emptive outage as it works to upgrade its equipment so blackouts affect fewer people.
Appearing before an emergency meeting of the California Public Utilities Commission, Johnson said the Oct. 9 outage was the right call but said the utility could have done much better executing it.
PG&E shut off power to more than 2 million people last week because of fears that dangerous winds could knock down utility equipment and spark wildfires.
Customers complained of overloaded call centers and a crashing website that made getting information difficult.
Top executives of California's largest utility are expected at an emergency meeting Friday to answer questions by state regulators about a massive pre-emptive power shutdown last week that has been criticized as poorly executed and unacceptable.
The meeting of the California Public Utilities Commission in San Francisco comes as outrage grows against Pacific Gas & Electric Co. The utility cut off power to more than 2 million people in northern and central California Oct. 9, saying that high wind forecasts could have damaged equipment and sparked wildfires.
In a letter to PG&E chief executive Bill Johnson issued earlier this week, commission President Marybel Batjer scolded the utility for an "unacceptable situation" and ordered a series of corrective actions, including a goal of restoring power within 12 hours, not the utility's current 48-hour goal.
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