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\begin{document}


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\title{CALIBRATION OF ELASTIC-PLASTIC MATERIAL MODEL FOR TIRE SHREDS}

\author{
Boris Jeremi{\'c}%
\thanks
{Department  of  Civil  and  Environmental  Engineering,  University  of
California, Davis, CA 95616. \texttt{jeremic@ucdavis.edu}} \ 
Member, ASCE, \\ 
James Putnam%
\thanks
{Department  of  Civil  and  Environmental  Engineering,  University  of
California, Davis, CA 95616.} \
Member, ASCE,  \\ 
Kallol Sett%
\thanks
{Department  of  Civil  and  Environmental  Engineering,  University  of
California, Davis, CA 95616.} \
Student Member, ASCE,\\ 
Dana Humphrey%
\thanks
{Department  of  Civil  and  Environmental  Engineering,  University  of
Maine, Orino, ME 04469.} \
Member, ASCE,  \\ 
Stacey Patenaude%
\thanks
{California Integrated Waste Management Board, Sacramento, CA 95812.}
}

%
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%\\
%and
%Younyee Kuhn%
%\thanks{Flourishing wife of same.},%
%\ Not a Member, ASCE
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\maketitle
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\begin{abstract}
In  this  paper  we present the development of an elastic-plastic material model
for  tire shreds material. In particular, our aim is to use large scale triaxial
tests  to  calibrate  the elastic and elastic-plastic model for this anisotropic
material.  The  main  effort  is  focused  on the dynamic behavior of tire shred
material and in particular the energy dissipation characteristics during dynamic
(seismic)  loading.  This  paper  presents results of a material model calibration
using wave propagation tests on a triaxially loaded tire shred specimen.
 
                                                                                                                                                             
The  immediate  goal  of  the  presented  work is the creation of a verif\/ied and
validated  elastic-plastic  material  model  that will then be used in numerical
assessment  of  seismic  behavior  of bridge abutments. The validation procedure
will ultimately include  simulations of two bridge abutments made of tire shreds performed
using a large centrifuge facility at UC Davis.

\end{abstract}
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\section{Introduction}
Civil  engineering  application is a growing market for scrap tires. Scrap tires
have  been  successfully  used  as  light-weight  f\/ill  \cite{grout:humphrey};
\cite{active:humphrey};   \cite{pavement:humphrey},   insulation  beneath  roads
\cite{insul:humphrey},    light-weight    backf\/ill    for    retaining   walls
\cite{active:humphrey},   overlay   of   underground   ramps  or  storage  space
\cite{ani:drescher},  and  as  stream  crossing \cite{msthesis:megan}, primarily
because of low unit weight, high permeability and good insulating properties.


The  placement  and  compaction  of  tire shreds leads to a layered, anisotropic
material behavior.
Isotropic   deformability   parameters  and  time-dependent  creep  behavior  of
tire-shreds    were    reported    in    the    past   by   several   researchers
\cite{grout:humphrey};       \cite{insul:humphrey};      \cite{active:humphrey};
\cite{deform:drescher};  however,  only  a  few studies have been carried out to
evaluate  anisotropic  properties  of  a tire shred model. Heimdahl and Drescher
(1999)  f\/irst attempted to develop an elastic anisotropic model for tire shred
behavior  using  a  biaxial  testing apparatus. They reported over a dozen plane
stress  tests  at two loading conditions and averaged the elastic constants thus
determined.  A large variation of in-plane Poisson's ratio was reported with the
possibility of negative values discussed. The authors acknowledged that the test
apparatus employed was not the best for determining Poisson's ratio.

This  paper  deals with the development of an elastic-plastic material model for
the  tire shred material. In particular, our aim was to use a large scale triaxial
testing  apparatus, developed at University of California, Davis  to  calibrate 
the elastic and elastic-plastic material model for 
anisotropic  material.  Our immediate goal was to create, verify and validate an
elastic-plastic  material model that can then be used in numerical assessment of
seismic  behavior  of bridge abutments. The validation procedure will ultimately
include  computational and experimental simulations of two bridge abutments made
of tire shreds, performed using
a large centrifuge facility at UC Davis.

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\section{Elastic Cross-Anisotropic Constitutive Law}
\label{ELASTIC_CONSTITUTIVE}
%
The linear, small-strain, elastic behavior of cross-anisotropic material
is governed by the following generalized Hooke's law \cite{ani:drescher}:
%
\begin{equation}
\epsilon_{11}=\frac{1}{E}\sigma_{11}
-
\frac{E-2G}{2GE}
\sigma_{22}
-
\frac{E^{\prime}-2G'}{2G'E'}\sigma_{33}
\label{eq01}
\end{equation}
%
\begin{equation}
\label{eqno0.2}
\epsilon_{22}=-\frac{E-2G}{2GE}\sigma_{11}+\frac{1}{E}\sigma_{22}-\frac{E'-2G'}{2G'E'}\sigma_{33}
\end{equation}
%
\begin{equation}
\label{eqno0.3}
\epsilon_{33}=-\frac{E-2G''}{2G''E}\sigma_{11}-\frac{E-2G''}{2G''E}\sigma_{22}+\frac{1}{E'}\sigma_{33}
\end{equation}
%
\begin{equation}
\epsilon_{23}=\frac{1}{2G''}\sigma_{23}
\;\;\; \mbox{;} \;\;\;
\epsilon_{13}=\frac{1}{2G''}\sigma_{13}
\;\;\; \mbox{;} \;\;\;
\epsilon_{12}=\frac{1}{2G}\sigma_{12}
\end{equation}
%
where, in equation (\ref{eq01}) $ E = E_{11} $ = in-plane Young's modulus; $ E' = E_{33} $ =
out-of-plane Young's modulus; $ G = G_{12} $ = in-plane shear modulus; $ G' =
G_{31} $ = out-of-plane shear modulus; $ G'' = G_{13} $ = out-of-plane
shear modulus. Note that ''in-plane'' refers to the 1-2 isotropy plane
(horizontal) and ''out-of-plane'' refers to the 3-direction (assumed vertical).
                                                                                                                                                             
                                                                                                                                                             
The  above  elastic  material  constants, $E$, $E'$, $G$, $G'$, and $G''$ can be
determined from three-dimensional P-wave and shear wave tests. The equation
relating   in-plane   Young's   modulus   ($E$)  and  in-plane  P-wave  velocity
($V_{c_{i}}$; $i=1, 2$) is,
%
\begin{equation}
\label{eqno0.7}
E=\rho V^2_{c_{i}}
\end{equation}
%
Similarly, the equation relating out-of-plane Young's modulus ($E'$) and
out-of-plane P-wave velocity ($V_{c_{3}}$) is,
%
\begin{equation}
\label{eqno0.8}
E'=\rho V^2_{c_{3}}
\end{equation}
%
The equation relating in-plane shear modulus ($G$) and in-plane shear wave velocity with horizontal polarization ($V^H_{s_{i}}$; $i=1, 2$) is,
%
\begin{equation}
\label{eqno0.9}
G=\rho (V^H_{s_{i}})^2
\end{equation}
%
where   wave   propagation   is  in  the  i-direction  (i=1,2)  with  horizontal
polarization. The equation relating out-of-plane shear modulus ($G'$)
and out-of-plane shear wave velocity ($V_{s_{3}})$is,
%
\begin{equation}
\label{eqno0.10}
G'=\rho (V_{s_{3}})^2
\end{equation}
%
where wave propagation is in the 3-direction with
horizontal polarization.
Finally, the equation relating out-of-plane shear modulus ($G''$) and
in-plane
shear wave velocity with vertical polarization ($V^V_{s_{i}}$; $i=1, 2$)
is,
%
\begin{equation}
\label{eqno0.11}
G''=\rho (V^V_{s_{i}})^2
\end{equation}
%
where wave propagation is in the i-direction ($i=1,2$)
with vertical polarization.\\

%
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\section{Elastic-Plastic Constitutive Law}
\label{ELASTIC-PLASTIC_CONSTITUTIVE}

%
The  most  general  form  of elastic-perfectly plastic incremental stress-strain
relationship can be expressed as \cite{plas:chen},
%
\begin{equation}
\label{eqno11}
d\sigma_{ij}=C_{ijkl}^ep d\epsilon_{kl} 
\;\;\;\;\;\; \mbox{where} \;\;\;\;\;\;
C_{ijkl}^ep
=
C_{ijkl}
-
\frac{C_{ijmn}
\displaystyle{ \frac{\partial f}{\partial \sigma_{mn}}}
\displaystyle{\frac{\partial f}{\partial \sigma_{pq}}}
C_{pqkl}}
{
\displaystyle{{\frac{\partial f}{\partial \sigma_{rs}}}}
C_{rstu}
\displaystyle{\frac{\partial f}{\partial \sigma_{tu}}}
}
\end{equation}
%
The coeff\/icient tensor $C_{ijkl}^ep$ is the elastic-plastic tensor of tangent moduli 
for an elastic-perfectly plastic material.
%
The tensor of elastic moduli, $C_{ijkl}$, can be expressed in terms of $E$
and $\nu$ as \cite{plas:chen},
%
\begin{equation}
\label{eqno13}
C_{ijkl}
=
\frac{E}{2(1+\nu)}\left[\frac{2\nu}{(1-\nu)}\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right]
\end{equation}
%
By  using   Hooke's law and additive decomposition of strain tensors in the elastic and
plastic parts \cite{Jeremic96b} we obtain
%
\begin{equation}
d\sigma_{ij}=C_{ijkl}d\epsilon_{kl}-d\lambda C_{ijkl}\frac{\partial f}{\partial \sigma_{kl}}
\label{eqno13b}
\end{equation}
%
By substituting Eq.~(\ref{eqno13}) into Eq.~(\ref{eqno13b}), and with
$\nu=(1/2)(3K-2G)/(3K+G)$ it follows
% %
% \begin{equation}
% \label{eqno14}
% d\sigma_{ij}=\frac{E}{1+\nu}d\epsilon_{ij}+\frac{\nu E}{(1+\nu)(1-2\nu)}d\epsilon_{kk}\delta_{ij}-d\lambda\left[\frac{E}{1+\nu}\frac{\partial 
% f}{\partial \sigma_{ij}}+\frac{\nu E}{(1+\nu)(1-2\nu)}\frac{\partial f}{\partial \sigma_{mn}}\delta_{mn}\delta_{ij}\right]
% \end{equation}
% %
% Substituting $\nu=\frac{\frac{1}{2}(3K-2G)}{3K+G}$ in Eq.~(\ref{eqno14}) we have,
%
\begin{equation}
\label{eqno15}
d\sigma_{ij}=2Gd\epsilon_{ij}+Kd\epsilon_{kk}\delta_{ij}-d\lambda\left[\left(K-\frac{2}{3}G\right)\frac{\partial f}{\partial \sigma_{mn}}\delta_{mn}\delta_{ij}+2G\frac{\partial f}{\partial \sigma_{ij}}\right]
\end{equation}
%
Assuming a Drucker-Prager material model, the yield function, $f$, is
given by,
$f(\sigma_{ij})=F(I_1,\sqrt{J_2})-K=0$ and hence it follows that
%
\begin{equation}
\frac{\partial f}{\partial \sigma_{ij}}=\frac{\partial f}{\partial I_1}\delta_{ij}+\frac{1}{2\sqrt {J_2}}\frac{\partial f}{\partial \sqrt{J_2}}S_{ij}
\label{eqno17}
\end{equation}
%
Substituting Eq.~(\ref{eqno17}) into Eq.~(\ref{eqno15}) we have,
%
\begin{equation}
\label{eqno18}
d\sigma_{ij}=2Gd\epsilon_{ij}+Kd\epsilon_{kk}\delta_{ij}-d\lambda\left(3K\frac{\partial f}{\partial I_1}\delta_{ij}+\frac{G}{\sqrt{J_2}}\frac{\partial
f}{\partial \sqrt{J_2}}S_{ij}\right)
\end{equation}
%
where $d\lambda$ has the form,
%
\begin{equation}
\label{eqno19}
d\lambda=\frac{3Kd\epsilon_{kk}\left(\frac{\partial f}{\partial I_1}\right)+\left(\frac{G}{\sqrt{J_2}}\right)\left(\frac{\partial f}{\partial
\sqrt{J_2}}\right)S_{mn}d\epsilon_{mn}}{9K\left(\frac{\partial f}{\partial I_1}\right)^2+G\left(\frac{\partial f}{\partial \sqrt{J_2}}\right)^2}
\end{equation}
%
Now, the Drucker-Prager yield condition,f, is given by
%
\begin{equation}
\label{eqno20}
f=\sqrt{J_2}+\alpha I_1-k=0
\end{equation}
%
Combining Eqs.~(\ref{eqno18}),~(\ref{eqno19}) \&~(\ref{eqno20}), we have
the most general form of Drucker-Prager elastic-perfectly plastic constitutive
relationship as,
%
\begin{eqnarray}
\label{eqno21}
d\sigma_{ij} &=&
\left[2G\delta_{im}\delta_{jn}+\left(K-\frac{2}{3}G\right)\delta_{ij}\delta_{mn}
\right.
-
\nonumber \\
& & \left.
\frac{\left(\frac{G}{\sqrt{J_2}}\right)S_{ij}
+
3K\alpha\delta_{ij}}{G+9K\alpha^2}\left(\frac{G}{\sqrt{J_2}}S_{mn}
+
3K\alpha\delta_{mn}\right)\right]d\epsilon_{mn}
\end{eqnarray}
%
%
This  relationship  will  be  used in assessing one dimensional wave propagation
characteristics of tire shred material described below.

 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Material and Methods}
\label{MATERIAL_METHODS}

The  concept  of  triaxial testing of tire shreds has been considered by some and
easily dismissed because of the material and equipment logistics required. At UC
Davis,    a large sample triaxial compression specimen was successfully developed
and  used for testing P-wave
and shear wave (S-wave) propagation characteristics.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Material}
\label{MATERIAL}

The  size of tire shreds, provided by the California Integrated Waste Management
Board  (CIWMB),  ranged from about 30 cm long by 5 cm wide by 0.8 cm thick (long
and  thin)  to  about  10 cm long by 5 cm wide by 5 cm thick (massive rubber) to
about  3 cm long by 2 cm wide by 0.8 cm thick (chips). A trace of tire crumbs (2
to  5  mm  size)  was  also  noted. These dimensions are typical of routine tire
shredding  operations.  The specif\/ic gravity of the tire shreds was determined
to  be  about  1.20.  The  variation  in  porosity  from  loosest to densest was
estimated as 0.68 to 0.49.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Methods}
\label{METHODS}

The  compaction  mold  for  a  large  triaxial  test specimen of tire shreds was
fabricated  from  a  1.2  m  inside-diameter  polypropylene culvert (provided by
CIWMB); 1.25 mm latex sheeting was used to fabricate the triaxial test membrane.
The  end platens were 25 mm (top platen) and 12.5 mm (bottom platen) thick steel
plates.  The  load  frame  was  an  existing  structural  steel  frame  used for
consolidating  large  soil samples at the National Geotechnical Centrifuge at UC
Davis. Vertical load was provided by dual-acting, 76 mm-diameter, hydraulic rams
with  a  76  cm-stroke  and a maximum operating pressure of about 20 MPa. Vacuum
loading  for  conf\/ining  pressure  was  provided  using  a  high-volume Thomas
Industries  H50A-60  vacuum  generator working off air pressure. Instrumentation
for   measurement   of  P-  and  S-waves  consisted  of  42  piezoelectric,  ICP
accelerometers by PCB Piezotronics.


After  the latex membrane was secured to the bottom platen, the mold was secured
in place and the combined apparatus was placed on a specially fabricated lifting
frame  to  provide  a  solid stable base for the bottom platen to rest on during
sample preparation. Two horizontal and two vertical rows of accelerometer arrays
were established as shown schematically in Fig.~\ref{sketch:Accel_Array}.

\begin{figure}[!htbp]
%\vspace*{-2.3cm}
\begin{center}
\includegraphics[width=15cm]{Accel_Array3.ps}
%\vspace*{-2.5cm}
\caption{Schematic of Accelerometer Arrays}
\label{sketch:Accel_Array}
%\vspace*{-1.0cm}
\end{center}
\end{figure}
 

The  tire  shreds  were  placed  in approximately 150 mm lifts, about 100 kg per
lift,
and compacted with a walk-behind vibrating plate compactor (Wacker model) with a
172  kg  static  weight.  As each lift was brought to the level of accelerometer
placement,  precise  accelerometer  locations were determined using a pre-marked
rod and plumb bob. The completed triaxial test set up, placed in the load frame,
is shown in Fig.~\ref{photo:test_setup}.

\begin{figure}[!htbp]
%\vspace*{-0.3cm}
\begin{center}
\includegraphics[width=7cm]{Test_Setup1}
%\vspace*{-0.5cm}
\caption{Triaxial Test Set Up}
\label{photo:test_setup}
%\vspace*{-1.0cm}
\end{center}
\end{figure}


As discussed earlier, the directional Young's moduli and shear moduli were obtained from three-dimensional P- and S- wave velocities. The
P- and S- waves were generated by striking appropriately positioned metal or PVC striker blocks (either welded to the top and
bottom platens or inserted during specimen preparation) with a 0.5 kg, double-headed 'dead blow' hammer. The P- and S- wave velocities were measured through the accelerometer arrays shown in Fig.~\ref{sketch:Accel_Array} and recorded via a data acquisition
system with a scan rate of 10,000 readings per second from each accelerometer. To model in situ conditions, the embankment prof\/ile shown in Table 1 was assumed.
The tire shred specimen was tested under four triaxial load conditions representing the top and bottom of each tire shred layer as tabulated in Table 1.
Table 2 presents the loading conditions modeled during our testing.

\begin{small}
\begin{table}[!htbp]
\begin{center}
\label{emban:pro}
\caption{Assumed Embankment Prof\/ile}
\begin{tabular}{|c|c|c|}
\hline
Thickness [m]&Material&Density [$Mg/m^3$]\\
\hline
1.2&Paving and compacted soil&2.0\\
\hline  
3&Tire shreds&0.8\\
\hline
1&Compacted soil&2.0\\
\hline
3&Tire shreds&0.8\\
\hline
\end{tabular}
\vspace*{-0.5truecm}
\end{center}
\end{table}  
\end{small}


\begin{small}
\begin{table}
\begin{center}
\label{load:cond}
\caption{Selected Loading Conditions for Tests}
\begin{tabular}{|c|c|c|c|}
\hline
load number&$\sigma_v$ [kPa]&$\sigma_h$ [kPa]&$K_0$\\
\hline
1&23.9&11.3&0.47\\ 
\hline
2&47.9&13.4&0.28\\
\hline
3&65.8&18.4&0.28\\ 
\hline
4&89.9&25.1&0.28\\
\hline
\end{tabular}
\vspace*{-0.5truecm}
\end{center}
\end{table}
\end{small}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Test Results and Discussions}
\label{TEST_RESULTS_DISCUSSIONS}

A  total  of  83  tests  were  performed  from different wave source locations.
Because  the  data  acquisition  system  was  limited  to  16  channels, several
unload-reload  cycles  were required for each load condition. Unfortunately this
load  cycling  resulted  in excessive deformation of the test specimen. Although
some accelerometer  response  was lost due to excessive deformation, useful data were
obtained  for  the  f\/irst  three  load  cases.  The  in-plane  Young's modulus
($E_{11}=E_{22}=E$)  and  the  out-of-plane  Young's  modulus ($E_{33}=E'$) were
calculated   from  Eq.~(\ref{eqno0.7})  and  Eq.~(\ref{eqno0.8}).  The  in-plane
Poisson's  ratio  ($\nu_{12}=\nu_{21}$)  was  calculated from the in-plane shear
modulus Eq.(~\ref{eqno0.9}) and in-plane Young's modulus as,

\begin{equation}
\label{eqnoex1}
\nu_{12}=\frac{E_{11}}{2G_{12}}-1
\end{equation}
The out-of-plane Poisson's ratios ($\nu_{31}$ and $\nu_{13}$) were calculated from the in-plane Young's modulus ($E_{11}$), out-of-plane 
Young's modulus ($E_{33}$), and out-of-plane shear moduli, $G''=G_{13}$ (Eq.~\ref{eqno0.11}) and $G'=G_{31}$ (Eq.~\ref{eqno0.10}) as 
%
\begin{equation}
\label{eqnoex2}
\nu_{31}=\frac{E_{33}}{2G_{31}}-1
\;\;\;\;\;\; \mbox{;} \;\;\;\;\;\;
\nu_{13}=\frac{E_{11}}{2G_{13}}-1
\end{equation}
%
To describe the behavior of a cross-anisotropic material, f\/ive independent elastic constants are required. All other constants can be def\/ined 
from these f\/ive independent constants. In this study the redundant constants are calculated either for use in the elastic-plastic material 
model Eq.~(\ref{eqno26}) or to cross-check with published data \cite{ani:drescher}. The test results are tabulated in Table 3.


\begin{small}
\begin{table}[h]
\begin{center}
\label{elas:cons}
\caption{Elastic Constants}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
load&$\sigma_v$&$\sigma_h$&$G_{31}$&$G_{13}$&$E_{33}$&$E_{11,22}$&$E_{11}/E_{33}$&$\nu_{12}$&$\nu_{13}$&$\nu_{31}$\\
\hline
1&23.9&11.3&1500&1820&3330&5000&1.5&0.11&0.37&0.11\\
\hline
2&47.9&13.4&2550&5320&3825&9000&2.35&0.22&-0.15&-0.25\\
\hline
3&65.8&18.4&4890&19440&3835&13000&3.39&0.33&-0.67&-0.6\\
\hline
\end{tabular}
\vspace*{-0.5truecm}
\end{center}
\end{table}
\end{small}


The  current  testing program shows that the computed $G_{31}$ is equal to the
computed  $G_{32}$,  as  would be expected for a cross-anisotropic material. The
testing  also shows that $E_{11}$ was greater than $E_{33}$, and that the ratio
appears  to  increase with increased loading. Thus the data appears to conf\/irm
the  cross-anisotropic  assumption  proposed  for  the layered tire shred model.
Indeed,  the results indicate a stress-induced anisotropy, i.e., the anisotropy
increases  with  increased  stress.  It is also noted that the measured negative
Poisson's    ratio,  while physically possible, may be open for discussion. This
will be investigated in the next set of tests that
are  forthcoming in early 2004. It is suspected that the increase of
conf\/inement  from the stretched  latex  membrane  may have added some  error  to calculating
Poisson's ratios.

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

\section{
Elastic-Plastic Model and Correlation with 
${\bf G/G_{max}}$
}
\label{MODEL_CALIBRATION}


The   general   form   of   three-dimensional   Drucker-Prager   elastic-plastic
constitutive   relationship   Eq.~(\ref{eqno21})  can  be  simplif\/ied  to  any
direction  of interest. Thus, knowing the current state of the stress tensor and
strain-increment  in  that  particular  direction,  the stress increment in that
direction   can   be   solved  for  incrementally  to  get  the  elastic-plastic
stress-strain relationship.


For  the  case  of  shear  wave  propagation  in the 3-direction with horizontal
polarization  Eq.~(\ref{eqno21})  can  be  simplif\/ied to the following form by
substituting  $i=3$;  $j=1$;  $m=3$  and  $n=1$, which after some tensor algebra
becomes:
%\begin{equation}
%\label{eqno22}
%d\sigma_{31}=\left[2G_{31}\delta_{33}\delta_{11}+\left(K-\frac{2}{3}G_{31}\right)\delta_{31}\delta_{31}-\frac{\left(\frac{G_{31}}{\sqrt{J_2}}\right)S_{31}+3K\alpha\delta_{31}}{G_{31}+9K\alpha^2}\left(\frac{G_{31}}{\sqrt{J_2}}S_{31}+3K\alpha\delta_{31}\right)\right]d\epsilon_{31}
%\end{equation}
%or
\begin{equation}
d\sigma_{31}=\left[2G_{31}-\frac{\frac{G_{31}^2}{J_2}}{G_{31}+9K\alpha^2}S_{31}S_{31}\right]d\epsilon_{31}
\label{eqno24}
\end{equation}


The  bulk  modulus,  K,  is  calculated  from $E_{11}$, $E_{33}$, $ \nu_{12}$, $
\nu_{31} $, using the relationship,
%
\begin{equation}
\label{eqno26}
K
=
\frac{1}{3
\displaystyle{\left(
\frac{1}{E_{11}}-\frac{\nu_{12}}{E_{11}}-\frac{\nu_{31}}{E_{33}}\right)}
}
\end{equation}
                                                                                                                                                             
                                                                                                                                                             
The Drucker-Prager material constant, $ \alpha $, is related to the friction angle, $ \phi $, as,
\begin{equation}
\label{eqno27}
\alpha=\frac{2\sin \phi}{\sqrt{3}(3-\sin \phi)}
\end{equation}
                                                                                                                                                             
                                                                                                                                                             
Knowing  $E_{11}$,  $E_{33}$,$  \nu_{12}$,  $\nu_{31}$  (Table  3)  and using 
$\phi=20^0$  \cite{active:humphrey}  Eq.~(\ref{eqno24}) was solved incrementally
and  the  predicted elastic-plastic stress ($ \tau_{31} $)-strain ($ \gamma_{31}
$) relationship is shown in Fig.~\ref{plot:stress-strain:G-Gmax}(a).


Eq.~(\ref{eqno24}) also represents the variation of $G/G_{max}$ which is usually
found  in  traditional analysis of dynamics of soils. The term inside
the bracket in Eq.~(\ref{eqno24}) is the elastic-plastic shear modulus, which is
dependent  on  the  stress  deviator and reduces with increase in stress/strain.
Fig.~\ref{plot:stress-strain:G-Gmax}(b)  shows  the predicted reduction of shear
modulus   in  terms  of  dimensionless  parameters  $G_{31}/G_{31  max}$  versus
$\gamma_{31}$.

\begin{figure}[!htbp]
%\vspace*{-1.0truecm}
\begin{center}
\includegraphics[width=15cm]{stress-strain_G-Gmax}
%\vspace*{-0.9truecm}
\caption{(a)      Predicted         Elastic  and  Elastic-Plastic  Stress-Strain
Relationship; (b) Predicted Variation of Shear Modulus with Strain}
\label{plot:stress-strain:G-Gmax}
%\vspace*{-1.0truecm}
\end{center}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
\section{Conclusions}
\label{SUMMARY}

Based on this experimental and theoretical study, the following conclusions can be advanced:
\begin{enumerate}

\item  Three-dimensional  triaxial  P-  and  S-wave velocity testing can be used
to  determine  the  f\/ive  elastic  constants  necessary  for  modeling the
behavior of anisotropic tire shred material.

\item    The  test results demonstrated the cross-anisotropic nature of the tire
shred  material.  It  was  also  observed  that anisotropy increased with stress
(stress-induced anisotropy).

\item    The  data  obtained  from  three-dimensional  triaxial  P-  and S- wave
propagation tests can be used to calculate an elastic-perfectly plastic material
model  for the tire shreds material. The model also represented the variation of
$G/G_{max}$,  that  is  usually  found  in  traditional  analysis of dynamics of
frictional media.

\item  Further testing is necessary to mitigate the excessive deformation caused
by  unload-reload  cycles  in the current program. Additional testing would also
target  redundant  measurements  of  six elastic constants so as to evaluate the
negative Poisson's ratio determinations from the current program.

\end{enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Acknowledgment}
\label{ACKNOWLEDGEMENT}
%
This  work  was  supported  by  the California Integrated Waste Management Board
Special  Waste  Division  (part  of  California Environmental Protection Agency)
award \# IWM-C0141/02-00296V.
%




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