Elastic properties of highly anisotropic thin poly(propylene) foams
Enis Tuncer\corauthref{cor}
In this letter, elastic properties of highly anisotropic cellular poly(propylene) films are reported. The material shows peculiar elastic properties compared to other foams in the literature. The data is displayed as the relative Young's modulus $E^*/E_s$ versus relative density $\rho^*/\rho_s$. Almost all the data from the literature are located on the region $E^*/E_s=(\rho^*/\rho_s)^n$ with $1\le n\le6$. The introduced material on the other hand have lower relative Young's modulus at high relative densities, $n\ge6$. %This characteristic is explained by the high anistropy in the material microstructure. % and verified by the computer simulations. %Such materials may be prefered when the material stores a soft materials
tuncerwegener.tex
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\begin{document}
\begin{frontmatter}
  \title{Elastic properties of highly anisotropic thin poly(propylene) foams}
  \author{Enis Tuncer\corauthref{cor}} \and
  \corauth[cor]{Corresponging author.}
  \ead{enis.tuncer@physics.org}
  \author{Michael Wegener}
%  \author{Reimund Gerhard-Multhaupt}
  \address{Applied Condensed-Matter Physics, Department of Physics, University of Potsdam, D-14469 Potsdam, Germany}

  \begin{abstract}
    In this letter, elastic properties of highly anisotropic cellular poly(propylene) films are reported. The material shows peculiar elastic properties compared to other foams in the literature. The data is displayed as the relative Young's modulus $E^*/E_s$ versus relative density $\rho^*/\rho_s$. Almost all the data from the literature are located on the region $E^*/E_s=(\rho^*/\rho_s)^n$ with $1\le n\le6$. The introduced material on the other hand have lower relative Young's modulus at high relative densities, $n\ge6$. %This characteristic is explained by the high anistropy in the material microstructure. % and verified by the computer simulations. %Such materials may be prefered when the material stores a soft materials 
  \end{abstract}
  \begin{keyword}
    Foams, cellular materials, homogenization, elastic properties,  Young's modulus%, micro-structural information
  \end{keyword}
\end{frontmatter}
Foams are solid-gas mixtures, in which the solid phase forms a continuous (percolating) network structure, and performs an action depending on the application, {\it i.e.}, supports load, conducts heat/electricity,  {\it etc}.   The ratio of effective Young's modulus of foams to the solid's modulus $E^*/E_s$ has been related to the volume fraction of the solid material $\rho^*/\rho_s$~\cite{CellularSolids}.
  \begin{equation}
    \label{eq:power}
    E^*/E_s=C\,(\rho^*/\rho_s)^n
  \end{equation}
where $C$ and $n$ are constants and superscript `$*$' and and subscript `$s$' denote the properties of the effective and the solid material, respectively. Yet, there is no unique theory that explains the effective properties of foams or any composite that takes into account the geometrical parameters of the material. However, equations based on the effective medium theory and empirical mixture formulas have been proposed~\cite{TorquatoBook,MiltonBook}. For foams Eq.~(\ref{eq:power}) has been extensively used, and the constant $n$ often gives information about the structure of material in hand %, {\eg}, for closed and open cell structures $n\sim1$ and $n\sim2$  are observed 
\cite{CellularSolids,Garboczi2002a,Garboczi2002b,Garboczi2001,TuncerEyeTruss2003}.%, respectively. 

In Table~\ref{tab:info}, data taken from the literature for various foams and characteristic exponent values of Eq.~(\ref{eq:power}) are presented. In the table, closed cell foams have the lowest exponent $n$ which is around $\approx3/2$~\cite{Morgan1981,Zwissler1983,walsh1965,Mills1999,Baxter1972,Clutton1991,Scanlon2001,Christensen1986,Sun2002,Kitazono2003,Baltsavias1999,CellularSolids,Ladd1997,Hellmich2002}. Numerical simulations performed by considering three-dimensional closed-cell structures also yield similar exponents~\cite{Garboczi2002a,Garboczi2001}. In contrast to closed cell foams, most of the open cell foams result in $E^{*}/E_s\propto (\rho^*/\rho_s)^{2}$ dependence~\cite{CellularSolids,Hagiwara1987,Han2003,Badiche2000,Warren1988,Silva1997,Lederman1971,Gent1959,Sun2002,Garboczi2002b}. There are some exceptions which show $n>5/2$ dependence~\cite{Gross1997,Pekala1990,Lui2003,Badiche2000,Garboczi2002a,Fazekas2002}, which, we believe in principle, could be due to the anisotropy in materials. 

The data from the given literature in Table~\ref{tab:info} are illustrated in Fig.~\ref{fig:others}. In the figure, the gray-region contains all the data presented in the literature (Table~\ref{tab:info}). Some of the newly published data are shown with symbols; ($\ast$) \cite{Ladd1997}, ($\triangle$) \cite{Baltsavias1999}, ($\star$) and ($\blacklozenge$) \cite{Badiche2000}, ($\blacktriangleleft$) \cite{Sun2002}, ($+$) \cite{Hellmich2002}, ($\bullet$) \cite{Kitazono2003}, ($\boxplus$) \cite{jong1997}, ($\boxdot$) \cite{evans1997b} and ($\circledcirc$) \cite{ong2001}. The data presented in this study are poly(propylene) foams and are plotted with ($\Box$); our data with errorbars and other performed experiments \cite{WegenerAPL2003}. The dashed lines (\kesik) shows Eq.~(\ref{eq:power}) for $C=1$ and $n=\{1,\,2,\,5/2,\,6\}$. All the data from the literature are located between $n\approx1$ and $n\approx 6$ region. The solid lines (\full) in the figure are calculated for a truss-like structure using the finite element method \cite{TuncerWegener2003}, which has voids as in the form of lozenge, with various void sizes, $a/b=\{1,\,2,\,4,\,8\}$. The parameters $a$ and $b$ define the void dimensions in the $x$- and $y$-dimensions, respectively--when the ratio $a/b$ is large the structure is highly anisotropic. At high concentrations of solid, the computer model based on the finite element method yield similar values as the experiments for $4<a/b<8$. 

The original films in this study were cellular poly(propylene) with typical thicknesses around $30-100\ \micro\meter$. They were stretched by the manufacturer (VTT Processes, Finland)\cite{PaajanenSensors2000} and the lateral dimensions were much larger than the thickness of the samples. The structure of samples resembles the puff pastry, some scanning electron micrographs have been presented in Ref.~\cite{TuncerWegener2003,WegenerPhysD2004,WegenerAPL2003,Wegener-CEIDP03}. The surface area of samples are larger than $1\ \centi\meter^2$  for our electromechanical applications. The average void structure is oblate with dimensions around $\sim50\ \micro\meter$ long and $\sim10\ \micro\meter$ high from the scanning electron micro-graphs, which indicates that there are at least $5$ voids in the thickness direction.  The original films were inflated with a post-processing procedure \cite{WegenerAPL2003}. Their elastic properties were measured after each inflation process with the dielectric spectroscopy method~\cite{AxelReview}. The inflation influences the relative density and average void (cell) dimension of the films. The change in the relative Young's moduli of the films is illustrated in the figure with open symbols. The relative Young's modulus and density were calculated by using $E_s=1\ \giga\pascal$ and $\rho_s=1\ \gram/\centi\meter^{3}$ for the solid poly(propylene). The letters {\sf A}, {\sf B} and {\sf C} present the different stages in the inflation process. The non-inflated films had the highest Young's moduli ({\sf A}). By inflating the samples, the stiffness of the films lowered down to $E^*/E_s\approx10^{-3}$ ({\sf B}). After this point, a further inflation of the films resulted an increase in the stiffness ({\sf C}). This was due to the shape of the voids, which became nearly-isotropic. The voids become rounder with $a/b\le4$--the anisotropy in the films were destroyed by the inflation.  The numerical simulations utilizing the finite element method by means of simple truss-like structure \cite{TuncerWegener2003} has showed a similar behavior (even simulations with more realistic geometries yield similar results \cite{TuncerEyeTruss2003}), and has verified the experimental observations. The `{\sf U}'-shaped behavior of our data could be explained by the solid materials' cell-wall thickness \cite{TuncerWegener2003,TuncerEyeTruss2003}. It should be mentioned that if the layered texture is exchanged to a fibrous one, one might obtain much softer materials with exponents $n>6$ in Eq.~(\ref{eq:power}). %One might also argue that the peculiar behavior of the present materials are due to their somewhat two-dimensional structure, physical strucuture changes only in two direction.

Almost all of our measurement points (only two of the data points fall in the region marked by $1<n<6$) are outside the region for the conventional foams. So we can conclude that as the films become more isotropic ($a/b\le2$),  they yield similar Young's modulus behavior as the foams presented in the literature, $n\le6$ in Eq.~(\ref{eq:power}). Although in the literature it is not clearly stated that foams with higher characteristic exponents in Eq.~(\ref{eq:power}) are anistropic, our observations affirm that anisotropic foams yield high exponent values for the power-law expression in Eq.~(\ref{eq:power}).

Finally, as a concluding remark, although poly(propylene) foam is widely used and has been known for decades, the presented low stiffness character of highly anistropic materials at high relative densities can be advantages and preferably selected for special applications in which a multifunctional material with soft elastic properties is needed. The solid phase in that case is utilized to possess an additional feature, such as storing charge, heat, magnetic/electric polarization \etc\ A discrete example is electret materials used in transducer applications~\cite{ReimundRev,ElectretPhysicsToday}. The solid phase in cellular electrets is used to store electrical charge, for this reason, in order to have high charge densities, high volume fraction of solid is simultaneously desired  together with low elastic constants. In addition, one can even tailor the elastic properties by using post processing techniques to change the micro-structure of the material. 

\begin{table}[tp]
  \caption{Effective Young's modulus relative density relation for different structures, Eq.~(\ref{eq:power}).\label{tab:info}}
  \centering
  \begin{tabular*}{6in}{l@{\extracolsep{\fill}}ll}
\hline\hline
 $1 < n < 2$ &   $3/2<n< 5/2$ &$n>5/2$ \\
\hline
 Closed-cell & Open-cells & Anisotropic-cells(?)\\
\hline
 Foamed glasses \citep{Morgan1981,Zwissler1983,walsh1965} &  Metal foams~\citep{CellularSolids,Hagiwara1987,Han2003,Badiche2000,Warren1988,Silva1997} & Aerogels~\citep{Gross1997,Pekala1990}\\
  Polymer foams~\citep{Christensen1986,Sun2002,Mills1999,Baxter1972,Clutton1991} & Polymer foams~\citep{Lederman1971,Gent1959,Sun2002,jong1997,evans1997b} & Honeycomb\citep{Badiche2000}\\
Metal foams~\citep{Kitazono2003,ong2001} & Numerical~\citep{Garboczi2002b} &Penrose network\citep{Badiche2000}\\
Composite food~\citep{Baltsavias1999,Scanlon2001} & & Various foams\citep{Lui2003}\\
Biological materials~\citep{CellularSolids,Ladd1997,Hellmich2002} & &Numerical~\citep{Garboczi2002a,Fazekas2002} \\
Numerical~\citep{Garboczi2002a,Garboczi2001} & &\\
\hline\hline
%& \citep{CellularSolids} & \citep{Badiche2000} &\citep{Warren1988} &\citep{Badiche2000}\\
  \end{tabular*}  
\end{table}


\begin{figure}[tp]
  \centering
  \includegraphics[width=5in]{tuncerwegenerFig1.eps}
  \caption{\label{fig:others} Relative effective Young's modulus versus relative density for different foams. The gray region marks the area of currently avaliable data present in the literature, Table~\ref{tab:info}. Some of the newly published data are shown with symbols; ($\ast$) \cite{Ladd1997}, ($\triangle$) \cite{Baltsavias1999}, ($\star$) and ($\blacklozenge$) \cite{Badiche2000}, ($\blacktriangleleft$) \cite{Sun2002}, ($+$) \cite{Hellmich2002}, ($\bullet$) \cite{Kitazono2003}, ($\boxplus$) \cite{jong1997}, ($\boxdot$) \cite{evans1997b} and ($\circledcirc$) \cite{ong2001}. The introduced and discussed poly(propylene) materials are plotted with ($\Box$) \cite{WegenerAPL2003} and ($\Box$) with errorbars.}
\end{figure}
  
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\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem{CellularSolids}
L.~J. Gibson, M.~Ashby, Cellular Solids: structure \& properties, 2nd Edition,
  Solid state science, Cambridge University Press, UK, 1997.

\bibitem{TorquatoBook}
S.~Torquato, Random Heterogeneous Materials: Microstructure and macroscopic
  properties, Vol.~16, Springer-Verlag, Berlin, 2001.

\bibitem{MiltonBook}
G.~W. Milton, The Theory of Composites, Vol.~6 of Cambridge monographs on
  applied and computational mathematics, Cambridge University Press, Cambridge
  UK, 2002.

\bibitem{Garboczi2002a}
A.~P. Roberts, E.~J. Garboczi, Computation of the linear elastic properties of
  random porous materials with a wide variety of microstructure, Proc. Royal
  Society 458~(2021) (2002) 1033--1054.

\bibitem{Garboczi2002b}
A.~P. Roberts, E.~J. Garboczi, Elastic properties of model random
  three-dimensional opel-cell solids, Journal of the Mechanics and Physics of
  Solids 50~(1) (2002) 33--55.

\bibitem{Garboczi2001}
A.~P. Roberts, E.~J. Garboczi, Elastic moduli of model random three-dimensional
  closed-cell cellular solids, Acta Materialia 49 (2001) 187--197.

\bibitem{TuncerEyeTruss2003}
E.~Tuncer, Effective elastic properties of two cellular structures in
  two-dimensions, {\tt arXiv:cond-mat/0404166} (2004).

\bibitem{Morgan1981}
J.~S. Morgan, J.~L. Wood, R.~C. Brandt, Cell-size effects on the strength of
  foamed glasses, Materials Science and Engineering 47~(1) (1981) 37.

\bibitem{Zwissler1983}
J.~G. Zwissler, M.~A. Adams, in: R.~C. Brandt, A.~G. Evans, D.~P.~H. Hasselman,
  F.~F. Lange (Eds.), Fracture mechanics of ceramics, Vol.~6, Plenum Press, NY
  USA, 1983, p. 211.

\bibitem{walsh1965}
J.~B. Walsh, W.~F. Brace, A.~W. England, Effect of porosity on compressibility
  of glass, Journal of American Ceramic Society 48~(12) (1965) 605--608.

\bibitem{Mills1999}
N.~J. Mills, H.~X. Zhu, The high strain compression of closed-cell polymer
  foams, Journal of Mechanics and Physics of Solids 47 (1999) 669.

\bibitem{Baxter1972}
S.~Baxter, T.~T. Jones, Physical properties of foamed plastics and their
  dependence on structure, Plastics and Polymers 40~(146) (1972) 69.

\bibitem{Clutton1991}
E.~Q. Clutton, G.~N. Rice, in: 34{th} Annual Technical Conference Proceedings,
  RAPRA Technology, Strewsbury, 1999, p.~99.

\bibitem{Scanlon2001}
M.~G. Scanlon, M.~C. Zghal, Bread properties and crumb structure, Food Research
  International 34 (2001) 841--864.

\bibitem{Christensen1986}
R.~M. Christensen, Mechanics of low density materials, Journal of Mechanics and
  Physics of Solids 34~(6) (1986) 563--578.

\bibitem{Sun2002}
H.~Sun, G.~S. Sur, J.~E. Mark, Microcellular foams from polyethersulfone and
  polyphenylsulfone: Preparation and mechanical properties, European Polymer
  Journal 38 (2002) 2373--2381.

\bibitem{Kitazono2003}
K.~Kitazono, E.~Sato, K.~Kuribayashi, Application of the mean-field
  approximation to the elastic-plastic behavior for closed-cell metal foams,
  Acta Materialia 51 (2003) 4823--4836.

\bibitem{Baltsavias1999}
A.~Baltsavias, A.~Jurgens, T.~van Vliet, Properties of short-dough biscuits in
  relation to structure, Journal of Cereal Science 29 (1999) 245--255.

\bibitem{Ladd1997}
A.~J.~C. Ladd, J.~H. Kinney, Deformation and failure in cellular materials,
  Physical Review E 55~(3) (1997) 3271--3275.

\bibitem{Hellmich2002}
C.~Hellmich, F.-J. Ulm, Are mineralized tissues open crystal foams reinforces
  by crosslinked collogen?--some energy arguments, Journal of Biomechanics 35
  (2003) 1199--1212.

\bibitem{Hagiwara1987}
H.~Hagiwara, D.~J. Green, Elastic behavior of open-cell alumina, Journal of
  Americal Ceramics Society 70 (1987) 811--815.

\bibitem{Han2003}
F.~Han, H.~Cheng, J.~Wang, Q.~Wang, Effect of pore combination on the
  mechanical properties of an open cell aluminum foam, Scripta Materialia 50
  (2004) 13--17.

\bibitem{Badiche2000}
X.~Badiche, S.~Forest, T.~Guibert, Y.~Bienvenu, J.-D. Bartout, P.~Ienny,
  M.~Croset, H.~Bernet, Mechanical propeties and non-homogeneous deformation of
  open-cell nickel foams; application of the mechanics and cellular solids and
  of porous materials, Material Science and Engineering A289 (2000) 276--288.

\bibitem{Warren1988}
W.~E. Warren, A.~M. Kraynik, The linear elastic properties of open-cell foams,
  Journal of Applied Mechanics 55~(2) (1988) 341--346.

\bibitem{Silva1997}
M.~J. Silva, L.~J. Gibson, The effect of non-periodic microstructure and
  defects on the compressive strength of two-dimensional cellular solids,
  International Journal of Mecnanical Sciences 39~(5) (1997) 549--563.

\bibitem{Lederman1971}
J.~M. Lederman, The prediction of the tensile properties of flexible foams,
  Journal of Applied Polymer Science 15 (1971) 693--703.

\bibitem{Gent1959}
A.~N. Gent, A.~G. Thomas, The deformation of foamed elastic materials, Journal
  of Applied Polymer Science 1 (1959) 107.

\bibitem{Gross1997}
J.~Gross, G.~W. Scherer, C.~T. Alviso, R.~W. Pekala, Elastic properties of
  crosslinked resoremol-formaldehyde gels and aerogels, Journal of
  Non-crystalline Solids 211 (1997) 132--142.

\bibitem{Pekala1990}
R.~W. Pekala, C.~T. Alviso, J.~D. Le{M}ay, Organic aerogels microstructural
  dependence of mechanical properties in compression, Journal of
  Non-crystalline Solids 125 (1990) 61--75.

\bibitem{Lui2003}
Z.~Lui, M.~G. Scanlon, Scaling young's modulus of cellular solids, Journal of
  Material Science Letters 22 (2003) 547--548.

\bibitem{Fazekas2002}
A.~Fazekas, R.~Dendievel, L.~Salvo, Y.~Br{\'e}chet, Effect of microstructural
  topology upon stiffness and strength of 2d cellular structures, International
  Journal of Mechanical Sciences 44 (2002) 2047--2066.

\bibitem{jong1997}
J.-P. Lee, K.-H. Lee, H.-K. Song, Manufacture of biodegradable packaging foams
  from agar by freeze-drying, Journal of Materials Science 32~(21) (1997)
  5825--5832.

\bibitem{evans1997b}
N.~Chan, K.~E. Evans, Microscopic examination of the microstructure and
  deformation of conventional and auxetic foams, Journal of Materials Science
  32~(11) (1997) 2797--2809.

\bibitem{ong2001}
T.~J. Lu, J.~M. Ong, Characterization of close-celled cellular aluminum alloys,
  Journal of Materials Science 36~(11) (2001) 2773--2786.

\bibitem{WegenerAPL2003}
M.~Wegener, W.~Wirges, R.~Gerhard-Multhaupt, M.~Dansachm{\"u}ller,
  R.~Schw{\"o}diauer, S.~Bauer-Gogonea, S.~Bauer, M.~Paajanen, H.~Minkkinen,
  J.~Raukola, Controlled inflation of voids in cellular polymer ferroelectrets:
  {O}ptimizing electromechanical transducer properties, Applied Physics Letters
  84 (2004) 392--394.

\bibitem{TuncerWegener2003}
E.~Tuncer, M.~Wegener, Elastic properties of cellular polypropylene films:
  Finite element simulations and their comparison with experiments, {\tt
  arXiv:cond-mat/0404165} (2004).

\bibitem{PaajanenSensors2000}
M.~Paajanen, J.~Lekkala, K.~Kirjavainen, Electro{M}echanical film ({EMF}i)--a
  new multipurpose electret material, Sensors and Actuators 84 (2000) 95--102.

\bibitem{Wegener-CEIDP03}
M.~Wegener, W.~Wirges, R.~Gerhard-Multhaupt, M.~Paajanen, H.~Minkkinen,
  J.~Raukola, Enhancing the cellular structure and the electromechanical
  response of ferroelectrets--gas diffusion expansion of voided polypropylene
  films, in: 2003 Annual Report, Conference on Electrical Insulation and
  Dielectric Phenomena CEIDP, IEEE Dielectrics and Electrical Insulation
  Society, 2003, pp. 36--39.

\bibitem{WegenerPhysD2004}
M.~Wegener, W.~Wirges, J.~Fohlmeister, B.~Tiersch, R.~Gerhard-Multhaupt,
  Two-step inflation of cellular polypropylene films: void-thickness increase
  and enhanced electromechanical properties, Journal of Physics D: Applied
  Physics 37~(4) (2004) 623--627.

\bibitem{AxelReview}
A.~Mellinger, Dielectric resonance spectroscopy: a versatile tool in the quest
  for better piezoelectric polymers, IEEE Transactions on Dielectrics and
  Electrical Insulation 10~(5) (2003) 842--861.

\bibitem{ReimundRev}
R.~Gerhard-Multhaupt, Less can be more: Holes in polymers lead to a new
  paradigm of piezoelectric materials for electret transducers, IEEE
  Transactions on Dielectrics and Electrical Insulation 9~(5) (2002) 850--859.

\bibitem{ElectretPhysicsToday}
S.~Bauer, R.~Gerhard-Multhaupt, G.~Sessler, Ferroelectrets: Soft electroactiv
  foams for transducers, Physics Today.

\end{thebibliography}

\end{document}

cond-mat0411538.extracted_bib
\begin{thebibliography}{10}
\expandafter\ifx\csname url\endcsname\relax
  \def\url#1{\texttt{#1}}\fi
\expandafter\ifx\csname urlprefix\endcsname\relax\def\urlprefix{URL }\fi

\bibitem{CellularSolids}
L.~J. Gibson, M.~Ashby, Cellular Solids: structure \& properties, 2nd Edition,
  Solid state science, Cambridge University Press, UK, 1997.

\bibitem{TorquatoBook}
S.~Torquato, Random Heterogeneous Materials: Microstructure and macroscopic
  properties, Vol.~16, Springer-Verlag, Berlin, 2001.

\bibitem{MiltonBook}
G.~W. Milton, The Theory of Composites, Vol.~6 of Cambridge monographs on
  applied and computational mathematics, Cambridge University Press, Cambridge
  UK, 2002.

\bibitem{Garboczi2002a}
A.~P. Roberts, E.~J. Garboczi, Computation of the linear elastic properties of
  random porous materials with a wide variety of microstructure, Proc. Royal
  Society 458~(2021) (2002) 1033--1054.

\bibitem{Garboczi2002b}
A.~P. Roberts, E.~J. Garboczi, Elastic properties of model random
  three-dimensional opel-cell solids, Journal of the Mechanics and Physics of
  Solids 50~(1) (2002) 33--55.

\bibitem{Garboczi2001}
A.~P. Roberts, E.~J. Garboczi, Elastic moduli of model random three-dimensional
  closed-cell cellular solids, Acta Materialia 49 (2001) 187--197.

\bibitem{TuncerEyeTruss2003}
E.~Tuncer, Effective elastic properties of two cellular structures in
  two-dimensions, {\tt arXiv:cond-mat/0404166} (2004).

\bibitem{Morgan1981}
J.~S. Morgan, J.~L. Wood, R.~C. Brandt, Cell-size effects on the strength of
  foamed glasses, Materials Science and Engineering 47~(1) (1981) 37.

\bibitem{Zwissler1983}
J.~G. Zwissler, M.~A. Adams, in: R.~C. Brandt, A.~G. Evans, D.~P.~H. Hasselman,
  F.~F. Lange (Eds.), Fracture mechanics of ceramics, Vol.~6, Plenum Press, NY
  USA, 1983, p. 211.

\bibitem{walsh1965}
J.~B. Walsh, W.~F. Brace, A.~W. England, Effect of porosity on compressibility
  of glass, Journal of American Ceramic Society 48~(12) (1965) 605--608.

\bibitem{Mills1999}
N.~J. Mills, H.~X. Zhu, The high strain compression of closed-cell polymer
  foams, Journal of Mechanics and Physics of Solids 47 (1999) 669.

\bibitem{Baxter1972}
S.~Baxter, T.~T. Jones, Physical properties of foamed plastics and their
  dependence on structure, Plastics and Polymers 40~(146) (1972) 69.

\bibitem{Clutton1991}
E.~Q. Clutton, G.~N. Rice, in: 34{th} Annual Technical Conference Proceedings,
  RAPRA Technology, Strewsbury, 1999, p.~99.

\bibitem{Scanlon2001}
M.~G. Scanlon, M.~C. Zghal, Bread properties and crumb structure, Food Research
  International 34 (2001) 841--864.

\bibitem{Christensen1986}
R.~M. Christensen, Mechanics of low density materials, Journal of Mechanics and
  Physics of Solids 34~(6) (1986) 563--578.

\bibitem{Sun2002}
H.~Sun, G.~S. Sur, J.~E. Mark, Microcellular foams from polyethersulfone and
  polyphenylsulfone: Preparation and mechanical properties, European Polymer
  Journal 38 (2002) 2373--2381.

\bibitem{Kitazono2003}
K.~Kitazono, E.~Sato, K.~Kuribayashi, Application of the mean-field
  approximation to the elastic-plastic behavior for closed-cell metal foams,
  Acta Materialia 51 (2003) 4823--4836.

\bibitem{Baltsavias1999}
A.~Baltsavias, A.~Jurgens, T.~van Vliet, Properties of short-dough biscuits in
  relation to structure, Journal of Cereal Science 29 (1999) 245--255.

\bibitem{Ladd1997}
A.~J.~C. Ladd, J.~H. Kinney, Deformation and failure in cellular materials,
  Physical Review E 55~(3) (1997) 3271--3275.

\bibitem{Hellmich2002}
C.~Hellmich, F.-J. Ulm, Are mineralized tissues open crystal foams reinforces
  by crosslinked collogen?--some energy arguments, Journal of Biomechanics 35
  (2003) 1199--1212.

\bibitem{Hagiwara1987}
H.~Hagiwara, D.~J. Green, Elastic behavior of open-cell alumina, Journal of
  Americal Ceramics Society 70 (1987) 811--815.

\bibitem{Han2003}
F.~Han, H.~Cheng, J.~Wang, Q.~Wang, Effect of pore combination on the
  mechanical properties of an open cell aluminum foam, Scripta Materialia 50
  (2004) 13--17.

\bibitem{Badiche2000}
X.~Badiche, S.~Forest, T.~Guibert, Y.~Bienvenu, J.-D. Bartout, P.~Ienny,
  M.~Croset, H.~Bernet, Mechanical propeties and non-homogeneous deformation of
  open-cell nickel foams; application of the mechanics and cellular solids and
  of porous materials, Material Science and Engineering A289 (2000) 276--288.

\bibitem{Warren1988}
W.~E. Warren, A.~M. Kraynik, The linear elastic properties of open-cell foams,
  Journal of Applied Mechanics 55~(2) (1988) 341--346.

\bibitem{Silva1997}
M.~J. Silva, L.~J. Gibson, The effect of non-periodic microstructure and
  defects on the compressive strength of two-dimensional cellular solids,
  International Journal of Mecnanical Sciences 39~(5) (1997) 549--563.

\bibitem{Lederman1971}
J.~M. Lederman, The prediction of the tensile properties of flexible foams,
  Journal of Applied Polymer Science 15 (1971) 693--703.

\bibitem{Gent1959}
A.~N. Gent, A.~G. Thomas, The deformation of foamed elastic materials, Journal
  of Applied Polymer Science 1 (1959) 107.

\bibitem{Gross1997}
J.~Gross, G.~W. Scherer, C.~T. Alviso, R.~W. Pekala, Elastic properties of
  crosslinked resoremol-formaldehyde gels and aerogels, Journal of
  Non-crystalline Solids 211 (1997) 132--142.

\bibitem{Pekala1990}
R.~W. Pekala, C.~T. Alviso, J.~D. Le{M}ay, Organic aerogels microstructural
  dependence of mechanical properties in compression, Journal of
  Non-crystalline Solids 125 (1990) 61--75.

\bibitem{Lui2003}
Z.~Lui, M.~G. Scanlon, Scaling young's modulus of cellular solids, Journal of
  Material Science Letters 22 (2003) 547--548.

\bibitem{Fazekas2002}
A.~Fazekas, R.~Dendievel, L.~Salvo, Y.~Br{\'e}chet, Effect of microstructural
  topology upon stiffness and strength of 2d cellular structures, International
  Journal of Mechanical Sciences 44 (2002) 2047--2066.

\bibitem{jong1997}
J.-P. Lee, K.-H. Lee, H.-K. Song, Manufacture of biodegradable packaging foams
  from agar by freeze-drying, Journal of Materials Science 32~(21) (1997)
  5825--5832.

\bibitem{evans1997b}
N.~Chan, K.~E. Evans, Microscopic examination of the microstructure and
  deformation of conventional and auxetic foams, Journal of Materials Science
  32~(11) (1997) 2797--2809.

\bibitem{ong2001}
T.~J. Lu, J.~M. Ong, Characterization of close-celled cellular aluminum alloys,
  Journal of Materials Science 36~(11) (2001) 2773--2786.

\bibitem{WegenerAPL2003}
M.~Wegener, W.~Wirges, R.~Gerhard-Multhaupt, M.~Dansachm{\"u}ller,
  R.~Schw{\"o}diauer, S.~Bauer-Gogonea, S.~Bauer, M.~Paajanen, H.~Minkkinen,
  J.~Raukola, Controlled inflation of voids in cellular polymer ferroelectrets:
  {O}ptimizing electromechanical transducer properties, Applied Physics Letters
  84 (2004) 392--394.

\bibitem{TuncerWegener2003}
E.~Tuncer, M.~Wegener, Elastic properties of cellular polypropylene films:
  Finite element simulations and their comparison with experiments, {\tt
  arXiv:cond-mat/0404165} (2004).

\bibitem{PaajanenSensors2000}
M.~Paajanen, J.~Lekkala, K.~Kirjavainen, Electro{M}echanical film ({EMF}i)--a
  new multipurpose electret material, Sensors and Actuators 84 (2000) 95--102.

\bibitem{Wegener-CEIDP03}
M.~Wegener, W.~Wirges, R.~Gerhard-Multhaupt, M.~Paajanen, H.~Minkkinen,
  J.~Raukola, Enhancing the cellular structure and the electromechanical
  response of ferroelectrets--gas diffusion expansion of voided polypropylene
  films, in: 2003 Annual Report, Conference on Electrical Insulation and
  Dielectric Phenomena CEIDP, IEEE Dielectrics and Electrical Insulation
  Society, 2003, pp. 36--39.

\bibitem{WegenerPhysD2004}
M.~Wegener, W.~Wirges, J.~Fohlmeister, B.~Tiersch, R.~Gerhard-Multhaupt,
  Two-step inflation of cellular polypropylene films: void-thickness increase
  and enhanced electromechanical properties, Journal of Physics D: Applied
  Physics 37~(4) (2004) 623--627.

\bibitem{AxelReview}
A.~Mellinger, Dielectric resonance spectroscopy: a versatile tool in the quest
  for better piezoelectric polymers, IEEE Transactions on Dielectrics and
  Electrical Insulation 10~(5) (2003) 842--861.

\bibitem{ReimundRev}
R.~Gerhard-Multhaupt, Less can be more: Holes in polymers lead to a new
  paradigm of piezoelectric materials for electret transducers, IEEE
  Transactions on Dielectrics and Electrical Insulation 9~(5) (2002) 850--859.

\bibitem{ElectretPhysicsToday}
S.~Bauer, R.~Gerhard-Multhaupt, G.~Sessler, Ferroelectrets: Soft electroactiv
  foams for transducers, Physics Today.

\end{thebibliography}