A comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai equations

Peter Wall

Applications of Mathematics (1997)

  • Volume: 42, Issue: 4, page 245-257
  • ISSN: 0862-7940

Abstract

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In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.

How to cite

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Wall, Peter. "A comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai equations." Applications of Mathematics 42.4 (1997): 245-257. <http://eudml.org/doc/32980>.

@article{Wall1997,
abstract = {In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.},
author = {Wall, Peter},
journal = {Applications of Mathematics},
keywords = {composite materials; homogenization; Hashin-Shtrikman bounds; Halpin-Tsai equations; effective properties; auxiliary partial differential equations; unidirectional elastic fiber composite; fitting parameter; auxiliary partial differential equations; unidirectional elastic fiber composite; fitting parameter},
language = {eng},
number = {4},
pages = {245-257},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai equations},
url = {http://eudml.org/doc/32980},
volume = {42},
year = {1997},
}

TY - JOUR
AU - Wall, Peter
TI - A comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai equations
JO - Applications of Mathematics
PY - 1997
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 42
IS - 4
SP - 245
EP - 257
AB - In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.
LA - eng
KW - composite materials; homogenization; Hashin-Shtrikman bounds; Halpin-Tsai equations; effective properties; auxiliary partial differential equations; unidirectional elastic fiber composite; fitting parameter; auxiliary partial differential equations; unidirectional elastic fiber composite; fitting parameter
UR - http://eudml.org/doc/32980
ER -

References

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  1. Analysis and Performance of Fiber Composites, second edn, Wiley Interscience, New York, 1990. (1990) 
  2. Homogenization:Averaging Processes in Periodic Media, Kluwer Academic Publishers, Dordrecht, 1989. (1989) MR1112788
  3. 10.1016/0022-5096(93)90006-2, J. Mech. Phys. Solids 41(5) (1993),  937–980). (1993),  937–980) MR1214022DOI10.1016/0022-5096(93)90006-2
  4. An Introduction to Γ -Convergence, Birkhäuser, Boston, 1993. (1993) Zbl0816.49001MR1201152
  5. 10.1002/pen.760160512, Polymer Engineering and science 16(5) (1976), 344–352. (1976) DOI10.1002/pen.760160512
  6. 10.1016/0022-5096(65)90015-3, J. Mech. Phys. Solids 13 (1965), 119–134. (1965) DOI10.1016/0022-5096(65)90015-3
  7. Analysis of composite materials-a survey, J. Mech. Phys. Solids 50 (1983), 481–505. (1983) Zbl0542.73092
  8. 10.1115/1.3629590, Journal of Applied Mechanics (1964), 223–232. (1964) DOI10.1115/1.3629590
  9. 10.1016/0022-5096(63)90060-7, J. Mech. Phys. Solids 11 (1963), 127–140. (1963) MR0159459DOI10.1016/0022-5096(63)90060-7
  10. Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin Heidelberg New York, 1994. (1994) MR1329546
  11. 10.1016/0022-5096(91)90046-Q, J. Mech. Phys. Solids 39(5) (1991), 663–681. (1991) MR1112738DOI10.1016/0022-5096(91)90046-Q
  12. 10.1016/0961-9526(95)00025-I, Composites Engineering 5(5) (1995), 519–531. (1995) DOI10.1016/0961-9526(95)00025-I
  13. 10.1080/00036819508840366, Applicable Analysis 58 (1995), 123–135. (1995) MR1384593DOI10.1080/00036819508840366
  14. On characterizing the set of possible effective tensors of composites: The variational method and the translation method, Communications on Pure and Applied Mathematics XLIII (1990), 63–125. (1990) Zbl0751.73041MR1024190
  15. 10.1016/0022-5096(88)90001-4, J. Mech. Phys. Solids 36(6) (1988), 597–629. (1988) MR0969257DOI10.1016/0022-5096(88)90001-4
  16. The Homogenization Method: An Introduction, Studentlitteratur, Lund, 1993. (1993) MR1250833
  17. Non Homogeneous Media and Vibration Theory, Lecture Notes in Physics 127, Springer Verlag, Berlin, 1980. (1980) Zbl0432.70002MR0578345
  18. Optimal Bounds on the Effective Properties of Multiphase Materials by Homogenization, Thesis 41L, Dept. of Appl. Math., Luleå University of Technology, 1994. (1994) 

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