Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Homogenization of stratified thermoviscoplastic materials

Authors: Nicolas Charalambakis and François Murat
Journal: Quart. Appl. Math. 64 (2006), 359-399
MSC (2000): Primary 74Q15, 74Q10, 35B27; Secondary 74C10, 74F05, 35Q72, 35M20, 35K55
DOI: https://doi.org/10.1090/S0033-569X-06-01017-3
Published electronically: May 22, 2006
MathSciNet review: 2243868
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the present paper we study the homogenization of the system of partial differential equations

\begin{displaymath} \begin{array}{l} \displaystyle{\rho^\varepsilon(x) {\partia... ...\partial v^\varepsilon \over \partial x}\right)^2,} \end{array}\end{displaymath}

posed in $ a < x < b$, $ 0 < t < T$, completed by boundary conditions on $ v^\varepsilon$ and by initial conditions on $ v^\varepsilon$ and $ \theta^\varepsilon$. The unknowns are the velocity $ v^\varepsilon$ and the temperature $ \theta^\varepsilon$, while the coefficients $ \rho^\varepsilon$, $ \mu^\varepsilon$ and $ c^\varepsilon$ are data which are assumed to satisfy

$\displaystyle 0 < c_1 \leq \mu^\varepsilon(x,s) \leq c_2, \quad 0 < c_3 \leq c^\varepsilon(x,s) \leq c_4,\quad 0 < c_5 \leq \rho^\varepsilon(x) \leq c_6,$    
$\displaystyle \displaystyle{- c_7 \leq {\partial \mu^\varepsilon \over \partial... ...t c^\varepsilon(x,s) - c^\varepsilon(x,s')\vert\leq \omega(\vert s - s'\vert).}$    

This sequence of one-dimensional systems is a model for the homogenization of nonhomogeneous, stratified, thermoviscoplastic materials exhibiting thermal softening and a temperature-dependent rate of plastic work converted into heat.

Under the above hypotheses we prove that this system is stable by homogenization. More precisely one can extract a subsequence $ \varepsilon'$ for which the velocity $ v^{\varepsilon'}$ and the temperature $ \theta^{\varepsilon'}$ converge to some homogenized velocity $ v^0$ and some homogenized temperature $ \theta^0$ which solve a system similar to the system solved by $ v^\varepsilon$ and $ \theta^\varepsilon$, for coefficients $ \rho^0$, $ \mu^0$ and $ c^0$ which satisfy hypotheses similar to the hypotheses satisfied by $ \rho^\varepsilon$, $ \mu^\varepsilon$ and $ c^\varepsilon$. These homogenized coefficients $ \rho^0$, $ \mu^0$ and $ c^0$ are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient $ c^0$ in general depends on the temperature even if the heterogeneous heat coefficients $ c^\varepsilon$ do not depend on it.

References [Enhancements On Off] (What's this?)

  • 1. J. Aboudi, M. Pindera, and S.M. Arnold, Higher-order theory for functionally graded materials, Composite Part B (Engineering) 30 (1999), 777-832.
  • 2. J. Aboudi, M. Pindera, and S.M. Arnold, Higher-order theory for periodic multiphase materials with inelastic phases, Int. J. Plasticity 19 (2003), 805-847.
  • 3. Y. Bansal and M.J. Pindera, A second look at the higher-order theory for periodic multiphase materials, J. Appl. Mech. 72 (2005), 177-195.
  • 4. R.C. Batra and B.M. Love, Adiabatic shear bands in functionally graded materials, J. Thermal Stresses 27 (2004), 1101-1123.
  • 5. T. Baxevanis, T. Katsaounis, and A. Tzavaras, A finite element method for computing shear band formation, in Proceedings of the International Hyperbolic Conference (Osaka 2004), Yokohama Publ., Yokohama, to appear.
  • 6. Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
  • 7. Nicolas Charalambakis and François Murat, Weak solutions to the initial-boundary value problem for the shearing of nonhomogeneous thermoviscoplastic materials, Proc. Roy. Soc. Edinburgh Sect. A 113 (1989), no. 3-4, 257–265. MR 1037731, https://doi.org/10.1017/S0308210500024124
  • 8. N. Charalambakis and F. Murat, Approximation by finite elements, existence and uniqueness for a model of stratified thermoviscoplastic materials, Ric. Mat. 55 (2006), to appear.
  • 9. C. M. Dafermos and L. Hsiao, Adiabatic shearing of incompressible fluids with temperature-dependent viscosity, Quart. Appl. Math. 41 (1983/84), no. 1, 45–58. MR 700660, https://doi.org/10.1090/S0033-569X-1983-0700660-8
  • 10. J. Hodowany, G. Ravichandran, A.J. Rosakis, and P. Rosakis, Partition of plastic work into heat and stored energy in metals, J. Exp. Mech. 40 (2000), 113-123.
  • 11. Z.H. Jin and R.C. Batra, Some basic fracture mechanics concepts in functionally graded materials, J. Mech. Phys. Solids 44 (1996), 1221-1235.
  • 12. P. Rosakis, A.J. Rosakis, G. Ravichandran, and J. Hodowany, A thermodynamical internal variable model for the partition of plastic work into heat and stored energy in metals, J. Mech. Phys. Solids 48 (2000), 582-607.
  • 13. P. Rosakis, A.J. Rosakis, G. Ravichandran, and J. Hodowany, On the conversion of plastic work into heat during high-strain-rate deformation, AIP Conference Proceedings 620 (2002), 557-562.
  • 14. Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
  • 15. L. Tartar, Homogénéisation et compacité par compensation, Séminaire Goulaouic-Schwartz (1978/1979), École Polytech., Palaiseau, 1979, pp. Exp. No. 9, 9 (French). MR 557520
  • 16. A. E. Tzavaras, Shearing of materials exhibiting thermal softening or temperature dependent viscosity, Quart. Appl. Math. 44 (1986), no. 1, 1–12. MR 840438
  • 17. Athanassios E. Tzavaras, Plastic shearing of materials exhibiting strain hardening or strain softening, Arch. Rational Mech. Anal. 94 (1986), no. 1, 39–58. MR 831769, https://doi.org/10.1007/BF00278242

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 74Q15, 74Q10, 35B27, 74C10, 74F05, 35Q72, 35M20, 35K55

Retrieve articles in all journals with MSC (2000): 74Q15, 74Q10, 35B27, 74C10, 74F05, 35Q72, 35M20, 35K55

Additional Information

Nicolas Charalambakis
Affiliation: Department of Civil Engineering, Aristotle University, GR 54124 Thessaloniki, Greece
Email: charalam@civil.auth.gr

François Murat
Affiliation: Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Boîte courrier 187, 75252 Paris Cedex 05, France
Email: murat@ann.jussieu.fr

DOI: https://doi.org/10.1090/S0033-569X-06-01017-3
Keywords: Homogenization, thermoviscoplastic materials
Received by editor(s): December 1, 2005
Published electronically: May 22, 2006
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

Brown University The Quarterly of Applied Mathematics
is distributed by the American Mathematical Society
for Brown University
Online ISSN 1552-4485; Print ISSN 0033-569X
© 2017 Brown University
Comments: qam-query@ams.org
AMS Website