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

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 for which the velocity and the temperature converge to some homogenized velocity and some homogenized temperature which solve a system similar to the system solved by and , for coefficients , and which satisfy hypotheses similar to the hypotheses satisfied by , and . These homogenized coefficients , and are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient in general depends on the temperature even if the heterogeneous heat coefficients do not depend on it.

**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.**A. Bensoussan, J.-L. Lions, and G. Papanicolaou,*Asymptotic analysis for periodic structures*, North-Holland, Amsterdam, 1978. MR**0503330 (82h:35001)****7.**N. Charalambakis and F. Murat,*Weak solutions to initial-boundary value problem for the shearing of non-homogeneous thermoviscoplastic materials*, Proc. Royal Soc. Edinburgh**113A**(1989), 257-265. MR**1037731 (91b:73008)****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), 45-58. MR**0700660 (84g:76026)****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.**E. Sanchez-Palencia,*Nonhomogeneous media and vibration theory*, Lecture Notes in Physics, vol. 127, Springer Verlag, Berlin, Heidelberg, 1978.MR**0578345 (82j:35010)****15.**L. Tartar,*Homogénéisation et compacité par compensation*, Cours Peccot, Collège de France, Paris, March 1977. Partly written in: F. Murat,*-convergence*, Séminaire d'Analyse Fonctionnelle et Numérique 1977-1978, Université d'Alger, Alger, multicopied, 34 pages. English translation: F. Murat and L. Tartar, -convergence, in*Topics in the mathematical modelling of composite materials*, ed. by A. Cherkaev and R.V. Kohn, Progress in Nonlinear Differential Equations and their Applications, vol. 31, Birkhäuser, Boston, 1997, pp. 21-43. MR**0557520 (81c:73012)****16.**A. Tzavaras,*Shearing of materials exhibiting thermal softening or temperature dependent viscosity*, Quart. Appl. Math.**44**(1986), 1-12. MR**0840438 (87m:76007)****17.**A. Tzavaras,*Plastic shearing of materials exhibiting strain hardening or strain softening*, Arch. Rat. Mech. Anal.**94**(1986), 39-58. MR**0831769 (87e:73054)**

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.