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
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Abstract: In the present paper we study the homogenization of the system of partial differential equations \[ \begin {array}{l} \displaystyle {\rho ^\varepsilon (x) {\partial v^\varepsilon \over \partial t} - {\partial \over \partial x} \left (\mu ^\varepsilon (x,\theta ^\varepsilon ) {\partial v^\varepsilon \over \partial x}\right ) = f,} \displaystyle {c^\varepsilon (x,\theta ^\varepsilon ) {\partial \theta ^\varepsilon \over \partial t} = \mu ^\varepsilon (x,\theta ^\varepsilon ) \left ({\partial v^\varepsilon \over \partial x}\right )^2,} \end {array} \] 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 \begin{gather*} 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 {- c_7 \leq {\partial \mu ^\varepsilon \over \partial s} (x,s) \leq 0, \quad |c^\varepsilon (x,s) - c^\varepsilon (x,s’)|\leq \omega (|s - s’|).} \end{gather*} 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.
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BP 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.
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APA2 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.
BP 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.
BL R.C. Batra and B.M. Love, Adiabatic shear bands in functionally graded materials, J. Thermal Stresses 27 (2004), 1101–1123.
BTZ 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.
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JB Z.H. Jin and R.C. Batra, Some basic fracture mechanics concepts in functionally graded materials, J. Mech. Phys. Solids 44 (1996), 1221–1235.
RoRoRaHo2 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.
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SaPa E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer Verlag, Berlin, Heidelberg, 1978.
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T1 A. Tzavaras, Shearing of materials exhibiting thermal softening or temperature dependent viscosity, Quart. Appl. Math. 44 (1986), 1–12.
T2 A. Tzavaras, Plastic shearing of materials exhibiting strain hardening or strain softening, Arch. Rat. Mech. Anal. 94 (1986), 39–58.
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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
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.