Homogenization of stratified thermoviscoplastic materials
Authors: Charalambakis Nicolas , Murat François 
Published: April 2015

In the present paper we study the homogenization of the system of partial differential equations
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row rho Superscript epsilon Baseline left-parenthesis x right-parenthesis StartFraction partial-differential v Superscript epsilon Baseline Over partial-differential t EndFraction minus StartFraction partial-differential Over partial-differential x EndFraction left-parenthesis mu Superscript epsilon Baseline left-parenthesis x comma theta Superscript epsilon Baseline right-parenthesis StartFraction partial-differential v Superscript epsilon Baseline Over partial-differential x EndFraction right-parenthesis equals f comma c Superscript epsilon Baseline left-parenthesis x comma theta Superscript epsilon Baseline right-parenthesis StartFraction partial-differential theta Superscript epsilon Baseline Over partial-differential t EndFraction equals mu Superscript epsilon Baseline left-parenthesis x comma theta Superscript epsilon Baseline right-parenthesis left-parenthesis StartFraction partial-differential v Superscript epsilon Baseline Over partial-differential x EndFraction right-parenthesis squared comma EndLayout">

ρ
ε

(
x
)

∂

v
ε

∂
t

−

∂

∂
x

(

μ
ε

(
x
,

θ
ε

)

∂

v
ε

∂
x

)

=
f
,

c
ε

(
x
,

θ
ε

)

∂

θ
ε

∂
t

=

μ
ε

(
x
,

θ
ε

)

(

∂

v
ε

∂
x

)

2

,

\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

a > x > b

,

0
>
t
>
T

0 > t > T

, completed by boundary conditions on

v
ε

v^\varepsilon

and by initial conditions on

v
ε

v^\varepsilon

and

θ
ε

\theta ^\varepsilon

. The unknowns are the velocity

v
ε

v^\varepsilon

and the temperature

θ
ε

\theta ^\varepsilon

, while the coefficients

ρ
ε

\rho ^\varepsilon

,

μ
ε

\mu ^\varepsilon

and

c
ε

c^\varepsilon

are data which are assumed to satisfy
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout 1st Row 0 greater-than c 1 less-than-or-equal-to mu Superscript epsilon Baseline left-parenthesis x comma s right-parenthesis less-than-or-equal-to c 2 comma 0 greater-than c 3 less-than-or-equal-to c Superscript epsilon Baseline left-parenthesis x comma s right-parenthesis less-than-or-equal-to c 4 comma 0 greater-than c 5 less-than-or-equal-to rho Superscript epsilon Baseline left-parenthesis x right-parenthesis less-than-or-equal-to c 6 comma minus c 7 less-than-or-equal-to StartFraction partial-differential mu Superscript epsilon Baseline Over partial-differential s EndFraction left-parenthesis x comma s right-parenthesis less-than-or-equal-to 0 comma StartAbsoluteValue c Superscript epsilon Baseline left-parenthesis x comma s right-parenthesis minus c Superscript epsilon Baseline left-parenthesis x comma s Superscript prime Baseline right-parenthesis EndAbsoluteValue less-than-or-equal-to omega left-parenthesis StartAbsoluteValue s minus s Superscript prime Baseline EndAbsoluteValue right-parenthesis period EndLayout">

0
>

c
1

≤

μ
ε

(
x
,
s
)
≤

c
2

,

0
>

c
3

≤

c
ε

(
x
,
s
)
≤

c
4

,

0
>

c
5

≤

ρ
ε

(
x
)
≤

c
6

,

−

c
7

≤

∂

μ
ε

∂
s

(
x
,
s
)
≤
0
,

|

c
ε

(
x
,
s
)
−

c
ε

(
x
,

s
′

)

|

≤
ω
(

|

s
−

s
′

|

)
.

\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

ε
′

\varepsilon ’

for which the velocity

v

ε
′

v^{\varepsilon ’}

and the temperature

θ

ε
′

\theta ^{\varepsilon ’}

converge to some homogenized velocity

v
0

v^0

and some homogenized temperature

θ
0

\theta ^0

which solve a system similar to the system solved by

v
ε

v^\varepsilon

and

θ
ε

\theta ^\varepsilon

, for coefficients

ρ
0

\rho ^0

,

μ
0

\mu ^0

and

c
0

c^0

which satisfy hypotheses similar to the hypotheses satisfied by

ρ
ε

\rho ^\varepsilon

,

μ
ε

\mu ^\varepsilon

and

c
ε

c^\varepsilon

. These homogenized coefficients

ρ
0

\rho ^0

,

μ
0

\mu ^0

and

c
0

c^0

are given by some explicit (even if sophisticated) formulas. In particular, the homogenized heat coefficient

c
0

c^0

in general depends on the temperature even if the heterogeneous heat coefficients

c
ε

c^\varepsilon

do not depend on it.