Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
On The Elliptic Variational Inequality Of The
First Kind For The Elasto-Plastic Torsion
Problem
Sahar Muhsen Jaabar
Dept. of Math., College of Education, Babylon University
Abstract
The elliptic variational inequality of the first kind for the " Elasto- Plastic torsion
problem" is considered. This elliptic variational inequality is related to second order
partial differential operator.The physical and mathematical interpretation and some
properties of the solution are given.
‫الخالصة‬
‫لقددا تبرنا ددم تليرةما ددي تلرامااقددي تل ملن د ي يددم تل ددلط ت لذ ليلددبلي تللددعال ةددال تليرةما ددي ل ددم ب لددي ةددملي ا‬
. ‫تلرفمضلع تلجزئع يم تلارةي تل م ي ال تلرفلاا تلاقمضع لتلفازقمئع لةعض خلتص تلحل لا تبلات‬
1. Introduction:
An important and very useful class of non- linear problems arising from
mechanics, physics etc. consists of the so- called variational inequalities. In this paper
we shall restrict our attention to the study of the existence, uniqueness, and properties
of the solutions of elliptic variational inequality (EVI) of the first kind .
1-1: Notations :
V : real Hilbert space with scalar product ( . , . ) and associated norm .
V * : The dual space of V .
a (. , .): VxV  IR is abilinear, continuous and V  elliptic mapping on VxV .
A bilinear form a (. , .) is said to be V  elliptic if there exists a positive constant 
2
such that a ( v, v )   v , vV .
In general we do not assume a (. , .) to be symmetric , since in some applications nonsymmetric bilinear forms may occur naturally (Lions , 1967).
L: V  IR continuous, linear functional.
K: is a closed, convex, non- empty subset of V .
j(.) : V  IR  IR   is a convex, lower semi-continuous (L.S.C.) and proper
functional.
(j( . ) is proper if < j( v )> vV and j   )
1.2: Elliptic Variationd Inequality of First kind ( EVI )
To find u V such that u is a solution of the problem:
a(u, v  u )  L(v  u ), v  k
p1 .....
u  k
1.3: Existence and Uniqueness Results for EVI of First kind
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1.3.1: A Theorem of Existence and Uniqueness (Lions , 1967 ; Chipot et al., 1987).
The problem p1 has one and only one solution.
Proof: 1- Uniqueness
Let u1 and u 2 be solutions of (p1). We have then:
v  k , u1  k (1)
a( u1 , v  u1 )  L ( v  u1 )
a( u2 , v  u2 )  L ( v  u2 )
v  k , u2  k
(2)
putting u 2 for v in (1) and u1 for v in (2) and adding we get, by using the V 
ellipticity of a ( .,. ).
2
 u 2  u1  a(u 2  u1 , u 2  u1 )  0
Which implies u1 = u 2 , since  >0
2- Existence: we will reduce the problem (p1) to a fixed point problem . By the Riesz
representation theorem for Hilbert spaces there exist .
A   (V ,V ), ( A  At if a(.,.) is symmetric and l V such that:u , v  V
( Au, v )=a( u, v )
and L( v )= (  , v )
vV ……….. (3)
Then the problem ( 1 ) is equivalent to finding u  V such that:v  k
(u   ( Au  l )  u, v  u )  o
( 4 )……. 
,  0
u  k
This is equivalent to finding u such that:
……….(5)
u  pk (u  p( Au  ), for some   0
Where pk denotes the projection operator from V to k in the . norm.
Consider the map W  : V  V defined by:
W  ( v )=pk( v   ( Av   )
……….(6)
let v1 ,v2  V , then since pk is a contraction we have
 w (v )  w (v )

1

2
2
 v2  v1   2 A(v2  v1 )
2
2
 2 a(v2  v1 , v2  v1 )
Hence we have
2
w (v1 )  w (v2 )  (1  2   2 A ) v2  v1
2
2
..........(7)
Thus w  is a strict contraction mapping if 0<  <
2
. By taking  in this
2
A
range we have a unique solution for the fixed point problem which implies the
existence of a solution for ( p1 ).
2- An Example of EVI of the first kind " The Elasto – plastic Torsion Problem"
2.1: Notations
 : a bounded domain in IR 2
 : 
X={ x1,x2} a generic point of 
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013
   

,


x
x 2 
1

Cm (  ): space of m- times continuously differentiable real valued functions for which
all the derivatives up to order m are continuous in 
C0m ()  v  C m () : sup p (v) is a compact subset of  

v
m , p ,
   m D  v
Lp (  )
for v  c m () where  ( 1 ,  2 ); 1 , 2 non  negative

int egers,    1   2 and D 



x1 1 x2
2
wm, p () : completion of C m () in the norm defined above .
wom , p () : completion of Com () in the above norm.
H m ()  wm, 2 ()
H 0m ()  w0m, 2 ()
2.2:The mathematical Interpretation of the problem
Let  be a bounded domain of IR 2 with a smooth boundary  , we consider the
following EVI of the first kind:
a(u, v  u )  L(v  u )  v  k
(8)....
u  k
where
k  v  H 01 () : v  1 a.e. in 
.........(9)


V = H 01 ()  v  H 1 () :v   trace of v on   0
a(u, v)   u.v, such that :

u.v 
u v u v
.

.
x1 x1 x2 x2
and L ( v )= < f , v > for f  V * =H-1 () and v V .
2.3: The physical Interpretation of the Problem
Let us consider on infinitely long cylindrical bar of cross-section  where  is
simply connected. Assume that this bar is made up of an isotropic, elastic, perfectly
plastic material whose plasticity yield is given by the von Mises criterion. ( Koiter ,
1987 ; Duvaut et al., 197]).
Starting from a zero stress initial state, an increasing torsion moment is applied to the
bar. The torsion is charcterised by c, which is defined as the torsion angle per unit
length (William , 2004). Then for all c, it follows from the Haar- karman principle
that the determination of the stress field is equivalent to the solution of the following
variational problem:-
748
Min
vk
2
1
v dx  c  vdx

2

..........(10)
This is a particular case of (2.2) with
L(v)  c  vdx
..........(11)

3. Existence and Uniqueness Results of the " Elasto-plastic Torsion Problem"
In order to apply theorem (1.3.1) , we only have to verify that k is a non- empty ,
closed, convex , subset of V .
K is non- empty because 0  K , and the convexity of K is obvious.
12To prove that K is closed , consider a sequence vn  in k such that vn  v
strongly in V .
Then there exists a subsequence v ni such that. lim vni  v a.e
i 
sin ce vni  1 a.e. we get v  a.e.
Therefore v  K . Hence K is closed.
4. Regularity Properties and exact solutions
4.1 Regularity results (Chipot et al., 1987)
4.1.1 Theorem
Let u be a solution of (8) and L(v)   fvdx

1Let  be a bounded convex domain of IR 2 with  lipschitz continuous and
f  Lp () with 1<p<  , Then we have
u  w2, p ()
2If  is a bounded domain of IR 2 with a smooth boundary,  if  Lp () with
1< p <  then u  w2, p () .
4.2 Exact solutions
In this section we are going to give an example of problems (8) for which
exact solutions are known .
Example :- we take  ={ x:0<x<1} and
1
L(v)  c  vdx with c  0.
0
Then the explicit form of (8) is
1
1

0
u l (v l  u l )dx  c  (v  u )dx
0
uk

v  k 



.........(12)
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Journal of Babylon University/Pure and Applied Sciences/ No.(3)/ Vol.(21): 2013

where k  v  H 01 () : v /  1 a.e. on and v / 
dv
dx
The exact solution of (12) is given by
c
u ( x)  x(1  x) x, if c  2
.........(13)
2
if c  2
1 1

 x if 0  x  2  c

1 1 
1 1
1 1
c 
u ( x)    x(1  x)  (  ) 2 
if   x  
2 c 
2 c
2 c
2 

1 1
if   x  1
1  x
2 c

5. An equivalent variational formulation (Lions , 1967)
If  is a bounded domain of IR 2 with a smooth boundary  and if
L(v)  c  v( x)dx
(c  0 for ins tan ce),

Then the solution of (8) is also a solution of
^

a
(
u
,
v

u
)

c
(
v

u
)
dx

v

k
,



^
u  k  v  H 1 (), v( x)  d ( x, )
0


..........(14)
Since a( . , . ) is symmetric , (14) is also equivalent to
^

v  k ,
 J (u )  J (v)

^
u  k
1
with J (v)  a (v, v)  c  v( x)dx
2

..........(15)
References
Chipot. M., Michaille G., (1987),"Uniqueness Results and Monotonicity Properties
for the Solutions of some Variational Inequalities" , contents of the Proceedings
FBP, Germany .
Duvaut. G., Lions J. L., (1972) ,"The Equations in Mechanics of Physics", Dunod,
Paris.
Koiter .W. I., (1960), " General Theorems for Elastic Plastic Solids". Progress in solid
mechanics, pp. (165-221). North- Holland.
Lions. J.L., Stampacchia G., (1967)," Variational Inequalities", comm.. pure Applied
Math., XX, pp.(493-519).
William G. Faris, (2004), "Lectures on Partial Differential Equations" , June.
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