Analysis of Microstructures and Phase Transition Phenomena contribution to Dynamic Days, Berlin,...
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Analysis of Microstructures and Phase Transition Phenomena contribution to Dynamic Days, Berlin, July 25-28, 2005 René J. Meziat, Jorge Villalobos Department of Mathematics, Universidad de los Andes, Bogotá, Colombia Acknowledgements: • Research Grant 1204-05-13627, CT 282-2003, COLCIENCIAS • Department of Physics, Universidad de los Andes • Fondo de Investigación, Facultad de Ciencias, Objectives • This work focus on the mathematical analysis and the numerical solution of non-convex, variational principles in non linear elasticity. As instance, consider the total energy of one dimensional elastic bodies whose general form is given by the following formulation: • We propose a general method to determine the theoretical microstructure in one dimensional elastic bars whose internal deformation energy is given by non-convex polynomials. 1 , 0 0 , 0 ' s.t. d , ' , min 1 0 u u x u x u x u x x u 1 , 0 , ' ' , 0 K c u x c u x K K k k k Considerations • u stands for the displacement of every point on the bar x [0, 1]. • u’ is the unitary deformation • represents the internal deformation energy • represents the potential of external forces and loads • See below the schematic curve of a typical deformation potential for a low carbon steel which has been calculated from the corresponding strain-stress curve: Non convex variational problems Uncertainty about the existence of minimizers Euler-Lagrange equations are not appropriate as we are analyzing a non-convex situation Minimizing sequences By using moments of Young Measures we obtain a convex variational relaxation Proper tool to analyze Young Measures Convex envelopes can be described by probability distributions Critical information about the existence of minimizers and minimizing sequences Method • The non-convex formulation can be transformed into a generalized problem defined over sets of parameterized measures called Young measures. • Each parametrized measure is a probability distribution supported on the real line. • Some features of the generalized formulation: Always has a minimizer (even if the original problem does not have any) Is a relaxation (both share the same infimum) 1 0 : :0 1 min , , s.t. 0 0, 1 x x v x Young Measure v x Generalized Variational F ormulation x d xu dx u x d u u x x s x every for * 0 every for 1 2 1 2 1 * 2 1 x p x x p x p x p x p j x s x s x • The values p j (x) and s j (x) determine the oscillatory behavior of the minimizing sequences and the existence of minimizers of the original non- convex formulation. • Given an arbitrary minimizing sequence u n of the original problem, its derivative is forced to follow one of the values s j (x) according to their proportions p j (x). • As , no parametrized measure can be supported on the negative semi- axis. Thus, we can reformulate the generalized problem by imposing this restriction on the support of the parametrized measures. 1 , 0 0 0, suport s.t. , , min 1 0 0 u u d x u dx u x d x x x x v • The polynomial form of the internal energy potential, allows us to transform the generalized problem in Young measures into a convex optimal control problem. • The new formulation is defined in vectorial control variables m, which form a convex control set whose elements are vectors composed by the algebraic moments of a probability measure supported on the positive semi-axis. • We can characterize this convex, control set by using the classical truncated Stieltjes Moment Problem. Thus, we must impose the positive semidefinite condition on the Hankel matrices formed with the entries of every parametrized vector m(x). , 1 , 1 , 0 0 , x s.t. , min 0 1 1 0 0 x m u u m x u dx u x x m x c K k k k m 2 , 0 1 2 1 , 0 0, and 0, even K i j ij K i j ij m x m x K • This method has been successfully applied to calculate the microstructure present in a plastic bar made of a particular material with two or more solid phases as has been observed in ductile steels in civil engineering. •In these cases the stress can be represented as the formal derivative of the deformation potential with respect to the unitary deformation . 64 . 1016 61 . 4664 87 . 6195 2 3 4 1 1 , 0 0 0 s.t. min 1 0 2 u u x u dx x u u u 1 , 5 . 0 for 5 . 0 , 0 for 158 . 0 0408 . 0 093 . 0 * x x x x x u 4 1 1 , 0 0 0 s.t. min 1 0 2 4 1 u u x u dx x x u u u 1 , 0 for 0.475 525 . 0 0.418 093 . 0 * x x Conclusions • We applied the theory of Young measures to find minimizers of non convex variational problems. • Young measures have been used to predict the optimal energetic balance of a one-dimensional metallic beam in the regime of non linear elasticity. '0 u
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Transcript of Analysis of Microstructures and Phase Transition Phenomena contribution to Dynamic Days, Berlin,...
Analysis of Microstructures and Phase Transition Phenomena
contribution to Dynamic Days, Berlin, July 25-28, 2005
René J. Meziat, Jorge VillalobosDepartment of Mathematics, Universidad de los Andes, Bogotá, Colombia
Acknowledgements:
• Research Grant 1204-05-13627, CT 282-2003, COLCIENCIAS
• Department of Physics, Universidad de los Andes
• Fondo de Investigación, Facultad de Ciencias, Universidad de los Andes
Objectives• This work focus on the mathematical analysis and the numerical solution of non-convex, variational principles in non linear elasticity. As instance, consider the total energy of one dimensional elastic bodies whose general form is given by the following formulation:
• We propose a general method to determine the theoretical microstructure in one dimensional elastic bars whose internal deformation energy is given by non-convex polynomials.
1 ,00
,0' s.t.
d ,', min1
0
uu
xu
xuxuxxu
1 ,0 ,' ',0
Kcuxcux K
K
k
kk
Considerations• u stands for the displacement of every point on the bar x [0, 1].• u’ is the unitary deformation• represents the internal deformation energy• represents the potential of external forces and loads• See below the schematic curve of a typical deformation potential for a low carbon steel which has been calculated from the corresponding strain-stress curve:
Non convex variational problemsNon convex variational problems
Uncertainty about the existence of minimizers Uncertainty about the
existence of minimizers
Euler-Lagrange equations are not appropriate as we are
analyzing a non-convex situation
Euler-Lagrange equations are not appropriate as we are
analyzing a non-convex situation
Minimizing sequencesMinimizing sequences
By using moments of Young Measures we obtain a convex variational relaxationBy using moments of Young Measures
we obtain a convex variational relaxation
Proper tool to analyze
Young Measures
Proper tool to analyze
Young Measures
Convex envelopescan be described by
probability distributions
Convex envelopescan be described by
probability distributions
Critical information about the existence of minimizers and minimizing sequencesCritical information about the existence of minimizers and minimizing sequences
Method• The non-convex formulation can be transformed into a generalized problem defined over sets of parameterized measures called Young measures.• Each parametrized measure is a probability distribution supported on the real line.• Some features of the generalized formulation:
Always has a minimizer (even if the original problem does not have any) Is a relaxation (both share the same infimum)
1
0
: : 0 1
min , ,
s.t.
0 0, 1
x
xv
x
Young Measure v x
Generalized Variational Formulation
x d x u dx
u x d
u u
xxsx every for *
0
every for 1
21
21*
21
xp
xxpxp
xpxp
j
xsxsx
• The values pj(x) and sj(x) determine the oscillatory behavior of the minimizing sequences and the existence of minimizers of the original non-convex formulation. • Given an arbitrary minimizing sequence un of the original problem, its derivative is forced to follow one of the values sj(x) according to their proportions pj(x).• As , no parametrized measure can be supported on the negative semi-axis. Thus, we can reformulate the generalized problem by imposing this restriction on the support of the parametrized measures.
1 ,00
0, suport s.t.
, ,min1
00
uu
dxu
dxuxdx
x
x
xv
• The polynomial form of the internal energy potential, allows us to transform the generalized problem in Young measures into a convex optimal control problem.• The new formulation is defined in vectorial control variables m, which form a convex control set whose elements are vectors composed by the algebraic moments of a probability measure supported on the positive semi-axis.• We can characterize this convex, control set by using the classical truncated Stieltjes Moment Problem. Thus, we must impose the positive semidefinite condition on the Hankel matrices formed with the entries of every parametrized vector m(x).
,1
,1 ,00
, x s.t.
,min
0
1
1
00
xm
uu
mxu
dxuxxmxcK
kkk
m
2, 0
12
1 , 0
0,
and 0, even
K
i j i j
K
i j i j
m x
m x K
• This method has been successfully applied to calculate the microstructure present in a plastic bar made of a particular material with two or more solid phases as has been observed in ductile steels in civil engineering.•In these cases the stress can be represented as the formal derivative of the deformation potential with respect to the unitary deformation .
64.101661.466487.6195 23
4
11 ,00
0 s.t.
min1
0
2
uu
xu
dxxuuu
1,5.0for
5.0,0for
158.00408.0
093.0*
x
x
x
xxu
4
11 ,00
0 s.t.
min1
0
2
41
uu
xu
dxxxuuu
1,0for 0.475525.0 0.418093.0* xx
Conclusions• We applied the theory of Young measures to find minimizers of non convex variational problems.• Young measures have been used to predict the optimal energetic balance of a one-dimensional metallic beam in the regime of non linear elasticity.
' 0u
