Institut Jean Le Rond d’Alembert Université Pierre et Marie Curie-

Paris, FRANCE

10ème Colloque Franco-Roumain de Mathématiques Appliquées

Poitiers, France, Août 2010

Gérard A. Maugin1 , Martine Rousseau1 and Misha Berezovski2 ,1Institut Jean Le Rond d’Alembert, UMR CNRS 7190, Université Pierre et Marie Curie, Paris, France,

2Centre for Nonlinear Studies, Tallinn Techn.Univ.,

Tallinn, Estonia.

Prolegomena to Studies on Dynamic Materials

We are interested in so-called dynamic materials. By these we understand materials whose characteristic properties (e.g., mass density, elasticity) may be made to vary in space or in time, or both, by an appropriate arrangement or control. Of course materially inhomogeneous materials abound in various forms, polycrystals, composites of the stratified type or so-called graded materials (with a more or less smooth gradient in their properties).

Materials inhomogeneous in time are not so frequent or are practically inexistent in natural conditions. We may conceive of some artificial means of causing these controlled changes in time, e.g., by the application of an external (non-mechanical) field, or through a phase transition.

To avoid any misunderstanding, we specify that this should be realized in a short or quasi-nil time lapse and over a sufficiently large material region if not over the whole specimen under consideration.

Difficulties of realization cannot be overlooked .

In fully dynamic elastic materials, both referential density and elastic potential W may also be explicit functions of both referential placement X and Newtonian time t. In small strains for a 1D motion we have the following wave equation

0

0,,0

x

utxE

xt

utx

t

inhgrowthxxtt ggutxcu ,2

With

 respectively, squared propagation velocity, force per unit mass due to « growth », and force per unit mass due to elastic inhomogeneities. 

x

E

tx

ug

ttx

ug

tx

txEtxc xinhtgrowth

,:,

,:,

,

,,

0

0

00

2

However, it is pertinent and simpler to consider the special case where depends only on x (case of inertial material inhomogeneity) and the elastic energy only explicitly on t (rheonomy of solid behavior) 

 so that the wave equation reduces to

 

0

onlytEEonlyx ,00

x

tEtxcwithutxcu xxtt

0

22 ,0,

In a 1D small-strain case, with an obvious notation, the motion equation yields (E=elasticity coefficient)

(2)

which can be written as the linear wave equation

(3)

xt utEuxpx

pt

,,0 0

xtEcutxcu xxtt 022 /,0,

This can also be written as the first-order system

(4)

which plays a fundamental role in discussing specific (boundary and initial) conditions in space and time. Lurié (2007) has studied the space-time homogenization and asymptotics of systems such as (3) or (4) for long times compared to the period of a repeated motif (e.g., a checkerboard) in 2D space-time.

txxt upuE 0:,:

System (4) presents a remarkable symmetry but we must distinguish between purely spatial transition layers (e.g., a fixed material interface ) and a time line (sudden change of value of E over a large spatial region). The situation is described in Figure 1.

xt

X

Σt

Σ

x

x0

T0

1a

1b1c

2+ 2-

1

I T

R

Figure 1.

At a line , we observe a change in wave number (classical condition; in 2D-space we would have a change of direction of rays and a possible concentration of energy) while at a line , it is frequency that will change and we should observe a kind of Doppler effect with capture of energy , i.e., an effect of the same class as the so-called Cherenkov effect.

x

t

Furthermore , we note that both energy and canonical (wave) momentum are not conserved in the present system. These (non)-conservation laws read at each point:

 

and  

where   

expl

x

tWu

ttE

Qx

Ht

22

pl ex

t

xKu

xxb

xP

t

2

2

HbuuPuuEvQutEuxH txtxxt ,,,21 22

As a partial conclusion, spatial inhomogeneity allows the convergence or divergence of wave by conservation of the momentum (an effect with vectorial properties in more than 1D), while a dynamic medium with time inhomogeneity allows capture of energy from the outside with a resulting change in frequency (a scalar effect in all cases).[the 1D case may be misleading]. The combination of the two in a true dynamic material may yield a concentration of energy although the system is fully linear.

With appropriate variations in space and time of and E, particular analytical solutions can be constructed that exhibit a concentration of trajectories and of energy in spite of the full linearity of the considered system. This is corroborated by numerical simulations [3]. This opens up new perspectives in newly developed (meta) materials.

0

Example :We consider the simple case where

(4.5)A simple algebra shows that u2 and u1 satisfy Bessel equations,

hence the solutions such that

(4.6)and

(4.7)where A, B, C, and D are real quantities, J0 and Y0 are Bessel

functions of order zero, and we have set and . Remark 4.1. With c decreasing along x and increasing in time, we have a situation that favors the capture of energy and the convergence of space-time trajectories.

texpEtE Kxexpx 000 ,

]/2[]/2[ 00002 t/2expYBt/2expJAtu

]Kx/2-/2[]/2/2[ 00001 expKkYDKxexpKkJCxu

0020 /Ec 000 kc

tuxutxu 21,

Reasoning in terms of frequency and wave number in the 1D case. In the latter case for a fixed we have continuity of the traction, i.e., in terms of jumps (see eqn.(4)), (5.1)hence

(5.2)In our special case, these two conditions reduce to

(5.3) where square brackets denote the usual jump.

x

xx atuE 0]ˆ[

tt atu 0]ˆ[ 0

ttxx atuatu 0][,0][

For a spatial interface , with an obvious notation we have

(5.4)with and (Figure 1b) since in general there is one incoming wave (on side 1), one transmitted wave (on side 2) and a reflected wave (on side 1). The transmission and reflection coefficients are given by

(5.5)

x02121 ,,, xxsayatuuuu xx

RI uuu 1 Tuu 2

2211

221121

2211

1112 ,

2cccc

Rcc

cT

Let the impedances. Of course, (5.6)

At , we have

(5.7)Of course, tuning the impedances means

(5.8)

Reflection effects disappear and the whole of energy is transmitted.

iii cz 1/ 1212

221 TzzR

0x

2

22

1

112121 ,,,

ck

ckcc

00,0][ 21 xatRz

For a timelike interface situated , say, at , we have (compare to (5.4))

(5.9)where region 1 is below in space-time and region 2 is above. Here we should pay attention to the wording because we are no longer speaking of propagating waves. A solution will exist at (cf. Lurie, 2007; Mkrtchyan et al, 2008; Ginzburg. and Tsytovich, 1979) only if we have one “signal” A1

coming from region 1 toward region 2, one signal continuing in region 2 and, what is more surprising, one signal in region 2, oriented from region 2 to region 1 (as if it were coming from the future). That is, we have (cf. Figure 1c):

t 0t

tt uuuu 221121 ,

t

t

2A

2A

uuuandu 21

A simple estimate yields (5.10)

Then at , we have (5.11)

together with the local balance at , (5.12)

With perfect tuning of the impedance , and (trivial case).

22

11221212

22

11221212 2

/,2

/c

ccAAT

cccAAT

t

t

212122

1212 //1 TzzTzz

tatz 0][ 012 T

112 T

211212121 /,, cccckkk

Numerical simulations (Tallinn)

I. Amplitude variations and speed of propagation

Fig.2. Pulse propagation in medium with increasing density

Fig.3. Pulse propagation in medium with increasing stiffness

Fig.4. Pulse propagation in medium with increasing stiffness and density (increasing stiffness is slower than increasing density)

Fig.5. Pulse propagation in medium with increasing stiffness and density (increasing in stiffness is the same as increasing in density).

Fig.6. Pulse propagation in medium with increasing stiffness and density (increasing in stiffness is faster than increasing in density).

II. Frequency changes and Doppler effect

Fig.7. Reference sinusoidal signal

Fig.8. Sinusoidal signal with increasing stiffness in time

Fig.9. Frequency shift for dynamic materials 

tcct 11 /

1

2

3

4

0 X1 Xc X

tc

t1

ρ1 , E1 ρ2 > ρ1 , E2 = E1

ρ3= ρ1 ,E3 > E1

ρ4 = ρ2 ,E4 = E3

Fig.10. Elementary cell in a checkerboard space-time (Lurié)

But by homogenization for long time (and space) propagation where both material coefficients are fast periodic functions of the characteristic right-running variable, it is shown that we obtain the balance of energy and canonical momentum in the source-free form (superimposed tilde corresponds to the zeroth-order asymptotic homogenized solution)

(7.1)which is tantamount to saying that the looked for effects disappear altogether by successive increases and decreases that compensate each other

0~~

,0~~

xb

tP

xQ

tH

THANK YOU FOR YOUR ATTENTION,

M.R., G.A.M. & M.B.