dynamics of networks with bistable links
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The Seventh International Conference on Vibration Problems. Dynamics of Networks with Bistable Links. Andrej Cherkaev, Elena Cherkaev Department of Mathematics, University of Utah, USA and Leonid Slepyan Department of Solid Mechanics, Materials and Structures, - PowerPoint PPT PresentationTRANSCRIPT
IPME, sPb, November 2007
Dynamics of Networks with Bistable LinksAndrej Cherkaev, Elena Cherkaev
Department of Mathematics,
University of Utah, USA
and
Leonid SlepyanDepartment of Solid Mechanics, Materials and Structures,
Tel Aviv University, Israel
The Seventh International Conference on Vibration Problems
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Acknowledgements
Professor Alexander Balk,
PhD student Seubpong Leelavanishkul,
Assistant Research Professor Alexander Vinogradov,
Assistant Research Professor Toshio Yoshikawa,
Assistant Research Professor Liya Zhornitzkaya.
The project is supported by ARO and NSF
The talk is partly based on the joint results
obtained in collaboration with the fellows
of Department of Mathematics, University of Utah
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Motivation: Find a structure that can withstand a dynamic impact
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Outline
• Waiting links and structures from them
• Waves of damage
• Smart structures from bistable links
• Dynamics of invertible phase transition
• Spontaneous waves
• Dynamic homogenization
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Situation:A relatively slow and massive projectile collides with the structure.
(“slow” compared to the speed of sound in the material, appox. 3000 m/sec for steel)
Failure is caused by stress localization: Most of the material stays undamaged after the structure fails!
Aim: delocalize the stress and stabilize the failure
The material itself can absorb he energy and resist until it melts
Observe that
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Tao of Damage
o Damage happens! o Dispersed damage absorbs energy; concentrated
damage destroys.
o Design is the Art of Damage Scattering
Tao -- the process of nature by which all things change and which is to be followed for a life of harmony. Webster
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Means of Delocalization
The propagation of “partial damage” can be achieved by using chains and lattices from locally unstable cells --waiting links -- with reserved capacity to resistance
(Ch, Slepyan, 1998)
The waiting links structures excite waves of partial damage that effectively dissipate and radiate the energy of an impact
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Elastic-brittle material(limited strength)
• The force-versus-elongation relation in a monotonically elongated bar from elastic-brittle material is
• Accounting for the prehistory, we obtain the relation
1,0)0,(
otherwise ,0
1, if ,),(
)),,(1(),(
Sxc
cxxStxc
txcxEtxf
f
c(x,t) is the damage parameter
f
f
xx
xxxExf
if ,0
if ,)(
fx
)(xf
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Waiting links• Consists of two “parallel” rods; one is
slightly longer. • The second (slack) rod starts to resist
when the elongation is large enough.
• Waiting links allow for inner instability and increase the interval of nondestuctive (nonfatal) deformations.
1 2 1 1 2 2( , , ) (1 ( , )) ( ) (1 ( , )) ( ) ( )f L c c c L t f L c L t f L H L c is the damage parameter
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Chain of rods
• Several elements form a chain ),(),( 111 kkkkkkk caxxfcaxxfxm
What happens when the chain is elongated?
Multiple breakings occur and “Partial damage” propagates along the rod.
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Quasistatic state and the energy
The chain behaves as a superplastic material.
The absorbed energy Ew is proportional to the number of “partially damaged” links;
The chain absorbs more energy before total breakage than a single rod of the
combined thickness; 1)1(1 NE
Ew
Breaks of basic links
Elongation
Final break
Force
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Waves in waiting-link structuresWhy study them?
• Breakages transform the energy of a slow impact to energy of intensive waves.
• Waves radiate and dissipate the energy and carry it away from the zone of contact.
• At the other hand, waves localize the energy that may damage or destroy the link.
After: L. Slepyan, A.Cherkaev, E.Cherkaev. JPMS 2004. Part1
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Effectiveness of waiting links
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Simulation
)()()()1(),(
.1,...,1,0)0(,)0(
),()(
),(
),()(
1
11
1
LHLfLfcLf
Nkxvx
caxxfxxM
caxxf
caxxfxxm
kN
NNNNN
kkk
kkkkk
Constitutive relation in links
a is the fraction of material used in the forerun “basic” link
A large slow-moving mass (7% of the speed of sound) is attached to one end of the chain, the other end is fixed.During the simulation, the mass of the projectile M was increased until the chain was broken.
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Results
a=1 (no waiting links)M = 150
M=375, a=0.25
M=700a=.25Small dissipation
M=750a=.25Small dissipation
Efficiency: 750/150=5
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Effectiveness of structure resistance
To measure the effectiveness, we use
• the ratio R of the momentum of the projectile after and before the impact. This parameter is independent of the type of structural damage.
• The minimal mass M that breaks the chain moving with a fixed speed.
Collision propagation rejection absorption
R 1 -1 0
v(0) v(T)
v(T)
Both parameters are proportional to the volume of a chain with waiting links, not to its crossection.
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Theory: Wave of transition in infinite chains
Analytic description and the speed of propagation.
L. Slepyan, A.Cherkaev, E.Cherkaev. JPMS 2004 Part 11
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Use of a linear theory for description of nonlinear chains
The force in the nonlinear link is viewed as a linear response to the elongation plus an additional external pair of forces, applied in the proper time when the elongation reaches the critical value.
Trick (Slepyan and Troyankina, 1988): Model the jump in resistance by an action of an external pair of forces
)()(
)()(
0mod
0
ttHCzzF
zzHCzzFreal
Force
ElongationBefore damage
(transition)After damage
(transition)
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Equations of chain dynamics
))()(2)(()(
),2()(
where
),()(
),(),()(
11
11
111
kkkk
kkkk
kk
kkkkkkkk
cHcHcHxn
xxxCxl
Zkxnxl
caxxfcaxxfxxm
k
Dynamics is described by an infinite system of nonlinear equations with coefficients that are discontinuous trigger-type functions of the solution
Can we integrate it? Yes (in some cases)!
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Generally, the approach is as follows:
A nonlinear system of DE of the type
where L is a linear diff. operator, H is the Heaviside function
is equivalently replaced by a system of linear inhomogeneous DE
where the instances are defined from an additional nonlinear algebraic conditions (turn-on – turn-off conditions)
: ( , ) 0ik i k ikx
ZktxHatxLtxN ki
ikk ,0),(),(),(
ZktHatxLtxM iki
ikk ,0),(),(
ik
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Wave motion: Ansatz• Boundary conditions: constant speed at the infinitely distant left end, constant
pressure at the infinitely distant right end.• The wave propagates with a constant but unknown speed v
Therefore, the introduced “pairs of forces” are applied at equally-distanced instances.
• Motion of masses is self-similar
The ansatz reduces the problem to a problem about a wave in a linear chain caused by running pairs of forces applied to its links.
axxzv
atztz
atxtx
nnnnn
nn
11
1
, where)()(
,)()(
.1 nn tt
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Scheme of solution
1. The equation for the zero-numbered mass is
2. The Fourier transform of it is
3. Return to originals,
4. Find the unknown speed from the closing nonlinear eq:
)]()(2)([))()(2)(( tHtHtHtztztzCzm
).1)(cos(2)(
),2)(cos()(
);()(
2
pCmppL
ppG
pGzpL F
)()()( 11 pGpLtz F
fxzz )0()(
where
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Additional considerations
1. One needs to separate waves originated by breakage from other possible waves in the linear system. For this, we use the “causality principle” (Slepyan) of the viscous solution technique.
2. The solution is more complicated when the elastic properties are switched after the transition. In this case, the Wiener-Hopf technique is applied.
3. In finite networks, the reflection of the wave is critical since the magnitude doubles.
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Results• Dependence of the waves speed
on the initial pre-stress v=v(p). Measurements of the speed in direct simulation of dynamics
Wave’s speed
p
time
Positionsof masses*
* *
*
p
Direction of the transition wave
Force
p
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Discussion
• The domino problem: Will the wave of transiton propagate? The propagation of the wave is contingent on its accidental initiation.
• 2D model: Similarity between crack propagation and the plane waves of transition (Slepyan IMPS, 2003)
• Homogenization: The speed of the wave can be found from atomistic model, as a function of stress.
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Spread of damage as phase transition
Chain model allows to find (i) Condition for wave initiation
(ii) Speed of phase transition
(iii) Magnitude: max{u(x, t)}
The right-hand side of
is a bistable function; equilibrium is nonunique.
( ) ( )tt x x tt D x xu Eu u E u s
( ( ))tt x xu f u
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Lattices with waiting links
Smart structures from bistable elements
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Green’s function for a damaged lattice:Statics
Green’s function: Influence of one damaged link:
F(k,m, N, N’)
( , , , ')
( ', ') : '
F k m N N
G k k m m N N
N
N’
31 cos cos( ) 4cos 6 cos cos( )2 2
3 3cos cos( ) cos cos( ) 22 2
Fq p q p q p q
Gq p q q p q
The lattice with waiting links is described in the same manner.
The damage in a link is mimicked by an applied pair of forces
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The state of a partially damaged lattice
F(k,m, N, N’)
damage
extern
damage
( , , )
( , ) :
( , , ) , .
i i ii
F k m N
G k k m m N N F
F k m N TH k m
N
N’
Q: How to pass from one permissible configuration to another?
The answer is ambiguous if dynamics is not considered
Condition of non-destruction of a
link
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Unstrained damaged configurations
• Generally, damage propagates like a crack due to local stress concentration.
• The expanded configurations are not unique.
• There are exceptional unstrained configurations which are the attraction points of the damage dynamics and the null-space of F.
damage ( , , ) 0F k m N
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Set of unstrained configurations
• The geometrical problem of description of all possible unstrained configuration is still unsolved.
• Some sophisticated configurations can be found.
• Because of nonuniqueness, the growth of damage problem requires dynamic consideration.
Random lattices: Nothing known
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Waves in bistable lattices
Slepyan (JPMS, 2003) analytically describes two types of waves in bistable triangular lattices:
– Plane (frontal) waves
– Crack-like (finger) waves
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Conservative bistable system
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Dynamic problems for multiwell energies
• Problems of description of damageable materials and materials under phase transition deal with nonmonotone constitutive relations
• Nonconvexity of the energy leads to nonmonotonicity and nonuniqueness of constitutive relations.
w
w
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Static (Variational) approach
The Gibbs variational principle is able to select the solution with the least energy that corresponds to the (quasi)convex envelope of the energy.
At the micro-level, this solution corresponds to the transition state and results in a fine mixture of several pure phases (Maxwell line)
w
w
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Dynamic problems for multiwell energies
• Formulation: Lagrangian for a continuous medium
If W is (quasi)convex
• If W is not quasiconvex
• Questions: – There are many local minima; each corresponds to an equilibrium.
How to distinguish them?– The realization of a particular local minimum depends on the existence of a path to it.
What are initial conditions that lead to a particular local minimum?– How to account for dissipation and radiation?
)(21)( 2 uWuuL
),,()(21)( 2 uuuuWHuuL D
Radiation and other
losses
Dynamic homogenization
???)(21)( 2 uQWuuL
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Paradoxes of relaxation of an energy by a (quasi)convex envelope.
• To move along the surface of minimal energy, the particles must:– Sensor the proper instance of jump over the barrier
– Borrow somewhere an additional energy, store it, and use it to jump over the barrier
– Get to rest at the new position, and return the energy
I suddenly feel that now it is the time to
leave my locally stable position!
Thanks for the energy! I needed it to get over the barrier.
What a roller coaster!
Stop right here! Break!Here is your energy.Take
it back.
Energy bank
Energy bank
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Waves in active materials
• Preconditions: Links store additional energy remaining stable. Particles are inertial.
• Trigger: When an instability develops, the excessive energy is transmitted to the next particle, originating the wave.
• Energy: Kinetic energy of excited waves takes away the energy, the transition looks like an explosion.
• Intensity: Kinetic energy is bounded from below
• Dynamic homogenization: Accounting for radiation and the energy of high-frequency modes.
Extra energy
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Dynamics of chains from bi-stable elements (exciters)
with Alexander Balk, Leonid Slepyan, and Toshio YoshikawaA.Balk, A.Cherkaev, L.Slepyan 2000. IJPMS
T.Yoshikawa, 2002, submitted
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Unstable reversible links
• Each link consists of two parallel elastic rods, one of which is longer.
• Initially, only the longer road resists the load.
• If the load is larger than a critical (buckling)value: The longer bar looses stability
(buckling), and the shorter bar assumes the load.
The process is reversible.
Force
Elongation
)1()( xHxxf
No parameters!
H is the Heaviside function
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Chain dynamics. Generation of a spontaneous transition wave
Initial position:
linear regime (N=0), but close to the critical point of transition
kaxk )1(
x
0
1 1
1 1
linear chain
1 1
nonlinear impacts
( ) ( ) ( ) ( )
( ) 2
( ) ( ) ( )
k k k k k k k
k k k k
k k k k k
mX f X X f X X L X N X
L X X X X
N X H X X a H X X a
f(z)
z
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Observed spontaneous waves in a chain
Under a smooth excitation, the chain develops intensive oscillations and waves.
Sonic wave
“Twinkling” phase “Chaotic” phase
Wave of phase transition
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“Twinkling” phase and Wave of phase transition (Small time scale)
After the wave of transition, the chain transit to a new twinkling (or headed) state.
We find global (homogenized) parameters of transition: Speed of the wave of phase transition “Swelling” parameter Period Phase shift
1 1
1 1
linear chain
1 1
nonlinear impacts
( ) ( ) ( ) ( )
( ) 2
( ) ( ) ( )
k k k k k k k
k k k k
k k k k k
mX f X X f X X L X N X
L X X X X
N X H X X a H X X a
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Periodic waves: Analytic integration
Nonlinearities are replaced by a periodic external forces; period is unknown
Self-similarity
Periodicity1
( ) (sin( ))
(sin( ));
( ) ( ) ;
( ) ( 2 );k k
k k
N t H t p
H t p
X t X t a
X t X t
System (*) can be integrated by means of Fourier series.
A single nonlinear algebraic equation
defines the instance
( ) ( ) 2 ( ) ( )(*)
( ) ( )T T
m x t x t x t x t
H t H t T
Approach, after Slepyan and Troyankina
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1. Stationary waves
Use of the piece-wise linearity of the system of ODE and the above assumptions
The system is integrated as a linear system (using the Fourier transform),
then the nonlinear algebraic equation
for the unknown instances of the application/release of the applied forces is solved.
Result: The dispersion relation
)()( 1 pXFtX
1)()( tXtX
p
pp
e
eepXpp
1)()(sin4
)(22
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2. Waves excited by a point source For the wave of phase transition we assume that k-th
mass enters the twinkling phase after k periods
The self-similarity assumption is weakened.
Asymptotic periodicity is requested.
otherwise,0
,2if),(~
)(~~ kttN
tN kk
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Large time range description
with Toshio Yoshikawa
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Problem of homogenization of a dynamic system
1 1
1 0
( ) ( ), 1,..., 1,
( ), 0.
1 , .
k k k k k
N N N
mx f x x a f x x a k N
M x f x x a x
t M mM
Consider a chain, fixed at the lower end and is attached by a “heavy” mass M=3,000 m
at the top.
Problem: Approximate the motion of the mass M by a single differential equation
Find
,),( kXLLLM
L
T >> 1/M, M >>m
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Result: numerics + averaging
Average curve is smooth and monotonic.
Minimal value of derivative is close to zero.
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Homogenized constitutive relation (probabilistic approach)
Coordinate of the “large mass” is the sum of elongations of many nonconvex springs that (as we have checked by numerical experiments) are almost uncorrelated, (the correlation decays exponentially, ) while the time average of the force is the same in all springs:
The dispersion is of the order of the hollow in the nonconvex constitutive relation.
1 1 0( ) erf , 1.f f
f F f E
1 1( ) ( )(1 ( ))k F p f F p f f
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Add a small dissipation:
Continuous limit is very different: The force becomes
The system demonstrates a strong hysteresis.
],[if ),sign( ul xxxxkΨ
)()( 11 kkkkkk XXfXXfXXm
],[if , ul xxxf(x)Ψ
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Homogenized model (with dissipation)
• Initiation of vibration is modeled by the break of a barrier each time when the unstable zone is entered.
• Dissipation is modeled by tension in the unstable zone.
vBrokenbarrier
Tension bed
Small magnitude:Linear elastic material
Brokenbarrier
Larger magnitude:Highly dissipative nonlinear material.
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Energy path
Slow motion
Initial energy
Energy of high-frequency
vibrations:
Dissipation
High-frequency vibrations.
The magnitudeof the high-frequency mode is bounded from below
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Method of dynamic homogenization
• We are investigating mass-spring chains and lattices, which allows to– account for concentrated events as breakage– describe the basic mechanics of transition– compute the speed of phase transition.
• The atomic system is strongly nonlinear but can be piece-wise linear.
• To obtain the macro-level description, we – analyze the solutions of this nonlinear system at micro-level – homogenize these solutions,– derive the consistent equations for a homogenized system.
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Thank you for your attention