Dynamics of Networks with Bistable Links

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Dynamics of Networks with Bistable Links
The Seventh International Conference on Vibration
Problems

• Andrej Cherkaev, Elena Cherkaev
• Department of Mathematics,
• University of Utah, USA
• and
• Leonid Slepyan
• Department of Solid Mechanics, Materials and
  Structures,
• Tel Aviv University, Israel

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Acknowledgements
The talk is partly based on the joint results
obtained in collaboration with the fellows of
Department of Mathematics, University of Utah

• 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

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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)
Observe that
The material itself can absorb he energy and
resist until it melts
Failure is caused by stress localization Most
of the material stays undamaged after the
structure fails! Aim delocalize the stress and
stabilize the failure
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Tao of Damage
Tao -- the process of nature by which all things
change and which is to be followed for a life of
harmony. Webster

• Damage happens!
• Dispersed damage absorbs energy concentrated
  damage destroys.
• Design is the Art of Damage Scattering

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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

c(x,t) is the damage parameter
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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.

c is the damage parameter
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Chain of rods

• Several elements form a chain

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
Elongation
Breaks of basic links

• 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
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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
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.
Constitutive relation in links
a is the fraction of material used in the
forerun basic link
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Results
Efficiency 750/1505
M700 a.25 Small dissipation
a1 (no waiting links) M 150
M750 a.25 Small dissipation
M375, a0.25
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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 -1 0
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

Before damage (transition)
After damage (transition)
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Equations of chain dynamics
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)

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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.

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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

where
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Additional considerations

• 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.
• The solution is more complicated when the
  elastic properties are switched after the
  transition. In this case, the Wiener-Hopf
  technique is applied.
• 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 vv(p).

Measurements of the speed in direct simulation
of dynamics
p
Positions of masses




Force
Waves speed
time
p
p
Direction of the transition wave
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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 maxu(x, t)
• The right-hand side of
• is a bistable function equilibrium is nonunique.

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Lattices with waiting links

• Smart structures from bistable elements

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Greens function for a damaged latticeStatics
The lattice with waiting links is described in
the same manner. The damage in a link is mimicked
by an applied pair of forces
F(k,m, N, N)
Greens function Influence of one damaged link
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The state of a partially damaged lattice
Q How to pass from one permissible
configuration to another? The answer is
ambiguous if dynamics is not considered
F(k,m, N, N)
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.

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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.

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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)

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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?

Radiation and other losses
Dynamic homogenization
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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.
?
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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
  Yoshikawa
• A.Balk, A.Cherkaev, L.Slepyan 2000. IJPMS
• T.Yoshikawa, 2002, submitted

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Unstable reversible links
Force

• 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.

Elongation
H is the Heaviside function
No parameters!
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Chain dynamics. Generation of a spontaneous
transition wave
x
0
Initial position linear regime (N0), but close
to the critical point of transition
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Observed spontaneous waves in a chain
Twinkling phase
Chaotic phase
Under a smooth excitation, the chain develops
intensive oscillations and waves.
Sonic wave
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

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Periodic waves Analytic integration
Approach, after Slepyan and Troyankina

• Nonlinearities are replaced by a periodic
  external forces period is unknown
• Self-similarity
• Periodicity

System () can be integrated by means of Fourier
series. A single nonlinear algebraic
equation defines the instance q.
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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 q of the
  application/release of the applied forces is
  solved.
• Result The dispersion relation

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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.

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Large time range description

• with Toshio Yoshikawa

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Problem of homogenization of a dynamic system
Consider a chain, fixed at the lower end and is
attached by a heavy mass M3,000 m at the
top.
T gtgt 1/M, M gtgtm
Problem Approximate the motion of the mass M
by a single differential equation Find
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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)
f

• 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.

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Add a small dissipation

• Continuous limit is very different The force
  becomes
• The system demonstrates a strong hysteresis.

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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.

v
Broken barrier
Broken barrier
Tension bed
Small magnitude Linear elastic material
Larger magnitude Highly dissipative nonlinear
material.
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Energy path
The magnitude of the high-frequency mode is
bounded from below
Initial energy
Slow motion
Energy of high-frequency vibrations
High-frequency vibrations.
Dissipation
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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