sma modeling
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SMA MODELING
Rohit S. Gajbhiye(08d01010)
Guide : Dr. Guruprasad

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INTRODUCTION
Shape Memory Alloys (SMA's) are novel materials whichhave the ability to return to a predetermined shape whenheated.
When an SMA is cold, or below its transformationtemperature, it has a very low yield strength and can be
deformed quite easily into any new shapewhich it willretain.
However, when the material is heated above its transformationtemperature it undergoes a change in crystal structure whichcauses it to return to its original shape.
This phenomenon provides a unique mechanism for remoteactuation.
It is a light weight alloy
It has numerous applications in medical and aerospaceindustries
Mostly they are NiTi or copper based alloys

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MECHANICAL PROPERTIES
SMA below its phase transformation
temperature possesses a low yield
strength crystallography referred to as
Martensite.
While in this state, the material can be
deformed into other shapes with
relatively little force.
The new shape is retained provided
the material is kept below its
transformation temperature.
When heated above this temperature,
the material reverts to its parent
structure known as Austenite causing it
to return to its original shape.

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T Af
< T < Af
As
One Way Memory Effect:When a shapememory alloy is in its cold state (below As), the metal can be bent or
stretched and will hold those shapes until heated above the transition temperature.Upon heating, the shape changes to its original. When the metal cools again it willremain in the hot shape, until deformed again.
Two Way Memory Effect: The twoway shapememory effect is the effect that thematerial remembers two different shapes: one at low temperatures, and one at the
hightemperature shape

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AEROSPACE APPLICATIONS
The Smart Wing Program: SMA wire
tendons were used to actuate hingelessailerons while an SMA torque tube wasused to initiate spanwise wing twisting ofa scaleddown F18
The SAMPSON Program: SMAs wereused to rotate the inlet cowl in order tochange its crosssectional area
Chevron Research Efforts: The SMAbeam elements are formed such that theyforce the chevron inward and mix the flowof gases (reducing noise) at low altitudes
and low speeds where the enginetemperature is high.
Lightweight Flexible Solar Array: Utilizedan SMA wireactuated stepper motor fororientation of its solar flaps.
Used for Morphing of Wings

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SMA MODELING
Modeling SMA response is broadly divided into two categories:
Macroscopic Approach Microscopic Approach
Macroscopic Modeling Approach:
Captures the SMA response at a macroscopic scale (typically >100 microns)
Models which draw heavily on phenomenology are termed phenomenological models,while those with a significant amount of thermodynamic framework are classified asfreeenergy based models.
Another class of models attempts to capture the hysteretic inputoutput responsewithout explicitly accounting for underlying physics.
Microscopic Modeling Approach
Models have a notion of unit cell or Representative Volume Element (RVE).Martensitic variants introduced into the model are crystallographic variants.
Another aspect is that generally, multiple true crystallographic variants are consideredin the microscale and after analysis, an equivalent single or multiple (reduced) variantmacroscopic description of the phenomena is provided

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MODELING OF INELASTICITY
OA is linearly elastic and hence
modeled by a linear spring i.e. once
stress is removed, strain comes back
to zero A is the yield point
From A to C, strain increases with
constant stress. Hence it is modeled as
friction block i.e. when stress isreleased its associative residual strain
is still there.
Whole response is modeled with
spring and friction block as shown

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The phenomenon of creep is modeled
here with linear dashpot i.e.
phenomenon to be modeled is the
deformations that may occur over time
due to sustained nature of the some ofthe loads such as dead load and mean
live loads in the structure. These
deformations take place over long
periods of time.
For a dashpot
is viscous coefficient
Dashpot and spring in parallel has
strain response as shown in the figure.
Initially dashpot offers major resistance
to stress but with time spring offers
major resistance.
Response of dashpot and spring in
series is shown in the figure

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LUMPED PARAMETER APPROACH Evolution of state satisfies mechanical dissipation equation
Rate of displacement consistent with non negative rate of dissipation and the
constraints is acceptable from 2nd law of thermodynamics.
Generally types of stimuli applied are Ramp, sinusoid, impulse and step.
Material characteristics
YieldHardening
Stress relaxation
Creep
Retardation
Suitable combination of masses, springs and dashpots are used for reproducing above phenomena
Commonly used models for inelastic behavior(a) Maxwell model (b) KelvinVoigt model (c) Standard solid model

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Types of springs
General powerlaw spring,
Linear ( = 1)
Softening ( < 1)
Stiffening ( > 1)
Types of dashpots
General powerlaw dashpot,
Linear ( = 1)
Thixotropic (
< 1) Stiffening ( > 1)
Frictional (KuhnTucker conditions)
KuhnTucker conditions
The above set of implicit equations need to be
solved to obtain
Dry friction problems trial and error solutions
Viscous drag problems is given explicitly
Response of springs (conservative)(a) Linear (b) Stiffening (c) Softening
Response of dashpots (dissipative)(a) Linear (b) Thickening (c) Thixotropic (d) Frictional

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ENERGYFORMULATIONOFEQUATIONSOFMOTION
Procedure: Assign thermodynamic states of the system displacements and
temperature
Specify either the Helmholtz potential or Gibbs potential as a function of the
states
Obtain state evolution equations
The following principles are used in the energy formulation:
Mechanical power theorem
Maximum rate of dissipation hypothesis Lagrange multiplier method for constrained maximization
Law of conservation of energy
Helmholtz potential

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ONE DIMENSIONAL APPROACH AND WORKHARDENING:
Energy Conservation Equation after substituting mass conservation, momentum conservationand Gibbs Potential Equations is:
Plastic deformation => Defects (dislocations) in crystal structure => Increase in resistance to
further plastic deformation
Resistance to further deformation is a function of total amount of plastic work done Frictional Dashpots: LPM
Equivalent plastic strain,
Hardening laws:
Linear hardening,
Swift law,
Voce law,
Metals without a sharp yield point

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Bauschinger effect Fatigue behavior
Cyclic loading => Small microscopic changes in internal structure => Formation ofcracks
Asymmetry between tension and compression (path dependent nature)
Typical behavior of materials to cyclic loading: Rachetting, Mean stress relaxation
and Shakedown
Microscale model Dislocations pinned at fixed locations bulge out, resist forwardmotion and assist reverse motion (like stretched spring)
LPM:

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CASE STUDYPASSIVEDAMPINGOFFRAMEUSING
SUPERELASTIC SMA BRACINGS
The structural arrangement is
shown in the figure alongside.
The dimensions of the frame and
loading conditions are provided
Since the SMA braces are primarily
under tensile loads, the response of
bracing material under uniaxial
tension needs to be modeled
Using the obtained model, the response of the frame with braces can be
obtained under dynamic loading conditions.

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SMA WIREBEHAVIORUNDERUNIAXIALTENSION
The response of SMA wireobtained experimentally is
shown alongside

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REFERENCES
Inelasticity of materials: An engineering approach and practical guide, WorldScientific Book
K. Otsuka, X. Ren. Recent developments in the research of shape memoryalloys (1999). Intermetallics 7: 511528.
D. Hartl, D. Lagoudas. Aerospace applications of shape memory alloys.Aerospace Engineering Department, Texas A&M University.
G.Song, N. Ma, H.N. Li. Applications of shape memory alloys in civilstructures (2006). Engineering Structures 28: 12661274.
A. Khandelwal, V. Buravalla. Models for shape memory alloy behavior (2009).International Journal of Structural Changes in Solids 1: 111148.
UG Course Notes , Smart Materials and Structures Prof. Mira Mitra,Aerospace Engineering Department, IIT Bombay.
J. V. Humbeeck. Nonmedical applications of shape memory alloys (1999).Materials Science and Engineering A273275: 134148.
Introduction to Shape Memory Alloys,
Link: www.tinialloy.com/pdf/introductiontosma.pdf
Shapememory alloy Wikipedia
Link: http://en.wikipedia.org/wiki/Shapememory_alloy

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THANK
YOU

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BACKUP SLIDES

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LUMPED PARAMETER APPROACH1. Evolution of state satisfies mechanical dissipation equation
2. Rate of displacement consistent with non negative rate of dissipation and the constraints isacceptable from 2nd law of thermodynamics.
3. Equation of motion with Gauss theory of least constraint is
Generalised force equation should satisfy non negativity of dissipation rate
4. Convenient way is to maximise subject to constraints. This allows to :
Obtain Equation of motionIncorporate constraint forces
Obtain constitutive relation for dissipative forces that satisfy non negativity of rate ofdissipation

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ONE DIMENSIONAL APPROACH
where x is new location and X is old location of 2 reference
points
Mass Conservation: mass per unit length
Momentum Conservation:
Energy Conservation:
Hence Using Gibbs Potential approach:
Substitute to get

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BAUSCHINGEREFFECT
LPM:
is the stress in the Bauschinger spring and is referredto as the back stress