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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 shape--which 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 Ni-Ti 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 shape-memory 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 two-way shape-memory effect is the effect that thematerial remembers two different shapes: one at low temperatures, and one at the

high-temperature 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 scaled-down F-18

The SAMPSON Program: SMAs wereused to rotate the inlet cowl in order tochange its cross-sectional 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 wire-actuated 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 asfree-energy based models.

Another class of models attempts to capture the hysteretic input-output 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) Kelvin-Voigt model (c) Standard solid model

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Types of springs

General power-law spring,

Linear ( = 1)

Softening ( < 1)

Stiffening ( > 1)

Types of dashpots

General power-law dashpot,

Linear ( = 1)

Thixotropic (

< 1) Stiffening ( > 1)

Frictional (Kuhn-Tucker conditions)

Kuhn-Tucker 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

Micro-scale 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: 511-528.

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: 1266-1274.

A. Khandelwal, V. Buravalla. Models for shape memory alloy behavior (2009).International Journal of Structural Changes in Solids 1: 111-148.

UG Course Notes , Smart Materials and Structures Prof. Mira Mitra,Aerospace Engineering Department, IIT Bombay.

J. V. Humbeeck. Non-medical applications of shape memory alloys (1999).Materials Science and Engineering A273-275: 134-148.

Introduction to Shape Memory Alloys,

Link: www.tinialloy.com/pdf/introductiontosma.pdf

Shape-memory alloy Wikipedia

Link: http://en.wikipedia.org/wiki/Shape-memory_alloy

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THANK

YOU

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BACK-UP 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