1

Module 10

2

M10.3 Environmental Effects on Composites

M10.2 Joining of Composites

M10.1 Testing of Composites

Learning Units of Module 10

M10.4 Recycling of Composites

3

Mechanical Testing of Composites

4

Tensile Testing

Figure: Typical tensile composite test specimen (all

dimensions in mm)

5

Compressive Testing

Figure: Celanese

compressive fixture and

specimen (all dimensions in

mm)

6

Intra-laminar Shear Testing

Figure: Schematic

representation of the

asymmetric four-point

bend shear fixture

7

Inter-laminar Shear Testing

Figure: The short beam shear test

8

(a) Mode-I Fracture Toughness

Figure: An in situ Double Cantilever

Beam (DCB) test

9

(b) Mode-II Fracture Toughness

Figure: An in situ End

Notch Flexure (ENF) test

10

Standard Adhesion Testing Techniques For Non-fibrous Materials

• PEEL TEST

Quantitative

Reproducible

FE analyses exist

Mode I

Example: exposure of teflon to 500 eVAr+ sputtering

11

• (NANO)SCRATCH TEST

Quantitative

Reproducible

Mode II

Diamond tip

(above a critical load)

12

• FULL TEST

Quantitative

Not reproducible

Interpretation is problematic

13

Fiber-matrix interfacial adhesion

• Two general methodologies:– Indirect (or macromechanical) testing - focuses on

the collective behavior of fibers in a polymer matrix [interfacial strength is interpreted via simplistic approximations; fast but questionable data collection]

– Direct (or micromechanical) testing – probes interfacial behavior of individual fibers interacting with a polymer matrix [more fundamental information; variability within and between techniques; issue of relevance to real-life composites (scaling-up)]

14

Indirect (or macro-mechanical) testing

1. The transverse tensile test

2. The short-beam shear (or interlaminar shear strength -ILSS) test

15

3. Asymetrical 4-point (Iosipescu) bend test

(And other macroscopic tests… )

CONCLUSION – They are simple to perform, but yield ambiguous, inaccurate data.

16

Direct (or micromechanical) testing

1. The single-fiber pull-out test (several versions)

F F

~ 1 mm ~ 1 µm

17

2. The single-fiber microcompression (or MIT) test

Tricky test, difficult to perform, fiber-matrix interfacial fracture is often only partial due to slight off-axis moments, loading must be exactly parallel to the fiber axis.

18

3. The single-fiber fragmentation testSingle fiber embedded in matrix

Fragmented fiber

Pre-conditions for successful test: (a) failure strain of the fiber is much smaller than that of matrix; (b) Interfacial bonding must be fair to good.

19

Importance of adhesive strength

Simple example: Unidirectional carbon/epoxy composite

The presence of interfaces, and their adhesive strength, is critical for various physical properties of composites

Weak interface leads to tough composites

Strong interface leads to brittle composites

20

The Classification of Technological Features

Joints

Combined adhesive-mechanical joints

Adhesive-riveted

Adhesive-boltedAdhesive-threadedAdhesive-mechainal

Moulded-on

Spot mechanical joints

Riveted

Bolted

Theaded

Pin-bolted

Adhesive

Welded

Adhesive-rubber

Moulded-on

Continuous adhesive joints

21

The Requirement of the Joint Design

22

Advantage and disadvantages of bonded fastened joints

Limits to thickness that can be joined with simple joint configuration.Inspection difficulty.Prone to environmental degradation.Requires high level of process control.Sensitive to peel and through-thickness stresses.Residual stress problems when joining dissimilar metals.Disassembly is impossible without component damage.Requires surface preparation.

No stress concentrationin adherents.Stiff connection.Excellent fatigueproperties.No fretting problems.Sealed against corrosion.Damage tolerant.Small weight penalties.Fewer pieces, lower weight, good load distribution.

Bonded JointsDisadvantagesAdvantages

23

Advantage & Disadvantages of Mechanically fastened joints

Considerable stress concentration.Relatively compliant connection.Relatively poor fatigue properties.Hole formation may cause damage to composite.Prone to fretting.Prone to corrosion.Large weight penalty.

Positive connection.No thickness limitations.Simple process.Simple inspection procedure.Simple joint configuration.Not environmentally sensitive.Provides through-thickness

reinforcement and not sensitive to peel stresses.

No residual stress problems.No surface preparation of

component required.Disassembly possible without

component damage.High tolerance to repeated loads.

Bonded JointsDisadvantagesAdvantages

24

Design Considerations

The behaviour of mechanically fastened joints is influenced by:Material parameters.Configurational parameters.Fastener parameters, e.g.

• Fastener type (screw, bolt, rivet).• Fastener size.• Clamping force.• Washer size.• Hole size.• Tolerance.

25

The Issues & Design Approaches to each Issue

Specially designed blind rivets.Verify joint strength with tests.

Damage induced by installation of blind fasteners and drive rivets

Avoid, if possible.Increased laminate thickness (locally).

Countersunk head

Larger fastener head.Washers (one or both sides).Limit on installation torque.

Preload relaxation

Larger fastener diameter.Insert (bushing).Increased laminate thickness (locally).

High local stresses

Closely controlled manufacturing operations.Inspection of drilled holes.

Drilling damage.ApproachIssue

26

Failure Criteria: Allowable StressesThe allowable stresses in each of these modes are

a function of the following:Geometry of the joint including the hole size, plate width and distance of the hole from the edge of the plate.The clamping area and pressure.The fibre orientations ply sequence.The moisture content and exposure temperature.The nature of stressing, e.g. tension or compression, sustained or cyclic and any out-of-plane loads causing bending.

27

Failure Modes

It is recommended that highly loaded structural joints be designed to fail in a bearing mode to avoid the catastrophic failures associated with net section failures.The failure stresses will depend on the degree of anisotropy at the hole and hence on the local fibre orientation.

28

The Failure Modes

29

Fastener Failure Modes

30

Fastener Selection

Fastener requirements for joining composite structures differ from those joining metallic structures.Fastener selection considerations for joining composites include corrosion compatibility, fastener material, strength, stiffness, head configuration, importance of clamp-up, lightning protection, etc.

31

Fastener Selection considerations

Corrosion Compatibility: Neither fibre glass nor aramid fibre reinforced composites cause corrosion problems when used with most fastener materials. Composites reinforced with carbon fibres are quite cathodic when used with materials such as aluminium or cadmium. Presence of galvanic corrosion between metallic fasteners and non-metallic composite laminates has eliminated several commonly used alloys from consideration. Conventional plating materials are also not being used because of compatibility problems. The choice of fastener materials for composite joints has been limited to those alloys, which do not produce galvanic reactions. The practice followed in aircraft industry is to coat the fasteners with anti-corrosion agent to alleviate galvanic corrosion.

32

Continued…..Fastener material: The materials currently used in design include alloys of titanium and certain corrosion resistant stainless steels with aluminium being eliminated. The choice is obviously governed by the make up of the composite materials being joined, weight, cost, and operational environment. Titanium alloy Ti-6Al-4V is the most common alloy used with carbon fibre reinforced composite structures.Bolt bending: Due to increased inter-laminar shear between the composite plies, bending of the bolt occurs more easily. High modulus and high tensile strength fastener material is desired where bending may occur. Susceptibility of bolt bending in composite structure introduces higher reaction loads on the fastener head, which requires more careful consideration of head configuration. Bending should also be considered in multiple component fasteners such as blind fasteners. A threaded core bolt resists bending much better than a smooth bore pull-type blind fastener as shown in Figure (in next slide).

33

Continued…..

34

Continued…..Head configuration: Composites are sensitive to high bearing loads than are metals. This means fastener heads should be designed with as much bearing surface area as practicable. The larger area improves pull-through and delamination resistance in composites, while reducing over-turning forces from bolt bending. Countersunk or flush head fasteners are frequently used on exterior surfaces of the aircraft where aerodynamic smoothness is required. Countersunk fasteners for composites include tension head fasteners having the large head depths and shear head fasteners having smaller head depths with head angles ranging from 1000 to 1300 as shown in Figure (in next slide).

35

Continued…..

Figure: Countersunk fasteners for composites with different head depths

36

Continued…..Countersunk fasteners: tend to bear against the surrounding element more unevenly through the thickness than protruding head fasteners do. Tension head fasteners are generally preferred over shear head fasteners due to greater strength against head pull-through. However, if the joint element is so thin that the countersunk depth is greater than 70% of the element thickness, the tendency towards uneven bearing pressure in tension head fasteners is too great and shear head fasteners are recommended in this case. Caution should be observed in the use of 1300 countersunk head fasteners. Although this type of fastener increases the bearing area of the fastener and permits it to be used in thin laminates, pull-through strength can be adversely affected. Although; close tolerance fit fasteners are desirable for use with composites, interference fit fasteners cannot be used due to potential delamination of plies at the fastener hole.

37

Continued…..

Clamp up: When tolerance fit holes are used, high clamp up appears to be beneficial for joint strength and fatigue life. The clamping forces, however, must be spread out over a sufficient area so that the compressive strength of the resin system is not exceeded and the composite crushed.

38

Module 10.2

39

Adhesive Bonding

Joining process whereby a filler material is used to hold two closely spaced parts together by surface attachment.Filler material (adhesive)

Usually non-metalUsually a polymer

CuringProcess (usually chemical) whereby the adhesive physical properties are changed from a liquid to a solid.

40

The criteria for selection of adhesiveThe adhesive must be compatible with the adherends and able to retain its required strength when exposed to in-service stresses and environment.The joint should be designed to ensure failure in one of the adherends rather than failure within the adhesive bond line.Thermal expansion of dissimilar materials must be considered. Due to large thermal expansion difference between graphite composite and aluminium, adhesively bonded joints between these two materials are likely to fail during cool down from elevated temperature cures as a result of the thermal stresses induced by their differential expansion coefficients.Proper joint design should be used avoiding tension, peel or cleavage loading. If peel forces cannot be avoided, a lower modulus adhesive having high peel strength should be used.

41

Continued……Surface preparation should be conducted carefully, avoiding contamination of the bond line with moisture, oil, etc.The adhesive should be stored at the recommended temperature.Use of adhesives that evolve volatiles during cure should be avoided.The recommended pressure and proper alignment fixtures should be used. The bonding pressure should be great enough to ensure that -the adherends are in intimate contact with each other.Traveller coupons should always be made for testing.The exposed edge of the bond joint should be protected with an appropriate sealing compound.

42

General Properties of some adhesives

AcrylicThermoplastic, quick setting, tough bond at r.t.Tennis racquets, metal parts

EpoxyThermoset, strongest engineering adhesiveMetal, ceramic, rigid plastic parts

CyanoacrylateThermoplastic, touch“Crazy Glue”

Hot MeltThermoplastic, quick setting, easy to applyBonds most anything –Packaging, book binding, metal can joints

43

PhenolicThermoset, strong, brittleBrake lining, clutch pads, honeycomb structures

SiliconeThermoset, slow curing, flexible, rubber likeGaskets sealants

Water base AnimalVegetableRubbers

Inexpensive, non-toxicWood, paper, fabricLeather, dry seal envelopes

Conitinued….

44

Joint Design

Usually not as strong as welding or brazing jointsDesign principles

Maximize joint contact areaJoints are strongest in shear and or tension so joints should be designed to accommodate thisJoints are weakest in cleavage or peel. Avoid these stresses

45

Adhesive bonding DisadvantagesJoints are not as strongAdhesive must be compatible with materials being joinedService temperatures are limitedCleanliness and surface preparation prior to adhesive application are importantCuring times can impose a limit on production ratesInspection of the bonded joint is limited.

46

Joints

Bolted and Bonded

47

Single Lap Joint

Bonded

Bolted

48

Bonded

Bolted

Double Lap Joint

49

Bonded

Single Doubler Joint

50

Bonded

Double Doubler Joint

51

Bonded

Stepped Lap Joint

52

Bonded

Scarf Joint

53

Bolted

Reinforced-Edge Joint

54

Bolted

Shimmed Joint

55

Bolted-Bonded Double Lap Joint

56

More Examples

57

58

ADHESIVE FAILURE

COHESIVE FAILURE

59

• Thinner materials can be joined– Weight Savings– Cost Savings

• Fewer Production Parts• Less Machining• Unskilled Work Force• High strength to weight ratio

Advantages to Bonded Joints

60

• Aerodynamic smoothness• Serves as a seal or corrosion barrier• Excellent electrical and thermal insulation• Superior fatigue resistance• Good damping and noise reduction• Can help in CTE mismatches

Continued…

61

With Adherends of unequal thickness, the maximumshear stress will occur near the end of the overlap on the side of the thinner part.

Single Lap Joint

62

• Adhesive stresses decrease with width, up to about a 4 in width. Then they remain constant.

• One in results can be used conservatively on wider specimens.

• max stresses do not decrease significantly with increasing bond area.

• Stresses decrease with increasing bond thickness

Single Lap Joint

63

• Maximum stress increases almost linearly with shear modulus.

• As stiffness of adherends increase, the resistance of the joint to bending increases, and maximum stresses decrease.

• Maximum stresses in the adhesive are insensitive to the Poisson’s ratio of the adherends

Single Lap Joint

64

Defined as the axial load divided by the nominal bonded area divided by the strength of the weaker adherend without the joint.

Joint Efficiency

65

Small deflection theory not valid in analyzing the joint under load. Some authors suggest using maximum stress theory for adhesive and isotropic adherends and maximum strain theory for composite adherends.

Nonlinear Problem

66

Eccentric Loading

67

Stress Reduction-Maximum Peel and Shear

• Combination of flexible and stiff bonds• Stiffer lap plates• Tapered plates

68

Ineffective Length

Defined as the lap length beyond which an increase in that dimension is ineffective in reducing peak values of peel and shear stresses.

69

• More than twice as efficient as single-lap joints• Symmetric, so bending effects are minimized• Practical design rule: Lap length to adhesive

thickness ratios of 30

Double Lap Joints

70

Important for repair applications where both sides of structure are not readily available

Single Doubler Joint

71

• Smooth joint• Requires careful machining• Scales up to any load • Approaches the ideals of strain compatibility

in the adherends and uniform stress in the adhesive

• Adhesive ductility is less important than in other types of joints

• Provides efficient use of full overlap length• Not bounded by an “ineffective length”

Scarf Joints

72

• Strength is not sensitive to number of steps• Light weight• Approximates the strain compatibility of scarf

joints• Scales up to any load • Provides efficient use of full overlap length• Not bounded by an “ineffective length”

Stepped Lap Joints

73

Q1 Q2

N1

N2

M2

M1

σu(x) τu(x)

σl(x) τl(x)

General Adherend Element

74

Net Tension Failure

Bolted Joint Failure Modes

75

Cleavage Tension Failure

Bolted Joint Failure Modes

76

Shear Out Failure

Bolted Joint Failure Modes

77

Bearing Failure

Bolted Joint Failure Modes

78

Bolt Pull Through

Bolted Joint Failure Modes

79

Bolt Failure

Bolted Joint Failure Modes

80

81

Environmental Effects on Composites

Composite usage has increased enormously mainly due to the advantages of lightweight, specific strength and stiffness, dimensional stability, tailor-ability of properties such as coefficient of thermal expansion and high thermal conductivity. Environmental effects on these properties may compromise a structure and must be considered during the design process.

82

Biological AttackBiological attack on composite materials may consist of fungal growth or marine fouling. Fungal growth does not appear to be as damaging as the wet conditions that promote growth. Fungicide has been mixed in with resins to retard this growth. Even though marine organisms will grow on composite surfaces, mechanical properties do not appear to be affected and the fouling can be removed by scraping. Composites with graphite fibres have been used in medical applications for both internal and external purposes.

Internal composite structures such as artificial joints or plates for bone fracture support must be bio-compatible or the material may degrade over time. External composite designs (such as artificial limbs or orthotic braces) may experience impact damage, flexural and torsional loading during use.

83

Fatigue

Fatigue, either through mechanical loads or acoustic vibrations, can cause crack growth or local defect formation. Fatigue designdepends not only on the load but also on the use temperature range and amount of moisture present. Very cold temperatures (below -50°C) may increase the stiffness of some composite materials thereby increasing the susceptibility to fatigue damage.

84

85

Hygrothermal Behavior

Study of environmental effects (moisture and temperature) on composite material properties

and stress-strain behavior.

86

ThermalThermaleffects are caused by

temperature

Hygroscopic Hygroscopic effects are caused by moisture.

Hygrothermal Hygrothermal effects are

caused by either temperature or

moisture.

Hygrothermal

87

Topics to be Discussed

1. Matrix Dominated Property Degradation2. Stress-Strain Behavior3. Micromechanical Models

88

Hygrothermal Degradationof Properties

1. Matrix dominated properties such as stiffness and strength under transverse, off-axis and shear loadings are altered.

2. Increased temperature causes a softening of the polymer matrix.

3. Glass transition effects.

89

Glass Transition Temperature

At a certain value of the temperature the matrix

materials transitions from a glass-like behavior to a

rubbery behavior.

GlassyRegion

RubberyRegion

Wet

Dry

Temperature

Stiff

ness

0gTgwT

90

Moisture is present in the operational environment in which a composite is manufactured and throughout its useful life.Water acts as a plasticizer when absorbed by the matrix, softening the material and reducing some properties of the laminate.Moisture may also migrate along the fibre-matrix interface thereby affecting the adhesion.Moisture in composites reduces matrix dominated properties such as transverse strength, fracture toughness and impact resistance.

Moisture Effects

91

The relation between Moisture content and Glass transition temperature

Lowering of the glass transition temperature may also occur in epoxy and polyimide resins with an increase in absorbed moisture (as shown in Figure). Debonding can occur due to formation of discontinuous bubbles and cracking in the matrix. Mechanical properties can be reduced even further if heat is present or if the composite is under-cured or has a large amount of voids.

92

Temperature EffectsTemperature effects on composite materials include cryogenic temperatures, elevated temperatures and thermal cycling between these extremes.Cryogenic temperatures do not appear to affect the mechanical properties of graphite/epoxies or graphite/polyimides significantly.Elevated temperatures for a prolonged period of time can seriously affect the properties of a composite, with even greater effect if moisture is present. Thermal cycling may induce micro-cracking in some composites thereby resulting in reduction of compressive and shear strength.

93

Temperature Effects

Temperature effects are not limited to the matrix materials. Extended operation at 350°C (660°F) and 450°C (840°F) can cause oxidation of low modulus PAN-based fibres and high modulus PAN- or Pitch-based fibres, respectively. Oxidation resistance can be improved with higher purity fibres. Thermal cycling conditions are common for a number of applications, including aircraft and spacecraft.

94

The effect of temperature and ageing on stiffness of composite Materials

Loss of stiffness with temperature and ageing is indicated in Figure; Susceptibility to matrix softening is not only dependent on the resin but also the lay-up.

95

Protection against temperature effects

Protection against temperature effects can be achieved at the design stage itself by:Selection of resin system with high glass transition temperature.Potential degradation taken into account in the analysis and fatigue test.Protection against moisture exposure.

96

Overheat Conditions

Heat generated by lightning strikes has been known to vaporize matrix resins and create large areas of delamination and fibre fracturing on composite rudders, ailerons, wing and stabilizer tips, nose domes and nacelle cowling. When exposed to hot gases over long periods, polymeric resin binders can become completely destroyed through a process of thermo-oxidation. Preventive methods may consist of application of heat resistant ablative coatings.

97

Glass Transition Temperature

etemperatur transition glasswet"

etemperatur transition glass dry"

etemperatur transition glass

"T

"T

T

gw

0g

g

98

Through-thickness Distributionof Temperature and Moisture

Plate

a

a

cT

a

a

cT

Thicknessh

Ta – Ambient Temperatureca – Ambient Moisture

Concentration

99

Through-thickness Distributionof Temperature and Moisture

T

c

Thicknessh

z

100

Temperature DistributionFourier Heat Conduction Equation

timedirection- in material of tyconductivi thermal

materialthe of heatspecific materialthe of density

==

==

∂∂

∂∂=

∂∂ρ

tzK

Cρ

zTK

ztTC

z

z

101

Moisture DistributionFick’s Second Law

direction- in material of ydiffusivit mass zD

zcD

ztc

z

z

=

∂∂

∂∂=

∂∂

Temperature change is orders of magnitude faster than moisture change.

102

Boundary and Initial Conditions

ionconcentratmoisture initiale temperatur ambient

compositethe of ionconcentratmoisture initialcompositethe ofe temperatur initial

a

a

i

i

ccTT

0thz,0z

ccTT

0tandha0

==

>==

==

≤<<

103

Series Solution

( )( ) ( )

( )a

m

htDj

j

im

i

cc

eh

zjj

cccc

z

to relatedmaterial ofsurface at ionconcentratmoisture

⎟⎟⎠

⎞⎜⎜⎝

⎛ π+−∞

=⎟⎠⎞

⎜⎝⎛ π+

+π−

=−

−

∑2

2212

0

12sin12

141

104

Average Concentration

( )( )

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+π−−== ∑∫

∞

=

⎟⎟⎠

⎞⎜⎜⎝

⎛ π+−

1j2

h

tD1j2

im

h

0 1j2e81ccdz)t,z(c

h1c

2z

22

105

Weight Percent Moisture

( )

( )

conditions ambient withsaturated fully henmoisture w percent weight

moisture percent weight

m

1j2

h

tD1j2

im

i

MM

1j2e81

MMMM

G2

z22

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+π−=

−−

= ∑∞

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛ π+−

106

Hygrothermal Degradation

21

00g

gw

0m TT

TTPPF ⎥

⎦

⎤⎢⎣

⎡

−−

==

Empirical formula for polymer resins

107

ndegradatiobefore strength) or (stiffness property mechanicalmatrix Reference

ndegradatio after strength) or (stiffness property mechanicalMatrix

ratio retention property mechanicalMatrix

=

==

0P

PFm

Pressure Terms

108

measured was whichate TemperaturP to ingcorrespond condition wet referenced fore temperatur transition Glass

condition dry referenced fore temperatur transition Glass

whichate Temperatur

00

0

PT

T

TT

gw

g

=

=

==

Temperature Terms

109

Glass Transition Temperature

( ) 0gr2rgw T0.1M10.0M005.0T +−=

Empirical formula for polymer resins

110

Properties

m0mmf1f1 EFEE vv +=etc.

111

Lamina Stress-Strain Behavior

Including Hygrothermal EffectsIncluding Hygrothermal Effects

112

Free Thermal Strains

(CTE) expansion thermal of tcoefficieni all fore where temperatur initial

etemperatur final)T-(Tchange e temperatur

0

0

===

==∆

=∆α==ε

α 0εT

TT

3,2,1iifT6,5,4iif0

Ti

Ti

113

Hygroscopic Strains

0 0c

c

3,2,1iifc6,5,4iif0

Mi

Mi

=ε=

=β

=

=

=β==ε

when:conditionReference

(CHE) expansionc hygroscopi of tcoefficien

volume unit in material dry of massvolume unit inmoisture of mass

ionconcentratmoisture

114

Hygrothermal Strains

⎪⎩

⎪⎨⎧

=

=β+∆α=ε

ε+ε=ε

4,5,6iif

1,2,3iif

0

cT iiHi

Mi

Ti

Hi

115

Transverse Isotropy

32

32

β=βα=α

116

Stress-Strain Behavior

cT

S

SS

SS

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

β

β

+∆

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

α

α

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

σ

σ

σ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γ

ε

ε

0000

0

0

2

1

2

1

12

2

1

66

2221

1211

12

2

1

117

Stress-Strain Behavior

{ } [ ]{ } { } { }cTS β+∆α+σ=ε

Matrix Notation

118

Stress-Strain Behavior

{ } [ ] { } { } { }( )cTS 1 β−∆α−ε=σ −

Invert

119

Unrestrained Hygrothermal Exposure

{ } { } { }cT β+∆α=ε

120

Completely Restrained Hygrothermal Exposure

{ } [ ]{ } { } { }

{ } [ ] { } { }( )cTS

cTS

β−∆α−=σ

β+∆α+σ==ε

−1

0

121

Stress-Strain Behavior

cT

SSS

SSS

SSS

xy

y

x

xy

y

x

xy

y

x

662616

262212

161211

xy

y

x

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

β

β

β

+∆

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

α

α

α

+

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

σ

σ

σ

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γ

ε

ε

122

Transformation

CTE’s and CHE’stransform like tensor

strains.

123

Tensor Strain Transformation

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γε

ε

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γε

ε

2

2

2

212

2

1

22

22

22

sccscs

cscs

cssc

xy

y

x

124

Tensor Strain Transformation

[ ]⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γε

ε

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

γε

ε−

2212

2

1

1Txy

y

x

125

Matrices

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

=−

22

22

22

1 2

2

sccscs

cscs

cssc

T

126

Matrices

[ ]⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−=22

22

22

2

2

sccscs

cscs

cssc

T

127

Transformation

[ ]⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

α

α

=

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

αα

α−

0

T

2

2

1

1

xy

y

x

128

( ) θθα−α=α

θα+θα=α

θα+θα=α

sincos2

cossin

sincos

21xy

22

21y

22

21x

129

1 -by scaled by normalized sCTE' All 1

xy

2

1

21

αα

=αα

Off axis thermal expansion

0

0.5

1

1.5

2

2.5

0 30 60 90

Angle θ

CTE

's axayaxy

xαyαxyα

130

1 -by scaled by normalized sCTE' All 1

xy

2

1

21

αα

−=αα

Off axis thermal expansion

-1.5-1

-0.50

0.51

1.52

2.53

3.5

0 30 60 90

Angle θ

CTE

's axayaxy

xαyαxyα

131

1 -by scaled by normalized sCTE' All 1

xy

2

1

51

αα

=αα

Off axis thermal expansion

0

1

2

3

4

5

6

0 30 60 90

Angle θ

CTE

's axayaxy

xαyαxyα

132

1 -by scaled by normalized sCTE' All 1

xy

2

1

51

αα

−=αα

Off axis thermal expansion

-2-101234567

0 30 60 90

Angle θ

CTE

's axayaxy

xαyαxyα