An Immediate Formula for the Radius of Curvature of A ... · 2. Timoshenko Formula From the...

An Immediate Formula for the Radius of Curvature of A Bimetallic Strip

G. D. Angel

School of Engineering and Technology,

University of Hertfordshire

G. Haritos

School of Engineering and Technology,

University of Hertfordshire

Abstract An alternative formula has been derived to enable a

close prediction of the radius of curvature of a thin

bimetallic strip that at initial ambient temperature, is

both flat and straight, but at above ambient

temperature, forms into an arc of a circle. The formula

enables the evaluation of the radius of curvature of the

strip as a function of heating or cooling. A formula for

calculating the radius of curvature of a bimetallic strip

already exists, and was produced by Timoshenko in his

paper on Bimetal Thermostats. The formula by

Timoshenko has been vigorously proven, tried and

tested and accepted in countless papers and journals

since its original publication. The formula introduced

by this work, very closely approximates to the

Timoshenko formula for equal thicknesses of the two

mating metals within the bimetallic. The drawbacks of

the Timoshenko formula are that it is both complex and

unwieldy to use, and requires some form of electronic

spreadsheet to enable its evaluation. The formula put

forward here is both simple and quick to use, making it

more immediate. For the correlation of the new

formula, Timoshenko’s formula is used as a datum, or

benchmark. From the simulation a good overall

correlation was shown to exist between the Timoshenko

generated values and the values generated by the new

formula put forward here. Key words: Timoshenko, bimetallic strip, radius of

curvature, formula, thin.

1. Introduction The original Timoshenko [1] bimetallic bending

formula was published in 1925 and since then it has

been applied in by multitudes of engineers and scientist

and referred to in many papers such as by Krulevitch

[2], Prasad [3] and books Kanthal [4]. Whilst it has

been proven and accepted to be the formula to evaluate

the hot radius of curvature of an initially flat bimetallic

strip, it is an unwieldy and a complex formula to

evaluate. This work introduces a new simpler, quicker

method of evaluating the radius of curvature of a

bimetallic strip from an initially flat ambient condition

that has been uniformly heated. When a bimetallic strip

is uniformly heated along its entire length, it will bend

or deform into an arc of a circle with a radius of

curvature, the value of which, is dependent on the

geometry and metal components making up the strip.

As will be seen later on, the nature of the bend as a

function of temperature change from ambient is

characteristically asymptotic. The new formula

introduced here, closely approximates to the

Timoshenko formula with the exception of

accommodating the change in the thicknesses of the

two mating metals making up the bimetallic strip. It is

important to note that in the majority of applications of

bimetallic strip, the ratio of the thickness of the two

constitute metals is normally one to one, i.e. of equal

thickness. This comes about due to the way that the

bimetallic strip is manufactured. The dominant method

in the mass production of commercially available

bimetallic strip involves either hot or cold rolling the

two separate metals under intense pressures to produce

interstitial bonding of the atoms at the bi- material

interface Uhlig [5]. Under such conditions, it is

expensive, because of setup costs, to make special

separate metal thicknesses unless specifically required.

Moreover, there is no data to support that different

thicknesses of the bimetals in the strip would have any

performance benefit over equal thickness bimetal strip.

Cladding of metals Haga[6] is used to provide a

product with a less expensive base metal that benefit

from a thinner skin for decorative and or surface

protection purposes, although clad metals are

technically bimetallic metals, clad bimetallic strip is not

being used for its functional bending qualities.

Therefore the need to cater for separate material

thicknesses is not required for most applications where

the bimetallic bending qualities of a bimetallic strip are

being exploited.

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International Journal of Engineering Research & Technology (IJERT)

Vol. 2 Issue 12, December - 2013

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2. Timoshenko Formula From the Timoshenko [1], the radius of curvature

of a bimetallic strip is given by:

eqn.(1)

Where ρ is the radius of curvature,

total thickness of the strip.

are the individual material thicknesses.

is the ratio of thicknesses.

is the ratio of the Young’s Moduli.

are the hot and cold temperatures states.

, is the linear Modulus of the two materials.

& are the coefficients of linear thermal

expansion for the two metals.

is assumed to be numerically larger than

Fig.1 shows a bimetallic strip in two states of

heating, at state 1, at ambient temperature, the strip

will be flat with no discernible radius of curvature

R. At state 2, uniformly distributed heating will

cause the strip to form into a radius of curvature.

Note that has a numerically higher coefficient

of linear thermal expansion and thus naturally

wants to extend further than the side with The

differences, leads to internal stresses, forces and

moments at the material interface, resulting in the

bending as shown at state 2.

Fig.1 Bimetallic strip in two states of

heating

3. Derivation of the Approximation

Formula The derivation is based upon the amalgamation of

two well established formulae, with the addition of

new correction relationships that are a combination

of the ratios, sums, and quotients of the coefficient

of linear expansion of the metals.

Where possible, the nomenclature employed in the

Timoshenko formula will be used in the new

formula.

It is commonly known that the internal force

developed within a metal bar by heating or cooling,

can be written as follows:

eqn.(2)

Where F is the force (N).

α is the coefficient of linear expansion of

the metal ( .

ΔT is the temperature change of the metal

from ambient (K).

A is the cross-sectional area of the bar(

).

E is the Youngs modulus of the material of

the bar ( ).

Eqn.(2) can be re-written in terms of the stress ,

since

, thus the internal stress due to heating:

eqn.(3)

By substitution of where y is assumed to be

the distance from bimetal interface to the outer

edge, this is also equal to half the total thickness of

the bimetallic strip.

From the well-known simple bending equation

substituting and re-arranging using the

first two terms of the simple bending equation, thus

;

eqn.(4)

Where:

R is the radius of curvature of the bimetallic strip

to the bimetallic joint center line (mm).

t is the total thickness of the bimetallic

strip(mm).

with the average coefficient of

linear expansion of both metals ( ).

Thus eqn.(5)

This is a first order estimate of the radius of

curvature of the strip as a function of change in

temperature that approximates to the Timoshenko

formula, see Fig.2.

The simple derivation resulting in eqn.5 and shown

in Fig.2, provides a rough or first order estimate of

the radius of curvature as a function of temperature

change in the strip. It should be noted that although

eqn.5 is an approximation to the Timoshenko

formula, the accuracy of eqn.5 tends to improve as

the temperature increases, or as the expression

becomes more asymptotic. Furthermore, the nature

the first order derivation, closes resembles the

Timoshenko formula and follows a similar

trajectory, that of an asymptotic curve.

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Fig. 2 Comparison of Timoshenko ρ vs. R

Using a similar approach to Timoshenko where the

ratios of the thicknesses of the metals and the sum

of the thicknesses in the metals, play an influential

part of the derivation. In the derivation put forward

here, the ratios of the coefficients of linear

expansions, sum and differences are used as a

correction factor with similar effect that ultimately

modifies the first order expression eqn.5, to a close

approximation of the Timoshenko formula. The

rationale for using the coefficients of linear

expansions to correct the first order curve, is based

upon the fact that in this derivation, the thicknesses

of the two constitute metals are assumed equal for

reasons explained earlier.

From Fig.2, it can be seen that the radius of

curvature of the approximate curve R, is slightly

larger numerically, than the Timoshenko line. Thus

a possible correction factor needs to multiply R by

a number slightly less than unity. Introducing a

proportional correction factor reduces the value of

R to a very close approximation of Timoshenko ρ.

Thus letting and this provides

an initial lowering of R.

and also letting and

be the proportional difference in the coefficients of

linear thermal expansions, then the proportional

correction influence on R is:

=

eqn.(6)

Eqn.6 is a very effective modifier when multiplied

by the eqn.5 see Fig. 3 for the very close

approximation of .

At the same time, the correction must take into

account the rapid change of curvature in the lower

temperature range of 20°C to 80°C. A further

correction factor is achieved if the ratios of the

sums and differences are also considered, thus

letting:

and and combining the ratios.

Thus that expands to

Combining and re-arranging and reducing, thus:

eqn.(7)

Fig.3 showing effects of the correction coefficients

Therefore adding derived components, to

eqn.5, enhances the accuracy of eqn.5 to that of

Timoshenko. Thus the radius of curvature R, as a

function of temperature is given by:

eqn.(8)

From eqn.8 it can be seen that in this derivation,

there is no requirement to know E, the Young’s

modulus of the two separate metals of the

bimetallic strip.

Also eqn.8 can be also be expressed directly as:

eqn.(9)

Where t is total thickness of the bimetallic strip

(m).

is the change in temperature of the strip from

ambient (°C ).

R is the radius of curvature of the strip (m).

is the correction factors multiplied by

(°C ).

Eqn.10 was used to generate data in the simulation.

comparison to eqn.1 by Timoshenko.

4. Simulation Data For the generation of comparison data, parameters

& were varied in the simulation; = 0.4, 0.8,

1.2, 1.6, 2, 4, 6, 8 and 10 mm, being the total

thickness of strip. It should be noted that in most

applications of bimetallic strip, the total thickness

is usually quite thin, up to 1mm thick for switching

applications [4].

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5. Simulation Data For the proof of the correlation between the new

formula proposed in this paper, and Timoshenko’s

original formula, two separate simulations were

performed.

The first simulation was based around a bespoke

Bimetallic strip SBC206-1 from Shivalik [7] ;

100mm long x 5mm wide x 0.4mm thick , these

were the starting values of the first simulation set.

For simulation set 1, the following data was

assumed;

The strip thickness t, was varied from 0.4mm thick

to 10mm thick.

= 213 ( ) ; Young’s modulus of Steel side

of the bimetallic strip.

= 145 ( ) ; Young’s modulus of Invar 36

side of the bimetallic strip.

= ( ) coefficient of linear thermal

expansion for Steel side of the strip.

= ( ) coefficient of linear thermal

expansion for Invar 36 side of the strip.

both equal, total thickness of

the strip.

= (20°C) assumed ambient temperature constant

throughout the simulation.

Input variable temperature (°C).

change in temperature , (°C).

= radius of curvature evaluated by eqn.(1)

Timoshenko formula (m).

R = radius of curvature evaluated by eqn.(10)

proposed new formula (m).

For the second simulation, a combination of

different materials within the bimetallic strip and

also a variety of thicknesses of strip were used. It

should be noted that the material combinations put

forward in the second simulation may not be

practical for the manufacture of bimetallic strip by

modern mass production methods of cold pressure

rolling, but they can be produced by other, older

fabrication methods such as by riveting the two

metals together.

The simulation data in set 2 have been included in a

random fashion to test the robustness of the new

formula.

The second simulation data set is as shown in Table

1. Note that Invar 36 is used as the common mating

material since it possesses a very low coefficient of

linear expansion as compared with all other

engineering materials.

As per simulation 1 data set, the ambient

temperature is assumed to be = 20 °C.

Ma

teri

al M

ixS

peci

fica

ton

Thic

kness

es

Co

eff

icie

nt

of

Lin

ea

r E

xpa

nsi

on

Yo

ung

s M

odulu

s R

efe

rence

Inva

r 36

B

388

-06

20

060

.254

mm

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Alu

min

ium

EN

AW

10

50A

H1

40

.254

mm

23.

5 x

10

-6m

/m

K6

9 G

Pa

[9]

Inva

r 36

B

388

-06

20

060

.3m

m

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Nic

kel

Silv

erB

S2

870

NS

103

0.3

mm

1

3 x

10

-6m

/m

K1

18 G

Pa

[10]

Inva

r 36

B

388

-06

20

060

.65m

m

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Bra

ssB

S 2

870

CZ

108

0.6

5mm

1

8.7 x

10

-6m

/m

K1

11G

Pa

[11]

Inva

r 36

B

388

-06

20

060

.8m

m

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Mild

Ste

elB

S E

N1

A0

.8m

m

11

x 10

-6m

/m

K1

95 G

Pa

[12]

Inva

r 36

B

388

-06

20

061

.3m

m

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Co

pper

B

S 2

879

C1

061

.3m

m

16.

6 x

10

-6m

/m

K1

17 G

Pa

[13]

Inva

r 36

B

388

-06

20

060

.4m

m

1.4

5 x

10

-6m

/m

K1

37 -

145

Gp

a [8

]

Sta

inle

ss S

teel

AIS

I B

S 3

04

0.4

mm

1

7.3 x

10

-6m

/m

K2

13G

pa

[14]

Table 1 Simulation set 2

6. Simulation Results

The two formulae of eqn.1 and eqn.10 were

used to generate data values of and R

respectively for both simulation sets. The

radii of curvature for simulation set 1 were

plotted against the change of temperature for

each thickness of bimetallic strip, see Fig.4

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Fig.4 Timoshenko Comparison range 0.4mm to

10mm

For simulation set 2, see Fig.’s 5,6,7,8,9,10.

Fig.5 Timoshenko Comparison Invar vs.

Aluminium 0.508mm thick

Fig.6 Timoshenko Comparison Invar vs. Nickel

silver 0.6mm thick

Fig.7 Timoshenko Comparison Invar vs. Brass

1.3mm thick

Fig.8 Timoshenko Comparison Invar vs. Mild

Steel 1.6mm thick

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Fig.9 Timoshenko Comparison Invar vs. Copper

2.6 mm thick

Fig.10 Timoshenko Comparison Invar vs. St.

Steel 0.8 mm thick

From Fig. 4 it is evident that for the 153 overall

data points from simulation set 1, yielding nine

different thicknesses of the strip, an excellent

correlation between the two formulae has resulted.

From simulation set 2, comparing the six different

materials types and thickness combinations

yielding 66 data points, a very good overall

correlation between the new formula and

Timoshenko was shown to exist.

7. Discussion of results From the 9 data tables generated from simulation 1,

the overall average error was 0.64%.

The break down in average error for each thickness

was as follows:

0.4mm thick = 0.9927 average 0.73%.




2.0 mm thick = 0.9912 average 0.88%.


6.0mm thick = 0.9960 average 0.40%

8.0mm thick = 0.9940 average 0.60%

10mm thick = 0.9940 average 0.60%

From the breakdown of average error it is evident

that the error fluctuates slightly as a function of the

thickness of the strip. The maximum fluctuation of

error lies between the 2mm thick and 6mm thick

test strips, was only 0.48%.

From simulation set 2, the error breakdown was as

follows:

0.508mm thick = 0.899 average 1.01% Invar vs.

Aluminium


Nickel Silver

1.3mm thick = 0.925 average 0.75% Invar vs. Brass

1.6mm thick = 1.158 average1.58% Invar vs. Mild

Steel


Copper

0.8mm thick = 0.948 average 0.52% Invar vs. St.

Steel

The maximum fluctuation of error of the function

was 1.18% which occurred between the 1.6mm and

2.6mm simulation data. It should be noted that the

second test was simultaneously subjecting the

formula to all changes of the data, i.e. different

thicknesses, different Young’s modulus, and

different coefficients of linear expansions.

The average error in simulation set 2 was 0.8% and

the maximum fluctuation error was 1.18%.

From simulation set 1 the average error was

0.622% and the maximum fluctuation error was

0.48%.

The derivation has shown, and the correlation of

the new formula to the Timoshenko formula has

proven, that the values of Young’s Modulus for

each metal within the bimetallic strip are not

required in the evaluation of the new formula. This

is very useful since it is not always quick and easy

to find the Young’s modulus of the metals, and the

value as used in the Timoshenko formula, takes the

average of both Young’s moduli which can only be

an approximation at best. It should also be noted

that this work assumes that Young’s modulus for a

bimetallic strip is the average of the two constitute

metals making up the strip, as per the original

Timoshenko formula.

8. Conclusions The results prove an acceptable overall maximum

error of 1.18%, and an overall average error of

0.64%. Thus it has been demonstrated that the

formula put forward here can be a useful, quick,

easier alternative to Timoshenko’s radius of

curvature formula for close estimates of the radius

of curvature as a function of temperature change.

Furthermore, it has been shown that the new

formula works without the requirement of first

knowing the Young’s moduli of the two metals

within the bimetallic strip. Most usefully, the

formula presented in this work can be evaluated

without the need of an electronic spread sheet or

program as is required with the more complex

Timoshenko’s formula, but can be easily used on a

hand held calculator at a fraction of the time.

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9. References [1] Timoshenko, S., Analysis of Bi-metal

Thermostats. JOSA, 1925. 11(3): p. 233-255.

[2] Krulevitch P, J.G.C., Curvature of a Cantilever

Beam Subjected to an Equi-Biaxial Bending

Moment, in Materials Research Society

Conference. 1998.

[3] Prasad, K., Principle and Properties of

Thermoststat. Journal of Materials, 1993.

[4] Kanthal, Kanthal Thermostatic Bimetal

Handbook. 2008, Box 502, SE-734 27

Hallstahammar, Sweden: Kanthal. 130.

[5] Uhlig, W., et al, Thermostatic Metal,

Manufacture and Application. 2nd, revised ed.

2007, Hammerplatz 1, D-08280,Aue/Sachsen:

Auerhammer Metallwerk GMBH. 198.

[6] Haga, T., Clad strip casting by a twin roll

caster. World Academy of Materials and

Manufacturing Engineering, 2009. 37(2): p.

117-124.

[7] Shivalik,S.Bimetallic

strip supplier. 2013; Available

from:

http://www.shivalikbimetals.com.

Accessed online Sept.2103.

[8 http://www.nickel- alloys.net

/invar_nickel_iron_alloy.html

Physical_properties,

WWW accessed Nov 2013.

[9]http://www.smithmetal.com

/downloads/1050A.pdf,


[10]

http://www.cuivre.org/contenu/docs/doc/pdf/CuNi

Zn/CuNi10Zn27.pdf


[11]http://www.copperinfo.co.uk/alloys/brass,


[12]http://www.matweb.com/ mild steel,


[13]

http://www.aalco.co.uk/datasheets/Copper~Brass~

Bronze_CW024A-C106_122.ashx


[14]Munday. A,J, and Farrar .R.A An Engineering

Data book , MacMillan, London,

ISBN 0 333 25829 0

11. Appendix Data tables Simulation set 1

m m

Tc(°C) Th(°C) ΔT Rt Rg Rt/Rg

20 30 10 1.525 1.535 0.993

20 40 20 0.763 0.767 0.995

20 50 30 0.508 0.512 0.992

20 60 40 0.381 0.383 0.995

20 90 70 0.218 0.219 0.995

20 120 100 0.152 0.153 0.993

20 150 130 0.117 0.118 0.992

20 180 160 0.095 0.096 0.993

20 210 190 0.080 0.081 0.994

20 240 220 0.069 0.070 0.986

20 270 250 0.061 0.061 0.993

20 300 280 0.054 0.055 0.982

20 330 310 0.049 0.050 0.994

20 370 350 0.044 0.044 0.995

20 400 380 0.040 0.040 0.993

20 430 410 0.037 0.037 0.995

20 470 450 0.034 0.034 0.997

0.993

t2+t1=0.4mm

m m


20 30 10 3.049 3.069 0.993

20 40 20 1.525 1.534 0.994

20 50 30 1.016 1.023 0.993

20 60 40 0.762 0.767 0.993

20 90 70 0.436 0.438 0.995

20 120 100 0.305 0.306 0.997

20 150 130 0.234 0.236 0.992

20 180 160 0.191 0.192 0.995

20 210 190 0.160 0.161 0.994

20 240 220 0.139 0.139 1.000

20 270 250 0.122 0.123 0.992

20 300 280 0.109 0.109 1.000

20 330 310 0.099 0.099 1.000

20 370 350 0.087 0.088 0.989

20 400 380 0.080 0.081 0.988

20 430 410 0.074 0.075 0.987

20 470 450 0.068 0.068 0.994

0.994

t2+t1=0.8mm

m m


20 30 10 4.575 4.604 0.994

20 40 20 2.287 2.302 0.993

20 50 30 1.525 1.535 0.993

20 60 40 1.144 1.151 0.994

20 90 70 0.653 0.657 0.994

20 120 100 0.457 0.460 0.993

20 150 130 0.352 0.354 0.994

20 180 160 0.286 0.288 0.993

20 210 190 0.241 0.242 0.996

20 240 220 0.208 0.209 0.995

20 270 250 0.183 0.184 0.995

20 300 280 0.163 0.164 0.994

20 330 310 0.147 0.148 0.993

20 370 350 0.131 0.131 1

20 400 380 0.120 0.121 0.992

20 430 410 0.112 0.112 1

20 470 450 0.102 0.102 1

0.995

t2+t1=1.2mm

m m


20 30 10 6.100 6.140 0.993

20 40 20 3.050 3.070 0.993

20 50 30 2.033 2.050 0.992

20 60 40 1.525 1.534 0.994

20 90 70 0.871 0.877 0.993

20 120 100 0.610 0.614 0.993

20 150 130 0.469 0.472 0.994

20 180 160 0.381 0.384 0.992

20 210 190 0.321 0.323 0.994

20 240 220 0.277 0.279 0.993

20 270 250 0.244 0.245 0.996

20 300 280 0.218 0.219 0.995

20 330 310 0.197 0.198 0.995

20 370 350 0.174 0.175 0.994

20 400 380 0.160 0.161 0.994

20 430 410 0.149 0.150 0.993

20 470 450 0.135 0.136 0.993

0.994

t2+t1=1.6mm

m m


20 30 10 7.625 7.674 0.994

20 40 20 3.812 3.837 0.993

20 50 30 2.542 2.558 0.994

20 60 40 1.906 1.918 0.994

20 90 70 1.089 1.096 0.994

20 120 100 0.762 0.767 0.993

20 150 130 0.586 0.590 0.993

20 180 160 0.476 0.479 0.994

20 210 190 0.401 0.404 0.993

20 240 220 0.346 0.348 0.994

20 270 250 0.305 0.307 0.993

20 300 280 0.272 0.274 0.993

20 330 310 0.246 0.247 0.996

20 370 350 0.218 0.219 0.995

20 400 380 0.201 0.212 0.948

20 430 410 0.186 0.187 0.995

20 470 450 0.169 0.170 0.994

0.991

t2+t1=2.0mm

m m


20 30 10 15.250 15.350 0.993

20 40 20 7.625 7.674 0.994

20 50 30 5.083 5.116 0.994

20 60 40 3.812 3.840 0.993

20 90 70 2.178 2.192 0.994

20 120 100 1.525 1.539 0.991

20 150 130 1.173 1.181 0.993

20 180 160 0.953 0.959 0.994

20 210 190 0.803 0.807 0.995

20 240 220 0.693 0.697 0.994

20 270 250 0.610 0.614 0.993

20 300 280 0.546 0.548 0.996

20 330 310 0.492 0.495 0.994

20 370 350 0.436 0.438 0.995

20 400 380 0.401 0.404 0.993

20 430 410 0.372 0.374 0.995

20 470 450 0.339 0.341 0.994

0.994

t2+t1=4.0mm

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m m

Tc(°C) Th(°C) ΔT ρ R ρ/R

20 30 10 22.875 23.024 0.994

20 40 20 11.437 11.512 0.993

20 50 30 7.625 7.674 0.994

20 60 40 5.718 5.755 0.994

20 90 70 3.268 3.289 0.994

20 120 100 2.287 2.302 0.993

20 150 130 1.759 1.771 0.993

20 180 160 1.429 1.438 0.994

20 210 190 1.204 1.211 0.994

20 240 220 1.039 1.046 0.993

20 270 250 0.915 0.920 0.995

20 300 280 0.817 0.822 0.994

20 330 310 0.738 0.742 0.995

20 370 350 0.653 0.657 0.994

20 400 380 0.602 0.606 0.993

20 430 410 0.577 0.562 1.027

20 470 450 0.508 0.511 0.994

0.996

t2+t1=6.0mm

m m


20 30 10 30.500 30.698 0.994

20 40 20 15.250 15.349 0.994

20 50 30 10.166 10.232 0.994

20 60 40 7.624 7.674 0.993

20 90 70 4.357 4.385 0.994

20 120 100 3.049 3.069 0.993

20 150 130 2.346 2.361 0.994

20 180 160 1.910 1.918 0.996

20 210 190 1.605 1.615 0.994

20 240 220 1.386 1.395 0.994

20 270 250 1.220 1.227 0.994

20 300 280 1.089 1.096 0.994

20 330 310 0.984 0.990 0.994

20 370 350 0.871 0.877 0.993

20 400 380 0.803 0.807 0.995

20 430 410 0.744 0.748 0.995

20 470 450 0.678 0.682 0.994

0.994

t2+t1= 8.0mm

m m


20 30 10 38.125 38.372 0.994

20 40 20 19.062 19.186 0.994

20 50 30 12.708 12.790 0.994

20 60 40 9.531 9.593 0.994

20 90 70 5.446 5.481 0.994

20 120 100 3.812 3.837 0.993

20 150 130 2.932 2.951 0.994

20 180 160 2.383 2.398 0.994

20 210 190 2.006 2.019 0.994

20 240 220 1.733 1.744 0.994

20 270 250 1.525 1.534 0.994

20 300 280 1.362 1.370 0.994

20 330 310 1.229 1.237 0.994

20 370 350 1.089 1.096 0.994

20 400 380 1.003 1.009 0.994

20 430 410 0.929 0.935 0.994

20 470 450 0.847 0.852 0.994

0.994

t2+t1= 10.0mm

Simulation Set 2; 0.5mm Simulation Set 2; 0.6mm

Invar 36 vs. Aluminium Invar 36 vs. Nickel silver

T ρ R ρ/R30 1.594 1.78 0.896

50 0.532 0.593 0.897

75 0.29 0.323 0.898

100 0.199 0.222 0.896

150 0.123 0.136 0.904

200 0.088 0.0989 0.89

250 0.07 0.077 0.909

300 0.057 0.063 0.905

350 0.048 0.0539 0.891

400 0.042 0.047 0.894

450 0.0371 0.041 0.905

0.899

T ρ R ρ/R30 3.472 3.3 1.052

50 1.157 1.1 1.052

75 0.631 0.6 1.052

100 0.434 0.41 1.059

150 0.267 0.252 1.06

200 0.193 0.182 1.06

250 0.151 0.143 1.056

300 0.124 0.12 1.033

350 0.105 0.1 1.05

400 0.091 0.086 1.058

450 0.081 0.076 1.066

1.054


Invar 36 vs. Brass Invar 36 vs. Mild Steel

T ρ R ρ/R30 5.046 5.489 0.919

50 1.68 1.829 0.919

75 0.917 0.998 0.919

100 0.631 0.686 0.92

150 0.4 0.422 0.948

200 0.28 0.304 0.921

250 0.219 0.238 0.92

300 0.18 0.196 0.918

350 0.153 0.166 0.922

400 0.133 0.144 0.924

450 0.12 0.127 0.945

0.925

T ρ R ρ/R30 11.23 9.7 1.158

50 3.743 3.23 1.159

75 2.042 1.76 1.16

100 1.4 1.21 1.157

150 0.864 0.745 1.16

200 0.623 0.538 1.158

250 0.488 0.421 1.159

300 0.401 0.346 1.159

350 0.34 0.294 1.156

400 0.295 0.255 1.157

450 0.261 0.225 1.16

1.158


Invar 36 vs. Copper Invar 36 vs. St. Steel

T ρ R ρ/R30 11.474 11.997 0.956

50 3.825 3.999 0.956

75 2.086 2.181 0.956

100 1.434 1.499 0.957

150 0.883 0.923 0.957

200 0.637 0.666 0.956

250 0.498 0.522 0.954

300 0.409 0.428 0.956

350 0.347 0.363 0.956

400 0.302 0.315 0.959

450 0.267 0.279 0.957

0.956

T ρ R ρ/R30 3.396 3.581 0.948

50 1.132 1.194 0.948

75 0.617 0.651 0.948

100 0.424 0.447 0.949

150 0.261 0.275 0.949

200 0.189 0.199 0.95

250 0.147 0.155 0.948

300 0.121 0.127 0.953

350 0.103 0.108 0.954

400 0.089 0.094 0.947

450 0.078 0.083 0.94

0.948

1319



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