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Dimensionally Stable Laminates Under Thermal Loading and Their Applications

Abstract

There exist many types of structures, which are required to have stable dimensions within a wide range of temperatures. The specific nature of composites allows finding special conditions when a laminate stacking sequence can provide zero thermal expansion coefficients in one or more directions. This allows the structure being designed to have the same dimensions in a wide range of temperatures. This work is aimed to find mathematical conditions, which guarantee in-plane zero CTE at least in one direction. As an application of thermally stable laminates a rotating disk is chosen. The mathematical model for such a disk is presented. Among investigated materials there was not found any of them, which can be used to layup a laminate with zero CTEs in two directions. However, all investigated materials can be used to layup many laminates with zero CTE in one or another direction. Moreover, it was discovered a laminate might have a zero CTE, if the lamina has zero or negative CTE at least in one direction. It was found the stresses, which appear in a laminated disk caused by centripetal forces, are insignificantly low in comparison to the thermal ones within the investigated ranges of angular velocity and temperature.

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Dimensionally Stable Laminates Under Thermal Loading and Their Applications

Author: Symonov, Volodymyr
Publisher: EDP Sciences
Year: 2019
DOI: 10.1051/matecconf/201930401001
Source: https://dspace.vut.cz/bitstreams/b0dc66f4-d147-4eac-8952-72248b473dd8/download
*
Co esponding au ho : symono[email p o ec ed]
Dimensionally s able lamina es unde he mal
loading and hei applica ions
Volodymy Symono *
B no Uni e si y o Technology, Ins i u e o Ae ospace Enginee ing, Ai c a design and es g oup,
616 69 B no, Technická 2896/2, Czech Republic
Abs ac . The e exis many ypes o s uc u es, which a e equi ed o ha e
s able dimensions wi hin a wide ange o empe a u es. The speci ic na u e
o composi es allows inding special condi ions when a lamina e s acking
sequence can p o ide ze o he mal expansion coe icien s in one o mo e
di ec ions. This allows he s uc u e being designed o ha e he same
dimensions in a wide ange o empe a u es. This wo k is aimed o ind
ma hema ical condi ions, which gua an ee in-plane ze o CTE a leas in one
di ec ion. As an applica ion o he mally s able lamina es a o a ing disk is
chosen. The ma hema ical model o such a disk is p esen ed. Among
in es iga ed ma e ials he e was no ound any o hem, which can be used
o layup a lamina e wi h ze o CTEs in wo di ec ions. Howe e , all
in es iga ed ma e ials can be used o layup many lamina es wi h ze o CTE
in one o ano he di ec ion. Mo eo e , i was disco e ed a lamina e migh
ha e a ze o CTE, i he lamina has ze o o nega i e CTE a leas in one
di ec ion. I was ound he s esses, which appea in a lamina ed disk caused
by cen ipe al o ces, a e insigni ican ly low in compa ison o he he mal
ones wi hin he in es iga ed anges o angula eloci y and empe a u e.
1 In oduc ion
Many s uc u al applica ions o lamina ed FRP ma e ials a e connec ed wi h he mal
loading, whe e special equi emen s a e applied o he s uc u e such as s abili y o i s
dimensions and/o shape wi hin a ce ain ange o empe a u es. Mainly, hese a e he space
applica ions, whe e he a ia ion o geome y and/o shape o a special s uc u e has s ic
limi a ions du ing i s ope a ion in o de o keep hei main cha ac e is ics. The examples o
such s uc u es can be space an ennas, elescopes, ins umen al pla o ms, e c. [1], [2], [3].
Howe e , he g ound applica ions a e possible also. Fo example, he p essu ized essels and
pipelines wi h a hea ed medium inside a e discussed in [4].
Ano he possible u u e applica ion could be a base-wheel and blades o a gas u bine
engine, which a e exposed o high empe a u es and whe e he blade ip clea ance ep esen
a sou ce o la ge loss in a u bine [5]. The less he gap he less loss. F om he o he hand, he
he mal expansion o he wheel and blades is one o he easons why i canno be designed
as small as possible o all engine modes.
The main eason o he lamina ed FRPs o be applied in abo e no ed s uc u es is an
abili y o con ol hei p ope ies ia s acking sequence a ia ion.
© The Au ho s, published by EDP Sciences. This is an open access a icle dis ibu ed unde he e ms o he C ea i e Commons
A ibu ion License 4.0 (h p://c ea i ecommons.o g/licenses/by/4.0/).
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
The ca bon-epoxy symme ically balanced angle-ply lamina es wi h almos null CTE in
one di ec ion i s ly we e ound in [6], whe e he laminas’ angles we e app oxima ely ±42°.
I was shown expe imen ally, he lamina CTEs a e signi ican ly changing wi hin wide
empe a u e ange. The e o e, he ound lamina e has ze o CTE only in na ow ange close
o oom empe a u e.
Fu he T.Ishikawa e al. [7] p oposed a echnique o designing lamina es, which ha e
almos ze o CTE in one di ec ion in a wide empe a u e ange. They used he moelas ic and
in-plane s i ness in a ian s while inding he ela ion be ween he lamina he mal and
mechanical p ope ies and he lamina e CTE in one di ec ion. The echnique is e y good and
simple o gene al case, when any ype o lamina e (o ho opic, aniso opic, e c.) is being
looked o . Howe e , he au ho s inally ansi om he gene al case o he symme ical
balanced lamina es like [0°, ±θ]s. In such a case i seems he ela ions and ze o CTE
lamina es could be ound in a mo e simple and e iden way.
2 Ze o CTE lamina es
2.1 Fo mula ion o he p oblem
The dimensionally s able lamina e exposed o he mal loading ( he mos able) is exp essed
by he nex sys em o equa ions:
��
�
=0,
�
�
=0,
(
1
)
whe e �
�
and �
�
– CTEs in he main di ec ions o lamina e’s o ho opy (see Fig. 1).
Fig. 1. Global lamina e and local lamina coo dina e sys ems.
I is equi ed o de elop a echnique, which will help inding he mos able lamina e(s) a leas
in one global di ec ion wi hin a na ow empe a u e ange close o oom empe a u e. The
lamina es being looked o a e limi ed o he balanced symme ical ones o [0°, 90°, ±φ]
ns
class, whe e φ ϵ (0°, 90°).
2.2 Gene al he mo-mechanical equa ions
The s esses in k- h lamina caused by mechanical and he mal loading a e gi en by [8]:
��
�
�
�
�
�
�
�
=���
�
���
�
�
�
�
�
�
�
���
�
�
�
0�
�
���,
(
2
)
whe e ���
�
– s i ness ma ix o he k- h lamina; �
�
,�
�
,�
�
- k- h lamina s ains in local
coo dina e sys em; �
�
,�
�
– CTEs o he k- h lamina in local coo dina e sys em.
x
y
2
θ
k
1
Hea ΔT
2
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
The ca bon-epoxy symme ically balanced angle-ply lamina es wi h almos null CTE in
one di ec ion i s ly we e ound in [6], whe e he laminas’ angles we e app oxima ely ±42°.
I was shown expe imen ally, he lamina CTEs a e signi ican ly changing wi hin wide
empe a u e ange. The e o e, he ound lamina e has ze o CTE only in na ow ange close
o oom empe a u e.
Fu he T.Ishikawa e al. [7] p oposed a echnique o designing lamina es, which ha e
almos ze o CTE in one di ec ion in a wide empe a u e ange. They used he moelas ic and
in-plane s i ness in a ian s while inding he ela ion be ween he lamina he mal and
mechanical p ope ies and he lamina e CTE in one di ec ion. The echnique is e y good and
simple o gene al case, when any ype o lamina e (o ho opic, aniso opic, e c.) is being
looked o . Howe e , he au ho s inally ansi om he gene al case o he symme ical
balanced lamina es like [0°, ±θ]s. In such a case i seems he ela ions and ze o CTE
lamina es could be ound in a mo e simple and e iden way.
2 Ze o CTE lamina es
2.1 Fo mula ion o he p oblem
The dimensionally s able lamina e exposed o he mal loading ( he mos able) is exp essed
by he nex sys em o equa ions:
��
�
=0,
�
�
=0,
(
1
)
whe e �
�
and �
�
– CTEs in he main di ec ions o lamina e’s o ho opy (see Fig. 1).
Fig. 1. Global lamina e and local lamina coo dina e sys ems.
I is equi ed o de elop a echnique, which will help inding he mos able lamina e(s) a leas
in one global di ec ion wi hin a na ow empe a u e ange close o oom empe a u e. The
lamina es being looked o a e limi ed o he balanced symme ical ones o [0°, 90°, ±φ]
ns
class, whe e φ ϵ (0°, 90°).
2.2 Gene al he mo-mechanical equa ions
The s esses in k- h lamina caused by mechanical and he mal loading a e gi en by [8]:
��
�
�
�
�
�
�
�
=���
�
���
�
�
�
�
�
�
�
���
�
�
�
0�
�
���,
(
2
)
whe e ���
�
– s i ness ma ix o he k- h lamina; �
�
,�
�
,�
�
- k- h lamina s ains in local
coo dina e sys em; �
�
,�
�
– CTEs o he k- h lamina in local coo dina e sys em.
x
y
2
θ
k
1
Hea ΔT
F om o he side, he lamina e s ains in lamina e’s XOY coo dina e sys em a e:
��
�
�
�
�
��
�=��
�
�
�
�
��
����
(
3
)
Using he o a ing ma ix o he k- h lamina ���
�
, we can ob ain lamina s ains om (3):
��
�
�
�
�
��
�
�
=����
��
�
��
��
�
�
�
�
��
����
(
4
)
Finally, by subs i u ing (4) o (2) and o a ing s esses in k- h lamina om local
coo dina e sys em 12 o lamina e’s XOY coo dina e sys em we will ob ain:
�
�
�
�
�
�
��
�
�
=
�
�
�
���
�
�
�
�
��
�
�
�
��
�
��
�
�
�
�
�
�
��
�
�
��
�
�
�
�
0
�
�
���� (
5
)
In eg a ion o (5) h ough he lamina e hickness will gi e he nex equa ion:
��
�
�
�
�
��
�=�� ���
��
�
��
�
�
�
���
��
�
�
�
�
��
����
�
���
��� ���
��
�
��
�
�
�
0�
�
��
�
�
�
���
���
�
���
,
(
6
)
whe e �
�
,�
�
,�
��
– he in-plane in e nal no mal and shea o ces, which appea in he
lamina e; N – he numbe o laminas in he lamina e; �
�
– he dis ance om he middle plane
o he lamina e o he lowe su ace o he k- h lamina; ��
��
�
=���
���
���
�
����
��
�
��
, ��
��
�
=
���
���
���
�
.
This equa ion can be w i en in he ma ix o m:
���=�����
��
������
�
�
,
(
7
)
whe e �
��
=����
��
�
�
��
�
��
���
�
�
���
, �
��
=����
��
�
�
��
�
��
���
�
�
���
.
I he s uc u e is ee o cons ains and mechanical loading he in e nal o ces a e equal
o ze o, he solu ion o he equa ion (7) is he nex :
��
��
�=���
��
��
�
�
�
(
8
)
The equa ion (8) has he nex un olded iew o o ho opic lamina es:
��
�
�
�
�
��
�=
��
����
��
��
�
��
��
��
�
0
��
��
��
�
�
��
�
0
0 0 �
��
���
����
�
��
��
�
��
0�� o
�
�
=�
��
�
��
�
��
��
�
��
��
�
,
�
�
�
=�
��
�
��
��
�
��
��
�
��
�
,
�
�
��
=0,
(
9
)
whe e H – is he o al hickness o he lamina e; �
�
,�
�
,�
��
,�
��
,�
��
– a e he elas ic moduli,
Poisson’s coe icien s and he in-plane shea modulus o he lamina e, espec i ely.
Finally, he he mos abili y condi ions o o ho opic ma e ials will ha e he nex shape:
3
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
�
��
�
��
�
��
��
�
��
��
�
=0,
�
��
�
��
��
�
��
��
�
��
�
=0,
o ��
��
�
��
��
��
�
��
=0,
�
��
�
��
��
��
�
��
=0� (
10
)
2.3 Themos abili y condi ions o [0°, 90°, ±φ]
ns
lamina es
The elemen s �
��
and �
��
in a gene al case can be expanded as ollows:
�
��
=��
�
���
��
���
�
�
�
����
��
�
���
���
�
�
�
���
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���
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�
���
,
�
�
��
=��
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�
����
��
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���
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�
�
�
���
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�
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�
�
��
���
���
�
��
�
�,
�
���
�
�
��
=��
�
���
��
���
��
����
�
��
�
���
��
�
���
����
�
�
�
����
�
�
�
���
���
���
�
��
�
�
�
���
,
�
(
11
)
and
�
��
=��
�
��
��
��
��
����
�
�
�
��
���
���
�
�
�
���
��
��
��
����
�
�
�
��
���
���
�
�
�
��
�
���
,
�
�
��
=��
�
��
��
��
��
����
�
�
�
��
���
���
�
�
�
���
��
��
��
����
�
�
�
��
���
���
�
�
�
���
�
���
(
12
)
whe e �
�
– he hickness o he k- h lamina; ��
��
=
�
��
���
���
�
���
,�=�,��
Howe e , o [0, 90°, ±φ]
ns
lamina es i can be signi ican ly simpli ied.
Le ’s in oduce he ela i e hicknesses o laminas g oup wi h he same o ien a ion:
�
�
=�
��
�,�
�
=�
���
�,
(
13
)
whe e �
��
,�
���
– he o al hicknesses o laminas g oup wi h o ien a ion o 0 and 90°,
espec i ely.
Thus, di iding he ela ionships (11) and (12) by he o al hickness o he lamina e and
a e some ma hema ical ans o ma ions we will ob ain:
�
��
=�
�
С
�
��
�
С
�
�����
�
��
�
���С
�
�С
�
����
�
��С
�
�,
�
��
=�
�
С
�
��
�
С
�
�����
�
��
�
���С
�
�С
�
����
�
��С
�
�,
�
��
=�
�
��
�
��
�
��
�
�����
�
��
�
����
�
���
�
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�
����
�
��
�
����
�
��,
�
��
=�
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��
�
�����
�
��
�
����
�
���
�
���
�
����
�
��
�
����
�
��,
�
�
��
=��
�
��
�
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�
�
��
�����
�
��
�
����
�
�
��
���
�
��
�
�����
�
�����
�
���,
(
14
)
whe e С
�
=�
�
��
�
��
�
��
�
�
��
,С
�
=�
�
��
�
��
�
��
�
�
��
,� �
�
=��
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�
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�
���
��
,�
�
�
=��
�
�
��
���
�
���
��
�
A e subs i u ion (14) o (10) and ma hema ical ans o ma ions, he he mos abili y
condi ions (10) ake he shape o wo quad a ic equa ions:
4
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
�
��
�
��
�
��
��
�
��
��
�
=0,
�
��
�
��
��
�
��
��
�
��
�
=0,
o ��
��
�
��
��
��
�
��
=0,
�
��
�
��
��
��
�
��
=0� (
10
)
2.3 Themos abili y condi ions o [0°, 90°, ±φ]
ns
lamina es
The elemen s �
��
and �
��
in a gene al case can be expanded as ollows:
�
��
=��
�
���
��
���
�
�
�
����
��
�
���
���
�
�
�
���
�
�
�
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���
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�
�
�
���
,
�
�
��
=��
�
���
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���
�
�
�
����
��
�
���
���
�
�
�
���
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�
�
���
��
���
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�
�
��
���
���
�
��
�
�,
�
���
�
�
��
=��
�
���
��
���
��
����
�
��
�
���
��
�
���
����
�
�
�
����
�
�
�
���
���
���
�
��
�
�
�
���
,
�
(
11
)
and
�
��
=��
�
��
��
��
��
����
�
�
�
��
���
���
�
�
�
���
��
��
��
����
�
�
�
��
���
���
�
�
�
��
�
���
,
�
�
��
=��
�
��
��
��
��
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�
�
�
��
���
���
�
�
�
���
��
��
��
����
�
�
�
��
���
���
�
�
�
���
�
���
(
12
)
whe e �
�
– he hickness o he k- h lamina; ��
��
=
�
��
���
���
�
���
,�=�,��
Howe e , o [0, 90°, ±φ]
ns
lamina es i can be signi ican ly simpli ied.
Le ’s in oduce he ela i e hicknesses o laminas g oup wi h he same o ien a ion:
�
�
=�
��
�,�
�
=�
���
�,
(
13
)
whe e �
��
,�
���
– he o al hicknesses o laminas g oup wi h o ien a ion o 0 and 90°,
espec i ely.
Thus, di iding he ela ionships (11) and (12) by he o al hickness o he lamina e and
a e some ma hema ical ans o ma ions we will ob ain:
�
��
=�
�
С
�
��
�
С
�
�����
�
��
�
���С
�
�С
�
����
�
��С
�
�,
�
��
=�
�
С
�
��
�
С
�
�����
�
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�
���С
�
�С
�
����
�
��С
�
�,
�
��
=�
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�
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=�
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��,
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=��
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�����
�
��
�
����
�
�
��
���
�
��
�
�����
�
�����
�
���,
(
14
)
whe e С
�
=�
�
��
�
��
�
��
�
�
��
,С
�
=�
�
��
�
��
�
��
�
�
��
,� �
�
=��
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�
���
��
,�
�
�
=��
�
�
��
���
�
���
��
�
A e subs i u ion (14) o (10) and ma hema ical ans o ma ions, he he mos abili y
condi ions (10) ake he shape o wo quad a ic equa ions:
����
�
������
��
�������������
��
���
��
=0,
����
�
������
��
�������������
��
���
��
=0,
(
15
)
whe e �=����
�
��
�
����С
�
�С
�
������
�
��
�
��С
�
�С
�
����
�
���
�
��,�
��
=���
�
�
��
�
���С
�
�С
�
����
�
���
�
���
�
����
�
���
�
�
��
���С
�
�С
�
�����
�
�
��
���
��
�����
�
�
�
����
�
���
�
��
�
�,��
��
=��
�
��
�
��
�
���
�
���
�
���
�
���
�
����
�
����
�
��
�
��
�
���
�
���
�
�
��
.
These equa ions can be easily sol ed apa o as a sys em. Thei oo s gi e he angle �
in dependence o he mechanical and he mal p ope ies o he lamina and o he ela i e
hicknesses �
�
and �
�
. The i s equa ion gi es 3 pa ame e s �
�
,�
�
and � a which he
lamina e has ze o CTE in OX di ec ion. The second equa ion gi es hese pa ame e s o he
case, when he lamina e has ze o CTE in OY di ec ion.
2.4 Theo e ical esul s
Many di e en ma e ials we e in es iga ed o abili y o build ze o CTE lamina es. The e
was no ound any ma e ial, which can be used o layup a lamina e wi h ze o CTEs in wo
di ec ions. Howe e , many in es iga ed ma e ials can be used o layup many lamina es wi h
ze o CTE in one o ano he di ec ion. The Fig. 2 shows dependencies �=���
�
,�
�
� o ze o
CTE ca bon-epoxy lamina es, whe e he lamina p ope ies a e gi en in he Table 1.
a) b)
Fig. 2. Dependencies
�=���
�
,�
�
�
o lamina es wi h: a)
�
�
=0
, b)
�
�
=0
.
Table 1. P ope ies o a unidi ec ional ca bon-epoxy lamina.
Ρ, kg/m
3
E
1
,
GPa E
2
, GPa G
12
, GPa μ
12
α
1
, 1/K α
2
, 1/K H, mm
1400 150 8 4 0.3 -2 40 0.12
3 Design o he mos able disks
3.1 Theo e ical equa ions
Le us ake an ideal case, when he sha does no in luence he disk and he s esses a e
no a ying h ough he hickness o he disk. The main s esses a e induced by cen ipe al
o ces and empe a u e loading. Le us assume he empe a u e ield is cons an wi hin whole

2 :
0 0.25 0.39
0.10 0.30 0.40
0.15 0.35
0.20 0.37

1
φ

1
φ

2 :
0 0.45 0.59
0.15 0.50 0.599
0.25 0.55
0.35 0.57
5
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019

disk, he e o e he he mal s esses a e cons an e e ywhe e also. The cen ipe al o ces
depend on he dis ance om he o a ion axis only. The e o e, he s esses, which appea in
he disk a e he unc ion o ha dis ance also.
The small sec o elemen o he disk wi h he dimensions o ����� (see Fig. 3a) is
loaded on he edges and wi h he olume ic cen ipe al o ces, which can be summa ized o
he esul an o ce:
��=�
����
�
��
�
����,
(
16
)
whe e
�
�
– he speci ic weigh o he ma e ial; H – he hickness o he disk.
The s esses’ equilib ium in he disk can be exp essed by he nex di e en ial equa ion:
���
�
����
�
��
�
��
��
�
�
�
=0�
(
17
)
The adial and ci cum e en ial s esses can be de ined h ough he co esponding s ains:
�
�
=��
�
��
�
��
�
�
��
�����
�
��
�
�
��
��,
�
�
=��
�
��
�
��
�
�
��
�����
�
��
�
�
��
��,
(
18
)
whe e ��
����
=
�
����
���
��
�
��
; �
�
,�
�
– adial and ci cum e en ial elas ic moduli o he lamina e;
�
�
, �
�
– adial and ci cum e en ial CTEs o he lamina e; �
��
,�
��
– Poisson’s coe icien s
o he lamina e.
The adial and ci cum e en ial s ains can be ound easily:
�
�
=��
��,�
�
=��,
(
19
)
whe e � – is he adial displacemen s wi hin he disk (see Fig. 3b).
A e subs i u ing (19) in o (18) and he esul s in o (17) and ma hema ical
ans o ma ions he nex di e en ial equa ion is ob ained:
��
�
�
�
�
������
�����
�
�
=��
���
�
��
�
�
��
����
�
��
�
�
��
�����
�
�
��
�
�,
(
20
)
whe e �=
��
�
��
�
�
The solu ion - he adial displacemen s in dependence o he adius is he nex :
����=�
��
������
���
�
���
�
����
�
�
��
�
�������
�
�
�
√�
��
���
�
�
�
�√�
,
(
21
)
whe e �
�
– he ou e adius o he disk; �
�
=�
�
��
�
�
��
; �
�
=�
�
��
�
�
��
; ��, �
� –
in eg a ion cons an s, which will be ound la e om he bounda y condi ions.
Finally, he adial and ci cum e en ial s esses can be ound by subs i u ion (21) in o (19)
and he esul in o (18):
�
�
=�
��
��
��
,
�
�
�
=�
��
��
��
,
(
22
)
6
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
disk, he e o e he he mal s esses a e cons an e e ywhe e also. The cen ipe al o ces
depend on he dis ance om he o a ion axis only. The e o e, he s esses, which appea in
he disk a e he unc ion o ha dis ance also.
The small sec o elemen o he disk wi h he dimensions o ����� (see Fig. 3a) is
loaded on he edges and wi h he olume ic cen ipe al o ces, which can be summa ized o
he esul an o ce:
��=�
����
�
��
�
����,
(
16
)
whe e
�
�
– he speci ic weigh o he ma e ial; H – he hickness o he disk.
The s esses’ equilib ium in he disk can be exp essed by he nex di e en ial equa ion:
���
�
����
�
��
�
��
��
�
�
�
=0�
(
17
)
The adial and ci cum e en ial s esses can be de ined h ough he co esponding s ains:
�
�
=��
�
��
�
��
�
�
��
�����
�
��
�
�
��
��,
�
�
=��
�
��
�
��
�
�
��
�����
�
��
�
�
��
��,
(
18
)
whe e ��
����
=
�
����
���
��
�
��
; �
�
,�
�
– adial and ci cum e en ial elas ic moduli o he lamina e;
�
�
, �
�
– adial and ci cum e en ial CTEs o he lamina e; �
��
,�
��
– Poisson’s coe icien s
o he lamina e.
The adial and ci cum e en ial s ains can be ound easily:
�
�
=��
��,�
�
=��,
(
19
)
whe e � – is he adial displacemen s wi hin he disk (see Fig. 3b).
A e subs i u ing (19) in o (18) and he esul s in o (17) and ma hema ical
ans o ma ions he nex di e en ial equa ion is ob ained:
��
�
�
�
�
������
�����
�
�
=��
���
�
��
�
�
��
����
�
��
�
�
��
�����
�
�
��
�
�,
(
20
)
whe e �=
��
�
��
�
�
The solu ion - he adial displacemen s in dependence o he adius is he nex :
����=�
��
������
���
�
���
�
����
�
�
��
�
�������
�
�
�
√�
��
���
�
�
�
�√�
,
(
21
)
whe e �
�
– he ou e adius o he disk; �
�
=�
�
��
�
�
��
; �
�
=�
�
��
�
�
��
; ��, �
� –
in eg a ion cons an s, which will be ound la e om he bounda y condi ions.
Finally, he adial and ci cum e en ial s esses can be ound by subs i u ion (21) in o (19)
and he esul in o (18):
�
�
=�
��
��
��
,
�
�
�
=�
��
��
��
,
(
22
)
whe e �
��
,�
��
- adial and ci cum e en ial he mal s esses, and �
��
,�
��
- adial and
ci cum e en ial s esses, caused by cen ipe al o ces, which can be calcula ed as ollows:
�
��
=��
�
���� �
���
��
�
�
�
�
��
�
���
�
���
��
�����
�
���
�
��
��
�√���
���√�
�
�√�
�
�
�
��
��
�√��√��
���√�
�
��√�
�,
�
��
=��
�
���� �
�����
�
��
�
��
�
���
�
�����
��
���
�
���
�
���√��
��
��
���√�
�
�√�
�
�
���√��
��
�√��
���√�
�
��√�
�,
�
��
=��
�
��
�
��
��
�√���
���√�
�
�√�
��
�
��
��
�√��√��
���√�
�
��√�
��
�
��
��
����,
�
��
=��
�
��
�
���√��
��
��
���√�
�
�√�
��
�
���√��
��
�√��
���√�
�
��√�
��
�
��
��
����,
(
23
)
whe e
�
�
=��� �
�
����
���√�
�
��
��
�√������
��√�
���
������
��√�
���
���√�
��
�
���
�
���
��
�����
�
�,
�
�
�
=��� �
�
��
���√�
��
��√�
�
��
��
�√������
��√�
�√���
������
��√�
���
���√�
��
�
���
�
���
��
�����
�
�,
�
�
=
�

����
���√�
��
�
��
��
���
��
��
�√������
�√�
�
, �
�
=
�

��
���√�
��
�√�
��
�
��
��
���
��
��
�√������
�√�
�
� – a e he in eg a ion
cons an s; �
�
=
��

�

���
�
�����
;��=
�

�
�
; �
�
– he inne adius o he disk.
a) b)
Fig. 3. Equilib ium o a disk elemen : a) s esses, which appea in he disk, b) displacemen s,
which appea in he disk.
3.2 An example o s esses calcula ion
The pa ame e s o he case being calcula ed a e shown in he Table 2.
dθ
d
σ
σθ
σ + dσ
θ
H
σθ
ΔT
d
u
u+du
dθ
7
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
The lamina ma e ial has he p ope ies, which a e shown in he Table 1 abo e. Two
s acking sequences [0°, 90°]
ns
a e in es iga ed. The i s one has 60% o 0° laye s and �
�
=
0. The second one has 40% o 0° laye s and �
�
=0.
Table 2. The pa ame e s o he case.
1
, m
2
, m ω, ps ΔT, °C
0.050 0.250 333.33 20
The dis ibu ion o adial and ci cum e en ial he mal, cen ipe al and o al s esses along
he disk adius a e shown in he Fig. 4 - Fig. 5.
a) b)
Fig. 4. S esses o he i s s acking sequence, 0° - 60%,
�
�
=0
: a) adial, b) ci cum e en ial.
a) b)
Fig. 5. S esses o he second s acking sequence, 0° - 40%,
�
�
=0
: a) adial, b)
ci cum e en ial.
The local lamina s esses in 1 and 2 di ec ions o he local coo dina e sys em o bo h
sequences a e shown below in he Fig. 6.
σ, Pa
σ
σ T
σ ω
, m σ, Pa
, m
, m
σ, Pa
σ
σ T
σ ω
σ, Pa
σθ
σθT
σθω
σθ
σθT
σθω
, m
8
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019
The lamina ma e ial has he p ope ies, which a e shown in he Table 1 abo e. Two
s acking sequences [0°, 90°]
ns
a e in es iga ed. The i s one has 60% o 0° laye s and �
�
=
0. The second one has 40% o 0° laye s and �
�
=0.
Table 2. The pa ame e s o he case.
1
, m
2
, m ω, ps ΔT, °C
0.050 0.250 333.33 20
The dis ibu ion o adial and ci cum e en ial he mal, cen ipe al and o al s esses along
he disk adius a e shown in he Fig. 4 - Fig. 5.
a) b)
Fig. 4. S esses o he i s s acking sequence, 0° - 60%,
�
�
=0
: a) adial, b) ci cum e en ial.
a) b)
Fig. 5. S esses o he second s acking sequence, 0° - 40%,
�
�
=0
: a) adial
, b)
ci cum e en ial.
The local lamina s esses in 1 and 2 di ec ions o he local coo dina e sys em o bo h
sequences a e shown below in he Fig. 6.
σ, Pa
σ
σ T
σ ω
, m σ, Pa
, m
, m
σ, Pa
σ
σ T
σ ω
σ, Pa
σθ
σθT
σθω
σθ
σθT
σθω
, m
a) b)
Fig. 6. Lamina s esses: a) o he i s s acking sequence, 0° - 60%,
�
�
=0,���
o he
second s acking sequence, 0° - 40%,
�
�
=0
.
The adial and ci cum e en ial s esses induced by hea in ela ion o he o al s esses o
hese wo sequences, a e shown below in he Fig. 7 and Fig. 8.
a) b)
Fig. 7. The mal s ess in ela ion o he o al s ess o he i s s acking sequence, 0° - 6
0%,
�
�
=0
: a) in adial di ec ion, b) in ci cum e en ial di ec ion.
a) b)
Fig. 8. The mal s ess in ela ion o he o al s ess o he second s acking sequence, 0° -
40%,
�
�
=0
: a) in adial di ec ion, b) in ci cum e en ial di ec ion.
, m
σ, Pa
Laye s 0°:
σ1
σ2
Laye s 90°:
σ1
σ2
, m
σ, Pa
Laye s 0°:
σ1
σ2
Laye s 90°:
σ1
σ2
ω, ps
�
��
�
�
ΔT = 20°C
ΔT = 40°C
ΔT = 60°C
ΔT = 80°C
ω, ps
�
��
�
�
ΔT = 20°C
ΔT = 40°C
ΔT = 60°C
ΔT = 80°C
ω, ps
�
��
�
�
ΔT = 20°C
ΔT = 40°C
ΔT = 60°C
ΔT = 80°C
ω, ps
�
��
�
�
ΔT = 20°C
ΔT = 40°C
ΔT = 60°C
ΔT = 80°C
9
MATEC Web o Con e ences 304, 01001 (2019) h ps://doi.o g/10.1051/ma eccon /201930401001
EASN 2019