Dimensionally Stable Laminates Under Thermal Loading and Their Applications

Symonov, Volodymyr

doi:10.1051/matecconf/201930401001

Dimensionally Stable Laminates Under Thermal Loading and Their Applications

Symonov, Volodymyr

Published: 2019-12-01
Publisher: EDP Sciences
DOI: 10.1051/matecconf/201930401001
Repository: dspace.vutbr.cz
Language: en

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.

Full text

* 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: � �� =�� , � � �� =�� , � �� =�� , � ( 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 � �, � �� =� � �� , � �� =� � �� , � � �� =�� , ( 14 ) whe e С � =� � �� ,С � =� � �� ,� � � =�� ,� � � =�� 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: � �� =�� , � � �� =�� , � �� =�� , � ( 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 � �, � �� =� � �� , � �� =� � �� , � � �� =�� , ( 14 ) whe e С � =� � �� ,С � =� � �� ,� � � =�� ,� � � =�� 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