Time-dependent modelling of thin poroelastic films drying on deformable plates
Abstract Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at...
Saved in:
Published in | European journal of applied mathematics Vol. 35; no. 1; pp. 62 - 95 |
---|---|
Main Authors | , , |
Format | Journal Article |
Language | English |
Published |
01.02.2024
|
Online Access | Get full text |
Cover
Loading…
Abstract | Abstract
Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from
$t$
to
$t^{1/2}$
, in agreement with experiments. |
---|---|
AbstractList | Abstract
Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to study these stresses is through the cantilever experiment, whereby a thin film is deposited onto the surface of a thin plate that is clamped at one end to a wall. The stresses that are generated in the film during drying are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Föppl-von Kármán (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the Péclet number increases, the time evolution of the plate deflection changes from
$t$
to
$t^{1/2}$
, in agreement with experiments. |
Author | Craster, Richard V. Hennessy, Matthew G. Matar, Omar K. |
Author_xml | – sequence: 1 givenname: Matthew G. orcidid: 0000-0002-5928-6256 surname: Hennessy fullname: Hennessy, Matthew G. – sequence: 2 givenname: Richard V. surname: Craster fullname: Craster, Richard V. – sequence: 3 givenname: Omar K. surname: Matar fullname: Matar, Omar K. |
BookMark | eNplkL1OBCEcxIk5E-9OH8COF1gFFhYozcWv5KKFZ71h4Y9iWNjANvf23qqdU8wUv8kUs0GrlBMgdE3JDSVU3r4RLTqpmWAtOaljZ2hNeacbzplYofWCm4VfoE2tX4TQlki9Ri-HMELjYILkIM14zA5iDOkDZ4_nz5DwlEuGaOocLPYhjhW7cvwpJOzA5zKaIQKeopmhXqJzb2KFq7_coveH-8Puqdm_Pj7v7vaNpZLMjVSKgrIDKGfE4CQbmDfWecu10qCMVFR5bpRmzDAuOiUpePBcDIqpk7dbRH93bcm1FvD9VMJoyrGnpF8O6f8d0n4DUgRWOQ |
Cites_doi | 10.1098/rspa.2010.0039 10.1002/aic.690480304 10.1002/aenm.202102233 10.1115/1.2901526 10.1016/j.colsurfa.2014.11.061 10.1140/epje/i2018-11639-2 10.1073/pnas.1005537107 10.1016/j.jpowsour.2018.04.097 10.1039/D0SM00252F 10.1039/D1SM01343B 10.1063/1.1712886 10.1021/acs.langmuir.2c00397 10.1103/PhysRevApplied.5.044020 10.1111/j.1151-2916.1990.tb05258.x 10.1016/j.cis.2021.102546 10.1103/PhysRevLett.91.224501 10.1039/C4NR02133A 10.1016/j.jpowsour.2016.04.018 10.1021/jp0536807 10.1002/app.1979.070230319 10.1039/D1SM00832C 10.1039/C5NR06266G 10.1103/PhysRevE.105.054602 10.1016/j.jpowsour.2017.01.117 10.1039/C9SM02294E 10.1021/la990256m 10.1016/j.cis.2020.102271 10.1021/acs.langmuir.1c03124 10.1017/jfm.2015.361 10.1023/A:1020886802632 10.1209/epl/i2003-00457-7 10.1039/C8SM02558D |
ContentType | Journal Article |
DBID | AAYXX CITATION |
DOI | 10.1017/S0956792523000062 |
DatabaseName | CrossRef |
DatabaseTitle | CrossRef |
DatabaseTitleList | CrossRef |
DeliveryMethod | fulltext_linktorsrc |
Discipline | Mathematics |
EISSN | 1469-4425 |
EndPage | 95 |
ExternalDocumentID | 10_1017_S0956792523000062 |
GroupedDBID | -1D -1F -2P -2V -E. -~6 -~N .DC .FH 09C 09E 0E1 0R~ 29G 3V. 4.4 5GY 5VS 6~7 74X 74Y 7~V 88I 8FE 8FG 8R4 8R5 9M5 AAAZR AABES AABWE AACJH AAGFV AAKTX AAMNQ AANRG AARAB AASVR AAUIS AAUKB AAYXX ABBXD ABBZL ABEFU ABGDZ ABITZ ABJCF ABJNI ABKKG ABMWE ABQTM ABQWD ABROB ABTCQ ABUWG ABVFV ABVZP ABXAU ABZCX ABZUI ACBMC ACETC ACGFO ACGFS ACGOD ACIMK ACIWK ACMRT ACUIJ ACYZP ACZBM ACZUX ACZWT ADCGK ADDNB ADFEC ADFRT ADKIL ADOVH ADOVT ADVJH AEBAK AEBPU AEHGV AEMTW AENCP AENEX AENGE AEYYC AFFUJ AFKQG AFKRA AFLOS AFLVW AFUTZ AGABE AGBYD AGJUD AGLWM AHQXX AHRGI AI. AIGNW AIHIV AIOIP AISIE AJ7 AJCYY AJPFC AJQAS AKZCZ ALMA_UNASSIGNED_HOLDINGS ALWZO AQJOH ARABE ARAPS ARZZG ATUCA AUXHV AYIQA AZQEC BBLKV BCGOX BENPR BESQT BGHMG BGLVJ BJBOZ BLZWO BMAJL BPHCQ BQFHP C0O CAG CBIIA CCPQU CCQAD CCUQV CDIZJ CFAFE CFBFF CGQII CHEAL CITATION CJCSC COF CS3 CTKSN DC4 DOHLZ DU5 DWQXO EBS EGQIC EJD F5P GNUQQ GROUPED_DOAJ HCIFZ HG- HST HZ~ I.6 I.7 I.9 IH6 IOEEP IOO IS6 I~P J36 J38 J3A JHPGK JQKCU K6V K7- KAFGG KC5 KCGVB KFECR L6V L98 LHUNA LW7 M-V M0N M2P M7S M7~ M8. NIKVX NMFBF NZEOI O9- OYBOY P2P P62 PQQKQ PROAC PTHSS PYCCK Q2X RAMDC RCA RIG ROL RR0 S0W S6- S6U SAAAG T9M UT1 VH1 VOH WFFJZ WQ3 WXU WYP ZDLDU ZJOSE ZMEZD ZYDXJ ~V1 |
ID | FETCH-LOGICAL-c170t-7881e8cbe8da5bd72b2facdfc4989e8a7818f4a8922a2456871efef45b82845b3 |
ISSN | 0956-7925 |
IngestDate | Thu Sep 26 18:22:52 EDT 2024 |
IsDoiOpenAccess | true |
IsOpenAccess | true |
IsPeerReviewed | true |
IsScholarly | true |
Issue | 1 |
Language | English |
LinkModel | OpenURL |
MergedId | FETCHMERGED-LOGICAL-c170t-7881e8cbe8da5bd72b2facdfc4989e8a7818f4a8922a2456871efef45b82845b3 |
ORCID | 0000-0002-5928-6256 |
OpenAccessLink | http://dx.doi.org/10.1017/S0956792523000062 |
PageCount | 34 |
ParticipantIDs | crossref_primary_10_1017_S0956792523000062 |
PublicationCentury | 2000 |
PublicationDate | 2024-02-00 |
PublicationDateYYYYMMDD | 2024-02-01 |
PublicationDate_xml | – month: 02 year: 2024 text: 2024-02-00 |
PublicationDecade | 2020 |
PublicationTitle | European journal of applied mathematics |
PublicationYear | 2024 |
References | S0956792523000062_ref1 S0956792523000062_ref2 S0956792523000062_ref27 S0956792523000062_ref29 Croll (S0956792523000062_ref8) 1978; 50 S0956792523000062_ref23 Landau (S0956792523000062_ref25) 1986 S0956792523000062_ref24 S0956792523000062_ref26 S0956792523000062_ref41 S0956792523000062_ref20 S0956792523000062_ref21 S0956792523000062_ref22 S0956792523000062_ref40 Stoney (S0956792523000062_ref36) 1909; 82 S0956792523000062_ref16 S0956792523000062_ref38 S0956792523000062_ref17 S0956792523000062_ref39 S0956792523000062_ref18 S0956792523000062_ref12 S0956792523000062_ref34 S0956792523000062_ref13 S0956792523000062_ref35 S0956792523000062_ref14 S0956792523000062_ref15 S0956792523000062_ref37 S0956792523000062_ref30 S0956792523000062_ref31 S0956792523000062_ref32 S0956792523000062_ref10 S0956792523000062_ref11 Coussy (S0956792523000062_ref7) 2004 Logg (S0956792523000062_ref28) 2012 Howell (S0956792523000062_ref19) 2009 S0956792523000062_ref9 S0956792523000062_ref5 Punati (S0956792523000062_ref33) 2022; 18 S0956792523000062_ref6 S0956792523000062_ref3 S0956792523000062_ref4 |
References_xml | – ident: S0956792523000062_ref37 doi: 10.1098/rspa.2010.0039 – volume-title: Poromechanics year: 2004 ident: S0956792523000062_ref7 contributor: fullname: Coussy – ident: S0956792523000062_ref26 doi: 10.1002/aic.690480304 – ident: S0956792523000062_ref41 doi: 10.1002/aenm.202102233 – ident: S0956792523000062_ref22 doi: 10.1115/1.2901526 – volume-title: Theory of Elasticity year: 1986 ident: S0956792523000062_ref25 contributor: fullname: Landau – volume: 82 start-page: 172 year: 1909 ident: S0956792523000062_ref36 article-title: The tension of metallic films deposited by electrolysis publication-title: Proc. Roy. Soc. A contributor: fullname: Stoney – volume-title: The FEniCS Book year: 2012 ident: S0956792523000062_ref28 contributor: fullname: Logg – ident: S0956792523000062_ref15 doi: 10.1016/j.colsurfa.2014.11.061 – ident: S0956792523000062_ref16 doi: 10.1140/epje/i2018-11639-2 – ident: S0956792523000062_ref11 – ident: S0956792523000062_ref40 doi: 10.1073/pnas.1005537107 – volume: 50 start-page: 33 year: 1978 ident: S0956792523000062_ref8 article-title: Internal stress in a solvent-cast thermoplastic coating publication-title: J. Coat. Technol. contributor: fullname: Croll – ident: S0956792523000062_ref13 doi: 10.1016/j.jpowsour.2018.04.097 – ident: S0956792523000062_ref30 doi: 10.1039/D0SM00252F – volume: 18 start-page: 214 year: 2022 ident: S0956792523000062_ref33 article-title: Modeling the drying of polymer coatings publication-title: Soft Matter doi: 10.1039/D1SM01343B contributor: fullname: Punati – ident: S0956792523000062_ref3 doi: 10.1063/1.1712886 – ident: S0956792523000062_ref27 doi: 10.1021/acs.langmuir.2c00397 – ident: S0956792523000062_ref29 doi: 10.1103/PhysRevApplied.5.044020 – ident: S0956792523000062_ref6 doi: 10.1111/j.1151-2916.1990.tb05258.x – ident: S0956792523000062_ref1 – ident: S0956792523000062_ref34 doi: 10.1016/j.cis.2021.102546 – ident: S0956792523000062_ref10 doi: 10.1103/PhysRevLett.91.224501 – ident: S0956792523000062_ref35 doi: 10.1039/C4NR02133A – ident: S0956792523000062_ref21 doi: 10.1016/j.jpowsour.2016.04.018 – ident: S0956792523000062_ref12 doi: 10.1021/jp0536807 – ident: S0956792523000062_ref9 doi: 10.1002/app.1979.070230319 – ident: S0956792523000062_ref5 doi: 10.1039/D1SM00832C – ident: S0956792523000062_ref23 doi: 10.1039/C5NR06266G – ident: S0956792523000062_ref17 doi: 10.1103/PhysRevE.105.054602 – ident: S0956792523000062_ref20 doi: 10.1016/j.jpowsour.2017.01.117 – ident: S0956792523000062_ref39 doi: 10.1039/C9SM02294E – volume-title: Applied Solid Mechanics year: 2009 ident: S0956792523000062_ref19 contributor: fullname: Howell – ident: S0956792523000062_ref32 doi: 10.1021/la990256m – ident: S0956792523000062_ref24 doi: 10.1016/j.cis.2020.102271 – ident: S0956792523000062_ref38 doi: 10.1021/acs.langmuir.1c03124 – ident: S0956792523000062_ref18 doi: 10.1017/jfm.2015.361 – ident: S0956792523000062_ref14 doi: 10.1023/A:1020886802632 – ident: S0956792523000062_ref31 doi: 10.1209/epl/i2003-00457-7 – ident: S0956792523000062_ref4 doi: 10.1039/C8SM02558D – ident: S0956792523000062_ref2 |
SSID | ssj0013079 |
Score | 2.3814335 |
Snippet | Abstract
Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. One way to... |
SourceID | crossref |
SourceType | Aggregation Database |
StartPage | 62 |
Title | Time-dependent modelling of thin poroelastic films drying on deformable plates |
Volume | 35 |
hasFullText | 1 |
inHoldings | 1 |
isFullTextHit | |
isPrint | |
link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1JSwMxFA5VL3oQV9zJwZMlpZNmtqOIWpTWi4q3kmQSFNpOmY4HvfnPfZlkFjdQL6FkQgbmfX35Xt6G0LEIYio9TolUMSMMTGYikighmsacsR7lSWCSkwfDoH_Hrh78h1brrZldkouOfP02r-Q_UoU5kKvJkv2DZKtNYQJ-g3xhBAnD-DsZP00UKdvY5rarzdiFMeePJj48zVIF_NhUZdVP48m8nWRFXlNqYmALvmoyp2Zjwzh_vKR3hJU7wjqpKr1WfLxv1PV8_mKzf4oO4u3LTuXfyHjZ_8Pl8bfvq4ew3MZ430x41r7uNK8hKCsjlxv3iQEJY5vF3FFWm4LtTRhzc07d2uokH2BldafTyvYUtp03v-h3VxTKvMy8C-wnc97S-jArHfifzrgq8tAGtYWjL1ssoCUaxn5YWuWlI6pbl2s0y0vHeFF1_NMWDWrT4Ci3a2jVGRf41CJlHbXUdAOtDGp5baLhR8zgCjM41dhgBjcwgwvMYIsZnE5xjRlsMbOF7i7Ob8_6xPXUINILu7kJHvVUJIWKEu6LJKSCai4TLVkcxSriIRA4zXgUU8qNTxzsaaWVZr4A0xzG3jZanKZTtYMwbBhw3u0xAX_znudHSofU0FcvElIGchedlJ9jNLOlU0Y_fv69vyzeR8s1BA_QYp49q0Pghrk4KqT3DtmAYRI |
link.rule.ids | 315,786,790,870,27957,27958 |
linkProvider | Directory of Open Access Journals |
openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Time-dependent+modelling+of+thin+poroelastic+films+drying+on+deformable+plates&rft.jtitle=European+journal+of+applied+mathematics&rft.au=Hennessy%2C+Matthew+G.&rft.au=Craster%2C+Richard+V.&rft.au=Matar%2C+Omar+K.&rft.date=2024-02-01&rft.issn=0956-7925&rft.eissn=1469-4425&rft.volume=35&rft.issue=1&rft.spage=62&rft.epage=95&rft_id=info:doi/10.1017%2FS0956792523000062&rft.externalDBID=n%2Fa&rft.externalDocID=10_1017_S0956792523000062 |
thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0956-7925&client=summon |
thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0956-7925&client=summon |
thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0956-7925&client=summon |