Abstract
This study investigates the problem of a circular elastic membrane clamped or adhered at its boundary and subjected to uniform transverse pressure. Many analytical solutions for this classical problem have been developed previously, using either a series-based approach (notably accurate but lengthy and implicit) or approximate kinematics (relatively simple yet lacking accuracy). Here, we seek new analytical solutions using a perturbed spherical cap to represent the shape of the pressurized membrane. Our approach yields simple, explicit solutions of remarkable accuracy for the deformed profile, pressure–deflection relation, strain distributions, and energy release rate, which are directly applicable to emerging ultrathin membrane systems.
Graphical Abstract Figure
References
1.
Beams
, J.
, 1959
, “Mechanical Properties of Thin Films of Gold and Silver,” Structure and Properties of Thin Films
, John Wiley and Sons
, New York
, p. 183
.2.
Freund
, L. B.
, and Suresh
, S.
, 2004
, Thin Film Materials: Stress, Defect Formation and Surface Evolution
, Cambridge University Press
, Cambridge
.3.
Mansfield
, E. H.
, 1989
, The Bending and Stretching of Plates
, Cambridge University Press
, Cambridge
.4.
Dai
, Z.
, Liu
, L.
, and Zhang
, Z.
, 2019
, “Strain Engineering of 2D Materials: Issues and Opportunities at the Interface
,” Adv. Mater.
, 31
(45
), p. 1805417
. 5.
Dai
, Z.
, Lu
, N.
, Liechti
, K. M.
, and Huang
, R.
, 2020
, “Mechanics at the Interfaces of 2D Materials: Challenges and Opportunities
,” Curr. Opin. Solid State Mater. Sci.
, 24
(4
), p. 100837
. 6.
Dong
, W.
, Dai
, Z.
, Liu
, L.
, and Zhang
, Z.
, 2023
, “Toward Clean 2d Materials and Devices: Recent Progress in Transfer and Cleaning Methods
,” Adv. Mater.
, p. 2303014
. 7.
Hencky
, H.
, 1915
, “Uber Den Spannungszustand in Kreisrunden Platten Mit Verschwindender Biegungssteifigkeit
,” Z. Math. Phys.
, 63
, pp. 311
–317
.8.
Dai
, Z.
, Sanchez
, D. A.
, Brennan
, C. J.
, and Lu
, N.
, 2020
, “Radial Buckle Delamination Around 2d Material Tents
,” J. Mech. Phys. Solids
, 137
, p. 103843
. 9.
Dai
, Z.
, and Lu
, N.
, 2021
, “Poking and Bulging of Suspended Thin Sheets: Slippage, Instabilities, and Metrology
,” J. Mech. Phys. Solids
, 149
, p. 104320
. 10.
Dai
, Z.
, Rao
, Y.
, and Lu
, N.
, 2022
, “Two-dimensional Crystals on Adhesive Substrates Subjected to Uniform Transverse Pressure
,” Int. J. Solids Struct.
, 257
, p. 111829
. 11.
Rao
, Y.
, Kim
, E.
, Dai
, Z.
, He
, J.
, Li
, Y.
, and Lu
, N.
, 2023
, “Size-Dependent Shape Characteristics of 2d Crystal Blisters
,” J. Mech. Phys. Solids
, 175
, p. 105286
. 12.
Gent
, A.
, and Lewandowski
, L.
, 1987
, “Blow-Off Pressures for Adhering Layers
,” J. Appl. Polym. Sci.
, 33
(5
), pp. 1567
–1577
. 13.
Fichter
, W.
, 1997
, “Some Solutions for the Large Deflections of Uniformly Loaded Circular Membranes
,” NASA Technical Reports
. Document ID: 19970023537
.14.
Jensen
, H. M.
, 1991
, “The Blister Test for Interface Toughness Measurement
,” Eng. Fract. Mech.
, 40
(3
), pp. 475
–486
. 15.
Yue
, K.
, Gao
, W.
, Huang
, R.
, and Liechti
, K. M.
, 2012
, “Analytical Methods for the Mechanics of Graphene Bubbles
,” J. Appl. Phys.
, 112
(8
), p. 083512
. 16.
Dai
, Z.
, Hou
, Y.
, Sanchez
, D. A.
, Wang
, G.
, Brennan
, C. J.
, Zhang
, Z.
, Liu
, L.
, and Lu
, N.
, 2018
, “Interface-governed Deformation of Nanobubbles and Nanotents Formed by Two-Dimensional Materials
,” Phys. Rev. Lett.
, 121
(26
), p. 266101
. 17.
Blundo
, E.
, Di Giorgio
, C.
, Pettinari
, G.
, Yildirim
, T.
, Felici
, M.
, Lu
, Y.
, Bobba
, F.
, and Polimeni
, A.
, 2020
, “Engineered Creation of Periodic Giant, Nonuniform Strains in Mos2 Monolayers
,” Adv. Mater. Interfaces
, 7
(17
), p. 2000621
. 18.
Wang
, G.
, Dai
, Z.
, Wang
, Y.
, Tan
, P.
, Liu
, L.
, Xu
, Z.
, Wei
, Y.
, Huang
, R.
, and Zhang
, Z.
, 2017
, “Measuring Interlayer Shear Stress in Bilayer Graphene
,” Phys. Rev. Lett.
, 119
(3
), p. 036101
. 19.
Wang
, P.
, Gao
, W.
, Cao
, Z.
, Liechti
, K. M.
, and Huang
, R.
, 2013
, “Numerical Analysis of Circular Graphene Bubbles
,” ASME J. Appl. Mech.
, 80
(4), p. 040905
. 20.
Johnson
, K. L.
, Kendall
, K.
, and Roberts
, A. D.
, 1971
, “Surface Energy and the Contact of Elastic Solids
,” Proc. R. Soc. Lond. A
, 324
(1558
), pp. 301
–313
.21.
Griffith
, A. A.
, 1921
, “Vi. The Phenomena of Rupture and Flow in Solids
,” Philos. Trans. Royal Soc. A
, 221
(582-893
), pp. 163
–198
.22.
Liechti
, K.
, 2019
, “Characterizing the Interfacial Behavior of 2d Materials: a Review
,” Exp. Mech.
, 59
(3
), pp. 395
–412
. 23.
Williams
, J.
, 1997
, “Energy Release Rates for the Peeling of Flexible Membranes and the Analysis of Blister Tests
,” Int. J. Fract.
, 87
(3
), pp. 265
–288
. 24.
Komaragiri
, U.
, Begley
, M.
, and Simmonds
, J.
, 2005
, “The Mechanical Response of Freestanding Circular Elastic Films Under Point and Pressure Loads
,” ASME J. Appl. Mech.
, 72
(2
), pp. 203
–212
.25.
Blundo
, E.
, Yildirim
, T.
, Pettinari
, G.
, and Polimeni
, A.
, 2021
, “Experimental Adhesion Energy in Van Der Waals Crystals and Heterostructures From Atomically Thin Bubbles
,” Phys. Rev. Lett.
, 127
(4
), p. 046101
. 26.
Lloyd
, D.
, Liu
, X.
, Christopher
, J. W.
, Cantley
, L.
, Wadehra
, A.
, Kim
, B. L.
, Goldberg
, B. B.
, Swan
, A. K.
, and Bunch
, J. S.
, 2016
, “Band Gap Engineering With Ultralarge Biaxial Strains in Suspended Monolayer Mos2
,” Nano Lett.
, 16
(9
), pp. 5836
–5841
. 27.
Sanchez
, D. A.
, Dai
, Z.
, and Lu
, N.
, 2021
, “2D Material Bubbles: Fabrication, Characterization, and Applications
,” Trends Chem.
, 3
(3
), pp. 204
–217
. 28.
Fang
, Z.
, Dai
, Z.
, Wang
, B.
, Tian
, Z.
, Yu
, C.
, Chen
, Q.
, and Wei
, X.
, 2023
, “Pull-to-Peel of Two-Dimensional Materials for the Simultaneous Determination of Elasticity and Adhesion
,” Nano Lett.
, 23
(2
), pp. 742
–749
. 29.
Wan
, K.-T.
, and Mai
, Y.-W.
, 1995
, “Fracture Mechanics of a New Blister Test With Stable Crack Growth
,” Acta Metall. Mater.
, 43
(11
), pp. 4109
–4115
. 30.
Koenig
, S. P.
, Boddeti
, N. G.
, Dunn
, M. L.
, and Bunch
, J. S.
, 2011
, “Ultrastrong Adhesion of Graphene Membranes
,” Nat. Nanotechnol.
, 6
(9
), pp. 543
–546
. 31.
Rao
, Y.
, Qiao
, S.
, Dai
, Z.
, and Lu
, N.
, 2021
, “Elastic Wetting: Substrate-Supported Droplets Confined by Soft Elastic Membranes
,” J. Mech. Phys. Solids
, 151
, p. 104399
. 32.
Wan
, K.-T.
, and Lim
, S.-C.
, 1998
, “The Bending to Stretching Transition of a Pressurized Blister Test
,” Int. J. Fracture
, 92
(4
), pp. 43
–47
. 33.
Sanchez
, D. A.
, Dai
, Z.
, Wang
, P.
, Cantu-Chavez
, A.
, Brennan
, C. J.
, Huang
, R.
, and Lu
, N.
, 2018
, “Mechanics of Spontaneously Formed Nanoblisters Trapped by Transferred 2d Crystals
,” Proc. Natl Acad. Sci. USA
, 115
(31
), pp. 7884
–7889
. 34.
Boddeti
, N. G.
, Koenig
, S. P.
, Long
, R.
, Xiao
, J.
, Bunch
, J. S.
, and Dunn
, M. L.
, 2013
, “Mechanics of Adhered, Pressurized Graphene Blisters
,” ASME J. Appl. Mech.
, 80
(4
), p. 040909
. 35.
Yu
, C.
, and Dai
, Z.
, 2023
, “Characterizing the Wetting Behavior of 2d Materials: A Review
,” J. Mater. Inf.
, 3
(3
), p. 20
.36.
Cao
, Z.
, Tao
, L.
, Akinwande
, D.
, Huang
, R.
, and Liechti
, K. M.
, 2016
, “Mixed-Mode Traction-Separation Relations Between Graphene and Copper by Blister Tests
,” Int. J. Solids Struct.
, 84
, pp. 147
–159
. 37.
Hinkley
, J.
, 1983
, “A Blister Test for Adhesion of Polymer Films to SiO2
,” J. Adhesion
, 16
(2
), pp. 115
–125
. 38.
Briscoe
, B.
, and Panesar
, S.
, 1991
, “The Application of the Blister Test to an Elastomeric Adhesive
,” Proc. R. Soc. A
, 433
(1887
), pp. 23
–43
.39.
Chen
, E.
, and Dai
, Z.
, 2023
, “Axisymmetric Peeling of Thin Elastic Films: A Perturbation Solution
,” ASME J. Appl. Mech.
, 90
(10
), p. 101011
. 40.
Wan
, K.-T.
, Guo
, S.
, and Dillard
, D. A.
, 2003
, “A Theoretical and Numerical Study of a Thin Clamped Circular Film Under an External Load in the Presence of a Tensile Residual Stress
,” Thin. Solid. Films.
, 425
(1–2
), pp. 150
–162
. 41.
Cotterell
, B.
, and Chen
, Z.
, 1997
, “The Blister Test–Transition From Plate to Membrane Behaviour for an Elastic Material
,” Int. J. Fracture
, 86
(3
), pp. 191
–198
. 42.
Arjun
, A.
, and Wan
, K.-T.
, 2005
, “Derivation of the Strain Energy Release Rate G From First Principles for the Pressurized Blister Test
,” Int. J. Adhes. Adhes.
, 25
(1
), pp. 13
–18
. 43.
Jensen
, H. M.
, 1998
, “Analysis of Mode Mixity in Blister Tests
,” Int. J. Fracture
, 94
(1
), pp. 79
–88
. 
