Abstract
In this paper we present an effective numerical algorithm for determining the equilibrium shapes of inflated elastic membranes susceptible to wrinkling. The use of a two-state constitutive law and the introduction of a suitable criterion allow for accounting for wrinkling of the membrane, although in an approximated way. In the active state, the material is able to transmit only tensile stresses; vice versa, in the passive state it is stress-free and can contract freely. Equilibrium of the membrane in the current inflated configuration is enforced by recourse to the minimum total potential energy principle, whereas the Lagrange multipliers method is used to solve the minimum problem by accounting for the aforesaid nonlinear constitutive law. We use an expressly developed iterative-incremental numerical algorithm, consistent with the established governing set of equations, for accurately monitoring the evolution of the stress field in the membrane during the inflation process. Specifically, we suppose that the membrane reaches its final shape at the end of a four-stage loading process corresponding to the temporary enforcement and the subsequent removal of a fictitious antagonist plane traction acting uniformly along its entire boundary. By this way it is possible to solve with great accuracy the set of governing equilibrium equations by means of a numerical procedure in which the membrane’s tangent stiffness is always kept different from zero. The soundness of the proposed algorithm is verified by comparing the results with well-known solutions available in the literature. In particular, for each specific value of pressure, the current configuration of the inflated membrane found by assuming that compressions are allowed is compared in details to the corresponding pseudo-deformed surface, obtained by assuming a tension-only response.
Similar content being viewed by others
References
Foster B, Mollaert M (2004) European design guide for tensile surface structures. TensiNet Ed, Brussels
Blandino JR, Johnston JD, Dharamsi UK (2002) Corner wrinkling of a square membrane due to symmetric mechanical loads. AIAA J Spacecr Rocket 39(5):717–724. doi:10.2514/2.3870
Tessler A, Sleight DW, Wang JT (2005) Effective modeling and nonlinear shell analysis of thin membranes exhibiting structural wrinkling. AIAA J Spacecr Rocket 42(2):287–298. doi:10.2514/1.3915
Tessler A, Sleight DW (2007) Geometrically nonlinear shell analysis of wrinkled thin-film membranes with stress concentrations. J Spacecr Rocket 44(3):582–588. doi:10.2514/1.22913
Suhey JD, Kim NH, Niezrecki C (2005) Numerical modeling and design of inflatable structures: application to open-ocean-aquaculture cages. Aquac Eng 33:285–303
Turner AW, Kabche JP, Peterson ML, Davids WG (2008) Tension/torsion testing on inflatable fabric tubes. Exp Tech 32(2):47–52
Galliot C, Luchsinger RH (2009) A simple model describing the non-linear biaxial tensile behaviour of PVC-coated polyester fabrics for use in finite element analysis. Compos Struct 90(4):438–447
Cavallaro PV, Johnson ME, Sadegh AM (2003) Mechanics of plain-woven fabrics for inflated structures. Compos Struct 61:375–393
Lee ES, Youn SK (2006) Finite element analysis of wrinkling membrane structures with large deformations. Finite Elem Anal Des 42:780–791
Reissner E (1938) On tension field theory. In: Proceedings of the Fifth International Congress for Applied Mechanics, pp 88–92
Ligarò SS, Barsotti R (2008) Equilibrium shapes of inflated inextensible membranes. Int J Solids Struct 45:5584–5598
Contri P, Schrefler BA (1988) A geometrically nonlinear finite element analysis of wrinkled membrane surfaces by a no-compression material model. Commun Appl Numer Methods 4:5–15
Ligarò SS, Barsotti R (2009) Membrane elastiche sottili fortemente pressurizzate. In Atti XIX Congress AIMETA 2009, Università Politecnica delle Marche. Facoltà di Ingegneria, Aras Edizioni, Ancona 14–17 Sept 2009
Ziegler R, Wagner W, Bletzinger KU (2003) A finite-element-model for the analysis of wrinkled membrane structures. Int J Space Struct 18:1–14
Pipkin AC (1993) The relaxed energy density for isotropic elastic membranes. IMA J Appl Math 50:225–237
Stein M, Hedgepeth JM (1961) Analysis of partly wrinkled membranes. NASA Technical Note D-813, Washington, DC
Wu CH (1974) Plane linear wrinkle elasticity without body force. Materials Engineering, University of Illinois, Chicago, Report Dept
Roddeman DG, Oomens CWJ, Janssen JD, Drukker J (1987) The wrinkling of thin membranes: part I-theory. ASME J Appl Mech 54:884–887
Mosler J, Cirak F (2009) A variational formulation for finite deformation wrinkling analysis of inelastic membranes. Comput Methods Appl Mech Eng 198:2087–2098
Jenkins CH, Haugen F, Spicher WH (1998) Experimental measurement of wrinkling in membranes undergoing planar deformation. Exp Mech 38(2):147–152
Wong YW, Pellegrino S (2006) Wrinkled membranes Part I: experiments. J Mech Mater Struct 1(1):3–25
Barsotti R, Vannucci P (2013) Wrinkling of orthotropic membranes: an analysis by the polar method. J Elast 113:5–26. doi:10.1007/s10659-012-9408-z
Pipkin AC (1994) Relaxed energy densities for large deformations of membranes. IMA J Appl Math 52:297–308
Bouzidi R, Le van A (2004) Numerical solution of hyperelastic membranes by energy minimization. Comput Struct 82:1961–1969
Luenberger DG (1984) Linear and nonlinear programming. Addison-Wesley, Reading
Schweizerhof K, Ramm E (1984) Displacement dependent pressure loads in nonlinear finite elements analyses. Comput Struct 18(6):1099–1114
Zienkiewicz OC (1983) The finite element method: third edition, 3rd edn. McGraw-Hill Book Co., New York
Barsotti R, Ligarò SS, Royer-Carfagni G (2001) The web bridge. Int J Solids Struct 38(48):8831–8850
Haseganu EM, Steigmann DJ (1994) Analysis of partly wrinkled membranes by the method of dynamic relaxation. Comput Mech 14(6):596–614
Jarasjarungkiat A, Wüchner R, Bletzinger KU (2008) A wrinkling model based on material modification for isotropic and orthotropic membranes. Comput Methods Appl Mech Eng 197:773–788
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Barsotti, R., Ligarò, S.S. Static response of elastic inflated wrinkled membranes. Comput Mech 53, 1001–1013 (2014). https://doi.org/10.1007/s00466-013-0945-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00466-013-0945-5