Contact problems of hyperelastic membranes: Existence theory. (English) Zbl 0977.74042
Summary: We describe the pressure-driven inflation of an incompressible isotropic hyperelastic membrane into a rigid mould by a variational inequality, and prove the existence of solution in the case of various, suitably modified, strain energy functions of the Ogden form. The variational inequality description is applicable to the case of perfect sliding contact of the membrane with the mould, and the modification to the strain energy functions is according to the tension field theory which rules out compressive stresses. The modified or relaxed strain energy functions obtained are shown, in our examples, to be polyconvex and in some cases convex.
MSC:
| 74M15 | Contact in solid mechanics |
| 74G25 | Global existence of solutions for equilibrium problems in solid mechanics (MSC2010) |
| 74K15 | Membranes |
| 74B20 | Nonlinear elasticity |
| 49J40 | Variational inequalities |
Keywords:
pressure-driven inflation; incompressible isotropic hyperelastic membrane; variational inequality; existence of solution; perfect sliding contact; tension field theory; relaxed strain energy functionsReferences:
| [1] | Sobolev Spaces. Academic Press: New York, 1975. · Zbl 0314.46030 |
| [2] | Constitutive inequalities and existence theorems in nonlinear elastostatics. In (ed.), Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, vol. 1, Pitman: London, 1977; 187-241. |
| [3] | Ball, Arch. Rat. Mech. Anal. 63 pp 337– (1977) · Zbl 0368.73040 · doi:10.1007/BF00279992 |
| [4] | Cesari, Bull. Am Math. Soc. 80 pp 467– (1974) · Zbl 0287.49003 · doi:10.1090/S0002-9904-1974-13452-X |
| [5] | Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Vol. 78, Springer-Verlag: Berlin, 1989. · Zbl 0703.49001 · doi:10.1007/978-3-642-51440-1 |
| [6] | Davies, J. Elasticity 26 pp 291– (1991) · Zbl 0759.73013 · doi:10.1007/BF00041893 |
| [7] | Foundations of Modern Analysis. Dover Publications Inc.: New York, 1982. |
| [8] | Numerical Analysis of Variational Inequalities, Studies in Maths and its Appls., Vol. 8, North-Holland: Amsterdam, 1976. |
| [9] | Glowinski, SIAM J. Appl. Math. 42 pp 400– (1982) · Zbl 0506.73045 · doi:10.1137/0142031 |
| [10] | Large Elastic Deformations, 2nd edn. Clarendon Press: Oxford, 1970. |
| [11] | Mathematical Theory of Non-Linear Elasticity. Ellis Horwood Limited: Chichester, 1985. |
| [12] | Hoppe, SIAM J. Numer. Anal. 31 pp 301– (1994) · Zbl 0806.65064 · doi:10.1137/0731016 |
| [13] | Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM: Philadelphia, 1988. · doi:10.1137/1.9781611970845 |
| [14] | Knowles, Arch. Rat. Mech. Anal. 63 pp 321– (1977) · Zbl 0351.73061 · doi:10.1007/BF00279991 |
| [15] | Kornhuber, Numerische Mathematik 72 pp 481– (1996) · Zbl 0861.65056 · doi:10.1007/s002110050178 |
| [16] | Morrey, Pacific J. Math. 2 pp 25– (1952) · Zbl 0046.10803 · doi:10.2140/pjm.1952.2.25 |
| [17] | Naghdi, Phil. Trans. R. Soc. Lond. A 287 pp 145– (1977) · Zbl 0373.73078 · doi:10.1098/rsta.1977.0143 |
| [18] | Introduction to the Theory of Nonlinear Elliptic Equations. Teubner: Leipzig, 1983. |
| [19] | Existence theorems and approximations in nonlinear elasticity. In Nonlinear Equations in Abstract Spaces, (ed.) Academic Press, Inc.: New York, 1978; 265-274. · doi:10.1016/B978-0-12-434160-9.50021-7 |
| [20] | Non-Linear Elastic Deformations. Ellis Horwood Limited: Chichester, 1984. |
| [21] | Non-axisymmetric pressure driven thermoforming. Scientific Report, CEC HCM Programme CHRX-CT93-0379, Brunel Institute of Computational Mathematics, Brunel University, Uxbridge, Middlesex UB8 3PH, U.K., 1995. |
| [22] | Non axisymmetric pressure driven thermoforming, (in preparation). |
| [23] | Pipkin, IMA J. Appl. Math. 36 pp 85– (1986) · Zbl 0644.73047 · doi:10.1093/imamat/36.1.85 |
| [24] | Pipikin, Arch. Rat. Mech. Anal. 121 pp 361– (1993) · Zbl 0768.73038 · doi:10.1007/BF00375626 |
| [25] | On tension field theory. In Proceedings of the 5th Int. Cong. on Appl. Mech., 1938; 88-92. |
| [26] | Le Tallec, Comp. Meth. Appl. Mech. Engng. 68 pp 67– (1988) · Zbl 0626.73028 · doi:10.1016/0045-7825(88)90107-7 |
| [27] | Thompson, J. Res. Nat. Bureau of Standards?B. Math. Sci. 75B pp 115– (1971) · Zbl 0265.15009 · doi:10.6028/jres.075B.007 |
| [28] | The non-linear field theories of mechanics. In Handbuch der Physik, (ed.), Vol. III/3. Springer: Berlin, 1965. |
| [29] | Ebene Blechwandtr?ger mit sehr d?nnem Stegblech. Z. Flugtechnik u. Motorluftschiffahrt, 1929. |
| [30] | The computational modelling of thermoforming for the creation of axisymmetric container structures. In Mathematical Modelling for Materials Processing, (eds), Clarendon Press: Oxford, 1993; 159-176. |
| [31] | Numerical simulation of non-axisymmetric pressure driven thermoforming. Scientific Report, CEC HCM Programme CHRX-CT93-0379, Brunel Institute of Comp. Math., Brunel University, Uxbridge, Middlesex UB8 3PH, U.K., 1996. |
| [32] | Functional Analysis. Springer: Berlin, 1980. · Zbl 0830.46001 · doi:10.1007/978-3-642-61859-8 |
| [33] | Nonlinear Functional Analysis and Its Applications: 4. Applications to Mathematical Physics, vol. IV. Springer: Heidelberg, 1986. · doi:10.1007/978-1-4612-4838-5 |
| [34] | Applied Functional Analysis, Applications to Mathematical Physics, Appl. Math. Sciences, vol. 108. Springer: Heidelberg, 1995. |
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