Deflections of a rubber membrane

Abstract This paper deals with the problem of the transverse deflection of a natural rubber membrane that is fixed along a circular boundary. Uniaxial experiments were performed in order to characterize the constitutive behaviour of the rubber material in terms of several constitutive models available in the literature. These constitutive models were used to develop computational estimates for the quasi-static load–displacement response of a rigid spherical indentor that deflects the rubber membrane in a controlled fashion and to determine the deflected shape of the membrane at specified load levels. Both axisymmetric and asymmetric deflections of the rubber membrane were investigated. The paper provides a comparison of the experimental results for the membrane deflections with results derived from computational simulations.

[1]  Yibin Fu,et al.  Nonlinear elasticity : theory and applications , 2001 .

[2]  F. S. Wong,et al.  Large plane deformations of thin elastic sheets of neo-Hookean material , 1969 .

[3]  D. Steigmann,et al.  Point loads on a hemispherical elastic membrane , 1995 .

[4]  D. W. Saunders,et al.  Large elastic deformations of isotropic materials VII. Experiments on the deformation of rubber , 1951, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[5]  I. L. Singer,et al.  Fundamentals of friction : macroscopic and microscopic processes , 1992 .

[6]  Y. Maoult,et al.  Elastomer biaxial characterization using bubble inflation technique. II: Numerical investigation of some constitutive models , 2001 .

[7]  R. Skalak,et al.  Strain energy function of red blood cell membranes. , 1973, Biophysical journal.

[8]  K. Kendall,et al.  Surface energy and the contact of elastic solids , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[9]  A. Roberts,et al.  A Guide to Estimating the Friction of Rubber , 1992 .

[10]  T. Charlton.  Progress in Solid Mechanics , 1962, Nature.

[11]  J. T. Tielking,et al.  The Inflation and Contact Constraint of a Rectangular Mooney Membrane , 1974 .

[12]  J. G. Simmonds,et al.  Large deformations under vertical edge loads of annular membranes with various strain energy densities , 1986 .

[13]  G. Marckmann,et al.  Inflation of elastomeric circular membranes using network constitutive equations , 2003 .

[14]  D. Steigmann.  Puncturing a thin elastic sheet , 2005 .

[15]  L. Treloar,et al.  Stress-strain data for vulcanised rubber under various types of deformation , 1944 .

[16]  M. Boyce,et al.  Constitutive models of rubber elasticity: A review , 2000 .

[17]  Christopher Y. Tuan.  Ponding on circular membranes , 1998 .

[18]  L. Mullins.  Effect of Stretching on the Properties of Rubber , 1948 .

[19]  Stephen P. Timoshenko,et al.  History of strength of materials : with a brief account of the history of theory of elasticity and theory of structures , 1983 .

[20]  S. Antman.  Nonlinear problems of elasticity , 1994 .

[21]  Sam F. Edwards,et al.  The theory of rubber elasticity , 1976, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[22]  Tube to annulus-an exact nonlinear membrane solution , 1970 .

[23]  Allen C. Pipkin.  Integration of an equation in membrane theory , 1968 .

[24]  W. Szyszkowski,et al.  Spherical membranes subjected to vertical concentrated loads: an experimental study , 1987 .

[25]  W. W. Feng,et al.  On the Contact Problem of an Inflated Spherical Nonlinear Membrane , 1973 .

[26]  Chien H. Wu.  On certain integrable nonlinear membrane solutions , 1970 .

[27]  P. Glockner,et al.  On the analysis of non-linear membranes , 1972 .

[28]  A. Spencer,et al.  The Static Theory of Finite Elasticity , 1970 .

[29]  A.J.M. Spencer,et al.  FINITE AXISYMMETRIC DEFORMATIONS OF AN INITIALLY CYLINDRICAL ELASTIC MEMBRANE , 1969 .

[30]  M. Beatty.  Nonlinear Elasticity: Hyperelastic Bell Materials: Retrospection, Experiment, Theory , 2001 .

[31]  Alan Muhr,et al.  Constitutive Models for Rubber , 1999 .

[32]  William W. Feng,et al.  On the general contact problem of an inflated nonlinear plane membrane , 1975 .

[33]  Wei H. Yang,et al.  Indentation of a Circular Membrane , 1971 .

[34]  A. Roberts,et al.  The adhesion and friction of smooth rubber surfaces , 1975 .

[35]  Millard F. Beatty,et al.  Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples , 1987 .

[36]  L. Treloar,et al.  The mechanics of rubber elasticity , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[37]  L. Hart-Smith,et al.  Large elastic deformations of thin rubber membranes , 1967 .

[38]  Reinhard Klette,et al.  Computer vision - three-dimensional data from images , 1998 .

[39]  J. M. Hill.  Nonlinear Elasticity: Exact Integrals and Solutions for Finite Deformations of the Incompressible Varga Elastic Materials , 2001 .

[40]  A. D. Kydoniefs.  FINITE AXISYMMETRIC DEFORMATIONS OF AN INITIALLY CYLINDRICAL MEMBRANE REINFORCED WITH INEXTENSIBLE CORDS , 1970 .

[41]  R. Rivlin,et al.  Experiments on the Mechanics of Rubber II: The Torsion, Inflation and Extension of a Tube , 1952 .

[42]  J. T. Tielking,et al.  The Application of the Minimum Potential Energy Principle to Nonlinear Axisymmetric Membrane Problems , 1974 .

[43]  A. Selvadurai,et al.  Second-order elasticity with axial symmetry—I General theory , 1972 .

[44]  D. Carlson,et al.  Proceedings of the IUTAM Symposium on Finite Elasticity: "Held at Lehigh University, Bethlehem, PA, USA August 10-15, 1980" , 2011 .

[45]  Ronald S. Rivlin,et al.  Some Topics in Finite Elasticity , 1997 .

[46]  J. Z. Zhu,et al.  The finite element method , 1977 .

[47]  M. Boyce,et al.  A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials , 1993 .

[48]  M. A. Johnson,et al.  The mullins effect in uniaxial extension and its influence on the transverse vibration of a rubber string , 1993 .

[49]  Michel Barquins,et al.  Adherence, Friction and Wear of Rubber-Like Materials , 1992 .

[50]  Werner Walter Klingbell,et al.  Some numerical investigations on empirical strain energy functions in the large axi-symmetric extensions of rubber membranes , 1964 .

[51]  W. W. Feng,et al.  On Axisymmetrical Deformations of Nonlinear Membranes , 1970 .

[52]  R. Ogden.  Elasticity and Inelasticity of Rubber , 2004 .

[53]  W. Yang,et al.  Stress concentration in a rubber sheet under axially symmetric stretching. , 1966 .

[54]  R. Rivlin.  Large elastic deformations of isotropic materials IV. further developments of the general theory , 1948, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[55]  Wing Kam Liu,et al.  Nonlinear Finite Elements for Continua and Structures , 2000 .

[56]  T. E. Tezduyar,et al.  Finite deformation of a circular elastic membrane containing a concentric rigid inclusion , 1987 .

[57]  Wei H. Yang,et al.  General Deformations of Neo-Hookean Membranes , 1973 .

[58]  G. I. Barenblatt,et al.  Collected Papers Of R S Rivlin , 1997 .

[59]  M. A. Johnson,et al.  A constitutive equation for the Mullins effect in stress controlled uniaxial extension experiments , 1993 .

[60]  Morton E. Gurtin,et al.  On the Nonlinear Theory of Elasticity , 1978 .

[61]  J. E. Adkins,et al.  Large elastic deformations of isotropic materials IX. The deformation of thin shells , 1952, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[62]  D. C. Pamplona,et al.  Large deformations under axial force and moment loads of initially flat annular membranes , 1992 .

[63]  Chien H. Wu.  Spherelike deformations of a balloon , 1972 .

[64]  J. E. Adkins,et al.  Large Elastic Deformations , 1971 .

[65]  Ronald S. Rivlin,et al.  The Solution of Problems in Second Order Elasticity Theory , 1953 .

[66]  Millard F. Beatty,et al.  The Mullins effect in equibiaxial extension and its influence on the inflation of a balloon , 1995 .

[67]  Mohamed Rachik,et al.  Elastomer biaxial characterization using bubble inflation technique. I: Experimental investigations , 2001 .

[68]  R. Ogden.  Large deformation isotropic elasticity – on the correlation of theory and experiment for incompressible rubberlike solids , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[69]  L. Treloar,et al.  The mechanics of rubber elasticity , 1976, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[70]  Mechanics and Thermomechanics of Rubberlike Solids , 2004 .

[71]  C. H. Wu.  ON THE CONTACT PROBLEMS OF INFLATED CYLINDRICAL MEMBRANES WITH A LIFE RAFT AS AN EXAMPLE , 1971 .

[72]  Michel Barquins,et al.  Friction and wear of rubber-like materials , 1993 .

[73]  William L. Ko,et al.  Application of Finite Elastic Theory to the Deformation of Rubbery Materials , 1962 .

[74]  S. Timoshenko,et al.  THEORY OF PLATES AND SHELLS , 1959 .

[75]  E. B. Spratt,et al.  Second-order effects in the deformation of elastic bodies , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[76]  P. M. Naghdi,et al.  The Theory of Shells and Plates , 1973 .

[77]  Ted Belytschko,et al.  An atomistic-based finite deformation membrane for single layer crystalline films , 2002 .

[78]  K. A. Grosch,et al.  The Relation between the Friction and Viscoelastic Properties of Rubber , 1964 .

[79]  J. B. Haddow,et al.  A finite element formulation for finite static axisymmetric deformation of hyperelastic membranes , 1995 .

[80]  Ray W. Ogden,et al.  A constitutive model for the Mullins effect with permanent set in particle-reinforced rubber , 2004 .

[81]  L. Treloar.  Stress-Strain Data for Vulcanized Rubber under Various Types of Deformation , 1944 .

[82]  A. Schallamach.  A theory of dynamic rubber friction , 1963 .

[83]  Finite elasticity in spatial description: Linearization aspects with 3-D membrane applications , 1995 .

[84]  J. G. Simmonds,et al.  The Nonlinear Theory of Elastic Shells , 1998 .

[85]  M. Mooney.  A Theory of Large Elastic Deformation , 1940 .

[86]  B. Persson.  On the theory of rubber friction , 1998 .

[87]  K. A. Grosch,et al.  The relation between the friction and visco-elastic properties of rubber , 1963, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[88]  J. T. Oden,et al.  Finite strains and displacements of elastic membranes by the finite element method , 1967 .

[89]  1996 Kluwer Academic Publishers. Printed in the Netherlands. , 1996 .

[90]  C. Truesdell,et al.  The Non-Linear Field Theories Of Mechanics , 1992 .

[91]  T. Mackin,et al.  Spherical indentation of freestanding circular thin films in the membrane regime , 2004 .

[92]  R. Rivlin.  Large Elastic Deformations of Isotropic Materials , 1997 .

[93]  O. Yeoh.  Some Forms of the Strain Energy Function for Rubber , 1993 .

[94]  P.D.S. Verma,et al.  Radial deformation of a plane sheet containing a circular hole or inclusion , 1978 .

[95]  A. Drozdov,et al.  Finite Elasticity and Viscoelasticity: A Course in the Nonlinear Mechanics of Solids , 1996 .

[96]  Hisaaki Tobushi,et al.  Analysis of the mechanical behavior of shape memory polymer membranes by nanoindentation, bulging and point membrane deflection tests , 2000 .

[97]  Chien H. Wu.  Large finite strain membrane problems , 1979 .

[98]  P. M. Naghdi,et al.  Large deformation possible in every isotropic elastic membrane , 1977, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[99]  B. Persson,et al.  Sliding Friction: Physical Principles and Applications , 1997 .

[100]  D. Tabor,et al.  Friction and molecular structure: the behaviour of some thermoplastics , 1972, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[101]  T. J. Lardner,et al.  Deformations of elastic membranes—Effect of different constitutive relations , 1978 .

[102]  A. Schallamach.  The frictional contact of rubber , 1975 .

[103]  M. Barquins.  Sliding friction of rubber and Schallamach waves: a review , 1985 .

[104]  Erwan Verron,et al.  An axisymmetric B-spline model for the non-linear inflation of rubberlike membranes , 2001 .

[105]  L. Mullins.  Softening of Rubber by Deformation , 1969 .

[106]  R. Ogden.  Non-Linear Elastic Deformations , 1984 .

[107]  A. Schallamach.  The Load Dependence of Rubber Friction , 1952 .