This paper presents a mixed finite element formulation approximating large deformations observed in the analysis of elastomeric butt-joints. The rubber has been considered as nearly incompressible continuum obeying the Mooney/Rivlin (M/R) strain energy density function. The parameters of the model were determined by fitting the available from the literature uniaxial tension experimental data with the constitutive equation derived from the M/R model. The optimum value of the Poisson ratio is adjusted by comparing the experimentally observed diametral contraction of the model with that numerically obtained using the finite element method. The solution of the problem has been obtained utilizing the mixed finite element procedure on the basis of displacement/pressure mixed interpolation and enhanced strain energy mixed formulation. For comparison purposes, an axisymmetric with two-parameter M/R model and a three-dimensional (3D) with nine-parameters M/R model of the butt-joint are formulated and numerical results are illustrated concerning axisymmetric or general loading. For small strains the stress and/or strain distribution in the 2D axisymmetric butt-joint problem was compared with derived analytical solutions. Stress distributions along critical paths are evaluated and discussed.
Issue Section:
Technical Papers
1.
Dorfmann
, A.
, Fuller
, K. N. G.
, and Ogden
, R. W.
, 2002, “Shear, Compressive and Dilatational Response of Rubber-Like Solids Subject to Cavitations Damage
,” Int. J. Heat Mass Transfer
0017-9310, 39
, pp. 1845
–1861
.2.
Obata
, Y.
, Kawabuta
, S.
, and Kawai
, H.
, 1970, “Mechanical Properties of NBR Vulcanizates Under Finite Deformation
,” J. Polym. Sci., Part A-2
0098-1273, 8
, pp. 903
–919
.3.
Ogden
, R. W.
, 1972, “Large Deformation Isotropic Elasticity—On The Correlation of Theory and Experiment for Incompressible Rubber Like Solids
,” Proceedings of the Royal Society
, London, Series A, 326
, pp. 565
–584
.4.
Hibbit
, H. D.
, Marcal
, P. V.
, and Rice
, J. R.
, 1972, “Finite Element Formulation for Problems of Large Strain and Large Displacement
,” Int. J. Heat Mass Transfer
0017-9310, 6
, pp. 1069
–1086
.5.
James
, A. G.
, and Green
, A.
, 1975, “Strain Energy Functions of Rubber. II. The Characterization of Filled Vulcanizates
,” J. Appl. Polym. Sci.
0021-8995, 19
, pp. 2319
–2330
.6.
Chen
, J. S.
, Satyamurthy
, K.
, and Hirsschfelt
, L. R.
, 1994, “Consistent Finite Element Procedures for Nonlinear Rubber Elasticity with a Higher Order Strain Energy Function
,” Comput. Struct.
0045-7949, 50
, pp. 715
–727
.7.
Herrmann
, L. H.
, 1964, “Elasticity Equations for Incompressible and Nearly Incompressible Materials by a Variation Theorem
,” AIAA J.
0001-1452, 2
, pp. 1333
–1336
.8.
Maniatty
, A. M. L.
, Klaas
, Y. O.
, and Shephard
, M. S.
, 2002, “Higher Order Stabilized Finite Element Method for Hyperelastic Finite Deformation
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 191
, pp. 1491
–1503
.9.
Pantuso
, D.
, and Bathe
, K. J.
, 1997, “On the Stability of Mixed Finite Elements in Large Strain Analysis of Incompressible Solids
,” Finite Elem. Anal. Design
0168-874X, 28
, pp. 83
–104
.10.
Pian
, T. H. H.
, and Sumihara
, K.
, 1984, “Rational Approach for Assumed Stress Finite Elements
,” Int. J. Numer. Methods Eng.
0029-5981, 20
, pp. 1685
–1695
.11.
Brezzi
, F.
, and Fortin
, M.
, 1991, Mixed and Hybrid Finite Element Methods
, Springer
, Berlin, Germany.12.
Pian
, T. H. H.
, 1995, “State-of-the-Art Development of Hybrid/Mixed Finite Element Method
,” Finite Elem. Anal. Design
0168-874X, 21
, pp. 5
–20
.13.
Chang
, W. V.
, and Peng
, S. H.
, 1992, “Nonlinear Finite Element Analysis of the Butt-Joint Elastomer Specimen
,” J. Adhes. Sci. Technol.
0169-4243, 6
, pp. 919
–939
.14.
Kakavas
, P. A.
, and Chang
, W. V.
, 1992, “An Extension of the General Measure of Strain in Hyperelastic Solids
,” J. Appl. Polym. Sci.
0021-8995, 45
, pp. 865
–869
.15.
Kakavas
, P. A.
, and Blatz
, P. J.
, 1991, “A Geometric Determination of Void Production in an Elastic Pancake
,” J. Appl. Polym. Sci.
0021-8995, 43
, pp. 1081
–1086
.16.
Simo
, J. C.
, and Hughes
, T. J. R.
, 1998, Computational Inelasticity
, Springer
, New York.17.
Truesdell
, C.
, and Noll
, W.
, 1965, “The Non-linear Field Theories of Mechanics
,” Encyclopedia of Physics
, S.
Flugge
, ed., Springer
, Berlin, Vol. III/3
.18.
Ogden
, R. W.
, 1984, Nonlinear Elastic Deformations
, Wiley
, New York.19.
Simo
, J. C.
, and Taylor
, R. L.
, 1991, “Quasi-Incompressible Finite Elasticity in Principal Stretches. Continuum Basis and Numerical Algorithms
,” Comput. Methods Appl. Mech. Eng.
0045-7825, 85
, pp. 273
–310
.20.
Valanis
, K. C.
, and Landel
, R.
, 1967, “The Strain Energy Function of a Hyperelastic Material in Terms of the Extension Ratios
,” J. Appl. Phys.
0021-8979, 38
, pp. 2997
–3002
.21.
Mooney
, M.
, 1940, “A Theory of Large Elastic Deformation
,” J. Appl. Phys.
0021-8979, 11
, pp. 582
–592
.22.
Rivlin
, R. S.
, 1984, “Forty Years of Nonlinear Continuum Mechanics
,” Proceedings of the 9th International Congress on Rheology
, Mexico, Sept. 4-7, pp. 1
–29
,23.
Salomon
, O.
, Olle
, S.
, and Barrat
, A.
, 1999, “Finite Element Analysis of Base Isolated Buildings Subjected to Earthquake Loads
,” Int. J. Numer. Methods Eng.
0029-5981, 46
, pp. 1741
–1761
.24.
Wilson
, E.
, 1974, “The Static Condensation Algorithm
,” Int. J. Numer. Methods Eng.
0029-5981, 8
, pp. 199
–203
.25.
Bathe
, K. J.
, 1982, Finite Element Procedures in Engineering Analysis
, Prentice–Hall
, Englewood Cliffs, NJ.26.
Treloar
, L. R. G.
, 1975, The Physics of Rubber Elasticity
, Clarendon
, Oxford, UK.27.
Peng
, S. H.
, and Chang
, W. V.
, 1997, “A Compressible Approach in Finite Element Analysis of Rubber-Elastic Materials
,” Comput. Struct.
0045-7949, 62
, pp. 573
–593
.28.
Kakavas
, P. A.
, 2000, “A New Development of the Strain Energy Function for Hyperelastic Materials Using a Log-Strain Approach
,” J. Appl. Polym. Sci.
0021-8995, 77
, pp. 660
–672
.29.
Criscione
, J. C.
, Humphery
, J. D.
, Douglas
, A. S.
, and Hunter
, W.
, 2000, “An Invariant Basis for Natural Strain which Yields Orthogonal Stress Response Terms in Isotropic Hyperelasticity
,” J. Mech. Phys. Solids
0022-5096, 48
, pp. 2445
–2465
.30.
Gent
, A.
, 1996, “A New Constitutive Relation for Rubber
,” Rubber Chem. Technol.
0035-9475, 69
, pp. 59
–61
.31.
Arruda
, E. M.
, and Boyce
, M.
, 1993, “A Three Dimensional Constitutive Model for the Large Deformation Stretch Behavior of Rubber Elastic Materials
,” J. Mech. Phys. Solids
0022-5096, 41
, pp. 389
–412
.32.
Horgan
, C. O.
, and Saccomandi
, G.
, 1999, “Pure Axial Shear of Isotropic Hyperelastic Nonlinearly Elastic Materials with Limiting Chain Extensibility
,” J. Elast.
0374-3535, 57
, pp. 159
–170
.33.
Beatty
, M. F.
, 2003, “A Stretch Averaged Full Network Model for Rubber Elasticity
,” J. Elast.
0374-3535, 70
, pp. 65
–86
.34.
Kakavas
, P. A.
, and Blatz
, P. J.
, 2005, “New Constitutive Equation for Unfilled Rubbers Based on Maximum Chain Extensibility Approach
,” Proceedings ECCMR05 Conference Stockholm
, Sweden, June 27
–29
.35.
Williams
, M.
, and Schapery
, R.
, 1965, “Spherical Flow Instability in Hydrostatic Tension
,” Int. J. Fract. Mech.
0020-7268, 1
, pp. 64
–66
.36.
Lin
, Y. Y.
, Hui
, C. Y.
, and Conway
, H. D.
, 2000, “A Detailed Elastic Analysis of the Flat Punch (Tack) Test for Pressure Sensitive Adhesives
,” J. Polym. Sci., Part B: Polym. Phys.
0887-6266, 38
, pp. 2769
–2784
.37.
Brown
, F.
, 1958, Introduction to Bessel Functions
, Dover
, New York.38.
Chalhoub
, M. S.
, and Kelly
, J. M.
, 1990, “Effect of Bulk Compressibility on the Stiffness of Cylindrical Base Isolation Bearings
,” Int. J. Solids Struct.
0020-7683, 26
, pp. 743
–760
,39.
Gent
, A. N.
, 2000, Engineering with Rubber: How to Design Rubber Components
, 2nd ed., Hanser
, Munich, Germany.Copyright © 2007
by American Society of Mechanical Engineers
You do not currently have access to this content.
