Combined effects of axial load and temperature on finite deformation of incompressible thermo-hyperelastic cylinder
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摘要
A finite deformation problem is examined for a cylinder composed of a class of incompressible thermo-hyperelastic Mooney-Rivlin materials under an equal axial load at its two fixed ends and a temperature field at its lateral boundary. Firstly, a thermomechanical coupling term is taken into account in the strain energy density function, and a governing equation of the problem is obtained. Secondly, an implicit analytical solution is derived by using the incompressibility and the boundary conditions. Significantly, numerical examples show that the middle portion of the cylinder undergoes almost a uniform radial deformation. However, the deformation near the two ends varies remarkably along the axial direction for relatively large axial loads. In addition, the rising temperature can increase the deformation of structures, and its influence is linear approximately. Specially,in the case of tensile load, the jump increase of the axial deformation may occur.
A finite deformation problem is examined for a cylinder composed of a class of incompressible thermo-hyperelastic Mooney-Rivlin materials under an equal axial load at its two fixed ends and a temperature field at its lateral boundary. Firstly, a thermomechanical coupling term is taken into account in the strain energy density function, and a governing equation of the problem is obtained. Secondly, an implicit analytical solution is derived by using the incompressibility and the boundary conditions. Significantly, numerical examples show that the middle portion of the cylinder undergoes almost a uniform radial deformation. However, the deformation near the two ends varies remarkably along the axial direction for relatively large axial loads. In addition, the rising temperature can increase the deformation of structures, and its influence is linear approximately. Specially,in the case of tensile load, the jump increase of the axial deformation may occur.
A finite deformation problem is examined for a cylinder composed of a class of incompressible thermo-hyperelastic Mooney-Rivlin materials under an equal axial load at its two fixed ends and a temperature field at its lateral boundary. Firstly, a thermomechanical coupling term is taken into account in the strain energy density function, and a governing equation of the problem is obtained. Secondly, an implicit analytical solution is derived by using the incompressibility and the boundary conditions. Significantly, numerical examples show that the middle portion of the cylinder undergoes almost a uniform radial deformation. However, the deformation near the two ends varies remarkably along the axial direction for relatively large axial loads. In addition, the rising temperature can increase the deformation of structures, and its influence is linear approximately. Specially,in the case of tensile load, the jump increase of the axial deformation may occur.
引文
[1]FU,Y.B.and OGDEN,R.W.Nonlinear stability analysis of pre-stressed elastic bodies.Continuum Mechanics and Thermo-Dynamics,11(2),141-172(1999)
[2]GENT,A.N.Elastic instabilities in rubber.International Journal of Non-Linear Mechanics,40,165-175(2005)
[3]REN,J.S.Elastic instability of pseudo-elastic rubber balloons.Computers,Materials and Continua,7(1),25-31(2008)
[4]WANG,F.F.and DAI,H.H.Asymptotic bifurcation analysis and post-buckling for uniaxial compression of a thin incompressible hyperelastic rectangle.IMA Journal of Applied Mathematics,75(4),506-524(2010)
[5]PEARCE,S.P.and FU,Y.B.Characterization and stability of localized bulging/necking in inflated membrane tubes.IMA Journal of Applied Mathematics,75(4),581-602(2010)
[6]HILL,J.M.and ARRIGO,D.J.New families of exact solutions for finitely deformed incompressible elastic materials.IMA Journal of Applied Mathematics,54(2),109-123(1995)
[7]CHEN,Y.and HAUGHTON,D.M.Existence of exact solutions for the eversion of elastic cylinders.Journal of Elasticity,49(1),79-88(1997)
[8]HILL,J.M.Exact integrals and solutions for finite deformations of the incompressible Varga elastic materials.London Mathematical Society Lecture Note Series,283,160-200(2001)
[9]DAI,H.H.,HAO,Y.,and CHEN,Z.On constructing the analytical solutions for localizations in a slender cylinder composed of an incompressible hyperelastic material.International Journal of Solids and Structures,45(9),2613-2628(2008)
[10]ROONEY,F.and EBERHARD,S.Exact solutions in finite compressible elasticity via the complementary energy function.Mathematics and Mechanics of Solids,21(9),1116-1125(2016)
[11]BAGHERI,A.,TAGHIZADEH,D.,and DARIJANI,H.On the behavior of rotating thick-walled cylinders made of hyperelastic materials.Meccanica,51(3),673-692(2016)
[12]ANANI,Y.and RAHIMI,G.H.Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials.International Journal of Mechanical Sciences,108,122-128(2016)
[13]KLINGBEIL,W.W.and SHIELD,R.T.Large-deformation analyses of bonded elastic mounts.Zeitschrift f¨ur Angewandte Mathematik und Physik,17(2),281-305(1966)
[14]ZIDI,M.and CHEREF,M.Finite deformations of a hyperelastic,compressible and fibre reinforced tube.European Journal of Mechanics-A/Solids,21(6),971-980(2002)
[15]MERODIO,J.and OGDEN,R.W.Extension,inflation and torsion of a residually stressed circular cylindrical tube.Continuum Mechanics and Thermodynamics,28(1-2),157-174(2016)
[16]KANNER,L.M.and HORGAN,C.O.On extension and torsion of strain-stiffening rubber-like elastic circular cylinders.Journal of Elasticity,93(1),39-61(2008)
[17]DAI,H.H.and BI,Q.Exact solutions for the large axially symmetric deformations of a neoHookean rod subjected to static loads.The Quarterly Journal of Mechanics and Applied Mathematics,54(1),39-56(2001)
[18]DAI,H.H.and WANG,F.F.On a three-dimensional axisymmetric boundary-value problem of non-linear elastic deformation:asymptotic solution and exponentially small error.International Journal of Engineering Science,45(12),951-967(2007)
[19]HILL,J.M.,PADUKKA,N.,and DAI,H.H.Asymptotic axially symmetric deformations for perfectly elastic neo-Hookean and Mooney materials.Journal of Elasticity,86(2),113-137(2007)
[20]ZHANG,W.Z.,YUAN,X.G.,ZHANG,H.W.,and REN,J.S.Deformation analysis of an incompressible composite cylindrical tube subjected to end axial loads and internal constraint.Science China Physics,Mechanics and Astronomy,57(1),113-121(2014)
[21]NICHOLSON,D.W.and NELSON,N.W.Finite-element analysis in design with rubber.Rubber Chemistry and Technology,63(3),368-406(1990)
[22]NICHOLSON,D.W.and LIN,B.Theory of thermohyperelasticity for near-incompressible elastomers.Acta Mechanica,116(1),15-28(1996)
[23]ALMASI,A.,BAGHANI,M.,and MOALLEMI,A.Thermomechanical analysis of hyperelastic thick-walled cylindrical pressure vessels,analytical solutions and FEM.International Journal of Mechanical Sciences,130,426-436(2017)
[24]BAGHERI,A.,DARIJANI,H.,and DARIJANI,A.On the effect of temperature gradient on the stability of circular tubes made of hyperelastic entropic material.International Journal of Non-Linear Mechanics,95,93-102(2017)
[25]MIEHE,C.Entropic thermoelasticity at finite strains:aspects of the formulation and numerical implementation.Computer Methods in Applied Mechanics and Engineering,120(3-4),243-269(1995)
[2]GENT,A.N.Elastic instabilities in rubber.International Journal of Non-Linear Mechanics,40,165-175(2005)
[3]REN,J.S.Elastic instability of pseudo-elastic rubber balloons.Computers,Materials and Continua,7(1),25-31(2008)
[4]WANG,F.F.and DAI,H.H.Asymptotic bifurcation analysis and post-buckling for uniaxial compression of a thin incompressible hyperelastic rectangle.IMA Journal of Applied Mathematics,75(4),506-524(2010)
[5]PEARCE,S.P.and FU,Y.B.Characterization and stability of localized bulging/necking in inflated membrane tubes.IMA Journal of Applied Mathematics,75(4),581-602(2010)
[6]HILL,J.M.and ARRIGO,D.J.New families of exact solutions for finitely deformed incompressible elastic materials.IMA Journal of Applied Mathematics,54(2),109-123(1995)
[7]CHEN,Y.and HAUGHTON,D.M.Existence of exact solutions for the eversion of elastic cylinders.Journal of Elasticity,49(1),79-88(1997)
[8]HILL,J.M.Exact integrals and solutions for finite deformations of the incompressible Varga elastic materials.London Mathematical Society Lecture Note Series,283,160-200(2001)
[9]DAI,H.H.,HAO,Y.,and CHEN,Z.On constructing the analytical solutions for localizations in a slender cylinder composed of an incompressible hyperelastic material.International Journal of Solids and Structures,45(9),2613-2628(2008)
[10]ROONEY,F.and EBERHARD,S.Exact solutions in finite compressible elasticity via the complementary energy function.Mathematics and Mechanics of Solids,21(9),1116-1125(2016)
[11]BAGHERI,A.,TAGHIZADEH,D.,and DARIJANI,H.On the behavior of rotating thick-walled cylinders made of hyperelastic materials.Meccanica,51(3),673-692(2016)
[12]ANANI,Y.and RAHIMI,G.H.Stress analysis of rotating cylindrical shell composed of functionally graded incompressible hyperelastic materials.International Journal of Mechanical Sciences,108,122-128(2016)
[13]KLINGBEIL,W.W.and SHIELD,R.T.Large-deformation analyses of bonded elastic mounts.Zeitschrift f¨ur Angewandte Mathematik und Physik,17(2),281-305(1966)
[14]ZIDI,M.and CHEREF,M.Finite deformations of a hyperelastic,compressible and fibre reinforced tube.European Journal of Mechanics-A/Solids,21(6),971-980(2002)
[15]MERODIO,J.and OGDEN,R.W.Extension,inflation and torsion of a residually stressed circular cylindrical tube.Continuum Mechanics and Thermodynamics,28(1-2),157-174(2016)
[16]KANNER,L.M.and HORGAN,C.O.On extension and torsion of strain-stiffening rubber-like elastic circular cylinders.Journal of Elasticity,93(1),39-61(2008)
[17]DAI,H.H.and BI,Q.Exact solutions for the large axially symmetric deformations of a neoHookean rod subjected to static loads.The Quarterly Journal of Mechanics and Applied Mathematics,54(1),39-56(2001)
[18]DAI,H.H.and WANG,F.F.On a three-dimensional axisymmetric boundary-value problem of non-linear elastic deformation:asymptotic solution and exponentially small error.International Journal of Engineering Science,45(12),951-967(2007)
[19]HILL,J.M.,PADUKKA,N.,and DAI,H.H.Asymptotic axially symmetric deformations for perfectly elastic neo-Hookean and Mooney materials.Journal of Elasticity,86(2),113-137(2007)
[20]ZHANG,W.Z.,YUAN,X.G.,ZHANG,H.W.,and REN,J.S.Deformation analysis of an incompressible composite cylindrical tube subjected to end axial loads and internal constraint.Science China Physics,Mechanics and Astronomy,57(1),113-121(2014)
[21]NICHOLSON,D.W.and NELSON,N.W.Finite-element analysis in design with rubber.Rubber Chemistry and Technology,63(3),368-406(1990)
[22]NICHOLSON,D.W.and LIN,B.Theory of thermohyperelasticity for near-incompressible elastomers.Acta Mechanica,116(1),15-28(1996)
[23]ALMASI,A.,BAGHANI,M.,and MOALLEMI,A.Thermomechanical analysis of hyperelastic thick-walled cylindrical pressure vessels,analytical solutions and FEM.International Journal of Mechanical Sciences,130,426-436(2017)
[24]BAGHERI,A.,DARIJANI,H.,and DARIJANI,A.On the effect of temperature gradient on the stability of circular tubes made of hyperelastic entropic material.International Journal of Non-Linear Mechanics,95,93-102(2017)
[25]MIEHE,C.Entropic thermoelasticity at finite strains:aspects of the formulation and numerical implementation.Computer Methods in Applied Mechanics and Engineering,120(3-4),243-269(1995)