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Abstract

This work considers the uniaxial compression of a solid circular cylinder of time-dependent material. Initially, the cylinder undergoes a homogeneous compression history. The purpose is to determine a time when this deformation history can form a new inhomogenous branch. Material time dependence is described by a nonlinear single-integral constitutive equation that relates the stress in the material to its deformation history. A criterion is developed for determining branching times using a Pipkin–Rogers constitutive equation for nonlinear incompressible isotropic viscoelastic solids. For the purpose of numerical examples, material parameters are chosen so that the material acts as a neo-Hookean material in its short-time response and a softer one in its long-time equilibrium response. Examples show that characteristic relaxation times and deformation histories influence the time to branch and the corresponding compressive stretch and compressive force.

Keywords

Deformation history branching branching criterion nonlinear viscoelasticity integral constitutive equations

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References

Love

AEH

The mathematical theory of elasticity. 4th ed. Cambridge: Cambridge University Press, 1927, Reprinted New York: Dover Publications, 1944.

Reynolds

. On the linearized theory of buckling for a compressed viscoelastic rod. Proc Royal Soc Edinburgh 1985; 99A: 371–386.

Sackman

. On the buckling of aging linearly viscoelastic beam-columns. AIAA J 1967; 5: 1726–1728.

Wilkes

. On the stability of a circular tube under end thrust. Quart J Mech Appl Math 1955; 8: 88–100.

Dorfmann

Haughton

. Stability bifurcation of compressed elastic cylindrical tubes. Int J Eng Sci 2006; 44: 1353–1365.

Wineman

. Bifurcation of response of a nonlinear viscoelastic spherical membrane. Int J Solids Struct 1978; 14: 197–212.

Wineman

. Branching of the radial expansion history during inflation of a nonlinear incompressible isotropic time dependent thick walled sphere. Math Mech Solids 2011; 16: 513–523.

Wineman

. The Treloar-Kearsley problem for elastomeric materials undergoing time dependent microstructural evolution - branching of extensional creep histories. Math Mech Solids 2012; 17: 300–316.

Wineman

. Branching of stretch histories in biaxially loaded nonlinear viscoelastic fiber reinforced sheets. Math Mech Solids 2019; 24: 807–827.

10.

Wineman

. Determining the time of bulge formation in an elastomeric tube as it inflates, elongates and alters chemorheologically. Math Mech Solids 2015; 20: 9–24.

11.

Wineman

. Bulge initiation in tubes of time dependent materials. Math Mech Solids 2017; 22: 636–648.

12.

Rajagopal

Srinivasa

. Inelastic behavior of materials Part I: theoretical underpinnings. Int J Plast 1998; 14: 945–967.

13.

Wineman

. Nonlinear viscoelastic solids: a review. Math Mech Solids 2009; 14: 300–366.

14.

Pipkin

Rogers

. A nonlinear integral representation for viscoelasticity. J Mech Phys Solids 1968; 16: 59–70.

15.

Coleman

Noll

. Foundations of linear viscoelasticity. Rev Modern Phys 1961; 33: 239–249.

16.

Wineman

Rajagopal

. Mechanical response of polymers: an introduction. Cambridge: Cambridge University Press, 2001.

17.

Knauss

Emri

. Volume change and nonlinearly thermo-viscoelastic constitution of polymers. Polym Eng Sci 1987; 27: 86–100.

18.

Shaw

Jones

Wineman

. Chemorheological response of elastomers at elevated temperatures and simulations. J Mech Phys Solids 2005; 53: 2758–2793.

Branching during axial compression histories of solid cylinders of time-dependent materials

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