Nonlinear Viscoelastic Elliptical Bar Under Combined Torsion with Tension
In [1] DARWISH and CANNON considered the problem of twisting of a cylindrical bar of elliptic cross-section. The bar was assumed to consist of a viscoelastic material. The principal cubic theory of nonlinear viscoelasticity was used to obtain a nonlinear Volterra integral equation of the second kind for the angle of twist θ(t) as a function of time t. In this work we consider a cylindrical bar of elliptic cross-section which consists of a viscoelastic material. At first, we subject the bar to torsion around the axis of the bar as we did in the previous work [1]. After an interval of time, we subject the bar to a longitudinal axial tensile force, in addition to torsion, for an additional period of time. We derive a system of two nonlinear Volterra integral equations of the second kind for the angle of twist θ(t) and the relative extension function γ(t). We prove the existence and uniqueness of the solution pair (θ(t), γ(t)) and analyze a numerical procedure for the approximate solution of the pair (θ(t), γ(t)). Results of a numerical study of (θ(t), γ(t)) are presented for both the linear and nonlinear theory. The behaviour of the normal axial stress and the shearing stresses are also investigated for each pair (θ(t), γ(t)), and the results for both the linear and nonlinear theory are presented.
References
D.M. DARWISH and J.R. CANNON, Twisting of elliptical bars of nonlinear viscoelastic material, Bull. Fac. Sc. Alex. Univ., 35, 17-36, 1995.
D.M. DARWISH, Comparison of the principal cubic theory of viscoelasticity with experimental data, Viestnik MGU, Moscow, USSR, Vol. 4, 89-97, 1976.
D.M. DARWISH, Stress distribution for nonlinear viscoelastic material under abrupt changes in the state of stress, Presentation at the ORSA/TIMS International Meeting, Nashville, Tennessee, USA May 12-15, 1991.
J.S.Y. LAI and W.N. FINDLEY, Behavior of nonlinear viscoelastic material under simultaneous stress relaxation in tension and creep in torsion, Transactions of the ASME, J. Applied Mech., 22-28, 1969.
K.G. NOLTE and W.N. FINDLEY, A linear compressibility assumption for the multiple integral representation of nonlinear creep of polyurethane, Transactions of ASME, J. Applied Mech., 441-448, 1970.
J.L. DING and W.N. FINDLEY, Simultaneous and mixed stress relaxation in tension and creep in torsion of 2618 aluminum, Transactions of the ASME, J. Applied Mech., 53, 529-536, 1986.
R.T. SHIELD, Extension and torsion of elastic bars with initial twist, Transactions of ASME, J. Applied Mech., 779-787, 1982.
I.S. SOKOLNIKOFF, Mathematical theory of elasticity, Mc Graw-Hill, N.Y. 1956.
J. R. CANNON, The one-dimensional heat equation, Encyclopedia of Mathematics and its Applications, Vol. 23, pp. 82-100, Addison-Wesley Publishing Company, Reading, Massachusetts 1984.
J. SCHAUDER, Der Fixpunktsatz in Funktionalräumen, Studia Math., 2, 171-180, 1930.