Rotating Variable-Thickness Inhomogeneous Cylinders: Part II—Viscoelastic Solutions and Applications

ABSTRACT

Analytical solutions for the rotating variable-thickness inhomogeneous, orthotropic, hollow cylinders under plane strain assumption are developed in Part I of this paper. The extensions of these solutions to the viscoelastic case are discussed here. The method of effective moduli and Illyushin's approximation method are used for this purpose. The rotating fiber-reinforced viscoelastic homogeneous isotropic hollow cylinders with uniform thickness are obtained as special cases of the studied problem. Numerical application examples are given for the dimensionless displacement of and stresses in the different cylinders. The influences of time, constitutive parameter and elastic properties on the stresses and displacement are investigated.

Analytical solutions for the rotating variable-thickness inhomogeneous, orthotropic, hollow cylinders under plane strain assumption are developed in Part I of this paper. The extensions of these solutions to the viscoelastic case are discussed here. The method of effective moduli and Illyushin's approximation method are used for this purpose. The rotating fiber-reinforced viscoelastic homogeneous isotropic hollow cylinders with uniform thickness are obtained as special cases of the studied problem. Numerical application examples are given for the dimensionless displacement of and stresses in the different cylinders. The influences of time, constitutive parameter and elastic properties on the stresses and displacement are investigated.

Cite this paper

nullA. Zenkour, "Rotating Variable-Thickness Inhomogeneous Cylinders: Part II—Viscoelastic Solutions and Applications,"*Applied Mathematics*, Vol. 1 No. 6, 2010, pp. 489-498. doi: 10.4236/am.2010.16064.

nullA. Zenkour, "Rotating Variable-Thickness Inhomogeneous Cylinders: Part II—Viscoelastic Solutions and Applications,"

References

[1] I. E. Troyanovskii and M. A. Koltunov, “Temperature Stresses in a Long Hollow Viscoelastic Cylinder with Variable Inner Boundary,” Mekhanika Polimerov, Vol. 5, No. 2, 1969, pp. 219-226.

[2] M. A. Koltunov and I. E. Troyanovskii, “State of Stress of a Hollow Viscoelastic Cylinder Whose Material Properties Depend on Temperature,” Mechanics of Composite Materials, Vol. 6, No. 1, 1970, pp. 72-79.

[3] E. C. Ting and J. L. Tuan, “Effect of Cyclic Internal Pressure on the Temperature Distribution in a Viscoelastic Cylinder,” International Journal of Mechanical Sciences, Vol. 15, No. 11, 1973, pp. 861-871.

[4] L. Kh. Talybly, “Deformation of a Viscoelastic Cylinder Fastened to a Housing under Non-Isothermal Dynamic Loading,” Journal of Applied Mathematics and Mechanics, Vol. 54, No. 1, 1990, pp. 74-82.

[5] W. W. Feng, T. Hung and G. Chang, “Extension and Torsion of Hyperviscoelastic Cylinders,” International Journal of Non-Linear Mechanics, Vol. 27, No. 3, 1992, pp. 329-335.

[6] V. G. Karnaukhov and I. K. Senchenkov, “ThermomeChanical Behavior of a Viscoelastic Finite Circular Cylinder under Harmonic Deformations,” Journal of Engineering Mathematics, Vol. 46, No. 3-4, 2003, pp. 299-312.

[7] D. Bland, “The Linear Theory of Viscoelasticity,” Pergamon Press, New York, 1960.

[8] D. Abolinsh, “Elasticity Tensor for Unidirectionally Reinforced Elastic Material,” Polymer Mechanics, Vol. 4, No. 1, 1965, pp. 25-59.

[9] A. A. Illyushin and B. E. Pobedria, “Foundations of Ma- thematical Theory of Thermo-Viscoelasticity,” in Russian, Nauka, Moscow, 1970.

[10] M. N. M. Allam and P. G. Appleby, “On the Plane Deformation of Fiber-Reinforced Viscoelastic Plates,” Applied Mathematical Modelling, Vol. 9, No. 5, 1985, pp. 341-346.

[11] M. N. M. Allam and P. G. Appleby, “On the Stress Concentrations around a Circular Hole in a Fiber-Reinforced Viscoelastic Plate,” Res Mechanica, Vol. 19, No. 2, 1986, pp. 113-126.

[12] M. N. M. Allam and A. M. Zenkour, “Bending Response of a Fiber-Reinforced Viscoelastic Arched Bridge Model,” Applied Mathematical Modelling, Vol. 27, No. 3, 2003, pp. 233-248.

[13] A. M. Zenkour and M. N. M. Allam, “Stresses around Filled and Unfilled Circular Holes in a Fiber-Reinforced Viscoelastic Plate under Bending,” Mechanics of Advanced Materials and Structures, Vol. 12, No. 6, 2005, pp. 379-389.

[14] A. M. Zenkour and M. N. M. Allam, “On the Rotating Fiber-Reinforced Viscoelastic Composite Solid and Annular Disks of Variable Thickness,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 1, 2006, pp. 21-31.

[15] M. N. M. Allam, A. M. Zenkour and H. F. El-Mekawy, “Stress Concentrations in a Viscoelastic Composite Plate Weakened by a Triangular Hole,” Composite Structures, Vol. 79, No. 1, 2007, pp. 1-11.

[16] A. M. Zenkour, K. A. Elsibai and D. S. Mashat, “Elastic and Viscoelastic Solutions to Rotating Functionally Graded Hollow and Solid Cylinders,” Applied Mathematics and Mechanics - English Edition, Vol. 29, No. 12, 2008, pp. 1601-1616.

[17] A. E. Bogdanovich and C. M Pastore, “Mechanics of Textile and Laminated Composites with Applications to Structural Analysis,” Chapman and Hall, New York, 1996.

[18] B. E. Pobedria, “Structural Anisotropy in Viscoelasticity,” Mechanics of Composite Materials, Vol. 12, No. 4, 1976, pp. 557-561.

[19] A. M. Zenkour, “Thermal Effects on the Bending Response of Fiber-Reinforced Viscoelastic Composite Plates using a Sinusoidal Shear Deformation Theory,” Acta Mechanica, Vol. 171, No. 3-4, 2004, pp. 171-187.

[20] M. N. M. Allam and B. E. Pobedria, “On the Solution of Quasi-Statical Problems of Anisotropic Viscoelasticity,” Izv. Acad. Nauk ArSSR, In Russian, Mechanics, Vol. 31, No. 1, 1978, pp. 19-27.

[1] I. E. Troyanovskii and M. A. Koltunov, “Temperature Stresses in a Long Hollow Viscoelastic Cylinder with Variable Inner Boundary,” Mekhanika Polimerov, Vol. 5, No. 2, 1969, pp. 219-226.

[2] M. A. Koltunov and I. E. Troyanovskii, “State of Stress of a Hollow Viscoelastic Cylinder Whose Material Properties Depend on Temperature,” Mechanics of Composite Materials, Vol. 6, No. 1, 1970, pp. 72-79.

[3] E. C. Ting and J. L. Tuan, “Effect of Cyclic Internal Pressure on the Temperature Distribution in a Viscoelastic Cylinder,” International Journal of Mechanical Sciences, Vol. 15, No. 11, 1973, pp. 861-871.

[4] L. Kh. Talybly, “Deformation of a Viscoelastic Cylinder Fastened to a Housing under Non-Isothermal Dynamic Loading,” Journal of Applied Mathematics and Mechanics, Vol. 54, No. 1, 1990, pp. 74-82.

[5] W. W. Feng, T. Hung and G. Chang, “Extension and Torsion of Hyperviscoelastic Cylinders,” International Journal of Non-Linear Mechanics, Vol. 27, No. 3, 1992, pp. 329-335.

[6] V. G. Karnaukhov and I. K. Senchenkov, “ThermomeChanical Behavior of a Viscoelastic Finite Circular Cylinder under Harmonic Deformations,” Journal of Engineering Mathematics, Vol. 46, No. 3-4, 2003, pp. 299-312.

[7] D. Bland, “The Linear Theory of Viscoelasticity,” Pergamon Press, New York, 1960.

[8] D. Abolinsh, “Elasticity Tensor for Unidirectionally Reinforced Elastic Material,” Polymer Mechanics, Vol. 4, No. 1, 1965, pp. 25-59.

[9] A. A. Illyushin and B. E. Pobedria, “Foundations of Ma- thematical Theory of Thermo-Viscoelasticity,” in Russian, Nauka, Moscow, 1970.

[10] M. N. M. Allam and P. G. Appleby, “On the Plane Deformation of Fiber-Reinforced Viscoelastic Plates,” Applied Mathematical Modelling, Vol. 9, No. 5, 1985, pp. 341-346.

[11] M. N. M. Allam and P. G. Appleby, “On the Stress Concentrations around a Circular Hole in a Fiber-Reinforced Viscoelastic Plate,” Res Mechanica, Vol. 19, No. 2, 1986, pp. 113-126.

[12] M. N. M. Allam and A. M. Zenkour, “Bending Response of a Fiber-Reinforced Viscoelastic Arched Bridge Model,” Applied Mathematical Modelling, Vol. 27, No. 3, 2003, pp. 233-248.

[13] A. M. Zenkour and M. N. M. Allam, “Stresses around Filled and Unfilled Circular Holes in a Fiber-Reinforced Viscoelastic Plate under Bending,” Mechanics of Advanced Materials and Structures, Vol. 12, No. 6, 2005, pp. 379-389.

[14] A. M. Zenkour and M. N. M. Allam, “On the Rotating Fiber-Reinforced Viscoelastic Composite Solid and Annular Disks of Variable Thickness,” International Journal for Computational Methods in Engineering Science and Mechanics, Vol. 7, No. 1, 2006, pp. 21-31.

[15] M. N. M. Allam, A. M. Zenkour and H. F. El-Mekawy, “Stress Concentrations in a Viscoelastic Composite Plate Weakened by a Triangular Hole,” Composite Structures, Vol. 79, No. 1, 2007, pp. 1-11.

[16] A. M. Zenkour, K. A. Elsibai and D. S. Mashat, “Elastic and Viscoelastic Solutions to Rotating Functionally Graded Hollow and Solid Cylinders,” Applied Mathematics and Mechanics - English Edition, Vol. 29, No. 12, 2008, pp. 1601-1616.

[17] A. E. Bogdanovich and C. M Pastore, “Mechanics of Textile and Laminated Composites with Applications to Structural Analysis,” Chapman and Hall, New York, 1996.

[18] B. E. Pobedria, “Structural Anisotropy in Viscoelasticity,” Mechanics of Composite Materials, Vol. 12, No. 4, 1976, pp. 557-561.

[19] A. M. Zenkour, “Thermal Effects on the Bending Response of Fiber-Reinforced Viscoelastic Composite Plates using a Sinusoidal Shear Deformation Theory,” Acta Mechanica, Vol. 171, No. 3-4, 2004, pp. 171-187.

[20] M. N. M. Allam and B. E. Pobedria, “On the Solution of Quasi-Statical Problems of Anisotropic Viscoelasticity,” Izv. Acad. Nauk ArSSR, In Russian, Mechanics, Vol. 31, No. 1, 1978, pp. 19-27.