Investigation of the axial buckling and dynamic stability of functionally graded microshells based on the modified coupled stress theory

Authors

Abstract

Based on the modified couple stress theory, a size-dependent first-order shear deformation shell model is developed to investigate the axial buckling and dynamic stability of functionally graded microshells. The newly presented shell model capable of capturing the size effects. The higher-order governing equations and corresponding boundary conditions are obtained by using Hamilton’s principle. Afterward, using the Navier solution and Bolotin’s method, the axial buckling behavior and dynamic instability regions of simply-supported microshells are determined. A parametric study is presented to study the effects of various parameters such as the static load factor, dimensionless length scale parameter, material property gradient index, length-to-radius and length-to-thickness aspect ratios on the axial buckling and dynamic stability responses of microshells. 

Keywords

References

[1] Meschet, M.J., Brown, J.Q., Guice, K.B., and Lvov, Y.M., "Polyelectrolyte Microshells as Carriers for Fluorescent Sensors: Loading and Sensing Properties of a Ruthenium-based Oxygen Indicator", Journal of Nanoscience and Nanotechnology, Vol. 2, pp. 411-416, (2002).
 
[2] Mescher, M.J., Houston, K., Bernstein, J.J., Kirkos, G.A., Cheng, J., and Cross, L.E., "Novel MEMS Microshell Transducer Arrays for High-resolution Underwater Acoustic Imaging Applications", Sensors, Vol. 1, pp. 541-546, (2002).
 
[3] Chong, A.C.M., and Lam, D.C.C., "Strain Gradient Plasticity Effect in Indentation Hardness of Polymers", Journal of Materials Research, Vol. 14, pp. 4103-4110, (1999).
 
[4] Fleck, N.A., Muller, G.M., Ashby, M.F., and Hutchinson, J.W., "Strain Gradient Plasticity: Theory and Experiment", Acta Metallurgica et Materialia, Vol. 42, pp. 475-487, (1994).
 
[5] Stolken, J.S., and Evans, A.G., "Microbend Test Method for Measuring the Plasticity Length Scale", Acta Metallurgical Materialia, Vol. 46, pp. 5109-5115, (1998).
 
[6] Ansari, R., Sahmani, S., and Arash, B., "Nonlocal Plate Model for Free Vibrations of Single-layered Graphene Sheets", Physics Letters A, Vol. 375, pp. 53-62, (2010).
[7] Ansari, R., Sahmani, S., and Rouhi, H., "Axial Buckling Analysis of Single-walled Carbon Nanotubes in Thermal Environments via Rayleigh-Ritz Technique", Computational Materials Science, Vol. 50, pp. 3050-3055, (2011).
 
[8] Ansari, R., Rouhi, H., and Sahmani, S., "Calibration of the Analytical Nonlocal Shell Model for Vibrations of Double-walled Carbon Nanotubes with Arbitrary Boundary Conditions using Molecular Dynamics", International Journal of Mechanical Sciences, Vol. 53, pp. 786-792, (2011).
 
[9] Ansari, R., Gholami, R., and Darabi, M. A., "A Nonlinear Timoshenko Beam Formulation Based on Strain Gradient Theory", Journal of Mechanics of Materials and Structures, Vol. 7, pp. 195-211, (2012).
 
[10] Ansari, R., Rouhi, H., and Sahmani, S., "Thermal Effect on Axial Buckling Behavior of Multi-walled Carbon Nanotubes Based on Nonlocal Shell Model", Physica E, Vol. 44, pp. 373-378, (2011).
 
[11] Ansari, R., Gholami, R., and Sahmani, S., "Free Vibration of Size-dependent Functionally Graded Microbeams Based on a Strain Gradient Theory", Composite Structures, Vol. 94, pp. 221-228, (2011).
 
[12] Sahmani, S., and Ansari, R., "On the Free Vibration Response of Functionally Graded Higher-order Shear Deformable Microplates Based on the Strain Gradient Elasticity Theory", Composite Structures, Vol. 95, pp. 430-442, (2013).
 
[13] Mindlin, R.D., and Tiersten, H.F., "Effects of Couple-stresses in Linear Elasticity", Archive for Rational Mechanics and Analysis, Vol. 11, pp. 415-448, (1962).
 
[14] Koiter, W.T., "Couple Stresses in the Theory of Elasticity I and II", Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, B, Vol. 67, pp. 17-44, (1964).
 
[15] Eringen, A.C., and Suhubi, E.S., "Nonlinear Theory of Simple Microelastic Solid-I", International Journal of Engineering Science, Vol. 2, pp. 189-203, (1964).
 
[16] Eringen, A.C., and Suhubi, E.S., "Nonlinear Theory of Simple Microelastic Solid-II", International Journal of Engineering Science, Vol. 2, pp.389-404, (1964).
 
[17] Mindlin, R.D., "Micro-Structure in Linear Elasticity", Archive for Rational Mechanics and Analysis, Vol. 16, pp. 51-78, (1964).
 
[18] Toupin, R.A., "Theory of Elasticity with Couple Stresses", Archive for Rational Mechanics and Analysis, Vol. 17, pp. 85-112, (1964).
 
[19] Mindlin, R.D., "Second Gradient of Strain and Surface Tension in Linear Elasticity", International Journal of Solids and Structures, Vol. 1, pp. 417-438, (1965).
 
[20] Mindlin, R.D., and Eshel, N.N., "On First Strain-gradient Theories in Linear Elasticity", International Journal of Solids and Structures, Vol. 4, pp. 109-124, (1968).
 
 
[21] Eringen, A.C., "On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocation and Surface Waves", Journal of Applied Physics, Vol. 54, pp. 4703-4710, (1983).
 
[22] Vardoulaksi, I., Exadaktylos, G., and Kourkoulis, S.K., "Bending of Marble with Intrinsic Length Scales: A Gradient Theory with Surface Energy and Size Effects", Journal De Physique, Vol. 8, pp. 399-406, (1998).
 
[23] Yang, F., Chong, A.C.M., Lam, D.C.C., and Tong, P., "Couple Stress Based Strain Gradient Theory for Elasticity", International Journal of Solids and Structures, Vol. 39, pp. 2731–2743, (2002).
 
[24] Xia, W., Wang, L., and Yin, L., "Nonlinear Non-classical Microscale Beams: Static Bending, Postbuckling and Free Vibration", International Journal of Engineering Science, Vol. 48, pp. 2044-2053, (2010).
 
[25] Asghari, M., Rahaeifard, M., Kahrobaiyan, M.H., and Ahmadian, M.T., "The Modified Couple Stress Functionally Graded Timoshenko Beam Formulation", Materials and Design, Vol. 32, pp.1435-1443, (2011).
 
[26] Ke, L.L., and Wang, Y.S., "Size Effect on Dynamic Stability of Functionally Graded Microbeams Based on a Modified Couple Stress Theory", Composite Structures, Vol. 93, pp. 342-350, (2011).
 
[27] Ma, H.M., Gao, X.-L., and Reddy, J.N., "A Microstructure-dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory", Journal of the Mechanics and Physics of Solids, Vol. 56, pp.3379-3391, (2008).
 
[28] Ke, L.L., Wang, Y.S., and Wang, Z.D., "Thermal Effect on Free Vibration and Buckling of Size-dependent Microbeams", Physica E, Vol. 43, pp.1387-1393, (2011).
 
[29] Ke, L. L., Wang, Y. S., Yang, J., and Kitipornchai, S., "Free Vibration of Size-dependent Mindlin Microplates Based on the Modified Couple Stress Theory", Journal of Sound and Vibration, Vol. 331, pp. 94-106, (2012).
 
[30] Papargyri-Beskou, S., Tsinopoulos, S. V., and Beskos, D. E., "Wave Propagation in and Free Vibrations of Gradient Elastic Circular Cylindrical Shells", Acta Mechanica, Vol. 223, pp. 1789–1807, (2012).
 
[31] Lazopoulos, K.A., and Lazopoulos, A.K., "Nonlinear Strain Gradient Elastic Thin Shallow Shells", European Journal of Mechanics A/Solids, Vol. 30, pp. 286-292, (2011).
 
[32] Papargyri-Beskou, S., and Beskos, D.E., "Stability Analysis of Gradient Elastic Circular Cylindrical Thin Shells", International Journal of Engineering Science, Vol. 47, pp. 1379–1385, (2009).
 
[33] Altenbach, J., Altenbach, H., and Eremeyev, V.A., "On Generalized Cosserat-type Theories of Plates and Shells: A Short Review and Bibliography", Archive of Applied Mechanics, Vol. 80, pp. 73–92, (2010).
 
[34] Ganapathi, M., "Dynamic Stability Characteristics of Functionally Graded Materials Shallow Spherical Shells", Composite Structures, Vol. 79, pp. 338–343, (2007).
 
[35] Fares, M.E., Elmarghany, M.K., and Atta, D., "An Efficient and Simple Refined Theory for Bending and Vibration of Functionally Graded Plates", Composite Structures, Vol. 91, pp. 296–305, (2009).
 
[36] Donnell, L.H., "Beam, Plates and Shells", McGraw-Hill, New York, USA, (1976).
 
[37] Bolotin, V.V., "The Dynamic Stability of Elastic Systems", San Francisco, Holden- Day, (1964).
 
[38] Lam, D.C.C., Yang, F., Chong, A.C. M., Wang, J., and Tong, P., "Experiments and Theory in Strain Gradient Elasticity", Journal of the Mechanics and Physics of Solids, Vol. 51, pp. 1477–1508, (2003).
Iranian Journal of Mechanical Engineering Transactions of ISME
Volume 16, Issue 1 - Serial Number 34
System Dyanamics and Solid Mechanics
June 2014
Pages 78-102

Files

History

  • Receive Date: 22 May 2014
  • Accept Date: 22 May 2014

Share

How to cite

Statistics