Imperfect Bifurcation and Chaos of Slightly Curved Carbon Nanotube Conveying Hot Pressurized Fluid Resting on Foundations

2020 ◽  
Vol 142 (11) ◽  
Author(s):  
Akintoye O. Oyelade ◽  
Ayo A. Oyediran
Keyword(s):  
Thermal Loading ◽  
Dynamic Instability ◽  
Fluid Velocity ◽  
Expansion Method ◽  
Pitchfork Bifurcation ◽  
Length Scale ◽  
Geometric Imperfection ◽  
Material Length Scale ◽  
Material Length ◽  
Material Length Parameter

Abstract Unintended slight curvature of a straight pipe and temperature variation in a pipe has been found to create uncertainties in tubes and pipes. Fluttering, divergence, and chaotic instabilities of slightly curved carbon nanotubes (SCCNT) conveying hot pressurized fluid are investigated in this paper. The SCCNT is modeled on the basis of large deformation strains. Their gradients are included in the strain energy expression and the velocity and its gradients in the kinetic energy derivation. In modeling the size effects, both the static and kinetic length scales in the energy equations were considered. Governing equation is derived using Lagrangian approach. The effects of geometric imperfection (which leads to cusp bifurcation), small length scale, and kinetic material length parameter on the static and dynamic instability characteristics of the pipes are studied. Analysis is performed using the eigenfunction expansion method. It is found that the material length scale parameter increase tends to shift instability to the lower fluid velocity while the kinematic material length parameter increase does not change the buckling point but lowers the frequency. In the nonlinear dynamic case, both the parameters lead to chaos of the nanotube beyond the critical fluid velocity. The thermal loading changes the sudden supercritical pitchfork bifurcation to cusp bifurcation. The increasing linear and nonlinear foundation stiffness leads the system to chaotic features after the critical point.

scholarly journals A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

2019 ◽  
Vol 40 (12) ◽  
pp. 1695-1722 ◽  
Author(s):  
Lu Lu ◽  
Li Zhu ◽  
Xingming Guo ◽  
Jianzhong Zhao ◽  
Guanzhong Liu
Keyword(s):  
Strain Gradient ◽  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Length Scale ◽  
Proposed Model ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

AbstractIn this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

A Modified Gradient Plasticity Theory for Micro-Bending and Micro-Torsion Size Effects

2004 ◽  
Author(s):  
George Z. Voyiadjis ◽  
Rashid K. Abu Al-Rub
Keyword(s):  
Size Effects ◽  
Scale Parameter ◽  
Plasticity Theory ◽  
Material Behavior ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Gradient Plasticity ◽  
Non Local ◽  
Material Length Scale ◽  
Material Length

The definition and magnitude of the intrinsic length scale are keys to the development of the theory of plasticity that incorporates size effects. Gradient plasticity theory with a material length scale parameter is successfully in capturing the size dependence of material behavior at the micron scale. However, a fixed value of the material length-scale is not always realistic and that different problems could require different values. Moreover, a linear coupling between the local and non-local terms in the gradient plasticity theory is not always realistic and that different problems could require different couplings. A generalized gradient plasticity model with a non-fixed length scale parameter is proposed. This model assesses the sensitivity of predictions in the way in which the local and non-local parts are coupled. The proposed model gives good predictions of the size effect in micro-bending tests of thin films and micro-torsion tests of thin wires.

Localization-Induced Band and Cohesive Model1

2000 ◽  
Vol 67 (4) ◽  
pp. 803-812 ◽  
Author(s):  
S. Hao ◽  
W. K. Liu ◽  
D. Qian
Keyword(s):  
Strain Gradient ◽  
Multiple Scale ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Finite Width ◽  
Opening Displacement ◽  
One Dimensional ◽  
Material Length Scale ◽  
Material Length

A localization-induced cohesive model has been proposed for shear band evolution, crack growth, and fracture. Strain gradient theory has been applied to establish the criterion of the onset of localization and the governing equation in the post-bifurcation stage. Analytical solutions in one-dimensional case are used to establish the “traction-separation” law, in which strain gradient and material intrinsic length scale present strong effects. In addition, the solution predicts a finite width for the localization-induced band. It is observed that a larger length scale contributes to the growth of a larger width of localization region and separation for softening materials. The proposed model provides a procedure to establish the fracture toughness analytically since the material length scale is taken into account. From the traction-separation analysis, it is found that damage decreases separation, whereas an increase in material length scale increases the opening displacement; however, the traction-normalized opening displacement curves (with respect to the material length scale) are identical. Based on the methodology of multiple scale analysis in meshfree method, a computational approach has been proposed to enrich the one-dimensional traction-separation law to define fracture. [S0021-8936(00)01104-1]

Study of Small Scale Effects on the Nonlinear Vibration Response of Functionally Graded Timoshenko Microbeams Based on the Strain Gradient Theory

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
R. Ansari ◽  
R. Gholami ◽  
S. Sahmani
Keyword(s):  
Strain Gradient ◽  
Beam Theory ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Length Scale ◽  
Scale Parameters ◽  
Linear And Nonlinear ◽  
Material Length Scale ◽  
Material Length

In the current study, the nonlinear free vibration behavior of microbeams made of functionally graded materials (FGMs) is investigated based on the strain gradient elasticity theory and von Karman geometric nonlinearity. The nonclassical beam model is developed in the context of the Timoshenko beam theory which contains material length scale parameters to take the size effect into account. The model can reduce to the beam models based on the modified couple stress theory (MCST) and the classical beam theory (CBT) if two or all material length scale parameters are taken to be zero, respectively. The power low function is considered to describe the volume fraction of the ceramic and metal phases of the FGM microbeams. On the basis of Hamilton’s principle, the higher-order governing differential equations are obtained which are discretized along with different boundary conditions using the generalized differential quadrature method. The dimensionless linear and nonlinear frequencies of microbeams with various values of material property gradient index are calculated and compared with those obtained based on the MCST and an excellent agreement is found. Moreover, comparisons between the various beam models on the basis of linear and nonlinear types of strain gradient theory (SGT) and MCST are presented and it is observed that the difference between the frequencies obtained by the SGT and MCST is more significant for lower values of dimensionless length scale parameter.

Cusp bifurcation of slightly curved tensioned pipe conveying hot pressurized fluid

2018 ◽  
Vol 25 (5) ◽  
pp. 1109-1121 ◽  
Author(s):  
Kehinde O. Orolu ◽  
Theophilus A. Fashanu ◽  
Ayo A. Oyediran
Keyword(s):  
Longitudinal Vibration ◽  
Equations Of Motion ◽  
Expansion Method ◽  
Pitchfork Bifurcation ◽  
Straight Pipe ◽  
Geometric Imperfection ◽  
Eigenfunction Expansion Method ◽  
Catastrophic Failures ◽  
Nonlinear Equations Of Motion ◽  
High Temperature High Pressure

In the course of operation of pipes conveying high temperature-high pressure fluid, unexpected behaviors leading to catastrophic failures have been observed. These have been attributed to uncertainties arising from issues not adequately addressed in the design. Sources of such uncertainties include geometric imperfection of the pipe and temperature variation. The perfectly straight pipe is assumed in most designs, but it is an idealization that does not exist in practice. In a bid to reduce the number of uncertainties in design and operation, a model governing nonlinear vibration of tensioned pipes conveying hot pressurized fluid that accounts for the geometric imperfection of the pipe is developed in this work. Coupled nonlinear equations of motion in both axial and transverse directions are obtained and solved using the eigenfunction expansion method. The influence of initial curvature, temperature, and the longitudinal vibration on the pipe are investigated. The results obtained show that a pipe with geometric imperfection exhibits cusp bifurcation and not supercritical pitchfork bifurcation.

scholarly journals Flow-induced vibration and stability analysis of carbon nanotubes based on the nonlocal strain gradient Timoshenko beam theory

2018 ◽  
Vol 25 (1) ◽  
pp. 203-218 ◽  
Author(s):  
Reza Bahaadini ◽  
Ali Reza Saidi ◽  
Mohammad Hosseini
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Strain Gradient ◽  
Size Effects ◽  
Length Scale ◽  
Governing Equations ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

A nonlocal strain gradient Timoshenko beam model is developed to study the vibration and instability analysis of the carbon nanotubes conveying nanoflow. The governing equations of motion and boundary conditions are derived by employing Hamilton’s principle, including the effects of moving fluid, material length scale and nonlocal parameters, Knudsen number and gravity force. The material length scale and nonlocal parameters are considered, in order to take into account the size effects. Also, to consider the small-size effects on the flow field, the Knudsen number is used as a discriminant parameter. The Galerkin approach is chosen to analyze the governing equations under clamped–clamped, clamped–hinged and hinged–hinged boundary conditions. It is found that the natural frequency and critical fluid velocity can be decreased by increasing the nonlocal parameter or decreasing the material length scale parameter. Furthermore, it is revealed that the critical flow velocity does not affected by two size-dependent parameters and various boundary conditions in the free molecular flow regime.

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