Analytical and numerical analyses of neo-Hookean thin-wall composite spheres

2018 ◽  
Vol 24 (6) ◽  
pp. 1800-1820
Author(s):  
Gidon Weil ◽  
Gal deBotton
Keyword(s):  
Power Series ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Stretch Ratio ◽  
Local Fields ◽  
Power Series Solution ◽  
Thin Wall ◽  
Shear Moduli ◽  
Infinitesimal Deformations ◽  
Traction Boundary Conditions

Thin-wall composite spheres (TWCSs) are very common in both natural and artificial structures. Their response to mechanical loading was investigated in the past almost solely in the limit of infinitesimal deformations. We examine, within the framework of finite deformation elasticity, the mechanics of incompressible TWCSs with neo-Hookean core and shell phases subjected to general homogeneous displacement and traction boundary conditions. We derive explicit general forms for the displacement and the pressure fields in both phases in terms of a power series about the shear and the tension magnitudes and the shell volume fraction. The predictions of the analytical solutions are analyzed and compared with corresponding results of finite element simulations for TWCSs with different ratios between the phases shear moduli. In addition to an extension of the work of Weil and deBotton [22] from simple shear to general homogeneous boundary conditions, we modify the power series solution and provide a reliable solution for any combination of phases shear modulus. We demonstrate that a relatively small number of terms in the series is required for a good agreement with the numerical simulations up to a stretch ratio of 1.5 when considering the local fields, and up to a stretch ratio of 2 when considering the average fields. The analysis emphasizes the interaction between the shell and the core and reveals the different roles of the coating under different boundary conditions. We highlight interesting similarities and dissimilarities between the spatial distributions of the local stresses and the variations of the average stresses developing in TWCSs with stiff and soft shells, under displacement and traction boundary conditions.

Identification of the string boundary conditions using natural frequencies

2016 ◽  
Vol 11 (1) ◽  
pp. 38-52
Author(s):  
I.M. Utyashev ◽  
A.M. Akhtyamov
Keyword(s):  
Inverse Problems ◽  
Power Series ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Natural Frequencies ◽  
Boundary Value ◽  
External Environment ◽  
Direct And Inverse Problems ◽  
Symmetric Potential ◽  
Modified Method

The paper discusses direct and inverse problems of oscillations of the string taking into account symmetrical characteristics of the external environment. In particular, we propose a modified method of finding natural frequencies using power series, and also the problem of identification of the boundary conditions type and parameters for the boundary value problem describing the vibrations of a string is solved. It is shown that to identify the form and parameters of the boundary conditions the two natural frequencies is enough in the case of a symmetric potential q(x). The estimation of the convergence of the proposed methods is done.

Elasticity of colloidal gels: structural heterogeneity, floppy modes, and rigidity

Soft Matter ◽  
2021 ◽  
Author(s):  
D. Zeb Rocklin ◽  
Lilian C Hsiao ◽  
Megan E Szakasits ◽  
Michael J Solomon ◽  
Xiaoming Mao
Keyword(s):  
Coordination Number ◽  
Volume Fraction ◽  
Structural Parameters ◽  
Structural Heterogeneity ◽  
Colloidal Gels ◽  
Shear Moduli ◽  
Rheological Measurements

Rheological measurements of model colloidal gels reveal that large variations in the shear moduli as colloidal volume-fraction changes are not reflected by simple structural parameters such as the coordination number,...

Optimal algebra and power series solution of fractional Black-Scholes pricing model

2021 ◽  
Vol 25 (8) ◽  
pp. 6075-6082
Author(s):  
Hemanta Mandal ◽  
B. Bira ◽  
D. Zeidan
Keyword(s):  
Power Series ◽  
Series Solution ◽  
Power Series Solution ◽  
Pricing Model ◽  
Black Scholes

scholarly journals Simulation of MASPn Experiments in MISTRA Test Facility with COCOSYS Code

2008 ◽  
Vol 2008 ◽  
pp. 1-7 ◽  
Author(s):  
Mantas Povilaitis ◽  
Egidijus Urbonavičius
Keyword(s):  
Boundary Conditions ◽  
Nuclear Power ◽  
Power Plants ◽  
Volume Fraction ◽  
Characteristic Length ◽  
Initial Conditions ◽  
Test Facility ◽  
Work Package ◽  
Acceptable Agreement

An issue of the stratified atmospheres in the containments of nuclear power plants is still unresolved; different experiments are performed in the test facilities like TOSQAN and MISTRA. MASPn experiments belong to the spray benchmark, initiated in the containment atmosphere mixing work package of the SARNET network. The benchmark consisted of MASP0, MASP1 and MASP2 experiments. Only the measured depressurisation rates during MASPn were available for the comparison with calculations. When the analysis was performed, the boundary conditions were not clearly defined therefore most of the attention was concentrated on MASP0 simulation in order to develop the nodalisation scheme and define the initial and boundary conditions. After achieving acceptable agreement with measured depressurisation rate, simulations of MASP1 and MASP2 experiments were performed to check the influence of sprays. The paper presents developed nodalisation scheme of MISTRA for the COCOSYS code and the results of analyses. In the performed analyses, several parameters were considered: initial conditions, loss coefficient of the junctions, initial gradients of temperature and steam volume fraction, and characteristic length of structures. Parametric analysis shows that in the simulation the heat losses through the external walls behind the lower condenser installed in the MISTRA facility determine the long-term depressurisation rate.

Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

2021 ◽  
pp. 109963622110204
Author(s):  
Xue-Yang Miao ◽  
Chao-Feng Li ◽  
Yu-Lin Jiang ◽  
Zi-Xuan Zhang
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Admissible Function ◽  
Free Vibration Analysis ◽  
Middle Layer ◽  
Two Dimensional ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

Static Analysis of Moderately Thick Functionally Graded Plates With Various Boundary Conditions

2010 ◽  
Author(s):  
Nastaran Shahmansouri ◽  
Mohammad Mohammadi Aghdam ◽  
Kasra Bigdeli
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Closed Form Solution ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Functionally Graded Plates ◽  
Various Boundary Conditions ◽  
Fg Plates ◽  
Ode Systems

The present study investigates static analyses of moderately thick FG plates. Using the First Order Shear Deformation Theory (FSDT), functionally graded plates subjected to transversely distributed loading with various boundary conditions are studied. Effective mechanical properties which vary from one surface of the plate to the other assumed to be defined by a power law form of distribution. Different ceramic-metal sets of materials are studied. Solution of the governing equations, including five equilibrium and eight constitutive equations, is obtained by the Extended Kantorovich Method (EKM). The system of thirteen Partial Differential Equations (PDEs) in terms of displacements, rotations, force and moment resultants are considered as multiplications of separable function of independent variables x and y. Then by successful utilization of the EKM these equations are converted to a double set of ODE systems in terms of x and y. The obtained ODE systems are then solved iteratively until final convergence is achieved. Closed form solution is presented for these ODE sets. It is shown that the method is very stable and provides fast convergence and highly accurate predictions for both thin and moderately thick plates. Comparison of the normal stresses at various points of rectangular plates and deflection of mid-point of the plate are presented and compared with available data in the literature. The effects of the volume fraction exponent n on the behavior of the normalized deflection, moment resultants and stresses of FG plates are also studied. To validate data for analysis fully clamped FG plates, another analysis was carried out using finite element code ANSYS. Close agreement is observed between predictions of the EKM and ANSYS.

scholarly journals Surface Wave Speed of Functionally Graded Magneto-Electro-Elastic Materials with Initial Stresses

2014 ◽  
Vol 44 (3) ◽  
pp. 49-64 ◽  
Author(s):  
Li Li ◽  
P. J. Wei
Keyword(s):  
Surface Wave ◽  
Boundary Conditions ◽  
Elastic Material ◽  
Short Circuit ◽  
Equations Of Motion ◽  
Open Circuit ◽  
Functionally Graded ◽  
Initial Stresses ◽  
Shear Surface ◽  
Traction Boundary Conditions

Abstract The shear surface wave at the free traction surface of half- infinite functionally graded magneto-electro-elastic material with initial stress is investigated. The material parameters are assumed to vary ex- ponentially along the thickness direction, only. The velocity equations of shear surface wave are derived on the electrically or magnetically open circuit and short circuit boundary conditions, based on the equations of motion of the graded magneto-electro-elastic material with the initial stresses and the free traction boundary conditions. The dispersive curves are obtained numerically and the influences of the initial stresses and the material gradient index on the dispersive curves are discussed. The investigation provides a basis for the development of new functionally graded magneto-electro-elastic surface wave devices.

Thermo-Solutal Convection of a Nanofluid Utilizing Fourier’s-Type Compass Conditions

2020 ◽  
Vol 9 (4) ◽  
pp. 362-374
Author(s):  
J. C. Umavathi ◽  
Ali J. Chamkha
Keyword(s):  
Brownian Motion ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Temperature Fields ◽  
Physical Parameters ◽  
Surface Boundary ◽  
Perturbation Scheme ◽  
Runge Kutta ◽  
The Impact

Nanotechnology has infiltrated into duct design in parallel with many other fields of mechanical, medical and energy engineering. Motivated by the excellent potential of nanofluids, a subset of materials engineered at the nanoscale, in the present work, a new mathematical model is developed for natural convection in a vertical duct containing nanofluid. Numerical scrutiny for the double-diffusive free and forced convection within a duct encumbered with nanofluid is performed. Buongiorno’s model is deployed to define the nanofluid. Robin boundary conditions are used to define the surface boundary conditions. Thermal and concentration equations envisage the viscous, Brownian motion, thermosphores of the nanofluid, Soret and Dufour effects. Using the Boussi-nesq approximation the solutal buoyancy effect as a result of gradients in concentration are incorporated. The conservation equations which are nonlinear are numerically estimated using fourth order Runge-Kutta methodology and analytically ratifying regular perturbation scheme. The mass, heat, nanoparticle concentration and species concentration fields on eight dimensionless physical parameters such as thermal and mass Grashof numbers, Brownian motion parameter, thermal parameter, Prandtl number, Eckert number, Schmidt parameter, and Soret parameter are calculated. The impact of these parameters are outlined pictorially. The velocity and temperature fields are boosted with the thermal Grashof number. The Soret and the Schemidt parameters reduces the nanoparticle volume fraction but it heightens the momentum, temperature and concentration. At the cold wall thermal and concentration Grashof numbers reduces the Nusselt values but they increase the Nusselt values at the hot wall. The reversal consequence was attained at the hot plate. The perturbation and Runge-Kutta solutions are equal in the nonappearance of Prandtl number. The (E. Zanchini, Int. J. Heat Mass Transfer 41, 3949 (1998)). results are restored for the regular fluid. The heat transfer rate is high for nanofluid when matched with regular fluid.

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