On the Hyperelastic Pressurized Thick-Walled Spherical Shells and Cylindrical Tubes Using the Analytical Closed-Form Solutions

This paper focuses on thick-walled spherical shells and cylindrical tubes made of the soft tissues and the rubber-like materials. These materials are characterized by high deformability in which their stress–stretch curves are arranged in the range of S-shaped to J-shaped forms. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In order to fulfill the main aims of this paper, among all existing energy functions including polynomial, power law, logarithmic and exponential functions, or a linear combination of them, we deduced to evaluate the performance of an Ogden-type model with only integer powers for the mechanical behavior modeling of the S-shaped to J-shaped materials. Most of all, this strain energy function because of its mathematical form can play a constructive role in presentation of the analytical closed-form solutions for the boundary value problems in the field of the finite deformation elasticity. This constitutive model due to the high performance in constitutive modeling and the simplicity of its mathematical form is applied to pressurized thick-walled spherical shells and cylindrical tubes in order to find a closed-form analytical solution for their analysis. Using these analytical solutions, a comprehensive study is done on vanishing circumstance of the snap-through instability that occurs in the inflation of internally pressurized spherical shells and cylindrical tubes. It was observed that the parameters such as shell thickness, the elastic material properties specially the materials with J-shaped mechanical behaviors and the absence and presence of axial forces in cylindrical tubes have significant influence on vanishing of the snap-through instability in the thick-walled pressurized spherical shells and cylindrical tubes.

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Elastic Fields in a Polygon-Shaped Inclusion With Uniform Eigenstrains

Journal of Applied Mechanics ◽

10.1115/1.2788920 ◽

1997 ◽

Vol 64 (3) ◽

pp. 495-502 ◽

Cited By ~ 79

Author(s):

H. Nozaki ◽

M. Taya

Keyword(s):

Composite Materials ◽

Closed Form ◽

Strain Field ◽

Strain Energy ◽

Convex Polygon ◽

Numerical Examples ◽

Elastic Fields ◽

Closed Form Solutions ◽

Arbitrary Convex ◽

Shaped Inclusion

In this paper the elastic fields in an arbitrary, convex polygon-shaped inclusion with uniform eigenstrains are investigated under the condition of plane strain. Closed-form solutions are obtained for the elastic fields in a polygon-shaped inclusion. The applications to the evaluation of the effective elastic properties of composite materials with polygon-shaped reinforcements are also investigated for both dilute and dense systems. Numerical examples are presented for the strain field, strain energy, and stiffness of the composites with polygon shaped fibers. The results are also compared with some existing solutions.

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Likely oscillatory motions of stochastic hyperelastic solids

Transactions of Mathematics and Its Applications ◽

10.1093/imatrm/tnz003 ◽

2019 ◽

Vol 3 (1) ◽

Cited By ~ 3

Author(s):

L Angela Mihai ◽

Danielle Fitt ◽

Thomas E Woolley ◽

Alain Goriely

Keyword(s):

Strain Energy ◽

Probability Distributions ◽

Time Instant ◽

Spherical Shells ◽

Mechanical Responses ◽

Elastic Systems ◽

Cylindrical Tubes ◽

Parameter Interval ◽

Isotropic Hyperelastic Material ◽

Quantities Of Interest

Abstract Stochastic homogeneous hyperelastic solids are characterized by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney–Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterized. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as ‘likely oscillatory motions’.

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The Ornstein-Uhlenbeck-Type Model with a Hybrid Dividend Strategy

Journal of Applied Mathematics ◽

10.1155/2013/467948 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7

Author(s):

Dan Zhu ◽

Chuancun Yin

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Ordinary Differential Equations ◽

Closed Form ◽

Generating Functions ◽

Type Model ◽

Value Functions ◽

Closed Form Solutions ◽

Time Value ◽

Moment Generating Functions

We consider the Ornstein-Uhlenbeck-type model. We first introduce the model and then find the ordinary differential equations and boundary conditions satisfied by the dividend functions; closed-form solutions for the dividend value functions are given. We also study the distribution of the time value of ruin. Furthermore, the moments and moment-generating functions of total discounted dividends until ruin are discussed.

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Analytical Estimation for Static Deformation of Wire Ropes With Fibrous Core

Journal of Applied Mechanics ◽

10.1115/1.2897617 ◽

1990 ◽

Vol 57 (4) ◽

pp. 1000-1003 ◽

Cited By ~ 5

Author(s):

K. Kumar ◽

J. E. Cochran

Keyword(s):

Closed Form ◽

Analytical Solutions ◽

Close Agreement ◽

Helix Angle ◽

Closed Form Solutions ◽

Wire Ropes ◽

Axial Forces ◽

End Conditions ◽

Analytical Estimation ◽

Rigidity Modulus

This papers develops closed-form solutions for the extension of twisted wire ropes with fibrous cores which are subjected to axial forces as well as axial moments. The analytical results are compared with the corresponding numerical results obtained by Costello and Phillips. A close agreement between the two establishes validity of the analytical solutions. Finally, an expression for the effective rigidity modulus of wire ropes with fibrous core is obtained in terms of the helix angle and the number of helical wires in the rope for each of the two end conditions.

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Closed-form solution for column buckling optimization

10.21203/rs.3.rs-857140/v1 ◽

2021 ◽

Author(s):

Vladimir Kobelev

Keyword(s):

Closed Form ◽

Dimensional Analysis ◽

Optimization Problem ◽

Closed Form Solution ◽

Elliptic Functions ◽

Isoperimetric Inequalities ◽

Form Solution ◽

Algebraic Equations ◽

Closed Form Solutions ◽

Axial Forces

Abstract An optimization problem for a column, loaded by axial forces, whose direction and value remain constant, is studied in this article. The dimensional analysis introduces the dimensionless mass and rigidity factors, which simplicities the mathematical technique for the optimization problem. With the method of dimensional analysis, the solution of the nonlinear algebraic equations for the Lagrange multiplier is superfluous. The closed-form solutions for Sturm-Liouville and mixed types boundary conditions are derived. The solutions are expressed in terms of the higher transcendental function. The principal results are the closed form solution in terms of the hypergeometric and elliptic functions, the analysis of single- and bimodal regimes, and the exact bounds for the masses of the optimal columns. The proof of isoperimetric inequalities exploits the variational method and the Hölder inequality. The isoperimetric inequalities for Euler’s column are rigorously verified.

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