Nonlinear Periodic Oscillation of a Cylindrical Microvoid Centered at an Isotropic Incompressible Ogden Cylinder

Abstract

We study the dynamic mathematical model for an infinitely long cylinder composed of an isotropic incompressible Ogden material with a microvoid at its center, where the outer surface of the cylinder is subjected to a uniform radial tensile load. Using the incompressibility condition and the boundary conditions, we obtain a second-order nonlinear ordinary differential equation that describes the motion of the microvoid with time. Qualitatively, we find that this equation has two types of solutions. One is a classical nonlinear periodic solution which describes that the motion of the microvoid is a nonlinear periodic oscillation; the other is a blow-up solution. Significantly, for the isotropic incompressible Ogden material, there exist some special values of material parameters, the phase diagrams of the motion equation have homoclinic orbits, which means that the amplitude of a nonlinear periodic oscillation increases discontinuously with the increasing load.

1. Introduction

Cylindrical structures are very common used in social productions and human lives. The researches on the dynamic oscillation problems of such structures composed of hyperelastic materials are of important significance. As is well known, such problems can be formulated as initial (boundary) value problems of nonlinear evolution equation(s). Knowles [1] firstly studied the free radial oscillation of an incompressible cylindrical tube composed of an isotropic Mooney-Rivlin material; in the limiting case of a thin walled cylindrical tube, the equation reduces to the Ermakov-Pinney equation. Then, Shahinpoor and Nowinski [2] and Rogers and Baker [3] used the nonlinear superposition principle for the Ermakov-Pinney equation to derive solutions. The works appeared in this area have been reviewed by Rogers and Ames [4]. In 2007, Mason and Maluleke [5] introduced the Lie point symmetry into this area and investigated the nonlinear radial oscillations of a transversely isotropic incompressible cylindrical tube subjected to time dependent net applied surface pressures; moreover, they proved that for radial and tangential transversely isotropic tubes the differential equations may be reduced to the Abel equations of the second kind. In addition, with the development of the mathematical theory, Yuan et al. [6] investigated the dynamic inflation problems for infinitely long cylindrical tubes composed of a class of transversely isotropic incompressible Ogden materials from the equation itself and discussed the influences of material parameters, structure parameter and applied pressures on the dynamic behaviors of the tubes in detail. Ren [7] studied the dynamical responses, such as motion and destruction of hyperelastic cylindrical shells subjected to dynamic loads on the inner surface. Other references on the dynamic responses for hyperelastic cylindrical structures may be found in Dai and Kong [8], Yuan et al. [9], and so on.

The purpose of this paper is to investigate the nonlinear periodic oscillation of a cylindrical microvoid centered at an infinitely long cylinder, where the cylinder is composed of an isotropic incompressible Ogden material [10] and its outer surface is subjected to a uniform radial tensile load. In Section 2, the basic governing equations, the boundary conditions and the initial conditions are presented. In Section 3, a second order nonlinear ordinary differential equation describing the motion of the microvoid is obtained. Then, in Section 4, some nonlinear dynamic analyses of the equation are performed in detail. Meanwhile, some numerical examples are given.

2. Mathematical Model

The mathematical model examined in this paper is listed as follows.

(a) Basic Governing Equations
In the absence of body force, the equilibrium differential equation, the incompressibility condition and the strain-energy function associated with the known Ogden material are given by 𝜕𝜎_𝑟(𝑟,𝑡)+1𝜕𝑟𝑟𝜎_𝑟(𝑟,𝑡)−𝜎_𝜃(𝑟,𝑡)=𝜌₀𝜕²𝑟(𝑅,𝑡)𝜕𝑡²,𝜆_𝑟𝜆_𝜃𝜇=1,𝑊=¹𝛼₁𝜆^𝛼1_𝑟+𝜆^𝛼1_𝜃+𝜇−2²𝛼₂𝜆^𝛼2_𝑟+𝜆^𝛼2_𝜃−2. (2.1)(2.2)(2.3)

(b) Boundary Conditions and Initial Conditions
The boundary conditions are given by 𝜎_𝑟(𝑟(𝐴),𝑡)=0,𝜎_𝑟(𝑟(𝐵),𝑡)=𝑝₀𝐵𝑟(𝐵). (2.4)
The initial conditions are as follows: 𝑟(𝑅,0)=𝑅,𝑟̇(𝑅,0)=0. (2.5)
In (2.1)~(2.5), 𝜎_𝑖(𝑟,𝑡)=𝜆_𝑖(𝜕𝑊/𝜕𝜆_𝑖)−𝑝(𝑟,𝑡), 𝑖=𝑟,𝜃 are the principal Cauchy stresses, 𝑝(𝑟,𝑡) is the hydrostatic pressure related to the incompressibility condition, 𝜆_𝑟=𝜕𝑟/𝜕𝑅, 𝜆_𝜃=𝑟/𝑅 are the radial and the circumference stretches, respectively, 𝑟=𝑟(𝑅,𝑡), 0<𝐴≤𝑅≤𝐵 is the radial deformation function with time to be determined, and 𝐴 and 𝐵 are the radius of the microvoid and the outer radius of the cylinder in the undeformed configuration, respectively. 𝜌₀ is the constant mass density of the material. 𝜇_𝑖,𝛼_𝑖(𝑖=1,2) are material parameters. The boundary conditions in (2.4) mean that the surface of the microvoid is traction free and the outer surface of the cylinder is subjected to a uniform radial tensile load, denoted by 𝑝₀. The initial conditions in (2.5) mean that the cylinder is in an undeformed state at time 𝑡=0.

3. Solutions

From the incompressibility condition (2.2) we find that 𝑟(𝑅,𝑡)=𝑅²−𝐴²+𝑟²₁(𝑡)^1/2, (3.1) where 𝑟₁(𝑡) is an undetermined radial motion function of the radius of the microvoid. From (3.1), it is easy to know that the radial motion of the cylinder can be completely described by 𝑟₁(𝑡).

From (3.1), it is not difficult to show that 𝜕²𝑟(𝑅,𝑡)𝜕𝑡²=𝜕𝑟𝜕𝑟¹𝑟̈₁ln𝑟+𝑟̇₁²1ln𝑟+2𝑟̇₁²𝑟₁𝑟². (3.2)

Substituting (3.2) into (2.1), integrating it with respect to 𝑟 from 𝑟(𝐴,𝑡) to 𝑟(𝐵,𝑡) and using the traction boundary conditions in (2.4), we obtain 12𝜌₀𝑟₁𝑟̈₁𝐵ln²−𝐴²+𝑟²₁𝑟²₁+12𝜌₀𝑟²₁𝐵ln²−𝐴²+𝑟²₁𝑟²₁+12𝜌₀𝑟̇²₁𝐴²−𝐵²𝐵²−𝐴²+𝑟²₁−𝑝₀𝐵𝐵²−𝐴²+𝑟²₁^1/2−_𝑟()^{𝑟(𝐵,𝑡)}_𝐴,𝑡𝜆_𝑟𝜕𝑊𝜕𝜆_𝑟−𝜆_𝜃𝜕𝑊𝜕𝜆_𝜃𝑑𝑟𝑟=0, (3.3) where 𝑟(𝐴,𝑡)=(𝐴²−𝐴²+𝑟²₁(𝑡))^1/2=𝑟₁(𝑡),  𝑟(𝐵,𝑡)=(𝐵²−𝐴²+𝑟²₁(𝑡))^1/2.

Obviously, (3.3) is a second order nonlinear ordinary differential equation with respect to 𝑟₁(𝑡). Next we study the qualitative properties of solutions of (3.3).

For convenience, we introduce the following dimensionless notations: 𝑅𝑘=𝑟⁻¹=𝑟1−²¹−𝐴²𝑟^2−1/2(𝑡)=𝑟,𝑥¹(𝑡)𝐵,𝑥̇(𝑡)=𝑟̇₁(𝑡)𝐵𝐵,𝛿=²−𝐴^21/2𝐵. (3.4)

Thus, the initial conditions in (2.5) reduce to 𝑥(0)=1−𝛿^21/2,𝑥̇(0)=0, (3.5) and (3.3) may be rewritten as 12𝜌₀𝐵²𝑥̈𝛿𝑥ln²+𝑥²𝑥²+12𝜌₀𝐵²𝑥̇²𝛿ln²+𝑥²𝑥²−12𝜌₀𝐵²𝑥̇²𝛿²𝛿²+𝑥²−𝑝₀1𝛿²+𝑥^21/2+𝐹(𝑥,𝛿)=0, (3.6) where 𝐹𝑥,𝛿,𝛼₁,𝛼₂,𝜇₁,𝜇₂=^{(𝛿2+𝑥2)1/2}_{𝑥/1−𝛿21/2}𝜇₁𝑘^−𝛼1+𝜇₂𝑘^−𝛼2−𝜇₁𝑘^𝛼1−𝜇₂𝑘^𝛼2𝑘𝑘²−1𝑑𝑘. (3.7)

4. Nonlinear Dynamic Analyses

Multiplying (3.6) by 𝑥𝑥̇, we obtain the following first integral: 𝑈(𝑥,𝛿)𝑥̇²+𝑉𝑥,𝑝₀,𝛿=0, (4.1) where 𝑈()=1𝑥,𝛿4𝜌₀𝐵²𝑥²𝛿ln²+𝑥²𝑥²,𝑉𝑥,𝑝₀,𝛿=^𝑥_1−𝛿21/2𝐹(𝑠,𝛿)−𝑝₀1𝛿²+𝑠^21/2𝑠𝑑𝑠. (4.2)

Attentively, 𝑈(𝑥,𝛿)>0 is valid for any 𝑥>(1−𝛿²)^1/2, which means that (4.1) has real solutions only when 𝑉(𝑥,𝑝₀,𝛿)<0. However, 𝑉(𝑥,𝑝₀,𝛿)<0 is equivalent to 𝑉(𝑥,𝑝₀,𝛿)_min<0. The stationary point of 𝑉(𝑥,𝑝₀,𝛿) can be obtained from 𝑉_𝑥(𝑥,𝑝₀,𝛿)=0, this leads to 𝑃=𝛿²+𝑥^{21/2(𝛿2+𝑥2)1/2}_{𝑥/1−𝛿21/2}𝑘^−𝛼1+𝜇𝑘^−𝛼2−𝑘^𝛼1−𝜇𝑘^𝛼2𝑘𝑘²−1𝑑𝑘=𝐺𝑥,𝛿,𝛼₁,𝛼₂,𝜇, (4.3) where 𝑃=𝑝₀/𝜇₁, 𝜇=𝜇₂/𝜇₁.

Interestingly, the equilibrium points of (3.6) can be determined by the roots of (4.3).

4.1. Influences of Parameters on the Solution of (3.6)

4.1.1. Influences of Material Parameters

The following conclusions can be obtained from (4.3).(i)For the given values of 𝜇 and 𝛿, if 0<𝛼₁, 𝛼₂<1, there exists a maximum point, written as (𝑥_𝑚,𝑃_𝑚). 𝑃 increases monotonically as 0<𝑥<𝑥_𝑚 and decreases monotonically as 𝑥>𝑥_𝑚.(ii)For the case that 𝛼₁=𝛼₂=1, we have lim_𝑥→+∞𝐺(𝑥,𝛿,1,1,𝜇)=(1+𝜇)(1−(1−𝛿²)^1/2), which means (4.3) has a horizontal asymptote, written as 𝑃_ha=(1+𝜇)(1−(1−𝛿²)^1/2). For the given values of 𝜇 and 𝛿, curves of 𝑃 versus 𝑥 are shown in Figure 1 for 0<𝛼₁, 𝛼₂≤1.(iii)If 𝛼₁,𝛼₂>1, we have lim_𝑥→+∞𝐺(𝑥,𝛿,𝛼₁,𝛼₂,𝜇)=+∞. 𝑃 increases strictly with the increasing 𝑥. Particularly, if 𝛼₁=𝛼₂=2, it is easy to prove that there is another asymptote, written as 𝑃_𝑎(𝑥)=−((1+𝜇)/2)ln(1−𝛿²)𝑥. For the given values of 𝜇 and 𝛿, Figure 2 shows the relationships of 𝑃 versus 𝑥 for 𝛼₁,𝛼₂>1.(iv)For the case that 𝛼₁>1, 0<𝛼₂<1 or 0<𝛼₁<1, 𝛼₂>1 (here, we only discuss the case that 𝛼₁>1, 0<𝛼₂<1), it can be proved that there exists a critical value of 𝜇, written as 𝜇_𝑐, such that 𝑃 increases monotonically if 0≤𝜇<𝜇_𝑐 and has a local maximum and a local minimum if 𝜇>𝜇_𝑐, written as 𝑃₁ and 𝑃₂, respectively. Curves of 𝑃 versus 𝑥 are given in Figure 3 for 𝛼₁>1, 0<𝛼₂<1 and for the given values of 𝛿.

Figure 1 

Curves of 𝑃∼𝑥 for 0<𝛼1, 𝛼2≤1.

Figure 2 

Curves of 𝑃∼𝑥 for 𝛼1, 𝛼2>1.

(a)

(b)

Figure 3 

Curves of 𝑃∼𝑥 for 𝛼1>1, 0<𝛼2<1.

4.1.2. Influence of Structure Parameter

Once the values of 𝛼₁, 𝛼₂, and 𝜇 are given, the influence of 𝛿 on the relationships of 𝑃 versus 𝑥 is shown in Figure 4.

Figure 4 

Curves of 𝑃∼𝑥 for different values of 𝛿.

4.2. Number of Equilibrium Points

(1)For the case that 0<𝛼₁, 𝛼₂<1, it can be seen from Figure 1 that there are two different roots of (4.3) as 0<𝑃<𝑃_𝑚, written as 𝑥₁ and 𝑥₂ (𝑥₁<𝑥₂). It means that (3.6) has two equilibrium points (𝑥₁,0) and (𝑥₂,0); moreover, (𝑥₁,0) is a center and (𝑥₂,0) is a saddle point. (2)If 𝛼₁=𝛼₂=1, (3.6) has a unique equilibrium point as 0≤𝑃<𝑃_ha, written as (𝑥₃,0); moreover, (𝑥₃,0) is a center. While 𝑃>𝑃_ha, (3.6) has no equilibrium point. (3)For the case that 𝛼₁,𝛼₂>1, (3.6) has only one equilibrium point as shown in Figure 2, written as (𝑥₄,0), and it is also a center.(4)If 𝛼₁>1, 0<𝛼₂<1, from the above analyses we know that 𝑃 has a local maximum 𝑃₁ and a local minimum 𝑃₂ as 𝜇>𝜇_𝑐, where 𝜇_𝑐 is a critical value of 𝜇. Equation (3.6) has a unique equilibrium point as 𝑃>𝑃₁ or 𝑃<𝑃₂, written as (𝑥₅,0), and it is a center. Equation (3.6) has exactly three equilibrium points as 𝑃₁>𝑃>𝑃₂, written as (𝑥₆,0), (𝑥₇,0), and (𝑥₈,0), respectively, where 𝑥₆<𝑥₇<𝑥₈; moreover, (𝑥₆,0) and (𝑥₈,0) are centers and (𝑥₇,0) is a saddle point. For the given values of 𝜇, 𝛼₁, 𝛼₂, and 𝛿, the phase diagrams of (4.1) are shown in Figure 5. It is found that the radial oscillation of the microvoid presents a nonlinear periodic oscillation; moreover, the amplitude of the oscillation increases gradually as 𝑃 increases from 0 to 𝑃_cr. However, the increase of the amplitude of the nonlinear periodic oscillation is discontinuous as 𝑃 passes through 𝑃_cr. Another interesting phenomenon occurs as 𝑃=𝑃_cr, namely, the phase diagram is a homoclinic orbit at the moment.

Figure 5 

Phase diagrams for different values of 𝑃 and for 𝜇=12, 𝛼1=1.5, 𝛼2=0.5, and 𝛿=0.9999.

5. Conclusions

In this paper, the nonlinear periodic oscillation of a cylindrical microvoid centered at the isotropic incompressible Ogden cylinder is examined. Some interesting nonlinear dynamic properties of the mathematical model are obtained. The main conclusions are as follows. (1)If 𝛼₁>1, 0<𝛼₂<1, it is proved that the motion of the microvoid is the nonlinear periodic oscillation for any given load, but there exist some special values of material parameters, the phase diagrams of the motion equation have homoclinic orbits, which means that the amplitude of the nonlinear periodic oscillation increases discontinuously with the increasing load. (2)For the case that 𝛼₁,𝛼₂≥1, the radial oscillation of the microvoid always presents a nonlinear periodic oscillation.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 10872045, 11232003); the Program for New Century Excellent Talents in University (no. NCET-09-0096); the Fundamental Research Funds for the Central Universities (no. DC12010112); the Program for Liaoning Excellent Talents in University (no. LR2012044).