1 Introduction

As typical hyperelastic materials, the constitutive relations of rubber and rubber-like materials can be completely described by their strain energy functions, and these materials have various excellent physical properties, especially the high elasticity and the finite deformation. Since rubber products, such as rubber pipes, washers and plates, are widely used in engineering fields, many studies focusing on the stability of structures composed of hyperelastic materials have been contributed, which may be found in Refs. [1]-[5]. In addition, the existence of the temperature field is almost inevitable in practical engineering applications. Therefore, it is necessary to analyze the temperature influence on the finite deformation of hyperelastic structures in a mechanical system.

Because of the inherent nonlinearity of hyperelastic materials, the studies related to analytical solutions are relatively few, especially, when the temperature influence is taken into consideration. In the review paper of Hill and Arrigo^[6], some assumptive solution forms with the theoretical significance were summarized. By considering the complexity of finite deformation problems, related theoretical studies mainly focused on simple structures and the load forms, and the axially symmetric one is very typical. The correlative analytical solution is usually obtained by transforming the original problem into a one-dimensional problem, which means that the axial stretch is constant. From this point, Chen and Haughton^[7] studied the existence of exact eversion solutions of an elastic cylinder. Hill^[8] presented a variety of exact results applied to the perfectly elastic incompressible hyperelastic Varga materials. For these materials, the author showed the governing equations based on the assumptions of plane strain, plane stress and axially symmetric deformations. Using a novel approach of coupled series-asymptotic expansions, Dai et al.^[9] investigated the localization phenomena in a slender cylinder composed of an incompressible hyperelastic material subject to the axial tension. By analyzing a first-order dynamical system and using the phase plane, the authors solved two boundary-value problems analytically, and obtained the explicit solution expressions. With the aid of the complementary energy and the Legendre transform of strain energy functions, Rooney and Eberhard^[10] established a method of obtaining closed-form solutions in isotropic hyperelasticity. The authors explored some of the implications of this formulation for the equations and obtained solutions for spherical and cylindrical inflation. Bagheri et al.^[11] studied the mechanical behavior of a hyperelastic rotating thick walled tube. The authors provided the foundation of existence of a closed-form solution for the boundary-value problems in the finite deformation elasticity. Anani and Rahimi^[12] analyzed the rotating thick-walled hollow cylindrical shell composed of functionally graded incompressible hyperelastic materials, and obtained the analytical solution for the axisymmetric plane strain state. Accordingly, the authors plotted the profiles of circumferential stretch, radial stress, circumferential stress, and longitudinal stress as a function of the radial direction.

For the studies related to the axial load for cylindrical structures, some authors adopted the solution forms that the axial stretch was constant, but the others who did not follow this principle also obtain some analytical solutions. Klingbeil and Shield^[13] firstly presented an analytical solution for the deformation of a fixed circular rubber plate subject to an axial load, and obtained the load-deflection curves. Furthermore, employing the numerical method to solve the relevant problem is still a common research approach. In the context of the finite elasticity theory, Zidi and Cheref^[14] investigated the finite torsion and axial tension problems of a circular tube composed of fiber reinforced compressible hyperelastic materials, solved the problem numerically by using a Runge-Kutta method, and presented the effects of the combined deformation on the stress distributions for different pre-stresses supported by the tube. Merodio and Ogden^[15] processed a boundary-value problem of the finite deformation of an elastic cylindrical tube with radial and circumferential residual stresses, and compared the results for two strain-energy functions. Kanner and Horgan^[16] studied the effect of strain hardening on the elastic response of a solid rubber material cylinder under the axial tension and torsion combined deformation, and the results were in sharp contrast with those for classical models such as the Mooney-Rivlin (and neo-Hookean) models, which predicted that the stretched circular cylinder always tended to further elongate on twisting. Within the framework of finite deformation theory, numerous studies on hyperelastic rods were carried out^[17-19]. Dai and Bi^[17] obtained the exact solution expressed in terms of the elliptic integrals for an incompressible neo-Hookean rod. The results show that, if the radius-length ratio is relatively small, the deformation is almost independent of the ratio, and the middle portion of the rod undergoes a uniform deformation, while the deformation close to ends changes at a very fast rate. By using the inverse method, Zhang et al.^[20] obtained the implicit solutions of finite deformation problems of a composite cylindrical tube under equally axial loads acting on its two ends, where the tube was composed of two different incompressible neo-Hookean materials, and their results coincided with the conclusion of Dai and Bi^[17].

There are large numbers of difficulties about the related studies involved in the mechanical behavior of hyperelastic materials in a temperature field, because of both the material nonlinearity and the field coupling. For simple problems, the coupling effect can be neglected, but the much more general strategy is to take the effect into consideration. The constitutive model of thermo-hyperelasticity for near-incompressible elastomers was studied^[21-22]. They formulated a thermo-hyperelastic constitutive model for near-incompressible elastomers in terms of the Helmholtz free energy density. Thermo-mechanical coupling occurs mostly as thermal expansion. Almasi et al.^[23] exhibited that the increase of the axial stretch in the inner radius direction of the pressure vessel could improve the stability. Furthermore, Bagheri et al.^[24] analyzed the stability of a hyperelastic thick-walled cylinder in the temperature field. They observed that the positive temperature gradients in comparison with environment temperatures improve the stability of the circular tubes composed of an entropic material. As a verification of the analytical solution, the comparison of the results of the finite element method (FEM) and the analytical method showed a good fit. These researchers studied the influence of temperature. However, their studies were rather limited when dealing with the axial load and a more general assumption of known (and simple) shape of the structure after deformation in the temperature field.

The mathematical model describing the finite deformation problem about the radial and axial directions of a thermo-hyperelastic cylinder is formulated in Section 2. By using the energy variational principle, a second-order nonlinear ordinary differential equation describing the deformation of the lateral surface of the cylinder is derived, and the implicit analytical solution to the problem is obtained in Section 3. With the aid of numerical methods, the analyses of the compression and the tension cases are presented in Section 4. Finally, some interesting conclusions obtained in the paper are proposed in Section 5.

2 Formulation

Consider the finite deformation problem of a hyperelastic cylinder composed of a class of incompressible hyperelastic materials, where the cylinder is subject to an axial load Q₀ at its two ends and to a boundary temperature T_B at its lateral surface (see Fig. 1).

The initial configuration of the cylinder is given by

(1)

where H and B are the initial length and the radius of the cylinder, respectively.

Under the assumption of the axially symmetric deformation, every cross-section in the axial direction is still planar and normal to the axis after the deformation, such as Zhang et al.^[20] and Dai and Bi^[17]. The general static deformation configuration of the cylinder can be described as

(2)

where x=(r, θ, z) is the cylindrical coordinate system after the deformation.

Based on the theory of continuum mechanics, the deformation gradient tensor F and the left Cauchy-Green strain tensor B are as follows:

(3)

In terms of Eqs. (2) and (3),

(4)

In Eq. (4), is the azimuthal stretch, and z_Z represents the axial stretch. Correspondingly, the principal invariants of the left Cauchy-Green strain tensor are

(5)

Substituting Eq. (4) into Eq. (5) results in

(6)

This paper considers a class of incompressible thermo-hyperelastic Mooney-Rivlin materials, and the corresponding strain energy function is given by^[21]

(7)

where C₁ and C₂ are material constants, T (= T(R)), T₀, and ρ are the temperature distribution function in the cylinder, the reference temperature, and the density of the material, respectively. Moreover, the relations are given by C₃ = C_e and C₄=-αG, where C_e, α, and G are the volume specific heat, volumetric thermal diffusivity, and the second Lame constant (shear modulus), respectively.

Since the lateral surface of the cylinder is subject to a uniform distribution of heat source, of which the volumetric heat generation rate is given by , the differential equation of the axially symmetric form of steady heat conduction in cylindrical coordinates is given by

(8)

where k is the thermal conductivity. In addition, assume that the cylinder along its axial line remains solid during the course of finite deformation, which leads to

(9)

Assume that the two ends of the cylinder are fixed, which means that the areas are invariant at , in other words, the displacement boundary condition is

(10)

Based on the energy principle, the elastic potential energy φ per unit length of the cylinder composed of a class of incompressible thermo-hyperelastic Mooney-Rivlin materials and the work done by the external force can be described as the following expressions, namely,

(11)

3 Solutions

In this section, we present the implicit integral solution describing the finite deformation problem by using the energy variational principle.

Firstly, for incompressible hyperelastic materials, the incompressibility constraint requires that I₃ = 1, with the notation ε = z_Z, which leads to

(12)

Integrating Eq. (12) with respect to R, with the inner boundary condition (9), we obtain

(13)

In order to simplify the derivations and calculations, let f = ε^{- 1/2}, and the incompressibility constraint becomes εf² = 1. Substituting Eqs. (6) and (13) and εf² = 1 into Eq. (7) yields

(14)

It is not difficult to obtain the general solution to Eq. (8),

(15)

where C and D are both integral constants. With the aid of temperature boundary conditions T|_{R = 0} ≠ ∞ and T |_{R = B} = T_B, the corresponding distribution of the temperature field in the cylinder is given by

(16)

For convenience, we introduce the following dimensionless notations:

(17)

where

The total potential energy of the system P is given by

(18)

Using the notations in Eq. (17) leads to

(19)

where

(20)

in which M is a constant. Based on the variational principle, it leads to

(21)

Furthermore, we have

(22)

Let

(23)

The governing equation (22) is reduced to

(24)

From the term χ₁ in Eq. (24), we know that the influence of the uniform internal heat source ξ on the solid incompressible thermo-hyperelastic cylinder is equivalent to that of the external boundary temperature T_B. Therefore, to simplify the analysis, set the volumetric heat generation rate , i.e., the dimensionless parameter ξ = 0. Under this condition, Eq. (16) indicates that the temperature field is uniform.

For the thermo-hyperelastic Mooney-Rivlin materials given by Eq. (7), γ = C₂/C₁ ≠ 0. Multiplying Eq. (24) by f_x and integrating it with respect to x, we get

(25)

According to Eq. (12), it is not difficult to show that ε_x = - 2f^{- 3}f_x and f_x² = ε^{- 3}ε_x²/4. Substituting them into Eq. (25) results in

(26)

where

(27)

The necessary physical and thermo-physical properties used in this paper are listed in Table 1.

By combining with Eq. (13), the end boundary condition (10) becomes

(28)

Obviously, s > 0 and ε > 0 are valid. For the axial compression, we consider the case that η > 0, and f_x ≤ 0, f > 0. Combining with the incompressibility constraint yields ε_x = - 2f^{- 3}f_x ≥ 0. By employing the method of separation of variables to deal with Eq. (26), with the aid of Eq. (28), the solution to the compression case is given by

(29)

Due to the fact that the cylinder is symmetric about the middle section along the axial direction, that is, the extreme value ε_max = ε^★ and ε_x = 0 will occur at x = 0. With the aid of Eq. (26), we obtain

(30)

and

(31)

Combining Eqs. (30) and (31), the extreme value ε^★ and the integral constant d₀ can be determined. Substituting d₀ into Eq. (29), and using the incompressibility constraint εf² = 1, we can obtain the implicit integral expression of the dimensionless deformation curve of the deformed cylinder.

Similarly, for the axial tension case, the same procedure may be easily adapted to obtain the implicit integral expression describing the dimensionless deformation curve. The solution to the tension case is as follows:

(32)

4 Analyses

The implicit solutions of the compression and tension deformation problems of the thermo-hyperelastic cylinder in the temperature field are presented by Eqs. (29)-(32). The numerical solutions of the problem can be given by the programming method.

Substituting Eq. (30) into Eq. (31), the nonlinear implicit equation for ε^★ can be calculated. The numerical method is used to solve the nonlinear equation, and the computational residual is denoted as

(33)

where ε^★★ is the calculation result of the extreme value solved by the present numerical method.

As shown in Fig. 2, the extreme value ε^★ can be solved when the residual error Δ is equal to zero. By substituting the extremum ε^★ into Eq. (30), the integral constant can be determined. Then, the implicit expression of (x, ε)-curve can be described by the following relation:

(34)

where

(35)

Furthermore, with the aid of εf² = 1, we can obtain the implicit expression of (x, f)-curve.

4.1 Axial compressive load

In the compression case, the expression of the axial stretch-load curve is given by

(36)

Obviously, Eqs. (35) and (36) and εf² = 1 provide the implicit solution to the (x, f)-curve. Since the cylinder is symmetric about x = Z/H = 0, we shall only consider the upper half, i.e., x ≥ 0.

It is necessary to point out that the whole problem depends on three parameters, the radius-length ratio s, the dimensionless external force η, and the boundary temperature T_B. Next, we will discuss the solution behavior as these three parameters take different values.

The curves shown in Fig. 3 present the relationship between the azimuthal stretch extreme value f^★ and the cylindrical radius-length ratio s with different axial compressive loads and boundary temperatures. From Fig. 3, we can see that the maximum approaches 1 with the increasing radius-length ratio. The deformation of the lateral surface of the cylinder decreases with the increasing radius-length ratio under the same load condition, which means that the shorter and thicker cylinder can reduce the radial deformation effectively. Furthermore, the influence of the temperature on the deformation decreases gradually with the increasing radius-length ratio. As shown in the subgraph of Fig. 3, there exists a critical value s^★, if s < s^★, the radius-length ratio is independent of the azimuthal stretch, and the ratio has no effect on the stretch. This phenomenon has been explained by Dai and Bi^[17] with the asymptotic analysis.

In Fig. 4, the solution curves for the azimuthal stretch extreme value f^★ and the compressive load η with different boundary temperatures and radius-length ratios are illustrated. It is shown that when the radius-length ratio is relatively small, i.e., the structure is slender, the relationship between f^★ and η is nonlinear. However, the relationship will evolve into linearity gradually, and the effect of the temperature on the deformation will decrease gradually with the increasing radius-length ratio. The latter is consistent with the analyses of Fig. 3.

The curves illustrated in Fig. 5 describe the relationship of the azimuthal stretch maximum f^★ and the boundary temperature T_B with the fixed radius-length ratio s=0.25 and different compressive loads. It is found that the azimuthal stretch maximum increases with the temperature and load. The thermal diffusivity coefficient of the rubber material is very small, and it is several orders smaller than other material parameters. Therefore, the temperature has a few effects on the deformation from the perspective of the numerical changes of f^★. Moreover, the effect of the temperature on the deformation is approximately linear according to those curves.

Figure 6 shows the relationship between the azimuthal stretch maximum and the boundary temperature with the fixed compressive load η= 1 and different radius-length ratios. When the radius-length ratio is small, the deformation will increase with the load, and the relationship is linear approximately. The influence of the temperature on the deformation will decrease gradually with the increasing radius-length ratio. It means that the shorter and thicker cylindrical structure can reduce the influence of the temperature on the deformation under the same load condition.

From Fig. 7, we know that the dimensionless deformation curves of the lateral surface of the cylinder are illustrated with the fixed radius-length ratio, different compressive loads and temperatures. As shown in Fig. 7(a), the effect of the temperature on the deformation is negligible while the structural parameter, namely, the radius-length ratio, is relatively large. On the contrary, as shown in Fig. 7(d), the corresponding effect is more obvious with the smaller ratio of the cylindrical structure and the larger load. In addition, the deformation changes significantly at the ends of the cylinder, while the middle portion changes slowly.

4.2 Axial tensile load

In the tension case, the expression of the stretch-load curve is given by

(37)

Similar to the compression case, we also discuss the solution behavior of Eq. (37) with the variation of these three parameters, namely, the radius-length ratio, the dimensionless external force, and the boundary temperature, respectively.

The curves in Fig. 8 show the relationship between the extreme value of the azimuthal stretch and the radius-length ratio of the cylinder with different axial tensile loads and boundary temperatures. In general, the phenomena of the tension case are similar to those of the compression case, but there are also some differences. Compared with Fig. 2, we can find that the significant effect of the load on the critical radius-length ratio, which is the critical value, will increase gradually with the load. Furthermore, the effects of the temperature on the deformation of compression and tension are almost uniform.

In Fig. 9, with the fixed radius-length ratio and different boundary temperatures, the solution curves for the azimuthal stretch extremum f^★ and the tensile load η are illustrated. In this case, it is found that the response of the structure may involve an interesting phenomenon of the increasing jump. In order to have an increase of radial deformation, the road is decreased till the curvature of equilibrium path reverses. Actually, when the equivalent tangent stiffness decreases gradually and approaches zero, the structure will snap through from a stable point to another new stable point. When reaching the limiting load, the visible and sudden jump from one equilibrium configuration to another nonadjacent equilibrium configuration is referred to the limit point instability^[24]. For the compression case, the similar phenomenon would not appear. Following the theory of finite deformation, the deformation has a positive feedback effect for the tensile case, i.e., the larger the deformation is, the slender the cylinder will be, which causes that the deformation will be produced more easily. While the compression case is the opposite, the negative feedback results in shortening and thickening which will make the cylinder much more difficult to generate the deformation. The effect of temperature on the critical tensile load of limit point instability can be explained from the perspective of rubber softening and material properties weakening caused by the rising temperature.

Figure 10 illustrates the relationship of the azimuthal stretch extremum and the tensile load with the fixed boundary temperature T_B = 300 K and different radius-length ratios. The instability phenomenon would not occur when the structural parameter is small. Only when the radius-length ratio is greater than a critical value, this type of instability can appear. The larger the radius-length ratio is, the greater the critical load is. Meanwhile, even if the load changes slightly, the azimuthal stretch maximum of the cylinder changes dramatically in the neighborhood of the critical point. Under these circumstances, the deformation of the hyperelastic cylinder becomes very sensitive to the external conditions, such as the load, the temperature, which is similar to the necking phenomenon in the tensile process of low carbon steel.

Figure 11 shows the relationship of the azimuthal stretch extremum and the boundary temperature with the fixed radius-length ratio and different tensile loads. Both the increasing temperature and the increasing load decrease the stretch extremum, which is opposite to the compression case. For the tension case, the smaller the azimuthal stretch is, the greater the deformation is, while the compression is opposite. On the aspect of finite deformation, the effect of temperature remains the same for both compression and tension, and is linear approximately.

The curves illustrated in Fig. 12 describe the relationship of the azimuthal stretch extremum and the boundary temperature with the fixed tensile load and different radius-length ratios. From Fig. 12, we see that the increasing boundary temperature will reduce the stretch extremum when the ratios are relatively small, and approximately behave a linear relationship. It is necessary to mention that, in the tension case, the smaller the stretch is, the smaller the deformed radius is, and the larger the deformation is, while compression can be an opposing situation, which is explained in the analyses of Fig. 11. With the increasing radius-length ratio, the effect of the temperature on the deformation will decrease gradually. This phenomenon indicates that the shorter and thicker cylinder structure may reduce the effect of the temperature on the finite deformation under the condition of the fixed load, which is consistent with the conclusion drawn from the compression case.

In Fig. 13, the dimensionless deformation curves of the cylindrical lateral surface are illustrated with the fixed radius-length ratio, different tensile loads and boundary temperatures. The radius-length ratios of each subgraph are different. It is shown that the effect of the temperature on the finite deformation is almost negligible for the larger structural parameter, the radius-length ratio and the smaller axial load. However, for the relatively large load, the effect of the temperature on the finite deformation is extraordinary obvious which can be explained by Figs. 9 and 10. In this case, the larger the radius-length ratio is, the more tensile instability the cylinder may exhibit, which causes that the finite deformation becomes extremely sensitive to the external factors near the instability point, and the influence of different temperatures can also be equivalent to different loads. Therefore, the effect of the temperature on the deformation will be highly obvious, which is due to the "amplification effect" around the instability point. Since those radius-length ratios are smaller, the other figures will not exhibit tensile instability, and the effect of the temperature on the deformation is consistent with the compression case. Moreover, the deformation of the structure shows a drastic change at the two ends, while the portion far away from the ends undergoes a uniform deformation.

4.3 Comparison of compression and tension deformation states

The compression and tension cases are discussed in Subsection 4.1 and Subsection 4.2, respectively. The deformation behaviors of the two cases will be compared and analyzed briefly as follows.

Compared with the normalized axial stretch case, the unnormalized case can be visually distinguished by the numerical changes of the axial deformation, as shown in Fig. 14.

It can be seen clearly in Fig. 15 that, if the radius-length ratio is relatively small, the deformation degrees for the compression and the tension cases will be quite similar associated with the symmetric axis f = 1. However, with the increasing load, the tensile deformation will show some significant differences. When the radius-length ratio becomes larger, the tensile deformation degree will be greater compared with the compression deformation degree, since the tensile situation tends to be unstable as shown in Fig. 16. When the radius-length ratio is smaller, the two cases will exhibit a sharp deformation at the ends and undergo a uniform deformation in the middle portion. In other words, the end effect is more pronounced in the case of tensile deformation.

5 Conclusions

Within the framework of continuum mechanics, the finite deformation problem about the radial and axial directions is investigated for a cylinder composed of incompressible thermo-hyperelastic Mooney-Rivlin materials with two fixed ends. Then, the implicit solution to the problem is presented, and the following three conclusions are obtained by analyzing the numerical results:

(ⅰ) In a stable temperature field, the rising temperature will weaken the mechanical properties of hyperelastic materials, which shows that, in the finite deformation, the deformation about the radial and axial directions will increase or decrease with the temperature. However, due to the small coefficient describing the thermal volume diffusion of the rubber material, it is several orders of magnitude smaller than other material parameters, and the effect of the temperature on the static deformation is not very significant.

(ⅱ) Based on the boundary conditions of two fixed ends, it is shown that the deformation styles of the cylinder under different axial loads exhibit significant effects, that is, the end deformation is severe, and the deformation away from the end tends to be uniform. At the same time, the effect is more obvious in the tensile process.

(ⅲ) For a hyperelastic cylinder subject to a tensile load, the tensile instability may occur, and the critical load is positively correlated with the radius-length ratio.