Research on mechanical properties of a polymer membrane with a void based on the finite deformation theory

1 Introduction

Generally, hyperelastic polymer materials play a significant role in the polymer, automobile, aerospace and biotechnology industries. The dominating mechanical behavior of a polymer membrane is its extreme deformation and almost full recovery in unloading. Polymer materials also exhibit a highly nonlinear behavior (1). Normally, values of the maximum extensibility of polymer materials can be very high and the typical stress-strain curve in tension is nonlinear so that Hooke’s law cannot be used. Therefore, it is not possible to assign a definite value to the Young’s modulus except in the region of small strain. Being subjected to large deformations, polymer membranes demonstrate nonlinear features inherently. Therefore, the selection of strain-energy density function is vital for theoretical research (2). Many attempts have been made to develop a theoretical stress-strain relation. Mooney (3) proposed a phenomenological model with two parameters based on the assumption of a linear relation between the stress and strain during simple shear deformation. In 1948, Rivlin (4, 5) put forward the strain energy function model to the isotropic hyperelastic materials. Later, Treloar (6, 7) published a model based on the statistical theory, the so-called neo-Hookean material model with only one material parameter. Mooney and neo-Hookean strain energy function have played an important role in the development of the nonlinear hyperelastic theory and its applications. In 1972, Ogden (8, 9) proposed a strain energy function expressed in terms of principal stretches, which is probably the best-known example for the principal stretch-based constitutive formulations consistent with the Valanis-Landel hypothesis.

In 1990, Gao (10) proposed a constitutive model from the physics point of view of shape change and volume change. Moreover, in 1997, Gao (11) put forward a strain energy function from the perspectives of tensile and compression of the materials and applied it to the material fracture research. However, under incompressible conditions, strain energy function cannot be simplified to Neo-Hookean material or to Mooney-Rivlin material, which has an effect on the experimental basis of the strain energy function application. In (12), the Neo-Hookean model, the Mooney-Rivlin model and the Ogden model are considered for chloroprene rubber. Three kinds of tests are performed for chloroprene specimens and through four kinds of test combinations, numerical simulations are carried out for the Neo-Hookean model, the Mooney-Rivlin model and the Ogden model. It is shown that the Mooney-Rivlin model and the Ogden model can be used for chloroprene rubber in the specific ranges for an isotropic hyperelastic model.

In practical production, a polymer membrane generates a void when the membrane is blown unevenly. The size of the void is much smaller than the size of the polymer membrane and the polymer membrane sustained tension. A circular polymer membrane with a void in the center has been researched and uniform tensile stress is put on the polymer membrane boundary. The structure will be destroyed due to larger deformation and stress concentration where a void occurs, with the effect of externally applied load, which has attracted the attention of the engineering field. Yang (13) studied the stress concentration of the circular membrane near the hole. Haughton (14) introduced the exact solution of a circular membrane containing a hole or inclusion. With the appearance of new materials, studies from a perspective on large deformation problems of membranes with a void have a great value on production practice (15–17). In the meantime, unstable growth of a void is a typical mechanism of material failure. When void instabilities occur in the material regions, they can lead to failure localization and crack propagation. Liu (18) simulated various membrane structures by using the inhomogeneous field theory of a polymeric network in equilibrium with a solvent and mechanical load or constraint with the Finite Element software Abaqus (Dassault Systèmes, France).

In addition, vibration of composite plates has also attracted the attention of researchers. Mahi et al. (19) proposed a new hyperbolic shear deformation theory and applied it to bending and free vibration analysis of isotropic, functionally graded, sandwich and laminated composite plates. By utilizing the trigonometric four variable plate theories, free vibration analysis of laminated rectangular plates supporting a localized patch mass was researched by Draiche et al. (20) in 2014. The free vibration of laminated composite plates on elastic foundations was also examined by Nedri et al. (21) using a refined hyperbolic shear deformation theory. This theory is based on the assumption that the transverse displacements consist of bending and shear components where the bending components do not contribute to shear forces.

Based on Gao’s constitutive model, a modified model has been proposed to describe the incompressible polymer membrane materials. With the modified model, finite deformation analysis of a polymer membrane containing a void has been researched. By deducing the basic governing equation for solving the problem, the paper figures out the stress distribution of different constitutive parameters and discusses the effects on membrane deformation by different parameters and the reasons for the failure of the membrane. The results show that the constitutive parametern has a major impact on the mechanical properties of a polymer membrane with void, which provides a reference for the design of polymer membranes.

2 Finite deformation theory

For isotropic materials, strain energy function can be represented in terms of invariants or principal stretches:

where W is the strain energy function and λ₁, λ₂, λ₃ are the principal stretch ratios. The invariants of the left Cauchy-Green deformation tensor B can be expressed as:

For an invariant based strain energy density function, the Cauchy stress can be expressed as:

in which:

Considering the conventional assumption of incompressibility for polymer membranes, Eq. [3] can also be expressed as follows:

where I is the identity tensor and p is the Lagrange multiplier associated with hydrostatic pressure.

Gao (11) proposed the following strain energy function in 1997:

where A and n are material parameters and I_-1=I₂/I₃.

Based on Gao’s constitutive model, a modified strain energy function (22) for the incompressible rubber-like materials has been proposed as follows:

From the new constitutive model, we can see that when n=1 and α=0, it transforms into Neo-Hookean model; when n=1 and α=1, it transforms into the Mooney-Rivlin model.

A polymer membrane with a void is taken into account, the radius of which is a. The uniform tensile stress imposes on the infinity of a polymer membrane, as illustrated in Figure 1.

Figure 1:

The diagram of polymer membrane with a void.

If its coordinate before deformation is (R, Θ, Z) and the coordinate after deformation is (r, θ, z), then the deformation pattern of the membrane can be expressed as:

where H and h are the polymer membrane thicknesses according to undeformed and deformed conditions, respectively. The principal stretch ratios are:

For the assumption of incompressibility λ₁λ₂λ₃=1, we can get λ₃=R/rr′. Consequently, the deformation gradient F is:

The left Cauchy-Green strain tensor B and B^-1 have the following forms:

Substituting Eq. [10] and Eq. [11] into Eq. [5], we can get:

where σ₁, σ₂ and σ₃ are the radical Cauchy stress, circumferential Cauchy stress and axial Cauchy stress of the polymer membrane, respectively.

Because plane stress condition of the polymer membrane is taken into account, from Eq. [12] we can get:

Substituting Eq. [13] into Eq. [12] we can get:

In the absence of body forces, the equilibrium equation is expressed as divT=0. From the equilibrium equation, only one component is not satisfied identically, namely, the radial component, which is:

From Eq. [9] dr=λ₁dR, if T_i=Hλ₃σ_i (i=1, 2), the balance equation (Eq. [15]) can be achieved as:

Substitute the strain energy function Eq. [7] into Eq. [14], and then the governing equation of solving the problem can be obtained with Eq. [16] as:

In which:

Based on the strain energy function Eq. [7], we can get:

When α=0 and n=1, the strain energy function transforms into the Neo-Hookean strain energy function and we can get the following governing equation from Eq. [17]. The equation is inconsistent with the result of the document (14).

When α=1 and n=1, the strain energy function transforms into the Mooney strain energy function and we can get the following governing equation from Eq. [17]. The equation is also inconsistent with the result of the document (14).

From Eq. [9], we can get:

From the Lame results of the small deformation, radical Cauchy stress of a polymer membrane is equal to circumferential Cauchy stress at infinity under uniform tensile force. From this we can get the boundary condition in the infinite distance as follows:

With uniform tensile force, polymer membrane with void can be deformed freely along the direction of diameter when R=r=a, from which we can get:

For a void, at the internal boundary deformation R=r=a, the radial Cauchy stress is zero. By utilizing the assumption of incompressibility λ₁λ₂λ₃=1, we can get the following boundary condition λ1=λ2-1/2. In order to simplify the process, dimensionless is adopted with R/α and r/α. The Runge-Kutta method has been utilized to solve the differential equation (Eq. [17]) and λ_∝ has been found which can meet the above boundary conditions.

In order to research the deformation of a polymer membrane from the influence of constitutive parameters α and n, two circumstances have been taken into account. That is, when α is fixed, Cauchy stress distribution with the change of n has been researched. Simultaneously, when n is fixed, Cauchy stress distribution with the change of α also has been researched.

Tables 1 and 2 illustrate the value of radial deformation and circumferential deformation at the external boundary and internal boundary with the variation of constitutive parameters. Table 1 shows the value of λ₁=λ_∝ at an infinite distance of a polymer membrane and the value of λ₁ at the internal boundary deformation R=r=a. In Table 2, simultaneously, λ₂=λ_∝ and λ₂ are the values of polymer membrane at infinite distance and at the internal boundary deformation R=r=a.

For fixed material parameters n=1, α=0.1, when infinite distance boundary of a polymer membrane is subject to different external loads, we can get the different principal stretch λ₁=λ₂=λ_∝. Accordingly, different principal stretch λ_∝ can be expressed by different external load. Figure 2 shows the deformation distribution curves of the first and second principal stretch. From that we can see that radial deformation increases gradually in the direction of the radius of the polymer membrane with the increase of external load. However, circumferential deformation decreases gradually in the direction of the radius of the polymer membrane with the increase of external load. From Figure 2, circumferential deformation is larger at the boundary of the void and tends to be constant at infinite distance of the polymer membrane. Figure 3 shows the radial and circumferential Cauchy stress distribution with the change of parameter n when α is fixed. Figure 4 shows the radial and circumferential Cauchy stress distribution and the change of membrane with the change of parameter α when n is fixed.

Figure 2:

The deformation distribution curves of (A) λ₁ and (B) λ₂.

Figure 3:

(A) Radial Cauchy stress and (B) circumferential Cauchy stress distribution with the parameter α.

Figure 4:

(A) Radial Cauchy stress and (B) circumferential Cauchy stress distribution with the parameter n.

From Figures 3 and 4, we can see that whatever the change of constitutive parameters, the value of circumferential stresses is higher than the value of radial stresses. Especially at the boundary of the void, the radial stress is zero and the value of circumferential stresses is higher than the value of radial stresses. At infinite distance in the direction from the radius of the polymer membrane, the stress difference becomes zero, and its value is the same as the value of external load, that is σ₁=σ₂=k₀. It can be seen from Figure 4 that the effect of constitutive parameter n has a major impact on the mechanical properties of the polymer membrane with a void. When the parameter α is fixed (α=0.1), the value of circumferential stress goes to infinity when n=1, which implies that the polymer membrane has been buckled at the boundary of the void. This phenomenon also occurs in the inflation of a spherical membrane. By utilizing the constitutive equation of (Eq. [7]), inflation of a spherical membrane has been proposed in the document of (22). Internal pressure monotonously increases with the increase of principal circumferential stretch when n>1.2, which means the inflation is stable. By contrast, there exists instability when n<1.2 and internal pressure monotonously decreases with the increase of principal circumferential stretch after internal pressure reaches maximum.

From Figure 3, we can also see that the constitutive parameter α has minor impact on the mechanical properties of polymer membranes with a void. When parameter n is fixed (n=3) and α=0.1, 0.3, 0.5, the value of radial stress almost has no change and there is a slight change on the value of circumferential stress; it is reasonable that the effect of the second principal invariant can be ignored when we analyze the polymer membrane with a void.

3 Conclusion

In summary, the material constitutive parameters n and α have important influence on the mechanical properties of a polymer membrane. 1. The value of circumferential stresses is higher than the value of radial stresses. Especially at the boundary of the void, the radial stress is zero and the value of circumferential stresses is higher than the value of radial stresses. 2. The effect of constitutive parameter n has a major impact on the mechanical properties of a polymer membrane with a void. When the parameter α is fixed (α=0.1), the value of circumferential stresses goes to infinity when n=1, which implies that the polymer membrane has been buckled at the boundary of void. 3. When the parameter n is fixed (n=3) and α=0.1, 0.3, 0.5, the value of radial stress almost has no change and there is a slight change to the value of circumferential stress, which means it is reasonable that the effect of the second principal invariant can be ignored when we analyze the incompressible polymer membrane.