Plane-stress analysis of a holed membrane at finite equibiaxial stretch

Abstract

An equibiaxially stretched thin neo-Hookean circular membrane with a hole at its center under plane-stress condition is analyzed within the framework of finite deformation elasticity. Initially, we introduce a novel form for the differential governing equation to the problem. This enables the introduction of a closed-form solution in the limit of infinite stretch. Comparison of this solution to corresponding finite element simulations reveals a neat agreement for stretch ratios larger than 2.5. In the practically important case of a small hole, at the circumference of the hole, the stress concentration factor is 4 and the tangential stretch ratio is twice the applied far-field stretch ratio. These values are double the corresponding ratios in the well-known limit of infinitesimal deformation.

1. Introduction

The problem of an equibiaxially stretched membrane with a hole at its center under a plane-stress condition is frequently encountered. The well-known axisymmetric solution in the limit of infinitesimal deformations [1] is crucial for analyzing failures due to stress concentration around the hole. Moreover, the asymptotic solution in the limit of a small hole is frequently used for estimating the stress in many other problems containing small holes. However, the known solution for infinitesimal deformations is not applicable for materials such as elastomers and tissues that undergo finite deformations. For these stretchable materials, this problem needs to be analyzed within the framework of finite deformation elasticity [2].

Previous analyses [3–5] dealt with a general Mooney–Rivlin material [6, 7], with the neo-Hookean material as a special case, and a wide range of boundary conditions. Exact analytical results were obtained only in the limit of infinitesimal deformations. Approximated expressions based on expansion series about these exact solutions were proposed and compared with corresponding numerical results for a narrow range of ratios between the hole and the membrane radii. Yang [4] and Wong and Shield [5] further pointed out how their approximations break down in the case of traction-free boundary conditions at the hole. Haughton [8] solved a similar plane-stress problem for the Varga [9] material, revealing that for this material the membrane thickness remains uniform. Various numerical analyses of similar problems and the related cavitation problems were conducted for different hyperelastic materials [10–13].More recently, Geng et al. [14] obtained asymptotic approximations for a few hyperelastic membranes with a small hole near the cavitation critical stretch ratio.

Herein, we tackle the plane-stress problem of an equibiaxially stretched incompressible neo-Hookean circular membrane with a traction-free hole of an arbitrary size at its center. Initially, we introduce a new non-dimensional, non-linear second-order ordinary differential equation governing the boundary value problem. This further leads to a closed-form solution in the limit of infinite stretching of the membrane. This solution is then compared to the corresponding finite element (FE) simulation of the problem, demonstrating that the closed-form solution provides an excellent approximation for stretch ratios larger than 2.5.

2. Background

The deformation of a three-dimensional body from a reference (undeformed) configuration to a current (deformed) configuration can be described by a bijection mapping of each material point at a reference position $\mathbf{X}$ to its corresponding current position $\mathbf{x}=\mathbf{\chi }(\mathbf{X})$ . The deformation gradient is $\mathbf{F}\equiv {\nabla }_{\mathbf{X}}\mathbf{\chi }$ , where ${\nabla }_{\mathbf{X}}$ is the gradient with respect to $\mathbf{X}$ (and similarly ${\nabla }_{\mathbf{x}}$ is with respect to $\mathbf{x}$ ). The change in the volume of a material element is

$$J\equiv det(\mathbf{F})=\frac{\mathrm{d}v}{\mathrm{d}V},$$

(1)

where $\mathrm{d}V$ is its referential volume and $\mathrm{d}v$ its current volume. The right Cauchy-Green deformation tensor is $\mathbf{C}={\mathbf{F}}^T\mathbf{F}$ .

If no body forces are at play, the equilibrium equation for linear momentum is

$${\nabla }_{\mathbf{x}}·\mathit{\sigma }=0,$$

(2)

where $\mathit{\sigma }$ is the Cauchy stress. The balance of angular momentum implies that $\mathit{\sigma }$ is symmetric. Cauchy stress is related to the nominal stress $\mathbf{P}$ by

$$\mathit{\sigma }=J^{-1}\mathbf{P}{\mathbf{F}}^T,$$

(3)

where $\mathbf{P}$ is frequently denoted as the first Piola–Kirchhoff stress. In terms of the nominal stress, the referential equilibrium equation is

$${\nabla }_{\mathbf{X}}·\mathbf{P}=0.$$

(4)

The constitutive behavior of hyperelastic materials can be expressed in terms of a Strain Energy Density Function (SEDF) $W$ such that

$$\mathbf{P}=\frac{\partial W(\mathbf{F})}{\partial \mathbf{F}}.$$

(5)

A simple constitutive relation that describes the behavior of many highly deformable materials in the small to intermediate deformation range is the incompressible neo-Hookean material, for which

$$W=\frac{\mu }{2}(\mathrm{tr}(\mathbf{C})-3),$$

(6)

where the incompressibility constraint is $J=1$ . Accordingly,

$$\mathbf{P}=\mu \mathbf{F}-p{\mathbf{F}}^{-T}\mathrm{and}\mathit{\sigma }=\mu \mathbf{F}{\mathbf{F}}^T-p1,$$

(7)

where $1$ is the identity tensor and the pressure-like term $p$ is a Lagrange multiplier associated with the incompressibility constraint.

3. The boundary value problem

Consider a thin circular incompressible neo-Hookean membrane with a hole at its center (see Figure 1(a)). In its referential state, the outer radius of the membrane is $R_{\mathrm{o}}$ and the radius of the hole is $R_{\mathrm{i}}$ . The membrane is subjected to biaxial stretch ${\lambda }_{\mathrm{o}}$ at the outer radius (see Figure 1(b)) such that the outer radius in the deformed state is $r_{\mathrm{o}}={\lambda }_{\mathrm{o}}{R}_{\mathrm{o}}$ . The circumference of the hole and the lateral faces of the membrane are stress-free. We recall that a well-known solution exists for the plane-strain case of a circular cylindrical tube subjected to extension and inflation [2].

Figure 1.

An equibiaxially stretched punctured membrane in the (a) reference configuration and (b) deformed configuration.

We examine this problem in polar coordinates $\{R,\mathrm{\Theta },Z\}$ in the reference configuration and $\{r,\theta ,z\}$ in the current configuration, where the axial direction ( $Z$ and $z$ ) is perpendicular to the membrane plane. Since the problem is axisymmetric, for any point $\mathbf{X}=R{\hat{\mathbf{E}}}_R+Z{\hat{\mathbf{E}}}_z$ in the reference configuration, we assume the mapping

$$\mathbf{x}=r(R){\hat{\mathbf{e}}}_r+Z\zeta (R){\hat{\mathbf{e}}}_z,$$

(8)

where ${\hat{\mathbf{e}}}_r={\hat{\mathbf{E}}}_R$ and ${\hat{\mathbf{e}}}_z={\hat{\mathbf{E}}}_Z$ are the unit vectors of the polar system in the current and the reference configurations, respectively.

For future reference, we define the nominal problem of a circular membrane (without any holes) subjected to an equibiaxial stretch. For this problem, there is a trivial solution,

$$r_0(R)={\lambda }_{\mathrm{o}}R\mathrm{and}{\zeta }_0(R)=\frac{1}{{\lambda }_{\mathrm{o}}^2},$$

(9)

where we adopt the $0$ subscript to denote quantities associated with this problem. The corresponding in-plane stress is ${\sigma }_0=\mu ({\lambda }_{\mathrm{o}}^2-{\lambda }_{\mathrm{o}}^{-4})$ .

Returning to the BVP at hand, the boundary conditions at the outer and inner radii are

$$r(R_{\mathrm{o}})={\lambda }_{\mathrm{o}}{R}_{\mathrm{o}},{\sigma }_{\mathrm{rr}}(r_{\mathrm{i}})={\sigma }_{\mathrm{rz}}(r_{\mathrm{i}})={\sigma }_{r\theta }(r_{\mathrm{i}})=0.$$

(10)

Since the lateral boundaries of the membrane are stress-free, on account of the small thickness of the membrane, we assume the plane-stress condition

$${\sigma }_{\mathrm{zz}}={\sigma }_{\mathrm{rz}}={\sigma }_{\theta z}=0,$$

(11)

within the membrane.

The deformation gradient of the assumed mapping is

$$\mathbf{F}=\left[\begin{array}{ccc}{r}_{,R}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& \frac{r}{R}\hfill & 0\hfill \\ \multicolumn{1}{c}{Z{\zeta }_{,R}}& 0\hfill & \zeta \hfill \end{array}\right].$$

(12)

We denote the radial, hoop, and axial stretches

$${\lambda }_r\equiv F_{\mathrm{rR}}=r_{,R},{\lambda }_{\theta }\equiv F_{\theta \mathrm{\Theta }}=\frac{r}{R},\mathrm{and}{\lambda }_z\equiv F_{\mathrm{zZ}}=\zeta ,$$

(13)

respectively, where we use the comma convention for the partial derivative. Thanks to the negligible thickness of the membrane, we set $F_{\mathrm{zR}}=Z{\zeta }_{,R}=0$ , and note that, on account of the membrane constitutive behavior, this satisfies the assumed plane-stress condition.

Incompressibility yields that $J=r_{,R}\frac{r}{R}\zeta =1$ , therefore the axial stretch is

$${\lambda }_z=\zeta =r_{,R}^{-1}\frac{R}{r}.$$

(14)

Next, the constitutive relations (7) lead to

$$\mathit{\sigma }=\mathrm{diag}\{\mu r_{,R}^2-p,\mu \frac{r^2}{R^2}-p,\mu {\zeta }^2-p\},$$

(15)

$$\mathbf{P}=\mathrm{diag}\{\mu r_{,R}-\frac{p}{r_{,R}},\mu \frac{r}{R}-\frac{\mathrm{pR}}{r},\mu \zeta -\frac{p}{\zeta }\}.$$

(16)

Plane-stress (11) implies $p=\mu {\zeta }^2$ , and substitution of equation (14) in the expressions for $W$ , $\mathit{\sigma }$ , and $\mathbf{P}$ gives

$$W(R)=\frac{\mu }{2}(r_{,R}^2(R)+\frac{r^2(R)}{R^2}+\frac{R^2}{r_{,R}^2(R)r^2(R)}-3),$$

(17)

$$\mathit{\sigma }(R)=\mu \mathrm{diag}\{r_{,R}^2(R)-\frac{R^2}{r_{,R}^2(R)r^2(R)},\frac{r^2(R)}{R^2}-\frac{R^2}{r_{,R}^2(R)r^2(R)},0\},$$

(18)

and

$$\mathbf{P}(R)=\mu \mathrm{diag}\{r_{,R}(R)-\frac{R^2}{r_{,R}^3(R)r^2(R)},\frac{r(R)}{R}-\frac{R^3}{r_{,R}^2(R)r^3(R)},0\}.$$

(19)

Substituting the resulting expression for $\mathbf{P}(R)$ into the equilibrium equation (4), we find that the $\theta$ and the $z$ components vanish identically. The radial component

$$P_{\mathrm{rR},R}+\frac{P_{\mathrm{rR}}-P_{\theta \mathrm{\Theta }}}{R}=0$$

(20)

leads to the non-linear differential equation

$$r_{,\mathrm{RR}}+\frac{r_{,R}}{R}-\frac{r}{R^2}+3\frac{R}{r^2{r}_{,R}^2}(\frac{R{r}_{,\mathrm{RR}}}{r_{,R}^2}+\frac{R}{r}-\frac{1}{r_{,R}})=0.$$

(21)

The associated boundary conditions (10) are

$$r(R_{\mathrm{o}})={\lambda }_{\mathrm{o}}{R}_{\mathrm{o}},\frac{r(R_{\mathrm{i}})}{R_{\mathrm{i}}}{r}_{,R}^2(R_{\mathrm{i}})=1,$$

(22)

where the conditions on ${\sigma }_{\mathrm{rz}}$ and ${\sigma }_{\theta z}$ are satisfied identically. Equations (21) and (22) define the governing boundary value problem (BVP) of the stretched membrane.

For later reference, we note that the boundary condition at the hole, together with the incompressibility constraint and the plane-stress condition, define the deformation gradient at the boundary of the hole up to a single degree of freedom ${\lambda }_{\mathrm{i}}={\lambda }_{\theta }(R_{\mathrm{i}})=r(R_{\mathrm{i}})/R_{\mathrm{i}}$ , such that

$$\mathbf{F}(R=R_{\mathrm{i}})=\left[\begin{array}{ccc}\frac{1}{\sqrt{{\lambda }_{\mathrm{i}}}}\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& {\lambda }_{\mathrm{i}}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & \frac{1}{\sqrt{{\lambda }_{\mathrm{i}}}}\hfill \end{array}\right].$$

(23)

This deformation gradient is identical to one of an incompressible strip under uniaxial tension.

4. The limit of infinite stretch

To solve the problem in the limit of infinite stretch, we represent the BVP in terms of the normalized dimensionless variables

$$\kappa \equiv \frac{R_{\mathrm{i}}^2}{R^2}\mathrm{and}\mathrm{\Lambda }(\kappa )\equiv \frac{r(R)}{{\lambda }_{\mathrm{o}}R}=\frac{{\lambda }_{\theta }}{{\lambda }_{\mathrm{o}}}.$$

(24)

For conciseness, we also define the hole expansion ratio

$$\rho \equiv \frac{{\lambda }_{\mathrm{i}}}{{\lambda }_{\mathrm{o}}}=\mathrm{\Lambda }(1).$$

(25)

In terms of $\kappa$ and $\mathrm{\Lambda }$ , the governing equation (21) takes the simpler form

$${\mathrm{\Lambda }}_{,\kappa \kappa }+\frac{3}{2{\lambda }_{\mathrm{o}}^6}\frac{{\left({\mathrm{\Lambda }}^2\right)}_{,\kappa \kappa }}{{\mathrm{\Lambda }}^3{(\mathrm{\Lambda }-2\kappa {\mathrm{\Lambda }}_{,\kappa })}^4}=0.$$

(26)

The corresponding boundary conditions at the outer boundary and at the hole are

$$\mathrm{\Lambda }({\kappa }_{\mathrm{o}}\equiv \frac{R_{\mathrm{i}}^2}{R_{\mathrm{o}}^2})=1\mathrm{and}\mathrm{\Lambda }(1){[\mathrm{\Lambda }(1)-2{\mathrm{\Lambda }}_{,\kappa }(1)]}^2={\lambda }_{\mathrm{o}}^{-3},$$

(27)

respectively.

To solve the problem in the limit of infinite stretch, we let ${\lambda }_{\mathrm{o}}\to \infty$ in equations (26) and (27). Assuming that the expression $\mathrm{\Lambda }-2\kappa {\mathrm{\Lambda }}_{,\kappa }$ remains finite for all ${\kappa }_{\mathrm{o}}\le \kappa <1$ in this limit, the associated BVP becomes the linear problem,

$$\{\begin{array}{cc}{\mathrm{\Lambda }}_{,\kappa \kappa }^{\infty }=0,\hfill & {\kappa }_{\mathrm{o}}\le \kappa <1\hfill \\ \multicolumn{1}{c}{{\mathrm{\Lambda }}^{\infty }({\kappa }_{\mathrm{o}})=1,}& \hfill \\ \multicolumn{1}{c}{{\mathrm{\Lambda }}^{\infty }(1)-2{\mathrm{\Lambda }}_{,\kappa }^{\infty }(1)=0,}& \hfill \end{array}$$

(28)

where the superscript $\infty$ denotes the infinite stretch limit. Note that the simplification of the boundary condition at the hole $(\kappa =1)$ stems from the requirement $\mathrm{\Lambda }(1)=\frac{{\lambda }_{\mathrm{i}}}{{\lambda }_{\mathrm{o}}}\ge 1$ , which is trivial for an incompressible membrane.

The solution of equation (28) is

$${\mathrm{\Lambda }}^{\infty }(\kappa )=\frac{1+\kappa }{1+{\kappa }_{\mathrm{o}}}.$$

(29)

This solution indeed satisfies the assumption we made above and is therefore the solution to the problem in the infinite stretch limit.

We further note that if we want to expand this solution for finite values of ${\lambda }_{\mathrm{o}}$ , we can consider the series expansion of the solution in terms of the small parameter $\frac{1}{\sqrt{{\lambda }_{\mathrm{o}}}}$ ,

$$\mathrm{\Lambda }(\kappa )={\mathrm{\Lambda }}^{\infty }(\kappa )+\sum _{n=1}^{\infty }{\mathrm{\Lambda }}_n(\kappa ){\left(\frac{1}{\sqrt{{\lambda }_{\mathrm{o}}}}\right)}^n.$$

(30)

The reasoning behind this choice for the expansion parameter is attributed to the deformation gradient at the boundary of the hole (23). Since the lowest non-vanishing term in this expansion corresponds to $n=3$ , we deduce that ${\mathrm{\Lambda }}^{\infty }$ approximates the exact solution for an extremely wide range of deformations.

Returning to the physical variables, the solution for ${\lambda }_{\theta }$ is

$$\underset{{\lambda }_{\mathrm{o}}\to \infty }{lim}\frac{{\lambda }_{\theta }^{\infty }}{{\lambda }_{\mathrm{o}}}=\frac{r^{\infty }}{r_0}=\frac{1+\frac{R_{\mathrm{i}}^2}{R^2}}{1+\frac{R_{\mathrm{i}}^2}{R_{\mathrm{o}}^2}}.$$

(31)

The hole expansion ratio in this limit is

$${\rho }^{\infty }=\underset{{\lambda }_{\mathrm{o}}\to \infty }{lim}\frac{{\lambda }_{\mathrm{i}}}{{\lambda }_{\mathrm{o}}}=\frac{2}{1+\frac{R_{\mathrm{i}}^2}{R_{\mathrm{o}}^2}}$$

(32)

and from equation (18) we have that the hoop stress is

$$\underset{{\lambda }_{\mathrm{o}}\to \infty }{lim}\frac{{\sigma }_{\theta \theta }}{{\sigma }_0}={\left(\frac{1+\frac{R_{\mathrm{i}}^2}{R^2}}{1+\frac{R_{\mathrm{i}}^2}{R_{\mathrm{o}}^2}}\right)}^2.$$

(33)

5. Application

We begin this section with the important case of an infinitesimally small hole. Equation (32) shows that in the limit of infinite stretch, the hole expansion ratio of an infinitesimally small hole is ${\rho }^{\infty }=2$ . That is, under plane-stress condition, the radius of the hole is twice that of a similar hole in a membrane subjected to plane-strain condition. Equation (31) implies that away from the hole, where $R>>R_{\mathrm{i}}$ , the solution rapidly converges to the solution of the nominal problem (9). At the other limit, of an extremely large hole (i.e., $R_{\mathrm{i}}/R_{\mathrm{o}}\to 1$ ), the hole expansion ratio is $\rho =1$ for any stretch ratio ${\lambda }_{\mathrm{o}}$ and the membrane acts as an uniaxially stretched strip, along the tangential direction.

For a finite ratio between the radii, $R_{\mathrm{i}}=0.1R_{\mathrm{o}}$ , Figure 2 compares the results of the simple expression in equation (31) to FE simulations for ${\lambda }_{\theta }/{\lambda }_{\mathrm{o}}$ at various values of ${\lambda }_{\mathrm{o}}$ . The FE simulation results were determined with the FEniCSx software package [15–17]. The comparison shows that for ${\lambda }_{\mathrm{o}}\ge 2$ the distribution of the tangential stretch ratio is practically identical to the distribution in the limit of infinite stretch. Away from the hole, the solution converges rapidly to the plane-strain solution, and the maximal difference between the solutions in the infinite and the finite deformation cases is at the hole circumference.

Figure 2.

The tangential stretch ratio normalized by ${\lambda }_{\mathrm{o}}$ for $R_{\mathrm{i}}=0.1R_{\mathrm{o}}$ and for several values of ${\lambda }_{\mathrm{o}}$ . The continuous curves were obtained from the finite element simulations and the dashed curve corresponds to equation (31).

A contour plot of the hoop stress resulting from an FE simulation is shown in Figure 3. Note that the stress reduces to the nominal stress at a rate similar to expression (33).

Figure 3.

The hoop stress determined by a finite element simulation of a membrane with $R_{\mathrm{i}}/R_{\mathrm{o}}=0.1$ subjected to equibiaxial stretch ratio ${\lambda }_{\mathrm{o}}=5.0$ . The left figure shows the entire membrane and the right figure is a close-up of around the hole.

The stress concentration factor in the infinite stretch limit is

$$\underset{\genfrac{}{}{0ex}{}{{\lambda }_{\mathrm{o}}\to \infty }{R\to R_{\mathrm{i}}}}{lim}\frac{{\sigma }_{\theta \theta }}{{\sigma }_0}=\frac{4}{{(1+\frac{R_{\mathrm{i}}^2}{R_{\mathrm{o}}^2})}^2}.$$

(34)

In particular, for an infinitesimally small hole, the stress concentration factor is 4. This is twice the well-known factor in the infinitesimal deformation limit [1].

Figure 4 compares the infinite stretch solution to FE simulations for the hole expansion ratios (Figure 4(a)) and the stress concentration factor (Figure 4(b)) as functions of the stretch ratio ${\lambda }_{\mathrm{o}}$ for several hole sizes. Both the hole expansion ratio and the stress concentration factor converge rapidly to the infinite stretch limit. From a practical viewpoint, as soon as ${\lambda }_{\mathrm{o}}>2.5$ , the closed-form expressions for the infinite stretch limit provide neat approximations for the finite deformation problem. Figure 4 also shows that in the limit of a large hole $(R_i\to R_{\mathrm{o}})$ , the tangential deformation and hoop stress in the membrane approach those in an uniaxially stretched strip.

Figure 4.

(a) The hole expansion ratio $\rho$ and (b) the stress concentration factor at the hole circumference as functions of the stretch ratio at the outer surface ${\lambda }_{\mathrm{o}}$ for several hole sizes. The continuous curves were obtained from the finite element simulations and the dashed lines correspond to the infinite stretch limit (equations (32) and (34), respectively). The dash-dotted curve corresponds to a strip uniaxially stretched by ${\lambda }_{\mathrm{o}}$ .

6. Conclusion

We analyze, within the framework of finite deformation, equibiaxial extension of a thin neo-Hookean circular membrane with a hole at its center under plane-stress condition. First, examining the problem in polar coordinates, we derived the non-linear radial equilibrium equation with the associated non-linear boundary conditions. Next, with an appropriate change of variables, we introduced a new representation for the governing problem. This new form lends itself to an exact closed-form solution in the limit of infinite stretch.

Comparison of this solution with corresponding finite element simulations revealed fine agreement for applied equibiaxial stretch ratios larger than 2.5 for any hole size. In the limit of an extremely large hole, we show that the deformation of the membrane approaches that of an uniaxially stretched thin strip. Finally, we reveal that at the limit of infinite stretch, at the circumference of an infinitesimal hole, the stress concentration factor is 4 and the tangential stretch ratio is twice the applied far-field stretch ratio. These values are double the corresponding stress concentration factor and hole expansion ratio in the limit of infinitesimal deformation.