A holed membrane at finite equibiaxial stretch
AA HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH
IDAN Z. FRIEDBERG AND GAL DEBOTTON
DEPT. OF MECHANICAL ENGINEERING, BEN-GURION UNIVERSITY,BEER-SHEVA 8410501, ISRAEL
Abstract.
The deformation and stress distribution in a stretched thin neo-Hookeancircular membrane with a hole at its center are analyzed within the framework of finitedeformation elasticity. Initially, we derive a simple form for the differential governingequation to the problem. This enables us to introduce a closed-form solution in the limitof infinite stretch. Subsequently, we propose approximate solutions for intermediate andlarge deformations. These approximations approach the exact solutions in the limitsof small and infinite stretches. The transition stretch at which the membrane behaviorswitches from the intermediate to the large deformation approximation is determinedtoo. Comparison of our solution and approximations to corresponding numerical resultsreveal a neat agreement for any stretch and ratio between the hole to the membraneradii.In the limit of large stretches and a small hole, the ratio of the hoop stress at thehole boundary to the nominal stress is 4, which is twice the corresponding ratio in thesmall deformation limit. Comparison of the strain energy stored in the membrane tothe one in a membrane without a hole reveals that only at finite stretches the differencebetween these energies becomes meaningful. This implies that it is likely that a flaw ina membrane will tear out only at a finite level of stretches. Introduction
The problem of an equibiaxially stretched membrane with a hole at its center is a fre-quently encounter one. The well known axisymmetric solution in the limit of infinitesimaldeformations (e.g. Shames (1997)) is crucial for analyzing failure due to stress concen-trations around the hole. Moreover, the asymptotic solution in the limit a small hole isfrequently used for estimating the stress in many other problems containing small holes.However, the known solution for infinitesimal deformations is not applicable for materialssuch as elastomers and tissues that undergo finite deformations. For these stretchablematerials this problem need to be analyzed within the framework of finite deformationelasticity
Ogden (1997). a r X i v : . [ c ond - m a t . s o f t ] J a n FRIEDBERG AND DEBOTTON, BEN-GURION UNIV.
Previous analyses (e.g., Rivlin and Thomas, 1951; Wong and Shield, 1969; Yang, 1967)dealt with a general Mooney-Rivlin material (Mooney, 1940; Rivlin, 1948), with the neo-Hookean material as a special case, and a wide range boundary conditions. Exact analyt-ical results were developed in the limit of small deformations. Approximated expressionsbased on expansion series about these exact solution were proposed and compared withcorresponding numerical results for a narrow range of ratios between the hole and themembrane radii. Yang (1967) and Wong and Shield (1969) further pointed out how theirapproximations break down in the case of traction free boundary condition at the hole.Haughton (1991) solved a similar plane-stress problem for the Varga (1966) material,revealing that for this material the membrane thickness remains uniform. Various nu-merical analyses of similar problems and the related cavitation problems were conductedwith different hyperelastic materials (e.g., Cohen and Durban, 2010; Haughton, 1990;Sang et al., 2015).Herein, we tackle the problem of a finitely stretched incompressible neo-Hookean cir-cular membrane with a traction free hole of an arbitrary size at its center. Initially, weintroduce a simple non-dimensional, non-linear second order ordinary differential equa-tion governing the boundary value problem. This further leads to a closed-form solutionfor infinite stretching of the membrane and approximations for intermediate and largedeformations. The approximations are compared to corresponding numerical solutions ofthe problem, confirming their accuracy. Additionally, the strain energy and stress distri-butions are analyzed and compared to the nominal cases of a membrane without a holeand a circumferentially stretched thin ring. The former comparison implies that a smallflaw in stretched membrane is likely to tear out only at finite strains.2.
Background
The deformation of a 3-dimensional body from a reference (undeformed) configuration B r ⊂ R to a current (deformed) configuration B ⊂ R , can be described by the bijectionmapping of each material point P at a reference position X ∈ B r to its correspondingcurrent position x = χ ( X ) ∈ B . Both configurations can be represented in a coordinatesystem. We denote the unit vectors along chosen referential and current coordinates ( α and i respectively) as ˆE α and ˆe i respectively.The deformation gradient is(1) F ≡ Grad( x ) , HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 3 where Grad is the gradient with respect to the reference position X . The change in thevolume of a material element is(2) J ≡ det( F ) = d v d V , where d V is its referential volume and d v its current volume. The right and left Cauchy-Green deformation tensors are(3) C = F T F and b = FF T , respectively.If no body forces are at play, the equilibrium equation for linear momentum(4) div( σ ) = , where σ is the Cauchy stress (or true stress) and div is the divergence with respect tothe current position. The balance of angular momentum implies that σ is symmetric.Using Nanson’s formula for the transformation of area elements, the force on a currentarea element d s is equated to the traction on a referential area element d S ,(5) σ T d s = J σ T F − T d S ≡ P d S , where P is defined as the Piola stress (or nominal stress). Substituting σ = J − FP T into (4) yields the referential equilibrium equation(6) Div( P ) = , where Div is the divergence with respect to the reference position.The constitutive behavior of hyperelastic materials can be expressed in terms of a strainenergy-density function (SEDF) W such that(7) P iα = ∂W∂F iα . A simple constitutive relation that describes the behavior of many materials in the smallto intermediate deformation range is the incompressible neo-Hookean material, for which(8) W = µ I −
3) and I ≡ , where I = tr( C ) and I = det( C ) = J . Accordingly,(9) P = µ F − p F − T and σ = µ b − p , where is the identity tensor and p is a Lagrange multiplier that represents a pressurelike term. FRIEDBERG AND DEBOTTON, BEN-GURION UNIV. Analysis
Consider a thin circular incompressible neo-Hookean membrane with a hole at its center(see Fig. 3.1a). In its referential state the outer radius of the membrane is R o and theradius of the hole is R i . The membrane is subjected to biaxial stretch λ o at the outerradius (see Fig. 3.1b and c) such that the outer radius in the deformed state is r o = λ o R o .The circumference of the hole and the lateral faces of the membrane are stress free. Werecall that a well-known solution exists for the case where the punctured membrane issubjected to plane-strain condition with the axial stretch being fixed (e.g., Ogden, 1997).We examine this problem in polar coordinates { R, Θ , Z } in the reference configura-tion and { r, θ, z } in the current configuration, where the axial direction ( Z and z ) isperpendicular to the membrane plane. Since the problem is axisymmetric, for any point X = R ˆE R + Z ˆE z in the reference configuration we assume the mapping(10) x = r ( R ) ˆe r + Zζ ( R ) ˆe z , where ˆe r = ˆE R and ˆe z = ˆE Z are the unit vectors of the polar system in the current andthe reference configurations, respectively.The boundary conditions at the outer and inner radii are(11) r ( R o ) = λ o R o , σ rr ( R i ) = σ rz ( R i ) = σ rθ ( R i ) = 0 . Since lateral boundaries of the membrane are stress free, on account of the smallthickness of the membrane we assume the plane stress condition (12) σ zz = σ rz = σ θz = 0 , within the membrane.The deformation gradient of the assumed mapping is(13) F = r ,R rR Zζ ,R ζ . We denote the radial, hoop and axial stretches(14) λ r ≡ F rR = r ,R , λ θ ≡ F θ Θ = rR and λ z ≡ F zZ = ζ, respectively. Thanks to the negligible thickness of the membrane we set F zR = Zζ ,R = 0,and note that this is in agreement with the assumed plane stress condition. HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 5 R o R i R Θ ˆ E R ˆ E Θ R o R i R Θ ˆE R ˆE Θ r o r θ t = r i ˆe r ˆe θ (a)(b) r ( R ) θ ˆe r ˆe θ t = r o = λ o R o (c) λ o > λ τ o r i = ρλ o R i r t λ o < λ τ o Figure 3.1.
A biaxially stretched punctured membrane with R i = 0 . R o in its (a) reference configuration, (b) moderately deformed configuration with λ o = 1 . < λ τ o and (c) severely deformed configuration λ o = 2 . > λ τ o . The tran-sition radius r t between the inner and the outer regions of the severely deformedmembrane is depicted in (c). Incompressibility yields that J = r ,R rR ζ = 1, therefore the axial stretch is(15) λ z = ζ = r − ,R Rr .
Substituting F into the constitutive relations (9) leads to σ = diag (cid:26) µr ,R − p, µ r R − p, µζ − p (cid:27) , (16) P = diag (cid:26) µr ,R − pr ,R , µ rR − pRr , µζ − pζ (cid:27) . (17)Plane stress (12) implies p = µζ , and substitution of (15) in the expressions for W , σ and P gives W ( R ) = µ (cid:32) r ,R ( R ) + r ( R ) R + R r ,R ( R ) r ( R ) − (cid:33) , (18) σ ( R ) = µ diag (cid:40) r ,R ( R ) − R r ,R ( R ) r ( R ) , r ( R ) R − R r ,R ( R ) r ( R ) , (cid:41) , (19) FRIEDBERG AND DEBOTTON, BEN-GURION UNIV. and(20) P ( R ) = µ diag (cid:40) r ,R ( R ) − R r ,R ( R ) r ( R ) , r ( R ) R − R r ,R ( R ) r ( R ) , (cid:41) . We note that both the θ and the z components of the equilibrium equation (6) vanishidentically, and the radial component is(21) ∂P rR ∂R + 1 R ( P rR − P θ Θ ) = 0 . Substituting P ( R ) into (21), the non-linear governing equation for the problem is(22) f ( r ( R ) , R ) ≡ r ,RR + r ,R R − rR + 3 Rr r ,R (cid:32) Rr ,RR r ,R + Rr − r ,R (cid:33) = 0 . The associated boundary conditions (11) are(23) r ( R o ) = λ o R o , r ( R i ) R i r ,R ( R i ) = 1 , and where the conditions on σ rz and σ θz are satisfied identically. Equations (22) and (23)define the boundary value problem (BVP) of the stretched membrane.In the limit of infinitesimal deformation, λ o = 1 + (cid:15) where (cid:15) (cid:28)
1. In this limit the wellknown solution is (e.g., Shames, 1997)(24) r = r S ( R ) + O ( (cid:15) ); r S ( R ) = AR + BR .
The constants(25) A = 1 + 11 + 3 R /R (cid:15), B = 3 R R /R (cid:15) are obtained from the boundary conditions.For later reference we define the ratio between the hole and the membrane radii in thedeformed configuration as the hole expansion ratio ,(26) ρ ≡ r i r o = λ i λ o , where r i ≡ r ( R i ) and λ i ≡ λ θ ( R i ) is the tangential stretch at the hole boundary. In thesmall deformation limit(27) ρ = ρ S ≡ R − R R + 3 R (cid:15). In order to solve the problem in the limit of infinite deformation it is useful to representthe boundary value problem in terms of the dimensionless variables(28) κ ≡ R R and Λ( κ ) ≡ r ( R ) λ o R = λ θ λ o . HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 7
Note that the hole expansion ratio is(29) ρ = Λ(1) . In terms of κ and Λ the governing equation (22) takes the simpler form(30) F (Λ( κ ) , κ, λ o ) = Λ ,κκ + 32 λ (Λ ) ,κκ Λ (Λ − κ Λ ,κ ) = 0 . The corresponding boundary conditions at the outer boundary and at the hole are(31) Λ (cid:18) κ o ≡ R R (cid:19) = 1 , and Λ(1) [Λ(1) − ,κ (1)] = λ − , respectively.In terms of κ and Λ, the solution at the small deformation limit is Λ S ( κ ) = a S + b S κ ,where(32) a S = Aλ o = 1 − κ o κ o (cid:15), and b S = Bλ o R = 31 + 3 κ o (cid:15). Adopting a similar form we define(33) Λ L ( κ ) = a + bκ, and substitute for Λ into equation (30), the resulting term is(34) F L (Λ L ( κ ) , κ, λ o ) = 3 b λ ( a − bκ ) ( a + bκ ) . Since F L (cid:54) = 0, (33) is generally not a solution to the problem. However, as λ o → ∞ we have that F L →
0, suggesting that (33) is the solution to the problem in the infinitedeformation limit. The boundary condition on the outer radius leads to(35) a = 1 − bκ o , and at the boundary of the hole, in the limit λ o → ∞ , we have that(36) lim λ o →∞ b = b ∞ ≡
11 + κ o , and a ∞ is obtained by substituting b ∞ in (35). This leads to the following expression forΛ L in the limit λ o → ∞ , namely(37) Λ ∞ ( κ ) = 1 + κ κ o . Before we proceed, we note that by substituting Λ ∞ back in (34) we have that(38) F (Λ ∞ ( κ ) , κ, λ o ) = 3(1 + κ o ) λ (1 − κ ) (1 + κ ) . FRIEDBERG AND DEBOTTON, BEN-GURION UNIV.
In the limit λ o → ∞ , this expression vanishes for any κ (cid:54) = 1, that is everywhere but atthe inner boundary. Thus, Λ ∞ is the solution for the problem in the infinite deformationlimit, with an identified singularity at the inner boundary. We further note that if weconsider a series solution for the problem in the form(39) Λ = a ∞ + b ∞ κ + b p ( κ ) (cid:18) λ o (cid:19) p , as λ o → ∞ the lowest non vanishing correction term is p = 3 /
2. This higher than linearcorrection term implies that Λ ∞ approximates the exact solution for an extremely widerange of deformations.Returning to physical variables, the solution for r ( R ) in the infinite deformation limitis(40) r ∞ ( R ) = λ o R o R / R i + R i / RR o / R i + R i / R o . The hole expansion ratio in this limit is(41) ρ ∞ = Λ ∞ (1) = 21 + κ o = 2 R R + R . Next, we propose an approximation for a range of deformations which are finite butrather moderate. This is accomplished by adding a quadratic term in κ to the smalldeformation solution. This leads to an expression in the form a Taylor expansion series.(42) ˜Λ S ( κ ) = a S + b S κ + γ S κ = ˜ ρ S + β S ( κ −
1) + γ S ( κ − , where we find the second expression in (42) more convenient for the subsequent analysis,where ˜ ρ S = a S − β S + γ S and β S = b S + 2 γ S . In order for ˜Λ S to approach Λ S in the smalldeformation limit we require that(43) lim λ o → γ S = 0 . Note that ˜Λ S (1) = ˜ ρ S is an approximation for the hole expansion ratio. Therefore, wehave the additional constraint(44) lim λ o → ˜ ρ S = 1 . The approximation we propose is constructed such that it will satisfy the differentialequation (30) at the inner boundary ( κ = 1), and we assume that F will remain smallaway from it. Thus, F ( ˜Λ S (1) , , λ o ) = (3 β S ) λ ( ˜ ρ S ) ( ˜ ρ S − β S ) + γ S (cid:18) λ ( ˜ ρ S ) ( ˜ ρ S − β S ) + 2 (cid:19) = 0(45) HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 9 leads to γ S = − β S ) ρ S [ λ ( ˜ ρ S ) ( ˜ ρ S − β S ) + 3] . (46)The boundary condition at the hole requires(47) β S = ˜ ρ S ± (cid:112) ˜ ρ S λ / . Since only the ( − ) option in (47) satisfies (43), we have that(48) β S = ˜ ρ S − (cid:112) ˜ ρ S λ / , γ S = − (cid:104) ( ˜ ρ S ) / λ / − (cid:105)
32( ˜ ρ S ) λ . In order to satisfy the boundary condition at the outer radius, λ o must be related to ˜ ρ S by(49) λ o = L κ o ( ˜ ρ S ) , where(50) L κ o ( ρ ) = (cid:32) ρ (1 − κ o ) (281 − κ o + 9 κ ) ρ − √ κ o − (cid:112) ρ (5 ρ − − (cid:33) . Since only one branch of L κ o ( ρ ) satisfies (44), ˜ ρ S can be determined uniquely by theinverse of the said branch,(51) ˜ ρ S = L − κ o ( λ o )We note that since lim λ o →∞ L − κ o ( λ o ) > ρ ∞ , at large deformations this approximate solutiondoes not converge to the solution of this problem.When the punctured membrane is subjected to large deformation we distinguish be-tween the responses in the inner and outer regions. Specifically, we assume that in thevicinity of the hole there is a band in which the dependence of the tangential stretchon R is small in comparison with the corresponding dependence in the outer part. Wefurther assume that this band shrinks as the applied stretch increases and finally, in thelimit λ o → ∞ , reduces to the singularity at the surface of the hole. Note that this modelresembles the partition of a wrinkled membrane to an inner wrinkly and an outer tautregions Wu (1978).According to our assumption r ∞ provides a good approximation for the solution in theouter region. Indeed near the outer boundary, in the region R o ≥ R (cid:29) R i ,(52) f ( r ∞ ( R ) , R ) = 1 λ (cid:32) (cid:18) R i R (cid:19) + O (cid:18) R i R (cid:19) (cid:33) (cid:28) is small as long as λ o is large. Therefore, r ∞ can serve as a basis for an approximatesolution near the outer boundary in the large deformation regime.Following this observation, our approximation in the outer region will be based on theinfinite deformation solution (37) with a quadratic correction term in κ . In the innerregion we approximate the solution with a quadratic polynomial in the form of (42).At the interface between the inner and outer section we require continuity of the radialdeformation and stress. These are fulfilled by requiring continuity of r ( R ) and its firstderivative.Specifically, in terms of κ and Λ, the proposed large deformations approximation is(53) ˜Λ ∞ = ˜Λ I = ˜ ρ ∞ + β ∞ ( κ −
1) + γ ∞ ( κ − , ≥ κ ≥ κ t ˜Λ O = κ κ o + c ( κ − κ o ) , κ t ≥ κ ≥ κ o , where κ t ≡ ( R i /R t ) and R t is the transition radius between the inner and outer sections.The proposed approximation involve six unknowns, the five constants and the transitionradius R t . In order to determine these unknowns, two equations are obtained by imposingthe governing equation to vanish at the boundary of the hole and the outer boundary.Additional two equations arise from the boundary conditions. The final two equationsare obtained from the continuity requirement on the deformation and its derivative.Note that ˜Λ I is subjected to the same conditions at the inner boundary as ˜Λ S . Hence,the relations between β S , γ S and ˜ ρ S in (48) apply for β ∞ , γ ∞ and ˜ ρ ∞ too. Similarly to˜ ρ S , ˜Λ ∞ (1) = ˜ ρ ∞ is an approximation for the hole expansion ratio.In the outer region the condition we impose on ˜Λ O is that F vanishes at the outerboundary ( κ = κ o ), assuming that it will remain small away from it. Thus, F (cid:16) ˜Λ O ( κ o ) , κ o , λ o (cid:17) = 2 c + 3( κ o + 1) (2 c ( κ o + 1) + 1) λ ( κ o − = 0(54)leads to c = − κ o + 1) λ ( κ o − + 6( κ o + 1) . (55)The requirement of the smooth transition between the two parts of the approximatesolution allows to obtain κ t as a function of λ o and ˜ ρ ∞ . Specifically, κ t = 1 c − γ ∞ (cid:18) β (1 + κ o ) −
11 + κ o + cκ o − γ ∞ (cid:19) . (56) HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 11
Finally, continuity of displacement dictates the relation between ˜ ρ ∞ and λ o via the im-plicit relation(57) ˜ ρ ∞ + β ∞ ( κ t −
1) + γ ∞ ( κ t − = 1 + κ t κ o + c ( κ t − κ o ) . Note that not all values of λ o ≥ ≤ ˜ ρ ∞ < / (1 + κ o ) suchthat (57) is satisfied. Moreover, not all pairs of λ o , ˜ ρ ∞ that satisfy (57) correspond to avalue of κ o ≤ κ t ≤
1. Next, we determine the range of λ o for which the large deformationapproximation is valid.If the applied stretch is gradually increased from λ o = 1 there is a transition stretch λ τ o at which the membrane response switches from intermediate to large. Moreover, when λ o = λ τ o then κ t = κ o and hence ˜Λ S = ˜Λ I as both sencod order polynomials satisfy thesame conditions. The corresponding hole expansion ratio at λ τ o is then ˜ ρ τ = ˜ ρ S ( λ o = λ τ o ) = ˜ ρ ∞ ( λ o = λ τ o ). The transition parameters λ τ o and ˜ ρ τ are found by solving thecontinuity (57) and smoothness (56) conditions on ˜Λ ∞ in the limit κ t → κ o . In this limit,(56) becomes 2(3 κ o − λ / √ ˜ ρ ∞ + 3 − κ o λ ( ˜ ρ ∞ ) + ˜ ρ ∞ (11 − κ o ) − κ o + 1 = 0 . (58)This quadratic equation has two solutions for λ τ o , out of which only(59) λ τ o = 3 / (1 − κ o ) / ˜ ρ τ (cid:32) − κ o − (cid:115) − κ o ˜ ρ τ (1 + κ o ) (cid:33) − , corresponds to λ τ o >
1. Substituting λ τ o in (57) and taking the limit κ t → κ o gives(60) κ o (cid:34) ρ τ (cid:32)(cid:115) − κ o ˜ ρ τ (1 + κ o ) − (cid:33) + 3 (cid:35) + 2 ˜ ρ τ (cid:32)(cid:115) − κ o ˜ ρ τ (1 + κ o ) − (cid:33) + 9 = 0 . From the two solutions for ˜ ρ τ , only(61) ˜ ρ τ = 3 κ o + 13 + (cid:112) − κ o ( κ o + 2)10( κ o + 1) , that corresponds to 1 < ˜ ρ τ < / (1 + κ o ) is feasible. Substituting ˜ ρ τ back to (59) resultsin(62) λ τ o = 10 · / ( κ o + 1)(1 − κ o ) / κ o + (cid:112) − κ o ( κ o + 2) + 13 (cid:32) − κ o − (cid:115) − κ o + (cid:112) − κ o ( κ o + 2)13 + 3 κ o + (cid:112) − κ o ( κ o + 2) (cid:33) − / . Figure 3.2.
The hole expansion ratio as a function of 1 /λ o in the limit of in-finitesimally small hole according to the numerical results (dots), the infinite andsmall deformation limits (dash-dotted and densely dashed lines, respectively),and the intermediate and large deformation approximations, (dashed and con-tinuous curves, respectively). Thus, we obtained a closed-form expression for the transition stretch from the intermedi-ate to the large deformation regime. This depends solely on the geometry of the membranein terms on the ratio between the hole and the membrane radii κ o = ( R i /R o ) .Of particular interest is the asymptotic limit of an infinitesimally small hole. In thelimit of infinite deformation, according to Eq. (41) the hole expansion ratio is ρ ∞ = 2.That is, under plane stress the radius of the hole is twice that of a similar hole in amembrane under the aforementioned plane strain condition. Yet, Eq. (40) implies thataway from the hole, where R (cid:29) R i , the solution rapidly converges to the plane-straincase.For small holes the transition stretch from intermediate to large deformations(63) λ τ o → .
628 + 1 . κ o . is obtained via a Taylor series expansion of (62). At this transition stretch the holeexpansion ratio is(64) ˜ ρ τ → . − . κ o . We note that the hole expansion ratio varies only slightly as the membrane stretch in-creases from λ τ o to infinity (form 1.883 to 2). HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 13
Fig. 3.2 shows the variation of the hole expansion ratio ρ as a function of 1 /λ o , accord-ing to the intermediate and large deformation approximations (dashed and continuouscurves, respectively), together with extensions of the solutions for the small and infinitedeformation limits (densely dashed and continuous lines, respectively). Also shown arethe corresponding results determined by numerical solution of the governing Eq. (30)(dots). Interestingly, we note that the form of the governing equation (30) enables to de-termine a numerical solution for the problem even in the limit R i → ≤ κ ≤ λ o = 1 .
628 and ˜ ρ = 1 . ρ S for λ o < λ τ o and ˜ ρ ∞ for λ o > λ τ o provides a neat approximationfor the numerical results for ρ throughout the entire range of deformations.4. Numerical Application
We begin this section with a through comparison between the proposed approximationand the corresponding numerical solution of the problem. For conciseness, in the sequelwe make use of the intermediate and the large approximations in their perspective rangesof the applied stretch.Fig. 4.1 shows the variation of the hole expansion ratio as a function of 1 /λ o for severalvalues of R i /R o , according to the proposed approximation (continuous curves) and nu-merical results (dots). The dashed curve is a parametric curve that marks the transitionbetween the intermediate deformation approximation ˜Λ S and the large deformation ap-proximation ˜Λ ∞ . We observe the agreement between the analytical approximation andthe numerical results throughout the entire range of stretch ratios and hole sizes.Fig. 4.2 compares between the proposed approximation and the numerical results forthe tangential and radial stretches in the membrane in the case √ κ o = R i /R o = 1 / λ o . The largest difference between the approximation and thenumerical solution, which is smaller than 5%, is exhibited at λ o = 1 .
3. Both Figs. 4.1and 4.2 demonstrate that the approximation proposed herein agrees with the numericalsolutions throughout the entire range of deformations and ratios between the hole andthe membrane radii.The distribution of radial stretch near a small hole ( R i = R o /
20) is shown for λ o = 5in Fig. 4.3a. This serves to demonstrate the change in the trend of the solution in thevicinity of the hole at large stretches. In the numerical results we observe an inflection Figure 4.1.
The hole expansion ratio as a function of 1 /λ o for several values of √ κ o = R i /R o . The continuous curves and dots correspond to the approximationand the numerical results, respectively. The dashed curve depicts the transitionstretch λ τ o as a function of R i /R o Figure 4.2.
Principal stretches in the punctured membrane for R i = R o / point at which the variation rate of the radial stretch is maximal. The lower slope of thecurve near the hole is captured by ˜Λ I . We further note that both the inflection point andthe transition radius R t approach the hole boundary as the applied stretch increases.In the limit of an infinitesimally small hole under infinite deformation, the axial stretchat the surface of the hole is λ z = 1 / √ λ o , and the axial stretch at the membrane outerboundary is λ out z = 1 /λ . Clearly, the membrane gets thinner everywhere. Yet, thethickness at the surface of the hole becomes unboundedly larger relative to the thickness at HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 15 away from the hole. Fig. 4.3b illustrates this phenomenon for a small hole for which R i = R o /
20. As was mentioned before, we note that at radii larger than 3 R i , the membranebecomes virtually flat, implying that plane strain condition is applicable away from thehole. Nonetheless, the rapid variation in the thickness near the hole is fundamentallydifferent from the uniform deformation determined by Haughton (1991) for the puncturedVerga membrane.Fig. 4.4 shows the ratio between the hoop stress at the outer boundary and the stress ina membrane with no hole (the nominal stress) as a function of 1 /λ o for different values of κ o . Here and elsewhere, we compare the analytical results in the limit R i /R o → R i /R o = 0 .
99. We note that for any hole size, the difference between σ θθ and σ out θθ becomes negligible for λ o > λ τ o . Furthermore for small holes, regardless ofthe applied stretch, the hoop stress at the outer boundary is practically identical to thestress in a membrane with no hole, that is(65) lim κ o → σ out θθ = σ θθ = µ (cid:0) λ − λ − (cid:1) . This is not the case for finite holes in the small deformation limit, where the hoop stressat the outer boundary depends on the hole size via the relation(66) σ out , S θθ = 1 + κ o κ o µ(cid:15). However, in the infinite deformation limit,(67) σ out , ∞ θθ = µλ , inflection point1.00 1.05 1.10 1.15 1.20 1.250.050.100.150.200.250.300.35 Figure 4.3.
Normalized radial (a) and axial (b) stretches in the vicinity of asmall hole of a punctured membrane ( R i = R o / λ o = 5. Figure 4.4.
The hoop stress at the outer boundary normalized by the nominalstress as a function of 1 /λ o for several values of R i /R o . The continuous curvesand dots correspond to the approximation and the numerical results, respectively.The dashed curve depicts the transition stretch λ τ o as a function of R i /R o . which is equal σ θθ in this limit independently of κ o . In the limit of a thin strip ( R i /R o → σ u = µ (cid:18) λ − λ o (cid:19) , which is nothing but the longitudinal stress in a uniaxially stretched strip.Fig. 4.5 shows the ratio between the hoop stress at the inner boundary and the nominalstress as a function of 1 /λ o for different values of κ o . For values of λ o slightly larger than λ τ o the difference between this ratio and its limit at infinite deformation becomes negligible.In the limit of a small hole, σ in θθ /σ θθ is known to be 2 in small strains. In the limit ofinfinite deformation the hoop stress at the surface of the hole is(69) σ in , ∞ θθ = (cid:18)
21 + κ o (cid:19) µλ . Comparing to the nominal stress, the stress concentration factor at the surface of thehole in this limit is(70) lim λ o →∞ σ in θθ /σ θθ = (cid:18)
21 + κ o (cid:19) . In the limit of a small hole this ratio approaches 4, twice the stress concentration factorin the limit of infinitesimal deformations.
HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 17
Figure 4.5.
The hoop stress at the surface of the hole normalized by thenominal stress as a function of 1 /λ o for several values of R i /R o . The continuouscurves and dots correspond to the approximation and the numerical results,respectively. The dashed curve depicts the transition stretch λ τ o as a function of R i /R o . We consider next the strain energy stored in the punctured membrane. In the limit ofsmall deformations, the dependence of the energy density function on the radius is(71) W S = µ (cid:15) (1 + 3 κ o ) (cid:18) R R (cid:19) . Therefore the total strain energy in this limit is(72) W S = (cid:90) V W S d V = V (1 − κ o ) 6 µ(cid:15) κ o , where V = πHR is the volume of a membrane without a hole and the same outer radiusand thickness. The referential volume of the hole is(73) V hole ≡ V − V = κ o V , where V is the volume of the punctured membrane.The energy density of a membrane without a hole under similar boundary conditionsis(74) W S0 = 6 µ(cid:15) , and the total energy stored in the whole membrane is(75) W S0 = (cid:90) V W S0 d V = 6 µV (cid:15) . Figure 4.6.
Strain energy difference normalized by µV hole as a function of λ in the limit R i /R o →
0. The continuous curve and dots correspond to theapproximation and the numerical results, respectively. The dashed curve depictsthe energy difference according to the infinite deformation solution, and thesquare and circular marks represent the inflection and transition points.
The difference between the energies is(76) ∆ SW ≡ W S0 − W S = 24 κ o κ o µV (cid:15) = 241 + 3 κ o µV hole (cid:15) . In the limit of a small hole this difference becomes(77) lim κ o → ∆ SW = 24 µV hole (cid:15) = 4 V hole W S0 . Following similar steps, in the infinite deformation limit the energy difference is(78) ∆ ∞ W ≡ W ∞ − W ∞ = 21 + κ o µV hole λ , and in the limit of a small hole,(79) lim κ o → ∆ ∞ W = 2 µV hole λ = 2 V hole W ∞ . Figure 4.6 shows the variations of the strain energy difference ∆ W as a function of λ in the limit of an infinitely small hole. For convenience we mark the values of λ o on thehorizontal axis. Note that even tough ∆ W increases with λ o , it is rather small at smallstrains. The derivative of ∆ W with respect to λ reaches a maximum at λ o = 1 . ≡ λ inflectiono , beyond which the slope decreases to the slope at the infinite deformation limit(the dashed line). This finding implies that, in the case of a flawed membrane, since atsmall stretches ∆ W is small the likelihood for the development of a hole is small too.However, at larger stretches the rate at which ∆ W increases is higher than quadratic HOLED MEMBRANE AT FINITE EQUIBIAXIAL STRETCH 19 rate. Thus, if ∆ W is required for overcoming the surface energy for generating a hole,this result suggests that a hole will be generated only at finite strains.5. Conclusion
We analyze, within the framework of finite deformation, equvi-biaxial extension of athin neo-Hookean circular membrane with a hole at its center. The goal is to study howthe shape and the stresses depend on the hole size and the applied stretch. First, exam-ining the problem in polar coordinates, we obtained the non-linear governing equationwith the associated non-linear boundary conditions. Next, by a change of variables, weintroduced a simpler representation for the governing problem. This new form lands itselfto an exact closed-form solution in the limit of infinite extension. We further proposean approximation to the solution of the problem. Specifically, we approximate the mem-brane responses under intermediate and large deformations, and determine the transitionstretch between them. We reveal that at large deformations there is a need to distin-guish between two regions, an outer region where the response is reminiscent of the oneunder infinitely stretched membrane, and an inner region in which the dependence of thestretches on the radius is weaker. This finding maybe correlate to the wrinkling phe-nomenon which is frequently observed near the boundary of a hole in finitely stretchedmembranes.Comparison of our solution and proposed approximations with corresponding numericalresults revealed fine agreement for any hole size and applied stretch. We find that thethickness of the membrane rapidly decreases away from the hole. We also find that forany hole size, the hoop stress at the outer boundary is almost identical to the nominalstress in a homogeneous membrane for applied stretches larger than the transition stretch.We reveal that in the at the surface of a small hole under an infinite deformation, thehoop stress at is 4 times the nominal stress and the tangential stretch is twice the appliedstretch. This is twice the stress concentration factor in the small deformation limit.Finally, we determined the strain energy stored in the punctured membrane. As ex-pected, we find that it is smaller than the strain energy stored in a membrane withouta hole. However, in the limit of a small hole and small deformations the difference israther small. Only at finite deformations the difference between the energies becomessubstantial, implying that a flaw in the membrane will tear out only at a finite level ofstretches.
Acknowledgment
The work was supported by the Israel Science Foundation founded by the Israel Acad-emy of Sciences and Humanities (Grant No. 1874/16).
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