4.2 Biaxial Extension of a Membrane
163
above relations give P y = 0 and P x as in Eq. (4.12). This serves as a useful check.
Finally, with J = 1, the Cauchy stresses are σ x = λ x P x and σ y = λ y P y .
Case 2: Compressible Membrane For this case, the constitutive relations are
again given by (4.14), and setting P z = 0 yields
λ z = (λ x λ y )
−ν/(1−ν)
(4.22)
as given by Eq. (4.15). Substitution into Eqs. (4.14) 1,2 gives
P x = μ
λ
−1+3ν
x
λ
2ν
y
1/(1−ν) − λ
−3
x
P y = μ
λ
2ν
x λ
−1+3ν
y
1/(1−ν) − λ
−3
y
.
(4.23)
Finally, with λ z given above, J = λ x λ y λ z = (λ x λ y ) (1−2ν)/(1−ν) , and the Cauchy
stresses are
σ x =
λ x
J
P x ,
σ y =
λ y
J
P y .
(4.24)
The uniaxial solution is recovered by setting λ y = λ −ν
x from (4.16) in the above
equations. Doing this gives P y = 0 and P x as in Eq. (4.17), as well as σ x of (4.18).
4.2.3 Illustrative Results
Since a thin membrane likely buckles when compressed, results for this problem are
given for tension only. For equibiaxial stretching (λ x = λ y ) of an incompressible
membrane with W given by Eq. (4.1), Fig. 4.4a shows plots of Cauchy stress σ x
in the fiber direction as a function of stretch ratio. Results are presented for neoHookean membranes (c 1 = 1, c 2 = 0) without fibers (isotropic, c 3 = 0), with linear
fibers (c 3 = 0.1, c 4 → 0), and with exponential fibers (c 3 = 0.1, c 4 = 0.01). Fibers
markedly increase membrane stiffness in the fiber (x) direction only; for all cases,
the curve for σ y (not shown) is the same as that for σ x in the isotropic membrane.
Results also are shown for an isotropic neo-Hookean membrane stretched in the
x-direction, while the length in the y-direction is held fixed at three values of λ y
(Fig. 4.4b). As λ x increases, the curve for σ x is relatively independent of λ y , while
σ y increases but remains nearly uniform.
The effects of compressibility are shown for equibiaxial stretch of a membrane
composed of isotropic material defined by Eq. (4.2). As in uniaxial extension
(Fig. 4.2b), the stiffness increases with ν (Fig. 4.5a). However, these effects are
considerably more dramatic for biaxial stretching as the membrane approaches
incompressibility (ν → 0.5; see Fig. 4.5b).
163
above relations give P y = 0 and P x as in Eq. (4.12). This serves as a useful check.
Finally, with J = 1, the Cauchy stresses are σ x = λ x P x and σ y = λ y P y .
Case 2: Compressible Membrane For this case, the constitutive relations are
again given by (4.14), and setting P z = 0 yields
λ z = (λ x λ y )
−ν/(1−ν)
(4.22)
as given by Eq. (4.15). Substitution into Eqs. (4.14) 1,2 gives
P x = μ
λ
−1+3ν
x
λ
2ν
y
1/(1−ν) − λ
−3
x
P y = μ
λ
2ν
x λ
−1+3ν
y
1/(1−ν) − λ
−3
y
.
(4.23)
Finally, with λ z given above, J = λ x λ y λ z = (λ x λ y ) (1−2ν)/(1−ν) , and the Cauchy
stresses are
σ x =
λ x
J
P x ,
σ y =
λ y
J
P y .
(4.24)
The uniaxial solution is recovered by setting λ y = λ −ν
x from (4.16) in the above
equations. Doing this gives P y = 0 and P x as in Eq. (4.17), as well as σ x of (4.18).
4.2.3 Illustrative Results
Since a thin membrane likely buckles when compressed, results for this problem are
given for tension only. For equibiaxial stretching (λ x = λ y ) of an incompressible
membrane with W given by Eq. (4.1), Fig. 4.4a shows plots of Cauchy stress σ x
in the fiber direction as a function of stretch ratio. Results are presented for neoHookean membranes (c 1 = 1, c 2 = 0) without fibers (isotropic, c 3 = 0), with linear
fibers (c 3 = 0.1, c 4 → 0), and with exponential fibers (c 3 = 0.1, c 4 = 0.01). Fibers
markedly increase membrane stiffness in the fiber (x) direction only; for all cases,
the curve for σ y (not shown) is the same as that for σ x in the isotropic membrane.
Results also are shown for an isotropic neo-Hookean membrane stretched in the
x-direction, while the length in the y-direction is held fixed at three values of λ y
(Fig. 4.4b). As λ x increases, the curve for σ x is relatively independent of λ y , while
σ y increases but remains nearly uniform.
The effects of compressibility are shown for equibiaxial stretch of a membrane
composed of isotropic material defined by Eq. (4.2). As in uniaxial extension
(Fig. 4.2b), the stiffness increases with ν (Fig. 4.5a). However, these effects are
considerably more dramatic for biaxial stretching as the membrane approaches
incompressibility (ν → 0.5; see Fig. 4.5b).