Inflation of a hyperelastic spherical balloon Problem is solved in spherical coordinate system Local orthonormal frame (basis vectors) is denoted (

Inflation of a hyperelastic spherical balloon

Problem is solved in spherical coordinate system

Local orthonormal frame (basis vectors) is denoted (𝐸_𝑟, 𝐸_𝜃, 𝐸_𝜑) in the undeformed reference configuration and (𝑒_𝑟, 𝑒_𝜃, 𝑒_𝜑) in the deformed configuration

No shear => (𝐸_𝑟, 𝐸_𝜃, 𝐸_𝜑) = (𝑒_𝑟, 𝑒_𝜃, 𝑒_𝜑)

𝐹(𝑟) = [

𝜕𝑟

𝜕𝑅 0 0 0 𝑟

𝑅 0 0 0 𝑟 𝑅]

Incompressibility assumption

𝑣 = 𝑉

𝑉 =4

3𝜋𝑅₂³−4 3𝜋𝑅₁³ 𝑣 =4

3𝜋𝑟₂³−4 3𝜋𝑟₁³ 𝑅₂³− 𝑅₁³= 𝑟₂³− 𝑟₁³

Initial thickness 𝐻, final thickness ℎ

(𝑅₁+ 𝐻)³− 𝑅₁³= (𝑟₁+ 𝐻)³− 𝑟₁³

(𝑅₁+ 𝐻)³= (𝑅₁+ 𝐻)(𝑅₁²+ 2𝑅₁+ 𝐻²) = 𝑅₁³+ 2𝐻𝑅₁²+ 𝑅₁𝐻²+ 𝐻𝑅₁²+ 2𝐻²𝑅₁+ 𝐻³ 𝑅₂³− 𝑅₁³= (𝑅₁+ 𝐻)³− 𝑅₁³= 3𝐻𝑅₁²+ 3𝑅₁𝐻²+ 𝐻³

𝑟₂³− 𝑟₁³= 3ℎ𝑟₁²+ 3𝑟₁ℎ²+ ℎ³

3𝐻𝑅₁²+ 3𝑅₁𝐻²+ 𝐻³= 3ℎ𝑟₁²+ 3𝑟₁ℎ²+ ℎ³ Divide by 𝐻𝑅₁²

3 + 3𝐻 𝑅₁+𝐻²

𝑅₁² = 3ℎ𝑟₁² 𝐻𝑅₁²+ 3 ℎ

𝑅₁ ℎ 𝐻

𝑟1

𝑅₁+ℎ² 𝑅₁²

ℎ 𝐻

We introduce:

𝜆 = 𝑟₁/𝑅₁ 𝜇 = ℎ

𝐻 3 + 3𝐻

𝑅₁+𝐻²

𝑅₁² = 3𝜆²ℎ

𝐻+ 3𝜆²ℎ 𝑟₁

ℎ

𝐻+ 𝜆²ℎ² 𝑟₁²

ℎ 𝐻

3 + 3𝐻 𝑅₁+𝐻²

𝑅₁² = 3𝜆²𝜇 + 3𝜆𝜇² 𝐻

𝑅₁+ 𝜇³𝐻² 𝑅₁²

1 + 𝐻 𝑅₁+ 𝐻²

3𝑅₁²= 𝜆²𝜇 + 𝜆𝜇² 𝐻

𝑅₁+ 𝜇³ 𝐻² 3𝑅₁²

𝐻 𝑅₁= 𝛾

Finally we obtain with incompressibility:

1 + 𝛾 +𝛾²

3 = 𝜆²𝜇 + 𝜆𝜇²𝛾 + 𝜇³𝛾² 3

Thin balloon: 𝛾 ≪ 1

𝜆²𝜇 = 1

𝐹(𝑟) = 𝐹 = [

𝜕𝑟

𝜕𝑅≈ 𝜇 0 0

0 𝑟

𝑅≈ 𝜆 0

0 0 𝑟

𝑅≈ 𝜆]

𝐹 = [

1/𝜆² 0 0

0 𝜆 0

0 0 𝜆

]

Cauchy stress:

𝜎 = [

𝜎_𝑟𝑟 0 0 0 𝜎_𝜃𝜃 0 0 0 𝜎_𝜑𝜑

]

−𝑝 ≤ 𝜎_𝑟𝑟≤ 0

𝜎_𝜃𝜃 = 𝜎_𝜑𝜑=?

Laplace law:

𝜎_𝜃𝜃= 𝜎_𝜑𝜑=𝑃𝑟₁

2ℎ =𝑃𝜆³𝑅₁ 2𝐻 =𝑃𝜆³

2𝛾

𝜎 = 𝑃 [

< 1 0 0 0 𝜆³

2𝛾 0 0 0 𝜆³

2𝛾]

Thin balloon

𝛾 ≪ 1 1 𝛾≫ 1 𝜎_𝑟𝑟≪ 𝜎_𝜃𝜃 𝜎_𝑟𝑟 ≪ 𝜎_𝜑𝜑

𝜎 ≈𝑃𝜆³𝑅1

2𝐻 [

0 0 0 0 1 0 0 0 1 ]

Incompressible Hyperelasticity

Strain energy density (equation 10.8 Beatty’s paper)

For Neo Hooke

𝛽 = 0 and ℎ(𝐼₁− 3) =^𝜇₂⁰(𝐼₁− 3)

𝜎 = 𝜇0𝐹𝐹^𝑇+ 𝑐𝐼

Equilibrium

𝜎 = [ 𝜇₀

𝜆⁴+ 𝑐 0 0

0 𝜇₀𝜆²+ 𝑐 0 0 0 𝜇₀𝜆²+ 𝑐

] =𝑃𝜆³𝑅1

2𝐻 [

0 0 0 0 1 0 0 0 1 ]

𝜎_𝑟𝑟≈ 0

𝑐 = −𝜇₀ 𝜆⁴

𝜎 = [

0 0 0

0 𝜇₀𝜆²−𝜇₀

𝜆⁴ 0

0 0 𝜇₀𝜆²−𝜇₀ 𝜆⁴]

= 𝜇₀(𝜆²− 1 𝜆⁴) [

0 0 0 0 1 0 0 0 1 ]

Finally:

𝑃𝜆³𝑅₁

2𝐻 = 𝜇₀(𝜆²− 1 𝜆⁴)

𝑃 = 𝜇₀2𝐻 𝑅₁ (1

𝜆− 1 𝜆⁷)

In Beatty’s paper:

There is a maximum pressure

Strain energy density of biological tissue 𝛽 = 0 and ℎ(𝐼₁− 3) =^𝜇_2𝛾⁰[𝑒^{𝛾(𝐼1−3)} − 1]

𝜎 = 2𝜕ℎ

𝜕𝐼₁𝐹𝐹^𝑇+ 𝑐𝐼

𝜎 = 𝜇₀𝑒^{𝛾(𝐼1−3)} 𝐹𝐹^𝑇+ 𝑐𝐼

𝜎 = [

𝜇₀𝑒^{𝛾(𝐼1−3)}

𝜆⁴ + 𝑐 0 0

0 𝜇₀𝑒^{𝛾(𝐼1−3)} 𝜆²+ 𝑐 0

0 0 𝜇₀𝑒^{𝛾(𝐼1−3)} 𝜆²+ 𝑐]

𝐼₁= 2𝜆²+ 1 𝜆⁴

𝑐 = −𝜇₀𝑒^𝛾(2𝜆2+

1 𝜆⁴−3)

𝜆⁴

𝜎 = 𝜇₀𝑒^𝛾(2𝜆2+

1 𝜆⁴−3)

(𝜆²− 1 𝜆⁴) [

0 0 0 0 1 0 0 0 1 ]

𝑃 = 𝜇₀𝑒^𝛾(2𝜆2+

1

𝜆⁴−3) 2𝐻 𝑅₁ (1

𝜆− 1 𝜆⁷)

When 𝜆 becomes large

𝑃 ≈ 𝜇₀𝑒^{𝛾(2𝜆2−3)} 2𝐻 𝜆𝑅₁