Inflation of an artery modelled as a hyperelastic cylinder Humphrey, J. D. (2013). Cardiovascular solid mechanics: cells, tissues, and organs. Springer Science & Business Media Holzapfel, Gasser, Ogden 2000

Inflation of an artery modelled as a hyperelastic cylinder

Humphrey, J. D. (2013). Cardiovascular solid mechanics: cells, tissues, and organs. Springer Science &

Business Media

Holzapfel, Gasser, Ogden 2000

𝐹 = 𝐹₂𝐹₁

𝐹₂?

If we neglect torsion/twisting: 𝑟𝛾 = 0

𝜆 is the axial stretch

𝐹₂= [

𝜌

𝑟𝜆 0 0 0 𝑟

𝜌 0 0 0 𝜆]

Cauchy stress

𝜎 = [

𝜎_𝑟𝑟 0 0 0 𝜎_𝜃𝜃 0 0 0 𝜎_𝑧𝑧

]

Thick wall, there is no simple expression relating 𝜎 to the loads:

Pressure 𝑃 Axial load 𝑁

Integrating local equilibrium, we get:

Incompressible Hyperelasticity:

Strain energy density (equation 10.8 Beatty’s paper)

𝛽 = 0

𝜎 = 2𝜕ℎ

𝜕𝐼₁𝐹₂𝐹₂^𝑇+ 𝑐𝐼

𝜎 = 2𝜕ℎ

𝜕𝐼₁ [ (𝜌

𝑟𝜆)

2

0 0

0 (𝑟 𝜌)

2

0 0 0 𝜆²]

+ 𝑐 [

1 0 0 0 1 0 0 0 1 ]

𝜎_𝑟𝑟= 2𝜕ℎ

𝜕𝐼₁(𝜌 𝑟𝜆)

2

+ 𝑐

𝜎𝜃𝜃 = 2𝜕ℎ

𝜕𝐼₁(𝑟 𝜌)

2

+ 𝑐

𝑃 = ∫ 2𝜕ℎ

𝜕𝐼1

[(𝑟 𝜌)

2

− (𝜌 𝑟𝜆)

2

]𝑑𝑟 𝑟

𝑟+ℎ 𝑟

We can complete with the expression of N and then find the relationship between P and the stretch How can we solve that?

Numerical Euler integration in Matlab

There is no analytical solution?

Yes with thin wall (h<<r) This is ok if h/r<0.1

For instance the human aorta h=1.5mm (in vivo, deformed) r=15 mm (in vivo, deformed) but if it is undeformed H = 2mm

R=10mm It does not work

Constant stress across the thickness (no integration)

𝜎 = 2𝜕ℎ

𝜕𝐼₁ [ (𝜌

𝑟𝜆)

2

0 0

0 (𝑟 𝜌)

2

0 0 0 𝜆²]

+ 𝑐 [

1 0 0 0 1 0 0 0 1 ]

𝜎 = 2𝜕ℎ

𝜕𝐼₁ [ (1

𝜇𝜆)

2

0 0 0 𝜇² 0 0 0 𝜆²]

+ 𝑐 [

1 0 0 0 1 0 0 0 1 ]

Laplace law

𝜎_𝜃𝜃=𝑃𝑟 ℎ

Radial equilibrium

−𝑃 ≤ 𝜎_𝑟𝑟≤ 0

Thin wall:

𝑟 ℎ≫ 1

𝑃𝑟 ℎ ≫ 𝑃

𝜎_𝜃𝜃 ≫ 𝜎_𝑟𝑟

𝜎 ≈ [

0 0 0

0 𝑃𝑟

ℎ 0

0 0 𝜎_𝑧𝑧 ]

Then:

𝜎_𝑟𝑟≈ 0

𝑐 ≈ −2𝜕ℎ

𝜕𝐼₁(1 𝜇𝜆)

2

𝜎 = 2𝜕ℎ

𝜕𝐼₁[

0 0 0

0 𝜇²− (1 𝜇𝜆)

2

0

0 0 𝜆²

]

𝑃𝜌

𝐻 𝜆𝜇²= 2𝜕ℎ

𝜕𝐼1

[𝜇²− (1 𝜇𝜆)

2

]

𝐻 is the initial thickness 𝜌 is the initial radius Then we get P

𝑃 = 2𝐻 𝜌

𝜕ℎ

𝜕𝐼₁ 1

𝜆[1 − ( 1 𝜇²𝜆)

2

] If we use the exponential density function of biological tissues:

ℎ(𝐼₁− 3) = 𝜇₀

2𝛾[𝑒^{𝛾(𝐼1−3)} − 1]

𝑃 =𝜇₀ 𝜆

𝐻

𝜌𝑒^{𝛾(𝐼1−3)} [1 − ( 1 𝜇²𝜆)

2

]

𝐼1= (1 𝜇𝜆)

2

+ 𝜆²+ 𝜇²

If we do not apply an axial stretch: 𝜆 = 1

𝑃 = 𝜇0

𝐻

𝜌𝑒^{𝛾(𝐼1−3)} [1 − 1 𝜇⁴]

If we apply an axial stretch: 𝜆 > 1 (arteries are always stretched axially)

𝑃 =𝜇₀ 𝜆

𝐻

𝜌𝑒^{𝛾(𝐼1−3)} [1 − ( 1 𝜇²𝜆)

2

]

How to solve for 𝐹1?

Incompressibility

𝜎 = 2𝜕ℎ

𝜕𝐼₁ [ (Θ₀𝑅

Λ𝜋𝜌)

2

0 0

0 (𝜋𝜌 Θ₀𝑅)

2

0

0 0 Λ²]

+ 𝑐 [

1 0 0 0 1 0 0 0 1 ]

We have 2 equations: P=0 and N=0 (load-free configuration) Solving these 2 equations will yield: 𝑐 and Λ

This problem can only be solved numerically using Euler integration