1) The document discusses modeling fiber-reinforced membrane materials using continuum mechanics. Two tasks are presented: modeling the stresses in a textile membrane used in lightweight roof constructions, and modeling stresses in the aorta under blood pressure. 2) The model uses a strain energy potential that accounts for the isotropic matrix and transversely isotropic behavior of the fiber reinforcement. Boundary conditions and simplifying assumptions are applied to solve for stresses and strains. 3) For the textile membrane, stresses are calculated using Barlow's formula and compared to failure values. For the aorta, analogous calculations are performed to see if an arterial tissue could replace the roof membrane.

1. Fakult¨at Bauingenieurwesen Institut f¨ur Mechanik und Fl¨achentragwerke
Modeling of Fiber-Reinforced Membrane Materials
Daniel Balzani
Faculty of Civil Engineering, Institute of Mechanics and Shell Structures
Acknowledgement: Anna Zahn
• Introduction / Motivation
• Continuum Mechanical Preliminaries
• Task 1: Textile Membrane of a Lightweight Structure
• Task 2: Aorta under Physiological Blood Pressure

2. Motivation: Fiber-Reinforced Materials
Engineering applications
• Light-weight roof constructions
• Facade cover design
• Weather-proof awnings
Roof construction at the ATP
tournement in Indian Wells
Roof construction of Dresden
main station
Textile membranes
• Composite material
• Woven network of stiff fibers
• Soft and isotropic matrix material
Soft biological tissues
• Conceptionally similar material composition
• Collagen fibers reinforce an isotropic ground substance
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

3. Continuum Mechanical Preliminaries I
Assumptions
• Idealization as thin membranes
• Small strain framework
• Representation of one fiber reinforcement by
transversely isotropic model; fiber direction a(a) z
ϕ
a(2)
a(1)
t r
Calculation of the stresses
The stress tensor σ can be calculated by the derivative of a strain energy
function ψ(ε) with respect to the classical strain tensor ε:
σ =
∂ψ(ε)
∂ε
(1)
A specific energy function ψ has to be constructed such that the resulting stresses
match experimental data.
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

4. Continuum Mechanical Preliminaries II
Strain energy function
Here, we consider the strain energy function
ψ =
1
2
λ J2
1 + µ J2
ψiso
+
2
a=1
1
2
α(a)
J
(a)
4
2
ψti
(a)
(2)
which is formulated in terms of the basic and mixed invariants
J1 = tr [ε], J2 = tr [ε2
] and J
(a)
4 = tr [εM(a)
]. (3)
The coefficients of the structural tensor M(a)
are
M
(a)
ij = a
(a)
i a
(a)
j , (4)
wherein a
(a)
i are the coefficients of the fiber orientation vectors a(a)
.
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

5. Continuum Mechanical Preliminaries III
Remarks for the solution of the tasks
• The Lam´e constants λ and µ are determined by the Young’s modulus E and
the Poisson ratio ν according to
λ =
Eν
(1 + ν) (1 − 2ν)
and µ =
E
2 (1 + ν)
(5)
• Rotation-symmetric structures are parameterized by polar coordinates (r, ϕ, z)
• Stresses/strains in radial direction are neglected and shear stresses/strains do
not occur
• Summing up these simplifications, 2 of 9 non-trivial equations remain from (1)
σii =
∂ψ(εii)
∂εii
with i ∈ [ϕ, z]. (6)
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

6. Task 1: Textile Membrane of a Lightweight Structure
ϕ
z
z
r
ϕ
pM
rM
tM
a
(2)
M
a
(1)
M
• Using the presented energy function, a system of equations for the unknown
quantities σϕ, σz, εϕ and εz can be determined based on σ = ∂εψ.
• From the boundary conditions we obtain εz = 0 and the stress σϕ can be
calculated from Barlow’s formula,
σϕ = pM
rM
tM
. (7)
• Solve the system of equations for εϕ and σz and compare with the ultimate
values εϕ,max and σz,max.
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

7. Task 2: Aorta under Physiological Blood Pressure
ϕ
z
z
r
ϕ
rA
pA
tA
β
(1)
A
a
(1)
A
β
(2)
A
a
(2)
A
Compared to the air-inflated membrane, the internal pressure of human arteries is
significantly higher and the fiber stiffnesses are relatively low.
• Compute analogously the values for εϕ and σz.
• Although technically impossible, check if the membrane in the roof construction
of Task 1 could be replaced by arterial tissue.
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke

8. Have fun!
c Prof. Dr.-Ing. habil. D. Balzani, Institut f¨ur Mechanik und Fl¨achentragwerke