Prof. Dr. Balzani for MOOC@TU9

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