Fakultat Bauingenieurwesen Institut fur Mechanik und Flachentragwerke
Modeling of Fiber-Reinforced Membrane Materials
Daniel BalzaniFaculty 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
Motivation: Fiber-Reinforced Materials
Engineering applications
• Light-weight roof constructions
• Facade cover design
• Weather-proof awnings
Roof construction at the ATPtournement in Indian Wells
Roof construction of Dresdenmain 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 fur Mechanik und Flachentragwerke
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 energyfunction 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 stressesmatch experimental data.
c© Prof. Dr.-Ing. habil. D. Balzani, Institut fur Mechanik und Flachentragwerke
Continuum Mechanical Preliminaries II
Strain energy function
Here, we consider the strain energy function
ψ =1
2λJ2
1 + µJ2︸ ︷︷ ︸ψiso
+
2∑a=1
[1
2α(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 fur Mechanik und Flachentragwerke
Continuum Mechanical Preliminaries III
Remarks for the solution of the tasks
• The Lame constants λ and µ are determined by the Young’s modulus E andthe 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 donot occur
• Summing up these simplifications, 2 of 9 non-trivial equations remain from (1)
σii =∂ψ(εii)
∂εiiwith i ∈ [ϕ, z]. (6)
c© Prof. Dr.-Ing. habil. D. Balzani, Institut fur Mechanik und Flachentragwerke
Task 1: Textile Membrane of a Lightweight Structure
ϕ
z
z
rϕ
pM
rM
tM
a(2)M
a(1)M
• Using the presented energy function, a system of equations for the unknownquantities 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 becalculated from Barlow’s formula,
σϕ = pMrM
tM. (7)
• Solve the system of equations for εϕ and σz and compare with the ultimatevalues εϕ,max and σz,max.
c© Prof. Dr.-Ing. habil. D. Balzani, Institut fur Mechanik und Flachentragwerke
Task 2: Aorta under Physiological Blood Pressure
ϕ
z
z
rϕ
rA
pAtA
β(1)A
a(1)A
β(2)A
a(2)A
Compared to the air-inflated membrane, the internal pressure of human arteries issignificantly 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 constructionof Task 1 could be replaced by arterial tissue.
c© Prof. Dr.-Ing. habil. D. Balzani, Institut fur Mechanik und Flachentragwerke