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

Have fun!

c© Prof. Dr.-Ing. habil. D. Balzani, Institut fur Mechanik und Flachentragwerke