Modeling the Respiratory Tree by Means of Mechanical Analogy - The Human Respiratory System - page 74

Chapter 6
Modeling the Respiratory Tree by Means
of Mechanical Analogy

6.1 Basic Elements

When a force F is applied to an object with initial length and cross-sectional area
A , a mechanical stress σ results. Consequently, a deformation occurs , which
leads us to define the strain ε :

F
A ;

σ = ε = (6.1)

The following relations can be defined between the stress and the strain, in which
E is the elasticity modulus and η the viscosity coefficient [ 23 ].
For a spring:

σ(t) = E · ε(t) (6.2)
denoting Hooke's Law and the linear elastic behavior of materials. Supposing a si-
nusoidal strain applied to the material: ε(t)

= ε 0 · sin (ωt) , then the stress is in phase

with the strain and its amplitude is given by E ε 0 . Observing the corresponding

stress-strain curve from Fig. 6.1 -left, the load and unload are following the same
path; therefore no loss of energy occurs. Hence, we conclude that elastic materials
do not show energy-dissipation phenomena.
For a damper :

·

d
dt ε(t)

σ(t) = μ · (6.3)

denoting Newton's Law and describing the viscous behavior of a linear flow. Apply-
ing a similar strain as above, the stress will lead the strain by 90 ◦ with an amplitude
equal to μ

ε 0 . The amplitude is therefore frequency dependent. When both sig-
nals are opposite in phase, as depicted in Fig. 6.1 -right, we see that all the energy is
used (equal hysteresis on both sides). Therefore, we conclude that viscous materials
show energy-dissipation phenomena.
Emerging theories of fractional calculus allowed the appearance of a novel term,
i.e. a spring-pot of order n (with 0

· ω ·

≤ n ≤ 1), characterized by the following relation:

d n
dt n ε(t)

σ(t) = η · (6.4)