Biomedical Engineering Reference
In-Depth Information
where the fourth invariant I 4 , c / s can be expressed as scalar product of the right
Cauchy-Green tensor C and Z c / s .
5.2.2 Balance Equations
Using classical non-linear continuum mechanics, a coupled problem of chemo-
mechanical SM contraction is formulated in terms of two primary field variables,
namely the placement ϕ ( X ,t) and the calcium concentration c( X ,t) . Consequently,
a chemomechanical state
S
of a material point X at the time t is defined as
S ( X ,t) = ϕ ( X ,t),c( X ,t) .
(5.4)
Spatial as well as temporal evolution of the primary field variables are governed by
two basic field equations: the balance of linear momentum and the diffusion-type
equation of excitation through calcium.
The balance of linear momentum in spatial form
+ b
div σ
=
0
in
B
(5.5)
describes the quasi-static stress equilibrium. Herein b is the given spatial body force
per unit reference volume. The operator div
indicates the divergence with respect
to the spatial coordinates x , and σ denotes the Cauchy stress tensor given as
[•]
2 J 1 F ∂Ψ( ϕ )
C
F T ,
σ
=
(5.6)
depending on the deformation measures C and F as well as on a strain-energy func-
tion Ψ( ϕ ) , see Sect. 5.2.3 . The mechanical problem is completed by essential and
natural boundary conditions,
= t
ϕ
= ¯
ϕ
on B ϕ and
t
on B σ .
(5.7)
The surface stress traction vector t , defined on B σ , is related to the Cauchy stress
tensor σ via the Cauchy stress theorem t
:=
σ n , where the outward surface normal
is specified as n .
The second field equation describes the calcium concentration inside the SM
tissue. The well-known Fick's second law
c
˙
=−
div q
in
B
(5.8)
predicts how diffusion causes the concentration field c to change with time. Herein,
the diffusion flux vector
q
=−
d ( ϕ )
x c
(5.9)
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