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Rigidity
Elastic strain
WOOD SCIENCE FOR CONSERVATION OF CULTURAL HERITAGE
t1
ε e () t
σ ( t) = σ ⋅H( t−t )
1 1
Moisture content
49
( )
k t t
min 1 ,
Fig. 2: Response of the hygro-lock model
σ ( t )
ko
ε ( t )
1 k
η1
σ ( t )
k(
t)
Fig. 3: Kelvin Voigt model with hygro-lock effect
According to the specific hygro-lock effect for each spring, the global behaviour can take this
following form:
t
t
∂σ
⎡ t 1
1 1 k ( τα , ) ⎤
ε () t = ∫ J( τ, t) ⋅ dτ
min
with J ( τ , t) = + exp⎢dα⎥dϑ
∂τ
o
0
−
k ( τ , t)
∫ ⋅ −
η1( ϑ) ⎢ ∫ (3)
η1( α)
⎥
min τ ⎣ ϑ
⎦
This last formulation can be compared with a non linear Boltzman’s formulation. In this case, the
classical Boltzman’s integral is generalized by considering two time integrals. The first expresses
loading effects and the second presents the coupling with moisture content effects. Analytic
resolutions are performed using an incremental formulation implemented in mathematical formal
software. In order to model long term behaviour taking into account more complex and more realistic
creep functions this model can be completed by N hygro-lock Kelvin Voigt cells added with a specific
swelling-shrinkage element, Fig. 4. In this context, the relation (3) can be updated as follow:
t t
∂σ∂w ε () t = ∫J( τ, t) ⋅ dτ + α⋅ dτ
∂τ ∫ ∂τ
0 0
with ( , )
( α) ⋅
p
( τ, α)
min
( τα , ) − ( α)
( )
Time
⎡ β p
k k
⎤
exp ⎢−dα⎥ N t ⎢ ∫ p p
1
k k ⎥
⎢ τ min
⎥
J t τ = +
⎣ ⎦
dβ
o ∑∫ p
(4)
kmin ( τ, t)
p=
1
η β
τ