Article title: Experimental study of the mechanosorptive behaviour of softwood in relaxation
Journal: Wood Science and Technology
Authors: Omar Saifouni 1,3, Jean-François Destrebecq 2,3, Julien Froidevaux 4, Parviz Navi 4
Clermont Université, IFMA, Institut Pascal, BP 10448, F-63000 Clermont-Ferrand, France
Clermont Université, Université Blaise Pascal, Institut Pascal, BP 10448, F-63000 Clermont-Ferrand,
France
3
CNRS, UMR 6602, Institut Pascal, F-63171 Aubière, France
4
Bern University of Applied Sciences, Biel, Switzerland
1
2
Corresponding author: [email protected] – Tel. +33 473 28 80 78 –
Fax +33 473 28 80 27
APPENDIX
A – Relaxation function in terms of the ambient humidity
In the frame of linear viscoelasticity, the relaxation behaviour can be expressed as a
Dirichlet’s series, equivalent to an analogue generalised Maxwell’s model (Bazant and Wu
1974; Jurkiewiez et al. 1999). Accordingly, the relaxation function in a fixed environment can
be written in a dimensionless form, given the boundary condition 𝑅(0) = 1, as follows
𝑟
𝑅(𝑡) = 1 − ∑ 𝜌𝜇 (1 − 𝑒 −𝛼𝜇 𝑡 )
(A. 1)
𝜇=1
where 𝑟 is the number of Maxwell’s chains in the analogue model. In this equation, 𝛼𝜇 and 𝜌𝜇
are two sets of material parameters to be determined from tests.
In order to account for the influence of the ambient humidity, the relaxation function is
supposed to depend in a linear way on the relative humidity. Hence, the dimensionless
relaxation function at a given intermediate relative humidity is expressed as follows
𝑅𝑅𝐻 (𝑡) = 𝛽𝑅𝐻 𝑅2 (𝑡) + (1 − 𝛽𝑅𝐻 )𝑅1 (𝑡)
(A. 2)
where 𝑅1 (𝑡) and 𝑅2 (𝑡) are two known dimensionless relaxation functions at relative humidity
levels 𝑅𝐻1 and 𝑅𝐻2 respectively. 𝛽𝑅𝐻 is a parameter which depends on the relative humidity,
as follows
𝛽𝑅𝐻 =
𝑅𝐻 − 𝑅𝐻1
𝑅𝐻2 − 𝑅𝐻1
(A. 3)
To use Equation (A.2), it is necessary to determine the values of the material parameters in
Equation (A.1) for the two functions 𝑅1 (𝑡) and 𝑅2 (𝑡). Given a set of 𝛼𝜇 parameters, the
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determination of the 𝜌𝜇 parameters is carried out by means of the mean square method. For a
fixed relative humidity, the error function to be minimized writes
𝑚
2
𝑆 = ∑(𝑅𝑗 − 𝑅̅𝑗 )
(A. 4)
𝑗=1
where 𝑅̅𝑗 = 𝑅̅ (𝑡𝑗 ) are 𝑚 reference values obtained from tests at times 𝑡𝑗 , and 𝑅𝑗 = 𝑅(𝑡𝑗 ) are
the 𝑚 corresponding analytical values given by Equation (A.1), respectively. The error
function 𝑆 reaches its minimum value under the following condition
∀ 𝜇 = 1. . . 𝑟 ∶
𝜕𝑆
=0
𝜕𝜌𝜇
𝑚
𝑚
𝜕𝑅𝑗
𝜕𝑅𝑗
⟹ ∑ 𝑅𝑗 (
) = ∑ 𝑅̅𝑗 (
)
𝜕𝜌𝜇
𝜕𝜌𝜇
𝑗=1
(A. 5)
𝑗=1
Given Equation (A.1), the above expression leads to a system of 𝑟 linear equations, which can
be written in a matrix form, as follows
[𝐴]{𝜌} = {𝐵}
(A. 6)
The components of the square matrix [𝐴] and the column matrix {𝐵} write
𝑚
𝐴𝜆𝜇 = ∑(1 − 𝑒 −𝛼𝜆𝑡𝑗 ) (1 − 𝑒 −𝛼𝜇 𝑡𝑗 )
𝑗=1
𝑚
(A. 7)
𝐵𝜆 = ∑(1 − 𝑒 −𝛼𝜆𝑡𝑗 )(1 − 𝑅̅𝑗 )
{
𝑗=1
The 𝜌𝜇 parameters are the 𝑟 components of the column matrix {𝜌}, to be determined by
solving Equation (A.6).
The method is used for the determination of the 𝜌𝜇 parameters for the two functions 𝑅1 (𝑡)
and 𝑅2 (𝑡) corresponding to 𝑅𝐻1 = 30% and 𝑅𝐻2 = 70% respectively. The parameters 𝛼𝜇
are chosen identical for the both functions, according to the following recursive form
𝛼𝜇 = 𝛼1 10𝜇−1 with 𝛼1 = 0.25 h-1
(A.8)
The result of the calibration procedure is shown in Table 1. The root mean squared error
RMSE = √𝑆/𝑚, where 𝑆 is given by Equation (A.4), is used to estimate the precision of the
procedure.
Table 1: Parameters of the Dirichlet’s series for 𝑅𝐻 = 30% and 𝑅𝐻 = 70%
Parameter:
RMSE:
2
𝜇
1
2
3
𝛼𝜇 (h-1)
0.25
2.5
25
𝜌𝜇 (RH=30%)
0.0669
0.0187
0.0251
1.32 10-3
𝜌𝜇 (RH=70%)
0.0919
0.0303
0.0439
0.56 10-3
Finally, given Equations (A.1) to (A.3) with parameter values in Table 1, it comes for the
dimensionless relaxation function, in terms of the relative humidity
𝑅𝑅𝐻 (𝑡) = (0.8893 − 0.0554𝛽𝑅𝐻 ) + (0.0669 + 0.0250𝛽𝑅𝐻 )𝑒 −0.25𝑡
+ (0.0187 + 0.0116𝛽𝑅𝐻 )𝑒 −2.5𝑡 + (0.0251 + 0.0188𝛽𝑅𝐻 )𝑒 −25𝑡
(A. 9)
where 𝛽𝑅𝐻 = 0.025(𝑅𝐻 − 30%).
B – Viscoelastic stress under variable ambient humidity
Figure B.1 illustrates the use of the principle of superposition to estimate the viscoelastic
stress 𝜎𝑣𝑒 (𝑡) in response to a constant strain 𝜀0 for a stepwise variation of relative humidity
𝑅𝐻2 ≠ 𝑅𝐻1 . From time 𝑡1 , the relaxation stress is estimated by using the relaxation function
𝑅1 (𝑡1 , 𝑡) given by Equation (A.2) for 𝑅𝐻 = 𝑅𝐻1, hence 𝜎𝑣𝑒 (𝑡) = 𝑅1 (𝑡1 , 𝑡)𝜀0 . At time 𝑡2 , a
variation of humidity occurs, i.e. 𝑅𝐻1 is substituted for 𝑅𝐻2 . In application of the principle of
superposition, the term (𝑅2 (𝑡2 , 𝑡) − 𝑅1 (𝑡2 , 𝑡))𝜀0 is added to the previous expression of
𝜎𝑣𝑒 (𝑡), where 𝑅2 (𝑡2 , 𝑡) is the relaxation function given by Equation (A.2) for 𝑅𝐻 = 𝑅𝐻2 .
This is equivalent to subtracting the effect of 𝜀0 under 𝑅𝐻1 from time 𝑡2 , while applying the
effect of 𝜀0 under 𝑅𝐻2 simultaneously. If relevant, this procedure is repeated for any
subsequent humidity variation.
Strain
(a)
+ε0
ε0
Time
t1
t2
-ε0
RH2
RH1
(b)
R1(t1,t)ε0
+R2(t2,t)ε0
Stress
σ(t)
Time
t1
t2
-R1(t2,t)ε 0
Figure B.1: Illustration of the principle of superposition to estimate the viscoelastic stress
evolution for a constant strain 𝜀0 combined with a stepwise humidity variation 𝑅𝐻2 > 𝑅𝐻1.
Accordingly, given 𝑅𝑖 (𝑡𝑖 , 𝑡) the relaxation functions for a number of successive humidity
levels 𝐻𝑅𝑖 occurring from times 𝑡𝑖 , the resulting viscoelastic stress 𝜎𝑣𝑒 (𝑡) caused at any
subsequent time 𝑡 by a constant strain loading 𝜀0 writes
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𝜎𝑣𝑒 (𝑡) = 𝑅̃ (𝑡) 𝜀0
(B.1)
where
𝑅̃ (𝑡) = 𝑅1 (𝑡1 , 𝑡) + ∑[𝑅𝑖 (𝑡𝑖 , 𝑡) − 𝑅𝑖−1 (𝑡𝑖 , 𝑡)],
∀ 𝑡 ∈ [𝑡𝑖 , 𝑡𝑖+1 [
(B. 2)
𝑖>1
Equation (B.2) is applied to the estimation of the evolution of the viscoelastic stress during
the two mechanosorptive tests. The results are shown on Figures B.2(a) and (b) during the
first period of testing (i.e. before unloading). On each figure, the thick solid line shows the
estimated viscoelastic stress. The dotted lines represent the reference relaxation functions
under constant relative humidity levels, and the thin solid line depicts the cyclic relative
humidity.
(3) R (70%)
180
150 R (RH
120 variable)
90
60
30
0
(2)
(3)
14
12
10
8
0
5
Time [h]
10
15
(1)
240 R (30%)
(2) R (50%)
210
180
(3) R (70%)
150
R (RH
120 variable)
90
60
30
0
18
(1)
16
(2)
(3)
14
12
RH [%]
(1)
16
(b)
Relaxation function [GPa]
(1) R (30%)
240
(2) R (50%)
210
RH [%]
Relaxation function [GPa]
(a)
18
10
8
0
5
10
Time [h]
15
20
Figure B.2: Simulated relaxation curve (thick solid line) for the two mechanosorptive tests
with variable relative humidity.
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