Article Title: Experimental Study of the Mechanosorptive Behaviour of Softwood in Relaxation

Article title: Experimental study of the mechanosorptive behaviour of softwood in relaxation

Journal: Wood Science and Technology

Authors: Omar Saifouni1,3, Jean-François Destrebecq2,3, Julien Froidevaux4, Parviz Navi 4

1Clermont Université, IFMA, Institut Pascal, BP 10448, F-63000 Clermont-Ferrand, France

2Clermont Université, Université Blaise Pascal, Institut Pascal, BP 10448, F-63000 Clermont-Ferrand, France

3CNRS, UMR 6602, Institut Pascal, F-63171 Aubière, France

4Bern University of Applied Sciences, Biel, Switzerland

Corresponding author: – 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 , as follows

where is the number of Maxwell’s chains in the analogue model. In this equation, and 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

where and are two known dimensionless relaxation functions at relative humidity levels and respectively. is a parameter which depends on the relative humidity, as follows

To use Equation (A.2), it is necessary to determine the values of the material parameters in Equation (A.1) for the two functions and . Given a set of parameters, the determination of the parameters is carried out by means of the mean square method. For a fixed relative humidity, the error functiontobe minimized writes

where are reference values obtained from tests at times , and are the corresponding analytical values given by Equation (A.1), respectively. The error function reaches its minimum value under the following condition

Given Equation (A.1), the above expression leads to a system of linear equations, which can be written in a matrix form, as follows

The components of the square matrix and the column matrix write

The parameters are the components of the column matrix , to be determined by solving Equation (A.6).

The method is used for the determination of the parameters for the two functions and corresponding to and respectively. The parameters are chosen identical for the both functions, according to the following recursive form

with h-1 (A.8)

The result of the calibration procedure is shown in Table 1. The root mean squared error , where is given by Equation (A.4), is used to estimate the precision of the procedure.

Table 1: Parameters of the Dirichlet’s series for and

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

where .

B – Viscoelastic stress under variable ambient humidity

FigureB.1 illustrates the use of the principle of superposition to estimate the viscoelastic stress in response to a constant strain for a stepwise variation of relative humidity . From time , the relaxation stress is estimated by using the relaxation function given by Equation (A.2) for , hence . At time , a variation of humidity occurs, i.e. is substituted for . In application of the principle of superposition, the term is added to the previous expression of , where is the relaxation function given by Equation (A.2) for . This is equivalent to subtracting the effect of under from time , while applying the effect of under simultaneously. If relevant, this procedure is repeated for any subsequent humidity variation.

Figure B.1: Illustration of the principle of superposition to estimate the viscoelastic stress evolution for a constant strain combined with a stepwise humidity variation .

Accordingly, given the relaxation functions for a number of successive humidity levels occurring from times , the resulting viscoelastic stress caused at any subsequent time by a constant strain loading writes

(B.1)

where

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 FiguresB.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.

Figure B.2: Simulated relaxation curve (thick solid line) for the two mechanosorptive tests with variable relative humidity.

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