Lomnitz-type viscoelastic behavior of clear spruce wood as identified by creep and relaxation experiments: influence of moisture content and elevated temperatures up to 80 °C

The strong dependence of viscoelastic behavior of wood on temperature and moisture content is well documented in the literature. In this paper, viscoelastic behavior of spruce wood is monitored experimentally and modeled with a (logarithmic) Lomnitz-type creep model. Creep tests are performed at different levels of temperature (25-80∘C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(25{-}80\,^{\circ }\hbox {C})$$\end{document} and moisture content (0 to ≈15\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\approx 15$$\end{document} mass-%). The variation of Lomnitz parameters as regards temperature and moisture content is well captured by Arrhenius-type dependencies. Furthermore, the uniqueness of the results is discussed and relaxation tests are performed in order to verify the so-obtained viscoelastic parameters.


Introduction
Wood may be idealized as a fiber-reinforced polymer, with the microfibrils of cellulose acting as fibers and reinforcing the wood in longitudinal direction, and the polymeric hemicellulose and lignin acting as a matrix material embedding the cellulose fibers (Holzer et al. 1989;Hofstetter et al. 2005;Bader et al. 2011). This matrix additionally contains distinct porous structures. As many other polymers, woodespecially the amorphous phases hemicellulose and lignin (Kelley et al. 1987)shows a pronounced time-dependent behavior (Navi and Stanzl-Tschegg 2009). In this regard, one needs to distinguish between transport-induced, load-induced, and pseudo (shrinkage and swelling) viscoelastic phenomena, respectively (Hunt 1999). and the Burgers model are best suited to describe creep of Pinus radiata at different levels of moisture content. Hering and Niemz (2012) used a Kelvin-Voigt model to describe the moisture content-dependent creep of beech wood. Mukudai (1983) used a Maxwell model and three Kelvin-Voigt models connected in series to model the viscoelastic behavior of Japanese cypress, whereas Hunt (2004) used a generalized Kelvin-Voigt model to estimate the creep curves of wood. Clouser (1959), Schniewind and Barrett (1972) and Hoyle et al. (1985) employed power-law models to model creep behavior of timber beams. Eitelberger et al. (2012) modeled the viscoelastic behavior of the lignin-hemicellulose matrix with a power-law model, whereas the macroscopic wood behavior was predicted to follow a fractional Zener model. King (1961) observed that the creep deformation of wood is proportional to time and hence introduced a logarithmic creep model to consider the viscoelastic behavior of wood, whereas Bach and Pentoney (1968) introduced a modified logarithmic creep law. In geology, a logarithmic creep model, the so-called Lomnitz model, has been introduced by Lomnitz (1956Lomnitz ( , 1957Lomnitz ( , 1962. So far it has been applied to model cement-based materials (e.g., in Bažant and Prasannan 1989;Bažant et al. 1997;Acker and Ulm 2001;Pichler et al. 2008;Pichler and Lackner 2009), polyurethane foams (Pichler et al. 2018) and metals (Nabarro 2001a, b).
For wood, on the other hand, logarithmic models have been introduced by King (1961) and Bach and Pentoney (1968), but have not been applied to the thermohygro-mechanical analysis of wood and wood structures. That may be (i) due to the missing expression for the relaxation function, which cannot be obtained in a straightforward manner and has only been derived recently by Mainardi and Spada (2012), and (ii) due to the lack of a physical interpretation of logarithmic creep laws, which has been given only recently by Pandey and Holm (2016) by linking timedependent Newtonian viscosity to the parameters of the Lomnitz model. With these recent developments, the Lomnitz model has become much more attractive, also on account of the smaller number of model parameters, as compared to other models (e.g., the generalized Kelvin-Voigt or the generalized Maxwell model), necessary for representation of test data. Therefore, the present paper readopts the logarithmic Lomnitz model for wood and considers, in addition to King (1961) and Bach and Pentoney (1968), the influence of temperature (Morlier 1994;Bekhta and Niemz 2003) and moisture content (Schänzlin 2010), as those parameters are commonly known to majorly influence the time-dependent behavior. For this purpose, the viscoelastic behavior of clear (knotfree) wood specimens of spruce at the 10 mm scale of observation is determined, disabling mechano-sorptive creep as well as shrinkage and swelling. Half-hour long creep tests were performed at different (constant) levels of temperature and moisture content.
The paper is structured as follows: in "The viscoelastic Lomnitz law" section, the logarithmic Lomnitz law is briefly reviewed. In the "Experimental work" section, the experimental methods are described, which give access to data employed for identification of the viscoelastic parameters of the Lomnitz law as described in the "Parameter identification" section. The uniqueness of the so-obtained parameters is discussed in the "Uniqueness analysis of fitted material parameters" section, while the results are discussed in the "Discussion" section. The paper closes with final remarks in the "Conclusion" section.

The viscoelastic Lomnitz law
The Lomnitz-type creep law, first introduced by Lomnitz (1956), assumes that the strain rate in a creep test is inversely proportional to time, i.e., d ∕dt ∝ 1∕t . The uniaxial creep compliance J, defined as (t)∕ , for a constant stress applied at t = 0 , i.e., (t) = H(t) 0 , reads with E as the materials Young's modulus, J log as a creep parameter and log as the characteristic time of the creep process. The creep compliance J can be split into an instantaneous part J 0 and a time-dependent part J v (t): where J 0 is the inverse of the Young's modulus, i.e., J 0 = 1∕E , and J v (t) can be expressed in terms of time-dependent bulk and shear moduli, respectively (Brinson and Brinson 2008): Inserting Eq. (3) into Eq.
(2) yields with J vol = 1∕K(t) and J dev = 1∕G(t) denoting the volumetric and deviatoric creep compliances, respectively. As viscoelastic deformations in polymers are assumed to be caused by a sliding mechanism at the microscale (bond breakage in shear directions), viscoelasticity is assumed to be purely deviatoric (Ward and Sweeney 2012). This assumption is commonly adapted, see for example, Brinson and Brinson (2008), Pichler et al. (2018), Hofer et al. (2018), Beijer and Spoormaker (2002), Idesman et al. (2001) and Singh and Rosenman (1974). As hemicellulose and lignin may be characterized as natural polymers, this assumption is also presupposed in the present paper, hence, the volumetric compliance J vol is set to zero. Considering the Lomnitz model [Eq.
(1)] for the deviatoric compliance in Eq. (4) results in the deviatoric Lomnitz model (Fig. 1a) as with J dev log as the deviatoric creep compliance parameter.
(1) Pichler et al. (2018) considered a finite loading ramp from t = 0 to t = t 0 as (t) = 0 t∕t 0 for 0 ≤ t ≤ t 0 giving the so-called ramp compliance J as In order to improve the versatility of the model (i.e., improve the quality of fit of model to the test data, c.f. "Uniqueness analysis of fitted material parameters" section), the Lomnitz model is extended in series by a Newtonian dashpot (Flügge 1975) (referred to as extended Lomnitz model in the remainder of this paper), resulting in with dev as the (deviatoric) viscosity of the dashpot (see Fig. 1b). In terms of the ramp compliance, Eq. (7) becomes The derivatives of Eqs. (6) and (8) are given as and respectively. The relaxation modulus R(t) is connected to the creep compliance J(t) via their respective Laplace-Carson transformations using the reciprocity principle: The Laplace-Carson transformation of Eq. (7) results in Uniaxial representation of Lomnitz model (a) and extended Lomnitz model (b) with (a, z) = ∫ ∞ z t a−1 exp(−t)dt as the incomplete gamma function. Since the inverse of J * (p) cannot be transformed into time domain by analytical means, no closed-form expression can be obtained for the relaxation modulus by exploiting the reciprocity principle. However, Mainardi and Spada (2012) derived an analytical expression for the relaxation modulus by relating the relaxation and compliance function by a Volterra integral equation (Pipkin 1986): By using Eq. (5) and expressing the relaxation modulus as R(t) = (t)E , Eq. (13) becomes with the solution of the associated Volterra equation obtained numerically employing (Press et al. 1992), and the (creep) kernel defined as

Experimental work
Clear wood specimens of spruce (Picea abies), with the dimensions B × H × L = 20 × 20 × 280 mm , are cut such that the grain direction and the annual rings are parallel to the surface, see Fig. 2.
The spruce specimens were conditioned at four different temperatures, T = 25 • C , T = 40 • C , T = 65 • C , T = 80 • C , and at different relative humidity conditions. The varying relative humidities were controlled either by subjecting specimens to an atmosphere associated with saturated salt solutions or direct conditioning in climate chambers. Oven-dried specimens (line #1 in Table 1) were dried at T = 105 • C in an oven before exposing them to the specified testing temperature. Note that ovendrying has only a negligible influence on the viscoelastic properties of dry wood, i.e., the microstructure of the wood is not altered by oven-drying (Stamm 1956;Placet et al. 2008;Bekhta and Niemz 2003). Specimens referring to lines #2-#5 in Table 1 were conditioned in glass boxes containing saturated salt solutions (see Table 2, Greenspan 1977); those boxes were then placed in climate chambers with Wood Science and Technology (2019) 53:765-783 T = 80 • C and relative air humidity of 90% . The specimens were conditioned until their weight changed less than 1% in two consecutive weighings (separated by more than one day). In total, 21 conditioning classes were employed, leading to average wood moisture contents as shown in Table 1. The specimens were assigned to the conditioning classes such that similar densities are evenly distributed among all classes. For each class, six specimens were conditioned, leading to a total amount of 126 specimens employed in 126 creep tests.
Three-point-bending creep tests with load application at half-span and length between the two supports of L s = 240 mm were employed to determine the viscoelastic behavior. A Shimadzu AG-X plus 10kN testing frame was used for all tests, exhibiting a characteristic tolerance of ± 0.3% as regards the applied force. A constant load F = 555 N , corresponding to an edge normal stress of about 25 MPa [far below the flexural strength of ≈ 95 MPa (Austrian Standards International 2003)], was applied with a loading rate of Ḟ = 20 N/s ; the load was kept constant for 1800 s. The mid-span vertical displacement u(t), representing the bending deflection of the beam and measured in the direction of the applied load, was monitored with an inductive displacement transducer (HBM WA20) with a characteristic tolerance of ± 1% . The data acquisition rate was set to 5 Hz throughout the experiments. After the 1800 s creep phase, the specimen was unloaded, whereupon small deformations remained. Each test was conducted at the temperature which has been prescribed previously during conditioning, i.e., the temperature given in Table 1. As an exception, the oven-dried specimens were conditioned at T = 105 • C but tested at lower temperatures ( T = 25 • C , T = 40 • C , T = 65 • C and T = 80 • C ). The transport of specimens from the conditioning domain, i.e., salt box or climate chamber, to the thermostatic chamber of the testing frame was done within less than a minute, hence the temperature remained almost constant during the transition from conditioning to testing. With the Shimadzu thermostatic chamber TCE-N300, the temperature was kept constant during testing, which was verified at start and end of the tests with a temperature sensor (Testo 177-T4, characteristic tolerance ± 0.3 • C ) directly through a hole in the specimen (outside of the supports). The spruce specimens were wrapped in diffusion-resistant film to prevent sample drying during the half-hour creep test, which was verified by weighing the specimen immediately before and after the tests. By keeping the moisture content constant throughout the test, it was ensured that mechano-sorptive creep processes were not active. After performing a test, the specimen was oven-dried at 105 • C , hence the moisture content w during testing can be obtained as with m wet as the mass of the wet specimen immediately after the test procedure and m dry as the mass after oven-drying.
In addition to the aforementioned creep tests, four three-point bending relaxation tests were performed at temperature levels of T = 25 • C , T = 40 • C , T = 65 • C , and T = 80 • C , respectively, serving as independent verification experiments as regards (17) w = m wet − m dry m dry material parameters back-calculated in the "Discussion" section. Test setup, equipment and testing procedure were exactly the same as for the creep tests, except that the specimens were subjected to a constant displacement u 0 = 1.21 mm at midspan and the force history was recorded. The displacement was applied at a rate of 0.2 mm/s (hence the duration for the load application was ≈ 6 s ) and kept constant for 1800 s.

Parameter identification
The experimentally obtained displacement histories are used to calculate the creep compliance J test as where linear viscoelastic behavior is assumed and the specimen is idealized as an Euler-Bernoulli beam, i.e., the beam is assumed to be slender (ratio L∕H = 12 ), hence, shear deformation can be neglected and the displacements are small, i.e., ≈ 1−2 mm as compared to the beam height of 20 mm. Note that the uniaxial compliance as obtained in this paper [Eq. (18)] is related to the bending stresses with the axis of bending (i) normal to fiber direction and (ii) in the plane tangential to the annual rings. The so-obtained creep compliances are fitted with the expression for the extended Lomnitz law with the respective ramp compliance given in Eq. (8). For identification of the parameters E, J dev log , log and dev , least-square fitting is employed, using the Levenberg-Marquardt algorithm, introducing the error with data points given every 0.2 s, fitting for the range t > 100 s.

Uniqueness analysis of fitted material parameters
The Lomnitz parameters obtained by least-square fitting were investigated for their uniqueness. The least-square error , normalized by its minimum value, is shown in Fig. 3 for a 80 • C test. In Fig. 3a, the least-square error below a certain threshold value is plotted for a parameter range of 9000 MPa < E < 10500 MPa, 10 −3 s < log < 1 s and 3 ⋅ 10 −6 MPa −1 < J dev log < 6 ⋅ 10 −6 MPa −1 , sampled with 200 × 200 × 200 equally spaced points (linearly spaced with regard to E and J dev log , logarithmically spaced with regard to log ), whereas the dashpot parameter dev was set constant to the value found by least-square fitting. Obviously, the least-square error is not converging to a distinct minimum, but stretched over a large area in the three-parameter space (Fig. 3a) and in the Elog space (Fig. 3b); hence, the three parameters E , J dev log and log are not independent. A similar dependency, most pronounced between parameters E and log , has previously been observed when analyzing creep experiments for polyurethane foams (Pichler et al. 2018). The reasoning given in Pichler et al. (2018) was followed, i.e., log being probably too small to be reasonably obtained by standard creep (or relaxation) experiments. Hence, log is set to a constant value of 1 s what ensures uniqueness of the remaining three parameters as obtained below. A distinct minimum is found in both the three-dimensional parameter space (Fig. 4a) and the two-dimensional sections ( Fig. 4b-d) when setting log constant. The solution previously found by the Levenberg-Marquardt algorithm (denoted by black circles in Fig. 4b-d) corresponds to the color-coded minimum least-square error. In order to assess the goodness of the employed least-square fitting procedure, the coefficient of determination R 2 is computed as with Ĵ as the average value of the test data points. Figure 5a shows the coefficients of determination when employing the extended Lomnitz model for parameter  , d). Note that the moisture contents in the legends refer to single test results and, thus, are not matching with the average values given in Table 1 identification, whereas Fig. 5b depicts the coefficients of determination when using the Lomnitz model without a dashpot, for all tests, respectively. Whereas the Lomnitz model without additional dashpot exhibits a minimum R 2 ≈ 0.972 , the extended Lomnitz model exhibits a minimum R 2 ≈ 0.996 , hence, fits the test data significantly better.
Some of the obtained creep compliance functions are shown in Fig. 6a, b, their respective derivatives are given in Fig. 6c, d. As the derivatives of the test data are obtained by numerical differentiation of the creep compliance functions and, hence, are rather noisy, a Savitzky-Golay filter (Savitzky and Golay 1964) (employing a second-order polynomial approximation) is applied to the experimental data, considering 100 (t < 160 s) , 400 (160 < t < 350 s) , and 800 (350 < t < 1800 s) left and right data points, respectively, for smoothing. The black lines represent the fitted curves of the extended Lomnitz model [Eq. (8)] and its derivative [Eq. (10)]. The extended Lommnitz models was fitted in the time domain (Fig. 6a, b) to each test, respectively, whereas the so-obtained Lomnitz parameters are also used for plotting the derivatives in Fig. 6c, d. Lomnitz parameters for varying moisture content for different temperature levels, including regression curves with corresponding coefficient of determination. Note: the coefficient of determination in (c, d) refers to the exponential fit of parameters determined in the "Uniqueness analysis of fitted material parameters" section. Although there is quite a scatter of data in (c, d), the obtained regression graphs are still suitable to interpret the general trend of the Lomnitz parameters with respect to temperature and moisture content

Discussion
In Fig. 7, the Lomnitz parameters obtained by least-square fitting are depicted as functions of the moisture content and temperature; Table 3 shows the mean values and standard deviations of all Lomnitz parameters with regard to the conditioning classes defined in Table 1. The results shown in Fig. 7c, d were least-square fitted by exponential functions (appearing as linear functions since the ordinate is plotted in logarithmic scaling), emphasizing the trend in Lomnitz parameters. Figure 7a, b shows that the increase in the compliance 1 / E with moisture content and temperature, respectively, is almost negligible when compared to the other parameters plotted in Fig. 7c, d, i.e., J dev log and 1∕ dev . Figure 7c, d shows the dependency of the Lomnitz parameters J dev log and 1∕ dev on the moisture content. Both parameters J dev log and 1∕ dev increase with increasing temperature and moisture content. The dependency of creep parameters J dev log and 1∕ dev is consistent with previous results, showing that increasing the temperature and moisture content leads to a more pronounced creep behavior (c.f. Jiang et al. 2009;Hering and Niemz 2012).

Conclusion
Based on half-hour long creep tests on clear spruce specimens performed at constant moisture content to deactivate potential mechano-sorptive creep phenomena, the viscoelastic behavior was found to obey a Lomnitz model extended by a Newtonian dashpot. Thereby, the influence of temperature and moisture content was taken into account. The extended Lomnitz model was shown to provide a good fit to the test data at all investigated levels of temperature and moisture content with a limited number of only three model parameters. The respective model parameters E, J dev log and dev were back-calculated using a least-square fitting algorithm, whereas the characteristic time log was set to a fixed value of log = 1 s . The uniqueness of parameters was shown by visualization of the least-square error in the three-parameter space. While the dependence of the elastic compliance 1 / E on temperature and moisture content was shown to be comparatively small, a pronounced dependency of the Lomnitz parameters J dev log and 1∕ dev on moisture content and temperature was observed, both increasing with increasing temperature and moisture content. Arrhenius plots gave access to a distinctively different temperature dependency of Lomnitz parameters J dev log and 1∕ dev . That may indicate the presence of at least two different creep processes at the microscale of the material, which could be caused by the different viscoelastic behavior of the amorphous wood constituents hemicellulose and lignin, respectively. In order to confirm the applicability of the Lomnitz law as regards time-dependent behavior of wood, the obtained material parameters were validated by relaxation tests.