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 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., \(\hbox {d}\varepsilon / \hbox {d}t \propto 1/t\). The uniaxial creep compliance J, defined as \(\varepsilon (t) / \sigma\), for a constant stress applied at \(t=0\), i.e., \(\sigma (t) = H(t) \sigma _0\), readswith E as the materials Young’s modulus, \(J_{\log }\) as a creep parameter and \(\tau _{\log }\) 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) yieldswith \(J^{\mathrm{vol}} = 1/K(t)\) and \(J^{\mathrm{dev}} = 1/G(t)\) 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 \(J^{\mathrm{vol}}\) 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) aswith \(J_{\log }^{\text{dev}}\) as the deviatoric creep compliance parameter.

Pichler et al. (2018) considered a finite loading ramp from \(t=0\) to \(t=t_0\) as \(\sigma (t) = \sigma _0 t/t_0\) for \(0 \le t \le t_0\) giving the so-called ramp compliance \({\widetilde{J}}\) asIn 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 inwith \(\eta ^{\text{dev}}\) as the (deviatoric) viscosity of the dashpot (see Fig. 1b). In terms of the ramp compliance, Eq. (7) becomesThe derivatives of Eqs. (6) and (8) are given asandrespectively. 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 inwith \(\varGamma \left( a, z \right) = \int _{z}^{\infty }t^{a-1} \exp (-t) {\mathrm{d}}t\) 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 \(R(t) = \phi (t) E\), Eq. (13) becomeswith the solution of the associated Volterra equationobtained numerically employing (Press et al. 1992), and the (creep) kernel defined as

Clear wood specimens of spruce (Picea abies), with the dimensions \(B\times H\times L = 20\times 20\times 280\,\hbox {mm}\), 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, \(T = 25\,^{\circ }\hbox {C}\), \(T = 40\,^{\circ }\hbox {C}\), \(T = 65\,^{\circ }\hbox {C}\), \(T = 80\,^{\circ }\hbox {C}\), 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 \(T = 105\,^{\circ }\hbox {C}\) in an oven before exposing them to the specified testing temperature. Note that oven-drying 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 temperatures of \(T = 25\,^{\circ }\hbox {C}\), \(T = 40\,^{\circ }\hbox {C}\), \(T = 65\,^{\circ }\hbox {C}\), \(T = 80\,^{\circ }\hbox {C}\), respectively. Specimens referring to line #6 of Table 1 were conditioned in a climate chamber at \(T = 80\,^{\circ }\hbox {C}\) 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 \(L_{s}=240\,\hbox {mm}\) 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 \(\pm \,0.3\%\) as regards the applied force. A constant load \(F=555\,\hbox {N}\), corresponding to an edge normal stress of about 25 MPa [far below the flexural strength of \(\approx 95\,\hbox {MPa}\) (Austrian Standards International 2003)], was applied with a loading rate of \({\dot{F}} = 20\,\hbox {N/s}\); 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 \(\pm \,1\%\). 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 \(T = 105\,^{\circ }\hbox {C}\) but tested at lower temperatures (\(T = 25\,^{\circ }\hbox {C}\), \(T = 40\,^{\circ }\hbox {C}\), \(T = 65\,^{\circ }\hbox {C}\) and \(T = 80\,^{\circ }\hbox {C}\)). 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 \(\pm \, 0.3\,^{\circ }\hbox {C}\)) 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 \(105\,^{\circ }\hbox {C}\), hence the moisture content w during testing can be obtained aswith \(m_{wet}\) 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 \(T = 25\,^{\circ }\hbox {C}\), \(T = 40\,^{\circ }\hbox {C}\), \(T = 65\,^{\circ }\hbox {C}\), and \(T = 80\,^{\circ }\hbox {C}\), respectively, serving as independent verification experiments as regards 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 \(u_0=1.21\,\hbox {mm}\) at mid-span 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 \(\approx 6\,\hbox {s}\)) and kept constant for 1800 s.

The experimentally obtained displacement histories are used to calculate the creep compliance \(J_{test}\) aswhere 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., \(\approx 1{-}2\,\hbox {mm}\) 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, \(J_{\log }^{\text{dev}}\), \(\tau _{\log }\) and \(\eta ^{\text{dev}}\), least-square fitting is employed, using the Levenberg–Marquardt algorithm, introducing the errorwith data points given every 0.2 s, fitting for the range \(t>100\,\hbox {s}.\)

The Lomnitz parameters obtained by least-square fitting were investigated for their uniqueness. The least-square error \(\varepsilon\), normalized by its minimum value, is shown in Fig. 3 for a \(80\,^{\circ }\hbox {C}\) test. In Fig. 3a, the least-square error below a certain threshold value is plotted for a parameter range of \(9000\,{\text{MPa}}<E<10500\,{\text{MPa}}, \, 10^{-3}\,{\text{s}}<\tau _{\log }<1\,{\text{s}}\) and \(3\cdot 10^{-6}\,\hbox {MPa}^{-1}<J_{\log }^{\text{dev}}<6\cdot 10^{-6}\,\hbox {MPa}^{-1}\), sampled with \(200\times 200\times 200\) equally spaced points (linearly spaced with regard to E and \(J_{\log }^{\text{dev}}\), logarithmically spaced with regard to \(\tau _{\log }\)), whereas the dashpot parameter \(\eta ^{\text{dev}}\) 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 E-\(\tau _{\log }\) space (Fig. 3b); hence, the three parameters \(E\), \(J_{\log }^{\text{dev}}\) and \(\tau _{\log }\) are not independent. A similar dependency, most pronounced between parameters E and \(\tau _{\log }\), 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., \(\tau _{\log }\) being probably too small to be reasonably obtained by standard creep (or relaxation) experiments. Hence, \(\tau _{\log }\) 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 \(\tau _{\log }\) 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 aswith \({\hat{J}}\) as the average value of the test data points. Figure 5a shows the coefficients of determination when employing the extended Lomnitz model for parameter 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 \(R^2 \approx 0.972\), the extended Lomnitz model exhibits a minimum \(R^2 \approx 0.996\), 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 \(100\;(t < 160\,\text{s})\), \(400\;(160< t < 350\,\text{s})\), and \(800\;(350< t < 1800\,\text{s})\) 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.

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., \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\). Figure 7c, d shows the dependency of the Lomnitz parameters \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\) on the moisture content. Both parameters \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\) increase with increasing temperature and moisture content. The dependency of creep parameters \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\) 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).

Figure 8 shows an Arrhenius plot for the Lomnitz parameters \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\), with the moisture content visualized by color coding. The black lines represent fitted Arrhenius equations to the test data, reading (Navi and Stanzl-Tschegg 2009)with \(\bullet\) as \(J_{\log }^{\text{dev}}\) or \(1/\eta ^{\text{dev}}\), respectively, A as a pre-exponential factor and \(E_a\;(\hbox {J}\,\hbox {mol}^{-1})\) as the activation energy of the creep process, \(R=8.314\,\hbox {J}\,\hbox {mol}^{-1}\,\hbox {K}^{-1}\) as the gas constant and T (K) as the absolute temperature. By taking the logarithm, Eq. (21) is written asi.e., linear in terms of the independent variable 1 / T; hence, a linear function is fitted to the logarithmic Lomnitz-type creep parameters \(\ln \left( J_{\log }^{\text{dev}}\right)\) and \(\ln \left( 1/\eta ^{\text{dev}}\right)\), respectively, vs. 1 / T (Fig. 8). The activation energy represents the slope in the Arrhenius diagram and is given by \(E_a=11.8\,\hbox {kJ/mol}\) and \(E_a=43.7\,\hbox {kJ/mol}\) for \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\), respectively. As all materials exhibit, in general, different activation energies (see e.g., Lomellini (1992) and Eckstein et al. (1997) for activation energies of polymers), the difference in activation energies for \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\) could indicate that different material phases are involved in the creep mechanism of wood. According to Kelley et al. (1987), amorphous lignin and hemicellulose exhibit a different viscoelastic behavior, both being strongly dependent on heat and moisture content. One may explain the different activation energies of the Lomnitz parameters in the macroscopic wood specimens with the differences in the viscoelastic behavior of those polymers at the microscale.

Relaxation tests (“Experimental work” section) are used for verification of the proposed viscoelastic model. As shown in Fig. 9a, the begin of the relaxation phase is influenced by the finite displacement application time, however, for \(t>\approx 20\,\hbox {s}\) the extended Lomnitz model fits the relaxation curves rather good. The obtained Lomnitz parameters are summarized in Table 4 and depicted in Fig. 9. Those parameters were not obtained by least-square fitting, since the relaxation function Eqs. (13)–(16) can only be expressed as a Volterra integral equation, which is solved numerically using the algorithm outlined in Press et al. (1992); hence, the Lomnitz parameters were determined visually (Fig. 9a). The so-obtained Lomnitz parameters 1 / E, \(J_{\log }^{\text{dev}}\) and \(1/\eta ^{\text{dev}}\) are shown in Fig. 9b–d as colored dots; the gray dots and the regression lines were previously determined from the creep tests (Fig. 7). The Lomnitz parameters determined from relaxation tests agree well with the parameters obtained from creep tests; hence, the relaxation experiments may be regarded as a verification of Lomnitz-type behavior identified from creep tests.