Comparative Study of Numerical Simulation on Short-Term Creep Behavior of Steam-Pretreated White Oak (Quercus alba L.) Wood

Abstract

This paper investigates the effects of steam pretreatment temperature (100~120 °C), test temperature (20~80 °C), and earlywood vessel belt on parameters associated with the bending creep properties of white oak (Quercus alba L.) wood. The Burger model, Five-parameter model, and Six-parameter model are used for short-term simulation and comparative analysis of the bending creep curve of steam-pretreated white oak wood, and creep fitting curves and viscoelastic parameters are obtained. The advantages and disadvantages of different viscoelastic mathematical models for fitting the bending creep curve of white oak are analyzed. The results indicate that the Six-parameter model is more consistent with the creep behavior of wood in simulating and predicting the creep behavior of wood than that of the Burger and Five-parameter model and can reflect the short-term deformation characteristics of wood.

1. Introduction

Wood, as a distinctive biomass material, exhibits viscoelasticity by responding to stresses in a manner that reflects the combined characteristics of both elastic solids and viscous fluids [1]. The investigation of wood’s viscoelasticity is a prominent research focus in the field of wood composite rheology, encompassing both static and dynamic viscoelasticity. Static viscoelasticity primarily explores creep, which refers to the gradual increase in strain on wood over time under a constant stress [2,3]. The phenomenon of wood creep has a significant impact on the quality of wood products, engineered components, and structural design safety. The creep behavior of wood not only has an impact on the manufacturing and processing of wood products, such as the hot pressing and drying process of wood-based panels, but also plays a crucial role in determining the overall service life of the product [4]. Accurately simulating and predicting this behavior poses a technical challenge that must be addressed in the wood processing industry [5]. Therefore, the development of mathematical models plays a crucial role in researching and predicting wood creep behavior [6,7].

Domestic and international researchers have recently conducted simulation studies to predict the bending creep properties of wood using mathematical models such as Maxwell, Kelvin, Burger, generalized Maxwell, and Five-parameter models. These studies have quantitatively analyzed the viscoelastic behavior of wood [8,9,10,11]. Among which, the effect of applied load and relative humidity on the bending creep of furfural-modified poplar (Populus tomentosa) was effectively fitted using the Burger model [9]. Wang et al. utilized both the Burger model and generalized Kelvin model to fit short-term creep curves of white oak wood under different grain orientations, providing theoretical references for the spreading and straightening process of warped white oak lumber under humid-thermal conditions [12]. In addition, Huč et al. discovered that the Kelvin model better predicted simulated bending creep characteristics of Douglas fir (Pseudotsuga Menziesii), Hinoki (Chamaecyparis obtusa), Norway spruce (Picea Abies), and European beech (Fagus sylvatica) [13]. Asyraf et al. found that the results of fitting Burger’s model to the bending creep of Balau (Shorea spp.) wood under a 30% stress level were highly consistent with experimental results [11]. In addition, the finite element model developed by Hu et al. [10] using ABAQUS 6.16 (2016) numerical simulation software can better predict the compressive stress relaxation behavior of beech (Fagus orientalis) wood. The report by Yin et al. also confirmed that the Five-parameter model fitted significantly better than the Burger model because of the introduction of nonlinear fitting coefficients, thus effectively predicting the bending creep behavior of wood [14]. Based on the above evidence, it can be concluded that generalized Kelvin, Burger, and Five-parameter models are capable of accurately predicting the bending creep properties of wood.

Although the Maxwell model could accurately describe the stress relaxation process of viscoelastic materials, it fails to fully reflect the creep deformation exhibited by wood [15,16]. Meanwhile, the Burger model could provide a better representation of both creep deformation and stress relaxation characteristics, but its assumption of constant viscous flow deformation leads to an inconsistency between part of the deformation rate and the actual final creep effect. Furthermore, its expression follows a linearly increasing functional equation, resulting in a linear trend observed in the fitted strain curve over time [17]. In recent years, there has been increasing interest in the numerical simulation of the rutting deformation of asphalt pavement based on a Six-parameter model. Chen presented a numerical simulation and analysis of rutting deformation in asphalt pavement based on a Six-parameter model [18]. Through the simulation, the viscoelastic parameters of typical graded asphalt mixtures were obtained, and it was demonstrated that the Six-parameter model can be used as a method for predicting rutting [19]. The research findings indicate that the Six-parameter model can effectively describe the deformation behavior of asphalt pavement under various conditions and provide a theoretical basis for optimizing the design and maintenance of asphalt pavement. However, there have been no reports on applying this model to wood viscoelasticity. Therefore, utilizing the Six-parameter model in quantitatively investigating the influence of earlywood vessel belts on static viscoelastic of wood and revealing the bending mechanism of wood from a static viscoelastic perspective is both feasible and innovative. This approach can provide a new method for understanding the mechanical behavior of wood and may lead to improvements in wood processing techniques and design methods.

Superheated steam pretreatment was environmentally friendly, as the pretreatment process involved no oxidation or combustion reactions, and there was no risk of fire or explosion under high-temperature conditions. Furthermore, the evaporated water itself could serve as the drying medium [20]. Wood subjected to steam treatment experiences reduced stress during the drying process and effectively exhibits improved dimensional stability [21]. However, to date, research on the short-term bending creep of wood under steam pretreatment conditions is incomplete. Steamboat pretreatment plays a critical role in the bending process of wood products. Most studies have focused on experiments involving different wood species, developmental processes, chemical compositions, and grain directions at various temperatures. Under steam pretreatment conditions, there have been few reports on the impact of wood anatomical structures (such as vessel bands) on its bending creep [22,23]. In addition, steam pretreatment plays a critical role in the bending process of wood products. There are few reports about numerical simulation on short-term creep behavior of steam-pretreated wood. In light of the above, the Burger model, Five-parameter model, and Six-parameter model were employed in this study to simulate and analyze the short-term bending creep of white oak (Quercus alba L.) wood before and after steam pretreatment. The optimal model simulation was compared and analyzed to predict the influence of the earlywood vessel belt of white oak wood on the creep characteristics of specimens under different steam pretreatment and test temperatures.

2. Materials and Methods

2.1. Materials

The white oak (Quercus alba L.) wood, characterized by its earlywood of ring porous wood, originates from the eastern part of the United States. It was sourced from the Nanxun building materials market in Huzhou City, Zhejiang Province, China. The selected trees had a minimum age of 20 years and a diameter at breast height (DBH) exceeding 25 cm. The annual growth rings ranged between 3 mm and 5 mm in width. The air-dry density was measured to be approximately 0.76 ± 0.05 g/cm³. The moisture content (MC) of the specimens was adjusted to 12.0 ± 1.0% using a constant temperature and humidity chamber (HTB HWHS 20, HONGJIN, Guangdong, China) prior to the test. Subsequently, samples were extracted from the same growth wheel and prepared with dimensions of 40 mm (L) × 12 mm (W) × 2.0 mm (T), which were obtained from the inner region of the third annual ring of white oak wood. Two specimens were prepared according to the distribution characteristics of the earlywood vessel belt, i.e., specimen A with the earlywood vessel belt located in the mesne layer of the specimen and specimen B without the earlywood vessel belt [23], as shown in Figure 1. Specimen preparation for the bending creep test requires steam treatment using saturated steam at 100 °C and superheated steam at 110 °C and 120 °C in a hydrothermal synthesis reactor (HTG-200, CHEMN, Anhui, China) as shown in

Table 1. Steam pretreatment of specimens.

Sample No.	Steam Pretreatment Temperature (°C)	Sample No.	Steam Pretreatment Temperature (°C)
A	The control	B	The control
A1	100	B1	100
A2	110	B2	110
A3	120	B3	120

. The specimen should be subjected to this pretreatment for 1 h. Afterwards, the moisture content of the specimen should be adjusted to 12.0 ± 1.0% using a constant temperature and humidity box.

2.2. Bending Creep Test

A dynamic thermomechanical analyzer (DMA-Q800, TA instrument, New Castle, DE, USA) with a double cantilever fixture was utilized for obtaining the creep strain curves of all specimens under the corresponding temperature. The bending creep characteristics of the test specimens were evaluated by applying a constant load radially in the double cantilever fixture (with a span of 35 mm) while maintaining a moisture content of 12% ± 0.1% throughout the test. The temperature ranged from 20 °C to 80 °C during the experiment, and corresponding relative humidity (RH) levels were controlled at 66%, 69%, 72%, 74%, 77%, 79%, and 81%, respectively. Once the temperature and humidity inside the testing furnace reached their set values, they were kept stable for an hour to ensure gradual stabilization of sample moisture content (MC). A constant load of 5 MPa was applied to prevent sample fractures during the experiment, enabling enhanced observation of creep behavior. Therefore, a consistent load of 5 MPa was applied and sustained for a 45 min duration throughout the test, followed by an additional 45 min period to maintain ambient conditions after removing the constant load. Bending creep data from the specimens were collected and recorded using dynamic thermo-mechanical analysis (DMA) [17,23]. Then, the numerical model simulation and figure preparation were completed by Origin 2022 in this study.

2.3. Mathematical Models

2.3.1. Burger Model

The Burger model has the dual advantages of the Maxwell and Kelvin models, which can reflect both the transient elasticity and permanent deformation properties of wood. It can also analyze the rheological properties and short-term relaxation phenomena of wood. The eigenstructural equations of the Burger model are [18]

$$\epsilon \left(t\right)={\sigma }_0\left[{\displaystyle \frac{1}{E_1}}+{\displaystyle \frac{1}{E_2}}\left(1-e^{{\displaystyle \frac{-t{E}_2}{{\eta }_1}}}\right)+{\displaystyle \frac{1}{{\eta }_2}}t\right],$$

(1)

where ε(t) is the strain; σ₀ is the stress; E₁ is the generalized elastic module in MPa, which reflects the response of ideal elasticity, i.e., transient elasticity; E₂ is the high elastic module in MPa; η₁ is the coefficient of viscosity, which reflects hysteresis elastic deformation; η₂ is the parameter that reflects the unrecoverable deformation left behind after removing external force during the recovery process; and t represents time.

2.3.2. Five-Parameter Model

In order to address the issue of linear growth in Burger’s model when simulating the viscous flow component of fitted wood creep and enhance accuracy in predicting wood bending creep behavior. Dinwoodie et al. proposed a Five-parameter model with nonlinear viscoelastic coefficients to fit wood creep behavior [24]. This model aims to comprehensively capture the actual deformation characteristics of wood by incorporating initial creep, steady creep, and accelerated creep stages based on Burger’s model, thereby providing a comprehensive representation of the entire deformation process of wood. The constitutive equation for this Five-parameter model is presented as follows [18]:

$$\epsilon (t)={\sigma }_0\left[{\displaystyle \frac{1}{E_1}}+{\displaystyle \frac{1}{AB}}\left(1-{{\displaystyle e}}^{-Bt}\right)+{\displaystyle \frac{1}{E_2}}\left(1-{{\displaystyle e}}^{-\tau \cdot t}\right)\right],$$

(2)

where $\tau ={\displaystyle \frac{E_2}{{\eta }_2}}$; ε(t) represents the strain; σ₀ represents the stress; E₁ is the generalized elastic module in MPa reflecting the response of ideal elasticity, i.e., transient elasticity; E₂ is the high elastic module in MPa; and η₂ is the bulk viscosity reflecting the unrecoverable deformation left behind after the completion of the recovery process of the withdrawal of the external force. A is a coefficient, and B is a parameter added additionally to solve the model simulation prediction of the viscous flow of the deformation of its viscous flow of the part of the problem of the nonlinear coefficients, and t is the time.

2.3.3. Six-Parameter Model

Compared to the Burger model and the generalized Kelvin model, the Six-parameter model more effectively reflects the changes in viscoelastic materials during different stages, including instantaneous deformation, initial creep, steady creep stage, and accelerated creep stage. The creep curve of the Six-parameter model more accurately describes the deformation characteristics of wood, thus conforming to the bending creep law of wood and serving as a suitable viscoelastic model for analyzing and calculating the permanent deformation of wood [25]. The Six-parameter model is based on the Five-parameter model with an added dashpot in series. Its intrinsic equation is [18]

$$\epsilon (t)={\sigma }_0\left[{\displaystyle \frac{1}{E_1}}+{\displaystyle \frac{1}{{\eta }_1}}t+{\displaystyle \frac{1}{AB}}\left(1-{{\displaystyle e}}^{-Bt}\right)+{\displaystyle \frac{1}{E_2}}\left(1-{{\displaystyle e}}^{-\tau \cdot t}\right)\right],$$

(3)

where $\tau ={\displaystyle \frac{E_2}{{\eta }_2}}$; ${\eta }_1=A{e}^{Bt}$; ε(t) is the strain; σ₀ is the stress; E₁ is the generalized elastic module in MPa, which reflects the response of ideal elasticity, i.e., transient elasticity; E₂ is the high elastic module in MPa; η₁ is the coefficient of viscosity, which reflects hysteresis elastic deformation; η₂ is the bulk viscosity, which reflects unrecoverable deformation left behind after removing external force upon completion of the recovery process; and t is the time.

3. Results and Discussion

3.1. Effect of Steam Pretreatment on Creep Characteristics of White Oak Wood

The effects of steam pretreatment temperature (100~120 °C), test temperature (20~80 °C), and earlywood vessel belt distribution on parameters associated with the bending creep properties of white oak (Quercus alba L.) wood are shown in Figure 2. Under the same test temperature conditions, the instantaneous strain and 45 min strain of samples subjected to steam pre-treatment show a significant decrease compared to untreated samples. Furthermore, the creep behavior of steam pre-treated samples is smaller compared to untreated samples. Under the same test temperature conditions, samples with earlywood vessel belts exhibit greater instantaneous strain and 45 min strain than earlywood samples under all steam pre-treatment conditions. Under the same steam pre-treatment conditions, as the test temperature increases, the instantaneous strain and 45 min strain of the samples increase. Under the 100 °C steam pre-treatment condition, the reduction in instantaneous elastic strain for samples with earlywood vessel belts becomes negative as the test temperature rises (70~80 °C), indicating that strain increases with temperature. In contrast, the reduction in instantaneous elastic strain for earlywood samples shows a decreasing trend. Under the 110 °C steam pre-treatment condition, the reduction in instantaneous elastic strain for samples with earlywood vessel belts gradually decreases across the test temperature range of 50~80 °C, with the smallest reduction in strain occurring at 80 °C. Under the 120 °C steam pre-treatment condition, the reduction in strain for all samples remains relatively consistent, making these samples more stable compared to those treated at other steam temperatures (100 °C and 110 °C). The reduction in 45 min strain for samples subjected to steam pre-treatment at 100 °C is much smaller than that for samples treated at 110 °C and 120 °C. Furthermore, the reduction in 45 min strain for samples treated at 120 °C is greater than that for samples treated at 110 °C, especially at higher test temperatures (50~80 °C), where they exhibit less creep behavior. This is because hemicellulose, an amorphous substance, is one of the main components of wood. It contains a large number of hydrophilic groups and has a strong water absorption capacity, making it one of the factors that contribute to strain in wood. During the pressure steam treatment process, the acetyl groups in hemicellulose are hydrolyzed by heat and removed from the hemicellulose, producing acetic acid, which increases the acidity of the treatment environment [26]. In addition, the degree of polymerization of hemicellulose gradually decreases during this hydrolysis process, resulting in the formation of oligosaccharides and monosaccharides [27]. The pentose sugars in the monosaccharides react to form furfural, while hexose sugars react to form hydroxymethylfurfural. This hydrolysis process further leads to the formation of acetic acid, which induces additional hydrolysis reactions, promoting further degradation of hemicellulose [28]. At the same time, steam pre-treatment significantly improves the efficiency of hemicellulose hydrolysis, thereby substantially reducing the content of free hydroxyl groups in wood. Steam pre-treatment lowers the equilibrium moisture content (EMC) of wood. For wood with an EMC of less than 15%, physical changes occur during thermal treatment (at temperatures above 100 °C), leading to a reduced ability of moisture to bond with the free hydroxyl groups in hemicellulose. Steam pre-treatment also has a softening effect on the samples, inducing creep and stress relaxation, which helps to effectively suppress cracking.

3.2. Viscoelastic Model Fitting Effect Evaluation

In the realm of rheological materials, the overall creep deformation typically falls into four stages: transient deformation ε₀ occurs when the material is subjected to transient stress; initial creep ε₁ follows as the initial stage of the deformation process, which occurs after a certain period of stress application; isochronous creep ε₂, characterized by a constant creep rate, follows thereafter; and finally, accelerated creep ε₃ leads to failure [18], namely

$$\epsilon ={\epsilon }_0+{\epsilon }_1+{\epsilon }_2+{\epsilon }_3$$

(4)

According to the first three stages of creep deformation, different models have been proposed to describe the wood deformation characteristics. Specifically, the instantaneous deformation ε₀ is expressed using a spring model, while the initial creep deformation characteristics are captured using the Kelvin model. Finally, the steady creep stage of deformation characteristics is described using a dashpot model.

Results of the Burger, Five-parameter, and Six-parameter models for simulating and fitting the bending creep curves of white oak wood specimens under the conditions of a testing temperature of 20 °C and moisture content of 12% for 45 min are presented in

Table 2. The creep fitting parameters of different mathematical models for the specimens.

Model Type	Parameter Value
	E1	E2	η1	η2	A	B	R2
Burger model	119.970	−3.930	−1.030	28,912.960	-	-	0.8624
Five-parameter model	134.100	1264.270	-	3116.350	13,863.250	0.040	0.9996
Six-parameter model	135.280	1646.960	80,775.850	2554.930	77.320	0.090	0.9998

. Comparing the fitting results obtained from different mathematical models reveals that both the Five-parameter model and Six-parameter model demonstrate superior short-term fitting performance compared to the Burger model when applied to white oak wood bending creep analysis (Figure 3). However, due to minimal discernible disparity between the experimental data and fitted curves, an investigation was conducted to analyze discrepancies between viscoelastic models and specimen curve strains in order to distinguish various stages of total creep deformation in wood.

The conventional residual values exhibit significant fluctuations in the time intervals of 0~5 min and 35~45 min, as depicted in Figure 4. Specifically, the fluctuation range of the conventional residuals is observed to be −0.004 to 0 and −0.002 to 0, respectively. However, during the time interval of 5~35 min, a relatively smooth fluctuation amplitude characterizes the conventional residual values with a range from 0 to 0.001. Consequently, for both initial creep and steady creep stages, superior agreement is achieved by employing the Burger model. This is because the Burger model assumes a constant viscous flow deformation rate and uses a linearly increasing functional equation to simulate the viscous flow deformation of wood, resulting in a linear trend of the fitted wood strain curve over time [17]. Therefore, the Burger model simulates the short-term bending creep portion of the wood so that the fitting effect of the third stage is higher than the experimental curve, which is inconsistent with the actual creep behavior of the wood, and it is more appropriate to carry out the short-term stage of the specimen of the creep finite element calculation.

The Five-parameter model, depicted in Figure 5, provides a more accurate representation of wood creep trends. In the time range from 0 to 8 min, conventional residual values exhibit relatively large fluctuations, ranging from −0.0002 to 0.0018. However, in other time ranges, conventional residuals display smoother fluctuations with a range of −0.0001 to 0.0001. It is evident that the Five-parameter model does not fit well during the initial stage and leads to significant deviations in the transient stage. The reason for this is that the nonlinear parameter (B) of the Five-parameter model is introduced into the equation, which results in a nonlinear trend of growth and change during the third stage of creep. This trend aligns more closely with the actual bending creep change observed in wood [29]. Therefore, the Five-parameter model provides a better fit and accurately reflects the short-term creep characteristics of white oak wood, making it suitable for short-term creep finite element analysis.

The Six-parameter model, as depicted in Figure 6, accurately captures the temporal variations across different stages. Within the time interval of 0 to 8 min, conventional residuals exhibit a relatively substantial fluctuation amplitude and range of −0.0001 to 0.0001, respectively. In other time intervals, the fluctuation amplitude of conventional residuals appears smoother with a range between −0.00005 and 0.00005. It is evident that compared to the Five-parameter model, both instantaneous and initial segments are better fitted by the Six-parameter model, which also demonstrates an improved fitting trend at the end segment. Consequently, this Six-parameter model effectively captures wood’s creep characteristics and proves suitable for short-term creep finite element analysis.

The Five-parameter and Six-parameter models have been demonstrated to be effective in simulating the total creep deformation of wood and the changes in each stage. However, in terms of the initial creep stage and the growth trend in the third stage, the Five-parameter model slightly lags behind the Six-parameter model. In summary, from best to worst in simulating the total creep deformation of wood, the optimal choice would be the Six-parameter model as it can accurately capture trends at various stages, closely followed by the Five-parameter model with a higher fitting effect. The Burger model, while fitting well for the initial creep and isotropic creep stages, fails to accurately simulate deformations in the third stage.

3.3. Analysis of Simulation Results

The mathematical model fitting analysis of the 144 sets of data collected from the test yields the following results: the average coefficient of determination (R²) for the Burger model is 0.893, for the Five-parameter model is 0.967, and for the Six-parameter model is 0.999. The results of Burger and the Five-parameter model for simulating the 45 min bending creep curves of the white oak wood specimens are also provided in the Support Information (Tables S1–S8) as a comparative analysis parameter for evaluating the results obtained from fitting the Six-parameter model in this thesis.

The results of the 45 min bending creep curves of the two white oak wood specimens fitted with the Six-parameter model at different test temperatures for the specimens without steam pretreatment, saturated steam treatment at 100 °C, and superheated steam treatment at 120 °C are given in

Table 3. The fitting creep parameters of the Six-parameter model at different test temperatures for samples without steam pretreatment.

Specimen No.	A	B	E1	E2	η1	η2	R2
30 °C-A	54.221	0.102	115.349	928.655	39,660.365	1248.809	0.99991
30 °C-B	58.707	0.103	123.938	950.310	41,619.989	1266.289	0.99992
40 °C-A	40.324	0.094	94.609	926.298	43,039.905	1185.257	0.99995
40 °C-B	39.444	0.090	113.712	906.642	36,264.232	1253.568	0.99996
50 °C-A	25.118	0.091	89.572	615.275	19,937.825	850.497	0.99996
50 °C-B	15.946	0.068	107.352	832.009	21,678.822	1305.139	0.99997
60 °C-A	21.004	0.093	89.298	489.570	11,631.957	682.544	0.99996
60 °C-B	23.906	0.094	90.884	533.276	15,007.150	743.102	0.99995
70 °C-A	18.498	0.097	85.191	369.137	8755.920	530.075	0.99995
70 °C-B	19.588	0.097	86.332	393.385	9702.867	562.958	0.99995
80 °C-A	13.730	0.090	61.824	360.487	7437.382	521.472	0.99997
80 °C-B	14.046	0.091	63.859	356.03101	7422.394	515.261	0.99997

,

Table 4. The fitting creep parameters of the Six-parameter model at different test temperatures for samples treated at 100 °C by saturated steam.

Specimen No.	A	B	E1	E2	η1	η2	R2
30 °C-A1	88.371	0.090	115.656	1394.104	63523.791	2130.707	0.99948
30 °C-B1	54.045	0.094	150.630	1493.337	41425.351	1717.062	0.99995
40 °C-A1	37.787	0.085	106.214	892.708	20468.742	1349.896	0.99992
40 °C-B1	23.613	0.072	141.630	981.412	21318.088	1588.244	0.99994
50 °C-A1	33.626	0.103	102.441	776.486	13250.061	873.656	0.99994
50 °C-B1	22.667	0.077	125.285	889.201	11962.896	1353.927	0.99998
60 °C-A1	22.382	0.083	89.036	689.961	28068.007	886.611	0.99996
60 °C-B1	44.302	0.116	99.485	792.504	21194.137	820.351	0.99993
70 °C-A1	20.235	0.093	69.569	628.445	12563.810	839.819	0.99997
70 °C-B1	12.183	0.077	90.315	425.789	13085.048	680.247	0.99995
80 °C-A1	19.727	0.090	59.222	483.005	8036.453	661.361	0.99997
80 °C-B1	16.643	0.097	82.999	328.618	7326.137	474.600	0.99995

,

Table 5. The fitting creep parameters of the Six-parameter model at different test temperatures for samples pretreated at 110 °C by superheated steam.

Specimen No.	A	B	E1	E2	η1	η2	R2
30 °C-A2	54.349	0.098	218.461	1375.815	218,130.449	1811.668	0.99993
30 °C-B2	100.421	0.105	227.052	1642.013	67,339.245	2586.697	0.99971
40 °C-A2	116.430	0.141	175.692	1665.473	45,243.702	1706.100	0.99990
40 °C-B2	95.095	0.115	208.503	1644.112	66,145.565	1765.508	0.99988
50 °C-A2	31.138	0.089	134.120	958.909	19,786.005	1360.331	0.99996
50 °C-B2	42.137	0.088	153.891	1225.784	31,359.811	1736.174	0.99990
60 °C-A2	20.446	0.084	120.440	635.552	9549.612	1035.023	0.99997
60 °C-B2	15.286	0.066	128.369	739.032	12,217.617	1394.188	0.99996
70 °C-A2	43.110	0.136	92.535	843.326	13,949.481	641.1787	0.99818
70 °C-B2	18.8107	0.075	95.475	695.201	20,426.802	995.035	0.99996
80 °C-A2	20.235	0.093	69.569	628.445	12,563.890	839.819	0.99997
80 °C-B2	27.411	0.102	72.344	533.948	10,154.6310	715.915	0.99996

and

Table 6. The fitting creep parameters of the Six-parameter model at different test temperatures for samples pretreated at 120 °C by superheated steam.

Specimen No.	A	B	E1	E2	η1	η2	R2
30 °C-A3	45.723	0.042	202.249	1926.874	506,716.105	5558.816	0.99970
30 °C-B3	1001.920	0.328	242.935	5349.853	120,988.861	2562.949	0.99994
40 °C-A3	20.206	0.054	149.284	1338.580	16,331.660	2844.861	0.99997
40 °C-B3	42.588	0.098	172.036	994.782	98,701.955	1416.815	0.99987
50 °C-A3	23.363	0.044	151.130	1811.770	1,170,040.000	4670.174	0.99971
50 °C-B3	64.561	0.073	223.5934	2550.546	66,972.513	2975.074	0.99986
60 °C-A3	16.919	0.044	149.722	1479.720	48,032.780	3288.001	0.99993
60 °C-B3	34.025	0.067	209.185	1499.225	25,045.821	2191.718	0.99997
70 °C-A3	16.487	0.048	149.107	1281.165	17,652.804	2895.157	0.99996
70 °C-B3	40.548	0.094	160.328	919.413	41,665.421	1272.908	0.99988
80 °C-A3	16.333	0.084	110.180	541.136	11,931.507	813.383	0.99997
80 °C-B3	18.547	0.090	112.232	587.215	12,777.809	811.412	0.99998

. The coefficient of determination (R²) obtained from simulating the Six-parameter model ranges from 0.99918 to 0.99998, indicating that the model is highly reliable and accurate in describing the creep behavior of white oak wood. Furthermore, Figure 7 and Figure 8 demonstrate that the simulated creep curves closely match with the experimental data, indicating that the Six-parameter model effectively describes the creep behavior of white oak wood specimens with and without earlywood vessel belt within a short period of time.

3.4. Effect of Testing Temperature on Creep Parameters

According to the predictive fitting results obtained from the Six-parameter model of the bending creep test data of white oak specimens in Table 3, Table 4, Table 5 and Table 6 and creep curves in Figure 7 and Figure 8, it can be observed that for specimen A, the elastic module (E₁) decreases with an increase in test temperature. For specimen B, under unstreamed, saturated steam at 100 °C and superheated steam at 110 °C treatments, the elastic module (E₁) also decreases with an increase in test temperature. The ductless tape specimens treated with superheated steam at 120 °C exhibit a decrease in elastic module (E₁) as the test temperature increases within the range of 30~40 °C. However, there is an increase in elastic module (E₁) with increasing test temperature within the range of 40~50 °C, followed by a gradual decrease within the temperature range of 50~80 °C. The high elastic module (E₂) of specimens pretreated with 120 °C superheated steam decreases as the test temperature increases in the range of 30~40 °C, increases in the range of 40~50 °C, and gradually decreases again as the test temperature continues to increase in the range of 50~80 °C. For other steam pretreatment conditions, the high elastic module (E₂) of specimens also decreases with increasing test temperature. The viscosity coefficient (η₁) gradually decreases as a whole with the increase in test temperature, indicating that elevated temperatures promote the transition of wood from an elastic state to a viscous state. Meanwhile, the bulk viscosity (η₂) of specimens also gradually decreases with increasing test temperature, while the viscosity coefficient increases, suggesting that higher temperatures induce a transition in wood from an elastic state to a viscous state. The nonlinear coefficients (B) obtained from the Six-parameter model fitting varied with testing temperature but remained stable within a specific range of 0.03 to 0.30, showing minor fluctuations. Additionally, different specimens exhibited varying nonlinear coefficients under different testing temperature conditions.

The following is an analysis of the reasons behind the findings: Firstly, the activation energy generated by the moving units in wood increases with an increase in the test temperature, leading to a decrease in the corresponding intermolecular interaction force. This in turn results in an increase in the intermolecular distance, which expands the active space of the moving units [30,31]. Secondly, an increase in test temperature leads to an expansion of the molecular occupation volume. Simultaneously, the temperature rise also induces an expansion of free volume within the lignin, leading to alterations in the lattice spacing of cellulose [32,33]. Thirdly, under a constant applied load of 5 MPa, the instantaneous elastic deformation, delayed elastic deformation, viscoelastic coefficient, and viscosity coefficient of the specimens increased as the test temperature increased. This indicates that the elastic module (E₁), high elastic module (E₂), viscous hysteresis coefficient (η₁), and bulk viscosity (η₂) decrease with increasing test temperature.

3.5. Effect of Steam Pretreatment on Creep Parameters

As shown in Table 3, Table 4, Table 5 and Table 6, steam pretreatment conditions at different temperatures have different degrees of influence on the common elastic modulus (E₁), the high modulus of elastic (E₂), the viscosity coefficient (η₁), and bulk viscosity (η₂) of wood samples. The elastic module (E₁₎ of samples pretreated with saturated steam at 100 °C was slightly higher than that of untreated specimens without steam pretreatment. Additionally, the elastic module (E₁) of specimens treated with superheated steam at both 110 °C and 120 °C was higher than that of untreated specimens without steam pretreatment. The high module of elasticity (E₂) showed no significant deviation from that of the non-steam-pretreated samples in the samples treated with saturated steam at 100 °C, while it increased in the superheated steam-treated samples at 110 °C and 120 °C compared to the non-steam-pretreated samples. The viscosity coefficient (η₁) of specimens without steam pretreatment exhibited an overall increasing pattern with the rise in steam temperature, while the bulk viscosity (η₂) of specimens did not show a clear trend. This can be attributed to hemicellulose, which is one of the main components in wood composition and contains numerous hydrophilic groups that contribute to its strong water absorption [34]. This water absorption is one of the factors causing strain in wood [32]. After undergoing pressure steam treatment, the hydrolysis of hemicellulose under acidic conditions reduces the number of free hydroxyl groups in the wood, thereby decreasing its EMC. Wood undergoes physical changes during heat treatment temperatures, resulting in a decrease in the ability of water to bind to free hydroxyl groups in hemicellulose [31]. Upon exposure to heat, a fraction of the polysaccharides presents in hemicellulose undergo cleavage, yielding aldehydes and sugars. Subsequently, these compounds engage in polymerization reactions leading to the formation of water-insoluble polymers. As a result, this process results in reduced water absorption capacity and enhanced dimensional stability of wood [35,36]. The studies have demonstrated that subjecting wood to saturated wet air or atmospheric pressure-saturated steam treatment at the appropriate temperature can effectively reduce the disparity in shrinkage coefficient and anisotropy. The studies also show that steam pretreatment induces creep and stress relaxation, thereby significantly inhibiting cracking [37]. Therefore, the elastic module (E₁), high elastic module (E₂), and viscous hysteresis coefficient (η₁) are larger in the steam-pretreated specimens compared to the non-steam-treated ones.

3.6. Effect of Earlywood Vessel Belt on Creep Parameters

As shown in Table 3, Table 4, Table 5 and Table 6 and Figure 7 and Figure 8, the earlywood vessel belt had different effects on the creep behavior of the wood samples at different test temperatures. Under identical conditions of steam pretreatment temperature and test temperature, specimens with earlywood vessel belts exhibit a smaller universal elastic module (E₁) compared to those without them as a whole. However, no clear pattern is observed for the high elasticity module (E₂) obtained from the Six-parameter model fitting. Under the conditions of no steam pretreatment, treatment with saturated steam at 100 °C and superheated steam at 110 °C, the bulk viscosity (η₂) of specimens containing an earlywood vessel belt was smaller than that of earlywood specimens without an earlywood vessel belt in the test temperature range of 40~50 °C. However, under the condition of treatment with superheated steam at 120 °C, the bulk viscosity (η₂) of specimens containing an earlywood vessel belt was higher than that of earlywood specimens without an earlywood vessel belt. This is because the vessel cell, as an important constituent cell type of broadleaf timber, plays a role in reducing stress concentration during bending and under stress. Therefore, the existence of earlywood vessel belt will weaken the resulting stress concentration effect, leading to a reduction in the creep behavior of white oak wood specimens containing earlywood vessel belt. In the test temperature of 80 °C, the weakening of the influence of temperature on the creep behavior of white oak wood specimens containing earlywood vessel belt is more obvious. This is because, as described in the problem of stress concentration in holes or circular holes in electrodynamics, under applied loads, the edges of the hole structure of an elastic material produce a stress concentration effect, and the stresses at the edges of the holes will be much greater than in the absence of holes [23]. The density of the specimen containing the earlywood vessel belt (0.726 g/cm³) is lower than that of the control specimen (0.734 g/cm³), resulting in a reduction in mechanical strength and an increase in strain. Consequently, the presence of the earlywood vessel belt leads to enhanced bending creep, instantaneous elastic deformation, and viscosity coefficient, ultimately causing a decrease in the elastic module (E₁) and bulk viscosity (η₂).

4. Conclusions

In this study, different viscoelastic mathematical models (i.e., the Burger model, Five-parameter model, and Six-parameter model) were employed to conduct short-term simulations and comparative analyses of the bending creep curve of white oak (Quercus alba L.) wood under the synergistic effects of different testing temperatures, steam pretreatment temperatures, and earlywood vessel belts. The advantages and disadvantages of different viscoelastic mathematical models in fitting the bending creep curve of white oak wood were analyzed. Experimental results showed that the Burger model, Five-parameter model, and Six-parameter model exhibited superior predictive capabilities for the bending creep behavior of white oak wood. In terms of fitting effectiveness, they ranked in the following order: Six-parameter model > Five-parameter model > Burger model. The mean coefficient of determination (R²) for the Six-parameter model was 0.999, for the Five-parameter model was 0.966, and for the Burger model was 0.893. For the Six-parameter model, conventional residuals fluctuated within a range of −0.0001~0.0001 during 0~8 min. However, in other time ranges, they displayed a more moderate fluctuation amplitude ranging from −0.00005 to 0.00005, respectively. Concerning the fluctuation amplitude of conventional residual values among these three models, it can be concluded that their magnitudes follow this order from largest to smallest: Six-parameter model > Five-parameter model > Burger model. Therefore, the Six-parameter model simulation predicts that the creep behavior of white oak wood is more in line with the actual creep process, which can reflect the short-term deformation characteristics of wood and is suitable for wood creep finite element analysis. Therefore, the Six-parameter model is the optimized suitable model for prediction of the creep behavior of steam-pretreated white oak wood.