Viscoelasticity of wood cell walls with different moisture content as measured by nanoindentation

Abstract

Instrumental indentation was performed on wood cell walls to measure the cell walls’ mechanical properties and creep behaviour over a range of various steady moisture conditions. The wood cell walls’ Young’s modulus and hardness were found to negatively correlate with moisture content. Theoretical modeling with Burgers model and finite element analysis with the Generalized Maxwell model were used to simulate the process of nanoindentation separately and to gain insight into the response of viscoelastic materials. The Young’s modulus extracted using Burgers model was lower than the values obtained using the Generalized Maxwell model and Oliver–Pharr method, indicating that creep happened during the indentation loading process. The extracted parameters from both models exhibited negative trends with increasing moisture content. Both models confirmed an intuitive understanding of creep in viscoelastic wood cell walls, which reflects previous basic information about the composite structure of wood cell walls and their rheological properties.

---

Introduction

Nanoindentation has been demonstrated to be an effective analytical technique for measuring the small scale mechanical properties of biomass, with materials widely ranging from grass,^1–3 to crop stalks,^4,5 wood,^6–8 bamboo,⁹ Lyocell fiber,¹⁰ lignocelluloses¹¹ and so forth. Generally, the Young’s modulus and hardness values obtained using commercial instruments are extracted using the approach of the Oliver–Pharr method, which is based on the assumption that the unloading curve is pure elastic.¹² In cases where materials have time-dependent behaviour (i.e. viscoelasticity), rather than elastic-plastic as in Oliver and Pharr’s paper, creep occurs during nanoindentation unloading where the continuous increase of displacement at the onset of unloading and the value of Young’s modulus will be lower than the real instantaneous one (t = 0 s). This has been ascribed to internal friction stress opposing slip.¹³ An extreme case of a viscoelastic material’s response is the presence of a “nose” shape on unloading curves, resulting in either an overestimation of Young’s modulus or negative contact stiffness as measured by Oliver and Pharr’s method. Common ways to overcome the effect of creep include allowing sufficient creep time and removing the creep effect on contact stiffness and contact area during data analysis.

Viscoelasticity can be described as a combination of viscous components and simultaneously elastic behaviour. Attempts have been made using several models to link nanoindentation test data to meaningful mechanical properties of viscoelastic materials.¹⁴ Radok et al. developed an analytical solution for a viscoelastic model based on a principle between elasticity and linear viscoelasticity, but limited the validity by positing a constant contact area.¹⁵ Cheng et al. have employed a three-element analytical model of flat-ended punch indentation to describe a polymer’s viscoelastic properties.¹⁶ Ngan et al. used a linear viscoelasticity model to uncover the details of the unloading portion of an indentation curve where a nose appears.¹³ Shortly after, they proposed a creep correction procedure based on power-law to successfully calculate the contact stiffness from commercial nanoindentation data, and they validated the formula by finite element analysis (FEM) simulation and by experiment.¹⁷ Schiffmann summarized and compared four rheology models using nanoindentation creep and stress relaxation tests to describe the viscoelastic behaviour of polycarbonate.¹⁸ Liu et al. developed a model based on the Burgers viscoelastic concept and made a correlation between the nanoindentation data and viscoelastic behaviour of a polymer material.¹⁹ Oyen et al. expanded a viscous-elastic-plastic (VEP) model to predict the indentation response of time-dependent materials by incorporating viscous, elastic, and plastic elements.²⁰ Liu et al. investigated the displacement of the tip during loading and the onset of unloading by indentation performed on a polymer material, showing that the derived solution consisted of elastic, viscous and plastic deformation.¹⁹

Wood is a complex composite with many components. It displays viscoelastic behaviour and also has water-sensitive properties. The mechanical and creep behaviour of wood subjected to diverse relative humidity environments has been widely studied experimentally and theoretically at the macro scale.^21–24 Mukudai proposed a viscoelastic bending model to interpret the characteristics of viscoelastic wood under moisture change cycles based on a hypothesis that a slippage exists between the S₁ and S₂ layers when wood is in the process of drying.²⁵ Padanyi predicted the difference between moisture-accelerated creep and constant relative humidity creep, a difference that depends on the amount of free volume in the fiber. It is now clear that moisture content has a negative effect on the stiffness and strength of bulk wood and wood cell walls below a saturated moisture content.²⁶ On the specific question of how the three main components in the cell wall respond to varying moisture content, a number of reports have demonstrated that hemicelluloses are more sensitive to moisture content than celluloses and lignin.²⁷ However, studies of how moisture content affects the viscoelastic properties of wood cell walls at the submicron-scale have been limited. In the current study, we performed nanoindentation tests on the S₂ layer of a wood cell wall and calculated the creep compliance of the wood cell wall subjected to various moisture content. The viscoelastic properties of the wood cell wall in various relative humidity conditions were investigated based on the Burgers model and the Generalized Maxwell viscoelastic concept. The parameters presented in the Burgers model are discussed. Finite element analysis (FEA) was conducted based on the Generalized Maxwell model and correlated with the nanoindentation experiments. As an extension of the experiments, the stress evolution due to the visco effect was obtained via FEA.

Experimental procedures

Materials

Loblolly pine (Pinus taeda L.) was obtained from southern Arkansas and Tennessee. A loblolly pine disk was cut 30 cm above the ground from a tree. The 32^nd annual ring was picked with the late wood located in the apex of the block. The microfibril angle was measured to be 31 degrees. Two sample preparation methods, with and without epoxy resin, were used in this research. The sample without embedding medium was used to investigate the cell wall swelling. A wood sample was cut into a block 8 mm in length, 8 mm in width, and 15 mm (ref. 28) in height. Four pieces of wood block with late wood located on the same annual ring were picked and cut into small blocks 1 mm in width, 1 mm in thickness, and 5 mm in length. Samples were embedded into epoxy resin without the influence of resin penetration into the wood cell wall. The detailed sample preparation procedure has been discussed in our previous research.²⁹ In brief, a cross-section of the wood cell wall was polished using an ultramicrotome equipped with a diamond knife. A smooth and well-polished surface with a roughness of less than 0.5 nm was obtained to ensure good contact. Samples were ready for performing indentation after being fixed to an aluminium mounting holder.³⁰ Industrial-grade glycerin solution (ASTM® D6584 Glycerin solution, Sigma Aldrich, MO, USA) and desiccant were used to control the relative humidity inside the chamber.

Control of humidity in the small environment chamber

Inside the indentation chamber, the relative humidity to which samples needed to be subjected was maintained constantly and adjusted by using various aqueous glycerin solutions. All of the glycerin solutions were maintained at a constant laboratory temperature of 23 ± 2 °C. The samples were stored in a pre-sealed TriboIndenter chamber and exposed to the environment 24 hours before the indentation test. The moisture content in the wood is ordinarily expressed as a fraction of the water mass and the mass of the oven dry wood.³¹ Oven-dry weight (samples were dried in an oven set at 103 ± 2 °C) and weight under each relative humidity (RH) were used to determine the percentage of moisture content according to eqn (1):

MC = (G_g − G_od)/G_od × 100% (1)

where G_g is the green mass of wood and G_od is the oven-dry mass.

For indentations performed in water, each wood sample was stored in a specially designed aluminum holder filled with water throughout the testing. For each wood sample, nanoindentation tests were finished within two hours to minimize the variation in the structure/mechanical properties. The relative humidity was monitored using a hygrometer. The specimens, denoted as A, B, C and D were conditioned in an oven, 18% RH, 37% RH and in water, separately. The corresponding moisture content of each of the samples were calculated to be 0%, 6%, 18% and 110%.

Measurement of wood cell wall hardness and elastic modulus by nanoindentation

Nanoindentation tests were performed using a Hysitron TriboIndenter system (Hysitron, Inc., Minneapolis, MN) equipped with a Berkovich indenter, which is a three-sided pyramidal point with a known area-to-depth function.¹² The fluid cell tip, which is a standard Berkovich tip mounted on a long shaft, was used throughout the test. The extended shaft, approximately 4 mm in length, allowed the end of the probe to be completely immersed in a liquid, while the probe holder and transducer remained in the air. When underwater indentation was performed, the tip approached the surface of the targeted cell wall with the sample partially immersed in the water. Water was added after this operation to fully cover the cell wall. A closed-loop feedback control was selected to accurately identify the sample surface, which is very important for shallow indentations. At each relative humidity, at least 25 tests were conducted with the fluid cell tip in load control and using a constant loading rate at room temperature. The trapezoidal load function was set up at maximum load P_max = 250 μN, with loading, holding and unloading times of 5 s, 5 s and 2 s, respectively. The holding times were set up, firstly, to avoid the effect of creep occurring in the viscous material during the unloading;¹⁹ and secondly to investigate the viscoelastic properties of the cell wall.

The configuration of the scanning probe microscopy (SPM) assembly in the TriboIndenter system makes it capable of accurately positioning the wood cell wall’s S₂ layer. By scanning the cell wall’s cross-section surface, interesting indent positions were marked on the SPM image. Indents were implemented and checked before and after scanning. Only indents in the middle of the cell wall’s S₂ layer were selected as valid data. Indents performed in the embedding epoxy or in the borders of cell walls were discarded. The load-displace curves that were obtained were used to calculate the reduced modulus E_r and hardness H based on Oliver and Pharr’s method (eqn (2) and (3)). The reduced modulus E_r was obtained from the unloading slope dP/dh evaluated at 70–90% of the maximum load P_max; the hardness H was calculated from the maximum load and project contact area A_hc at peak load.¹²

where P is the indentation load and A_hc is the projected contact area.

Rheological models

Viscoelastic material during nanoindentation. Viscoelastic material exhibits both viscous and elastic behaviour. For the linear elastic properties, the material reacts as a spring and deformation will recover to its original dimensions when stress is removed. This can be described by Hooke’s law:

σ = Eε (4)

where E is the elastic modulus in the nanoindentation case. The deformation is not a pure shear deformation but a complex superposed normal and shear deformation. However, the material retains the relationship between σ and ε no matter the type of deformation. On the other side, the viscous component is modeled as a dashpot, which is analogous to the energy dissipation. The stress and strain rate relationship can be expressed as follows:

σ = ηdε/dt (5)

in which η is the viscosity of a material and dε/dt is the time derivative of the strain.

For a creep test, the stress is held constant, σ = σ₀, so the creep compliance J(t) is defined by the following:

J(t) = ε(t)/σ₀ (6)

The load P, contact area A and indentation depth h are three important parameters in nanoindentation. The tip used in all of the experiments was a pyramidal indenter (Berkovich tip). Kirsten Ingolf Schiffmann¹⁸ has mentioned in his research that the representative stress is given by σ = P/A, and that the representative strain is defined by the following:

thus the creep compliance is given by:where, c = 2(1 − ν²)tan δ , ν is the Poisson ratio, and δ is the half opening angle of the indenter (70° in Berkovich). Load P = P(0) was able to be obtained directly by the instrument. However, the contact area A(t) had to be calculated from the area function of the tip, which is described by a polynomial:

A = C₀h_c² + C₁h_c + C₂h_c^1/2 + C₃h_c^1/4 + C₄h_c^1/8 + C₅h_c^1/16 (9)

where C_i is the tip constant and h_c is the contact depth, which can be related to the maximum indentation depth h_max in Oliver and Pharr’s method by the following:where α is the geometrical constant and S is the contact stiffness, which can be derived from dP/dh.

In the creep test, the holding load is constant and the contact area changes with time. Thus, h_c is given by

where h(t) is the actual indentation depth during the holding period and S_f is the final stiffness calculated from the unloading segment.

Burgers model and Generalized Maxwell model. For different viscoelastic materials, various models, including the Maxwell model and the Kelvin–Voigt model, were developed to predict the deformation of different viscoelastic materials.¹⁸ In our analysis, both the Burgers model and the Generalized Maxwell model were applied to rationalize the experimental results and to investigate the wood cell wall with different humidities separately.

With reference to the Burgers model, a combination of a Maxwell and a Kelvin element were taken into consideration with the schematic diagram shown in Fig. 1(a). The Maxwell element is represented by a purely viscous damper and a purely elastic spring connected in series. The Kevin element is a purely viscous damper and a purely elastic spring connected in parallel. The relationship between stress σ and the deformation ε of the viscoelastic material is given by the following:

σ + p₁[small sigma, Greek, dot above] + p₂[greek sigma with two dots above] = q₁[small epsi, Greek, dot above] + q₂[greek epsilon with two dots above] (12)

where [small sigma, Greek, dot above], [greek sigma with two dots above], [small epsi, Greek, dot above], [greek epsilon with two dots above] are defined as the first and second time derivatives of stress and deformation and can be expressed aswhere E^B_e, E^B_d are the spring constants and η₁, η₂ are viscosities in Fig. 1(a).

Fig. 1 Schematic images of (a) the four-element Burgers model and (b) the Generalized Maxwell Model.

The stress σ is constant in the case of the creep test, so the first and second derivatives of stress are zero. Consequently, eqn (12) can be constructed into

σ = q₁[small epsi, Greek, dot above] + q₂[greek epsilon with two dots above] (14)

By applying this equation to eqn (6), the creep compliance formula, a solution can be written as the form of creep compliance:

J(t) = J₀ + J₁t + J₂[1 − exp(−t/τ^B₀)] (15)

where J₀ = 1/E^B_e, J₁ = 1/η^B₁, J₂ = 1/E^B_d and τ^B₀ = η^B₂/E^B_d. Here τ^B₀ is the retardation time, which describes the retarded elastic deformation of the Kelvin model in the Burgers model, and it is an important parameter that reflects the viscoelastic properties of a material.

The Generalized Maxwell model is another of the most general forms of linear models of viscoelasticity. With n + 1 constituent elements in parallel, n Maxwell models, and one isolated spring element to warrant solid behaviour in parallel as shown in Fig. 1(b), the combination can simulate the viscoelastic properties of many different materials.³²

It is clear that the shear modulus G(t) decreases with time as follows:

G(t) = g(t)RG₀ (16)

where g_R(t) is the time-dependent dimensionless relaxation modulus, only related to time t, and G₀ is the instantaneous shear modulus. The dimensionless relaxation modulus can be expressed by an N-terms Prony series expansion:in which τ_i^G is the retardation time for the i^th dashpot and g_i^G is the material constant that represents the influence of the i^th Maxwell element. For early times (t ≪ τ_i), g_R(t) = 1, and the shear modulus is G₀ which means that all of the dashpots are frozen and cannot move. After that, g_R(t) is gradually decreased and finally reaches the long-term shear relaxation modulus G_∞. For more details of the Generalized Maxwell model, we refer the reader to the Abaqus documentation collection.³³

Results and discussion

Effects of the moisture content on the elastic modulus and hardness

The nanoindentation load–displacement curves (P–h curves) for samples A, B, C and D, with the same loading and unloading rates and with constant holding times at peak load, are plotted in Fig. 2(a), which includes a comparison of the wood cell walls with different moisture content. Obvious differences among the indentation responses of the four samples include the displacements during the loading and holding segments. When the moisture content was high, the curves shifted from the left-hand to the right-hand side and caused an increase in the penetration depths at the end of loading, residual displacement, and displacement during the constant holding segment. The atomic force microscope images in Fig. 2(b) illustrate the residual indentation marks after unloading. There was no distinct size change in the residual marks between samples A and B. However, obviously larger indent marks were clearly seen on sample C and vaguely identified on sample D under wet conditions.

Fig. 2 (a) Typical indentation curves with fluid cell indenter with different moisture content with a peak load of 250 μN and 5 s holding time at maximum load for creep, and (b) atomic force microscope images showing the residual indentation marks on the cell walls. In D, the triangles show residual indentation marks.

Fig. 3 illustrates the relationship between moisture content and elastic modulus/hardness in the longitudinal direction of the wood cell wall. The trends highlighted by these two figures show that moisture content has a negative effect on both the elastic modulus and hardness; this can be inferred from the box-and-whisker plots in Fig. 3. The samples have more variation in elastic modulus than they do in hardness. All pairwise comparisons of the mean values of the wood cell walls’ elastic modulus and hardness with moisture content were conducted using the Student’s t-test. Statistical analysis (connecting letters report shown in the insets of Fig. 3(a) and (b)) using the Student’s t-test further revealed that significant differences existed between the different conditions. The percentage reduction of the modulus from the oven-dried state to the 18% moisture content was 32.4%. The percentage reduction of hardness from the oven-dried state to the 18% moisture content state was 46.7%. These trends agree well with Kojima’s results, which illustrated the phenomenon of moisture content dependence and were verified by simulation experiments.³⁴

Fig. 3 Effect of moisture content on (a) elastic modulus and (b) hardness of wood cell walls. The insets in the top right of each graph show the connecting letter report for Student’s t comparison between each pair. Levels not connected by the same letter are significantly different.

The effects of moisture content on the nanoindentation displacement rate were investigated and the results are presented in Fig. 4. The nanoindentation displacement rate could be described by the equation:

ḣ = dh/dt

where h is the instantaneous indenter displacement and t is the time.³⁵ We plotted the data separately for two regions of loading and holding. In the indentation’s rising region, the displacement rate diminished significantly for each moisture content with the same amount of applied force and time. As the moisture content increased, the indentation displacement rate underwent a rapid increase as well, resulting in a large number. It is also evident from Fig. 4(b) that the displacement rates for each moisture content had the tendency to become constant after 6 seconds. These observations indicate that high moisture content leads to a softening of the wood cell wall, which consequently weakens the mechanical properties of the cell wall.

Fig. 4 Indentation creep rate as a function of time with different moisture content under nanoindentation with a Berkovich indenter (a) during indentation for 5 seconds; (b) during indentation creep under a fixed load of 250 μN.

Cell wall swelling

An interference microscope with a dimension readout function in the SPIP system was used to investigate the cell walls’ swelling behaviour as affected by water. The cell walls’ line profile and the 3D surface topography are plotted in Fig. 5. An obvious dimension change (1.7143 μm in thickness difference) was detected between the air-dry sample and the wet sample. The amount of amorphous area and the number of dissociative hydroxyl groups dominated the cell walls’ water absorption ability.³⁶ It has been demonstrated that large amounts of hydroxyls and other oxygen-containing groups exist in amorphous cellulose, and that hemicellulose and lignin have the ability to attract water.³⁷ When the water is introduced, dissociative hydroxyl groups in the amorphous area start to attract water molecules. The original hydrogen bonds existing in the amorphous region of the wood constituents are broken, causing the formation of new hydrogen bonds between water molecules and hydroxyl groups, increasing the distance between the molecule chains, and consequently leading to the expansion of the cell wall in the transverse direction. The swelling of the wood cell wall leads to a decrease in the cellulose microfibril numbers per unit area underneath the indentation; consequently, the stiffness of the wood cell wall is reduced.

Fig. 5 Line profile and optical micrograph of air-dried and wet loblolly pine cell walls using an interference surface mapping microscope.

The wood cell walls’ ultrastructure can be described as a complicated fiber-reinforced composite, wherein hierarchically oriented microfibrils are surrounded by a matrix of hemicellulose and lignin. Typically, in the S₂ layer, the thickest part of the secondary cell wall layer where these nanoindentations were performed, it has been argued that cellulose microfibrils aggregate into fibril bundles and are mainly surrounded by amorphous hemicellulose, xylan, glucomannan and lignin. However, it is evident that only the hemicellulose has a pronounced effect on the mechanical properties of moist wood.³⁸ Hemicellulose has the strongest water absorption ability in the cell wall due to its structure and distribution in the natural cell wall. It is now clear that hemicellulose in the S₂ layer continuously surrounds the cellulose microfibril, preferentially oriented along the cell wall axis. In addition, the hemicellulose has a random, amorphous structure with little strength. The negative linear relationship between moisture content and the glass transition temperature (T_g) of hemicellulose indicates that high moisture content (moisture content = 20%) dramatically drags the T_g down to room temperature, in contrast to the high T_g of 150 to 220 °C under dry conditions.³⁹ In the case of wet conditions where the cell wall is subject to varying moisture content, numerous spherical water molecules are initially stored in the expanded volume caused by the cell wall’s swelling. When nanoindentation is performed, the aforementioned water molecules existing in the wood cell wall underneath the indenter make it easier to push the indenter into the material. For this reason, water acts as a type of plasticizer in the wood cell wall by softening the hemicellulose and the amorphous cellulose.

Effect of moisture content on creep behavior via Burgers model

The viscoelastic behaviour of wood cell walls was investigated by fitting the calculated creep compliance to the viscoelastic Burgers model. The benefit of this model is that fit parameters can be interpreted physically as an instantaneous elastic deformation, followed by a viscoelastic deformation and a viscous deformation. Based on Burgers model, ideally, the stress–strain–time behaviour of a material is modelled when the material is subjected to a constant stress over time plus an initial strain. Hence, the investigation was restricted to 5 s of creep and the parameters were determined by means of the indentation holding data. Fig. 6 shows the experimental creep compliance data for the wood cell walls with varying moisture content, together with Burgers model predicted creep compliance, shown as red lines. The comparative study of viscoelastic behaviour shows excellent agreement, with a high correlation coefficient of 0.99, indicating that Burgers model is appropriate for predicting the wood cell walls’ viscoelastic behaviour. The creep compliance at the onset of unloading increased from 0.265 GPa⁻¹ for the oven-dried condition to 1.12 GPa⁻¹ under water. The creep compliance percentages increased during the 5 s; creep compliance percentage is defined as the compliance differences between the beginning and end of the holding period, divided by the creep compliance at the beginning of holding. These were compared among the four samples. The results showed that the creep compliance percentages increased by 9.0%, 12.1%, 17.8%, and 17.0% for samples A, B, C, and D, respectively. In other words, the samples with higher moisture content deformed more when subjected to the constant load.

Fig. 6 Creep compliance J(t) and Burgers model fit (red line) of wood cell walls, (a) sample A with a moisture content of 0%, (b) sample B with a moisture content of 6%, (c) sample C with a moisture content of 18% and (d) sample D in water.

In a polymer chain, the induction of moisture content tends to increase the number of water molecules at the interface between amorphous hemicellulose and crystal cellulose and to enlarge the inter-chain distance, which weakens the structural continuity and reduces the resistance of the wood cell wall to creep.

Simulated results using the Burgers model

The viscoelastic behaviour of a wood cell wall in terms of stress and strain can be described by the four-parameter Burgers model by means of the observed J(t).

The parameters in the Burgers model include E^B_e, E^B_d, η^B₁, η^B₂ and τ^B₀: modulus of elasticity, modulus of viscoelasticity, coefficients of plasticity, viscoelasticity, and retardation time, respectively, which were obtained by fitting the creep compliance curve to eqn (15). The effects of moisture content on the viscoelastic behaviour in terms of the fitting parameters are plotted in Fig. 7. Apparently, the viscoelastic behaviour is sensitive to the moisture content. It was observed that in the moisture range of the current study, the scale of the parameters E^B_e, E^B_d, η^B₁ and η^B₂ coincided almost linearly with moisture. Wood cell walls immersed in water were found to have the lowest values for the four parameters fitted from Burgers model of any of the cell walls under varying moisture content conditions. A similar trend has been observed in the tensile creep behaviour of wood.²²

Fig. 7 Effect of moisture content on the fitted parameters from the Burgers model; (a) spring constant E_e, (b) spring constant E_d, (c) viscosity η₁ and (d) viscosity η₂; note: the insets in the upper right corners show letter reports for the Student’s t comparison between each pair. Levels not connected by the same letter are significantly different.

The Young’s modulus can be converted from the reciprocal start value of the creep compliance using E = 2(1 + ν)/J(0). The estimated Young’s moduli from the Burgers model for the various moisture content values are summarized in Table 1. These values are smaller than the Young’s modulus obtained from Oliver and Pharr’s standard method. This is to be expected, since, according to the solution in eqn (8), creep compliance is directly proportional to the contact area, which is related to the square of displacement at the onset of holding. In fact, the 5 s loading during nanoindentation induced a large starting displacement, consequently shifting the creep compliance curves up and leading to a decrease in the Young’s modulus. In spite of the value decreasing, the effect of moisture content was still pronounced and consistent with the conclusions from Oliver and Pharr’s calculation.

Table 1 Comparison of Young's modulus calculated from different methods^a

Moisture content (%)	Creep compliance (GPa^−1)	E* from the Burgers model (GPa)	E from the O–P method (GPa)	E from the Generalized Maxwell model (GPa)
a E*: Young's modulus calculated from the Burgers model is based on a holding segment when t = 5–10 s.
0	0.25	10.5	13.9	14.4
6	0.31	8.3	12.4	11.5
18	0.4	6.3	9.4	9
Water	0.91	2.9	4.5	N/A

---

Retardation time spectrum

The creep behaviour of a viscoelastic material during a creep test exhibits a creep compliance that can be described in terms of continuous retardation multiplied by the spectrum L(τ). Accordingly, the retardation time spectrum resulting from the creep compliance is given by:

The details of this formula’s derivation are described in ref. 18. The curve plotted using eqn (18) can be directly compared with experimental data as shown in Fig. 8. The result clearly shows that the Burgers model gives a good approximation of the experimental time spectrum, with one single peak predicted at the first second of holding, although a small variation exists. In fact, the time at the single peak represents the retardation time τ₀, which is associated with the parallel spring-dashpot system in the Burgers model. The Burgers model yields average retardation times of 1.36 s, 1.21 s, 1.09 s and 1.12 s for each moisture content, ranging from the oven-dry state to the in-water state. A negative trend for the retardation time was observed for wood cell walls that were subjected to the raised moisture content in air. This observation is consistent with the negative trend obtained from the longitudinal tensile creep tests in other research.²² In addition, the values seem to keep decreasing until they approach a plateau value around 1.1 s, where water starts to dominate the viscoelastic property.

Fig. 8 Retardation time spectrum calculated from numerical differentiation of the creep compliance overlaid with the time spectrum derived from the analytical expression of the Burgers model.

Finite elemental analysis

In order to further interpret the test results from the nanoindentation that was presented in the previous section and in order to establish a relationship between the wood cell wall properties and the indentation response, finite element analysis (FEA) was carried out using a rigid body Berkovich indenter with a cone semi-angle at 70.3°. The boundary value problem was resolved with the commercial finite element package ABAQUS 6.10 (3DS SIMULIA).³³ An axisymmetric model was used in the simulation due to the load condition, the boundary condition, geometry features, and constitutive behaviour of wood, as shown in Fig. 9. The area close to the contact area was finely meshed and the fine mesh was migrated to a coarse one. Contact between the indenter and the wood sample was assumed to be frictionless. Following the real experiment in the lab, the deformation had two successive steps: applying 250 μN loads for 5 s and holding the load for 5 s to fit the P–h curve and to obtain the material parameters of the wood samples at different moisture contents and evaluate the stress. The bottom of the wood sample was fixed (i.e. u_i = 0, i = 1, 2…6). Considering that the size of the cell wall is significantly larger than the size of the indenter contact area in the experiment, the deformation was quite localized and the boundary effect was negligible. As a result, the depth of the sample in the simulation was more than 10 times the displacement of the indenter, in order to eliminate the boundary effects. A load along the symmetrical axis was applied on the indenter and the remaining five degrees of freedom were fixed. The significant advantage of the axisymmetric analysis was that the computational cost was substantially reduced without losing accuracy, compared with a full three-dimensional analysis.

Fig. 9 Mesh pattern, boundary condition and load condition used for axisymmetric finite element modeling of the nanoindentation process.

To deal with the previously stated significant size difference between the indenter and sample, the wood cell wall in the simulation was considered to be an isotropic viscoelastic according to the Generalized Maxwell model. For woodmaterials, there are two different relaxation behaviours that have been observed in experiments: a quick relax mode (low frequency) and a slow relax mode (high frequency).⁴⁰ Therefore, the dimensionless relaxation modulus in eqn (17) can be expressed as:

For simplicity, following suggestions in previous literature, it was assumed that the two modes have almost the same influence on the decay of the shear modulus (i.e. g₁ = g₂ = 0.5).⁴¹ For this reason, finite element analysis was used to determine three parameters, G₀ (or E₀ for E₀ = 2G(1 + ν)), τ₁ and τ₂, which link the indentation response parameter with the material properties, by fitting the P–h curves. In fact, only the P–h curves of samples A, B and C were successfully fitted. Fig. 2 shows that sample D’s (in-water) P–h curve shape exhibits significant differences to the other three curves, with a nearly linear loading curve. This implies that a wood cell wall immersed in water behaves more like a fluid than a solid. Despite the fact that the Generalized Maxwell model contains dashpots, which is the mechanical analog for a Newtonian fluid, the whole model is based on the assumption of solid mechanics. Consequently, the model fails to catch the essence of the pure fluid and was found to be inappropriate for the case of sample D.

The parameters obtained via simulations are listed in Table 2. It is of interest to note that the instantaneous elastic module was dependent on the moisture content. The values decreased from 14.4 GPa to 9 GPa with an increase in the moisture content. A similar trend was observed for the relaxation time (both τ₁ and τ₂). It can be speculated that the wood cell wall with low moisture content is harder in the loading region but has a better ability to resist the further deformation in the holding region, which is consistent with experimental observation. Moreover, the calculated values from FEA are consistent with the ones from Oliver and Pharr’s method.

Table 2 Summary of the fitting parameters from the Generalized Maxwell model using FEA

Moisture content (%)	E0 (GPa)	τ1 (s)	τ2 (s)
0%	14.4	0.13	45
6%	11.5	0.05	30
18%	9	0.01	16

---

To further correlate the moisture content with the cell wall’s ability to creep, the stress evolution during the time at constant load was investigated. Von Mises stress distributions at t = 10 s, normalized by the maximum of von Mises stress at t = 5 s, are presented in Fig. 10 and compare the three conditions in Fig. 10(a)–(c). It can be easily observed that the decay of stress was greatest in sample C with a value of 80.93%, followed by sample B with a value of 88.5%, and sample A with a value of 92.02%. This trend implies that the dry sample has better creep resistance, which agrees with our previous experimental analysis.

Fig. 10 Normalized stress distributions at the onset of unloading (t = 10 s) for (a) dry wood, (b) 6% moisture content, and (c) 18% moisture content (the axisymmetrical results are swept into full three-dimensional contours).

Conclusions

Water is considered to act as a plasticizer to soften and thicken wood cell walls, making it easier to break the hydrogen bonds between microfibrils, leading to a decrease in the cell walls’ mechanical properties. Nanoindentation was performed on wood cell walls to investigate the effects of moisture content on the mechanical and rheological properties of cell walls. Negative relationships between instantaneous elastic modulus/hardness with moisture content were experimentally observed from indentations in the cell walls. Based on the viscoelastic concept, the moisture dependence of wood cell creep was investigated using the Burgers model and the Generalized Maxwell model. The parameters with physical interpretation extracted from both models were validated to be sensitive to moisture content. The microstructure analysis on the moisture-dependent creep agreed well with the model simulation, from which parameters were extracted to explain the rheological properties of the wood cell walls. Retardation time spectra were determined from creep compliance values calculated from experiment results and were found to give a good approximation with the fitted spectra obtained from the Burgers model. The elastic and viscous parameters tended to decrease with increasing moisture content. In all, both the Burgers model and the Generalized Maxwell model have been proven to be appropriate for investigating the moisture dependence of wood cell walls’ viscoelastic creep. In spite of the fact that the Burgers model worked well for predicting wood cell walls’ viscoelastic behaviour, the calculated elastic modulus from fitting parameters exhibited smaller values compared with Oliver and Pharr’s method, due to the creep phenomenon observed during the loading segment of the nanoindentation. FEA with the Generalized Maxwell model was applied to overcome this problem. Its simulation yielded similar results to those using the Burgers model but with a more accurate approximation of the elastic modulus.

Acknowledgements

The authors gratefully acknowledge financial support from the Zhejiang Key Level 1 Discipline of Forestry Engineering, Wood Utilization Grant and Forest Products Lab at Madison, the support of this research by USDA Special Wood Utilization Grants R11-0515-041 and R11-2219-510, the University of Tennessee, Department of Forestry, Wildlife and Fisheries, Center for Renewable Carbon, and the Agricultural Experiment Station McIntire-Stennis Grant TENOOMS-101.

References

1. Q. Wu, Y. Meng, K. Concha, S. Wang, Y. Li, L. Ma and S. Fu, Ind. Crops Prod., 2013, 48, 28–35.
2. C. Liao, Y. Deng, S. Wang, Y. Meng, X. Wang and N. Wang, Wood Fiber Sci., 2012, 44, 63–70.
3. X. Wang, Y. Deng, S. Wang, C. Liao, Y. Meng and T. Pham, BioResources, 2013, 8, 1986–1996.
4. Y. Wu, S. Wang, D. Zhou, C. Xing, Y. Zhang and Z. Cai, Bioresour. Technol., 2010, 101, 2867–2871.
5. X. Li, S. Wang, G. Du, Z. Wu and Y. Meng, Ind. Crops Prod., 2013, 42, 344–348.
6. C. Xing, S. Wang and G. Pharr, Wood Sci. Technol., 2009, 43, 615–625.
7. M. A. Ermeydan, E. Cabane, N. Gierlinger, J. Koetz and I. Burgert, RSC Adv., 2014, 4, 12981–12988.
8. A. Jäger, K. Hofstetter, C. Buksnowitz, W. Gindl-Altmutter and J. Konnerth, Composites, Part A, 2011, 42, 2101–2109.
9. L. Zou, H. Jin, W.-Y. Lu and X. Li, Mater. Sci. Eng., C, 2009, 29, 1375–1379.
10. S. H. Lee, S. Wang, G. M. Pharr, M. Kant and D. Penumadu, Holzforschung, 2007, 61, 254–260.
11. S. H. Lee, S. Wang, G. M. Pharr and H. T. Xu, Composites, Part A, 2007, 38, 1517–1524.
12. W. C. Oliver and G. M. Pharr, J. Mater. Res., 1992, 7, 1564–1583.
13. A. H. W. Ngan and B. Tang, J. Mater. Res., 2002, 17, 2604–2610.
14. J. Zhang, C. Wang, F. Yang and C. Du, RSC Adv., 2014, 4, 41003–41009.
15. E. Lee and J. R. M. Radok, J. Appl. Mech., 1960, 27, 438–444.
16. L. Cheng, X. Xia, W. Yu, L. E. Scriven and W. W. Gerberich, J. Polym. Sci., Part B: Polym. Phys., 2000, 38, 10–22.
17. A. H. W. Ngan, H. T. Wang, B. Tang and K. Y. Sze, Int. J. Solids Struct., 2005, 42, 1831–1846.
18. K. I. Schiffmann, Int. J. Mater. Res., 2006, 97, 1199–1211.
19. C. K. Liu, S. Lee, L. P. Sung and T. Nguyen, J. Appl. Phys., 2006, 100, 9.
20. M. L. Oyen and R. F. Cook, J. Mater. Res., 2003, 18, 139–150.
21. C. Skaar, Wood-water relations, Berlin Heidelberg New York, 1988.
22. Y. Kojima and H. Yamamoto, J. Wood Sci., 2005, 51, 462–467.
23. C. Takahashi, Y. Ishimaru, I. Iida and Y. Furuta, Holzforschung, 2006, 60, 299–303.
24. H. Qing and L. Mishnaevsky, Comput. Mater. Sci., 2009, 46, 310–320.
25. J. Mukudai and S. Yata, Wood Sci. Technol., 1987, 21, 49–63.
26. Y. Yu, B. Fei, H. Wang and G. Tian, Holzforschung, 2011, 65, 121–126.
27. W. J. Cousins, Wood Sci. Technol., 1978, 12, 161–167.
28. W. T. Y. Tze, S. Wang, T. G. Rials, G. M. Pharr and S. S. Kelley, Composites, Part A, 2007, 38, 945–953.
29. Y. Meng, S. Wang, Z. Cai, T. M. Young, G. Du and Y. Li, Appl. Phys. A: Mater. Sci. Process., 2013, 110, 361–369.
30. G. B. d. Souza, C. E. Foerster, S. L. R. d. Silva and C. M. Lepienski, Mater. Res., 2006, 9, 159–163.
31. ASTM Standard D4442-07, Standard test methods for direct moisture content measurement of wood and wood-base materials, ASTM International, West Conshohocken, PA, 2007.
32. R. Christensen, Theory of viscoelasticity: an introduction, Elsevier, 2012.
33. Documentation and ABAQUS, Dassault Systèmes, 2010.
34. Y. Kojima and H. Yamamoto, Wood Sci. Technol., 2004, 37, 427–434.
35. C. Su, E. G. Herbert, S. Sohn, J. A. LaManna, W. C. Oliver and G. M. Pharr, J. Mech. Phys. Solids, 2013, 61, 517–536.
36. S. Zauscher, D. F. Caulfield and A. H. Nissan, Tappi J., 1996, 79, 178–182.
37. F. F. Wangaard, J. Educ. Modules Mater. Sci. Eng., 1979, 1, 437–534.
38. L. Salmén, C. R. Biol., 2004, 327, 873–880.
39. A. M. Olsson and L. Salmen, ACS Symp. Ser., 2004, 864, 184–197.
40. D. Husken, E. Steudle and U. Zimmermann, Plant Physiol., 1978, 61, 158–163.
41. C. M. Hayot, E. Forouzesh, A. Goel, Z. Avramova and J. A. Turner, J. Exp. Bot., 2012, 63, 2525–2540.

---