Show Summary Details
More options …
New at De Gruyter

# Nordic Pulp & Paper Research Journal

### The international research journal on sustainable utilization of forest bioresources

Editor-in-Chief: Lindström, Tom

IMPACT FACTOR 2017: 1.131
5-year IMPACT FACTOR: 1.262

CiteScore 2017: 1.18

SCImago Journal Rank (SJR) 2017: 0.623
Source Normalized Impact per Paper (SNIP) 2017: 0.555

Online
ISSN
2000-0669
See all formats and pricing
More options …
Volume 33, Issue 1

# The effect of the through-thickness moisture content gradient on the moisture accelerated creep of paperboard: Hygro-viscoelastic modeling approach

Joonas Sorvari
• Corresponding author
• Centre of Computational Engineering and Integrated Design (CEID), Lappeenranta University of Technology, PO Box 20, FI-53851 Lappeenranta, Finland
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Teemu Leppänen
• LUT Savo Sustainable Technologies, Lappeenranta University of Technology, Varkaus unit, Opiskelijankatu 3, FI-78210 Varkaus, Finland
• Email
• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Jukka Silvennoinen
Published Online: 2018-05-23 | DOI: https://doi.org/10.1515/npprj-2018-3001

## Abstract

Paper-based materials are viscous materials, the time-dependent behavior of which depends strongly on moisture content. Particularly the creep of paperboard containers under compressive forces is greatly affected by changes in the relative humidity. In the present paper, we examine the creep behavior of paperboard under cyclic humidity conditions using the finite element method. Especially the shape and rate of the through-thickness moisture content gradient on moisture accelerated creep are studied. An isotropic hygro-viscoelastic constitutive law is used for paperboard. The results of the simulations are compared with experiments. It is concluded that the through-thickness moisture gradients have a great impact on the moisture accelerated creep of paperboard. Furthermore, the results show that depending on the direction of external load the through-thickness moisture content gradient may increase or decrease creep rate.

## Introduction

Corrugated board containers and packages are widely used in the shipping and storage of many products. Due to their use, containers must be designed to resist compressive loads. The box compression strength depends on the loading type and on environmental conditions. During lengthy storage, packages often experience creep-type loading which decreases compression strength. The situation is even worse if humidity conditions fluctuate. Under cyclic humidity conditions, the creep rate of board materials increases. The moisture accelerated creep can cause a sudden collapse of packages at unexpectedly low levels of load. According to Considine et al. (1989), in cyclic humidity conditions the acceptable load level of paperboard is less than 30 % of the short-term compressive strength. Papers by Haslach Jr. (2000) and Coffin (2005) provide excellent reviews of moisture accelerated creep.

The creep of paper and board has been extensively studied since the pioneering work of Brezinski (1956). However, it took several years before the influence of cyclic humidity on creep was recognized (Byrd 1972). Since then, several authors have contributed to the subject, and various explanations for the causes of accelerated creep have been proposed including viscous softening, physical aging and through-thickness and in-plane variation of the stresses. Vlahinic et al. (2012) suggest that the viscous softening and the moisture accelerated creep are mainly originating from the nanoscale movement of water enhancing the internal lubrication. In their study the creep rate factor was calculated for the data published by Alfthan (2004) and it was concluded that both the basic and accelerated creep are of common origin. Padanyi (1991) proposed that the physical aging representing the movement towards the thermodynamic equilibrium and the mechano-sorptive effects are actually same phenomena. Padanyi (1993) concluded that aging is a parameter as important as temperature and relative humidity for creep behavior.

It has also been suggested that the moisture accelerated creep is due to nonlinear creep in combination with transient stresses produced by inhomogeneous hygroexpansion (Habeger and Coffin 2000, Alfthan et al. 2002). Material inhomogeneities and moisture gradients can cause inhomogeneous hygroexpansion in the material. Moisture gradients will cause nonuniform expansion/shrinkage, which in turn will introduce internal stresses. Although several studies have been performed to clarify the effect of moisture gradients on the moisture accelerated creep the results are not totally consistent. The recent work of Lindström et al. (2012) suggested that in the tension the through-thickness moisture content gradients do not accelerate the creep significantly.

The objective of this work is to clarify the effect of through-thickness moisture content gradient on the moisture accelerated creep of paperboard. Especially the shape and the rate of the through-thickness moisture content gradient are considered. For this purpose, finite element (FE) simulations of paperboard creep under cyclic humidity conditions were carried out. The paperboard was modeled as an isotropic hygro-viscoelastic material. The material model was determined from experimental creep tests and validated by comparing FE simulations with experimental data. The simulated results obtained in cyclic humidity conditions also give a potential explanation for the diverge behavior of the moisture accelerated creep under external tension and compression.

## Isotropic hygro-viscoelastic constitutive law

We modeled paperboard as isotropic hygro-viscoelastic material in which Poisson’s ratio was constant. In addition, the paperboard was assumed to be in a plane stress state. In reality, paperboard is an anisotropic material and is typically modeled using orthotropic material models. However, due inadequate measurement data and the main forces acting in one direction, the isotropic model was utilized. The material model of homogeneous linear hygro-viscoelastic material can be expressed using the tensile compliance $\mathit{D}\mathrm{\left(}\mathit{t}\mathrm{\right)}$ and Poisson’s ratio ν as $εxx=∫0tDψt−ψτ(σ˙xxτ−νσ˙yyτ)dτ+εh$(1)$εyy=∫0tDψt−ψτ(σ˙yyτ−νσ˙xxτ)dτ+εh$(2)$εxy=∫0tDψt−ψτ1+νσ˙xyτdτ,$(3) where the hygroexpansion strain is given by $εh=βc−c0,$(4) where β is the coefficient of hygroexpansion, c is the current moisture content and ${\mathit{c}}_{0}$ is the reference moisture content. The reduced time is given by $ψt=∫0tdηacη,$(5) where a is the time-shift parameter which is a function of the moisture content. The creep compliance was modeled using the Prony series $Dt=D0+∑i=1NDi1−e−t/τi,$(6) where ${\mathit{D}}_{0}$ is the instantaneous compliance and ${\mathit{\tau }}_{\mathit{i}}$ is the retardation time associated with the compliance ${\mathit{D}}_{\mathit{i}}$. For detailed information, see Tschoegl (1997).

Figure 1

Measurement instrument (left) and CCT sample holder and sample after test (right).

## Experimental

Compressive creep tests were conducted in the cross-machine direction (CD) for commercial corrugated medium made of virgin fiber with a basis weight of $120\phantom{\rule{0.1667em}{0ex}}{\text{g/m}}^{2}$. The width, height and thickness of the paperboard samples were 152 mm, 12.7 mm and 0.17 mm, respectively. Paperboard samples were conditioned according the ISO Standard ISO 187 (20187). The creep tests were conducted at room temperature and at a relative humidity (RH) of 50 %, 70 % and 90 %. The measurements were performed in a specially constructed measurement instrument placed in an environmental cabin, see Figure 1. The instrument consists of the following elements: base plate with attached magnetic distance measurement probes and with housing for a corrugated crush test (CCT) holder, standard CCT sample holder, side beams, loading plate with magnets installed parallel with the base plate (supported by side beams), and loading plates. All parts were installed on the base plate. The base plate was insulated against any possible vibrations by gel-filled cushions. The test loads were chosen so that the paperboard would not collapse but would exhibit clear creep deformation. The test loads were 6.01 kg, 10.09 kg and 12.01 kg for relative humidity levels of 90 %, 70 % and 50 %, respectively. Each test was repeated four times.

In addition to creep tests at a constant humidity, creep tests in cyclic humidity were conducted. In the cyclic tests, loading was applied to the unconditioned specimen at 90 % RH, and the specimen was then subjected to three-hour humidity cycles between 50 % RH and 90 % RH by altering the climate in the cabin. At the beginning of the test, the moisture content of the sample was very low, and during the first three hours, the moisture content of the sample increased significantly. The test loads were 2.63 kg and 2.83 kg.

The hygroexpansion of the sample was measured using an optical method and a humidity controlled cabinet (OPTIDIM). The measurements follow the principles described in standard ISO 8226-2. The moisture expansion coefficient was determined by dividing the length change obtained between 50 % RH and 90 % RH by the corresponding moisture content change. The value of the hygroexpansion coefficient was found to be 0.135 in CD. Based on the hygroexpansion experiment, it was estimated that 50 % RH, 70 % RH and 90 % RH corresponded to equilibrium moisture contents of 0.071, 0.094 and 0.135, respectively.

## Determination of material parameters

In a creep test with the constant moisture content ${\mathit{c}}^{\ast }$, when loaded in the y-direction, the tensile compliance is approximately given by $Dtac∗=εyytσ0,$(7) where ${\mathit{\sigma }}_{0}$ is the applied constant stress and ${\mathit{\epsilon }}_{\mathit{y}\mathit{y}}$ is the measured strain. When the Prony series is used to represent the compliance, we obtain $εyytσ0=D0+∑i=1NDi1−e−t/(ac∗τi).$(8) The retardation times can be chosen a priori, for example, equidistantly on the logarithmic time axis one or two per decade of experimental data. Since the length of the creep tests was around ${10}^{5}\phantom{\rule{0.1667em}{0ex}}\text{s}$, the following retardation times were used: ${\mathit{\tau }}_{1}={10}^{0}$, ${\mathit{\tau }}_{2}={10}^{1}$, ${\mathit{\tau }}_{3}={10}^{2}$, ${\mathit{\tau }}_{4}={10}^{3}$ and ${\mathit{\tau }}_{5}={10}^{4}$. The compliances ${\mathit{D}}_{\mathit{i}}$ were determined by fitting the Prony series simultaneously to all creep tests. This yielded also the values of the time-shift parameter at different moisture contents. MATLAB and its built-in nonlinear least squares fitting function (lsqnonlin) was used in the fitting. The discrete values of the time-shift parameter were fitted to exponential form $ac=eAc+B,$(9) where A and B are constants with values $\mathit{A}=-98.7$ and $\mathit{B}=15.04$. The determined values of the Prony series constants are given in Table 1.

Table 1

Prony series constants of paperboard.

For paper and board materials, typical values of the in-plane Poisson ratio ${\mathit{\nu }}_{\mathit{x}\mathit{y}}$ are around 0.15–0.3 (Baum et al. 1981). Based on this, the value of the Poisson ratio ν was estimated to equal 0.25.

## Implementation

The isotropic material model was implemented in the FE code ABAQUS using the user subroutine UMAT (Abaqus 2013). In the user subroutine, the stresses and solution-dependent state variables must be updated to their values at the end of the increment, and the material Jacobian matrix must be computed. The mechanical strain increments and values of the state variables at the beginning of the considered increment are known.

The uniaxial integration scheme is given in Appendix. By using exactly the same integration scheme for the plane stress model, we obtain that the increments of the mechanical strain components are given by $ΔεxxmΔεyymΔεxym=D∞−∑i=1NDi1−e−Δψ/τiΔψ/τi·1−ν0−ν10001+νΔσxxΔσyyΔσxy−∑i=1NDie−Δψ/τi−1ϕitnDie−Δψ/τi−1φitnDie−Δψ/τi−1γitn,$(10) where ${\mathit{\varphi }}_{\mathit{i}}$, ${\mathit{\phi }}_{\mathit{i}}$ and ${\mathit{\gamma }}_{\mathit{i}}$ are state variables. The recursive relations for the state variables are given by $ϕitn+1=e−Δψ/τiϕitn+1−e−Δψ/τiΔψ/τi·Δσxx−νΔσyy$(11)$φitn+1=e−Δψ/τiφitn+1−e−Δψ/τiΔψ/τi·Δσyy−νΔσxx$(12)$γitn+1=e−Δψ/τiγitn+1−e−Δψ/τiΔψ/τi·1+νΔσxy.$(13) By inverting Equation 10, we obtain that the stress increments are given by $ΔσxxΔσyyΔσxy=D∞−∑i=1NDi1−e−Δψ/τiΔψ/τi−1·1−ν2−11ν0ν10001−ν·ΔεxxmΔεyymΔεxym+∑i=1NDie−Δψ/τi−1ϕitnDie−Δψ/τi−1φitnDie−Δψ/τi−1γitn$(14) Equation 14 was used to calculate the values of the stresses at the end of the increment. After the stresses were updated, the state variables were updated using the recursive relation given in Equation 1113. The (consistent) Jacobian matrix is given by $∂Δσ∂Δε=D∞−∑i=1NDi1−e−Δψ/τiΔψ/τi−1·1−ν2−11ν0ν10001−ν.$(15) The time-integration scheme used is similar to that used by for example Henriksen (1984), Haj-Ali and Muliana (2004) and many other authors. For linearly viscoelastic materials, a piecewise linear approximation of the excitation history leads to a second-order accurate scheme (Sorvari and Hämäläinen 2010).

In the FE simulations, shell elements were used. When UMAT is used to define shell material behavior, the transverse shear stiffness must also be specified. The transverse shear stiffness should be defined as the initial linear elastic stiffness of the shell in response to pure transverse shear strains (Abaqus 2013): $Kxxts=Kyyts=5Et121+ν,Kxyts=0,$(16) where t is the thickness of the shell and E is the elastic modulus. In the model, formulas given in Equation 16 were used with $\mathit{E}=1\mathrm{/}{\mathit{D}}_{0}$.

## Comparison with experiments

The compressive creep behavior of the determined isotropic hygro-viscoelastic material model at a constant humidity was studied using FE simulations. The Prony series constants of the material model are given in Table 1 and the rest of the material parameters have been summarized in Table 2. In the FE simulations, a rectangular sample was used with the same dimensions as in the experiments. The sample was meshed with plane stress shell elements. Compressive force was applied to the top of the sample and the opposite edge was completely fixed. The same loads were used as in the experiments.

Table 2

Material parameters of paperboard.

Figure 2

Comparison between modeled and experimental creep curves.

The creep curves of the material model at 50 %, 70 % and 90 % RH obtained from FE simulations are compared with the experimental data in Figure 2. The modeled creep curves are, as expected, in good agreement with the experimental ones.

## Sample setup, numerical solution and simulation procedure

We considered the compressive creep behavior of a paperboard sample with an x-directional width of 98.16 mm, y-directional height of 12.7 mm, and thickness of 0.170 mm under cyclic humidity conditions. The wave amplitude was 4.75 mm and the wavelength 8.55 mm. The total length of the paperboard in the x-direction was 152 mm. The geometry of the sample is shown in Figure 3. The isotropic hygro-viscoelastic material model presented in the previous section was used for the paperboard.

Figure 3

Sample and coordinate system used in the simulations.

Figure 4

Element S4R used in the simulations. In the element, the material is divided into eight layers with equal thicknesses.

In the FE model, shell elements (ABAQUS/Standard element type S4R (Abaqus 2013)) were used with a composite structure (see Figure 4). The in-plane size of the elements was approximately $1\phantom{\rule{0.1667em}{0ex}}\text{mm}×1\phantom{\rule{0.1667em}{0ex}}\text{mm}$. In the thick-ness direction, each element was divided into eight material layers with equal thicknesses. The thickness directional moisture content (MC) was defined in the element locations 1–9 shown in Figure 4; similar approach has been used in Bosco et al. (2017).

The creep load was modeled by applying a y-directional compressive load into nodes located in the plane $\mathit{y}=12.7\phantom{\rule{0.1667em}{0ex}}\text{mm}$ with equation constraints. The equation constraints applied unifies the y-directional displacement of the edge nodes. All displacements and rotations were restrained in the nodes located in the plane $\mathit{y}=0$.

In all simulations, the initial moisture content of the sample was 13.5 % and the moisture content was kept constant for the first three hours before initiating the moisture cycling. The total time of each cycle was six hours. In the first three hours of the cycle, the moisture content was decreased to 7.1 %, and in the last three hours of the cycle, the moisture content was restored to 13.5 %.

The time at which the moisture content changes were attained was varied, as explained later. One intermediate state was used to define the progression of the moisture content change. Between the stages defining the moisture content, the moisture content was changed linearly with respect to the time in every location.

## Constant through-thickness moisture content change

We first considered the cyclic humidity creep without the through-thickness moisture content gradient. Thus, the moisture content in the thickness direction was the same in all material layers. The moisture content change from 13.5 % to 7.1 % and vice versa occurred in 7200 s (see Figure 5).

Figure 5 presents the moisture content change used in the simulations and the experimental and simulated results of the cyclic test. The simulation is performed without through-thickness moisture content gradient and due to that the experimental situation does not correspond to the simulated situation; in reality there always exist through-thickness moisture content gradients. The results show that for both the measured and the simulated results, the moisture expansion is notable. However, although the moisture expansion is clearly greater in the simulated results, it did not accelerate the creep. The creep rate was practically the same as in the constant humidity creep test. The experimental results show that during the moisture content change, the compressive strain changed nonlinearly. This was not observed in the simulated results where the change was linear. It should be noted, as already discussed in the previous section, that in the measurements the moisture content during the first three hours was not constant as in the simulations, but was rather increasing. Therefore, at the beginning of the experimental test, extensional strain can be observed.

## Through-thickness moisture content gradient

Figure 5

Moisture content change used in the simulation approach when no thickness directional moisture content gradient exists (top), and simulated and measured creep (bottom).

Figure 6

Through-thickness moisture content gradients used in the simulations (from top to bottom): Fast-Gentle, Fast-Steep, Slow-Gentle and Slow-Steep. The locations on the horizontal axis are clarified in Figure 4.

The development of the moisture content and through-thickness moisture content gradients on paper and paperboard has been studied by several authors, such as Dano and Bourque (2009), Lavrykov et al. (2004) and Östlund (2006). In this study, the effect of the through-thickness moisture content gradient on the accelerated creep was studied by four different approaches (see Figure 6). These approaches are called Fast-Gentle, Fast-Steep, Slow-Gentle and Slow-Steep; the first term describes the rate of the moisture content change and the second term the steepness of the moisture content gradient. The moisture content change was attained in 1200 s in the fast case and in 7200 s in the slow one. In the gentle case, the maximum moisture content difference between the outer and inner part of the sample was 4.4 %, while in steep case it was 6.4 %. The moisture content gradients used are suggestive; they are based on the earlier published studies.

Figure 7

Measured and simulated creep when different through-thickness moisture content gradients were used in the simulations.

Figure 7 presents experimental and simulated results when the thickness directional moisture content gradients (Figure 6) were applied. In contrast to the case without a through-thickness moisture gradient, an increase in creep rates can be observed. The differences in the creep rates between different approaches are notable. The slower and steeper the moisture content gradient is, the higher the creep rate is. In addition, in the slow case the shapes of the simulated creep curves are closer to the experimental curves than previously. The y-directional stresses of the approaches Fast-Steep and Slow-Steep are presented in Figure 8. The internal stresses arising from the hygro-expansion are strongly affected by the rate of the moisture content gradient. Although the maximum stress values are lower with Slow-Steep than Fast-Steep the time and trough-thickness distribution of the stresses in Slow-Steep increases the creep rate.

Figure 8

Stresses in the layers 1 – 4 when Fast-Steep (top) and Slow-Steep (bottom) approaches are used in the simulation. Corresponding strains are presented in Figure 7.

Also, the moisture dependency of the material is essential for the creep acceleration. When the moisture dependency of the material was ignored by setting $\mathit{A}=\mathit{B}=0$ in Equation 9 the creep acceleration vanished, see Figure 9.

Figure 9

Simulated creep with constant moisture content (13.5 %), with Slow-Steep and with Slow-Steep when A and B are zero in Equation 9.

Figure 10

Simulated creep for different through-thickness moisture content gradients and loads.

The effect of load on the accelerated creep was also studied. As assumed, the creep rate increased with the load in each approach, see Figure 10. However, also in this case the through-thickness moisture content gradient had a major effect on the creep rates especially in the slow case.

## Through-thickness moisture content gradient under external tension

Figure 11

Simulated creep in tension for different through-thickness moisture content gradients and loads.

Figure 11 presents the effect of through-thickness moisture content gradients and loads on the simulated creep under external tension. Stronger the “acceleration” has been under external compression, greater “deceleration” is obtained under tension. Under external tension the load and internal forces generated by the through-thickness moisture content gradients are affecting into opposite directions. When the most “accelerating” through-thickness moisture content gradient in compression, Slow-Steep, is applied with the external tensile load 2.73 kg a negative strain is obtained. That is, the internal compressive forces generated by the through-thickness moisture content gradient are greater than the tensile forces generated by the external load. However, it should noted that the material model is calibrated by measurements performed under compression and the real material behavior diverges between the tension and compression at some level.

## Moisture expansion coefficient

Figure 12

Simulated creep without a thickness directional moisture content gradient (top) and with Slow-Steep gradient (bottom) when the level of the moisture expansion coefficient was modified. An 8 kg load was applied in the simulations.

The effect of the moisture expansion coefficient on the accelerated creep was studied by varying the overall value of the coefficient. The simulated results with and without the moisture content gradient are presented in Figure 12. Without the moisture content gradient, the increase in the value of the moisture expansion coefficient did not affect the creep rate although the hygroexpansive strain increased. The results are quite different when the moisture content gradient was applied during the cycles. A clear increase in creep rates can be observed. In addition, the creep rate increased slightly with the moisture expansion coefficient.

## Conclusions

The simulated results suggest that the through-thickness moisture content gradient is an important factor affecting the moisture accelerated creep. The rate of the moisture content change and the steepness of the gradient were shown to affect the creep rate greatly. The creep rate was especially shown to increase when the moisture content gradient was used with a slowly changing moisture content. In addition, the creep rate increased with the steepness of the gradient. Also the applied creep load was shown to accelerate the creep. Without a through-thickness moisture content gradient or moisture content dependency of the material, no acceleration in the creep was observed. The results suggest that the internal stresses generated by hygroexpansion may increase the creep rate when material layers with different moisture content exist. Furthermore, the level of the moisture expansion coefficient influenced the creep rate only when the moisture content gradient was present. In that case, the creep rate was shown to increase slightly with the moisture expansion coefficient.

Results presented by Strömbro and Gudmundson (2008) show that a great difference in the creep rate is obtained between external tension and compression in the cyclic humidity conditions; significantly higher rate is obtained in the compression. The simulated results presented here show a parallel behavior; depending on the direction of the external load the through-thickness moisture content gradient may increase or decrease the creep rate. However, it should be noted that under external tension the simulated results probably overestimate the effect of the through-thickness moisture content gradient and the same material parameters were used for tension and compression. Also, the effects of the viscous softening and physical aging on the moisture accelerated creep are not considered in the presented approach.

The isotropic linear hygro-viscoelastic model used is a simplified model which neglects the anisotropic and nonlinear behavior of paperboard. As it has been suggested, the nonlinear creep behavior may have an important role in accelerated creep. Nevertheless, the results show that the accelerated creep can be observed even with a linear model. Furthermore, the presented model and modeling approach may be useful in analyzing the performance of paperboard packages under cyclic humidity conditions.

## Acknowledgments

The simulations were performed by the commercial software ABAQUS, which was licensed to CSC (the Finnish IT center for science).

## Appendix

We begin by considering the uniaxial version of the hygro-viscoelastic model. For the uniaxial model, the mechanical strain is given by $εmt=∫0tDψt−ψτσ˙τdτ.$(17) When the Prony series is used to represent the compliance, the strain at time ${\mathit{t}}_{\mathit{n}+1}$ is given by $εmtn+1=D∞σtn+1−∑i=1NDiσi∗tn+1,$(18) where $D∞=D0+∑i=1NDi,$(19)$σi∗tn+1=∫0tn+1e−ψtn+1−ψτ/τiσ˙τdτ$(20) are the equilibrium compliance and internal stress (state variable) associated with the compliance ${\mathit{D}}_{\mathit{i}}$, respectively. In the time interval $\left[{\mathit{t}}_{\mathit{n}}\mathrm{,}\phantom{\rule{2.38387pt}{0ex}}{\mathit{t}}_{\mathit{n}+1}\right]$, the time-shift parameter is assumed to be constant and the value is taken to be $\mathit{a}\left(\mathit{c}\left({\mathit{t}}_{\mathit{n}+1}\right)\right)$. Then, the reduced time is given by $ψtn+1=∫0tn+1dηa=∫0tndηa+∫tntn+1dηa≈∫0tndηa+Δtactn+1≡ψtn+Δψ,$(21) where $\mathrm{\Delta }\mathit{t}={\mathit{t}}_{\mathit{n}+1}-{\mathit{t}}_{\mathit{n}}$ is the time increment. The internal stress is now given by $σi∗tn+1=∫0tn+1e−ψtn+1−ψτ/τiσ˙τdτ=∫0tne−ψtn+1−ψτ/τiσ˙τdτ+∫tntn+1e−ψtn+1−ψτ/τiσ˙τdτ=e−Δψ/τiσi∗tn+∫tntn+1e−ψtn+1−ψτ/τiσ˙τdτ.$(22) If the stress is assumed to vary linearly over the considered time interval, we can evaluate the integral in Equation 22 analytically since the time-shift parameter is constant in the considered time interval. We obtain the recursive relation $σi∗tn+1=e−Δψ/τiσi∗tn+1−e−Δψ/τiΔψ/τiΔσ,$(23) where $\mathrm{\Delta }\mathit{\sigma }=\mathit{\sigma }\left({\mathit{t}}_{\mathit{n}+1}\right)-\mathit{\sigma }\left({\mathit{t}}_{\mathit{n}}\right)$ is the stress increment. The internal stress increment is given by $Δσi∗=e−Δψ/τi−1σi∗tn+1−e−Δψ/τiΔψ/τiΔσ.$(24) The strain increment is thus given by $Δεm=D∞Δσ−∑i=1NDiΔσi∗=D∞−∑i=1NDi1−e−Δψ/τiΔψ/τiΔσ−∑i=1NDie−Δψ/τi−1σi∗tn.$(25) We obtain that the stress increment is given by $Δσ=Δεm+∑i=1NDie−Δψ/τi−1σi∗tn·D∞−∑i=1NDi1−e−Δψ/τiΔψ/τi−1,$(26) which can be used to calculate the stress at the end of the increment. The internal stresses can be then updated using the recursive relation.

## References

• Abaqus (2013) Documentation, Dassault Systemes. Providence, RI, USA. Google Scholar

• Alfthan, J. (2004) The effect of humidity cycle amplitude on accelerated tensile creep of paper. Mech. Time-Depend. Mater. 8:289–302.

• Alfthan, J., Gudmundson, P., Östlund, S. (2002) A micro-mechanical model for mechano-sorptive creep in paper. J. Pulp Pap. Sci. 28:98–104. Google Scholar

• Baum, G.A., Brennan, D.C., Habeger, C.C. (1981) Orthotropic elastic constants of paper. Tappi J. 64:97–101. Google Scholar

• Bosco, E., Peerlings, R.H.J., Lomans, B.A.G., van der Sman, C.G., Geers, M.G.D. (2017) On the role of moisture in triggering out-of-plane displacement in paper: from the network level to the macroscopic scale. Int. J. Solids Struct. .

• Brezinski, J.P. (1956) The creep properties of paper. Tappi J. 39:116–128. Google Scholar

• Byrd, V.L. (1972) Effect of relative humidity changes on the compressive creep response of paper. Tappi J. 55:1612–1613. Google Scholar

• Coffin, D.W. (2005) The creep response of paper. In: Advances in Paper Science and Technology, Transactions of the 13th Fundamental Research Symposium. Cambridge, UK. pp. 651–747. Google Scholar

• Considine, J.M., Gunderson, D.E., Thelin, P., Fellers, C. (1989) Compressive creep behavior of paperboard in a cyclic humidity environment – exploratory experiment. Tappi J.. 72:131–136. Google Scholar

• Dano, M.L., Bourque, J.P. (2009) Deformation behavior of paper and board subjected to moisture diffusion. Int. J. Solids Struct. 46:1305–1316.

• Habeger, C.C., Coffin, D.W. (2000) The role of stress concentration in accelerated creep and sorption-induced physical aging. J. Pulp Pap. Sci. 26:145–157. Google Scholar

• Haj-Ali, R.M., Muliana, A.H. (2004) Numerical finite element formulation of the Schapery non-linear viscoelastic material model. Int. J. Numer. Methods Eng. 59:25–45.

• Haslach Jr., H.W. (2000) The moisture and rate-dependent mechanical properties of paper: a review. Mech. Time-Depend. Mater. 4:169–210.

• Henriksen, M. (1984) Nonlinear viscoelastic stress analysis – a finite element approach. Comput. Struct. 18:133–139.

• Lavrykov, S., Ramarao, B., Lyne, Å. (2004) The planar transient hygroexpansion of copy paper: experiments and analysis. Nord. Pulp Pap. Res. J. 19:183–190.

• Lindström, S.B., Karabulut, E., Kulachenko, A., Sehaqui, H., Wagberg, L. (2012) Mechanosorptive creep in nanocellulose materials. Cellulose 19:809–819.

• Östlund, M. (2006) Modeling the influence of drying conditions on the stress buildup during drying of paperboard. J. Eng. Mater. Technol. 128:495–502.

• Padanyi, Z.V. (1991) Mechano-sorptive effects and accelerated creep in paper. In: Proceedings of International Paper Physics Conference. Kona, Hawaii, US. pp. 397–411. Google Scholar

• Padanyi, Z.V. (1993) Physical aging and glass transition: effects on the mechanical properties of paper and board. In: Proceedings of the 10th Pulp and Paper Fundamental Research Symposium. Oxford, UK. pp. 521–545. Google Scholar

• Sorvari, J., Hämäläinen, J. (2010) Time integration in linear viscoelasticity – comparative study. Mech. Time-Depend. Mater. 14:307–328.

• Strömbro, J., Gudmundson, P. (2008) Mechano-sorptive creep under compressive loading – a micromechanical model. Int. J. Solids Struct. 45:2420–2450.

• Tschoegl, N.W. (1997) Time dependence in material properties: an overview. Mech. Time-Depend. Mater. 1:3–31.

• Vlahinic, I., Thomas, J.J., Jennings, H.M., Andrade, J.E. (2012) Transient creep effects and the lubricating power of water in materials ranging from paper to concrete and kevlar. J. Mech. Phys. Solids 60:1350–1362.

## About the article

Accepted: 2017-12-06

Published Online: 2018-05-23

Published in Print: 2018-05-23

Conflict of interest: The authors do not have any conflicts of interest to declare.

Citation Information: Nordic Pulp & Paper Research Journal, Volume 33, Issue 1, Pages 122–132, ISSN (Online) 2000-0669, ISSN (Print) 0283-2631,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.