# Coupled two-dimensional modeling of viscoelastic creep of wood

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## Abstract

Three coupled two-dimensional viscoelastic creep models for orthotropic material are analyzed. The models of different complexity are mathematically formulated and implemented in a finite element software. Required viscoelastic material parameters are determined by calibration procedure, where numerical results are compared against experimentally obtained viscoelastic strains caused by tensile or shear loading. Finally, a comparison method is used to evaluate the accuracy of strain predictions of each particular model. The analysis shows that all the models are able to accurately predict viscoelastic creep simultaneously in two perpendicular directions for various periods of time and wood species. Calculated numerical values of the viscoelastic material parameters suitable for the three models and wood species, i.e., Douglas fir (*Pseudotsuga menziesii*), Norway spruce (*Picea abies*), Japanese cypress (*Chamaecyparis obtusa*), and European beech (*Fagus sylvatica* L.), under constant tensile loading are also given.

## Introduction

Long-term performance of wood is strongly affected by a process of creep. It is understood as an increment of strain over time due to applied constant or changing load in a constant or changing environment. Timber structures are usually exposed to all four combinations during their service life, making designing timber structures challenging, and when accounting for wood as natural, heterogeneous, and anisotropic material, the task becomes even more so. Therefore, significant research efforts to improve mathematical, empirical, and numerical expressions for describing time-dependent behavior of wood have been undertaken. For simplicity, wood sections far enough from the stem are considered as orthotropic material. Viscoelastic creep is a pure creep response solely inflicted by a load’s magnitude and its duration. Even though pure creep for load-carrying wood in ordinary varying climate is small in magnitude when compared to, for example, mechano-sorptive creep (Toratti and Svensson 2000), its formulation is a fundamental basis for the description of wood behavior, impacted by all sorts of environment variation (Hanhijärvi 1995; Bonfield et al. 1996; Hunt and Gril 1996; Hanhijärvi and Hunt 1998; Dubois et al. 2005, 2012; Colmars et al. 2014; Reichel and Kaliske 2015b). Several series of experimental data on viscoelastic creep strains in different orthotropic directions of wood under constant tensile loading are reported in Schniewind and Barrett (1972), Hayashi et al. (1993), Taniguchi and Ando (2010), and Ożyhar et al. (2013). Experimental data of viscoelastic shear strains is reported in Schniewind and Barrett (1972). The researchers point out the importance and difficulties of isolating the viscoelastic creep from other types of excitations that influence long-term behavior of wood. Constant environmental conditions are, therefore, required during the experiments in order to eliminate possible additional strains. Schniewind and Barrett (1972) and Hayashi et al. (1993) support their experimental work by theoretical considerations on creep compliance components. Taniguchi and Ando (2010) study time dependence of Poisson’s ratios based on their experimental results. Frandsen (2007) discusses the existence of Poisson’s ratios in orthotropic creep models and mathematically proves that diagonal components of the creep compliance matrices cannot scale proportionally to the diagonal components of the elastic compliance matrix as a ratio. Additionally, he shows that the off-diagonal components scale disproportionally to the diagonal components of all creep compliance matrices introduced in an orthotropic model. These findings are quite revolutionary and not easy to establish since they are violated in several creep models, for example, in Fortino et al. (2009), and Hassani et al. (2015). The coupled formulation of creep by Frandsen (2007) accounts for the necessary requirements about the compliance matrix. Another approach, where the requirements of the creep compliance matrix symmetry are fulfilled with defining relative creep functions, is found in Ormarsson (1999) and Reichel and Kaliske (2015a). On the other hand, Ando et al. (2013) and Jiang et al. (2016) have experimentally obtained the asymmetric creep compliance matrix of wood. In the present paper, three constitutive creep models of different complexity based on Frandsen's (2007) coupled two-dimensional formulation of creep for orthotropic material are studied. The models are implemented in a finite element software where numerical solution of time-dependent strain state is obtained. Viscoelastic material parameters (from now on referred to as material parameters, unless otherwise stated) of the models are calibrated against series of experimental data of different wood species. Extensive analysis and calibration of the proposed coupled two-dimensional formulation of creep of wood (Frandsen 2007) is novel and presents an easy and efficient way of determining the characteristic material parameters for each wood species, i.e., Douglas fir (*Pseudotsuga menziesii*), Norway spruce (*Picea abies*), Japanese cypress (*Chamaecyparis obtusa*), and European beech (*Fagus sylvatica* L.).

## Methods and modeling

*Q*and dashpot viscosities by

*η*. In Fig. 2, coupling of the longitudinal and transverse directions is graphically illustrated as a diamond geometry containing the rheological assembly. Since it is positioned in parallel, it enables response in transverse direction when the longitudinal excitation is applied (Fig. 2a), and vice versa (Fig. 2b). Obviously, the direction of excitation and the transverse response are opposite in signs.

*ε*

_{L}and

*ε*

_{T}, respectively, and the total shear strain (

*γ*) are formed from a sum of elastic and viscoelastic parts

*i*= 0 and the viscoelastic part by

*i*= 1 for model K and

*i*= 1, 2 for models KD and KK. In the rheological models (K, KD, and KK), it is assumed that viscoelastic strains are induced by the equilibrating stresses

*σ*

_{LL}and

*σ*

_{TL}for loading in L direction, and

*σ*

_{TT}and

*σ*

_{LT}for transverse loading.

*σ*

_{LL}and

*σ*

_{LT}equilibrate the stress in the spring and the stress in Kelvin element for longitudinal and transverse strains, respectively. Furthermore, in KD model,

*σ*

_{LL}and

*σ*

_{LT}equilibrate the stress in the spring, Kelvin element, and the dashpot for longitudinal and transverse strains, respectively. Similar consideration is applied for a transverse loading and KK model. Reader should be aware that

*σ*

_{LL},

*σ*

_{LT},

*σ*

_{TT}, and

*σ*

_{TL}do not have any physical meaning and are introduced due to derivation of the rheological models. Comprehensive description and derivation can be found in Frandsen (2007). The three rheological models are implemented in a commercial finite element software, where the corresponding constitutive equations are as follows

*τ*the shear stress and the index

*i*is not associated with a summation. The elastic material parameters, i.e., components of the elastic stiffness matrix in Eq. (3) are determined from relations of elastic moduli and Poisson’s ratios for orthotropic material. The viscoelastic response in each model is prescribed in a form from partial differential Eqs. (4) and (5). For clarity, all models constitute Eq. (3); additionally, the K model (Fig. 2) constitutes Eq. (5) for

*i*= 1, the KD model Eqs. (4) and (5) for

*i*= 1, and the KK model Eq. (5) for

*i*= 1 and 2. The formulation of the K model is identical to the one proposed by Frandsen (2007). The KD model has already been applied to the results of Taniguchi and Ando (2010) by Kawahara et al. (2015). Formulation of the model was based on an analytic approach, considering only the loading phase. With that approach, numerical values of material parameters in Eqs. (3)–(5) were impossible to find. To the authors’ knowledge, the K and KK models are calibrated against experimental data in the current study for the first time.

In Eqs. (3)–(5), shear stress–strain state in terms of rheological elements is also included. Shear behavior of the material is not coupled to the longitudinal or transverse response of the material. Therefore, the material’s viscoelastic response in shear can be formulated separately with the use of basic viscoelastic elements in one dimension. The shear models are directly associated with schematic illustrations of Fig. 1. The set of Eqs. (3)–(5) present a complete mathematical description of the two-dimensional orthotropic behavior of wood. The solution can be obtained elegantly by means of the numerical finite element software.

In general, the more complex the rheological model, the better the fit to experimental data, and the higher the number of unknown material parameters (*Q*, *η*). For viscoelastic compliance matrix to be symmetric, the symmetry of 3 × 3 matrices in Eqs. (4) and (5) is required. Therefore, the simplest two-dimensional K model includes six material parameters. The KD and KK models require 9 and 12 material parameters, respectively. The values of the material parameters dictate the magnitude of the equilibrating stresses that induce strains in longitudinal and transverse directions [Eq. (2)] when the material is stressed in one or another direction.

Researchers and wood species tested in relation to abbreviations used in this manuscript

In the present paper, the complexity of the models according to the number of material parameters needed is tried to be evaluated against the accuracy of numerically obtained strain development due to tensile and shear loading in comparison to available experimental data. The accuracy of a particular model is evaluated by an average of absolute value of relative errors (RE) of predicted strains at observed times, and by a method presented in Annex D in EN (1990). The evaluation parameters are determined for the strain development predicted by each model in the longitudinal and the transverse directions separately, as well as for both directions together, L + T. In total, the comparison method analyzes 42 different cases that cover the three viscoelastic constitutive models calibrated to five sets of experimental data in loading phase, including shear and constant tension in longitudinal and transverse direction separately and combined. The method presented in Annex D in EN (1990) includes an estimation of the mean value correction factor *b* that is the “least-squares” best fit to the slope, and the coefficient of variation, *V* _{δ}, of the lognormally distributed error terms. In regression analysis, the term *b* is recognized as the slope of the regression line that tells the amount by which the experimental variable increases or decreases, on average, when the numerical variable increases by one. Practically, that means if the value of *b* is larger than one, the numerically predicted values are smaller than experimental values of creep strains on average. In that case, the particular constitutive model underestimates the experimental data on average. On the contrary, when the *b* value is smaller than one, the constitutive model overestimates the creep strains on average. Coefficient of variation, *V* _{δ}, is a measure of an average scatter of the numerical value from the experimental value at each point compared. Hence, the smaller the coefficient of variation, the better the constitutive model predicts the experimentally observed values of creep strains.

## Results and discussion

*abbrev.*exp) two-dimensional viscoelastic response of four different wood species. The abbreviations for each set of data (Table 1) are used in combination with the label of the type of the creep model, explained earlier (K, KD, and KK), which ends up in 12 combinations in tension and three in shear. Reason for making those combinations of labels is related to different values of material parameters determined by the fitting procedure of particular creep model to a particular set of experimental data. The calculated values of the material parameters (given in Tables A1 and A2 in Online Resource) characterize each type of analyzed creep model fitted to specific experimental data. The examined wood specimens are loaded under constant tension in longitudinal (Figs. 3b, c, 4, 5, 6) and transverse directions (Fig. 3a). Taniguchi and Ando (2010) also presented strain response after unloading for a period of extra 24 h (Fig. 5). Experimental data of pure viscoelastic shear behavior of Douglas fir are extracted from Schniewind and Barrett (1972). Time interval of monitored viscoelastic creep response in loading phase differs among the experimental data, from 16.67 h (SB exp), 100 h (HA exp) to 24 h (TA exp and OZ exp). The longitudinal strain,

*ε*L, and the transverse strain,

*ε*T, that is the strain in radial orthotropic direction of wood, are related to the data HA, TA, and OZ. In the data related to SB, the longitudinal strain is denoted by

*ε*LL_L and the transverse, that is the strain in tangential orthotropic direction of wood, by

*ε*TT_L, when the specimen is loaded in longitudinal direction. When the specimen is loaded in the transverse direction, that is the tangential direction, the transverse strain is denoted by

*ε*TT_T. No data of longitudinal strain are available for the latter loading mode. Total shear strain measured in tangential–longitudinal plane is denoted by

*γ*in Fig. 7. The predicted curves of viscoelastic strains in longitudinal and transverse directions, and in shear, present the best fit of particular creep model to the experimental data in loading phase.

*b*(Table 2), all three constitutive models on average underestimate the experimental data SB exp in the longitudinal and transverse direction, except the K model which on average overestimates the transverse strain,

*ε*TT_L, in SB K case where the tensile load is applied in longitudinal direction. The coefficient of variation,

*V*

_{δ}, indicates that the case SB KK gives the closest fit of measured strains in longitudinal and transverse directions separately and together, followed by the cases SB KD, and SB K. Exception is noticed in the longitudinal direction, where SB K gives better fit of

*ε*LL_L than SB KD. The case HA KK gives the best fit of experimental results HA exp in all strain directions separately and together (Table 3). According to the values of

*V*

_{δ}, the case HA KD fits the longitudinal strain,

*ε*L, better than the case HA K. The opposite is noticed by the strain in transverse direction,

*ε*T, and when the evaluation method is applied to the predicted strains in both directions together. The K model on average underestimates the experimental results TA exp in the case TA K in all directions considered separately and together. The comparison of the magnitude of

*V*

_{δ}in TA cases shows that the KK model is the best in predicting the transverse strain alone, while the longitudinal strain and strains in both directions together are better predicted by the KD model. When comparing the predicted and the experimental data OZ exp (Table 3), it can be seen that the KK model gives the best fit in longitudinal, transverse, and both directions together.

Comparison of the three coupled two-dimensional models (K, KD, and KK) on predicted viscoelastic strains in longitudinal and transverse direction separately and together against experimentally obtained data from Schniewind and Barrett (1972)

Case | SB K | SB KD | SB KK |
---|---|---|---|

| |||

LL_L + TT_L + TT_T | 1.0011 | 1.0031 | 1.0003 |

LL_L | 1.0011 | 1.0037 | 1.0004 |

TT_L | 0.9999 | 1.0001 | 1.0001 |

TT_T | 1.0075 | 1.0024 | 1.0002 |

| |||

LL_L + TT_L + TT_T | 1.12 | 0.77 | 0.25 |

LL_L | 0.21 | 0.55 | 0.18 |

TT_L | 0.87 | 0.34 | 0.13 |

TT_T | 1.64 | 1.18 | 0.39 |

RE (%) | |||

LL_L + TT_L + TT_T | 0.72 | 0.61 | 0.19 |

LL_L | 0.19 | 0.56 | 0.16 |

TT_L | 0.59 | 0.25 | 0.08 |

TT_T | 1.37 | 1.03 | 0.32 |

Comparison of the three coupled two-dimensional models (K, KD, and KK) on predicted viscoelastic strains in longitudinal and transverse direction separately and together due to constant tensile loading in longitudinal direction against experimentally obtained data from Hayashi et al. (1993) (HA), Taniguchi and Ando (2010) (TA), and Ożyhar et al. (2013) (OZ)

Case | HA K | HA KD | HA KK | TA K | TA KD | TA KK | OZ K | OZ KD | OZ KK |
---|---|---|---|---|---|---|---|---|---|

| |||||||||

L + T | 0.9999 | 0.9992 | 1.0004 | 1.0027 | 1.0000 | 0.9996 | 1.0046 | 1.0012 | 1.0015 |

L | 0.9999 | 0.9991 | 1.0004 | 1.0028 | 1.0000 | 0.9995 | 1.0050 | 1.0014 | 1.0018 |

T | 1.0003 | 0.9999 | 1.0001 | 1.0024 | 0.9997 | 1.0000 | 1.0014 | 0.9999 | 0.9999 |

| |||||||||

L + T | 0.78 | 1.47 | 0.48 | 1.27 | 0.71 | 0.80 | 1.47 | 1.39 | 0.50 |

L | 0.96 | 0.84 | 0.62 | 1.24 | 0.44 | 1.14 | 1.56 | 1.24 | 0.47 |

T | 0.60 | 1.96 | 0.31 | 1.36 | 0.93 | 0.18 | 1.43 | 1.59 | 0.52 |

RE (%) | |||||||||

L + T | 0.63 | 1.05 | 0.34 | 1.03 | 0.54 | 0.50 | 1.24 | 1.07 | 0.34 |

L | 0.77 | 0.60 | 0.46 | 0.91 | 0.35 | 0.88 | 1.33 | 0.91 | 0.27 |

T | 0.49 | 1.50 | 0.22 | 1.15 | 0.73 | 0.12 | 1.15 | 1.23 | 0.41 |

Magnitudes of the predicted and the experimental strains at final time of loading phase together with the strain growth trend are interesting to observe, in particular, in cases, when the predicted strain rates are significantly smaller than the experimentally measured. This type of underestimation of prediction will be severe when extrapolated. The strain predictions of K models is such a case, where the strain rate at the end of predicted results in loading phase is negligible, while the strain rate of experimental results is still evident. Besides the observed variations in strains growth trend, the highest RE at final time of loading phase is noticed in the K model prediction of the transverse strain *ε*TT_L (Fig. 3c), RE (*t* = 16.67 h) = 2.0% in the case SB K. The KD model overestimates the strains at final observed time of loading phase in the cases of the longitudinal strains *ε*L (HA KD, Fig. 4a and TA KD, Fig. 5a), the transverse strain *ε*T (TA KD, Fig. 5b), and the transverse *ε*TT_T and the longitudinal strain *ε*LL_L (SB KD, Fig. 3a, b, respectively). The most remarkable overestimation of the KD model is noticed in the case TA KD (*t* = 24 h) and results in RE = 1.7% when predicting the strain in the transverse direction *ε*T (Fig. 5b). In all other cases, KD model underestimates the values of strains at final time. The largest underestimation is noticed in the prediction of the transverse strain *ε*T in the case OZ KD (Fig. 6b), which results in RE = 1.7% at *t* = 24 h. The KK model tends to slightly underestimate the strains at final observed times. The maximum underestimation is noticed in the predictions of the longitudinal strain *ε*L at *t* = 24 h in the case TA KK (Fig. 5a) and amounts to RE = 1.6%. It can be concluded that the K and KK models have a tendency to underestimate the strains at final time of loading phase, which could potentially be unsafe, while the KD model can both under and overestimate the strains at final observed time of loading phase of wood specimen under tension.

*V*

_{δ}= 0.21%. The worst fit is obtained by the K model, where the viscoelastic shear creep strain reaches its limit value

*γ*= 1.31 × 10

^{−2}at

*t*= 6.67 h. In that case, the relative error of predicted shear strain at final time equals to RE = 2.3%.

Comparison of the three different constitutive models for shear (K, KD, and KK) on numerically predicted shear strains against experimentally obtained data from Schniewind and Barrett (1972)

Shear | SB K | SB KD | SB KK |
---|---|---|---|

| 1.0030 | 1.0004 | 1.0001 |

| 1.49 | 0.48 | 0.21 |

RE (%) | 1.25 | 0.38 | 0.18 |

Generally, the results in Figs. 3, 4, 5, 6, and 7 are presented on a large scale that makes the behavior of the constitutive models’ predictions quite visible and could on some occasions give false impression on the accuracy of the results. Though there are some visible differences in models predictions, RE of predicted strains in all cases is less or equal to 1.50%. The error can be considered as very small taking into account the reasonable likelihood that some amount of error has already been introduced by measuring and plotting the results, and finally by extracting the data from pictorial in the papers, since no original measurements are available to the authors.

The experimental data by Taniguchi and Ando (2010) include not only the viscoelastic strain response of a wood specimen under constant tensile load in longitudinal direction but also the response after unloading (Fig. 5). The unloading and the following creep return are predicted by the three constitutive models where the material parameters determined by fitting procedure of the loading phase are used (Table A.1 in Online Resource). It can be seen (Fig. 5) that the models predict the strain recovery after unloading relatively poor. The elastic return is predicted perfectly by all models, followed by complete strain recovery by the K and KK models in both directions, since they cannot retain non-recoverable strain. On the other hand, the KD model is able to simultaneously predict unrecoverable part of the total strain in longitudinal and transverse direction. It is obvious that after 24 h of recovery, the longitudinal strain prediction is much better than the transverse strain prediction of the KD model.

The predicted values of the material parameters applicable to all three models are given in Tables A1 and A2 in Online Resource and present another significant outcome of the performed analysis. Due to requirements of the formulation of the models, the material parameters are positive values of the magnitude up to 10^{17}. They are given with precision of 15 digits, since it is observed that a relatively small variation (on a 14th decimal place) of some parameters can cause a noticeable change in the strain response. Sensitivity of the material parameters is not studied in the present paper; therefore, it is decided to show all the digits of the material parameters as they are derived in the analysis.

## Conclusion

The paper shows applicability and formulation of the three coupled two-dimensional creep models for orthotropic material based on the theory presented by Frandsen (2007). Complexity of the models differs and is connected to the number of material parameters needed in the particular formulation. The models are calibrated against the experimental data and the material parameters are determined for four wood species when loaded in constant tension, namely Douglas fir, spruce, Japanese cypress, and European beech. Additionally, the material parameters for Douglas fir exposed to constant shear load are given. Therefore, this paper delivers formulations of the three completely coupled two-dimensional constitutive orthotropic models together with numerical values of the material parameters for Douglas fir wood species. According to the comparison method for experimental and numerical data, it can be concluded that all the models are capable to predict viscoelastic creep of wood for the time periods studied. Nevertheless, the comparison method shows that the most complex model (KK) also gives the best predictions of strains in the majority of analyzed cases; when accounting for relatively small errors already, the simplest model (K) can be considered as useful enough for describing viscoelastic creep of wood in two dimensions. That may prove as important when additional phenomena connected to the long-term behavior of wood, i.e., mechano-sorptive creep, are included in the model, and at the same time, the number of unknown material parameters is desired to be kept as few as possible. Application of the constitutive models in the finite element software and simultaneous fitting procedure to the available experimental data allows obtaining numerical values of the material parameters for different wood species.

The presented extensive analysis of the three coupled two-dimensional models of viscoelastic creep of wood initiated some ideas for future work. To get more detailed insight into models’ functioning and find the degree of influence of a particular material parameter on final output in different time periods, a sensitivity analysis is needed. It could be helpful in finding potential relations between the parameters and reducing the number of digits required as well. Studying the ability of the models to predict mechano-sorptive creep, and their improvement and validation for cyclic loading might also be interesting. A great challenge would present an upgrade of the models’ formulations to three dimensions that are able to describe a complete long-term behavior of timber elements under changing environmental conditions. Certainly, the presented rheological constitutive models have a potential to complement or reduce experimental work needed for finding accurate and sufficient values of the material parameters of various wood species that will allow optimal design of timber structures.

## Notes

### Acknowledgements

The financial support by Gunnar Ivarson’s Foundation (Gunnar Ivarsons Stiftelse för Hållbart Samhällsbyggande, GIS) made this work possible.

## Supplementary material

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