Hygro-mechanical long-term behaviour of spruce, pine and lime wood: parameter identification and model validation

Abstract

Lime wood, spruce and pine are investigated with regard to its hygro-mechanical long-term behaviour. Experiments are conducted for an identification of model parameters and for model validation. Swelling and shrinkage coefficients, dry density, sorption characteristics and parameters for visco-elasticity, visco-plasticity and mechano-sorption are determined for the main material directions. Supplemented by literature values, a complete set of parameters for long-term hygro-mechanical modelling of wood species is found. Constrained swelling and shrinkage are analysed and the origin of the stress development is investigated. It is demonstrated, that creep phenomena lead to significant stress reduction by relaxation, in case of moisture changes especially due to mechano-sorption. The influence of different model parts is investigated. A numerical parameter study shows the influence of several material parameters on the stress evolution. Experimental material investigations such as those presented here are essential for the application of numerical simulation methods for the prediction of material behaviour and for the assessment of deformations, stresses and damage potential of climatically loaded timber structures.

Introduction

As naturally grown raw material, wood is increasingly in demand due to its sustainability, but it is also difficult to characterise due to its inhomogeneous and anisotropic structure. Wood is especially known for its pronounced hygroscopicity, leading to considerable swelling and shrinkage strains induced by moisture variations. Moisture gradients and swelling-/shrinkage-constraints lead to moisture-induced stresses. As a result, the ambient climate has a substantial influence on the mechanical state of wood and thus can be an origin of damage.

Stresses resulting from constrained hygroexpansion are studied experimentally for several decades, mostly regarding swelling pressure, but also shrinkage tension. An overview on the first publications in this field is found in Perkitny and Kingston (1972). Experiments are published investigating swelling pressure for single humidification steps (Perkitny 1961; Krauss 2004; Virta et al. 2005; Koponen and Virta 2004) as well as for moisture cycles (Rybarczyk and Ganowicz 1974; Mazzanti et al. 2014) or considering temperature influence (Bolton et al. 1974). The results cannot be explained solely by hygroexpansion and the mechanical short-term behaviour of wood (i.e. elasticity, plasticity), because relaxation is observed. This behaviour is in Rybarczyk and Ganowicz (1974), Virta et al. (2005), Svensson and Toratti (2002), Koponen and Virta (2004), Mazzanti et al. (2014) attributed to viscous creep and the mechano-sorptive effect, leading to lower swelling pressure caused by simultaneous mechanical constraints than would be required to reverse free swelling afterwards. According to Svensson and Toratti (2002), Virta et al. (2005), mechano-sorption has the largest influence on the described relaxation.

For a good understanding of these relationships, knowledge of creep behaviour of wood is required. Compression creep tests are reported in Toratti and Svensson (2000), Bengtsson et al. (2022), Svensson and Toratti (2002), tension creep tests in Toratti and Svensson (2000), Dubois et al. (2005), Svensson and Toratti (2002), showing visco-elastic and visco-plastic creep at long-term loads. Mechano-sorption, i.e. the effect of enlarged creep deformations at mechanical load and simultaneous moisture change is firstly reported in Armstrong and Christensen (1961). Further studies, e.g. Toratti and Svensson (2000), Svensson and Toratti (2002), Huang (2016), investigate the creep strain development at moisture cycles, discussing approaches to explain the effect. But due to the high degree of complexity of the phenomenon and large effort for comprehensive experimental investigations, mechano-sorption is still not sufficiently described. Experimental investigations are rare, which is crucial with respect to its large influence on the long-term behaviour.

Experimental investigations of creep effects often go hand in hand with modelling approaches and numerical simulations. Wood is mostly represented by a rheological formulation containing serial elements for elasticity, visco-elasticity and hygroexpansion, partly supplemented by plastic, visco-plastic, mechano-sorptive and additional irrecoverable parts. Visco-elasticity is described by a Maxwell-element (Afshar et al. 2020; Bengtsson et al. 2022) or Kelvin-element (Hanhijärvi 2000; Hassani et al. 2013; Huč et al. 2018; Reichel and Kaliske 2015a; Dubois et al. 2005). For mechano-sorption, different approaches exist. Rybarczyk and Ganowicz (1974) proposes a simple mathematical description, Hanhijärvi (2000), Fortino et al. (2009), Hassani et al. (2013), Huč et al. (2018) introduce a further moisture change dependent Kelvin-type element and Reichel and Kaliske (2015b), Dubois et al. (2005) describe mechano-sorption as enlargement or modification of visco-elastic creep. Short-term plasticity is added to the model in Hassani et al. (2013) and visco-plasticity in Reichel and Kaliske (2015a). Due to the significance of realistic moisture content simulations for these multi-physical processes, multi-Fickian moisture transport is considered in coupled hygro-mechanical analysis as sequential (Fortino et al. 2019a, b) or monolithic algorithm (Konopka and Kaliske 2018; Stöcklein and Kaliske 2023). The latter combined with the viscous and mechano-sorptive creep model in Reichel and Kaliske (2015a, 2015b) is used in this study for the material model parameter determination and is shortly presented in Sect. Constitutive material model.

Focus of this study is the experimental investigation of wood material behaviour at constrained swelling and shrinkage, accompanied by simulations supporting the understanding of mechanisms in wood. The numerical models are in general introduced in previous publications. This study contains a combined analysis of free swelling/shrinkage and creep experiments in order to find a model parameter set for hygroexpansion, visco-elasticity, visco-plasticity and mechano-sorption, as well as constrained swelling and shrinkage tests to validate the model and to analyse the influence of model contributions. The selection of materials is based on two wooden cultural heritage objects studied in the research project CultWood in cooperation with the Dresden State Art Collections. An altarpiece by L. Cranach the Elder (1506) of St. Catherine from the castle church Wittenberg (today Dresden State Art Collections), constructed of lime wood panels and a support system of pine and spruce (Gebhardt et al. 2018; Stöcklein and Kaliske 2023), and a painted cupboard from the 18th century with Upper Lusatian provenance (today Dresden State Art Collections) are used as examples. The hygro-mechanical behaviour of wooden artwork is due to its sensitivity to deformation of special interest.

This paper can be seen as pre-study for the investigation of the complex wooden cultural heritage objects. In order to describe the material sufficiently, small wood samples are tested in the three principal anatomical directions. Standard swelling and shrinkage tests are carried out to determine the hygroexpansion properties as well as sorption characteristics. Uniaxial compression creep tests are conducted in constant and in variable relative humidity (RH) to identify parameters of visco-elastic and visco-plastic creep and mechano-sorption. Two further experiments are carried out regarding moisture-induced stress evolution by constrained swelling and shrinkage. By this, the effect of creep and especially of mechano-sorption on the stress magnitude in constrained wood exposed to climatic changes is demonstrated.

Materials and methods

Materials

Three different types of wood are investigated in this study: scots pine (Pinus sylvestris), European spruce (Picea abies) and small-leaved lime wood (Tilia cordata). The board and plank materials used for the experiments come from a regional sawmill that obtains its logs from the eastern Erzgebirge (Ore Mountains, Germany). The exact location of growth could no longer be identified. Intended for use in monument conservation, restoration of art objects and sculpting, the material is not conditioned in a drying chamber to the required working moisture content $\omega$, as is common today, but is dried traditionally in an open air stack. Very evenly grown parts of the boards with closely spaced annual rings ($\ge$ 10/10 mm) and free of defects are chosen. Before and during the manufacturing process, all raw materials for the production of the wooden test specimens are conditioned to 65 % RH at 22$\,^\circ$C room temperature, as these are the climatic conditions in the workshop.

The average dry density $\rho _0$ of the present wood types are $491\,\hbox {kg}\,\hbox {m}^{-3}$ for spruce, $521\,\hbox {kg}\,\hbox {m}^{-3}$ for lime wood, $501\,\hbox {kg}\,\hbox {m}^{-3}$ for the sapwood (sw.) of pine and $570\,\hbox {kg}\,\hbox {m}^{-3}$ for the heartwood (hw.) of pine.

The sample geometries for the different experiments are described in Sect. Experimental methods and presented in Figs. 1, 2, 3 and 4.

Experimental methods

Firstly, swelling and shrinkage experiments are carried out in order to use reliable hygroexpansion parameters and sorption characteristics for the moisture dependent modelling. Then, creep experiments are conducted to investigate the long-term behaviour and to determine viscosity parameters for the numerical simulations. Furthermore, moisture-induced stress evolution and relaxation in simple wood specimens are investigated by constrained swelling and shrinkage experiments in order to investigate and validate the hygromechanically coupled behaviour.

Swelling and shrinkage

The swelling and shrinkage experiment is carried out according to DIN 52184. The sample geometry and the climatic regime are shown in Fig. 1. A cuboid sample with 20 mm in radial (R) and tangential (T) direction and 10 mm in longitudinal (L) direction is tested to obtain the hygroexpansion coefficients across the fibre direction, a sample with the same radial and tangential dimensions and 100 mm length is tested to obtain the coefficient in longitudinal direction. The samples are exposed to the climate steps shown in Fig. 1 until the equilibrium moisture content is reached. The equilibrium is defined as reached, when the mass change is $< 0.1$ % for 24 h. After each step, weight and dimensions are measured. The last steps are water storage and oven drying. By the mass and dimension values after drying, the dry density is determined, which allows the moisture content at each humidity step to be calculated. The differences of moisture and dimensions lead to the hygroexpansion coefficients, the moisture content provides information about the sorption properties.

Fig. 1

Setting of the swelling and shrinkage experiment: a samples for the determination of ① radial/tangential and ② longitudinal swelling and shrinkage; b climatic conditions

Creep experiments

The aim of the creep experiments is to identify parameters for a comprehensive model describing all relevant creep phenomena. Firstly, observable characteristics are introduced, while the modelling is explained in Sect. Constitutive material model. One quantity to describe the visco-elastic behaviour of the first creep phase is the creep limit $\varphi _\infty ^{ve} = \varepsilon _\infty ^{ve}/\varepsilon ^{el}$, which is the ratio of visco-elastic strain after infinite loading time $\varepsilon _\infty ^{ve}$ to elastic strain $\varepsilon ^{el}$. A measure of the rate of visco-elastic strain development is the half-value time $t^{1/2}$, defined by ${\varepsilon ^{ve}(t^{1/2})=0.5\cdot \varepsilon _\infty ^{ve}}$. In terms of the modelling (Sect. Constitutive material model), the half-value time is converted to the relaxation time $\tau ^{ve} = t^{1/2}/\ln (2)$. In the second creep phase, additionally visco-plastic creep strains develop, characterised by the plastic viscosity $\eta ^{vp}$, which is the increment of visco-plastic strain per time. Visco-plastic strain only occurs above a stress level threshold, therefore, a visco-plastic yield stress $\sigma _y^{vp}$ is introduced as threshold. The limit of linearity $L_L=\sigma _y^{vp}/\sigma _y$ characterises the beginning of visco-plastic strain development depending on the (moisture-dependent) short-term yield stress $\sigma _y$. Mechano-sorptive creep can be described as enlargement of visco-elastic creep in case of simultaneous moisture content change. It is characterised by the enlargement factor $\kappa$. Cuboid samples with a geometry given in Fig. 2a are loaded with a constant compression force for 2 weeks followed by an unloading time of again two weeks. The load is generated by weights and a lever construction, see Fig. 2c. One side of each sample has a speckle pattern, which is observed by a camera (MixMart HD USB Digital Microscope) in a region of interest of about 15 by 15 mms (Fig. 2d), taking a picture about every seven minutes. The pictures are evaluated by digital image correlation (DIC) using the open source software DICe by Turner et al. (2015) to obtain the strain in loading direction.

Fig. 2

Setting of the creep experiments: a samples loaded in ① longitudinal, ② radial and ③ tangential direction; b FE-models for simulations at changing climate and constant climate with loading in radial/tangential direction and for simulations with loading in longitudinal direction; c schematic side view of one test stand; d speckle pattern for digital image correlation and the area covered by the camera; e all ten test stands with ten cameras in the climate chamber; f mechanical and hygric boundary conditions for experiments with constant climate and changing climate

The test bench contains 10 test stands, each with one weight, lever construction and camera, respectively. One specimen is unloaded, three are loaded at around 20%, three at 40% and three at 60% of the short-term yield stress in loading direction as technical compression stress. The short-term yield stress is taken from Konopka et al. (2024) for lime wood and Reichel and Kaliske (2015a) for spruce and pine. For tangential and radial loading direction, the yield stresses differ rarely, therefore, the same pressure is applied. In contrast, for longitudinal direction, the yield stress and, thereby, the loading pressure is much larger. Nine test runs are carried out, differing in the tested wood species, humidity and loading direction. Figure 2f shows the humidity and load regime. The details are given in Table 1, where the test runs are specified.

Table 1 Creep experiment: specification of the samples and boundary conditions of the nine different experimental runs

For the data processing, the software DICe produces a point cloud depending on the speckle pattern and calculates the strain at the points. In order to model clear wood as a homogeneous material on a macroscopic scale, the strain is determined as an average over the region of interest. The evaluation of the photos provides also strains $\gg 1$ or equal to zero, therefore, the maximum and minimum 10 % of the values and zeros are cut off and the average is calculated of the remaining data. Finally, a time-strain characteristic is obtained for every test. To get the visco-elastic parameters $\varphi _\infty ^{ve}$ and $\tau ^{ve}$, the unloading paths of all constant climate tests for each wood type and loading direction, respectively, are normalised by elastic strain $\varphi ^{ve}=\varepsilon /\varepsilon ^{el}-1$ in order to fit a function ${\varphi ^{ve}(t)=(1-\exp (-t/\tau ^{ve}))\cdot \varphi _\infty ^{ve}}$. For the visco-plastic parameters, the elastic and visco-elastic strain is subtracted from the loading paths of the same tests. The remaining visco-plastic strain at the end of the loading period is divided by the loading time, evaluated for each load level. Then, 3 data points at 3 different load levels are used to fit the linear function $\varepsilon ^{vp}/t = \sigma /\eta ^{vp} + C_1$, where $C_1$ leads to the visco-plastic yield stress $\sigma _y^{vp}=-C_1\cdot \eta ^{vp}$. The parameter $\kappa$ for the mechano-sorptive effect cannot be extracted directly, because the effect is complex and combined with moisture content change. It is fitted by inverse analysis simulating the experimental run with changing climate. Due to the enormous effort of space and energy consumption, not every possible combination of wood type and loading direction is tested. Therefore, the not tested quantities are estimated by assuming the same R/T- or L/T-ratio as lime wood, which is tested for all loading directions. Mechano-sorption is only tested for lime wood in tangential direction. In the simulations, the same mechano-sorptive behaviour is assumed for all wood types and loading directions.

For comparison, the experimental runs are simulated by the finite element method (FEM) using the determined parameters of the experiments. The material model is described in Sect. Constitutive material model. The structural models can be seen in Fig. 2b. A quarter of the specimens is simulated, taking advantage of symmetry. The geometry is meshed with hexahedral 20 node elements using quadratic shape functions and 27 integration points per element. For test run six, a finer mesh is necessary due to the changing climatic conditions and the resulting moisture gradients.

Moisture-induced pressure

Aim of the third experiment is the validation of the material modelling for moisture-induced pressure. After drying cuboid test specimens (see Fig. 3a) for $65 - 67\,\hbox {h}$ at $103\,^{\circ }\textrm{C}$, they are installed in a rigid frame, preventing expansion in the direction of $60\,\hbox {mm}$ length. Then, the RH is conditioned to 85 % in order to remove the sample after 48 h, measure it and finally dry it again. During the experiment, the reaction force is measured by a Burster load cell of type 8524-6010-V206 continuously in the direction, where swelling deformation is prevented. For validation, the experiment is simulated using the material modelling described in Sect. Constitutive material model and the material parameters obtained in the creep experiment. The FE discretisation is shown in Fig. 3d. Utilising the symmetry, only a part of the specimen is simulated. The nodal forces at the fixed top surface are added up and multiplied by four to obtain the quantity measured by the load cell in the experiment.

Fig. 3

Setting of the moisture-induced pressure experiment: a samples for prevented swelling in ① radial and ② tangential direction; b schematic setup; c climatic boundary conditions; d FE-model of a sample part, symmetry planes in grey; e experimental setup

Moisture-induced tension

By the fourth experiment, the third experiment is adapted for moisture-induced tension. Shape and dimensions of the samples can be seen in Fig. 4a and b. They are fixed in a test rig in which contraction is prevented (Fig. 4c, d). The test specimens are conditioned in a climate of 85 % RH and installed in the test rig with a pre-stressing force of $20\,\hbox {N}$. Afterwards, the specimens are dried to 20% RH. The drying process is measured for about 36 h. Analogously to the experiments of moisture-induced pressure, the reaction force is measured by an Ahlborn load cell of type K-25 continuously in the direction, where shrinkage deformation is prevented. The FE-discretisation is shown in Fig. 4f. Utilising the symmetry, only one eighth of the specimen is simulated. The nodal forces at the fixed surface are added up and multiplied by 4 to obtain the quantity measured by the load cell in the experiment.

Fig. 4

Moisture-induced tension experiment: a samples for tension in ① radial and ② tangential direction; b samples with blocks as holding for the constraint; c schematic setup; d setup with 3 test stands; e climatic boundary conditions; f FE-model of a sample part, symmetry planes in grey

Constitutive material model

All simulations are carried out with the in-house FE-software WoodFEM. The implemented material models that are applied in this study are briefly presented subsequently. Further implementation details are given in the Supplementary Material, part B. A detailed description can be found in Saft and Kaliske (2011), Reichel and Kaliske (2015a) and Konopka and Kaliske (2018).

The constitutive material model describes elasticity, visco-elasticity, visco-plasticity, hygroexpansion, mechano-sorption, moisture transport and moisture exchange. Wood is assumed to be orthotropic. The material directions radial (R), tangential (T) and longitudinal (L) are perpendicular to each other at every material point. Wood behaves strongly anisotropic, therefore, material tests for the parameter identification have to be conducted at least in all principal directions. The material behaviour of wood is illustrated in Reichel (2015) for the one-dimensional case by the rheological model shown in Fig. 5. It consists of a spring, a Kelvin-element, a Bingham-element and elements for hygroexpansion and mechano-sorption, connected in series. With this serial connection, the total strain is split additively into elastic, visco-elastic, visco-plastic, hygroexpansion and mechano-sorptive parts.

Fig. 5

One-dimensional rheological model for the material behaviour of wood and the accompanying characteristic (moisture dependent) material parameters

Elasticity, represented by the spring in series in the rheological model and characterised by the Young’s modulus $E^{el}$ is described by Hooke’s law, which reads in three-dimensional form

$$\begin{aligned} {{\underline{\varvec{\sigma }}}}={{\underline{\underline{\varvec{C}}}}}^{el}(\omega ,\rho ):{{\underline{\varvec{\varepsilon }}}}^{el}. \end{aligned}$$

(1)

The elasticity tensor ${{\underline{\underline{\varvec{C}}}}}^{el}(\omega ,\rho )$ contains moisture- and density-dependent Young’s moduli $E_R, \, E_T, \, E_L$, shear moduli $G_{RT}, \, G_{TL}, \, G_{RL}$ and Poisson’s ratios $\nu _{RT},\,\nu _{TL},\,\nu _{RL}$ for the radial (R), tangential (T) and longitudinal (L) direction. For orthotropic material, the material tensor is described e.g. in Reichel (2015) in Voigt notation as

$$\begin{aligned} {{{\underline{\underline{\varvec{C}}}}}^{el}}^{-1}=\left[ \begin{array}{c c c c c c} \frac{1}{E_R} &{} -\frac{\nu _{RT}}{E_T} &{} -\frac{\nu _{RL}}{E_L} &{} 0 &{} 0 &{} 0 \\ -\frac{\nu _{RT}}{E_T} &{} \frac{1}{E_T} &{} -\frac{\nu _{TL}}{E_L} &{} 0 &{} 0 &{} 0 \\ -\frac{\nu _{RL}}{E_L} &{} -\frac{\nu _{TL}}{E_L} &{} \frac{1}{E_L} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{G_{RT}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{G_{TL}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{G_{RL}} \\ \end{array}\right] . \end{aligned}$$

(2)

The moisture- and density-dependent elastic parameters are described by the quadratic equation

$$\begin{aligned} X(\rho,\omega ) = X_{ref}\cdot (a_1\cdot \rho ^2 +a_2\cdot \rho +a_3)(b_1\cdot \omega ^2+b_2\cdot \omega +b_3), \qquad X \in \{E,\,G\}, \end{aligned}$$

(3)

which is valid for

$$\begin{aligned} \rho<0.85{\frac{\textrm{g}}{\textrm{cm}^{3}}},\;0.05<\omega <0.25, \; T=20^\circ C. \end{aligned}$$

(4)

Literature values for the parameters of Eq. (3) and for Poisson's ratios are given in the Supplementary Material, part A.

The Kelvin-element for the visco-elastic behaviour consists of an elastic spring, characterised by the stiffness $E^{ve}$ that is derived by the creep limit $\varphi _\infty ^{ve}=E^{ve}/E^{el}$, and a dashpot, characterised by the viscosity $\eta ^{ve}$ that is derived by the relaxation time $\tau ^{ve}=\eta ^{ve}/E^{ve}$. In contrast to Reichel (2015), the viscous stiffness tensor

$$\begin{aligned} {{{\underline{\underline{\varvec{C}}}}}^{ve}}^{-1}=\left[ \begin{array}{c c c c c c} \frac{1}{E_R^{ve}} &{} -\frac{\nu _{RT}}{E_T^{ve}} &{} -\frac{\nu _{RL}}{E_L^{ve}} &{} 0 &{} 0 &{} 0 \\ -\frac{\nu _{RT}}{E_T^{ve}} &{} \frac{1}{E_T^{ve}} &{} -\frac{\nu _{TL}}{E_L^{ve}} &{} 0 &{} 0 &{} 0 \\ -\frac{\nu _{RL}}{E_L^{ve}} &{} -\frac{\nu _{TL}}{E_L^{ve}} &{} \frac{1}{E_L^{ve}} &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \frac{1}{G_{RT}^{ve}} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{G_{TL}^{ve}} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{G_{RL}^{ve}} \\ \end{array}\right] \end{aligned}$$

(5)

is obtained inserting the elastic moduli depending on the creep limit as described. The creep limit is chosen as material parameter, because it is easy to determine by an experimentally tested creep strain evolution. The tensor of the relaxation times

$$\begin{aligned} {{\underline{\underline{\varvec{T}}}}}^{ve}=\left[ \begin{array}{c c c c c c} \tau ^{ve}_R &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} \tau ^{ve}_T &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \tau ^{ve}_L &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} \tau ^{ve}_{RT} &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \tau ^{ve}_{TL} &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \tau ^{ve}_{RL} \\ \end{array}\right] \end{aligned}$$

(6)

is assumed to be a diagonal tensor, which means, that there is only one viscosity and relaxation time for each creep direction, independent of the direction of load. The differential equation for the visco-elastic strain

$$\begin{aligned} \dot{{{\underline{\varvec{\varepsilon }}}}}^{ve}+{{{\underline{\underline{\varvec{T}}}}}^{ve}}^{-1}:{{\underline{\varvec{\varepsilon }}}}^{ve} ={{{\underline{\underline{\varvec{T}}}}}^{ve}}^{-1}:\left( {{\underline{\underline{\varvec{C}}}}}^{ve}\right) ^{-1}:{{\underline{\varvec{\sigma }}}} \end{aligned}$$

(7)

is discretised with respect to time and transformed in order to obtain the stress–strain relation based on the known state of the last time step. The parameters $\varphi _\infty ^{ve}$ and $\tau ^{ve}$ are investigated in the creep experiment.

The Bingham-element for the visco-plastic behaviour consists of a St. Venant-element representing the onset of plastic deformation when the limit of linearity is exceeded, defined by the yield stress $\sigma _y^{vp}$, and a dashpot, defined by the visco-plastic viscosity $\eta ^{vp}$. According to Reichel (2015), the visco-plastic viscosity tensor ${{\underline{\underline{\varvec{\eta }}}}}^{vp}$ is assumed to be diagonal with one viscosity for each strain direction. The stress–strain relation of the Bingham-element is

$$\begin{aligned} {{\underline{\varvec{\sigma }}}}={{\underline{\varvec{\sigma }}}}_{SV} + {{\underline{\underline{\varvec{\eta }}}}}^{vp}:\dot{{{\underline{\varvec{\varepsilon }}}}}^{vp}, \end{aligned}$$

(8)

where ${{\underline{\varvec{\sigma }}}}_{SV}$ is the stress of the St. Venant-element. To the St. Venant-element applies the flow condition

$$\begin{aligned} f={{\underline{\varvec{\sigma }}}}_{SV}:{{\underline{\underline{\varvec{b}}}}}:{{\underline{\varvec{\sigma }}}}_{SV}-1 \end{aligned}$$

(9)

with

$$\begin{aligned} {{\underline{\underline{\varvec{b}}}}}=\left[ \begin{array}{c c c c c c} 1/\sigma _{y_{t/c,R}}^{{vp}^2} &{}0&{}0&{}0&{}0&{}0\\ 0&{}1/\sigma _{y_{t/c,T}}^{{vp}^2}&{}0&{}0&{}0&{}0\\ 0&{}0&{}1/\sigma _{y_{t/c,L}}^{{vp}^2}&{}0&{}0&{}0\\ 0&{}0&{}0&{}1/\sigma _{y_{v,RT}}^{{vp}^2}&{}0&{}0\\ 0&{}0&{}0&{}0&{}1/\sigma _{y_{v,TL}}^{{vp}^2}&{}0\\ 0&{}0&{}0&{}0&{}0&{}1/\sigma _{y_{v,RL}}^{{vp}^2}\\ \end{array}\right] , \end{aligned}$$

(10)

the Kuhn-Tucker inequality

$$\begin{aligned} f({{\underline{\varvec{\sigma }}}}_{SV})\le 0,\quad \Delta \gamma \ge 0, \quad f({{\underline{\varvec{\sigma }}}}_{SV})\cdot \Delta \gamma =0, \end{aligned}$$

(11)

and the flow rule

$$\begin{aligned} \dot{{{\underline{\varvec{\varepsilon }}}}}^{vp}=\Delta \gamma \cdot \frac{\partial f({{\underline{\varvec{\sigma }}}}_{SV})}{\partial {{\underline{\varvec{\sigma }}}}_{SV}}. \end{aligned}$$

(12)

In these equations, $\sigma ^{vp}_{y_{i,j}}$ is the visco-plastic yield stress, which is expressed related to the short-term elastic limit f

$$\begin{aligned} \sigma ^{vp}_{y_{i,j}}={L_L}_{i,j}\cdot {f_{i,j}}\quad i\in \{t,c,v\},\,j \in \{R,T,L,RT,TL,RL\}. \end{aligned}$$

(13)

A simple quadratic moisture-dependent formulation is used for ${f}$,

$$\begin{aligned} {f} = a\cdot \omega ^2 + b\cdot \omega + c. \end{aligned}$$

(14)

The parameters for Eq. (14) can be found in the Supplementary Material, part A. $\Delta \gamma$ is the consistency parameter for the evolution of visco-plastic strain. If the flow condition is met, the visco-plastic strain is updated iteratively. The visco-plastic viscosity, which is developed and implemented in WoodFEM in Reichel and Kaliske (2015a), is modelled depending on the stress level $SL=\sigma /{f}$ and on the strain energy density e

$$\begin{aligned} \eta ^{vp}_{SL}= &\, {} \eta ^{vp}_{start}\cdot \left[ a_1+\frac{1-a_1}{2}-\frac{1-a_1}{\pi }\cdot {\textrm{arctan}}((SL-0.5)\cdot b_1)\right] , \end{aligned}$$

(15)

$$\begin{aligned} \eta ^{vp}= & \,{} \eta ^{vp}_{SL}\cdot \left[ a_2+\frac{1-a_2}{2}-\frac{1-a_2}{\pi }\cdot {\textrm{arctan}}\left( (e-0.9\cdot e_{crit})\cdot \frac{b_2}{e_{crit}}\right) \right] . \end{aligned}$$

(16)

The parameters $a_1$, $b_1$, $a_2$, $b_2$ and the critical strain energy density for creep failure $e_{crit}$ are not investigated within this study and are taken from Reichel and Kaliske (2015a), see the Supplementary Material, part A. In the experiments, the limit of linearity $L_L$ and the start value of the visco-plastic viscosity $\eta ^{vp}_{start}$ are determined.

For diffusion, a multi-Fickian model introduced by Frandsen (2007), based on Krabbenhøft and Damkilde (2004) considering two phases of moisture, bound water and water vapour, is implemented in WoodFEM in Konopka and Kaliske (2018). The diffusion of both phases is described by Fick’s law

$$\begin{aligned} \frac{\textrm{d} c_b}{\textrm{d}t}= & \,{} \nabla \cdot ({{\underline{\varvec{D}}}}_b\cdot \nabla c_b)+{\dot{c}}, \end{aligned}$$

(17)

$$\begin{aligned} \frac{\textrm{d}c_v}{\textrm{d}t}= & \,{} \nabla \cdot ({{\underline{\varvec{D}}}}_v\cdot \nabla c_v)-{\dot{c}}, \end{aligned}$$

(18)

for the concentration of bound water $c_b$ and water vapour $c_v$, respectively. With the water vapour pressure $p_v$ as state variable, as it is common in multi-Fickian modelling, and related to the lumen fraction by the porosity $\phi$, Eq. (18) is formulated as

$$\begin{aligned} \phi \frac{\textrm{d}p_v}{\textrm{d}t}=\phi \nabla \cdot ({{\underline{\varvec{D}}}}_v\cdot \nabla p_v)-\frac{R\cdot T}{M_{H_2O}}\cdot {\dot{c}}. \end{aligned}$$

(19)

The diffusion tensor ${{\underline{\varvec{D}}}}_b$ is described in Siau (1984) and ${{\underline{\varvec{D}}}}_v$ in Frandsen et al. (2007), see Supplementary Material, part A. The sorption rate ${\dot{c}}$ is the phase transition between water vapour and bound water. It is described in Frandsen et al. (2007) by

$$\begin{aligned} {\dot{c}}=H\cdot ({c_b}_{eq}-c_b) \end{aligned}$$

(20)

with an exponential function H for the sorption velocity, see Supplementary Material, part A. ${c_b}_{eq}$ is the bound water concentration which is in equilibrium with the water vapour pressure $p_v$. For the equilibrium of bound water and water vapour, two different models, giving a relationship between the moisture content $\omega =c_b/\rho _0$ and the relative humidity $RH=p_v/p_{sat}$, are calibrated by experimental results. The first model is an averaged sorption isotherm using the Anderson-McCarthy (Anderson and McCarthy 1963) model

$$\begin{aligned} \omega =-\ln (\ln (1/RH)/f_1)/f_2. \end{aligned}$$

(21)

The second one is a model by Frandsen (2007) that takes hysteresis into account, using sorption curves as in Eq. (21) for ad- and desorption, indices a and b, respectively, with temperature-dependent parameters

$$\begin{aligned} f_i^\alpha =b_{i0}^\alpha +b_{i1}^\alpha \cdot T, \qquad i\in \{1,2\}, \qquad \alpha \in \{a,d\}. \end{aligned}$$

(22)

For the transition between ad- and desorption state, scanning curves

$$\begin{aligned} s(RH)={\left\{ \begin{array}{ll} -1+2^{\left( \frac{1-RH}{1-RH_d^0}\right) ^{\left( \frac{d_1}{\ln (d2(1-RH_d^0))}\right) }} &{} {\dot{RH}}>0\wedge s_0>0 \\ 2-2^{\left( \frac{RH}{RH_a^0}\right) ^{\left( \frac{d_1}{\ln (d2\cdot RH_a^0)}\right) }} &{} {\dot{RH}}<0\wedge s_0<1 \\ 0 &{} {\dot{RH}}>0\wedge s_0=0 \\ 1 &{} {\dot{RH}}<0\wedge s_0=1 \\ \end{array}\right. } \end{aligned}$$

(23)

are defined, where $RH_a^0$ and $RH_d^0$ are the contact points of the scanning curve with the ad- and desorption curve. The moisture content is then calculated by

$$\begin{aligned} \omega _b = \omega _a+s\cdot (\omega _d-\omega _a). \end{aligned}$$

(24)

By the swelling and shrinkage experiments, the model parameters $f_1$ and $f_2$ of the Anderson-McCarthy model and the parameters $d_1$, $d_2$, $b_{10}^a$, $b_{20}^a$, $b_{10}^d$ and $b_{20}^d$ of the hysteresis model are determined. Temperature-dependency is not investigated.

Swelling and shrinkage strains are calculated by the equation

$$\begin{aligned} {{\underline{\varvec{\varepsilon }}}}^{he}={{\underline{\varvec{\beta }}}}\cdot (\omega -\omega _{ref}). \end{aligned}$$

(25)

The mechano-sorptive effect is modelled as enlargement of the visco-elastic strain depending on the moisture change increment

$$\begin{aligned} \dot{{{\underline{\varvec{\varepsilon }}}}}^{ms}={{\underline{\underline{\varvec{\kappa }}}}}:\dot{{{\underline{\varvec{\varepsilon }}}}}^{ve} \cdot \Vert \textrm{d}\omega \Vert \quad {\textrm{if}} \,\, \textrm{sign} (\textrm{d}\omega )= \textrm{sign} ({{\underline{\varvec{\varepsilon }}}}-{{\underline{\varvec{\varepsilon }}}}^{he}-{{\underline{\varvec{\varepsilon }}}}^{ms}). \end{aligned}$$

(26)

The surface emission of water vapour is described by the boundary layer theory as in Frandsen (2005), see Supplementary Material, part A.

Results and discussion

Swelling and shrinkage

The swelling and shrinkage experiments provide the length difference and weight of each sample and moisture step. By the equations of DIN 52184, the differential swelling parameters q are calculated. They are converted to the differential shrinkage coefficients $\beta$, given in Table 2 together with the anisotropy ratios. Perpendicular to grain, spruce shows much larger shrinkage coefficients compared to the other wood types. $\beta _T$ fits to literature values (Reichel and Kaliske 2015b; Kollmann and Côté 1984), while $\beta _R$ is significantly larger than $16\,\%$ given in Reichel and Kaliske (2015b), Kollmann and Côté (1984). The differential shrinkage coefficients of lime wood and pine are in the same range and comparable to literature values of Konopka et al. (2024) and Kollmann and Côté (1984). It should be noted, that there is no large difference between the heartwood and sapwood of pine, except for longitudinal shrinkage.

The volume dimensions and mass of the samples in dry state lead to the dry density $\rho _0$ also given in Table 2. Spruce shows the lowest mean dry density of $491\,\hbox {kg}\,\hbox {m}^{-3}$, but it has to be noted, that the variance is very large ($> 12\,\%$) for this wood type. The mean value is large compared to literature values (Saft and Kaliske 2013; Kollmann and Côté 1984; Niemz 1993), where a range of $350-410\,\hbox {kg}\,\hbox {m}^{-3}$ is found. This might be explained by close annual rings of the investigated material, which lead to a larger density for softwood. Lime wood and pine show lower variance and are comparable to literature values (lime wood: $530\,\hbox {kg}\,\hbox {m}^{-3}$ (Konopka et al. 2024) and pine: $440-580\,\hbox {kg}\,\hbox {m}^{-3}$ (Kollmann and Côté 1984; Niemz 1993)). Since pine shows very different dry density values for heartwood and sapwood ($14\,\%$ difference), both types are investigated within this study.

By the wet density at each humidity step and the dry density, the moisture content of each sample and humidity step is determined. The mean values of each wood type and humidity step are used to fit the sorption models described in Sect. Constitutive material model by the least squares method. Figure 6 depicts the experimental results as boxplots showing the median, quartiles, extrema and outliers, and the fitted sorption curves. The model parameters are given in Table 3. It is pointed out, that there is only one climate series consisting of a desorption path between $65\,\%$ and $35\,\%$ and a scanning curve towards the adsorption path. Especially to find an adequate adsorption curve, a second climate series beginning with adsorption or a wider RH range would be helpful.

Table 2 Differential shrinkage coefficients and dry density

Table 3 Sorption model parameters

Fig. 6

Experimentally determined RH-$\omega$ relation and sorption characteristics fitted to models by Frandsen et al. (2007) and Anderson and McCarthy (1963): a spruce; b lime wood; c pine sapwood; d pine heartwood

Creep experiment

The visco-elastic, visco-plastic and mechano-sorptive parameters resulting from the creep experiments are given in Table 4 as well as the estimated but not tested parameters. The creep limit $\varphi _\infty ^{ve}$ in tangential direction is for all wood types in a range of $0.37-0.51$, the largest value is shown by lime wood and the smallest by pine. Lime wood exhibits also the largest relaxation time, which is several times larger than these of the softwoods while spruce has the lowest relaxation time. Visco-plastic behaviour is detected for lime wood at larger stress levels than for the softwoods. Between the sapwood and heartwood of pine, the main difference is that sapwood shows visco-plastic behaviour for almost all stress levels while heartwood has a limit of linearity of $7\,\%$. For the three material directions tested with lime wood, it can be noted that $\tau ^{ve}_R<\tau ^{ve}_T < \tau ^{ve}_L$. Visco-plastic behaviour starts at a much lower stress level in radial and tangential than in longitudinal direction. For the longitudinal direction, no visco-plastic viscosity could be determined, therefore, a literature value is added to Table 4.

Figure 7 shows the experimental results compared to the simulation results, using the obtained material parameters, as strain versus time relations. In general, the simulations fit well in the range of experimental results. For some experiments, especially at larger loads, the variation of results is quite large. The experimental run with changing RH shows also buckling of the samples with the highest load level. More samples (here three per load level) per experimental run are necessary for reliable statistical analysis. The simulations of lime wood at different constant RH values are run with the same moisture-independent creep parameters. Due to dependency of the creep simulation on moisture dependent short-term parameters, also the viscous creep result is moisture dependent. Figure 7a, d and e show that this approach is reasonable.

As can be seen, the experiment of lime wood in longitudinal direction cannot be evaluated properly, especially for the unloading time. It is assumed, that this error occurs since the region of interest is moving towards the camera, which falsifies the results. This error can hardly be prohibited for this measuring system and material direction, since the camera must be positioned in radial or tangential direction, where large lateral strains occur. Errors can also be seen for pine heartwood at 40 % load level, as the positive strains in the unloading time have no physical meaning.

Table 4 Creep model parameters

Fig. 7

Results of creep experiments and comparison to simulations: strain vs. time of experimental runs 1 to 9, see Table 1

Moisture-induced pressure

The force measurements of the moisture-induced pressure experiments are shown as time-force diagrams in Fig. 8. In every case, the pressure firstly increases strongly towards the maximum reaction force value around $500\,N$, which is reached after 1 - $4\,\hbox {h}$ depending on the wood species and material direction. The decrease slows down in time, so that the force tends towards a final value. It is obvious from the diagrams that the material response is not purely elastic, because the moisture is monotonically increasing, which would lead to monotonically increasing stress.

Fig. 8

Experimental and simulation results of the experiments of moisture-induced pressure

The experimental results are used for a validation of the model by a simulation (FEA I). For comparison, simulation results using a model without consideration of mechano-sorption (FEA II) and a purely elastic model (FEA III) are added. FEA III leads to monotonically increasing reaction forces where the maximum is largely reached after $10-20\,\hbox {h}$ and is much larger (around factor 10) than measured in the experiments. FEA II (without mechano-sorption) results in a reduced maximum of the reaction force which decreases after the maximum towards a final value related to the creep limit. By relaxation, swelling strain is split into an increasing viscous part and a decreasing remaining mechanically effective elastic strain. This resembles the behaviour observed in the experiments, but the force is still simulated as far too large. FEA I (with mechano-sorption) results in a further strong reduction of the reaction force compared to FEA III and FEA II. Due to the large moisture content change, the relaxation is enlarged, which has an immense effect for this experiment. The discrepancy to the experiment is the lowest for lime wood, but the simulated reaction force is still larger by factor two. However, the shape of the time-force diagram with the peak within 5 h and a very flat progress after 20 h is qualitatively represented by FEA I. Possible sources of error are the compliance of the load cell and of the test stand, which are not taken into account in the simulation, the not tested parameters (elasticity, sorption rate, diffusion properties), which may differ from literature values and influence the results, and also the wide range of humidity, which is wider than in the parameter determination experiments. For the softwoods, the simulation shows larger discrepancy with the experimental results. The authors suppose, that the predominating effect, mechano-sorption, has to be investigated for all wood types, as the use of the same creep strain enlargement parameter has not succeed.

Fig. 9

Parameter study for lime wood tangential: influence of the parameters $\varphi _{\infty ,T}^{ve}$, ${{\underline{\underline{\varvec{C}}}}}$ and $\tau _T^{ve}$ on the moisture-induced pressure simulation

A parameter study is conducted for lime wood in tangential direction, investigating the influence of the elastic stiffness tensor ${{\underline{\underline{\varvec{C}}}}}^{el}$, the visco-elastic creep limit $\varphi _\infty ^{ve}$ and the relaxation time $\tau ^{ve}$, see Fig. 9. A variation of $\varphi _\infty ^{ve}$ does not affect the initial slope and slightly influences the peak stress. The main influence of $\varphi _\infty ^{ve}$ is on the final stress level, since it marks the creep strain limit and the strain is directly related to the stress. At higher $\varphi _\infty ^{ve}$, the final stress is lower. A reduction of ${{\underline{\underline{\varvec{C}}}}}^{el}$ leads to a generally lower stress during the experiment. The peak stress is influenced as well as the final stress. The third investigated parameter is the relaxation time $\tau ^{ve}$. The variation does not affect the final stress and the initial slope, but the peak stress and the time period, when the peak stress is reached.

The diffusion and sorption velocity can be mentioned as further influence properties on the time-force relationship, which have not been studied for the specific wood species within this project.

Moisture-induced tension

The force measurements of the moisture-induced tension experiments are shown as time-force diagrams in Fig. 10. The tension force increases at the beginning towards a peak, which is reached faster than in the compression experiment after up to two hours due to the small thickness and, thus, shorter diffusion distance. Then, the force decreases slightly to a local minimum, which is reached after ca. 10 h. Subsequently, the force increases again very slightly and, finally, tends towards a final value. The samples 1 and 2 of spruce tested in tangential direction show sudden stress drops due to (partial) fracture. The simulations show again a large reduction of reaction force by creep relaxation and mechano-sorption in FEA I compared to FEA III (elastic), but the final values are mostly still too large, also for lime wood. It has to be noted that the creep experiments are only conducted as compression tests. A new parameter set for tension is not investigated. Apparently, the creep limit is too low, as the final reaction force is overestimated, and the relaxation time is too large. It is necessary to investigate the influence of different parameters for tension tests. In addition to the error sources mentioned for the swelling pressure experiment, sliding or yielding of the adhesive joint could occur and distort the results.

Fig. 10

Experimental and simulation results of the experiments of moisture-induced tension

Fig. 11

Parameter study for lime wood tangential: influence of the parameters $\varphi _{\infty ,T}^{ve}$, ${{\underline{\underline{\varvec{C}}}}}$ and $\tau _T^{ve}$ on the moisture-induced tension simulation

A parameter study is conducted for lime wood in tangential direction (Fig. 11) according to the study for moisture-induced pressure. A higher $\varphi _\infty ^{ve}$ results in a sharper peak and a lower final stress. A lower ${{\underline{\underline{\varvec{C}}}}}^{el}$ results in a generally lower stress, a lower initial slope and a flatter peak. A lower $\tau ^{ve}$ leads to a lower stress in the beginning and, thus, a lower peak. From the comparison of the experimental and simulation results, it can be stated, that in the experiments on lime wood and pine, the simulated final force is too high. Also, for all experiments, the simulated peak is reached too late. Therefore, it is concluded, that for tension, $\varphi _\infty ^{ve}$ is higher and $\tau ^{ve}$ is lower than for pressure, which should be proved in future experiments.

Conclusions and outlook

In this study, hygro-mechanical long-term behaviour of lime wood, spruce and pine in the principal anatomical directions was investigated. Experiments were conducted to analyse the swelling and shrinkage, sorption and compression creep behaviour and a parameter set for the comprehensive material model was determined. This parameter set was used for the simulation of validation experiments of constrained swelling and shrinkage investigating the coupled hygro-mechanical behaviour. The own FE-code in the in-house software WoodFEM was used for all simulations.

Compression creep tests were conducted for the described wood types at different loading directions and RH. Not every possible setup was investigated, therefore, the behaviour of not tested setups was estimated by the tested ones assuming analogous behaviour. The creep model includes visco-elastic creep, characterised by a creep limit and a relaxation time, visco-plastic creep, characterised by a limit of linearity and a viscosity, and mechano-sorption, characterised by an enlargement factor for the viscous strain. The model parameters were determined by the experiments.

Validation experiments were conducted for moisture induced pressure and tension. The results show that the reaction force due to constrained swelling and shrinkage is significantly reduced by creep relaxation. The mechano-sorptive effect has the largest influence on this stress reduction. Purely elastic simulations mislead to multiple times larger reaction forces. By a parameter study, the influence of selected parameters of the elastic and visco-elastic material model are shown. A comparison of constrained compression and tension experiments show that effects of creep are the same in both cases but the parameters of compression creep cannot be directly adopted for tension.

The described findings and parameters are useful for the three-dimensional hygro-mechanical simulation of wood, considering viscous behaviour and mechano-sorption. This allows the non-destructive assessment of deformations, stresses and the damage potential for wooden objects due to climatic scenarios. By fitting numerical models with material data to increase the accuracy of simulation results, experiments also play an important role for more advanced applications such as Digital Twins. Therefore, there is still a high general demand for creep experiments, but especially for (hardwood) species, not commonly used as construction material. More mechano-sorption experiments and ideally a standard creep test procedure would be highly appreciated.