STUDY ON COMPRESSIVE STRESS RELAXATION BEHAVIOR OF BEECH BASED ON THE FINITE ELEMENT METHOD

The compressive stress relaxation behaviors in three directions of beech (Fagus orientalis) were studied by experiments, and predicted by mechanical model and finite element method. Firstly, shortterm (3 hours) compressive stress relaxation experiments were carried out in longitudinal (L), radical (R) and tangential (T) directions of beech, and then the experimental data was fitted by mechanical model with two single Maxwell bodies in parallel connection. Secondly, the method of predicting the long-term (12 hours) stress relaxation behaviors of beech based on the finite element method was studied using the experimental data of short-term stress relaxation. Finally, the long-term stress relaxation behaviors in three directions of beech were investigated by experiments, mechanical model and finite element method respectively, and the results of them were compared. The results showed that stress relaxation behaviors of beech were different in three directions, and the short-term stress relaxation in L was much smaller than those in R and T directions under the same load. Besides, the mechanical model with two single Maxwell bodies in parallel connection was capable of predicting the short-term relaxation behaviors of beech in three directions with correlation coefficients beyond 0,99 but it did not work in long-term relaxation. In addition, the errors of FEM were smaller than those of the mechanical model compared with the results of experiments in the long-term stress relaxation, and the errors of the FEM were approximately 8% in L and 20% in R and T directions, which were all accepted in the field of wood engineer. This study will contribute to predict the long-term relaxation behaviors of wood products and wooden structures based on the FEM.


INTRODUCTION
Wood is a natural biomass material, which has complex physical and mechanical properties, including the elasticity of solid and of viscosity of fluid, so it is called viscoelastic material.Although creep and stress relaxation are two main parts of viscoelasticity, there is relation between them.The stress relaxation can be regarded as creep in different stress levels (Zhou et al. 2001).The viscoelasticity of wood plays an important role in durability of solid wood structures and wood products, which are able to decrease the strength of wood beam and wood joints (Figueroa et al. 2012, Daniela et al. 2013).
There are many factors influencing the viscoelasticity of wood, such as temperature (Peng et al. 2017a), humidity (Jiang et al. 2017) and stress levels (Peng et al. 2017b, Zhang et al. 2017).However, it is difficult to measure long-term viscoelastic behavior of wood for it is time consuming.Thus, many researchers have studied how to predict the long-term creep and stress relaxation of wood by short-term experimental data.Empirical method (Mukudai et al. 1978, Mukudai andSakamoto 1978) and mechanical method (Taniguchi et al. 2010, Chang et al. 2013) are two most used of them.The former used empirical equation to predict long-term viscoelasticity of wood, and the latter predicted viscoelastic behavior though using fitting curves of mechanical models, such as Maxwell, Kelvin, and Burgers.Hunt  2008) investigated MS strains which generated in strips of rectangular section stressed in torsion with the humidity cycled between 38% and 84%.After a number of humidity cycles the specimen was unloaded, the applied stress was reversed and further humidity cycles were imposed, as well as all the initial strains were recovered and reversed strain was developed.It was concluded that the large MS strain produces no permanent change of the features of the cell wall Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version structure that determine the shear modulus.There were close series between the mechanism generating the MS strains and that responsible for the large plastic strains (Keckes et al. 2003).Hering and Niemz (2012)

Materials and equipment
The material used in experiment was beech (Fagus orientalis Lipsky) wood bought from local wood commercial supplier (Nanjing, China).The average density was 0.629g/cm 3 , and the moisture content was conditioned to and held at 10.8% before and during experiments.The main equipment used in the experiment was a 20kN universal testing machine (AGS-X, SHIMDZU, Japan).

Specimens preparation
All specimens used in this work were processed by computer numerical control

Test method
An universal testing machine was used to load with loading rate 1mm/min.The proportional limit load of beech in three directions of beech were measured in advance to make sure that the load imposed on specimens was smaller than the minimum proportional limit load of beech in three directions.After pretests, the proportional limit load of T direction was 1796N, which is minimum in three directions of beech wood.Thus, the load 1000N was selected in this study to keep the specimens in elasticity when loaded.

Finite element method of stress relaxation
In this paper, the ABAQUS 16.4-1(ref) was used to establish the finite element model.Generally, a finite element model includes following aspects, geometric model, mechanical properties of material, loads and constrains.In this paper, the stress relaxation parameters needed in ABAQUS were the key points to the study.
The geometric model is shown in Figure 1.The steel loading head and steel base Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version were regarded as rigid, which were not deformation during loading process, and the beech specimen was loaded as the procedures described in testing method of experiment.Besides, the specimen was meshed with C3D8R element, and totally 12000 elements were used in the model.
The elastic constants and yield stress of beech were measured in the previous study (Hu andGuan 2017b, 2017c).For orthotropic beech wood, required inputs for elastic properties were three moduli of elasticity, three moduli of rigidity, and six Poisson's ratios as orthotropic material.The stress relaxation parameters of beech used in ABAQUS were the key.In ABAQUS, the visoelasticity was defined by a new constitutive equation expressed by Prony series parameters (Zhou et al. 2017) normalized shear modulus ( ) and parameter ( ) shown in Eq. 2. (2) The methods of getting the parameters used in ABAQUS are as followings.

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Table 1 shows the parameters needed in ABAQUS in three directions of beech, Ahead of Print: Accepted Authors Version which are figured out according to the calculating method described above.In this study, the second order Prony expansion parameters are used to predict the stress relaxation of beech in three directions.

Comparison and analysis
Figure 6 shows the entire processes of stress relaxation behaviors of beech based on FEM.The aim of paper is to put forward a method of predicting the long-term stress relaxation with short-term experimental data.Compared with other methods, the FEM can presents the processes of stress relaxation intuitively, and the accuracy is more higher.Besides, the mechanical properties of small specimen can be used to analyze Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version the stress relaxation of wood joint made by the same material based on FEM (Hu et al. 2018b).Although only the effect of directions of beech on stress relaxation behaviors were considered in this study, this method can also be used in other factors that influencing the stress relaxation of wood, such as temperature and humidity.In this work, only 3 hours experimental data was used to predict 12 hours relaxation by FEM, since the time and equipments were limited.The longer the relaxation time is, the more obvious the advantage of the FEM is.Two orders Prony expansion parameters (Table 1) were enough in ABAQUS to predict 12 hours relaxation in this paper.
However, the longer the relaxation time is, the more orders Prony expansion parameters are needed.

CONCLUSIONS
The method of using FEM to predict long-term stress relaxation of beech with short-term relaxation data was studied, and compared with mechanical model and experiment.The following conclusions were drawn.
1) The stress relaxation of beech in L direction is smaller than that in R and T directions, and the compressive force decreases by 20% in L and 40% and 50% in R and T directions respectively.
2) The mechanical model with two single bodies in parallel connection is able to predict the short-term stress relaxation of beech, but the errors increased as the time went by.
3) Relative error of FEM are smaller than those of mechanical model, and the Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version error of FEM are within 8% in L direction and nearly 20% in R and T directions.
In conclusion, the stress relaxation of beech in three directions are different from each other, and the relaxation behavior of wood must be considered in wood products and wood constructions.The FEM is able to predict the long-term stress relaxation of beech with short-term experimental data, which will contribute to the structure design of wood products and wood constructions.However, further studies are also supposed to be conducted, such as the FEM to simulate the influences of temperature, humidity and load levels on the relaxation.
(2004) studied the method of using short-term creep data to evaluate long-term viscoelastic behavior of spruce in bending and beech in tension.In a carefully Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version controlled creep experiment at constant humidity the results after the first few hours appear as a straight line, whether plotted as creep versus log time or as logarithm of creep rate versus creep.Then the curve was fitted and used to predict long-term creep behavior.Ozyhar et al. (2013) investigated the time dependency of the orthotropic compliance for beech wood by tensile and compressive creep experiments.In fact, wood is usually used in a dynamic environment of temperature and moisture.Mechano-sorptive (MS) creep is the deformation of loaded wood under changing climatic conditions.Salin (1992) studied on the dependence of the amount of checking on shrinkage induced stress level during timber drying with a new MS model and compared theoretically calculated maximal stress with observed amount of checking in a set of drying tests.Zhang et al. (2007) investigated the effects of temperature on MS creep of delignified hinoki wood (Chamaecyparis obtusa Endl.) using L and R specimens during adsorption and desorption over the temperature range of 20°-80°C.It was found that the effects of temperature on the MS creep of delignified specimens were more remarkable than for untreated specimens.Entwistle and Zadoroshny ( analysed the pure creep behaviour of European beech wood in the longitudinal direction at three different moisture contents.The moisture-dependent creep compliance was identified using a four-point bending test device.The creep behaviour was ascertained to be linear with moisture content and quantified by means of a Kelvin-Voigt model approach.Huang (2016) carried out an experiment to investigate the influences of load and moisture content on MS creep of wood.It was found that the wood specimens exhibited a partial recovery during all the adsorption phase and deflection increase during all desorption phase when low load level was applied, and an amplified load effect existed within the creep under cyclic moisture changes.With developments of computer science and finite element theories, FEM has been widely used in wood structures and wood products (Mackerle 2005).The strength of wood mortise-and-tenon joint furniture by FEM were studied by Hu and Guan (2017a) and Hu et al. (2018a), but the viscoelasticity of wood were not considered.It is known to us the relaxation of wood joint will decrease the strength of joint significantly, which then influences the strength of the whole structure of wood products, since joints are the critical parts of wood structures.Beech wood is widely used in wood products in China.In this paper, the short-term stress relaxation behaviors of beech in L, R and T directions were studied by experiments, and the data Maderas-Cienc Tecnol 21(1):2019 Ahead of Print: Accepted Authors Version was analyzed by mechanical model and FEM respectively.Then the long-term relaxation were predicted by mechanical model and FEM, and the results were compared with those of long-term experiments.The aim of the study was to provide an efficient FEM to predict long-term stress relaxation of beech with short-term stress relaxation experimental data.This study will contribute to structure design of wood constructions and wood products based on FEM.
Figure 1.Dimensions of beech wood specimens in L (a), R (b) and T(c) directions.

Figure 2
Figure2shows the testing method of experiment.The specimens were

Figure 2 .Figure 3 .
Figure 2. The setup to determine stress relaxation behaviors of beech.
Figure4shows experimental results of compressive force relaxation and

228 Figure 4 .
Figure 4. Curves of compressive force and time (left) and curves of relaxation

Figure 5 .
Figure 5. Compressive force and relaxation modulus in three directions of beech Figure 6-a indicates the initial state without load imposed on the specimen, and Figure 6-b presents the state that the load reaches 1000N, as well as Figure 6-c suggests the stress distributions of specimen decreases gradually during relaxation.

Figure 6 -Figure 6 .Figure 7 .
Figure 6-d shows the stress distributions of specimens at the end of test.Obviously,

Table 1 .
Dimensionless parameters used in FEM to simulate stress relaxation of beech.