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Evaluation of the applicability of different viscoelasticity constitutive models in bamboo scrimber short-term tensile creep property research

  • Sun Songsong and Wan Maosong EMAIL logo
Published/Copyright: July 13, 2021
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Science and Engineering of Composite Materials
From the journal Science and Engineering of Composite Materials Volume 28 Issue 1

Abstract

Bamboo scrimber is a new natural fiber-reinforced composite material in modern industry. In this paper, the tensile creep characteristics of this material were chosen as the object of the study. First, axial tensile fracture experiments were conducted on different test specimens to determine the corresponding strength data. Then six sets of standard tensile creep experiments were conducted under different given stress levels. Finally, the Maxwell constitutive model was applied in fitting the relationship between strain and time. The results showed that the traditional Maxwell viscoelasticity constitutive model will result in some errors in the fitting results, while the combined fractional and Maxwell model can provide much higher accuracy in this study, thus it is more suitable for engineering applications. This paper provides a solid foundation for a better understanding of the mechanism of the bamboo scrimber creep behavior.

Keywords: bamboo scrimber; fractional derivative; tensile creep; viscoelasticity constitutive model

1 Introduction

Bamboo scrimber is a new natural fiber-reinforced composite material that has been developed in recent decades [1]. This material has several advantages (e.g., low cost, green, clean, and low weight) that make it usable in the modern industry [2,3,4]. Thus, the correct determination of the mechanical properties of this material becomes important during the design stage.

To solve this problem, creative work has been conducted in recent years. For example, He Wen investigated the mechanical performance of bamboo scrimber made from a given type of moso bamboo and treated it with hot oil [5]. Naresworo Nugroho chose zephyr strands from another type of moso bamboo to make a structural composite board [6,7]. Huang et al. examined how the accelerated aging method and aging resistance influence the performance of bamboo scrimber and conducted mechanic experiments on a double cantilever beam and an end-notched beam [8,9,10]. Li and Wei conducted an experimental study on the deformation and failure mechanism of bamboo scrimber and proposed an axial stress–strain model [11,12,13]. According to previous research, a primary conclusion can be proposed that bamboo scrimber is a typical kind of plywood. According to the theory of viscoelastic mechanics, this kind of material always shows obvious creep characteristic, which has an obvious impact on the service life [14,15,16]. While in a previous study, this material is usually considered to be an anisotropic elastic material, special creep property research of the bamboo material experimental results has rarely been discovered in the published documents.

In this paper, axial tensile fracture experiments were conducted on different test specimens to determine the corresponding strength data. Then six sets of standard tensile creep experiments were conducted under different given stress levels. Finally, the Maxwell viscoelasticity constitutive model was applied in fitting the relationship between the strain and serving time. The results showed that the traditional Maxwell viscoelasticity constitutive model will result in some errors in the fitting results, while the combined fractional and Maxwell model can provide much higher accuracy in this study, and thus is more suitable for engineering applications.

2 Method

In the present short-term creep experiments of the fiber-reinforced composites, the load applied on the specimen is always determined by the limit strength of the material (usually the range is no more than 70%). According to a previous study, some of the mechanical property parameters of the bamboo scrimber such as the tensile strength or young’s modulus always show obvious dispersion. As a result of this, it is necessary to conduct a statistical analysis to obtain the distribution property of the tensile strength before the creep experiment. So the whole process of the research can be divided into five steps:

Step 1: Conduct the standard tensile fracture experiment on a set of a specimen to determine the tensile strength in each case.

Step 2: Choose three commonly used distribution models to fit the distribution function between the tensile stress and the survival rate. Then the tensile stress under 50% survival rate based on the most accurate function is selected to be the tensile strength.

Step 3: Conduct the standard short-term tensile creep experiment under six different stress levels (from 10 to 60% of the tensile strength) and record the strain during the experiment.

Step 4: Choose different viscoelastic constitutive models to fit the variations of the strain with time to make a comparative study. In this way, the most accurate model can be determined for further study.

3 Results

3.1 The tensile fracture experiment results

In this paper, the bamboo scrimber material was manufactured by a cold-pressing technology. In the first stage of this approach, the fibers were immersed in the phenol resin under high pressure. Then in the second stage, the fibers were dried and put into the steel box and compressed to a standard density. The pressure in this stage is 22 MPa. Finally, in the third stage, the whole mold was fixed by a set of bolts and placed in an oast house for 12 h. In this way, the fibers were recombined into a piece to provide the manufacturing material of the specimen. The source of the fiber is a kind of phyllostachys pubescens. The density of this material is 1.08 × 103 kg/m3 and the moisture content is 5.6%.

According to the analysis in the previous chapter, the first step of the statistical analysis is to conduct the fracture experiment of the bamboo scrimber. Figure 1 shows the structural features of the tensile test equipment, from which it can be found that the whole piece was fixed at both ends, and a pair of strain gages were affixed on both sides of the specimen. During the experiment process, a steady increased tensile force was applied on it until the final fracture.

Figure 1 The experimental equipment.
Figure 1

The experimental equipment.

During the whole experimental process, the messages such as the load and displacement were recorded using a computer. The experiment was conducted using the UTM5504-GD microcomputer-controlled test equipment and the serial number of the experiment standard applied in this case is ASTM D143-09. During the experiment, the temperature is 25°C and the relative humidity is set to 60%. Using this equipment to conduct the tensile test on a set of given specimens, the results are presented in Table 1:

Table 1

The experimental results of the tensile fracture test

Case number Width (mm) Thickness (mm) Limit tensile load (N) Tensile strength (MPa)
1 9.05 5.00 5,239 110.3
2 9.21 5.04 8,314 175.1
3 9.49 5.15 9,380 197.5
4 9.33 5.06 6,336 133.4
5 8.97 5.12 6,430 135.4
6 9.21 5.05 7,882 165.9
7 9.46 5.08 8,751 184.2
8 9.40 5.10 7,749 163.1
9 9.39 5.09 6,949 146.3

According to ref. [17], the primary factor of deciding whether the data can be taken into further analysis is the coefficient of variation. According to the experiment standard demands, the value of C V in a set of given data should not be more than 20% before analysis. In this paper, the value of this parameter is 17.7%, which can fulfill the demands of accuracy.

3.2 Statistical analysis of the tensile strength

In this paper, three kinds of usual distribution functions in actual engineering applications are applied to make a comparative study. From the perspective of the best fitting effect, the constraint function is equivalent to varying the fitting parameters to obtain the maximum value of the fitting correlation coefficient. Table 2 and Figure 2 show the fitting results based on these three functions, from which it can be discovered that the fitting errors of the three models are all less than 5%, which is sufficient for actual engineering application. In addition, the estimated values of statistical parameters based on different models are nearly the same, especially for the expected values (the relative difference is less than 2%). So the tensile strength under the 50% survival rate is determined to be 155 MPa.

Table 2

Fitting functions of failure rate distribution of the tensile strength

Model type Distribution function Estimated value of a statistical parameter Correlation coefficient
Normal y = Φ − x − 155.7 29.2 μ = 155.7 0.983
σ = 29.2
Lognormal y = Φ − ln x − 5.03 0.2 μ = 152.9 0.966
σ = 29.4
Three-parameter Weibull y = 1 − e x p 1 − x − 93.9 72.9 3.11 μ = 156 0.979
σ = 28.6
Figure 2 Fitting results of failure rate distribution of the tensile strength.
Figure 2

Fitting results of failure rate distribution of the tensile strength.

3.3 Tensile creep experiment results

Based on the strength parameter obtained in the previous section, the creep experiment can be conducted. During the experiment, the temperature and humidity are fixed to be the same as those in the tensile fracture experiment. Before the creep experiment, two load cycles have been applied on the specimen for a pretreatment, during which the load was increased from 0 to 7 kN with the speed of 200 N/s and then reduced to 0. Then the static creep load was applied to the specimen. For each specimen, the load lasts for 6 h.

As shown in Figure 3, the strain was recorded by an extensometer fixed at the middle part of the specimen with the smallest cross-section. The standard of the creep experiment applied, in this case, is ASTM D2990-17. During the experiment, the sampling frequency is set to be 1 min.

Figure 3 The creep experiment equipment.
Figure 3

The creep experiment equipment.

Table 3 shows the detailed load information of the creep experiment. Based on the above stress level, six groups of experiments were conducted. Corresponding experimental results are shown in Figure 4 and Table 4. The initial strain refers to the strain recorded at the beginning of the creep experiment, while the final strain refers to the strain recorded at the end of the experiment. From these six curves, a clear conclusion can be proposed that the curves of the low-stress level (10 and 20% of the tensile strength) are nearly horizontal. The relative increments in both cases are less than 2%, which means that the whole amount of strain was mainly made up of the elastic strain and the creep behavior under this stress level is not obvious. In addition, the initial strain and the final strain are almost correlated linearly with the stress. While for the 30, 40, and 50% levels, the creep strain increases more obviously with higher slopes. The relative creep increments within these cases are much higher than those of the first two cases. This means that the creep resistance of the material under these stress levels becomes weaker. For the highest stress level (60%), the increasing rate of the strain with the given stage becomes unsteady. The unstable accelerated creep appears although the fracture has not happened so far. Generally speaking, the creep property of the bamboo scrimber is influenced by the stress level. The values of the creep resistance under the low-stress conditions are more obvious than that under the high-stress conditions.

Table 3

Load parameters of the creep experiment

Case number Width (mm) Thickness (mm) Stress level (%) Stress (MPa)
1 9.24 5.03 10 15.5
2 9.67 5.01 20 31
3 9.34 4.99 30 46.5
4 9.09 4.99 40 62
5 9.38 5.02 50 77.5
6 9.60 5.01 60 93
Figure 4 The strain history during the creep experiment.
Figure 4

The strain history during the creep experiment.

Table 4

The initial and final strain state of the creep experiment

Case number Initial strain (10−3) Final strain (10−3) Relative increment (%)
1 1.01 1.02 1.2
2 2.10 2.14 1.8
3 3.09 3.19 3.2
4 4.62 4.82 4.3
5 5.18 5.41 4.4
6 6.5 7.09 9.1

3.4 Creep model analysis

As mentioned above, the creep behavior of the bamboo scrimber changes with the stress level applied to it. In a previous study, several viscoelasticity constitutive models were proposed to research this property of composite materials. Among which the Maxwell model is considered to be an effective model in analyzing the creep behavior [18,19]. This model can exhibit previous rheological behavior, which is similar to the creep process. As shown in Figure 5, this model is made up of a series combination of a spring model and a Newtonian dashpot model. The stress–strain relationship of this model can be expressed as follows:

(1) σ 1 = σ 2 = σ 0 ,

(2) σ 1 = E 0 ε 1 ( t ) ,

(3) σ 2 = η 0 d ε 2 ( t ) d t ,

where σ 0 is the static stress generated by the load, t is the time, E 0 is the elastic modulus of the material and η 0 is the viscosity coefficient. σ 1 and σ 2 are the stress generated by the spring and the Newtonian dashpot model, respectively. ε 1 ( t ) and ε 2 ( t ) are the strain response from the spring and the Newtonian dashpot model, respectively. According to the previous study, the strain responses of this model can be expressed as follows:

(4) ε 1 ( t ) = σ 0 E 0 ,

(5) ε 2 ( t ) = σ 0 η 0 t ,

(6) ε ( t ) = ε 1 ( t ) + ε 2 ( t ) = σ 0 E 0 + σ 0 η 0 t .

Figure 5 The structure features of the Maxwell model.
Figure 5

The structure features of the Maxwell model.

According to the definition of the traditional Maxwell model, both the parameters E 0 and η 0 can be treated as material constants. Based on the experiment results above, the parameters of the model can be determined by a fitting approach. The theoretical basis of the process can be expressed as follows:

  1. The initial values of the strain obtained based on the experiment data and the response equation are the same.

    (7) ε ( t = 0 ) = σ 0 E 0 .

  2. The sum of the relative difference percentage between the experiment data and the response equation is the minimum. The definition of the relative difference percentage is:

(8) f = ∑ i = 1 n ε ( t i ) − ε e i ε e i ,

where f is the relative difference percentage and ε ( t i ) and ε e i are the values of the strain obtained by the response equation and the experiment at the ith time node. Based on this assumption, the model parameters in each set can be determined. The results are shown in Table 5 and Figures 6–11.

Table 5

Model parameters based on the conventional maxwell model

Case number E 0 (MPa) η 0 (GPa/min) Correlation coefficients
1 15,346 392,305 0.915
2 14,832 331,197 0.823
3 15,048 129,960 0.697
4 13,420 88,787 0.782
5 14,961 92,781 0.719
6 14,308 48,261 0.939
Figure 6 Fitting results based on the conventional Maxwell model and 10% stress level.
Figure 6

Fitting results based on the conventional Maxwell model and 10% stress level.

Figure 7 Fitting results based on the conventional Maxwell model and 20% stress level.
Figure 7

Fitting results based on the conventional Maxwell model and 20% stress level.

Figure 8 Fitting results based on the conventional Maxwell model and 30% stress level.
Figure 8

Fitting results based on the conventional Maxwell model and 30% stress level.

Figure 9 Fitting results based on the conventional Maxwell model and 40% stress level.
Figure 9

Fitting results based on the conventional Maxwell model and 40% stress level.

Figure 10 Fitting results based on the conventional Maxwell model and 50% stress level.
Figure 10

Fitting results based on the conventional Maxwell model and 50% stress level.

Figure 11 Fitting results based on the conventional Maxwell model and 60% stress level.
Figure 11

Fitting results based on the conventional Maxwell model and 60% stress level.

As shown in Figures 6–11, a clear conclusion can be proposed that the values of the strain obtained by the experiment and the response function are quite different from each other. In addition, the parameters in each set are obviously different from those in other set, which can be attributed to the diversity of the material. Besides, some of the correlation coefficients of the fitting results are mainly less than 90%, which means that the relative error in these sets are more than 10%. This accuracy can not fulfill the actual engineering demands. The main reason for this phenomenon may be the definition of the viscosity coefficient η 0 . In the traditional application of the Maxwell model, this parameter is usually considered to be a material constant. While in recent years, some experts discovered that this parameter has a time-variant characteristic.

According to previous research, the fractional-order model usually has the time-variant characteristic. In a previous study, some experts applied this theory to research the creep behavior of some geologic materials such as rock and clay [20,21]. The application of this approach in bamboo material has rarely been reported. In addition, the creep behavior research based on the fractional-order theory is usually long-term type, the applicability of this theory in short-term creep behavior is still unclear. In this paper, we applied this theory to fit the short-term creep strain curve to make a comprehensive comparative study.

Up to now, there are several fractional models such as the Riemann–Liouville (RL) model, the Caputo model, and so on. According to the previous study, the RL model seems to be appropriate for viscoelastic materials [22]. So in this paper, we applied this model in bamboo scrimber short-term creep behavior research. The definition of this model can be expressed as refs. [23,24,25,26]:

(9) d α d t α [ f ( t ) ] = 1 Γ ( 1 − α ) d d t ∫ 0 t f ( τ ) ( t − τ ) α d τ

where Γ represents the gamma function and α ( 0 < α < 1 ) ) is the order. According to this model, the strain response during the creep stage can be determined. The stress–strain relationship of the Maxwell model can be expressed as follows:

(10) ε 1 ( t ) = σ 0 E 0 ,

(11) ε 2 ( t ) = σ 0 η 0 t α Γ ( α + 1 ) ,

(12) ε ( t ) = ε 1 ( t ) + ε 2 ( t ) = σ 0 E 0 + σ 0 η 0 t α Γ ( α + 1 ) .

Compared with the conventional Maxwell model, this modified model has one more parameter in all. Based on this response function and the experimental results, the parameters of the model can be determined. Figures 12–17 and Table 6 show the fitting results based on this modified Maxwell model, a clear comparison can be found that compared with the traditional model, this fractional order-defined model can exhibit much higher accuracy in expressing the creep strain curve. The values of the strain obtained based on the response function are quite near to those from the experiment data at the same time nodes. Among the six groups, the correlation coefficients are all over 98.5%, which makes it completely enough for engineering applications. Besides, the values of the parameter α in every set based on the fitting results are located within its definition range (from 0 to 1). Thus, this model is more suitable for actual engineering applications.

Figure 12 Fitting results based on the fractional defined Maxwell model and 10% stress level.
Figure 12

Fitting results based on the fractional defined Maxwell model and 10% stress level.

Figure 13 Fitting results based on the fractional defined Maxwell model and 20% stress level.
Figure 13

Fitting results based on the fractional defined Maxwell model and 20% stress level.

Figure 14 Fitting results based on the fractional defined Maxwell model and 30% stress level.
Figure 14

Fitting results based on the fractional defined Maxwell model and 30% stress level.

Figure 15 Fitting results based on the fractional defined Maxwell model and 40% stress level.
Figure 15

Fitting results based on the fractional defined Maxwell model and 40% stress level.

Figure 16 Fitting results based on the fractional defined Maxwell model and 50% stress level.
Figure 16

Fitting results based on the fractional defined Maxwell model and 50% stress level.

Figure 17 Fitting results based on the fractional defined Maxwell model and 60% stress level.
Figure 17

Fitting results based on the fractional defined Maxwell model and 60% stress level.

Table 6

Model parameters based on the fractional defined Maxwell model

Case number α Correlation coefficients
1 0.324 0.985
2 0.607 0.987
3 0.431 0.998
4 0.474 0.997
5 0.378 0.998
6 0.646 0.999

In a previous study, some experts found that the Burgers viscoelasticity constitutive model which has four parameters according to its definition could get a high accuracy in fitting the tensile creep strain curve. Compared with this model, the modified Maxwell model proposed in this paper can fit the strain curve well with fewer parameters. This makes it superior to the other models in engineering applications.

4 Discussion and conclusion

The short-term tensile creep behavior of the bamboo scrimber is selected as the object of this study. First, several sets of standard tensile fracture experiments were conducted to obtain the tensile strength of the material. Then, six sets of short-term tensile creep experiments were done to obtain the strain history throughout the whole process. Finally, different models were applied in analyzing the creep property of the material. Corresponding conclusions are shown as follows:

  1. The tensile strength experiment results show obvious randomness property. Based on the commonly used distribution models, the tensile strength under the 50% survival rate can be determined. The results from different models are nearly the same.

  2. Compared with the traditional model, the combined fractional and Maxwell model can provide much higher accuracy in fitting the creep strain curve, thus it is more suitable for engineering applications.

This study mainly focuses on researching the effects of stress levels on the tensile creep properties of bamboo scirmber and the short-term strain predictive model under certain conditions of temperature and humidity. However, many factors, such as temperature and humidity, affect the creep of bamboo scrimber. Additionally, the parameters based on different fitting results show an obvious random property. So more properties of the creep behavior of the bamboo scirmber will be the focus of the next phase of research.

  1. Conflict of interest: Authors state no conflict of interest.

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Received: 2021-01-19
Accepted: 2021-06-02
Published Online: 2021-07-13

© 2021 Sun Songsong and Wan Maosong, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/secm-2021-0034
Keywords for this article
bamboo scrimber; fractional derivative; tensile creep; viscoelasticity constitutive model
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