Viscoelastoplastic modeling of compressional behaviors of kapok fibrous assembly

Abstract

Kapok fiber is a kind of cellulosic fiber harvested from the kapok fruit and has many unique properties and potential applications owing to its high degree of hollowness. It is important to understand the mechanical properties of the kapok fiber under transverse compression because its hollow structure can be squashed easily. In this research, kapok fibers were carded slightly to form a kapok fibrous assembly (KFA) in which the fibers were straightened and parallel. A KFA was considered as an approximately isotropic matrix material in the transverse direction. The fiber arrangement in a KFA was geometrically modeled with a pipe-piling structure. The viscoelastoplastic model and its constitutive equations were established to characterize the mechanical response of the KFA under transverse compression. Three compressional stages (A – the viscoelastic stage, from the initial point 0 to the yielding point 1; B – the viscoelastoplastic stage, from the yielding point 1 to the point 2; and C – the senior viscoelastic stage, from the yielding point 2 to the point at the maximum compressional load) were observed from the stress–strain curve, and four parameters were determined to describe the elastic, viscoelastic, and viscoplastic behaviors under each compression cycle performed on the Instron compression tester. The results indicate that the variable elasticity of the KFAs exists throughout the total compression, viscoelasticity appeared only in stage C, and the viscoplastic property was evident in stage B. The KFAs did not exhibit viscoelastic behavior in stages A and B because the viscoelastic element of the Kelvin model failed to work in these two stages. The influence of conditioning humidity on the parameters was also investigated.

Keywords

kapok fiber fiber assembly viscoelastoplastic model transverse compression

Kapok is a single-cell cellulose fiber with extremely high degree of hollowness (80–90%) that possesses many unique properties relatively new to engineering materials.¹ Kapok appears in a cylindrical shape with round or oval cross-sections and has a smooth surface without distinct convolution.^2,3 Because of its high degree of hollowness, kapok fibers are light, buoyant, and allow the cellulose capsule to retain still air, which make them exceptionally good for thermal insulation,⁴ noise, and vibration control.^5,6 However, hollow kapok fibers are easily squashed and broken during processing and use, and the performance of the end-use products that take advantage of kapok's hollowness may not be sustained for the intended period. Therefore, it is necessary to further understand the mechanical properties of a kapok fiber assembly (KFA) under transverse compression.

Many researchers have investigated the compressional mechanical behavior of fibrous assemblies with classical theories, models, and numerical methods to estimate their compressional performance. Van Wyk proposed a model based on the bending of randomly distributed fiber elements and derived the pressure–volume relationship for the compression stroke and the number of contacts in the assembly.⁷ Stearn deduced the compressed volume and derived the correction factor for the number of contacts in an initially random assembly.⁸ Komori and Makishima generalized Stearn's approach by introducing an orientation density function.⁹ Pan proposed the microstructure theory and steric restriction on a fibrous assembly.^10,11 Later, Lee and coworker's analytical expression for compressional strain and Poisson ratio,^12,13 Carnaby and Pan's slippage standard,¹⁴ Lee and Carnaby's energy method,¹⁵ and other works on the fibrous assembly greatly advanced the study of compressional properties of fibrous assemblies.^16–23 Meanwhile, many solid mechanical theories have been applied to study polymer physics. Wolcott,²⁴ Easterling,²⁵ and Lenth and Kamke applied the cellular solid theory to describe the nonlinearity of the transverse compression of solid wood.^26,27 Smith generalized the linear theory to a large strain based on the Maxwell model.²⁸ Kitagawa et al. adapted the differential equation of the standard linear solid to study polypropylene.²⁹ Our previous studies on KFA compressibility and resilience had demonstrated experimentally the nonlinear compressional behavior of KFAs under different treatments.³⁰

In this research, kapok fibers in the KFA were carded slightly to make fibers straightened and parallel and the KFA was considered as an approximately isotropic matrix material of the fiber. The fiber arrangement in the KFA was geometrically modeled with a pipe-piling structure. The viscoelastoplastic model and constitutive equations based on the compressibility and resilience of the KFA were built to obtain the stress–time functions in three characteristic stages. Then the stress–time functions were expressed through the Laplace transform, and the model parameters were estimated by using actual compression stress–strain data obtained from the Instron compression tester. The proposed viscoelastoplastic model with the estimated parameters was used to characterize the KFA's mechanical response under the corresponding compression. The influence of the conditioning humidity on these model parameters was also investigated.

Experimental details

The kapok fibers used in the study were selected from the family of Ceiba pentandra grown in Indonesia. For the transverse compressional test, the KFAs were prepared using the following steps:

Kapok fibers were pre-conditioned at standard temperature 20 ± 2℃ and relative humidity (RH) of 65 ± 5% for 24 h.

The fibers were carded slightly into bundles so that fibers were straightened and parallel (see Figure 1(a)).

A specimen was weighed to be 100 mg using a torsion balance, and uniformly packed in a 30 mm × 30 mm × 15 mm paper container to achieve a planar density approximately 1.11 g/mm²; the bottom of the container had an opening of 20 mm × 20 mm that permits only fibers to be compressed in the test (see Figure 1(b)).

The experimental work was performed on an Instron compression tester in a temperature and humidity-controlled laboratory (20 ± 2℃, 65 ± 5%) (see Figure 1(c)).³¹ KFA samples were heated in the oven at a temperature of 45 ± 2℃ for 1 h to first reduce moisture content, and then treated in two different humidity conditions: dry (15% RH) and wet (99% RH) treatments. The different viscoelastoplastic model parameters were estimated in the different humidity treatments and the results used to test the sensitivity of the model since the KFA compressibility was changed with the conditioning humidity.³⁰ The two dry specimens and the two wet specimens were labeled as d-1 to d-2 and as w-1 to w-2, respectively. These specimens were all given a pre-tension of 0.006 kgf/cm² and compressed in 50 compressing cycles (see Table 1). Typical operating conditions are summarized in Table 2.

Figure 1.

Sample preparation and photograph of the compression test at the initial position.

Table 1.

Sample treatments

Specimen		Condition
Treatment	Numbers	RH (%)	Compressing cycles
Dry	d-1,d-2	15	50
Wet	w-1,w-2	99	50

Table 2.

Operating conditions used in the compression test and the simulation³²

Condition	Description
Temperature of the treatments performed	20 ± 2℃
Area of the test feeler	15 cm²
Gauge length	2.5 cm
Test speed	60 mm/min
Compressional load range	0–214 kgf
Interval between two compressions	5 min

Pipe-piling structure and viscoelastoplastic model

The hollow structure of the kapok fiber was observed using a scanning electron microscope (SEM), as shown in Figure 2. It shows both the cross-section of the kapok fiber body and the cross-section of the KFA. In Figure 2(a), the uncompressed kapok fiber can be seen to be circular with an exceedingly hollow cross-section and a uniform longitudinal surface. Figure 2(b) shows the image of the cross-section of the KFA, which undergoes a deformation caused by the cross-sectioning. The compact arrangement of kapok fibers in the KFA can apparently be observed. The inter-fiber space was squashed out and filled with the deformed fiber walls, and the intra-fiber space from the initial hollow structure took up the main area of the KFA's cross-section.

Figure 2.

Scanning electron microscope (SEM) observation of the kapok fiber and the kapok fiber assembly (KFA): (a) cross-section of the fiber body; (b) cross-section of the KFA.

It is necessary to establish a simple structure to describe only geometrical changes of a KFA when being compressed. Kapok fibers in an assembly were assumed to be ideally parallel, tightly packed, and uncompressed. The structure of KFAs under transverse compression can be simulated by a simple pipe-piling structure, as depicted in Figure 3(a).³⁰

Figure 3.

Schematic representation of the pipe piling structure on KFA: a-the compression on KFA modeled in a pipe-piling structure, b-the column j selected as a representative unit.

According to the compressibility and resilience of the KFA reported in the two previous papers,^30,32 the properties of the KFA in transverse compression are highly nonlinear, as the compressional stress–strain (S-S) curve reveals in Figure 4. At the beginning of the compression, the KFA is in a low volume fraction, and has few contact fibers points and plenty of air among the fibers. The inter-fiber space is easily squeezed out when subject to an external load. As the external load increases, it compresses the assembly to some compactness level in which the fibers can be considered as piled pipes. The KFA was transferred into elastic buckling and exhibited nonlinear elasticity, which led the plateau of the S-S curve to begin to climb up and reach the yielding point 1 (see in Figure 4(b)). With the load continued, the hollow cells were crushed, the walls began to touch, and plenty of the fibers of the KFA started to be squashed as ribbon-like and even brittle crushed. Then the S-S curve rose rapidly after the yielding point 2 because of the densification of the KFA, as shown in Figure 4(b).

Figure 4.

The compressional stress–strain curves of the kapok fibre assembly (KFA): (a) a complete compressional stress–strain curve of the KFA; (b) characteristic phases of the stress–strain curve of the circled area of (a).

Figure 5 shows the compressional resilience of the KFAs under 50 compressing cycles. It was found that the KFA could not recover to the original position and the compression strains of the S-S curves consistently shift rightward, changing from 99.11% to 65.77% for the dry sample and 98.03% to 28.22% for the wet sample after the 50 compressing cycles.³² The plateau of the S-S curve occurred in the same manner in each compression. This phenomenon indicates that the samples lost some of their thickness in each of the cycles, and the start of the viscoelastoplastic compressional deformation on the S-S curves was delayed as the repeated compressing cycles continued, especially during the first 25 cycles. This behavior can be seen more clearly in the S-S curves of every five compressing cycles, as shown in the second row of Figure 5.

Figure 5.

The stress–strain curves of the (a) dry- and (b) wet-treated kapok fiber assemblies (KFAs) with 50 compressing cycles.

Therefore, the compressional rheology of the KFA can be expected to be time-dependent and a model used to describe the rheology of the KFA in the transverse compression should capture the yielding points (points 1 and 2 in Figure 4(b)) and the mechanical stiffness. The standard linear solid (SLS) model revealed the rheology of the material at a comparable level of accuracy and performed well in the overstress test.³³ To represent the yielding characteristics of the KFA in compression, this study proposed a viscoelastoplastic model named the Nishihara model by introducing the viscoplastic element named Bingham model to the SLS model, as shown in Figure 6. The Nishihara model takes into account the viscoelasticity via the SLS model and the viscoplasticity by the Bingham model. In Figure 6, E₁ is the modulus of the spring in the elastic element, η₁ refers to the viscosity of the dashpot in the Bingham model (the viscoplastic element), E₂ indicates the modulus of the spring, and η₂ is the viscosity of the dashpot in the Kelvin model (the viscoelastic element). These four coefficients of the Nishihara model are the key parameters that characterize the complex mechanical response of the KFA in transverse compression.

Figure 6.

Mechanical analog of the Nishihara model.

The Bingham model with a dashpot and a friction element in parallel tends to work as a nonlinear viscosity when σ ≥ σ_c. The coefficient of the viscosity can be expressed as

η_{1} = \frac{E_{1}}{θ \times T} [T + t (1 - \frac{σ - σ_{c}}{σ_{s}} \times θ)] 2

where θ is the dimensionless parameter, T is the processing time from the initial to the yielding point 1 (see in Figure 4(b)), σ_c is the yielding strength of the Bingham model, and σ_s is the instant strength of the Bingham model.³⁴ However, when σ > σ_c + σ_s/θ, the coefficient of the viscosity (η₁) gradually descends to zero with the increase of the compressing time, the Bingham model is going to fail rapidly, and finally the Nishihara model turns into a senior SLS model. Therefore, the compressional mechanical response of the Nishihara can be divided into three stages: A – the viscoelastic stage (when σ < σ_c), B – the viscoelastoplastic stage (when σ_c ≤ σ ≤ σ_c + σ_s/θ), and C – the senior viscoelastic stage (when σ > σ_c + σ_s/θ).

To model the KFA of an approximate transversely isotropic matrix material of the fiber and investigate its overall compressional mechanical properties, the pipe-piling structure is represented by the parallel columns shown in Figure 3 based on the following assumptions: (1) The consolidation of the KFA occurred in the thickness direction. (The fibers in the KFA were not bonded, so the KFA was not like a cellular solid, in which the shearing may exist as cells crush. In the transverse compression, the compressional stress was much higher than the shear and side pressure. Therefore, the deformations resulting from the compressional shear and the side pressure among fibers were negligible.) (2) The fibers in the KFA were isotropic in the in-plane direction and behaved as parallel columns. (3) The average strain of each column in the KFA was equal and the applied stress on the each column was equal. (4) It was assumed that each individual column deformed at a specific strain and independently of the remaining columns that constituted the control domain. (5) For the sake of model simplification, the friction among each column was transferred into the viscosity within the column by the nonlinear dashpots. Therefore, each column, as its starting point of the total network of the KFA shared the exterior compressional load.

In order to generate a more realistic and reliable material response, q Nishihara models were arranged in parallel to construct the network, as shown in Figure 7. The external compressional load was shared by q Nishihara models as the stress and the density increases. However, in the real KFA the columns were not even at the thickness and the stress would be concentrated on the thicker ones. As the compression progressed, the compressed area rapidly increased and the stresses were continuously redistributed. Therefore, the total applied load was the summation of the loads applied for the compression of individual columns.

Figure 7.

Schematic representation of the Nishihara model in compression.

As for the strain, the total strain of the KFA network was given the same value as the individual strain of the column, in agreement with the criteria defined for a parallel arrangement of the constitutive elements. Each column was subjected to the same stress in this transversely isotropic matrix material and worked as a Nishihara model which determined the KFA's mechanical behavior.

Therefore, the total deformation of each column unit, as the summation of the deformation of the fibers in the column, mechanically consisted of the elastic, viscoelastic and viscoplastic deformations, and could be expressed as follows:

ɛ = ɛ_{e} + ɛ_{ve} + ɛ_{vp}

(1)

Mechanical constitutive equation

As the Nishihara model described, the mechanical constitutive equations of the KFA under the compressional load were developed in the three stages individually.

(a) When σ < σ_c, the Nishihara model worked as the SLS model which performed the viscoelasticity. The mechanical constitutive equations were developed in the Laplace transform and inverse Laplace transform to solve the differential equations with the great benefit of producing easily solvable algebraic equations.³⁵

In the SLS model, the equations of the stress σ(t) and the strain ɛ(t) are:

σ = σ_{e} = σ_{ve}

(2)

ɛ = ɛ_{e} + ɛ_{ve}

(3)

The constitutive equations express the relationship between the stress of the jth unit and the related strain. Thus, the strain of a unit which is composed of its viscoelastic and elastic parts could be written by the function of two variables: time t and stress σ. According to the mechanical constitutive of the Kelvin model, the viscoelastic strain ɛ_ve can be expressed as follows:

ɛ_{ve} = \frac{σ_{ve}}{E_{2}} (1 - e - t / η_{2} / E_{2}),

(4)

and the elastic strain ɛ_e is

ɛ_{e} = \frac{σ_{e}}{E_{1}}

(5)

Therefore, the constitutive equation of the SLS model based on equation (3) can be expressed as:

ɛ = ɛ_{e} + ɛ_{ve} = σ [\frac{1}{E_{2}} (1 - e - t / η_{2} / E_{2}) + \frac{1}{E_{1}}] .

(6)

The Laplace transform was applied to equations (2)–(6), and the equations were arranged as:

- {\overset{\land}{ɛ}}_{e} - {\overset{\land}{ɛ}}_{ve} + {\overset{\land}{σ}}_{e} (\frac{E_{1} + E_{2}}{E_{1} E_{2}} \cdot \frac{1}{s} - \frac{1}{E_{2}} \cdot \frac{1}{s + \frac{E_{2}}{η_{2}}}) = 0

(7)

{\overset{\land}{ɛ}}_{e} - \frac{{\overset{\land}{σ}}_{e}}{E_{1}} = 0

(8)

{\overset{\land}{ɛ}}_{e} + {\overset{\land}{ɛ}}_{ve} - \overset{\land}{ɛ} = 0

(9)

{\overset{\land}{σ}}_{e} - \overset{\land}{σ} = 0

(10)

{\overset{\land}{σ}}_{e} - {\overset{\land}{σ}}_{ve} = 0

(11)

{\overset{\land}{σ}}_{ve} - \overset{\land}{σ} = 0

(12)

Hence, the matrix notation of the six equations (7)–(12) is:

Ax = 0

where 0 is a nil vector, x is a vector of variables:

x = {{\overset{\land}{ɛ}}_{e} {\overset{\land}{ɛ}}_{ve} {\overset{\land}{σ}}_{e} {\overset{\land}{σ}}_{ve} \overset{\land}{σ} \overset{\land}{ɛ}} T

(13)

and A is a matrix:

| - 1 - 1 \frac{E_{1} + E_{2}}{E_{1} E_{2}} \cdot \frac{1}{s} - \frac{1}{E_{2}} \cdot \frac{1}{s + \frac{E_{2}}{η_{2}}} 000010 - \frac{1}{E_{1}} 0011000 - 1 0010 - 1 0001 - 1 000001 - 1 0 |

Applying the Gaussian elimination method to the matrix,³⁵ A takes the following form:

| - 1 \frac{- η_{2} s 2 + (η_{2} - E_{2}) s + E_{1} + E_{2}}{η_{2} s 2 + E_{2} s} 000001 - \frac{1}{E_{1}} 000001 - 1 000001 - 1 00000 \frac{η_{2} s + E_{1} + E_{2}}{(η_{2} s + E_{2}) E_{1} s} - 1 000000 |

It can be found that the last equation is equal to zero. In accord with the fifth row of the matrix A , the relationship between the strain and the stress of the model in this stage can be expressed in the Laplace transform as follows:

\overset{\land}{σ} (s) = E_{a} (s) \times \overset{\land}{ɛ} (s), and E_{a} (s) = \frac{(η_{2} s + E_{2}) E_{1} s}{η_{2} s + E_{1} + E_{2}}

(14)

Given the stress function in the constant strain test, the strain is $ɛ (t) = K_{ɛ} t, and \overset{\land}{ɛ} (s) = \frac{K_{ɛ}}{s 2}$ in the Laplace transform, where K_ɛ is the constant strain. Then

\overset{\land}{σ} (s) = E_{a} (s) \times \overset{\land}{ɛ} (s) = \frac{(η_{2} s + E_{2}) E_{1} s}{η_{2} s + E_{1} + E_{2}} \cdot \frac{K_{ɛ}}{s 2}

(15)

Hence, equation (15) can be rewritten in the inverse Laplace transform and the stress–time function to reflect the mechanical response of the material can be expressed as follows:

σ_{a} (t) = E_{a} (t) \times K_{ɛ} = K_{ɛ} \times [\frac{E_{1} E_{2}}{E_{1} + E_{2}} + \frac{E_{1}^{2}}{E_{1} + E_{2}} e \frac{- (E_{1} + E_{2}) t}{η_{2}}]

(16)

(b) When σ_c ≤ σ ≤ σ_c + σ_s/θ, the overall stress reached the yielding stress of the Bingham model, and it starts to work as a nonlinear viscosity. The Nishihara model was used to reflect the compressional mechanical response of the kapok fiber in the KFA. The equations of the stress σ(t) and the strain ɛ(t) are:

σ = σ_{e} = σ_{ve} = σ_{vp}, ɛ = ɛ_{e} + ɛ_{ve} + ɛ_{vp}

(17)

where

ɛ_{e} = \frac{σ_{e}}{E_{1}}, ɛ_{ve} = \frac{σ_{ve}}{E_{2}} - \frac{η_{2}}{E_{2} ɛ} {\overset{\cdot}{ɛ}}_{ve}, ɛ_{vp} = \frac{(σ_{vp} - σ_{c})}{η_{1}}

Therefore, the mechanical constitutive equation is derived as follows:³⁶

\frac{η_{2}}{E_{2}} \overset{··}{ɛ} + \overset{\cdot}{ɛ} = \frac{η_{2}}{E_{1} E_{2}} \overset{··}{σ} + \frac{1}{E_{2}} (1 + \frac{E_{2}}{E_{1}} + \frac{η_{2}}{η_{1}}) \overset{\cdot}{σ} + \frac{σ - σ_{c}}{η_{1}}

(18)

Then the stress–time function in this viscoelastoplastic stage could be obtained by solving the equation (18) via the Laplace transform and inverse Laplace transform:

σ_{b} (t) = K_{ɛ} \times E_{b} (t) + \frac{σ_{c}}{u_{2}} [u_{2} + \frac{u_{2}}{\sqrt{u_{1}^{2} - 4 u_{2}}} \times (\frac{e - α t}{α} - \frac{e - β t}{β})]

(19)

where

E_{b} (t) = \frac{E_{1}}{α - β} [(- \frac{E_{2}}{η_{1}} + α) e - α t + (\frac{E_{2}}{η_{1}} - β) e - β t], u_{1} = \frac{η_{2}}{E_{1}} + \frac{η_{1} + η_{2}}{E_{2}}, u_{2} = \frac{η_{1} η_{2}}{E_{1} E_{2}}, α = \frac{(u_{1} + \sqrt{u_{1}^{2} - 4 u_{2}})}{2 u_{1}}, β = η_{2} \times \frac{(u_{1} - \sqrt{u_{1}^{2} - 4 u_{2}})}{2 u_{1}}

(c) When σ > σ_c + σ_s/θ, the compressional stress rapidly increases and accelerates the linear creep of the Bingham model. This makes the Bingham model fail instantly and the coefficient of the viscosity (η₁) of the viscoplastic element descend to zero as the compressing time increases. The Nishihara model is turned into a senior viscoelastic stage and works as the SLS model.

Therefore, the stress–time function in this stage can be derived from the constitutive relationship of the SLS model and be written as follows:

σ_{c} (t) = E_{c} (t) \times K_{ɛ} = K_{ɛ} \times [\frac{E_{1} E_{2}}{E_{1} + E_{2}} + \frac{E_{1}^{2}}{E_{1} + E_{2}} e \frac{- (E_{1} + E_{2}) t}{η_{2}}]

(20)

As illustrated in Figure 6, the KFA can be simulated by the parallel connection of q Nishihara models. Based on this assumption, in the overall network there are:

ɛ (t) = ɛ_{j}

(21)

σ (t) = σ_{1} + σ_{2} + \dots + σ_{j} + \dots + σ_{q} = \sum_{j}^{q} σ_{j} = q \times σ_{j}

(22)

where j is the number of columns, and q can be estimated from the width of the KFA (30 mm) and the diameter of the kapok fiber (about 2 × 10⁻² mm) according to the statistics in the pipe-piling structure section.⁴ In this work, q is approximately equal to 1.5 × 10³. Hence, the overall stress (

σ (t))

for the KFA can be expressed as follows:

σ (t) = {q \times σ_{a} (t), σ < σ_{c} q \times σ_{b} (t), σ_{c} \leq σ \leq σ_{c} + σ_{s} / θ q \times σ_{c} (t), σ > σ_{c} + σ_{s} / θ

(23)

Numerical solution

Equation (23) includes four material parameters (E₁, E₂, η₁, and η₂), one system parameter (q), and the constant of the strain (K_ɛ), in which q is estimated to be 1.5 × 10³ and K_ɛ is 60 mm/min.

The stress–time data were collected in compression tests directly and the statistical regression analyses were solved in the algorithm of curve-fitting based on least-squares. As demonstrated by the mechanical constitutive equations of the respective stages, both the stress–time functions $σ_{a} (t)$ and $σ_{c} (t)$ can be simplified to the same formula as follows:

y = f (x) = y_{0} + A_{1} \times exp (\frac{x}{T_{1}})

(24)

where

y_{0} = {qK}_{ɛ} \frac{E_{1} E_{2}}{E_{1} + E_{2}}, A_{1} = {qK}_{ɛ} \frac{E_{2}^{2}}{E_{1} + E_{2}}, T_{1} = - \frac{η_{1}}{E_{1} + E_{2}}

. The stress–time function

σ_{b} (t)

can be written as:

y = f (x) = y_{0} + A_{1} \times \exp (\frac{x}{T_{1}}) + A_{2} \times \exp (\frac{x}{T_{2}})

(25)

where

A_{1} = [\frac{{qK}_{ɛ} E_{1}}{α - β} (- \frac{E_{2}}{η_{1}} + α) + \frac{σ_{c}}{α \times \sqrt{u_{1}^{2} - 4 u_{2}}}], A_{2} = [\frac{{qK}_{ɛ} E_{1}}{α - β} (\frac{E_{2}}{η_{1}} - β) - \frac{σ_{c}}{β \times \sqrt{u_{1}^{2} - 4 u_{2}}}], y_{0} = σ_{c}, T_{1} = - \frac{1}{α}, T_{2} = - \frac{1}{β}, u_{1} = \frac{η_{2}}{E_{1}} + \frac{η_{1} + η_{2}}{E_{2}}, u_{2} = \frac{η_{1} η_{2}}{E_{1} E_{2}}, α = \frac{(u_{1} + \sqrt{u_{1}^{2} - 4 u_{2}})}{2 u_{1}}, β = η_{2} \times \frac{(u_{1} - \sqrt{u_{1}^{2} - 4 u_{2}})}{2 u_{1}}

The least-square algorithm of curve-fitting was used to find the coefficients y₀, A₁, A₂, T₁, and T₂ in every compressional stage of the KFA and to solve the following problem:

min f (y_{0}, A_{1}, A_{2}, T_{1}, T_{2}, x) - y 2 = min \sum_{i} [f (y_{0}, A_{1}, A_{2}, T_{1}, T_{2}, x_{i}) - y_{i}] 2

where x_i is the time variable and y_i is the stress variable, both of which could be recorded in the compression test. The values of y₀, A₁, A₂, T₁, and T₂ were found to solve this optimization problem.³⁷ Hence, the material parameters E₁, E₂, η₁, and η₂ for the KFA in each compression could be numerically simulated by solving the system of algebraic equations from the stress–time functions, i.e. equations (24) and (25).

Results and discussion

Calculation of the parameters of KFA's Nishihara model

The parameters of the viscoelastoplastic model in the stage of each compression should be first estimated by the numerical solution. Since the KFAs were examined in a constant strain test, the stress–time curves (see Figure 8(a)) can be transferred from the compressional stress–strain curves (see Figure 4(a)), and three characteristic stages in circled area could also be observed clearly. σ_c and σ_c + σ_s/θ as the stress at the yielding points 1 and 2 in the S-S curve. respectively, (see Figure 4(b)) divided the stress–time curve into the three corresponding stages. In each stage, the experimental data of stress and time were nonlinear curve-fitted by the derived stress–time functions, i.e. equations (24) and (25).

Figure 8.

The experimental stress–time curves of the kapok fiber assembly (KFA): (a) a complete compressional stress–time curve of the KFA; (b) characteristic stages of the stress–time curve of the circled area of (a).

The 12th compression test results of two dry-treated samples (d-1 in green and d-2 in red) were randomly selected to show the parameter calculation. The two yielding points and their respective stresses could be identified according to the stress-time data. As shown in Figure 8, three segments of the stress–time curve were obtained by the nonlinear curve-fitting method separately, in which the stress–time relationship in each stage was coded by the corresponding curve-fitting function. Then the fitting functions in each stage could be worked out. as listed in Table 3 and as the black curves portray in Figure 9.

Figure 9.

The nonlinear curve fitting in the three stages: (a) stage A; (b) stage B; (c) stage C.

Table 3.

The compressional stress–time curve fitting functions and adjusted R-square values in the stages

Sample	Stage	The stress–time curve fitting function	Adj. R-square
d-1	A	$y = 0.0027 + 1.0656 \times 10 - 22 * exp (\frac{x}{0.3870})$	0.9179
	B	$y = 0.0531 - 9.3500 \times 10 - 57 * exp (\frac{x}{0.1438}) + 8.0171 \times 10 - 58 * exp (\frac{x}{0.1412})$	0.9740
	C	$y = - 11.6437 + 9.6034 \times 10 - 6 * exp (\frac{x}{1.3466})$	0.9991
d-2	A	$y = 0.0042 + 1.3907 \times 10 - 22 * exp (\frac{x}{0.3794})$	0.9620
	B	$y = 0.0411 - 4.0971 \times 10 - 47 * exp (\frac{x}{0.1708}) + 1.0494 \times 10 - 48 * exp (\frac{x}{0.1651})$	0.9968
	C	$y = - 9.1264 + 1.0393 \times 10 - 6 * exp (\frac{x}{1.1572})$	0.9983

According to the calculating methods proposed in the numerical solution, the model parameters E₁, E₂, η₁, and η₂ were finally obtained by solving the system of linear equations in the corresponding stage. In the case of the 12th compression test result on d-1 and d-2, E₁, E₂, η₁, and η₂ were calculated individually, as listed in Table 4.

Table 4.

Material parameter calculations in the 12th compression test on dry-treated KFAs

Sample	Stage	E₁ (kgf/cm²)	E₂ (kgf/cm²)	η₁ (kgf s/cm²)	η₂ (kgf s /cm²)
d-1	A	4.5667E-05	1.1743E + 15	/	4.5443E + 14
	B	5.6617E-04	1.6657E + 61	1.4307E-02	2.0875E + 60
	C	1.9406E-01	2.3529E + 05	/	3.1685E + 05
d-2	A	6.9833E-05	2.1039E + 15	/	7.9827E + 14
	B	4.6667E-04	5.8275E + 54	1.1579E-02	7.9160E + 53
	C	1.5211E-01	1.3358E + 06	/	1.5457E + 06

Therefore, the four parameters of the viscoelastoplastic model in the three characteristic stages of the all samples in 50 compression cycles could be worked out via the solution, and then the transverse compressional behavior of the KFA and the viscoelastoplastic model modulus of kapok fiber were characterized and simulated numerically.

Analysis on the parameters of KFA's Nishihara model

Figures 10 –12 depict the relationships between the compressing-cycle and the model parameters E₁, E₂, η₁, and η₂ in the three stages. The E₁ and η₁ were the data calculated directly from the above solution, and the E₂ and η₂ were the testing data smoothed by the Savitzky–Golay filter method which could capture the important pattern in the data and leave out noise, such as the parameters of E₂ and η₂ in stage A shown in Figure 10(c). In general, the model parameters had an acceptable agreement on the samples with the same treatment and characterized the transverse compressional behavior of the KFAs well.

Figure 10.

Estimations of the parameters E₁, E₂, η₁, and η₂ estimations vs. compressing cycle in stage A. (a) E1 in stage A of the dry-treated KFAs, (b) E₁ in stage A of the wet-treated KFAs, (c) E₂ and η₂ in stage A of the dry-treated KFAs and (d) E₂ and η₂ in stage A of the wet-treated KFAs.

According to the viscoelastoplastic model derivation, the KFA was supposed to work as a SLS model in stage A. The four model parameters are changing with the compressing-cycle in this stage as shown in Figure 10. The E₁ values of the dry samples were unsteady and low at the beginning cycles and gradually approached to a stable level at the late cycles because the dry KFAs were loose and they were in an unsteady structure (Figure 10(a)). The E₁ values of the wet KFAs were consistent in all the cycles because of the relatively steady structure, as shown in Figure 10(b). For the dry KFAs, the E₁ values (average 5.9277E-05 kgf/cm², with a coefficient of variation 18.42%) are smaller and more variable than those of the wet ones (7.0161E-05 kgf/cm², 6.67%). Meanwhile, the E₂ and η₂ values of the wet KFAs also possessed a better reproducibility, as shown in Figure 10(c) and (d). This was mainly owing to the steady mechanical response from the more stable structure of the wet KFAs. Compared with the values of E₁, in terms of magnitude E₂ and η₂ were much larger. Therefore, in stage A the external compressional load was too small to deform the dashpot of the Kelvin model and the E₁ modulus of the spring was the dominant modulus used to reflect the transverse compressional response of the unit column.

During stage B, the compressional stress was larger than the yielding strength of the Bingham model (σ_c), but smaller than σ_c + σ_s/θ. This made the Bingham model work as a viscoplastic element. The E₁ and η₁ values were calculated, as shown in Figure 11(a) and (b). It was found that the wet KFAs have a better repeatability on these two parameter values in stage B. For E₂, there is a similar change both on the dry- and wet-treated KFAs as the solid red and black curves depict in Figure 11(c) and (d), but the magnitude of the values on the dry KFAs is almost 10²⁷ times larger than those of the wet ones. In this stage, E₂ and η₂ were also much larger than the values of E₁ and η₁. Consequently, the viscoelastoplastic model actually worked as the series connection of the elastic and the viscoplastic elements in stage B. E₁ as the determining modulus was used to reflect the main transverse elasticity, and η₁ as the primary viscosity was used to mirror the main viscous property of each unit column.

Figure 11.

Estimations of the parameters E₁, E₂, η₁, and η₂ vs. compressing cycle in stage B. (a) E1 and η₁ in stage B of the dry-treated KFAs, (b) E₁ and η₁ in stage B of the wet-treated KFAs, (c) E₂ and η₂ in stage B of the dry-treated KFAs and (d) E₁ and η₂ in stage B of the wet-treated KFAs.

In stage C, plenty of the fibers of the KFA were squashed as ribbon-like and even brittle crushed. The compressional stress rapidly increased. For the model, the linear creep of the Bingham model soared and made the viscoplastic element fail instantly. Then the KFA was supposed to act as a senior SLS model. As shown in Figure 12, the E₁ values of the dry KFAs are slightly higher than those of the wet samples, and the E₂ and η₂ values of all the samples plunge after the first 25 repetitive compressions. However, the KFA could only start to work as a real SLS model once the compressional stress was large enough to deform the dashpot of the viscoelastic element. Since the maximum compressional load was 214 kgf, the dashpot of the viscoelastic element could be deformed at the 13th and 11th compressions for the dry- and wet-treated KFAs, respectively, as shown by the η₂ values in Figure 11(c) and (d). As a result, after 13(11) compressing cycles on the dry (wet) KFAs, the determining modulus became the modulus of the working SLS model.

Figure 12.

Estimations of the parameters E₁, E₂, and η₂ vs. compressing cycle in stage C. (a) E1 in stage C of the dry-treated KFAs, (b) E₁ in stage C of the wet-treated KFAs, (c) E₂ and η₂ in stage C of the dry-treated KFAs and (d) E₂ and η₂ in stage C of the wet-treated KFAs.

On average, it took 1.5 s (1.2 s) to complete stage C on the dry (wet) KFAs according to the statistics in the compressions. At the 50th compression, the modulus of the SLS model could be estimated at 0.2230 kgf/cm² for the dry KFA and 0.1898 kgf/cm² for the wet KFA in accordance with the parameter calculations of the 50th compression and the function $E (t) = \frac{E_{1} E_{2}}{E_{1} + E_{2}} (1 + \frac{E_{1}}{E_{2}} e - \frac{{tE}_{2}}{η_{2}}$ ). Therefore, the transverse modulus of the KFA, composed of q (1.5 × 10³) model columns in a parallel arrangement, was calculated to be 344.50 kgf/cm² (3.28 GPa) for the dry and 284.70 kgf/cm² (2.79 GPa) for the wet KFA. These two calculations on the transverse modulus of the KFA in the final stage were in a good agreement with the transverse modulus of the matrix material of the cellulose fiber reported in previous research.^38–40

In summary, the Nishihara model, including elastic, viscoelastic, and viscoplastic elements, not only provided a good characterization on the transverse compressional behavior of the KFA, but also demonstrated their various rheology properties in the different stages. In compression, the variable elasticity of the KFA existed throughout the total process; the viscoelasticity was excluded in stages A and B because the viscoelastic element of the Kelvin model failed to work, but it appeared in stage C; and the viscoplastic property was effectively captured by the Bingham model in stage B. The four parameters of the model can quantitatively characterize the changes of the transverse compressional properties.

Influence of conditioning humidity on KFA's Nishihara model

The influence of the conditioning humidity on the Nishihara model of the KFA was investigated by examining the KFAs under different conditioning humidity treatments. The conditioning humidity generally improved the consistency and the repeatability of the model parameters. It reduced the viscosity of the Bingham model in stage B (see Figure 11(a) and(b)) and greatly increased the viscoelasticity of the KFAs in stage C (see Figure 12(c) and (d)).

The effects of conditioning humidity on the yielding compressional stress of the KFA are shown in Figure 13. The compressional stresses at the yielding point 1 (σ_c) of both the dry- and wet-treated KFAs are at the same level and could keep a relative good consistency within the 50 compressing cycles. In contrast, the compressional stresses at the yielding point 2 (σ_c + σ_s/θ) gradually decrease as the compression continues. In the final compression, the stresses at the point 2 of the wet KFAs were much larger than those of the dry ones. The processing time from the point 1 to 2 (in stage B) of the wet KFAs were also much more than those of the dry ones, as shown in Figure 14. Combined with the statistical analysis on the viscoplasticity of the KFAs in stage B, this indicates that the conditioning humidity increased the instant strength of the Bingham model in terms of the plasticity of the KFAs, but reduced the viscosity of the Bingham model.

Figure 13.

The compressional stress at the yielding point of the stress–time curve: (a) the point 1; (b) the point 2.

Figure 14.

Processing time in stage B.

Conclusions

A viscoelastoplastic model of the transverse compressional behavior of a KFA based on the pipe-piling structure has been presented in this study. The model includes the elastic (the spring model), viscoelastic (Kelvin model), and viscoplastic elements (Bingham model), and can be used to characterize the transverse compressional properties of the KFA in the three characteristic stages according to the experimental results from repetitive transverse compressions.

In a compression, four material parameters were worked out by the numerical solution of the mechanical constitutive equations. From the analysis of the parameters, it was found that the variable elasticity of the KFAs existed throughout the total compression, the viscoelasticity appeared only in stage C, and the viscoplastic property was effectively captured by the Bingham model in stage B. The KFAs did not exhibit viscoelastic behavior in stages A and B because the viscoelastic element of the Kelvin model failed to work in these two stages.

The conditioning humidity for the KFAs generally improved the consistency of the material parameters, and greatly increased the viscoelasticity of the KFAs in stage C. In addition, it increased the instant strength of the Bingham model regarding the plasticity of the KFA, but lowered the viscosity of the Bingham model in stage B. With some assumptions, the viscoelastoplastic mechanical model proposed in the study seems to work well in representing the properties of the KFA under repetitive transverse compression. It can reflect the variable rheology of the KFA in different stages of the changing structure.

Footnotes

Funding

The research was partially supported by a scholarship from China Scholarship Council (NO.2011663014).

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