**3.3. Maxwell, Kelvin, and four parameter models**

deformation behavior of the biocomposites, depending on time and temperature. A schematic diagram of flow behavior (creep) can be seen in **Figure 3**; given load shows a configuration of four point bending biocomposite. The weight or load, along with gravity, provides a constant effort in biocomposite. After 5 days in this condition no significant unfavorable deformation occurs. However, after 7 months deformation caused by the effort has increased, and deforms

**Figure 3.** CREEP response of a biocomposite beam subjected to bending at four points, an illustrative estimate given by

The biocomposites polymer matrix has significant sensitivity to use a function of time and temperature, resulting in a limited use of structural applications demanding applications or in dimensional stability value. When the biocomposite is subjected to high stresses, it can result in the material, and excessive deformations that may cause the product to lose its functionality, one could get the material to the tertiary region where creep occur until fracture. This is called the upper region and is also known as the acceleration phase of CREEP. The importance of the tertiary region for normal operation and design to CREEP is also important, since parts of polymer or compound should be designed to avoid this area; safety factors must ensure away

The experimental response of a dynamic test to tension, bending, or compression creep compliance of a biocomposite can be modeled with provisions of springs and dampers, where the springs represent the elastic solid behavior and cushion the behavior of a viscous liquid. It represents the Hooke spring deformation force that is proportional to the applied stress and the damper flow proportional to the strain rate Newtonian. To model mathematically one biocomposite, the stress, strain and time, you can relate to the constant characteristics of the mechanical elements [57]. The mechanical model mimics the actual behavior of biocomposites, although the elements themselves may not have direct analogies with real material. However, these models represent a mathematical understanding of the problems of viscoelastic performance of biocomposites, studied by accelerated tests in the laboratory, which can easily be articulate studies of continuum mechanics means to solve even more complex models with the

from this region over the lifetime of the products developed with biocomposites.

further after 2 years.

316 Composites from Renewable and Sustainable Materials

the author.

**3.2. Mathematical models**

**Figure 4(a)** shows the Maxwell model, which is represented by a spring connected in series with a damper and **Figure 4(b)** shows Kelvin model (or Voigt), which is represented by a spring connected in parallel with a damper; in both cases an approximation of a system characterized by time dependent and the ratio of *η* viscosity (damper) with the modulus of elasticity *E* represented by the spring is obtained, which they can be approximated to describe the viscoelastic behavior of a biocomposite. The parameter *η* leads to model a related response delay time for the Kelvin model, and the relaxation time for the Maxwell model.

**Figure 4.** Models: (a) Maxwell, (b) Kelvin.

To analyze the flow behavior (CREEP) of biocomposites, it is possible to apply the model of four parameters successfully, which is derived from a series combination of the Maxwell model and Kelvin model, as shown in **Figure 5**.

The four-facing model fits the response obtained experimentally for the controlled creep test, based on the technique of dynamic mechanical analysis (DMA). **Figure 6** depicts a curve of biocomposite creep subjected to a three-point bending at a constant effort in the linear viscoelastic region. The fraction of O to A shows the rapid response of the initial deformation on the flow curve, i.e., it occurs an instantaneous elastic response. This behavior is followed by a region of creep from A to B, where the shear rate decreases at a constant rate introduced in Section B to C. Once the stress is removed, the instantaneous elastic O to A response is fully retrieved from C to D, i.e., the distance a' = a. Then the curve drops from D to E in a slower recovery. However, this recovery is not complete due to the initial state by increasing c' = c. This response is completely unrecoverable, and is a measure of plastic flow [58].

**Figure 5.** Four-parameter model.

**Figure 6.** Four-parameter model and yield curve (CREEP).

**Figure 5** represents the elastic response changes in A and A'. A convenient to model this response element is the Hooke law. The Kelvin-Voigt model corresponds to changes in b' (change symbol) in the region of creep; it represents a damping plastic flow c'. **Figure 6** shows the mechanical models for the purpose of better illustrating the appearance of the flow curve. The four-parameter model is an assembly of a Maxwell element and Kelvin-Voigt element, where the latter component is time dependent. **Figure 7** shows that (zone 1) the system is idle effortless. When it exposed to constant stress to three-point bending, in **Figure 7** (zone 2), the spring system with a constant amount of *E*1 extends instantaneously to "a" *σ*/*E*1 = a. Then, in **Figure 7** (zone 3), the fluence rate decreases with a gradual increase in load bearing spring *E*2, until fully extended and the damping *η*2 no longer carry any load. As the spring is now *E*2 fully extended, the creep ratio to a solid phase, corresponding to the plastic flow of the linear viscoelastic region represented by the constant *η*1 damper. The damper deforms until the load is removed, as illustrated in **Figure 7** (zone 4), leaving permanent deformation. Now, the spring retracts quickly *E*1 to a' and the recovery period is b'. During this time, the damper *η*2 is forced to retreat to its initial position by spring *E*2 representing a delayed or anelasticity elastic response. The damper position *η*3 remains in the extended state, since the spring cannot influence its final position; this can be seen in **Figure 7** (zone 5). Thus, the nonrecoverable plastic flow is equal to c' = *σ t*/*η*3. The model fully represents the elastic, inelastic, and viscous behaviors of biocomposites, indicating that if possible with fillers or natural fibers affecting the interfacial relationship, fiber matrix polymer or cross-linked polymer, the variable *η*1 increase, which it is reflected in a decrease in permanent deformation c.

**Figure 7.** Response of four-parameter model adjusted to fit a creep curve.

recovery. However, this recovery is not complete due to the initial state by increasing c' = c.

**Figure 5** represents the elastic response changes in A and A'. A convenient to model this response element is the Hooke law. The Kelvin-Voigt model corresponds to changes in b' (change symbol) in the region of creep; it represents a damping plastic flow c'. **Figure 6** shows the mechanical models for the purpose of better illustrating the appearance of the flow curve.

This response is completely unrecoverable, and is a measure of plastic flow [58].

**Figure 5.** Four-parameter model.

318 Composites from Renewable and Sustainable Materials

**Figure 6.** Four-parameter model and yield curve (CREEP).

Given the balance of forces occurring in the four-parameter model, we can write the following expression, with respect to the effort and deformation:

$$
\sigma = \sigma\_1 = \sigma\_2 = \sigma\_3 + \sigma\_4 \tag{3}
$$

$$\mathfrak{e} = \mathfrak{e}\_1 + \mathfrak{e}\_2 + \mathfrak{e}\_{3,4} = \mathfrak{e}\_1 + \mathfrak{e}\_2 + \mathfrak{e}\_4, \dots, \dots, \mathfrak{e}\_3 = \mathfrak{e}\_4 \tag{4}$$

where *εk* is the strain response of the Kelvin-Voigt model, the equations representing different relationship stress-strain are:

$$
\sigma\_1 = E\_1 \varepsilon\_1,\\
\sigma\_2 = \eta\_2(d\varepsilon\_2 \mid dt),\\
\sigma\_3 = E\_3 \varepsilon\_3,\\
\sigma\_4 = \eta\_4(d\varepsilon\_4 \mid dt) \tag{5}
$$

And the equation that relates and models the behavior of biocomposite visco-elastic-plastic holistically can be arranged as follows:

$$
\left[\frac{\eta\_1 \eta\_2}{E\_1 E\_2}\right] \frac{d^2 \sigma}{dt^2} \bigg| + \left[\frac{\eta\_1}{E\_1} + \frac{\eta\_1 + \eta\_2}{E\_2}\right] \frac{d \sigma}{dt} + \sigma = \frac{\eta\_1 \eta\_2}{E\_2} \frac{d^2 \sigma}{dt^2} + \eta\_1 \frac{d \sigma}{dt} \tag{6}
$$

Equation (7) can be solved for flow conditions and/or relaxation; response to deformation is:

$$\varepsilon(t) = \frac{\sigma\_0}{E\_1} + \frac{\sigma\_0}{\eta\_1}t + \frac{\sigma\_0}{E\_2} \left[1 - e^{-\frac{E\_2 t}{\eta\_2}}\right] \tag{7}$$

Response to creep recovery is:

$$\varepsilon(t) = \frac{\sigma\_0}{\eta\_1} t + \frac{\sigma\_0}{E\_2} \left[ e^{-\frac{E\_2 t}{\eta\_2}} - 1 \right] e^{-\frac{E\_2 t}{\eta\_2}} \dots t > t\_1 \tag{8}$$

where *ε*(*t*) is the yield strength, *σ*0 is the initial applied stress, *t* is time, *E*1 and *E*2 are the elastic modulus of the springs of Maxwell and Kelvin, respectively, and *η*1 and *η*2 are the viscosities of the Maxwell and Kelvin dampers. *η*2/*E*2 is usually denoted as *τ*, the delay time required to generate 63.2% strain on the Kelvin unit [59].

In Equation (7), the first term is the instantaneous elastic strain. The second term is the early stage of creep deformation, and is due to mechanisms such as relaxation, extension of the molecular chain, and biocomposites closely related to the performance of the fiber matrix micromechanic relationship. The last term represents the long-term creep deformation, and is due to the overall performance of the biocomposite. The parameters *E*1, *E*2, *η*1, and *η*2 can be obtained by adjusting the equation to the experimental data and can be used to describe the creep behavior. The strain rate of the linear viscoelastic region of the biocomposite is possible to calculate, if we derive Eq. (7), and obtain the strain rate:

$$\stackrel{\bullet}{\sigma} = \frac{\sigma\_0}{\eta\_1} t + \frac{\sigma\_0}{\eta\_2} e^{-\frac{E\_2 t\_1}{\eta\_2}} \tag{9}$$

For all models of linear viscoelasticity, it is important to note that the response of deformation creep is independent of the level of effort, giving the opportunity to study and compare with other systems evaluated under the same environmental conditions, must obtain curves and models of creep. The equation for calculating the compliance *D*(*t*) is defined by:

$$D\_{(t)} = \frac{1}{E\_1} + \frac{t}{\eta\_1} + \frac{1}{E\_2} \left[ 1 - e^{-\frac{E\_2}{\eta\_2}} \right] \tag{10}$$

The viscoelastic behavior of a system biocomposite presents different delay times; therefore, it is possible to model more precisely repeating *n* Kelvin-Voigt models, which are particularly adjusted in the delayed elastic region or annalistic, which has default a typical nonlinear region. In **Figure 8**, you can see a diagram showing this behavior.

**Figure 8.** Multiple models for adjusting creep curves.

1 11 2 2 2 3 3 3 4 4 4

 s

2 2

1 2 1 12 1 2

1 2 1 2 2

 es

> hh

> > 2

*E t*

h

é ù -

ê ú ë û

s

 e

1

é ù é ù <sup>+</sup> ê ú + + += + ê ú ë û ë û (6)

 h  e

(7)

(8)

And the equation that relates and models the behavior of biocomposite visco-elastic-plastic

2 2 1

Equation (7) can be solved for flow conditions and/or relaxation; response to deformation is:

00 0 <sup>2</sup>

2 1 2

 h

is the yield strength, *σ*0 is the initial applied stress, *t* is time, *E*1 and *E*2 are the elastic

2 1

= + (9)

*E t*

h

0 0 <sup>2</sup> 1 2

For all models of linear viscoelasticity, it is important to note that the response of deformation creep is independent of the level of effort, giving the opportunity to study and compare with

*t e*

 s

· -

 h

s

h

e

*Et Et*

 s

=+ + - ê ú

11 2 ( ) 1

0 0 2 2

*t t e e tt*

h

é ù - - =+ - > ê ú ê ú ë û

modulus of the springs of Maxwell and Kelvin, respectively, and *η*1 and *η*2 are the viscosities of the Maxwell and Kelvin dampers. *η*2/*E*2 is usually denoted as *τ*, the delay time required to

In Equation (7), the first term is the instantaneous elastic strain. The second term is the early stage of creep deformation, and is due to mechanisms such as relaxation, extension of the molecular chain, and biocomposites closely related to the performance of the fiber matrix micromechanic relationship. The last term represents the long-term creep deformation, and is due to the overall performance of the biocomposite. The parameters *E*1, *E*2, *η*1, and *η*2 can be obtained by adjusting the equation to the experimental data and can be used to describe the creep behavior. The strain rate of the linear viscoelastic region of the biocomposite is possible

( ) 1 .....

1 2

s

h

to calculate, if we derive Eq. (7), and obtain the strain rate:

e

generate 63.2% strain on the Kelvin unit [59].

*E*

 s h

*t te E E* ss

*d d dd E E dt E E dt E dt dt*

 h e

== = = *E d dt E d dt* , ( / ), , ( / ) (5)

s

320 Composites from Renewable and Sustainable Materials

holistically can be arranged as follows:

hh

Response to creep recovery is:

where *ε*(*t*)

 es

> s

 h e

 h h h s

e

Fundamental viscoelastic properties such as creep compliance module or relaxation can be found by solving the differential equation that represents including for the appropriate load if necessary. For example, the compliance fluence can be determined using the conditions for a creep test and would be represented by:

$$D\_{(i)} = \frac{1}{E\_1} + \frac{t}{\eta\_1} + \sum\_{i=2}^{n} \frac{1}{E\_i} \left[ 1 - e^{-\frac{t}{\tau\_i}} \right] \pi\_i = \frac{\eta\_i}{E\_i} \varepsilon\_{(i)} = D\_{(i)} \sigma \tag{11}$$

The determination of initial conditions is achieved by inspection of the physical model. Since the input force is constant for the creep test, the change in stress is zero. The solutions of differential equations for relaxation conditions, constant deformation, or changes in stress and other conditions can be obtained similarly.

#### **3.4. Time-dependent deformation behavior of biocomposite**

A way to study the effect of a reinforcement or filler natural fibers to manufacture biocomposites is to show the interdependence of the stress, strain, and time through curve creep and recovery creep performed in short tests using the DMA technique dynamic mechanical analysis. To perform the creep tests and recovery (creep and creep recovery), it is necessary to set a DMA with the force of flow required to maintain the constant effort during testing of creep at a given time, and rezero when it reaches the estimated time then deformation is recorded by another equal time. For example, for a biocomposite made of polyethylene aluminum fiber fique, we can observe the response to creep and creep recovery in **Figure 9**, a maximum stress of 1.2 MPa was applied in a bending test at three points at 25°C, 1.2 MPa effort is an attempt that was obtained after identifying the linear viscoelastic region, a biocomposite test strain sweeps at 25°C in a DMA (RSAIII).

**Figure 9.** Creep curves and creep recovery for biocomposite LDPE-Al-Fique 10% reinforcement of natural fibers.

**Figure 10.** Creep curves for biocomposite LDPE-Al-Fique 30%.

As presented schematically in **Figure 6**. For an adjustment with a four-parameter interpretation is the same, it is important to note that the effect of reinforcement on a biocomposite can be studied by this method, especially when amending fiber volume, nanoreinforcements, fillers, filler, or some surface modification is made to the fibers or fillers in order to improve performance micromechanical, and default creep performance is used. In **Figure 10**, one can see a behavior of an LDPE manufactured biocomposite-Al-Fique 30% fiber volume. Being possible to observe the effect of increased volume is positive with respect to creep, decreasing the speed of decoration and to increase enforcement effort in the linear viscoelastic region by nearly 60%.
