8.1.1 Basics of Reinforced Polymers, Composites, and Their Testing

Engineering materials have undergone a great evolution in history. Depending on the technology available at a given time, certain preferences were made for materials as well. This relative importance of materials is seen in Figure 8.1. As is seen, due to the work and success of material researchers, more and more new polymer and composites materials have been developed in the last decades.

Besides polymers, hundreds of new materials exist with regard to ceramics or metals. As a result, traditional classifications – metal, polymer, ceramic, and composites – changed to application based classifications (aerospace materials, electronic materials, smart materials, etc.). To understand hybrid natural fiber composites (NFCs) better, a brief introduction about polymers is necessary.

8.1.2 Classification of Polymers

The industrial history of polymers started with rubber in the twentieth century. The first spurt in the application of different polymers was around the time of the Second World War. For the last 20–30 years the applications and the types of polymers increased continuously. It must be noted that polymers can be synthetic or natural, but this classification only refers to the source, not to the structure.

Polymers are organic materials that have specific macromolecular structures. Monomers are the basic units, which can repeat 10 000 times in a single polymer chain. The simplest monomer is ethylene (C₂H₄), which has double bonds between carbon atoms.

The following chemical process has to be carried out in order to create a simple polymer chain: the double bonds connecting two neighboring monomers must be broken first, and a new single bond connection must be created (Figure 8.2) to achieve a chain.

There are thousands of monomers that create different macromolecular structures as well. Based on these structures the polymers can be thermoplastics or thermosets. The basic difference between the two structures is the cross‐link. In the case of thermoplastics there are no cross‐links between the chains (Figure 8.3a); therefore, these materials can remold after solidification. The thermosets have cross‐links; thus during the chemical reaction when they are formed, they cannot be remolded any more (Figure 8.3b).

In thermoplastic structures, there are only secondary chemical bonds between the polymer chains. If the temperature rises to a limited point, the secondary chemical bonds break and the polymer chains can slip next to each other. Primary chemical bonds need much more energy to break so that the polymer chains could remain complete. After cooling, the new position of the chains becomes fixed by the new secondary chemical bond. Thermosets cannot melt since there are only primary chemical bonds in their structure. At much higher temperatures, both the chain and the cross‐link can be broken.

Naturally, in this elevated temperature the thermoplastic is also degraded, but there is still a technological range in the temperature where they can melt and a new shape can be created (by injection molding, extrusion, etc.).

Thermoplastic polymers have two different subtypes in their structures. If the molecules solidify in a random arrangement, then the polymers become amorphous (Figure 8.4a). Amorphous thermoplastics are generally transparent. If polymer chains show some dimensional order locally, then they are called semi‐crystalline polymers (Figure 8.4b).

The degree of crystallinity is one of the important parameters that have direct connection to the material property of the polymer. The degree of crystallinity can vary from 1% to 100%.

Thermoset polymers have cross‐links between the chains that provide better heat resistance. Based on the amount of cross‐links two groups can be differentiated: slightly cross‐linked thermosets or elastomers (e.g. rubber) and highly cross‐linked thermosets or duromers (e.g. epoxy resin).

8.1.3 Classification of Composites

Composites have different definitions. Nowadays, if a material has at least two additives, then companies in material science business term it a “composite,” but mostly this is only for marketing reasons. The polymer industry uses the so‐called fillers since the beginning of material production. Originally, if a filler can reinforce the polymer and improve its mechanical performance, then it can be a composite. In the case of polymers, the most typical reinforcing material is probably glass fiber, and it is widely used in thermoplastics and thermosets as well.

Fibers mostly have improved mechanical behaviors than the polymer (matrix). The purpose of the fiber is to make stronger connection between two neighboring areas in the material. Here, something must be made clear. In engineering, Young’s modulus is likely the most important parameter, rather than tensile strength. If the material has high modulus, then the resistance of the deformation is also high (high stiffness). This consideration is the main theory behind reinforced composites. Fibers also have the role of transferring load if they have appropriate connection with the matrix. The fibers connect to the matrix through chemical adhesion. If the adhesion is strong enough, then the composite will have adequate stiffness as well. If the adhesion is low, the fibers tend to loosen (pull out) and they create a dramatic decrease in the stiffness (Figure 8.5).

If the matrix has only one fiber reinforcement, the maximum effect can only be achieved if the fiber breaks (tensile strength – σ). This also means that the adhesion must be strong enough, which depends on the surface and the shear strength (τ) of the fiber–matrix connection. Based on a mechanical equation (if the fiber has perfect cylindrical surface) the critical length (L_c) can be calculated:

(8.1)

If the fiber is longer than the critical length, then it works effectively. In many cases, it is easier to achieve shorter fibers, which also makes working with it easier (e.g. injection molding), than enhancing the chemical adhesion to a higher level. There are different treatments to achieve better chemical connections that can improve the shear strength between the fiber and the matrix.

If the fiber is long enough and the deformation remains in the linear region of the material, then Hooke’s law can be applied with regard to the parallel fibers:

(8.2)

where E_c is the Young’s modulus of the composites, E_f is the Young’s modulus of the fiber, E_m is the Young’s modulus of the matrix, and V_f is the volume of the fiber. To achieve real reinforcement by applying fibers, usually the Young’s modulus and the tensile strength must be kept relatively high while the density is low. Table 8.1 includes some typical fibers used for reinforcement and their properties [2].

Different plants or animals form the source of natural fibers. It is generally true that these fibers are biodegradable and they are from renewable sources. Table 8.2 lists some typical natural fibers and their properties [3].

Composite materials can be classified according to their length and their orientation. Based on the length, there are two types: short fiber/discontinuous fiber (longer than the critical length) and continuous fiber (c. thousand times longer than the critical length). The orientations are also of two types: random (fibers have different directions) and aligned (all the fibers are in the same direction).

In two dimensions (sheet forms), there can be continuous aligned structures (Figure 8.6a), short aligned structures (Figure 8.6b), short random structures (Figure 8.6c), and continuous random structures (Figure 8.6d). In addition, different fiber weaves/fabrics also exist (Figure 8.6e).

Composite structures can be more complex in three dimensions. Depending on the technologies, usually there are more layers laid on each other. 2D aligned structures become random structures if the different layers have free directions. It must be noted that 3D fabrics exist that have oriented structure in 3D as well. These types of structures allow engineers to design composite parts where the reinforcement completely fits the stress distribution and the directions.

Additional need for optimizations and special properties led the researchers in composite technology to create innovative solutions. Hybrid composites are an answer to this need. Hybrid composites contain at least two types of fibers, where the different fibers can be in the same sheet (2D), or the different layers can create a hybrid system in 3D.

8.1.4 Basics of Tribo‐testing

Tribology is a science that deals with friction, wear, and lubrication. In reality, two interacting surfaces create a complex phenomenon with regard to tribology. In order to obtain a clear view about this contact phenomenon in relation to tribological behaviors, a schematic tribological system was created (Figure 8.7).

Basically, all tribological systems include two bodies, the motion of which can be described with the law of relative kinematics. Usually, the intermediate material plays the role of a lubricant and the wear debris. The rest of the parameters (e.g. temperature, load, sliding distance, surface roughness, materials) belong to the tribological environment. During an experiment, the researchers attempt to identify the main parameters and to be able to control them in the future experiments. One of the basic equations in tribology is the ratio of the coefficient of friction:

(8.3)

where F_F is the friction force and F_N is the normal force. Both the friction force and the normal force are influenced by many parameters that are included in the tribological environment. Owing to these parameters and their statistical deviance, high deviations are expected during repeated tribological measurements, even though the parameters are seemingly fixed. Figure 8.8 contains the classification of tribo tests.

Tests can be performed in almost endless number of ways. Typical factors are running time, accurate control of test conditions, isolated wear mechanisms, and so on. Generally, a test model is used to better identify the main tribological behavior. Testing a simplified model is considerably easier, quicker, and cheaper under well‐controlled test conditions, but it also correlates less with reality. Creating an appropriate model to address a real tribological problem is probably the most challenging part of the research. Researchers need to deal with basic effects (approximately the seven more important ones) to simulate the tribological conditions.

These effects are as follows:

1. Contact (tribocouples: ball on plate, block on ring, block on plate, pin on disk, etc.)
2. Load (forces, Hertzian contact, etc.)
3. Displacement (sliding, rolling, spinning, flowing, oscillating, impact, etc.)
4. Relative motion (constant, periodical, magnitude, etc.)
5. Track (clean surface, transfer film, lubrication, debris, etc.)
6. Surface roughness (R_a, R_z, etc.)
7. Temperature

Based on the earlier results, typical wear and friction models have been already established. The principle wear mechanisms are recognized as adhesive wear, abrasive wear, erosive wear, corrosive wear, and fatigue wear. Mostly, the wear debris changes the tribological environment by altering its complexity and deviations. In addition, in real applications there are no clear conditions that bring forth more complex wear phenomena.

8.1.5 Hybrid Natural Fiber Composites and Their Possible Use in Total Knee Replacements (TKR)

As was mentioned earlier, fibers are mostly classified based on their source, which can be related to minerals, plants, and animals. Biologically, these materials are made up of either cellulose or proteins depending on the source. Lately, there has been a rapid growth in R&D in the NFC area.

The particular interest in this material is due to the advantages in their mechanical properties compared to synthetic fiber composites, not mentioning the low environmental impact cost as well. It must be also mentioned that usually higher performance can be credited to plant fibers than the available animal fibers with regard to their higher strengths and stiffness.

Among the numerous possible applications, companies involved in biomedical part production are also interested in this new type of materials as a possible substitution for UHMWPE manufactured elements. Naturally, these new materials can only enter the market if they are able to provide acceptable wear characteristics and resilience compared to the materials being used today.

Undeniably, problems related to wear in total knee replacements (TKRs) have decreased in the past 10 years [5] but not disappeared. In the design‐related factors, wear is still the second most important mechanical factor (first is the periprosthetic fracture) that limits the lifetime of TKRs and it also greatly depends on the local kinematics of the knee.

On studying the wear properties of these materials, it becomes quite evident that NFCs have significant difference in their wear behavior. An important parameter that is widely used in analytical and numerical modeling is the specific wear coefficient (denoted as k), which is summarized in Table 8.3.

It must be also noted that these materials were tested approximately with the same sliding speed, but the load is significantly less than in the study by Patten et al. [10], where the authors tested materials that are actually being used in TKR production.

By looking at these results, it is apparent that the specific wear coefficient of the Lantana camara and the hybrid PTFE are exceptional, but it is unknown yet whether they could perform as well as the UHMWPE if the load is elevated not only to 200 N but significantly beyond. The palm fiber reinforced polyester is in the same range regarding the specific wear coefficient compared to the standard UHMWPE materials used in TKRs.

Since the load in the TKR can be up the six to seven times the body weight (BW) during squatting and two to three BW during gait, more experimental data are needed before these materials could enter the market not only as substitutional choices but also as prime material possibilities.

8.3.1 Wear Modeling

The instantaneous volume of material removed from the tibiofemoral surface of the TKR due to mild wear is predicted by Archard’s law [22]:

(8.4)

where k is the specific wear rate (mm³ N⁻¹ m⁻¹), which is a material‐dependent quantity; F_N is the normal force acting between the contact surfaces, which in this case is the contact surface of the tibia and the femur (N); and ds is the infinitesimal length of sliding (m). There are two parameters in this equation that are to be further enhanced: the force and the sliding length wherein the slide–roll ratio can be defined.

The wear mechanism between the connecting surfaces is supposed to be abrasion based (two‐body). This means that during contact, the harder metallic femoral part ploughs into the softer polyethylene tibial part, and in the connecting point(s) reaction force(s) appear (Figure 8.9). As a first step, we can approximate these forces with the Coulomb law:

(8.5)

where μ_k is the coefficient of kinetic friction.

In the abrasive wear mechanism, the frictional component is responsible for creating such a shear stress in the upper surface of the material that it begins to lose small debris. Therefore, it provides us a more precise approximation if the friction force, deduced from the tibiofemoral force, is introduced in our wear equation:

(8.6)

8.3.2 Force Modeling for Wear Equation

The normal force in Equation (8.6) that acts between the connecting surfaces is considered constant during wear tests. Nevertheless, in case of the tibiofemoral connection during squat movement, this force is a flexion angle dependent quantity and is called tibiofemoral force. It has been deduced analytically for standard (no horizontal movement of the center of gravity is permitted) and non‐standard [27] (horizontal movement of the center of gravity is allowed as it is in real life) squat motion (Figure 8.10), which makes it an adequate choice to incorporate this force into our model. The function of the tibiofemoral force can be described as follows [27]:

(8.7)

while the ϕ(α)‐function is described as [27]

(8.8)

The parameters of the equation are summarized in Table 8.4.

If Equation (8.7) is multiplied by an arbitrarily chosen BW then the tibiofemoral force is obtained. In the abovementioned analytical expressions, λ₁ function describes the horizontal movement of the center of gravity during squat. If the linear function is used as given in Table 8.4, then a more realistic, so‐called non‐standard squat is carried out where the torso leans forward, and it helps the knee to release a considerable amount of moment (Figure 8.11). This is the squat type that is practiced by most people in everyday life.

The so‐called standard squat is defined for tests and simple mechanical modeling. Nonetheless, it does not consider the movement of the torso during the motion; therefore, higher knee moment, tibio‐ and patellofemoral forces act in the knee if this motion is considered. With this model, both motions, and therefore the two types of tibiofemoral forces, are analyzed with their impact on wear.

When the non‐standard squat is being examined the λ₁ function is constant with a value of 1. This means that the torso does not lean forward (Figure 8.11, left drawing). In case the non‐standard squat is exercised, the torso leans forward, which is modeled by a flexion angle dependent λ₁(α) function (Figure 8.11, right drawing).

In the augmented wear equation, the tibiofemoral force will be included as follows:

(8.9)

If the tibiofemoral force is indexed as F_{tf−nst CoG} then the horizontal movement of the center of gravity is also included, while with the index of F_{tf−st CoG}, the forward lean of the torso, the horizontal motion of the center of gravity, is excluded.

8.3.3 Slide–Roll Modeling for Wear Equation

Slide–roll is known to be an important parameter of wear; however, its quantitative effect has not been studied earlier. By knowing the slide–roll function, exact conclusions can be drawn about the features of the motion. If the ratio is 0 then pure rolling is present, while 1 describes pure sliding. If the ratio is between 0 and 1, the movement is characterized as partial rolling and sliding. For example, a sliding–rolling ratio of 0.4 means 40% of sliding and 60% of rolling. Fekete et al. [20] introduced the slide–roll ratio by having the arc lengths on both connecting bodies as follows:

(8.10)

where

are the corresponding incremental differences of the connecting arc lengths.

A positive ratio shows the slip of the femur compared to the tibia. By this definition, the author derived an averaged slide–roll ratio with a standard deviation (SD = 0.136) based on several TKRs as a function of flexion angle [20]:

(8.11)

This function is used further on in the calculations, and it is assumed that the function is applicable for flexion and extension movement as well. With respect to the slide–roll ratio, there is an accordance between the authors that sliding is not dominant up to 65–67° of flexion angle [17, 18], which practically represents the domain of gait swing.

Nägerl et al. [19] and Fekete et al. [20] derived lower ratios between 0 and 65–67 flexion angle (Figure 8.12), and above this region they presume that the slide–roll ratio can reach 95% of sliding as well [19]. The interpretation for why these ratios have such discrepancies is likely to be credited to the simplifications, e.g. frictionless contact, two‐dimensional, simple circle or simplified knee geometry that the earlier authors applied in their models [17, 18]. Nägerl et al. [19] used the first‐time three‐dimensional geometry to derive the slide–roll ratio, while Fekete et al. [20] included the effect of friction besides three dimensionality in their three‐dimensional models.

Owing to these enhancements, the latter two authors revealed a nonlinear phenomenon in the trend of the slide–roll ratio. Since slide–roll is known, it must be involved in the wear calculation to quantify its effect. To include the slide–roll ratio into the equation, let us rewrite the length of sliding as the product of sliding velocity (m s⁻¹) and infinitesimal time (s).

(8.12)

The definition of slide–roll according to Fekete et al. [20] can be expressed by the tangential velocities instead of the arc lengths:

(8.13)

where v_CTt and v_CFt are the tangential velocities in the instantaneous contact points. The difference between these velocities (v_CTt − v_CFt = v_sliding) provides the so‐called sliding velocity. If Equation (8.13) is rearranged to v_sliding then it can be included in our equation and we obtain the augmented Archard’s law:

(8.14)

v_CTt velocity is kept constant similar to a wear test parameter [10] as an approximation. The original function of F_tf is the function of flexion angle [20], but it can be transformed into time domain. This has been carried out from the study of Fekete et al. [20], where the data were obtained also in time domain during the squat simulation. Thus, the final equation is as follows:

(8.15)

The parameters for the equation are summarized in Table 8.5.

In the following, six total wear volumes will be calculated, each of them during one squat cycle. One cycle in squat is defined between 0° and 120° of flexion while the complete range where the propagation of wear monitored is 15 years. A usual knee wear simulation period is 3.5 million cycles [30, 31]; however, squat and deep squat are less frequently exercised in everyday life.

If 3.5 million cycles were broken down over 15 years, it would mean 640 complete deep squat cycles carried out daily. This number of cycles is clearly overestimated; thus 50 deep squats per day will be assumed in this chapter. The total wear during one cycle can be calculated by integrating the wear functions over time. During deep squat, the total cycle will be expanded to 120° of flexion angle where the time interval goes from 0 to 3.06 s during the squat motion (Figure 8.13).

The total wear volumes are calculated in the following way: firstly, the slide–roll ratio is kept constant with the number 0.4 according to the study of McGloughlin and Kavanagh [17] together with Hollmann et al. [18] while a standard squat is carried out (the tibiofemoral force does not have the effect of the horizontally moving center of gravity). Secondly, the slide–roll ratio varies according to the function of Fekete et al. [20] while a standard squat is carried out as well. Thirdly, the slide–roll ratio is kept constant with the number 0.4, but non‐standard squat is considered (horizontally moving center of gravity), while fourthly, the slide–roll ratio varies during the non‐standard squat.

In the last two calculations, similarly to a wear test on a pin‐on‐disk or ball‐on‐disk configuration, the slide–roll ratio is kept constant at 0.4 in one case, while in the other case it varies.

Regarding the force, only the simple Coulomb law (F_s = μ_k F_N) is considered, which results in a constant load. The general method to calculate total wear during one squat cycle is as follows:

(8.16)

(8.17)

In the first calculations, both models use the tibiofemoral force where the horizontal movement of the center of gravity is included (non‐standard squat “nst”). Thus, the total wear with varying slide–roll ratio is

(8.18)

while the total wear with constant slide–roll ratio is

(8.19)

In the second calculation, both models use the tibiofemoral force where the center of gravity does not move horizontally (standard squat “st”).

• Total wear with varying slide–roll ratio but without the horizontal movement of the center of gravity:

(8.20)

• Total wear with constant slide–roll ratio:

(8.21)

• Total wear if constant load (BW) and constant slide–roll ratio are used:

(8.22)

• And, at last, total wear if constant load (BW) and variable slide–roll ratio are used:

(8.23)