Ceramic-matrix Composite Materials and Their Bending Properties

Source: Sina Shahbazmohamadi and Peiman Shahbeigi-Roodposhti-Roodposhti, School of Engineering, University of Connecticut, Storrs, CT

Bones are composites, made of a ceramic matrix and polymer fiber reinforcements. The ceramic contributes compressive strength, and the polymer provides tensile and flexural strength. By combining ceramic and polymer materials in different amounts, the body can create unique materials tailored for a specific application. As biomedical engineers, having the ability to replace and replicate bone due to disease or traumatic injury is a vital facet of medical science.

In this experiment we will create three different ceramic-matrix composites with plaster of Paris (which is a calcium sulfate compound), and allow them to undergo three-point bending test in order to determine which preparation is the strongest. The three composites are as follows: one comprised only of plaster of Paris, one with chopped glass shards mixed in a plaster matrix and lastly a plaster matrix with a fiberglass network embedded within it.

When a given material needs to be tested, one of the premier methods of testing the strength of less ductile materials is a three-point bending test. The three-point bending test is a method that allows a given sample to experiences a combination of forces (compressive and tensile) as well as a plane of shear stress in the middle of the material that is representative of most of the forces human bones are consistently subjected to. With the results of this experiment a better understanding of composite materials can be achieved, along with the scope and limitations to these biomaterials.

In the 3-point bend test, the bottom of the sample is in tension, the top is in compression, and there is a shear plane in the middle of the sample (Figure 1).

Figure 1: Schematic representation of the 3-point bend test.

Living bone can remodel and restructure itself to accommodate these forces. For example, in rib bones there is a high concentration of mineral phase on the inside of the curve (where there are compressive forces) and a high concentration of collagen fibers on the outside of the curve (where there are tensile forces).

The properties of a composite are based on the properties of its matrix and filler materials. Several formulas have been developed to calculate the overall strength and modulus of a composite as a function of the type and amount of fillers. The simplest of these is the "rule of mixtures", which gives the maximum theoretical value of the property in question. The rule of mixtures for flexural strength is given below:

σ_comp = σ_mV_m + σ₁V₁ + σ₂V₂ + ...  (1)

Where:

σ_comp = maximum of theoretical strength of the composite

σ_m = strength of the matrix

σ_1, σ₂ ... = strengths of the filler materials 1, 2, etc.

V_m, V₁, V₂,.. = volume fractions of the matrix and fillers.

1. Making one plain plaster sample

1. Obtain a blue rubber mold from the instructor. Each mold can make 3 bar-shaped samples, the size of the each bar is roughly about 26 mm in the width, 43 mm in the length, and 10 mm in the thickness.
2. Weigh 40 grams of dry plaster powder into a paper cup. Slowly add 20 ml of deionized water, and stir the slurry with a wooden stick, until a smooth consistency is achieved. Proceed immediately to step 3! The plaster starts to harden in ~5 minutes.
3. Pour the resulting slurry into one of the compartments of the mold. Fill the mold completely, and smooth it over with the wooden stick. Throw away the cup and any excess plaster; keep the stick for future use.

2. Making two composite samples

1. Prepare the sample made with chopped fiber reinforcement:
   a.) Weigh 4 grams of chopped glass fibers into a paper cup.
   b.) Weigh 40 grams of plaster powder into the same cup.
   c.) Slowly add 20 ml of deionized water, and stir the slurry with the wooden stick, until the fibers are thoroughly mixed in, and a smooth consistency is achieved.
   d.) Pour the slurry into one of the mold compartments. Fill the mold completely, and smooth it over with the wooden stick.
2. Prepare the sample made with fiberglass tape:
   a.) Cut 2 strips of fiberglass tape, about 5 inches long. Weigh the strips.
   b.) Weigh 40 grams of dry plaster powder into a paper cup. Slowly add 20 ml of deionized water, and stir the slurry until a smooth consistency is achieved.
   c.) Pour about one third of the plaster into the mold. Place one strip of fiberglass tape on top of the plaster, and press it down with the wooden stick. Make sure that the plaster thoroughly wets the fiberglass tape.
   d.) Pour about half of the remaining plaster on top of the fiberglass tape. Place the second strip of tape on top of the plaster, and press it down with the wooden stick.
   e.) Pour the rest of the plaster on top of the second strip, and press it down with the wooden stick. Make sure that the plaster thoroughly wets the fiberglass tape, and squeeze out any air bubbles.

3. Performing experiments

1. Measure the average length, thickness and width of each bar Measure L (span length in the figure below) on the 3-point test fixture, use calibrated calipers for the measurement.
2. Use a displacement speed of 5 mm/min for all tests. (The UTM then should be zeroed and initiated at a displacement speed of 5mm/min). For the plain plaster and chopped fiber sample, run the test until the sample fails. For the fiberglass tape sample, run the test until the deflection is 6 mm.
3. Use the LabVIEW program on the computer to collect the data from each test into a text file.

4. MATLAB Program

1. Create a MATLAB program that will do the following:
2. Read a single column text file and separate the readings into force and deflection data. Convert the raw data into force and deflection using the following conversion factors:
   Force = (Load Cell Maximum Value / 30000) * Number generated by UTM (2)
   Deflection = 0.001mm * Number generated by UTM (3)
3. Calculate the flexural strength and flexural strain of each sample:
   Flexural strength σ_f = (3FL)/(2wt²)   (4)
   Flexural strain ε_f = (6Dt)/(L²)   (5)
4. Plot a stress-strain curve for each sample. Let ε_f be the horizontal axis and σ_f be the vertical axis.
5. Find the maximum σ_f and ε_f values for each sample. For the composite samples, select the ε_f value that corresponds to the maximum σ_f value.
6. Find the flexural modulus E_f by calculating the slope of the curve in the elastic region.
7. Find the area under each stress-strain curve.

5. Data Analysis

1. Comparison of the flexural strength and modulus of the composite samples to that of the plain plaster sample
   Since the UTM generates a single column text file, for both force and deflection, MATLAB interface has to sort the corresponding values into different arrays. Thus, to determine both the force and deflection needed for Equations 4 and 5, Equations 2 and 3 should be implemented into MATLAB.
   Using a Load Cell Maximum of 1000, the determination of flexural strength and strain is the combination of all equations. Since MATLAB also generates the stress-strain curve of each sample, the flexural modulus was ascertained by calculating the slope of the elastic region. Using Equation 6, the flexural modulus will be calculated with respect to the two selected points on the stress-strain plot:
     (6)
   Examining a sample data, we will see that as different forms of reinforcement are added, the strength of the samples will be increased, with fiberglass tape providing the greatest additional strength. In the terms of ductility, (which can be considered as the "most plastically deformable") the fiberglass tape reinforced specimen will be the greatest as well.
   Also, fiber length and orientation drastically affect the properties of composite samples. For example, maximum reinforcement can only be achieved when the fiberglass tape is set parallel to the surfaces of the specimen. In doing so, this spatial orientation allows the fiberglass tape to withstand additional forces as the plaster matrix fails. In addition, it can also be concluded that longer strips of fiberglass tape would prove to provide more strength than shorter strips. Longer pieces would allow for maximum traction under the conditions of a 3-point bending test, as there is more plaster surrounding the fiberglass reinforcement.
2. Energy absorption during bond test
   The area under the stress-strain curve represents the energy a material absorbs before failure. According to the results we will achieve, it will be shown that the fiberglass reinforced specimen absorbs the greatest amount of energy. In addition, since toughness corresponds to the ability of a material to absorb energy and plastically deform without fracturing and the fiberglass sample proved to be the most ductile by absorbing the greatest amount of energy; the fiberglass specimen is inherently the toughest amongst the three. Hence, toughness is the balance between strength and ductility, and the fiberglass sample had the largest area beneath its stress strain curve.
3. Calculation of the theoretical strength of the chopped fiber and fiberglass tape composites using the "rule of mixtures" formula (the relevant material properties are listed in Table 1).
   The theoretical strength of the composite can be calculated through Equation 1, where:
   V_F = volume fraction of fiber = (volume of fiber)/(total volume of the sample)
   Volume of fiber = (mass of fiber)/(density of fiber)
   Volume fraction of plaster = V_P = 1- V_F .

Table 1. Material properties.

The overall objective of the series of aforementioned tests is to compare the different physical characteristics between various composite bone substitutes. Flexural strength and strain needs to be calculated using Equations 4 and 5, respectively. The stress and strain for each sample will be plotted in MATLAB. From this, the maximum flexural strength and the corresponding flexural strain can be found for each data set. The stress (σ_f1, σ_f2) and strain (ε_f1, ε_f2) for each data point will then be used in Equation 6 in order to determine the flexural modulus for each sample.

This experiment was designed to study flexural strength on three different kind of composite material. We fabricated three specimens with different reinforcement materials. The matrix was plaster of Paris (a calcium sulfate compound), and we used chopped glass fibers and fiberglass tape as reinforcements. We performed 3-point bending test on the fabricated specimens, and analyzed the achieved data, comparing the properties of composites made with long, oriented fibers vs. short random fibers.

Bones inherently have a strong composite structure, an adaptation to the many different forces the body has to withstand on a consistent basis. The composite structure can be described as a ceramic matrix interspersed with polymer fibers. The ceramic aspect provides for high compressive strength, while the polymer fibers give rise to increased flexural strength. Evidently, as biomedical engineers, having the ability to replace and replicate bone due to disease or traumatic injury is a vital facet of medical science. Moreover, synthesizing suitable replacement tissues from various metals, polymers, or ceramics is a viable alternative. Bioengineered replacements must match the functionality of their biological counterparts, and the critical analysis and testing of different biomaterials becomes increasingly important.