**4. The measurement and evaluation of cell body surfaces and the whole surfaces**

#### **4.1 Measurement strategy**

Cell body contains different fiber bundle orientations, and the whole surface is composed of cell bodies; thus standard procedures designated to 2D profile sampling at fiber bundle are in general not applicable for 3D topography measurement for cell body and the whole surfaces, because 2D measurement is of directionality which mainly reflects the damage between fibers, the fiber, and the matrix. Whereas for cell body and the whole surfaces, the scales are bigger, thus the measurement and evaluation mainly reflect the damage between fiber bundles and the matrix, which cannot consider the integrated effect of the fiber orientation and the processing direction simultaneously. Therefore, a 3D surface measurement and evaluation method should be adopted at these two grades.

#### **4.2 Determination of MaxSS on cell body surfaces**

A proper sampling step can make the surface information of a cell body extracted accurately and meanwhile save the cost of data collection and processing. MaxSS refers to the balance point of the accuracy and sampling cost. If a sampling step larger than the MaxSS is adopted, the information of a surface is distorted; if a sampling step larger than the MaxSS is adopted, unnecessary data sampling cost is spent. Therefore, how to determine the MaxSS on a cell body surface is very important. This section proposes a method for this topic, based on the principle of residual estimation.

For a cell body surface, the following steps can be executed to determine the MaxSS:

Sample the surface using a small sampling step of the measurement device which is at least one third of the WCMC fiber diameter.

Calculate *Sa*, *Sq*, *Ssk*, and *Sku* based on the sampling results, and set them as surface standard values *θ* (the standard value here does not refer to the ideal measurement result which has no error, rather, it means a standard which can be used to check whether other measurement results are acceptable).

Generally speaking, this standard value *θ*, containing measurement error, can be regarded as a random variable, which obeys Normal distribution, so, *θ N* (*μ*1, *σ*<sup>1</sup> 2 ). A measurement result *θ* that is obtained from a larger sampling step obeys Normal distribution as well, so, *θ N* (*μ*2, *σ*<sup>2</sup> 2 ). Because *θ* and *θ* are both the measurement results of the same surface, their expectation is equal to the ideal real value (with no errors) of the surface. Therefore,

*Surface Measurement and Evaluation of Fiber Woven Composites DOI: http://dx.doi.org/10.5772/intechopen.90813*

$$
\mu\_1 = \mu\_2 \tag{15}
$$

Set *e* as the difference between *θ* and *θ*,

process decreasing the height of the troughs. Thus, it can be inferred that *Rskave* is able to reflect the damage type or degree of the surface. A larger negative *Rskave* value is caused by a trough-dominant surface, and a larger positive *Rskave* value is caused by a crest-dominant surface. The surface state can be inferred by combining

*Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

**Figure 12** shows the fiber bundle surface topographies of three processing methods. **Figure 13** shows the microscopic surface topography of six surfaces. It is clear that the values of the four proposed indexes have a strong and direct connection with the surface damage and, thus, have a good feasibility and interpretability

**4. The measurement and evaluation of cell body surfaces and the whole**

Cell body contains different fiber bundle orientations, and the whole surface is composed of cell bodies; thus standard procedures designated to 2D profile sampling at fiber bundle are in general not applicable for 3D topography measurement for cell body and the whole surfaces, because 2D measurement is of directionality which mainly reflects the damage between fibers, the fiber, and the matrix. Whereas for cell body and the whole surfaces, the scales are bigger, thus the measurement and evaluation mainly reflect the damage between fiber bundles and the matrix, which cannot consider the integrated effect of the fiber orientation and the processing direction simultaneously. Therefore, a 3D surface measurement and

A proper sampling step can make the surface information of a cell body extracted accurately and meanwhile save the cost of data collection and processing. MaxSS refers to the balance point of the accuracy and sampling cost. If a sampling step larger than the MaxSS is adopted, the information of a surface is distorted; if a sampling step larger than the MaxSS is adopted, unnecessary data sampling cost is spent. Therefore, how to determine the MaxSS on a cell body surface is very important. This section proposes a method for this topic, based on the principle of

For a cell body surface, the following steps can be executed to determine the

Sample the surface using a small sampling step of the measurement device

Calculate *Sa*, *Sq*, *Ssk*, and *Sku* based on the sampling results, and set them as surface standard values *θ* (the standard value here does not refer to the ideal measurement result which has no error, rather, it means a standard which can be

Generally speaking, this standard value *θ*, containing measurement error, can be regarded as a random variable, which obeys Normal distribution, so, *θ N* (*μ*1, *σ*<sup>1</sup>

). Because *θ* and *θ* are both the measurement

A measurement result *θ* that is obtained from a larger sampling step obeys Normal

results of the same surface, their expectation is equal to the ideal real value (with no

2

2 ).

evaluation method should be adopted at these two grades.

**4.2 Determination of MaxSS on cell body surfaces**

which is at least one third of the WCMC fiber diameter.

distribution as well, so, *θ N* (*μ*2, *σ*<sup>2</sup>

errors) of the surface. Therefore,

used to check whether other measurement results are acceptable).

*Rskave* and *Rkuave*.

**surfaces**

residual estimation.

MaxSS:

**220**

for a surface evaluation.

**4.1 Measurement strategy**

$$
\underline{e} = \overline{\theta} - \theta \tag{16}
$$

Since *θ* and *θ* are independent identically distributed (IID), their difference *e* obeys Normal distribution too:

$$
\sigma \sim \mathcal{N}\left(\mu\_2 - \mu\_1, \sigma\_1^2 + \sigma\_2^2\right) \tag{17}
$$

Combining Eqs. (16) and (17) together,

$$
\sigma \sim N\left(0, \sigma\_1^2 + \sigma\_2^2\right) \tag{18}
$$

Based on the analysis above, for a set of measurement results obtained from different sampling steps, the Residual Errors (*RE*s) between each of them and the standard value *θ* are IID to Normal distribution, so

$$RE \sim N(\mathbf{0}, \sigma^2) \tag{19}$$

For every actual engineering question, it is reasonable to find an acceptable range of *RE* according to the actual requirements of measurement. For example, if the �15% smallest *RE*s are acceptable, the acceptable measurement results fall into the range of

$$\left[\theta \text{-} 0.39\sigma, \theta + 0.39\sigma\right] \tag{20}$$

When the sampling step is small enough, the measurement result is in the range above. However, if the sampling step grows larger, the measurement result will go out of the range sooner or later. The largest sampling step that holds the measurement result within the range of Eq. (20) can be defined as the MaxSS for cell body surface measurement.

Here we take the measurement of a cell body of the Cf/SiC with processing angle of 90° as an example. The sampling steps of 1–45 μm were adopted to measure the cell body. The measurement results of 1 μm were set as the standard values. The rest of the results were compared with the standard values to calculate the *RE*s. The acceptable ranges of measurement results are available through Eq. (20).

The changing trends of the measurement results of the four evaluation indexes under different sampling steps are illustrated in **Figure 14**. The red and blue lines refer to the boundaries of the acceptable ranges calculated from Eq. (20). It is clear that when the sampling steps exceed a certain value (in red blocks), the corresponding results begin to go out of the ranges. Then the MaxSS can be determined for each index. Combining the four MaxSSs together, the MaxSS is available and for this material, it is 7 μm.

It can also be proved that the MaxSS of a cell body is approximately equal to the diameter of its reinforcing fiber.

#### **4.3 Relationship between the measurement of cell body and the whole surface**

The whole surface of a WCMC consists of many cell bodies. Some cell bodies nearby each other faced the same fabrication and machining process and may perform similar surface quality. Therefore, the measurement and evaluation of a

#### *Composite and Nanocomposite Materials - From Knowledge to Industrial Applications*

cell body can be used to estimate the surface quality state of a certain area nearby it. It has been proved that, for the exampled Cf/SiC with processing angles of 0°, 30°, 45°, and 90°, the measurement results of the four indexes of one cell body have the similar values with the results of the nearby four cell bodies (shown in **Figure 15**).

*Surface Measurement and Evaluation of Fiber Woven Composites*

*DOI: http://dx.doi.org/10.5772/intechopen.90813*

This chapter aims at providing a grading surface measurement and evaluation system for woven ceramic matrix composites. The system contains four grading of fiber, fiber bundle, cell body, and the whole surface. The main conclusions are as

1.The type and degree of the damage on fibers influence the processing quality and property of the surface. The diameter and the direction of the fibers determine the measurement parameters when sampling fiber bundle or cell

parameters, including sampling length, number, step, and direction should be

evaluation indexes, namely, *Ra*, *Rq*, *Rsk*, and *Rku*, are usable for fiber bundle

3.3D measurement should be adopted on cell body surfaces. Maximum sampling step can be determined with the principle of residual estimate. *Sa*, S*q*, *Ssk*, and

4.The whole surface is consist of many cell bodies. Therefore, a small number of cell bodies can be used to represent a larger area nearby. This idea can help reduce the workload when measuring and evaluating a large area of WCMC

Special thanks the National Natural Science Foundation of China (Nos. 51375333

All the authors listed have approved the manuscript, and no interest of any third

2.2D measurement should be adopted on fiber bundle surfaces. Sampling

determined carefully to balance the accuracy and the efficiency. Four

**5. Conclusions**

body surfaces.

surface evaluation.

surface.

**Acknowledgements**

**Conflict of interest**

parties is infringed.

**223**

*Sku* are usable on this grade.

and 51805366) for financial assistance.

follows:

**Figure 14.**

*The changing trends of (a) Sa (b) Sq (c) Ssk and (d) Sku with sampling steps on cell body surface [28].*

#### **Figure 15.**

*Evaluation parameters with processing angles on cell body surface of Cf/SiC (a) Sa, (b) Sq, (c) Ssk, and (d) Sku.*

cell body can be used to estimate the surface quality state of a certain area nearby it. It has been proved that, for the exampled Cf/SiC with processing angles of 0°, 30°, 45°, and 90°, the measurement results of the four indexes of one cell body have the similar values with the results of the nearby four cell bodies (shown in **Figure 15**).
