Study of Residual Stresses from the Cutting Method

The present work studies the presence of Residual Stresses in a composite material manufactured using Filament Winding (FW). The composite material consists of Carbon Fibers (CF) in an epoxy matrix. Various parameters of the FW process were analyzed, and different techniques were employed to obtain and evaluate the results: micrographic analysis, instrumentation using strain gauges, tensile tests, and finite element analysis.

This work is a small excerpt from my thesis titled "Influence of Manufacturing Parameters on Residual Stresses in Carbon-Epoxy Fiber Tubes." See my full thesis at the following link: https://tesis-gfe-tensiones-residuales.netlify.app/ (in spanish) and a small resume (https://youtu.be/e4vdb0lk43c).

Application.

Residual stresses (TR) can jeopardize the service life of a structural component if not considered during mechanical design. Together with stresses due to external loads (forces, moments, gravitational acceleration, general accelerations, vibrations, etc.), they subject the structure to stresses that can lead to failure. In particular, it is important to quantify the RS of tubes made from Carbon Fiber (CF) and Epoxy Matrix, manufactured using the Filament Winding (EF) technique.

State of the Art

From previous works related to the measurement of residual stresses (TR), we can cite the following:

- Cohen and M. Schlottermüller et al. conducted similar studies investigating the manufacturing and design variables that affect the quality, strength, and stiffness of containers made from composite materials. They evaluated variables such as winding tension, stacking sequence, winding tension gradient, winding time, and mandrel temperature. The responses measured included, for example, the circumferential strength of the tubes, circumferential stiffness, distribution of fiber and pore volume fraction through the thickness, residual stresses, and interlaminar shear strength. The experimental design in both studies was fractional, employing a ? factorial design.The analysis of the responses found that the strength of the composite was significantly affected by the stacking sequence of the laminate, winding tension, winding tension gradient, winding time, and the interaction between the winding tension gradient and winding time. The mechanism that increases the strength of the composite was related to the strong correlation between the volume of fiber in the composite and the strength of the container. Cylinders with a high volume of fiber in the circumferential layers tended to offer high fiber strength. For more details, see (Cohen, 1997) and (M. Schlottermüller et al., 2004).

- P. Casari et al. conducted an interesting comparison between experimental variables and a simplified theoretical model. They measured residual stresses (TR) using hole drilling, layer removal method, and the crack compliance method (MFG). According to this study, the comparison between experimental results and the model shows good agreement (P. Casari et al., 2006).

- J. W. Kim & D. G. Lee manufactured composite tubes using pressed prepregs for rocket nozzles. They studied residual stresses in composites using the cutting method along with strain gauges to measure the residual deformation caused by these stresses. To calculate the TR, they employed the calculation method on which this thesis is based (J. W. Kim & D. G. Lee, 2005). Subsequently, C. Kang and others manufactured fiberglass/epoxy tubes using filament winding (EF) and contrasted their proposed theoretical model for predicting TR against tubes subjected to the cutting method and the calculation method also used in this thesis (C. Kang et al.,2018). Finally, Seif M. A. fabricated graphite/epoxy system tubes using EF and measured the TR with the cutting method, employing the same calculation method. The novelty of his proposal was the use of image analysis to measure deformations, highlighting the advantages of this technique over the use of strain gauges (Seif M.A., 2002).

- S. Akbari et al. conducted a comparison between two methods of measuring TR: the Slitting Method (MC) and the Crack Compliance Method (MFG). The stress results obtained from the two methods followed the same trend regarding the thickness of the ring and led to identical bending force and moment. However, the maximum and minimum values of the circumferential TR were significantly different (S. Akbari et al., 2012).

- An important compilation regarding TR in Composite Materials can be found in the book by Shokrieh. It is a summary with references to various research works on TR in composite materials (Shokrieh, M.M., 2014).

Sample Preparation.

Once all the manufacturing stages, including oven curing, have been completed, the tube is sectioned. The chosen method was disk cutting using a sensitive cutting saw (see Figure 1). A disk with a diameter of 310 mm and a thickness of 3 mm was used, which is sufficient to cut tubes with a diameter of at least 4 inches (101.6 mm), as is the ones in this work. The time taken to manufacture the tubes and prepare them for cutting is approximately three days: on the first day, the sealant and release agent are applied to the mandrel; on the second day, the fabrication and autoclave curing take place; and on the third, the extraction process is performed using a pulley extractor.

It is worth noting that cutting with a band saw was considered, but it had the drawback that the saw is intended for steel. Since cooling cannot be applied when cutting a composite, the cutting speed needed to be extremely low The quality of the cut was not adequate, and the width of the cut disk had significant variation. A homemade device was also tested that combined a grinder with a diamond disk, but it had the disadvantage that it could not cut the thick tubes to their full depth. Ultimately, the author this work opted for the sensitive saw due to its simplicity of operation, speed, and adequate repeatability in the width of the test specimen disks. The cuts to obtain the samples were carried out over two days.

The cuts were made by removing the ends of the composite cylinder (once they had been extracted from the mandrel, of course), resulting in cylinders approximately 300 mm long. These were then sectioned into 10 or 11 parts to obtain all the specimens needed for testing. One specimen was set aside for the study of the volume percentage of fiber, resin, and pores; another specimen for tensile testing using a split disk; and one specimen for thermal conductivity tests (TR). In the case of Run 7, three specimens were tested for TR measurement to evaluate if there was any significant dispersion in the measurement along the length of the tube.

The author observed some variation in thickness when cutting the disks; therefore, different quantities of specimens were obtained from the tubes. Figure 2 shows all the tubes prior to cutting, Figure 3 illustrates how each specimen was labeled after cutting, and Figure 4 displays all the tubes that have been cut and labeled.

The quality parameters of the manufactured tubes are summarized below in Table 1.

Samples for Measuring the Residual Strains (DR).

Figure 5 shows how a sample is extracted from the original tube to conduct the DR measurement test using the split disk method. The top left displays the starting piece and the coordinate axes of the piece. The bottom left shows the cross-section of the starting piece. From a cylinder that is 300 mm long, it is sectioned into approximately 10 smaller pieces, each about 25 mm long. A sample is taken from the central area to measure the DR. The exception to this is the replicated samples from Run 7 (7A, 7B, and 7C), from which 3 samples were extracted to evaluate whether there is any dispersion of values in the DR. One sample is taken from the beginning of the tube, another from the center, and another from the end. The aim of this is to assess whether there is any statistically significant dispersion of the measured DR values throughout the tube.

Procedure for Extracting a Sample for DR Measurement from the central part of the tube manufactured by EF. In the image on the right, the location where the strain gauge is attached using adhesives is highlighted in orange. The blue arrow indicates the length of the gauge.

Evaluation of the DR Using the Cutting Method.

In the Cutting Method, the residual stresses (TR) present in the material are relaxed by removing material. In this case, a strip is cut from a cylindrical sample, resulting in the creation of free edges that deform until they reach a state of zero stress. Opposite to the area where the strip will be cut, a strain gauge is placed, which amplifies the signal using a Wheatstone bridge. A data acquisition system records the resulting strain field from the sample. Subsequently, the TR can be calculated.

For the remainder of the discussion, the of the cylindrical coordinate axes that will be maintained throughout this work is clarified. The main axis is the X-axis or longitudinal axis (in red); the Y-axis is the circumferential axis (in blue); and the Z-axis the radial axis, in the direction of the of the piece (in green). The general schematic of the measurement was previously shown in Figure 5. The top a piece is cut using a Dremel in two places to remove a strip of material, ensuring the measurement of free deformation without interference in the piece itself; below a unidirectional strain gauge is placed in the direction of the circumferential axis (Y-axis), as these are the most significant stresses: those that would bear load if the composite cylinder were pressurized (in any case, the longitudinal stresses were already released when the tube was extracted, and the in the thickness direction, along the Z-axis, are always assumed to be zero for composite materials, whether for a plate or a revolute piece). In this way, the DR is measured.

The DR is then obtained by mounting astone bridge circuit in a quarter configuration. Figure 6 shows the physical wiring of the bridge in the data acquisition terminal. The black wire is always connected to the terminal P+, the white wire to the terminal S+, and the red wire to the terminal D350 (the relates to the resistance values that complete the bridge). Figure 7 shows physical wiring of the strain gauge, which is already bonded to the sample using Loctite? 414 (the adhesive must be of a quality that ensures the strain field of the piece is correctly transmitted to the gauge to produce deformation in it).

Finally, the actual measurement experiment is conducted. The piece is held using pointed pliers, and two cuts are made, as indicated in Figure 8. Figure 8 shows the Dremel used for the cut and a "tent" made of nylon over the box where the cutting was performed. The Dremel ensures a fine, precise, and maneuverable cut compared to other instruments that were tested for the cutting (manual saw and grinder). This is because the dust generated from the cut is very harmful to the human body, and measures must be taken to prevent the dust from dispersing into the environment (the "tent" is opened for the experimenter to make the cut and then closed, thus containing the dust). Additionally, the experimenter must wear eye protection (glasses), hand protection (latex gloves), body protection (using a lab coat to avoid dust on clothing), and respiratory protection (a mask with a dust filter).

In Figure 9, the measurement result is shown: in the central part, there is the segment of the disk that has been cut, while the screen is recording a value (in microstrains). It is necessary to wait 3 to 5 minutes for the measurement to stabilize, as there are load compensation effects inherent to the instrument and temperature effects due to the cutting of the piece, which cause the measurement to fluctuate until it stabilizes at a value. The measurement taken during the experiment is the one in which there is no variation on the recording equipment's screen for more than one minute.

Calculation of the Residual Stresses.

The calculation of the TRs from the DRs using elasticity equations has been derived and applied in the studies related to TR detailed below: Timoshenko and Goodier (Timoshenko y Goodier, 1951), Cohen (Cohen, 1997), Schlottermüller (M. Schlottermüller et al., 2004), Casari (P. Casari et al., 2006), Seif (Seif M.A., 2002), J. W. Kim & D. G. Lee (J. W. Kim & D. G. Lee, 2005), and C. Kang et al. (C. Kang et al., 2018).

The main effect when the TRs are released is the release of a Residual Moment (MR), which tends to open or close the piece. What the strain gauge measures is the extension or contraction that the deformation field of the piece experiences, a Residual Strain (DR), produced by a Residual Moment (MR). The physical phenomenon is explained in Figure 10. This figure shows a cylinder from which a section has been removed (in black) to measure its DRs. Firstly, it will be observed whether the cylinder of the sample tends to open or close. This is associated with the direction of the Residual Moments (MR), shown in red in the same figure. In the first case, the RMs will tend to open the ring, resulting in a tensile response in the strain gauge (in violet). Case two is the opposite, where if the circular section tends to close, the strain gauge tends to contract. This behavior can be understood by recalling the mechanical behavior of a beam, shown at the bottom of that figure. Moments such as those in Case 1 produce tension (denoted by green dashed arrows) on the upper face, and contraction (denoted by orange dashed arrows) on the lower face. Case 2 is the exact opposite.

The relationship between MR and DR is given by the following equation:

The obtained moment is in units of Newtons, as it is a moment per unit width. ExB (in GPa) is the elastic modulus of bending in the X direction (which will be specified later in Equation 5), a (in mm) is the inner radius of the section, and b (in mm) is the outer radius. δ (in mm) is the displacement obtained by calculation, knowing the length l (in mm) of the gauge, the measured strain value, and the following relationship:

The length of the gauge is obtained considering the manufacturer's data. Knowing that the identification code of the gauge is EA-00-125AC-350/LE and from the information contained in the Vishay? catalog, the length of the gauge in mils is 125 mil ≡ 3.175 mm.

Once the moments are calculated, the TR evaluated at the vector radius of the section is found:

Where N (in mm4) is obtained from the following relationship:

The value of r (in mm) is the vector radius value, understood to be between a and b. Figure 11 clarifies this concept. Since M is calculated in units of Newtons, σz and σz are in N/mm, and therefore in units of MPa.

The modulus?(in GPa) is calculated as follows:

Where t (in mm) is the thickness of the piece and D11, D22 and D12 are calculated as follows:

Where k s the k-th layer of the laminate, tk is the thickness of that layer, and z'k ?s the coordinate of that layer (See Figure 12). The constants Dij are in GPa mm3.

The values of Φ11, Φ22 and Φ12 ?are calculated as shown below:

The values of Q11, Q22, Q12?and Q66?are calculated as shown below:

For each of the values Q11, Q22, Q12 and Q66, it is necessary to calculate the values of E1, E2, G12 and ν12 that depend on the fiber volume. Each batch of the material will have a different measured fiber volume. Therefore, all the tubes will have different modules E1, E2, G12 and ν12 (as well as different thicknesses and different angles per layer, which means they will also be calculated differently in each case). Regarding the units, the modules E1, E2 and G12 are usually expressed in GPa. Thus, the Qij values are in GPa, and the Φij values are also in GPa.

Model of the proposed finite element model.

Below is the proposed finite element model to evaluate the TR. The model was created using the graphical interface for composite materials in the ANSYS Workbench? software, called ACP PRE. Initially, for Run No. 1, possible types of candidate elements for the model were evaluated, referred to as SHELL181, SHELL281, SOLID185, SOLID186, and SOLSH190 (see the reference “Ansys Mechanical APDL Element Reference”). These types of elements are suitable when addressing a finite element problem, as they allow a configuration for laminated orthotropic materials by adding the relevant properties to each layer of the material (see the Verification Model 'VM144: Bending of a Composite Beam' from the reference “Ansys Mechanical APDL Verification Manual”). In a first approximation, linear order elements (SHELL181, SOLID185, and SOLSH190) were used.

The model used (see Figure 13) has a motion constraint at the midpoint of the geometric representation of the specimen, with two equal and opposite moments applied at the ends, as shown in Figure 11. At least until ANSYS? Version 2022-R1, the ACP-PRE module does not support the application of symmetry (which would simplify the model and save time and computational resources).

The cylindrical coordinate system is observed at the center (X is the radial axis, Y is the circumferential axis, and Z is the longitudinal axis), and the Global coordinate system is at the bottom right.

A small iteration is carried out: initially, the moments are unitary, and then, taking into account the experimental DR values, a simple rule of three is applied (using the principle of superposition on which the finite element method is based) to isolate the moment that would produce the deformation reported in that table.

That is:

The circumferential displacement being modeled is measured along the Y-axis of cylindrical coordinate system shown in Figure 13 at the two end points that simulate the extension or contraction of the strain gauge (see Figure 14).

For the calculation deformation, a consultation was made with Dr. Michael B. Prime (Los Alamos National Laboratory, New Mexico, USA), author of several works on the Contour for measuring Residual St, regarding which is most accurate method for measuring deformations using finite elements simulating a strain gauge. Following his advice, the method developed by Gary S. Schajer and Michael B. Prime was implemented.

Then, the obtained moment is applied, and the Von Mises stresses, the circumfer stresses, and the stresses are calculated using the cylindrical coordinate system. A simulation must be performed for each run, in which the material must be configured according to the engineering constants of each tube. The unit moment is then applied, and subsequently, the moment is applied according to the calculation of Equation 9.

Results of Residual Strain Measurements.

The data of the residual strains (RS) obtained according to the procedure described above are presented in Table 2. Below is Figure 15. As mentioned, the rings tend to close or open depending on the present residual stresses at the moment of releasing the stresses using the cutting method (see Figure 10). Figure 15 shows on the left the Sample from Run 7B-Start with a tendency to open and on the right the Sample from Run 12 with a tendency to close. Almost all samples exhibited the latter behavior, except for the Samples from Run 7B. The reason for this exception is unclear. Regarding the dispersion of values, the Samples from Runs 7B are discarded for the reason mentioned earlier. For the Sample from Run 7A, an average of -78 με was obtained with a standard deviation of 20.22 με, and for the Sample from Run 7C, an average of -67.67 με with a standard deviation of 21.36 με. Considering both Runs 7A and 7C, they have an average of -72.83 με with a standard deviation of 19.45 με, this last deviation being the expected one when manufacturing the same tube under the same conditions.

Calculation of Calculated Residual Stresses (TR) from Elasticity Equations.

Before calculating the TR, various constants must be known, such as the Qij, Dij, flexural moduli, and bending moments, in order to ultimately calculate the TR.

From Table 2, and considering Equations 8, the constants Q11, Q22, Q12, and Q66 can be evaluated. The values of these constants are summarized below in Table 3.

Similarly to the case of the Q constants, the Φ constants can be evaluated from the values in Table 3. Taking into account Equations 7, the constants can be evaluated for the angles used in the layers of the tubes in this thesis, which are the angles of 5o, 45o, and 89o. The values of these constants are summarized below in Table 4.

Table 3: Qij Constants for the Calculation of the Residual Stresses (TR).

From the constants Φij in Table 4 and the summarized quality results in Table 2, the constants D11, D22, and D12 can be evaluated using Equation 6. The values of these constants are summarized in Table 5.

Using the values of the constants Dij from Table 5, the flexural moduli can then be evaluated with Equation 5, which will later allow us to calculate the residual stresses (MR). The values of the flexural moduli are summarized in Table 6.

Table 4: Constants Φij for the calculation of the Residual Stresses (TR).

Table 5: Constants Dij for the calculation of the Residual Stresses (TR)

Table 6: Flexural moduli Eyb.

Based on the data from the residual stresses (TR) summarized in Table 2, the values of the flexural moduli in Table 6, and the quality parameters summarized in Table 1, the MR values can be evaluated using Equation 1. The results are summarized in Table 7.

The TR values are obtained by evaluating Equations 3 for each run using the MR values from Table 7. It is also necessary to evaluate Equation 4 to obtain the constant N for each run. Table 8 summarizes the information for the maximum values of σy and σz. The maximum values for σy occur when the radius vector r is at the extremes, that is, r=a and r=b. On the other hand, for σz, the maximum value occurs at the center. Equations 3 were evaluated at 11 points to obtain the typical distribution of the TR. The results are shown in Figure 16 for σz and in Figure 17 for σy. It can be observed that the stresses in the Z direction are almost negligible and present a very low value.

Table 7: Residual Moments (MR).

Table 8: TR calculated by elasticity.

Calculation of TR according to the finite element model proposed.

According to the finite element model described in the previous section, 10 runs were simulated, obtaining the MR, Von Mises TR (or total), Circumferential TR, and Radial TR.

For this, it is necessary to consider the micro-mechanical properties according to the Rule of Mixtures for each run, the patterns used in each run, and the geometric dimensions of the specimens for each run. The measurement of the quality parameters of the specimens, diameter, and thickness were summarized in Table 1. A length of 25.4 mm was used for their width. Finally, the unit moment must be applied, and then the total moment is applied according to the calculation of Equation 9, taking into account the values from Table 7.

However, prior to this, an analysis should be performed to determine the type of element that is most optimal in terms of result quality, simplifies the problem, and leads to lower computational costs. To do this, the problem was solved for Run 1 while varying the type of element involved.

Table 9 compares the number of nodes and elements used in the model for Run No. 1 for each linear element, as well as the Von Mises TR (Total) obtained for each element (SHELL181, SOLID185, or SOLSH190). It can be observed in this table that the values obtained for each type of element are similar. The same system was also modeled with the SHELL281 quadratic element type but with the same average element size in the gauge area of 1 mm, and finally, a finer mesh (element size in the gauge area of .5 mm) with the same type of element. These results are also shown in Table 9. Since the values obtained by the quadratic element SHELL281, with a size in the gauge area of .5 mm, produce an intermediate approximation between the TR values of the SHELL181 (1 mm) and SHELL281 (1 mm) element types, while using fewer computational resources than the SOLID or SOLSH elements, the SHELL281 element type with a mesh size in the gauge area of .5 mm was chosen to model the simulations for the remaining runs. The mesh representation for this type of element is shown in Figure 18.

To represent the angles of the layers for each run, layers of thickness were created divided into 20. The angles for each run were configured as shown in Table 10. Fortunately, in ANSYS Workbench ACP-PRE?, it is quite simple to ensure that the angles of the layers have been configured correctly, as shown in Figure 19 for the angles of Run 1 (Corrida 1).

Table 9: Data from different finite element models.

Different types of elements were used to determine the most suitable model for simulating and obtaining the TR using ANSYS Workbench? software. The element type chosen for the simulation is highlighted in gray.

Created with SHELL281 elements and 0.5 mm element size in the strain gauge area.

Table 10: Configuration of the angles of the different layers.

It is verified that the angles are correctly configured in ANSYS Workbench ACP-PRE?. A very coarse mesh, with an average element size of 5 mm, is shown to facilitate the visualization of the arrows. The normal to the element is indicated in pink, and the fiber direction in green.

Taking all these considerations into account, Table 11 shows the MR, the average Von Mises (or total) TR, the average Circumferential TR, and the average Radial TR.

Table 11: Summary of all simulations from each run.

A final comment regarding the finite element model used is that it perfectly predicts the behavior before and after the application of the moments. Additionally, with SOLID and SOLSH elements, it perfectly predicts the behavior of the strain gauge on the inner face, in compression, and on the outer face, in tension. All of this holds true except for Run 7B, which shows the opposite behavior. For this, see Figure 20. It shows the undeformed wireframe representing the strain gauge area, where on the left is the side of the inner face (the undeformed wireframe is larger than the deformed area), clearly in compression, and on the right, the outer face, clearly in tension (the undeformed wireframe is smaller than the deformed area). Furthermore, in the same Figure 20, the color bands represent the displacement in the circumferential Y-axis of the cylindrical coordinate axes previously described.

Conclusions

Based on reference literature, the residual stresses of carbon fiber tubes manufactured using the Filament Winding technique were studied. A population sampling was carried out, and the experimental results of the Residual Strains were obtained. The results of the evaluation of these Residual Strains were measured and compared to obtain the Residual Stresses. First, through elasticity equations. Second, using a finite element model created with ANSYS.

Gustavo Francisco Eichhorn: Master Degree in Mechanical Engineer and Materials Science and Technology.

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