## Journal of the Brazilian Society of Mechanical Sciences

##
*Print version* ISSN 0100-7386

### J. Braz. Soc. Mech. Sci. vol.22 n.2 Campinas 2000

#### http://dx.doi.org/10.1590/S0100-73862000000200006

**Modelling of Mechanical Properties of CRFC Composites under Flexure Loading**

**Luiz Claudio Pardini**

Centro Técnico Aeroespacial. Instituto de Aeronáutica e Espaço. Divisão de Materiais (AMR).

12228-904.São José dos Campos.SP.Brazi

pardini@iae.cta.br

**Flamínio Levy Neto**

Universidade de Brasília. Departamento de Engenharia Mecânica. UnB-FT-ENM

70910-900 Brasília. DF. Brazil

levy@enm.unb.br

**Brian McEnaney**

University of Bat.Department of Engineering and Materials Science.Claverton Down.Bath.BA2 7AY.England.UK

Carbon Fibre Reinforced Carbon (CFRC) Composites are increasing their applications due to their high strength and Young’s Modulus at high temperatures in inert atmosphere. Although much work has been done on processing and structure and properties relationship, few studies have addressed the modelling of mechanical properties. This work is divided in two parts. In the first part, a modelling of mechanical properties was carried out for two bi-directional composites using a model based on the Bernoulli-Euler theory for symmetric laminated beams. In the second part, acoustic emission (AE) was used as an auxiliary technique for monitoring the failure process of the composites. Differences in fracture behaviour are reflected in patterns of AE.

Keywords:Carbon-Carbon Composites, mechanical properties, acoustic emission.

Introduction

The design of a structural component needs a previous knowledge of the properties of the material it is made. Compared with other engineering materials, composites are those exhibitting the highest strength/density ratio. As important sub-class of composite materials, the carbon fiber reinforced carbon (CFRC) composites have usually been exploited as structural elements for thermo-mechanical purposes. In recent years, these materials are being extensively investigated in terms of fabrication processes, structure, mechanical properties and oxidation behaviour (Buckley & Edie – 1993). The microstructure of these materials after processing exhibit a large number of inhomogeneities due to the fabrication process, such as voids, cracks, matrix pockets, fibre/matrix debonding, fiber bundle misalignments and consequently one can expect that these microstructural features induce a difficulty in the design in comparison with the classical composite mechanics (Savage – 1994). These features causes an appreciable scatter in strength, which is commonly superior to conventional materials (King, 1988). But until now there is a lack of studies addressing the correlation between experimental information given by mechanical tests and theoretical calculations given by the strength of materials approach. Therefore, the main purpose of this work is to present the experimental results obtained under flexure loading, as well as those calculated by using the classical laminate beam theory (Gibson – 1994), and to analyse the correlation between them. Flexural tests were used primarily in this study, and the structural design of composite beams normally requires a careful assessment of the bending stresses acting on the cross section. Bending tests are very useful to evaluate the mechanical behaviour of a material because they are simple and they can be performed with a modest consumption of test material. When pure flexural loads occur in a structural element both tensile and compressive normal stresses are simultaneously present along the thickness of the material. In addition, in flexural tests, transverse shear loads, along the cross section of the specimens also occur. Depending on the span-to-depth ratio, the fracture behaviour of these beams under these two load components can be associated either to the normal stresses, caused by pure flexure, or to delamination, caused by shear, or to both mechanims.

Theory of Mechanical Modelling

In order to predict the effective bending stiffness, Ef , the mid span deflection, Wm , the maximum transverse shear, tzx , and the maximum bending stress, sx; of the CFRC laminates tested to failure under three point bending, a mathematical model based on the Bernoulli-Euler theory for symmetric laminated beams was adopted (Gibson, 1994).

A side view of the laminated CFRC rectangular beam, of thickness t and width b, is shown in Figure 1. The coordinate x is associated with the longitudinal direction of the specimen, and y and z are taken along its width and depth, respectively. The laminated beam is symmetric about the (x,y) plane, i.e., the mid surface.

Since the CFRC beams were reinforced with woven fabrics with the warp and weft directions aligned with the coordinates x and y, i.e., fibre orientations are either 0o or 90o, respectively, there were no shearing-stretching coupling effects on the composite plies.

In this case, for an even number of plies (N) of uniform thickness zj = j.t/N, the effective flexural modulus of the beam, Ef (Gibson, 1994), is given by:

(1)

where:

j = index number of a generic ply, taken from the middle surface of the beam,

Ex = longitudinal Young’s Modulus of each ply (j)

zj = the far end distance of each ply (j) from the middle surface of the beam, as shown in Figure 2; and N is the total number of plies.

The mid span deflection, Wm, of the CFRC beam subjected to three point bending, as shown in Figure 1, can be obtained integrating the moment-curvature equation of the beam according to simple procedures. These procedures can be found in strength of materials textbooks. The mid span deflection, Wm, depends on the applied load (P), the geometry of the beam (L, t, and b), and its effective flexural modulus, Ef, as given by the following expression:

(2)

If (Ex)j = Ex is the same for all the layers, it can be shown from equation (1) that Ef=Ex. The distribution of normal stresses due to bending, at each layer, (sx)j , in the cross section at the middle of the beam is given by:

(3)

Equation (3) is valid for all layers of any symmetric CFRC composite beam subjected to three point bending, in which the reinforced fibres are orientated at 0o or 90o with respect to its longitudinal direction, i.e. coordinate x. For a situation in which all the layers have the same ply orientation (i.e. (Ex)j = Ex = Ef), and only the maximum stress (sx) is required, the following equation can be used:

(4)

where: P = applied load, L = span, b= beam width, t = beam thickness.

Equation (4) gives the maximum tensile (+) or compressive (-) normal stresses, sx, which occur at the top and bottom surfaces in the mid-span section of the beam (z = ± t/2). Combining equations (2) and (4), which relates the flexural stress and the mid-span deflection of a beam, it follows:

(5)

where: Wm = beam deflection, L = span, t = beam thickness, Ef = modulus.

In addition to the flexural stresses (sx)j given by Equation (3) , the CFRC composite laminate beam tested under three point bending is also subjected to transverse shear forces. The CFRC material also undergoes the action of two pairs of transverse shear stresses (txz = tzx ) in the (x,z) plane, as illustrated in Figure 3.

Depending on the span-to-depth ratio of the beam, and the mechanical properties of the material, the transverse shear stresses (tzx) can eventually cause delamination of the CFRC composite. So, the failure mode of the beam can be controlled either by the bending normal stresses, if (sx)j exceeds fracture strength of the CFRC material, or by the transverse shear stresses (tzx) if they are large enough to cause delamination.

The estimation of tzx in CFRC beams is important due to the fact that both the Young’s Modulus as well as the strength of the material, along the direction of the carbon fibres, are normally significantly higher than the transverse shear modulus and strength of the laminae.

According to Gibson (1994) for a large number of plies in a laminate (N > 6) the shear stress distribution, (tzx)j , through the thickness of a composite beam is expected to approach a parabolic shape in which the maximum peak, (tzx)max , occurs at the middle of the beam thickness. Therefore, the equation for the transverse shear stress distribution through the thickness of the composite beam is similar to that for homogeneous isotropic beam, and it is given by:

(6)

where: V = transverse shear force (V = ± P/2), b = beam width, t = beam thickness, Ef = effective flexural modulus of the beam, and

(7)

Experimental

Two CFRC Composites were used in this work:

a) CFRC-4HS is a 8 ply laminated composite (~ 3.2 mm thick) of a Rayon-based carbon fibres in a 4-harness satin weave, in a carbon matrix derived from a resin/pitch blend. Rayon-based carbon fibers have a Young’s modulus of 40 GPa (Matthews – 1994). The fabric pattern of this material is shown in Figure 4A. The CFRC-4HS has the trade name of KKarb Type A and was provided by Kaiser Aerotech Inc (Crocker, 1991). According to manufacture’s data for CFRC-4HS the volume fraction of fibers is 45%. Total open pore volume fraction obtained by water penetration method (ASTM D-20) for CFRC-4HS was 7,5%.

b) CFRC-TWILL is a 10 ply laminated composite (~ 2.9 mm thick) with PAN-based carbon fibres, in a hopsack weave, in a resin carbon matrix. Resin carbon matrices are usually obtained from phenolic resin pyrolysis. PAN-based carbon fibers have a Young’s modulus of 230 GPa (Matthews – 1994). The fabric pattern of this material is shown in Figure 4B. CFRC-TWILL has the trade name of Sigri CC1501G and was provided by Hoechst plc, (Crocker, 1991). According to manufacture’s data for CFRC-TWILL the volume fraction of fibers is 38%. Total open pore volume fraction obtained by water penetration method (ASTM D-20) for CFRC-TWILL was 14%.

Properties of these materials taken from manufacture’s data sheets are shown in Table I. Flexural tests were performed in an Instron 1122 with a cross head speed of 0.5 mm/min with a span/depth ratio of 35:1. Previous results found that, for uni and bi-directionalCRFC composites, the flexural modulus in three point bending increases progressively until it remains approximately constant for span/depth ratios greater than ~ 25:1 (Crocker – 1991). A span/depth ratio of 35:1 was used so that the AE transducer could be attached without interference from the load points. Flexural tests are of great convenience and are extremely useful in obtaining preliminary data on systems not yet fully developed. Unless specified, the experimental values are an average of at least 5 samples.

Optical microscopy was used to observe the microstructural features of the samples using a Zeiss ICM405 microscope. The samples were mounted in epoxy resin and polished in a automatic Motopol M12 polishing machine, using a standard polishing routine (Pardini, 1994).

Acoustic Emission (AE) responses were detected during bending tests of CFRC-4HS and CFRC-TWILL composites. The experimental AE system is shown schematically in Figure 5. The AE events are detected by the response of a piezoelectric transducer coupled to the surface of the stressed specimen using a thin layer of petroleum jelly. The transducer converts the stress waves into an electrical signal of low amplitude, which are passed through a 60 dB pre-amplifier. The characteristics of the AE events are measured using a Marandy MR1004 system. The peak amplitude of each event is sorted into one of 25 amplitude channels, each of which is 2.4 dB wide relative to the threshold amplitude (10 mV) (Russel-Floyd, 1990).

The most frequent method for evaluating structural damage by AE monitoring is to count the signals emitted during the deformation of the material, and plot the results as count rate or total count as a function of some measure such as stress (Stone and Dingwall, 1977).

AE events were counted up to the failure stress. Examinations of the relative number of counts at different amplitudes can provide a useful mean of distinguishing between individual failure mechanisms or between specimens of different quality (Stone and Dingwall, 1977). To describe the distribution of peak amplitude Pollock (1973) used a function, N(a), which defines the fraction of the emission population whose peak amplitude exceeds a, and suggested that AE amplitude distributions can often be fitted to an empirical power law of the form:

(8)

The Pollock plots in the form of log N(a) vs. amplitude level, or channel number (equivalent to amplitude gain), were taken at the failure stress. The exponent b is convenient to characterise the amplitude distribution (Pollock, 1973).

Results

Microstructure of CRFC composites

The microstructural features of the CFRC-TWILL and CFRC-4HS composites were explored by Crocker (1991). The CRFC-4HS material, Figure 6, is reinforced with Rayon-based carbon fibre bundles in a 4 harness satin weave. The matrix is formed from a blended mixture of pitch and resin and is graphitised. Optical micrography, as shown in Figure 6, revealed that there are many regularly-spaced, intrabundle cracks, ~ 200 mm long, some of which extend across the whole width of the bundle (C), and porosity is present. Also, a few small pores (P), ~ 50 mm wide, can be seen.

The CFRC-4HS, Figure 7, has a resin carbon matrix and PAN-based carbon fibre bundles in a hopsack weave with very large pores, mainly at bundle/bundle interfaces that can be easily seen by optical micrographs. Possibly, these pores have formed because the resin has shrunk, during carbonisation, and/or bubbles of volatiles or air have been trapped on curing (McEnaney & Mays – 1993). Cross bundle cracks, similar to those found for the CFRC-4HS are also seen in Figure 7.

Mechanical Modelling

Experimental results for flexural tests measured on CFRC-4HS and CFRC-TWILL are shown on Table 2. Figure 8 shows experimental bending curves, stress as a function of mid-span deflection, related to results from Table 2, for both CFRC composites. Results from measurements and theoretical calculations for apparent rigidity of the CFRC composites studied are shown on Table 3. Three different cases were considered in the comparisons between theoretical and experimental results for apparent rigidity, as following:

Case 1 - Equation 5 (sx/Wm = 6tE/L2).

Case 2 - linear slope taken from stress x deflection curve (A), in the elastic region, as shown schematically in Figure 9.

Case 3 - linear slope taken from the stress x deflection curve (B), from the onset of loading up to the failure stress, as shown schematically in Figure 9.

The experimental and calculated mid-span deflection for both composites are shown in Table 4. The theoretical calculations for mid-span deflection assumed validity of Equation 2.

Typical plots of stress-deflection and sketches of the crack patterns after fracture for CFRC-4HS and CFRC-TWILL materials at a span/depth ratio of 35:1 are shown on Figure 8. The stress-deflection curve for CFRC-4HS material has a catastrophic reduction in stress in two steps, and the crack fracture pattern is usually characterized by delaminating cracks in the plies connected to the tensile face region. For CFRC-TWILL material the stress-deflection curve shows that it fails by a series of catastrophic steps and the crack fracture pattern is dominated also by delaminating cracks, mainly in the compressive region. Similar crack fracture patterns were found also by Crocker (1991). In addition, Crocker (1991) found similar values of flexural strength (200± 40 MPa), which were independent of span/depth ratio in the range 10-50. Crocker (1991) also found, for the CFRC-TWILL composite that the flexural strength was independent of span/depth ratio in the range 20-50, although her value of flexural strength was somewhat lower (182.6± 25 MPa). This is not a surprise since batch-to-batch variations during the processing of CFRC composites are expected.

When the Young’s Modulus given in Table 1 were used in equations (1), (6), and (7) it was obtained, for both, CFRC-4HS and CFRC-TWILL specimens, that Ef » S. As a result, (tzx)max is given by:

(9)

Depending on the relative magnitudes of (sx)max and (tzx)max , the onset of failure of the CFRC beam can be controlled either by the bending stresses or by the transverse shear stresses. So, dividing Equation (4) by Equation (9) one obtains:

(10)

The values of strength properties of a CFRC beam, (sx)max and (tzx)max , that satisfy Equation (10) give the threshold of the change on the failure mode mechanism. If the CFRC composite beam under three point bending presents (L/t) > (L/t)crit , the failure is controlled only by bending stresses, i.e. (sx)max is greater than the tensile and/or compressive strength of the CFRC laminate, and, if (L/t) < (L/t)crit failure by delamination takes place, i.e (tzx)max overcomes the interlaminar shear strength of the laminate.

Acoustic Remission

Typical plots of cumulative acoustic emission counts and applied stress plotted against mid-span deflection, for CFRC-TWILL and CFRC-4HS composites, are shown in Figure 10. For the CFRC-4HS material AE is detected from the onset of applied stress and it builds up continuously to produce ~ 40.000 AE counts at fracture. This indicates that deformation and failure of CFRC-4HS composite is accompanied by a substantial amount of sub-critical activity.

The development of AE on stressing the CFRC-TWILL composite is quite different from that found for the CFRC-4HS material, and it can be divided in two stages. In the first stage there is a steady increase of AE until the onset of ply delamination at ~ 160 MPa, when the stress-deflection curve deviates from elastic behaviour, and, at that point, the second stage begins in which there is a higher rate of AE counts until failure.

Typical Pollock plots of CFRC-4HS and CFRC-TWILL composites are shown on Figure 11. The figure shows that AE from both materials conforms to the Pollock equation in the range N(a)= 102-104 (amplitude levels 2-15), but there is a greater number of low amplitude AE events from the CFRC-4HS material. This is reflected in the higher value of the Pollock exponent, b, for CFRC-4HS, Table 5.

Discussion of the Results

If the compressive modulus and the tensile modulus for the composites studied have values significantly different, the apparent rigidity calculated (Equation 5), and that obtained from the experimental stress x deflection curves, would disagree. In the case of CFRC-TWILL composite the results given by Equation 5 on Table 3 (~ 106 N/mm3), and from the experimental linear slope of the stress x deflection curve (~ 107 N/mm3), in the elastic region are in a very good agreement, indicating that similar values for compressive and tensile modulus for CFRC-TWILL material are expected.

The failure stress slope taken from the stress x deflection curve of CFRC-TWILL composite, Case 3, shows that a lower value for apparent rigidity, approximately ~ 10%, is obtained, which indicates the decrease on material’s stiffness when aproaching the failure stress.

For CFRC-4HS material the value of apparent rigidity (sx / Wm ) calculated from Equation 5, ~ 23 N/mm3, is much lower than the experimental values, > 28 N/mm3. This indicates that compressive modulus of CFRC-4HS composite is higher than its tensile modulus. For this theoretical calculation, Equation 5, it was assumed for CFRC-4HS material the highest value for tensile modulus (15 GPa) from Table 1. The failure stress slope taking from the stress x deflection curve of CFRC-4HS composite, Case 3, shows that a lower value for apparent rigidity, approximately 10%, is obtained, following a trend observed for CFRC-TWILL material.

The results suggests that there was some influence of the fibre type in the composite properties. In a certain degree, it is expected that the PAN-based carbon fibres, which the CFRC-TWILL composite is made, are more strongly bonded to the carbon matrix than are ex-rayon carbon fibres, which the CFRC-4HS material is composed (Savage – 1994). The nature of bonding in these composites reflects in differences in compressive and tensile modulus, as mentioned before. Moreover, the properties of CFRC composites are strongly dependent on materials microstructure, which can reveal pores and cracks.

The failure mode of CFRC-4HS composite is mainly characterised by ply delamination cracks, which develop in the tensile side of the specimen, Figure 8. The stress-deflection curve consist of an increase in stress to a single peak, followed by a catastrophic reduction in stress mainly in two steps. Kowbel (1992) and Crocker (1991) found that, for woven CFRC composites tested in flexure at span/depth ratios higher than 30:1, the tensile stresses between the fibre bundles increase, eventually pulling them apart in a delaminating mode. These interfaces are weaker than the transverse fibre bundles, and it is therefore easier for cracks to delaminate plies rather than to propagate across bundles.

The failure mode of CFRC-TWILL composite, Figure 8, is characterised as a combination of delamination and cross bundle shear cracks running between pores in the compressive stress field, even though large longitudinal cracks are more dominant. The stress-deflection curves have multiple peaks which is indicative of subcritical cracking, and the reduction in stress after the peak value involves a multi-step process, Figure 8. These differences in behaviour between the CFRC-4HS and CFRC-TWILL composites are probably influenced by the greater extent of porosity in the CFRC-TWILL material.

For both CFRC composites, CFRC-4HS and CFRC-TWIL, the experimental and calculated mid-span deflections are in a very good agreement. A good prediction for mid-span-deflection was achieved from Equation 2, and confirms all the above considerations.

By using Equation (10) is possible to evaluate if the failure mode of the CFRC beams tested were controlled by the maximum bending stresses, (sx)max , or by delamination, (tzx)max. The interlaminar transverse shear strength of 2D CFRC laminates varies in the range of 8 to 15 MPa (Pollock, 1973, Kowbel – 1992). Taking these values and the measured flexural strength of the CFRC-TWILL, (sx)max = 209 MPa, and CFRC-4HS, (sx)max = 180 MPa, from Table 2, the critical values of the (L/t)crit ratios, below which delamination can take place, can be calculated. Using 8 MPa as a lower bound for (tzx)max , and 15 MPa as an upper bound, it is possible to found if the failure mechanism occurs by delamination by estimating the L/t ratio. For the CFRC-TWILL composite the L/t ratio is 7 – 13 and for the CFRC-4HS composite is 6 – 12. So, the composites would fail by interlaminar shear if they were tested in a span-to-depth ratio preferably around 6. Probably a complex failure mode would be found when testing these materials at a L/t ratio in between 7 – 15. Results from Crocker (1973) show that an assymptotic value for the flexural modulus is also found for a L/t > 20 when testing bi-directional composites.

In the present work, all the CFRC beams were tested with a L/t ratio of 35. So, according to the above calculations, the failure of the specimens under a L/t ratio=35 is mainly controlled by the maximum bending stresses, i.e. tensile and compressive normal stresses.

Conclusions

The properties of CFRC composites are a consequence of their inherent microstructure, which in turn are derived from the type of carbon fibre, the carbon matrix precursor used and the heat treatment schedule utilized during manufacture.

The beam laminte theory gives reasonable good approximations for an initial evaluation of material’s flexural modulus. The extension of accuracy in this prediction will depend on the nature of fibre/matrix bonding, the span over thickness ratio (L/t), and also from the point where the slope from the stress x deflection curve is taken.

Using simple mechanics of materials approach it is possible to estimate the boundaries of span-to-depth ratios that control the shear failure and bending failure mechanisms. These results suggest that CFRC composites with similar compressive and tensile modulus give rise to better predictions for flexural modulus. The magnitudes of these predictions are in good agreement with experimental values.

It has also been demonstrated that AE is a viable tool to study and correlate the microstructure, mechanical behaviour, and failure modes of 2D CFRC composites. For example, between the two different materials, CFRC-4HS and CFRC-TWILL composites, the highest strain and toughest material was the one that presented the highest number of AE counts, i.e. S AE (CFRC-4HS) > S AE (CFRC-TWILL).

References

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Manuscript received: April 1999. Technical Editor: Hans Ingo Weber.