Experimental and numerical evaluation of bending and tensile behaviour of carbon-fibre reinforced SiC

Composites: Part A 43 (2012) 1877–1885

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Experimental and numerical evaluation of bending and tensile behaviour of carbon-fibre reinforced SiC S. Hofmann ⇑, B. Öztürk, D. Koch, H. Voggenreiter German Aerospace Center, Institute of Structures and Design, Pfaffenwaldring 38-40, 70569 Stuttgart, Germany

a r t i c l e

i n f o

Article history: Received 18 September 2011 Received in revised form 11 July 2012 Accepted 15 July 2012 Available online 2 August 2012 Keywords: A. Ceramic–matrix composites (CMCs) B. Fracture toughness B. Strength C. Finite element analysis (FEA)

a b s t r a c t The tensile and bending strength of the Liquid Silicon Infiltrated (LSI) ceramic–matrix composite (CMC), C/C–SiC, were investigated in varying orientations relative to the 0°/90° woven carbon fibres. The ratio of bending to tensile strength was about 1.7–2 depending on the loading direction. The non-linear stress– strain behaviour under tensile load and the linear elastic behaviour under compression load were included in the finite element analysis (FEA) of bending behaviour. The bending failure of the CMC-material was modelled by Cohesive Zone Elements (CZE) accounting for the directional tensile strength and Work of Fracture (WOF). The WOF was determined by Single Edge Notched Bending (SENB) tests. Comparable results from FEA and bending test were achieved. It was demonstrated that the failure of C/C–SiC at room temperature may be described by a macroscopic fracture mechanical FE-approach. The presented approach could also be adapted for the design of CMC-components and structures. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction It is a common observation, that the bending strength of CMCmaterials is much higher than the strength in tensile loading. This has been shown experimentally in a number of publications [1–4]. In the past different explanations were found. Marshall and Evans [2] showed for an uni-directional SiC-fibre glass–ceramic composite that fracture mechanical analysis is not appropriate to explain the bending strength because failure does not occur by one single planar-crack and mainly shear failure was observed in SENB-testing. Therefore Marshall and Evans [2] proposed that micromechanical considerations of interface stress and statistical strength of fibres have to be considered. Hild et al. [5] followed the idea from Marshall and Evans and considered additionally the theory of global load sharing. Hild et al. [5] showed that two factors are influencing the ratio of bending to tensile strength. Those are the tensile-to-compressive secant modulus caused by matrix cracking, leading to the shift of the neutral plane, and further the Weibull modulus of fibre strength. Following this approach, the tensile and bending strength for three different Nicalon-fibre reinforced CMCs were closely predicted. The micromechanical properties: fibre strength, its Weibull modulus and the interfacial sliding stress were necessary for the prediction of the tensile and bending strength. In contrast, the present work is using macroscopic parameters only to predict the tensile and bending strength in varying loading directions. Steif and Trojnacki [6] demonstrated that the ratio of bending ⇑ Corresponding author. Tel.: +49 711 6862 745; fax: +49 711 6862 227. E-mail address: [email protected] (S. Hofmann). 1359-835X/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compositesa.2012.07.017

to tensile strength depends mostly on the stress–strain behaviour after reaching ultimate tensile strength. Especially composites failing non-brittle in tensile testing show a comparatively high bending strength. For materials behaving linear elastic-ideal plastic in tension and linear elastic in compression a ratio of 3 can be reached. The factor of 3 was demonstrated earlier also by the theoretical work of Laws and Ali [7]. Steif and Trojnacki [6] investigated the influence of the behaviour after peak tensile stress and the influence of slope change reaching proportional limit under tensile loading. The effect of the slope change caused by matrix cracking under tensile load was shown to have only modest influence on the bending-tensile strength ratio. Mainly the stress–strain behaviour beyond ultimate tensile stress was influencing the bending-tensile strength ratio. The main problem about the work from Steif and Trojnacki [6] is that the slope after reaching ultimate tensile strength cannot be determined for most CMC-materials since they fail rather brittle under tensile load. Also the herein investigated C/C–SiC is failing brittle under tensile load. Therefore the WOF was determined in SENB tests and used to model the behaviour beyond ultimate tensile strength as linear stress–separation curve with Cohesive Zone Elements (CZE). In contrast to the results from Marshall and Evans [2] SENBtesting was successful with the investigated C/C–SiC material: failure occurred by one single planar crack. Shear failure was not observed. The present approach is similar to the work from Fink [8] who showed in his theoretical work that the bending strength could be much higher than the tensile strength for CMC-materials if critical energy release rates are considered.

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Table 1 Sample geometries for tensile, 3-point, 4-point bending and SENB test. Test

Test geometries

Investigated orientations

3rd dimension (mm)

Tensile

0°/90°, 15°/75°, 30°/60°,45°, 90°/0°

2.3

3-Point Bending

0°/90°,15°/75°,30°/60°, 45°

10

4-Point Bending

0°/90°, 45°

10

SENB

0°/90°, 45°, 90°/0° (in-plane: L = 22.5 mm; W = 4.4 mm), 0°/ 90° (translaminar: L = 14 mm; W = 3 mm)

2.3 in-plane; 10 translaminar

F/2

In the following the bending behaviour under varying loading directions will be modelled considering the differing stress–strain behaviour under tensile and compression load. Failure is modelled by Cohesive Zone Elements accounting for the directional ultimate tensile strength values and the WOF as determined in SENB-testing. It will be shown that the ratio of bending to tensile strength can only be partially explained by the differing behaviour under tensile and compression load. However, the bending strength will be modelled in good agreement with the experimental results by additionally taking the fracture toughness of the CMC-material into consideration.

2. Experimental setup 2.1. Material and sample preparation The investigated material, C/C–SiC, was produced from HTA carbon fibre fabrics of twill style 2/2 in-house at the German Aerospace Centre by LSI-process. The microstructure of C/C–SiC is built up by a dense SiC-matrix, surrounding blocks of C/C-fibre bundles, containing carbon fibres and residual pyrolytic carbon. The SiC-matrix typically shows cooling cracks from thermal mismatch of C-fibres and SiC-matrix. Further details on processing and microstructure can be found elsewhere [9,10]. Because of the two different matrices, carbon and SiC the mechanical behaviour of C/C–SiC lies somewhat between a weak matrix and a weak interface CMC-material [11]. A C/C–SiC plate of 300  300  3 mm3 was produced of 13 layers of HTA woven fabric. The weft and warp directions of the fabric were switched after each layer to get similar mechanical properties in 0° and 90° direction. After silicon infiltration the plate was grinded to a thickness of 2.3 mm to get even surfaces. Dog-bone specimens for tensile testing were cut by water-jet. The samples for 3-point bending and in-plane SENB-testing were produced by abrasive cutting from the same plate. Samples from another plate of C/C–SiC had to be taken for 4-point bending and translaminar SENB-testing. The thickness of the plate was about 3 mm, again.

2.2. Mechanical test set-up: tensile, 3-point-, 4-point bending and SENB All mechanical tests were performed in a universal testing machine (Zwick) at room temperature. The tensile tests were performed with strain gauges in longitudinal and transverse direction. In 4-point bending strain gauges were applied in between the inner span in longitudinal direction on the top and bottom side of the samples. In 3-point bending strain gauges could only be applied at the bottom side, opposite to the point of load transmission. The tensile and bending tests were performed following DIN-EN standard 6581 and respectively 658-3. Table 1 gives an overview about the performed mechanical tests, their geometries and fibre orientations. The SENB tests were conducted similarly to the work from Kuntz [12]. The starter-crack a0 was introduced by abrasive cutting with a diamond plate of 300 lm thickness. For in-plane SENB-testing the samples had a height of about 4.4 mm, the width was 2.3 mm and the outer span was set to 22.5 mm. Additionally, samples with pre-notches perpendicular to the fabric layers were produced. Those samples are denoted as translaminar SENB samples in the following. The samples had a height of about 3 mm and the outer support distance was set to 14 mm. To check if the sawed pre-notches are equivalent to cracks the sawed translaminar samples were loaded until first failure. Then the created crack length was determined under light microscope and the samples were loaded again. Fig. 4b shows that both, the sawed notches and the pre-cracks correspond well with the solution from LEFM. The in-plane tests were therefore performed only with prenotched samples. The KIc-value, that is the critical stress intensity factor, was determined from the maximum load following the equations from Tada et al. [13]. Additionally, the WOF was determined from the area under the load–displacement curves as described by Nakayama [14]. Five samples per orientation for tensile testing, four samples per orientation for 3-point and 4-point bending, were evaluated. Since the C/C–SiC material is showing relatively high compression and interlaminar shear strength only valid failure, i.e. tensile failure on the lower side of the sample, was observed in bending test.

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3. Results from mechanical testing

200 0°

15°

30°

Load / N

3.1. Results from tensile, 3-point and 4-point bending test

45°

160

120

80

40

0 0

0,5

1

1,5

2

2,5

3

Displacement / mm Fig. 1. Load–displacement behaviour from 3-point bending test in different orientations.

250 0º

Bending

15º

30º

Stress / MPa

200 45º 150 100

Loading direction α

Tensile 50 0 0

4

8

12

16

20

Strain / o/oo

(a) 250 compression

tensile

200

Stress / MPa

0º 45º

In the following the results from mechanical testing are summarised. Fig. 1 shows all load–deflection curves recorded during 3-point bending test in different directions. The scattering of results was rather low, as also indicated by the standard deviations in Fig. 11b. As expected the stiffness decreases with increasing angle between fibre and loading direction. The load–displacement response in 0° and 15° direction relative to the fibres is almost linear, in contrast to the 30° and 45° directions, which are showing a strongly non-linear behaviour. All curves show a sudden steep load drop, after reaching failure load, indicating rather brittle failure. The sample surfaces showed a crack through at least 30% of the sample thickness directly after failure. It was not possible to stop the testing machine at the load maximum, or to drive slowly enough, to see single ply failure. All of the bending samples failed by tensile failure at the lower side of the sample. Shear or compression failure was not observed. Fig. 2a compares the best average stress–strain curves from tensile and bending tests in different orientations to the 0°/90° fabric. The shown tensile stress–strain curve in 0° orientation is the best average curve of both orientations 0° and 90°. The Young’s modulus and tensile strength were about 10% lower in 90°-direction. The bending stress–strain curves correlate with the load–deflection curves from Fig. 1. The strain was measured by strain gauges on the tensile side. The load drop is therefore not as clearly indicated as by the load–deflection data in Fig. 1. Since beam theory does not describe accurately the true bending stress, as will be explained below, the load–deflection raw data is also shown. Even so, beam theory was used to calculate the bending stresses in Fig. 2a and b to show the discrepancy between tensile and bending stress–strain curves. There are mainly three significant results in Fig. 2a: first, in all directions the tensile strength lies about 100 MPa below the respective calculated bending strength. Further, the elastic behaviour from tensile and bending test is similar. And finally the nonlinearity observed in tensile testing is more distinctive leading to failure at only about half of the bending failure strain. The strain measurements on the 4-point bending samples revealed that the stress–strain behaviour on the compression side is almost linear compared to the strongly non-linear behaviour under tension

150

100

90 45º 80

50 70 60 -8

-4

0

4

8

12

Strain / o/oo

(b) Fig. 2. Stress–strain behaviour from tensile test and 3-point bending (a) and 4point bending (b) in different orientations, the stress values were calculated assuming beam theory. The 4-point bending strains were measured both, on tensile and compressive side.

Load / N

0

50 40

0º

30 Linear Extrapolation

90º 20 WOF - area 10 0 0,0

In SENB-testing 4–5 samples per direction were analysed. The exact number per orientation can be taken from Fig. 4b. All samples showed failure due to one single planar crack. Shear failure was not observed in any of the samples.

0,2

0,4

0,6

0,8

1,0

Displacement / mm Fig. 3. Typical load–displacement curves from in-plane SENB tests in 0°, 45° and 90° direction with similar initial relative crack length: a0/W = 0.41, 0.41 and 0.45, respectively.

3.2. Results from SENB test The SENB tests were performed to investigate the fracture toughness in four different crack orientations relative to the carbon-fibre fabric. The in-plane fracture toughness with fibre orientations of 0°/90°, 45°, 90°/0° relative to the crack and the translaminar fracture toughness for a crack, orientated perpendicular to the 0°/90° plies, were determined. The in-plane SENB tests showed similar load–displacement behaviour for the varying orientations (Fig. 3). The curves show linear behaviour up to failure load, followed by a small load drop and a section of rather stable crack propagation. In contrast to bending tests the 45° SENB test is showing almost linear elastic behaviour up to failure. This observation indicates that the chosen test set-up is well suited to suppress non-linear effects and to determine the fracture toughness also in 45° fibre orientation. The SENB tests were evaluated regarding the KIc-value and the WOF. Fig. 4a shows that the KIc-values and also the WOF, determined in the different directions, are almost constant. This result is confirmed by plotting the bending strength of the variously orientated SENB samples over the relative crack length in Fig. 4b. There is a good accordance with the results from LEFM, calculated for an overall average KIc value of 5.4 MPa m1/2, using the equations from Tada et al. [13]. 4. Finite element modelling: set-up The commercial code ANSYS WB 12.1 was used for all FE-simulations. The results from 4-point bending test (Fig. 2b) and the compression test results from Fink [8] showed that the upper half, under compression load acts mainly linear elastic, the lower half under tensile load shows, depending on the fibre orientation, strongly pseudo-plastic, non-linear behaviour. That is why the bending sample was divided into an upper and a lower half body for FE-modelling (Fig. 5a). The residual sample height was divided into an upper linear elastic and lower non-linear section for modelling the SENB test, as shown in Fig. 5b. The non-linear, tensile section was modelled with the generalised anisotropic Hill yield

7

7

6

6

5

5

4

4

3

3

WOF in plane WOF translaminar KIc translaminar KIc in plane

2 1

K Ic / MPa(m)1/2

(Fig. 2b). This result agrees with the relatively linear compression test results in 0° and 45° orientation, published by Fink [8], in comparison to the strongly non-linear tensile test curves. The elastic properties under compression and tensile load are comparable, but only for a low stress regime up to about 20 MPa for 45°. The tensile and compression curves from 0° orientation are similar up to 50 MPa. The difference in stress–strain behaviour under tensile and compression load shows that classical beam theory cannot be applied to calculate accurately the stresses in the bending sample. The comparison of the theoretical stress–strain curves under tensile and bending load shows that the linear elastic compression side supports the non-linear tensile side and leads to decreased stress values at the lower tensile side. This is clearly shown by the 45° bending stress–strain curve in Fig. 2a: the tensile failure strain is reached at a calculated bending stress of about 150 MPa, while the tensile failure strain of the tensile test is reached at about 100 MPa. This shows that the apparent stress at the tensile side must be about one third lower than the theoretical stress value. It can be demonstrated that the theoretical bending-tensile strength ratio must be at least 1.5 for 45° and about 1.15 for 0° direction, just by comparing the failure strains. Nevertheless the remaining difference in bending to tensile strength cannot be explained solely by the differing stress–strain behaviour. That is why additionally the effect of crack propagation under bending load was investigated by SENB-testing.

WOF / N/mm

S. Hofmann et al. / Composites: Part A 43 (2012) 1877–1885

2 1

0

0 0

45

90

Angle / °

(a) 200

0° out translam. pre-cracked translam. of plane pre-cracked 0° translam. pre-notched out of plane pre-notched 160

Strength / MPa

1880

0° in plane crack 45° in plane crack

120

90° in plane crack LEFM

80

40

0 0

0,2

0,4

0,6

0,8

1

relative Crack Length ao/W

(b) Fig. 4. WOF and KIc values measured in SENB test in various orientations (a), and bending strength from LEFM and SENB tests (b).

function with bilinear isotropic hardening [15]. Linear hexahedron elements were used for the 3D bending- and linear rectangular elements for the 2D SENB-model, see Fig. 5. The Hill yield model was selected since the Hill failure criterion was describing the tensile strength in satisfying accuracy (Fig. 6). It was expected that the generalised Hill yield function implemented in ANSYS [15] would also give results in good agreement with the experimental non-linear behaviour. The results from tensile testing in different orientations (see also Table 4) are compared with the Hill and Tsai-Wu failure criterions, which were fitted for the C/C–SiC material, see Fig. 6. The major tensile and shear stress fractions for each tensile orientation were calculated by the Cartesian transformations [16]. All necessary data to determine the Hill and Tsai-Wu parameters, as plotted in Fig. 6, are listed in Table 2. The tensile strength is an average value of the 0° and 90° tensile strength, that is why the standard deviation is rather high. The shear and compression strength were taken from an existing database. The mathematical expressions for the Hill and Tsai-Wu criterions can be found in Literature [17,18]. The selected hardening model implemented in ANSYS [15] is using the bilinear anisotropic hardening model from Valliappan et al. [19] and the generalised anisotropic Hill yield potential as presented by Shih and Lee [20]. The equivalent stress for the generalised Hill yield criterion by Shih and Lee [20] is:

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Displacement Linear-elastic

2.3 mm

Bonded Contact Anisotropic-plasticity

Cohesive Zone

Displacement 3 mm

Linear-elastic Bonded Contact Anisotropic-plasticity

a Cohesive Zone with pre-crack Fig. 5. FE-model for bending (a) and SENB test (b).

60 Table 2 Parameters for Hill and Tsai-Wu criterion.

Hill

Shear Stress / MPa

50

Strength (MPa)

Tsai Wu

45°

Avg. tensile 0° and 90° Shear 0°a Tensile 45° Compression 0°a

30°

40

Tensile Strength

30 15°

a

Unpublished internal measurements.

20

10 90°

With M11 = 1, K is defined in the (x, y)-fibre coordinate system to be:

0°

0 40

60

80

100

120

140

K ¼ rþx rx

160

Normal Stress / MPa

T

T

ðrÞ ½MðrÞ  ðrÞ fLg ¼ K

2 ð1Þ

The formulation is similar to the anisotropic Hill criterion but with the extension (r)T{L} for differing tensile and compression yield properties. The 0°/90°-reinforced C/C–SiC has three orthogonal planes of symmetry. The plastic behaviour therefore can be described by the three element coordinate directions and the three shear stress–strain curves. M has the form:

2

M 11 6M 6 12 6 6 M 13 M¼6 6 0 6 6 4 0

3

M 12

M 13

0

0

0

M 22

M 23

0

0

M 23

M 33

0

0

0

0

M 44

0

0 0

0 0

0 0

M 55 0

0 7 7 7 0 7 7 0 7 7 7 0 5

0

ð2Þ

M jj ¼

K

L1

3

6L 7 6 27 6 7 6 L3 7 7 fLg ¼ 6 607 6 7 6 7 405

ð5Þ

0 Table 3 shows that identical yield stresses were applied as input for tensile and compression yielding. The compression behaviour is nearly linear elastic. Nevertheless similar yield stresses had to be defined to fulfil the consistency Eq. (10). The entries of {L} are calculated as described in Eq. (6). {L} is equal to zero when using the input yield stresses from Table 3. That means no differences in initial tensile and compression yield stresses were defined.

Lj ¼ Mjj ðrþj  rj Þ

M66

ð6Þ

Assuming plastic incompressibility yields the following off axis relations:

With Mjj for j = 1–3 and 4–6, respectively:

K

ð4Þ

The strength differential vector {L} describes the differences in tensile and compressive yield stress and has the form:

Fig. 6. Tensile strength results compared with Hill and Tsai-Wu model.

M jj ¼

131.6 ± 13.3 86 99.4 ± 1.7 300

ð3Þ

M12 ¼ 1=2ðM11 þ M 22  M33 Þ

ð7Þ

r+j is the tensile yield stress, rj is the compressive yield stress in j direction. The compressive yield stress is handled as a positive number here. sj is the respective shear yield stress.

M13 ¼ 1=2ðM11  M 22 þ M33 Þ

ð8Þ

M23 ¼ 1=2ðM11 þ M 22 þ M 33 Þ

ð9Þ

rþj rj

;

s2j

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Finally M was calculated from the input data in Table 3:

2

1 0:5 0:5 0 0 6 0:5 1 0:5 0 0 6 6 6 0:5 0:5 1 0 0 M¼6 6 0 0 0 2:56 0 6 6 4 0 0 0 0 2:84 0

0

0

0

0

0 0 0 0 0

Table 4 Average tensile strength values used for cohesive zone failure model.

3 7 7 7 7 7 7 7 7 5

rmax (MPa) 0° 15° 30° 45°

35:78

Only the shear stress ratios deviate from the isotropic ratios: M44 = M55 = M66 = 3. The following consistency equation had to be fulfilled due to the requirements of the plastic incompressibility:

rþx  rx rþy  ry rþz  rz þ þ ¼0 rþx rx rþy ry rþz rz

120 100

ð11Þ

Since the yield stress changes with plastic strain, those criterions limit also the range of tangent moduli ET, which may be chosen. Due to the restrictions for differing tensile and compressive tangent moduli, ET, it was not possible to model the linear elastic compression side and the non-linear tensile side by using the presented bilinear hardening model and the parameters from Table 3 all over the bending beam. Because of the consistencyrestrictions the bending beam had to be divided into an upper linear elastic and lower bilinear part. Nevertheless the work hardening rule, as presented by Valliappan et al. [19], was successfully used to model the non-linear tensile behaviour. The subsequent yield strength increases with the amount of plastic work done in that direction, as described by Valliapan et al. [19]. The plastic work is equated to get an equivalent change in all directions. In that way the subsequent yield strengths in the fibre coordinate system may be calculated from any adopted loading direction. The total plastic work in x direction, W px , is e.g. [19]:

W px ¼ 1=2epx ðr0x þ rx Þ

ð12Þ

p x

with plastic strain e and the initial yield stress r0x and the updated yield stress rx.

epx ¼ ðrx  r0x Þ=Epx

ð13Þ

Epx is the plastic modulus:

Ex ETx Ex  ETx

ð14Þ

ET is tangent modulus from bilinear uniaxial stress–strain input; as determined in Fig. 7 and summarised in Table 3. The updated yield stress for the bilinear stress–strain behaviour in x direction is:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2W px Epx þ r20x

ð15Þ

Stress / MPa

M 211 þ M 222 þ M 233  2ðM 11 M 22 þ M 22 M 33 þ M 11 M33 Þ < 0

rx ¼

140

ð10Þ

Furthermore the yield stress has to describe a closed elliptical envelope. So the following criterion had to be met:

Epx ¼

140 ± 10.2 118 ± 3.8 106 ± 1.6 99.4 ± 1.7

E Tx 80 σ0x

60

G Txy

Ex

40

τ0xy

20 G xy 0 0

10

20

30

Fig. 7. Determination of E and ET (Table 3) for the bilinear hardening model from representative rx and sxy stress–strain curves. The input data for the other orientations were determined accordingly.

The last equation determines the updated yield stresses by equating the amount of plastic work done on the material to an equivalent amount of plastic work in each of the directions [19]. As the C/C–SiC material did not show a distinctive yield point in any direction (Fig. 2a) virtual yield stresses were defined as half of the ultimate tensile and shear strength from best average curves, see Fig. 7 and Table 3. A best average stress–strain curve of 0° and 90° tensile test was selected for the tensile input in x and y direction. The secant slopes from 0 MPa to the virtual yield point and from there onto failure stress were determined, as shown in Fig. 6 for the uniaxial tensile stress strain curve in x direction and the xy shear stress strain curve. The input data for the bilinear behaviour and yield stresses are summarised in Table 3. For the linear elastic compression side only the first slopes were used as Young’s moduli without yielding. The failure of C/C–SiC in SENB and 3-point bending test was modelled by introducing a cohesive zone in the middle of the sample at the location of maximum tensile and compressive stresses. The cohesive zone model describes the debonding of the material as a linear contact stress–separation curve, see Fig. 8. The separation starts at the maximum stress value rmax followed by linear softening up to the critical separation value uc. The area under the curve is equivalent to the critical fracture energy. The main

Table 3 Input data in fibre coordinate system (x, y) for bilinear hardening model from representative stress–strain curves, compare Fig. 7.

x y z xya yza xza a

Strength rmax (MPa)

Compressive yield stress r0j (MPa)

Young’s modulus Ej (GPa)

m

rmax (MPa)

Tensile tangent modulus E+Tj (GPa)

Compressive tangent modulus ETj (GPa)

128 128 – 80 76 21.4

64 64 64 40 38 10.7

64 64 64 – – –

58 58 20 5.14 6.6 9.06

– – – 0.01 0.1 0.1

44 44 19 1.6 2.54 4

56 56 19 – – –

Yield stress r+0j = 1/2

Unpublished measurements from Iosipescu test.

40

Strain / o/oo

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σ max

Critical fracture energy or WOF

uc Separation Fig. 8. Bilinear cohesive zone model with linear softening [15].

transverse

longitudinal

160

0°

140 120

Stress / MPa

0°

15°

15°

30°

30°

100 45°

45°

80 60

FEA Exp.

40 20 0 -8

-6

-4

-2

0

2

4

6

8

10

Strain / o/oo Fig. 9. Comparison of longitudinal and transverse stress–strain behaviour from tensile test and FEA with generalised Hill yield model and bilinear hardening.

input values for the cohesive zone model are the WOF as determined from SENB-testing and the directional ultimate tensile strength rmax. Since the WOF was determined to be almost independent of crack orientation (Fig. 4a), an average value of 2.5 N/mm was applied for all loading directions. The maximum stress value rmax was defined corresponding to the average tensile strength in the respective direction, see Table 4 and Fig. 11b. The initial contact stiffness was defined as 10E6 MPa/mm, which can be considered as ideal rigid contact.

The bending and tensile stress–strain curves from experiment and FEA are directly compared for 45° loading direction. This comparison proves, as mentioned above, that the differing stress–strain behaviour at the lower and upper side of the bending sample leads to decreased effective stress and strain values on the tensile side. In the next step the FE-models were extended with CZEs, enabling failure modelling. First the translaminar SENB test was modelled linear elastically with an introduced cohesive zone in the middle of the sample. The maximum stress value rmax was set to 140 MPa, compare Table 4. The critical fracture energy was 2.5 N/ mm, which is the average WOF in all loading directions (Fig. 4a). This loading case is similar to the normal bending case in 0°/90° direction but with shorter support distance and an inserted notch. Fig. 10b shows again that the experimental results are in good agreement with the LEFM. The results from linear elastic FE-modelling also correspond with LEFM, but the stress level is located slightly below the curve from LEFM. In contrast, the results from the partial non-linear FEA show a highly accurate fit to the LEFM and the SENB test results. Only for small relative crack length of 0.1 and lower the FEA deviates from LEFM. The resulting strength of an un-notched short-beam bending sample is again in good agreement with the experimental bending strength. This result is decisive, because it shows that the bending strength can be reproduced with a fracture mechanical approach and cohesive zone modelling. Finally this approach was applied for the 3-point long-beam bending samples in different loading directions. The average WOF, 2.5 N/mm, as used in SENB-modelling, and the directional tensile strength (Table 4) were used for the

250 15 °

0°

30 °

Bending 45°

200

Stress / MPa

Stress

150 45° Tensile

100

FEA Exp.

50 0 0

4

5. Results from FEA and mechanical testing

12

16

20

(a) 250

Experiment

200

Linear-Elastic Stress / MPa

In the first step the generalised Hill yield model with bilinear hardening was evaluated by comparing the longitudinal and transverse stress–strain curves from tensile testing and FEA. The stress– strain curves from experiment and FE-modelling show good agreement in longitudinal as well as in transverse direction (Fig. 9). The slopes are especially in the low stress regime in 30° and 45° orientation slightly below the experimental slopes. This can be understood by considering the bilinear input data (Fig. 7), which is always softer than the experimental data. Nevertheless the nonlinear behaviour is overall well reproduced by the bilinear hardening model. Since the tensile behaviour in the different orientations could be reproduced with the bilinear material model, it was applied for the tensile side of the 3-point bending model, too. The 4-point bending test was not analysed by FE-modelling. Nevertheless, as mentioned above, the 4-point bending test proved that the compressive side of the bending beam behaves almost linear elastic, see Fig. 1b. Fig. 10 a shows the resulting strains from FEA and experiment plotted together with the theoretical 3-point bending stresses, calculated from the applied load. The stress–strain behaviour measured by strain gauges on the tensile side and the strains from FEA at the corresponding location are in good agreement.

8

Strain / o/oo

Non-Linear

150

LEFM 100

50

0 0

0,2

0,4

0,6

0,8

1

relative Crack Length a0/W

(b)

Fig. 10. Strain results from FEA and experiment evaluated at the tensile side of the sample; the stress values were calculated from load reactions assuming beam theory, for a comparison 45° tensile test results are plotted (a), and bending strength from SENB test, LEFM, linear and non-linear FEA (b).

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difference of bending and tensile strength the characteristic brittle failure, which is not a ‘ply by ply failure’, was modelled with cohesive zone elements and linear softening. A relative error between numerical and experimental bending strength of about 8% (0°) to 17% (30°) remains. The deviations may be explained by the local strength of the material which can be even higher than the averaged tensile strength of mixed weft and warp orientations. Another explanation is the short-coming of the simple bilinear hardening model. Especially for off-axis loadings the non-linear stress–strain behaviour is not exactly reproduced. This circumstance may explain why the relative deviation of numerical and experimental results is increased for off-axis orientations.

200

FEA Exp .

0°

15°

160

45°

Load / N

30° 120

80

40

7. Conclusions 0 0

0,5

1

1,5

2

2,5

Displacement / mm

(a)

250

Strength / MPa

200

150

100

Tensile Strength Bending Strength Bending Strength by FEA

50 0

15

30

45

Angle / °

(b) Fig. 11. Load–displacement from bending test and combined non-linear cohesive zone failure model (a), and calculated failure stresses from FEA and experiments assuming beam theory (b).

cohesive zone modelling of 3-point bending failure. The resulting load–displacement behaviour is in good agreement with the experimental curves, as shown in Fig. 11a. The brittle failure indicated by a steep load drop is well reproduced by the FE-models. In average, the failure load from FEA remains about 12% below the maximum load from bending test. Overall, Fig. 11b shows that, using the bilinear hardening model in combination with the cohesive zone model, the bending strength is predicted with an accuracy of 8% (0°) to 17% (30°).

The bending strength is influenced first by the differing tensile and compression stress–strain behaviour, as already shown in literature [5–8], further by the ultimate tensile strength and finally by the WOF, being necessary to create a macroscopic crack, leading to failure under bending load. The failure of C/C–SiC was modelled considering two material properties: the tensile in the respective directions and the WOF, which is almost independent of crack orientation. In the future work it will be investigated, if the bendingtensile strength ratio can be explained by this approach also for other CMC-materials. It is expected that the presented approach is useful for other LSI-materials and rather brittle WIC- or WMCmaterials. The bending test is always a useful method to decide which failure models have to be applied. For materials, showing stepwise load decrease after first bending failure, like typical WMCs, simple strength or strain criteria are sufficient. If rather brittle failure occurs, like for the investigated C/C–SiC, fracture mechanical models might have to be applied. Furthermore, this modelling approach will be extended for the design of structural C/C–SiC-parts, as it allows a much better recovery of the material than ordinary strength criteria. The design of a simple bending beam with linear elastic properties and a tensile strength failure criterion would lead to an underestimation of the load carrying capacity by a factor of 1.7, at least. This corresponds to the bending tensile strength ratio in 0° direction. With the non-linear modelling and the fracture mechanical failure criterion, presented here, the ultimate load could be predicted with an average accuracy of 12%. Acknowledgements The financial support by the DLR program directorate for space research and technology is gratefully acknowledged. The authors would like to thank Bernhard Heidenreich, DLR Stuttgart, for enabling the work and Harald Kraft, DLR Stuttgart, for his support and sharing of mechanical test experience. References

6. Discussion The good agreement of experimental and numerical results shows that the bending strength of C/C–SiC can be modelled numerically if the main characteristics of the material (ultimate tensile strength, stiffness up to yield stress and beyond and the WOF) are considered. First of all the difference in stress–strain behaviour under tensile and compression load was considered. The linear compression behaviour leads to a support of the nonlinear tensile side of the sample. This effect leads to a shift of the neutral plane and a decrease in stress on the tensile side as well as an increase of stress on the compression side. In that way already an increase of theoretical bending strength relative to tensile strength can be explained. In order to explain the remaining

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