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Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling
Engineering Fracture Mechanics xxx (2017) xxx–xxx
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Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling S. Hofmann German Aerospace Centre, 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 January 2016 Received in revised form 1 May 2017 Accepted 15 May 2017 Available online xxxx Keywords: Cohesive zone modelling Ceramic matrix composite Carbon fibre reinforced silicon carbide Interlaminar fracture Finite element analysis
a b s t r a c t Delaminations are critical defects to the structural integrity of ceramic matrix composite (CMC) structures. They may be induced during processing due to differing shrinkage of fibres and matrix. The focus of this work is on the crack onset in order to determine conservative load limits for locally delaminated samples and components. Therefore, the critical energy release rate is determined for carbon fibre reinforced silicon carbide, C/C-SiC, at varying initial crack lengths. Double cantilever beam testing and a number of data reduction methods are applied on the fabric-reinforced C/C-SiC material. The resulting critical energy release rates are compared. The validation of the experimental energy release rate is done by cohesive zone modelling, using an average critical energy release rate in combination with a maximum stress criterion and a linear softening rule. A good agreement of numerical and experimental crack onset loads was obtained for varying initial crack lengths. Ó 2017 Published by Elsevier Ltd.
1. Introduction Various data reduction methods exist for the determination of critical energy release rate under mode I loading for fibre reinforced polymers [1–3]. The following work is going to apply the evaluation methods from ASTM [1] and Airbus [2] for a fibre reinforced ceramic material. The ISO [3] definitions are similar to the ASTM standard. Further evaluation methods may be found for fracture of adhesives [4,5]. Since the fracture toughness of adhesives under mode I loading is usually measured by double cantilever beam (DCB) testing, similar equations are applied. The recommendation for adhesives from British standard [5] is identical to ASTM [1] for testing of uni-directional composites, e.g.. The methods recommended by testing standards will be compared by the data reduction method proposed by Tamuzs [6] and Sorensen [7]. Tamuzs and Sorensen applied a simple formula for data reduction which does not require the measurement of crack length and is especially applicable for materials showing fibre bridging effects. In the field of cohesive zone modelling a lot of work is found for fracture of fibre reinforced polymers and fracture of adhesives. The origin of the cohesive zone modelling is going back to Dugdale [8] and Barenblatt [9], who described the debonding of material by the introduction of the cohesive law. The cohesive law determines the behaviour of the elements at the crack tip. In general the cohesive law considers an increase of traction with separation (initial stiffness) followed by the maximum stress (cohesive strength) and then followed by the softening up to complete traction free surfaces.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2017.05.018 0013-7944/Ó 2017 Published by Elsevier Ltd.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Nomenclature a E GIc h t F b u C D N L’ n A1 A ap
rmax
ss sc Kn dn I
crack length Young’s Modulus critical energy release rate under mode I height of sample vertical distance between load introduction and middle of one cantilever beam load width of sample total load point displacement compliance intersection with x-axis in C(1/3) over crack length plot compliance correction due to loading blocks Half-length of loading blocks slope of log (C) over log (a) plot slope of a/h over C^(1/3) integrated area under load –deflection curve propagation crack length maximum stress value in cohesive zone model, here: interlaminar tensile strength separation at maximum stress value separation at completion of debonding initial contact stiffness debonding or damage parameter geometrical moment of inertia of one cantilever beam
Abbreviations CMC Ceramic Matrix Composite DLR German Aerospace Centre C/C-SiC Carbon fibre reinforced Silicon carbide by LSI LSI Liquid Silicon Infiltration DCB Double Cantilever Beam LEFM Linear Elastic Fracture Mechanics ASTM American Society for Testing and Materials AITM Airbus Industry Test Methods FE Finite Element BT Beam Theory MBT Modified Beam Theory CC Compliance Calibration MCC Modified Compliance Calibration AM Area Method LBC Loading Block Corrected
The shapes of varying cohesive zone laws were investigated in detail e.g. by Alfano et al. [10] who compared the exponential model, the trapezoidal model and the bilinear model on its impact on the load-displacement curves for DCB-samples of adhesive joints. Alfano et al. [10] demonstrated for adhesive joints that the shape of the cohesive law is less decisive and that each approach may be fitted to the experimental load-deflection behaviour by fitting the cohesive zone parameters. That is GIc and cohesive strength. Alfano et al. [10,11] did also show that the numerical load-displacement curves show a strong non-linear behaviour before reaching maximum load, especially when the cohesive strength is relatively low compared to the GIC. This is one effect which will be discussed for the carbon fibre reinforced SiC material later on and is important when comparing with solutions from beam theory based linear-elastic fracture mechanics (LEFM). In contrast to adhesive joints, fibre reinforced composites are showing a tendency for fibre bridging during interlaminar fracture which makes the GIc geometry dependent [6,7]. The fibre bridging phenomena leads to an increase of GIc with crack propagation. In order to model the fibre bridging phenomena Tamuzs as well as Sorensen [6,7] introduced an extended cohesive zone law which is an extension of the bilinear model with a bridging zone. That means traction does get zero only for very high separations. Tamuzs as well as Sorensen [6,7] demonstrated that this bridging model enables modelling of load plateaus in the loaddisplacement curves of DCB samples as well as modelling of R-curve behaviour. Therewith the load-deflection-curves differ
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
S. Hofmann / Engineering Fracture Mechanics xxx (2017) xxx–xxx
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strongly from beam theory based LEFM propagation curves where crack extension is accompanied by distinct load decrease. Another effect which will also be discussed for carbon fibre reinforced SiC. Considering carbon fibre reinforced SiC materials (so called C/SiC or C/C-SiC), some literature is found on the crack propagation through the fibre reinforcement [12–15]. In the following, the interlaminar crack propagation (in between the laminate layers) will be investigated in detail for a carbon fibre reinforced SiC material from Liquid Silicon Infiltration (LSI) processing. Since carbon fibre reinforced carbon is the preliminary state of C/C-SiC produced by LSI, the work of Krause et al. [16] is mentioned here. Krause et al. [16] demonstrated the application of DCB standards on carbon fibre reinforced carbon. The load-displacement curves showed a distinct load drop after crack initiation. Further details on processing, microstructure and applications of C/C-SiC from LSI processing can be found at Heidenreich [17,18]. Due to its excellent thermal shock resistance and low density, typical applications of C/C-SiC are thermal protection systems for re-entry, components for rocket engines (nozzles, combustors, jet vanes) and automobile break discs. The present work is adapting DCB test standards for carbon fibre reinforced silicon carbide, C/C-SiC. In addition, the experimentally determined critical energy release rate is used for finite element modelling in order to prove if a reliable prediction of stiffness and crack onset load is possible for varying initial crack lengths. A maximum stress criterion with linear softening rule was applied for the cohesive zone elements within the commercial FE code ANSYS [19]. The experimental load-deflection curves of 0/90°-fabric reinforced C/C‐SiC material are additionally compared with beam theory based fracture mechanical solutions which were applied by Allix et al. and Szekrenyes [20,21] for fibre reinforced polymers. 2. Experimental set-up 2.1. Investigated material The investigated carbon fibre reinforced silicon carbide material, C/C–SiC, was produced from HTA carbon fibre fabrics of twill style 2/2 by LSI processing in house at the German Aerospace Centre. Since delaminations are usually formed during pyrolysis, filled with silicon during melt infiltration and then reopened during desiliconization [22], the following investigations are focused on desiliconized C/C-SiC where delaminations are most critical. The desiliconization step, that is removal of residual free silicon, is done for application temperatures above the melting point of silicon, e.g. in re-entry applications. A C/C-SiC plate of 300 300 5 mm3 was produced. During stacking of the 21 pre-preg layers, the weft and warp direction was switched after each layer to get similar mechanical properties in 0 and 90° direction. After curing in the autoclave, pyrolysis and silicon infiltration, the plate was grinded to a thickness of 4.9 mm to get even surfaces. Finally the desiliconization was conducted. The samples for DCB tests were taken by cut-off grinding. The microstructure of C/C–SiC is built up by a dense SiC-matrix, surrounding blocks of C/C-fibre bundles which contain carbon fibres and residual pyrolytic carbon, see Fig. 1a and b. The desiliconized C/C-SiC material shows almost linear-elastic behaviour in 0/90° tensile loading direction, see Fig. 2a. The following investigations are focusing on this loading direction. The evolution of hysteresis Modulus (secant slope between minimum and maximum stress/strain data points per unloading/reloading cycle) and pseudo-plastic strain for 0/90° tensile tests with unloading cycles are shown in Fig. 2b. A linear degradation of Modulus and a linear increase of plastic strain with reached stress level are observed. The degradation of stiffness is about 20% at 100 MPa of tensile stress. The plastic strain is 11% of the total strain at 100 MPa, which is rather close to tensile strength of about 120 MPa. Due to the almost linear-elastic behaviour under compression load in 0/90° direction, the non-linear effects are further reduced under bending load [12,22,23]. In-plane material non-linearities are neglected in the following computations. However, slight deviations from linearity should be kept in mind when experimental results are compared with computations from linear-elastic frac-
Fig. 1. SEM images of microstructures from C/C-SiC in siliconized (a) and desiliconized state - showing delamination from processing (b); the present work is focusing on the desiliconized state.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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120
Tensile stress / MPa
100 80 60 40 20 0 0,0
0,5
1,0
2,0
2,5
Strain / ‰
80
0,3
70
0,225
60
0,15
50
0,075
Plasc Strain / o/oo
Middle Hysteresis Modulus /GPa
(a)
1,5
Hysteresis Modulus Plasc Strain
40
0 0
(b)
20
40
60
80
100
120
Max. Stress reached / MPa
Fig. 2. Tensile-stress strain response from desiliconized C/C-SiC in 0/90° loading direction (a); evolution of hysteresis Modulus and plastic strain with tensile stresses for 0/90° loading (b).
ture mechanics and linear-elastic orthotropic finite element modelling. The C/C-SiC material is approximated linearelastically with a Young’s Modulus of 58 GPa in 0/90° orientation for the following finite element computations. 2.2. DCB test set-up Double cantilever beam tests were performed similar to the ASTM D 5528 – 94a standard [1] for the determination of mode I interlaminar fracture toughness of unidirectional fibre-reinforced polymer matrix composites. An artificial precrack was introduced by abrasive cutting with a diamond plate of 300 µm thickness. A sharp pre-crack was then introduced by opening the sawed crack front with a scalpel. Aluminum blocks with drilled holes were glued with two component adhesive X60 from supplier HBM on the outer surfaces of the two cantilevers. A steel pin connected the aluminum blocks on each side to the load introductions of the Zwick machine (see Fig. 3). The ASTM standard [1] is describing different methods to determine the onset load point for the computation of critical energy release rate under mode I loading, GIc: – Deviation from linearity in load-deflection curve – Visual observation of crack onset – 5%-offset or maximum load A visual resolution of the crack onset was not possible by the used GOM Aramis camera system, due to very fine interlaminar cracks. However the crack position during crack propagation could be estimated by strain evaluations with Aramis software, see Fig. 8. The DCB samples were showing non-linear behaviour from the very beginning, that is why the 5%-offset method was finally applied. In that case the intersection point of the load-deflection curve with a 5%-increased compliance curve was determined to calculate the GIc value. Additionally, the GIc value was determined by an area method following the Airbus Industry Test Method (AITM) [2] and by the data reduction method applied by Tamuzs et al. and Sorensen et al. [6,7]. The samples had a height of 4.9 mm, 10 mm of width and about 50 mm length. So, the test geometry deviated from the ASTM [1] standard geometries (length > 125 mm, width = 20–25 mm) in order to reduce the amount of needed CMC material. Nevertheless the geometrical guidelines of the ASTM standard were fulfilled [1]. Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
S. Hofmann / Engineering Fracture Mechanics xxx (2017) xxx–xxx
F
5
a aartificial asharp
h t
F
L’ Fig. 3. Test geometry for DCB testing.
Loading block corrections from ASTM standard were applied due to the initial crack length a being shorter than 50 mm [1]. The initial crack length was varied between 15 and 30 mm. It was not possible to produce longer cracks by cut-off grinding since the blade tended to drift away, resulting in non-straight notches. The only ASTM guide line which was not fulfilled is that a /h should be larger 10 when loading blocks are used. a /h was varying due to the varying initial crack length between 3 to 6. In order to further reduce loading block effects, typical loading blocks from CFRP-samples were glued with the short side of only 15 mm edge length on the CMC surface. In this way it was assumed that loading blocks would not influence the sample’s load-displacement behaviour – as will be proven later on by FE-modelling, see Section 5, Fig. 14. The test speed was set to 1 mm/min. The deformation by the test set-up and the Zwick machine itself were corrected by subtracting the displacement of two glued aluminum blocks at given loads from the displacement of the DCB samples. During testing the GOM Aramis System for optical strain measurement was used to take images at defined load steps in order to determine the current crack length. Therefore, all samples were painted with black and white speckle pattern before testing. All evaluation methods from ASTM standard were applied to calculate the GIc values for crack onset from 5%-offset loads: First the beam theory was used [1]:
GIc ¼
3Fu 2ba
ð1Þ
with F being the 5%-offset load, u the total load point displacement and b the width of the sample. As described in ASTM standard [1], this value is overestimating the GIc since the beam is usually not perfectly built in and so rotation may occur at the delamination front. This can be corrected by compliance calibration methods. The ASTM standard [1] is recommending the modified beam theory MBT because this method delivered the most conservative results for fibre reinforced polymers during round robin tests. The MBT is considering a slightly longer delamination a + D, where D may be determined by plotting the compliance C1/3 over the crack length a, see exemplary curves in Fig. 4. D is the intersection with the x-axis. The stiffening of the loading blocks had to be corrected, following ASTM standard [1], because the distance between load line and delamination front was less than 50 mm. This was the case in all samples, so the compliance in Fig. 4 was corrected by C/N. N was calculated by the equation from ASTM with the geometrical information from Fig. 3. The GIc was calculated following Modified Beam Theory (MBT) by Eq. (2). The GIc was finally multiplied by 1/N in order to correct the influence of loading blocks:
GIc ¼
3Fu 2bða þ jDjÞ
ð2Þ
Additionally, the Compliance Calibration (CC) and Modified Compliance Calibration methods (MCC), as described by ASTM standard [1], were applied for C/C-SiC. The loading block corrections were done as explained for MBT. The critical energy release rate by CC method is computed similarly as by Beam Theory, but the factor 3 in Eq. (2) is replaced by n. n is the slope of a log (C) over log (a) plot, respectively log (C/N), if loading block correction is done. The GIc was computed by MCC with the following equation:
GIc ¼
3F 2 C 2=3 2A1 bh
ð3Þ
A1 is the slope of the linear curve from a/h over C^1/3. Loading block correction (LBC) was performed as described for the MBT and CC. Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Compliance^1/3 / (mm/N)^1/3
0,4
y = 0,0112x + 0,0393 2 R = 0,954
0,3
y = 0,0116x + 0,023 0,2
2
R = 0,9653
MBT
0,1
MBT corrected 0 14
16
18
20
22
24
26
28
30
Crack length /mm Fig. 4. Determination of D (intersection with x-axis) for Modified Beam Theory (MBT) following ASTM standard with ((C/N)^1/3) and without loading block correction (C^1/3).
Since samples with varying initial crack length were investigated and crack propagation curves were recorded additionally, two different approaches could be considered for the determination of compliance calibration values |D|, n and A1 (necessary for MBT, CC and MCC with and without LBC): Calibration I: |D|, n and A1 were determined once from the initial compliances of all six samples with varying initial crack lengths. One calibration value was determined and applied for the evaluation of all samples. This approach is affected by specimen variability – the compliance calibration must be considered as average calibration of the initial stiffness of all samples. Calibration II: |D|, n and A1 were determined and applied for each crack propagation curve of the three fully delaminated samples separately (this is the recommendation from ASTM standard [1]). Further the Airbus Industry Testing Method [2] was applied to three samples (number 8, 9 and 12) which reached total delamination. Three other samples showed interlaminar crack onset and crack propagation but then the crack started to run upwards leading to bending failure of the upper beam. The Airbus standard is an area method. The determination of onset GIc and R-curves is not possible. The following equation was used:
GIc ¼
A bap
ð4Þ
A is the integrated area under the load-deflection curve, see Fig. 5 and ap is the propagation crack length (final crack length minus initial crack length).
60
3
4
2
50
5 6
Load / N
40
A 30
20
1
10
0 -0,5
0,0
0,5
1,0
1,5
2,0
2,5
Displacement / mm Fig. 5. Exemplary load-deflection curve (sample 9) of DCB test with integrated area (the exceeding work due to offset-corrections was not subtracted) defined as energy to achieve the total propagated crack length following AITM standard [2]; numbers 1–6 refer to Fig. 8a.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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The results from testing standards were finally compared with the data reduction method suggested by Tamuzs [6] and Sorensen [7]. The following Eq. (5) does not require a crack length and therefore eliminates errors by crack length measurement. In addition it was demonstrated that it works even for short crack length and for materials which show fibre bridging effects:
GIc ¼
2=3 F 2 2EIu bEI 2F
ð5Þ
E is the longitudinal Young’s Modulus (58 GPa) and I is the moment of inertia of one cantilever arm, see Eq. (8) below. 3. Finite element set-up The DCB test of desiliconized C/C-SiC was modelled as 2D plane strain state with linear-elastic properties. The elastic properties are summarized in Table 1. The z direction is perpendicular to the fabric plane. The interlaminar fracture of C/C-SiC under mode I loading was modelled with cohesive zone elements. The cohesive zone was introduced in the middle of the sample. The chosen cohesive zone model describes the material with a bi-linear contact stress-separation curve, see Fig. 6. The debonding starts at the maximum stress value rmax followed by linear softening up to the critical separation value sc. The area under the curve is equivalent to the critical fracture energy GIc. dn is the debonding or damage parameter. The debonding parameter for mode I is defined as [19]:
dn ¼
s s s s c s sc ss
ð6Þ
with ss the separation at maximum stress rmax. The initial contact stiffness Kn was defined as 106 MPa/mm which can be considered as ideal rigid contact. The two main input variables for the cohesive zone model are therefore the critical energy release rate as determined from DCB testing, see Section 4, and the cohesive strength rmax. Fig. 7 shows the 2D-FE-set-up with aluminum loading blocks for DCB modelling. The aluminum parts were modelled as isotropic linear-elastic with E = 70 GPa and m = 0.34. Further on the test was also modelled without loading blocks in order to quantify the stiffening effect of loading blocks; then the displacement was directly applied at the sample surface. Failure was modelled by cohesive zone elements, as described in Fig. 6. rmax was defined as 5 MPa, that is the interlaminar tensile strength from an DLR internal data base. The impact of deviating rmax values will be demonstrated in Section 5. The critical fracture energy of 0.16 N/mm was used, as will be determined in Section 4. A convergence study was performed to reach constant load-deflection response which is independent of substep number and mesh size. Linear rectangle elements with
Table 1 Input for linear elastic orthotropic material model in ANSYS; z is perpendicular to the fabric plane. Index
Modulus/GPa
m
xx yy zz xy yz xz
58 58 20 5.14 4 4
– – – 0.01 0.1 0.1
Stress σmax
dn=0 Critical fracture energy
Slope=Kn
Slope=K n (1-dn) dn=1 ss
s c Separation
Fig. 6. Cohesive zone model used with linear softening for interlaminar crack propagation [19].
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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S. Hofmann / Engineering Fracture Mechanics xxx (2017) xxx–xxx
Initial crack length Displacement Fixed vertex
Al
Al
Cohesivezone
Fig. 7. FE-model of DCB-test with interlaminar strain distribution under load.
0.5 mm edge length and 1000 substeps for the total displacement of about 1.5 mm were found to give reliable numerical results. 4. Experimental results The test results were evaluated in regard of stiffness, critical energy release rates and compared with crack propagation curves from beam theory based LEFM [20,21]. Fig. 8 a) shows the images and optical strain measurements of sample 9 at varying load points 1–6, see Fig. 5. The optical strain measurements are of qualitative interest to detect the current crack length. The quantitative values are not of significance. It is hardly possible to detect the crack length at the grey-scale images of Fig. 8. The strain measurements indicate the
Fig. 8. Grey-scale images and qualitative interlaminar tensile strain distribution at load steps 1–6, see Fig. 5, from sample 9 (the initial crack length is marked white in image 1); failure surfaces after crack propagation (b).
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
S. Hofmann / Engineering Fracture Mechanics xxx (2017) xxx–xxx
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crack position at the right end of the high strain areas (green). Fig. 8a and b show strong crack deflection and fibre bridging due to the waviness of the fabric. Overall six samples were investigated in DCB set-up. Fig. 9 shows the width-normalized load-deflection curves of four selected samples with varying initial crack lengths and the 5%-reduced stiffness curves for the determination of crack onset loads. The samples 8, 9 and 12 reached full delamination over the complete sample length (50 mm). The interlaminar crack started to run upwards leading to bending failure of the upper cantilever arm for all other samples. All load deflection curves show a first kink below 1 N/mm width-normalized load. The kink is assumed to be caused by the alignment and rotation of pins together with the aluminum loading blocks. First, the stiffness of the six DCB‐samples with varying initial crack lengths was evaluated. An effective Young’s Modulus was evaluated from the initial load‐deflection curves after the kink (due to set-up alignment) following beam theory, as cited by Allix et al. [20]:
E¼
dF 2a du 3I
ð7Þ
with a, initial crack length, and I, geometrical moment of inertia of one cantilever beam:
I¼b
3 h =12 2
ð8Þ
b is the width, h is the height of the sample. The weakening of the cantilever beams by 0.15 mm on each side due to grinding of the notch was neglected. The average effective Modulus was determined to be 24.5 ± 4.5 GPa. Since this evaluation does not consider crack root deflection and rotation, the ASTM standard for Modified Beam Theory was also applied: same equation as (7) but with (a + |D|)3. MBT gave 33.1 ± 5.3 GPa and 41.1 ± 6.4 GPa with LBC. |D| was determined by Calibration I, see Section 2.2. Fig. 10 shows the width‐normalized stiffness from experiment plus theoretical curves for effective Moduli of 24.5 GPa and 58 GPa. 58 GPa is the hysteresis Modulus of desiliconized C/C-SiC at about 90 MPa maximum stress level, compare Fig. 2 b, Section 2.1. The computed stiffness, Eq. (7), for a Young’s modulus of 24.5 GPa shows a good fit to the experimental results. The strong deviation in bending stiffness indicates limitations of beam theory based methods for the investigated CMCmaterial and sample geometries. Therefore all beam theory based results in Fig. 11 need to be considered carefully (only Area Method is not beam theory based). The FE-computations in Section 5 will reinforce the observed limitations of beam theory based methods. The effective bending stiffness of about 24.5 GPa will be successfully reproduced by a Young’s Modulus of 58 GPa when crack tip behaviour is taken into account by Cohesive Zone Elements. A variety of methods were applied to calculate the GIc values for crack onset load (5%-offset) and propagation load points: The equations from Beam Theory (BT), Modified Beam Theory (MBT), Compliance Calibration (CC) and Modified Compliance Calibration (MCC) with and without loading block corrections (LBC) as well as the Area Method (AM), see equations in section 2.2. Fig. 11 shows the GIc results for the different methods from test standards. Two approaches were considered for the determination of compliance calibration values |D|, n and A1 (necessary for MBT, CC and MCC with and without LBC): Calibration I and Calibration II, as explained in Section 2.2.
Fig. 9. Width-normalized load-deflection curves from selected DCB samples (sample number and initial crack length are indicated) with 95%-slopes for the determination of crack onset load points.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Normalized Stiffness / N/mm²
35 58 GPa
30
24,5 GPa Experiment
25 20 15 10 5 0 14
16
18
20
22
24
26
28
30
Crack length / mm Fig. 10. Width-normalized stiffness from experiment and beam theory computations for different Young’s Moduli at varying initial crack lengths.
Onset, all samples
Propagaon, fully delaminated only
Onset, fully delaminated only
0,30
Calibration I/II
GIc / N/mm
0,25 0,20 0,15 0,10 0,05 0,00 BT
MBT
MBT LBC
CC
CC LBC
MCC
MCC LBC
AM
Fig. 11. Averaged GIc values determined from onset and propagation load points; different calibration approaches are indicated. BT and AM are not compliance calibrated.
Overall Fig. 11 shows that the GIc results lie in a similar range when scatter bars are considered. However, three trends become rather clear: 1. Beam theory (BT) gave the highest GIc value in almost all cases (exception: the MCC LBC average onset value of all samples is similar). 2. The loading block corrections led to slightly (<5%) increased GIc values. 3. Calibration method II is reducing more significantly the average GIc values than calibration method I. After compliance calibration II the average GIC values are much lower than the original values from beam theory (compare BT with MBT, CC and MCC). The reason for the strong reduction of GIc with compliance calibration II may be found in load-deflection curves, see Fig. 9. An increase of GIc with crack propagation is indicated by the load plateaus after crack onset. Since the compliances are deviating from the ideal compliances if the fracture toughness is increasing during crack propagation, the compliance calibration leads to an overall reduction of GIc onset and propagation values. The calibration values |D| and A1 from propagation curves (calibration II) are much higher than from all onset compliances (calibration I). n is respectively lower for the propagation curves (calibration II) than from all onset compliances (calibration I). In that way the calibration values II lead to a strong reduction of original GIc from beam theory, see equations in Section 2.2. The evolution of fracture toughness with crack propagation was computed from MBT with calibration II in Fig. 12. The very low onset GIc values (about 0.12 N/mm) need to be mentioned here. The average GIc from MBT with calibration I was determined to be 0.163 N/mm at onset load; loading block corrected (LBC) MBT gave 0.169 N/mm. The area method (AM) from AITM standard gave a similar value of 0.167 ± 0.04 N/mm. The area method approach is simple; compliance calibration is not possible.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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0,3
Sample 9 Sample 8
GIc / N/mm
0,25
y = 0,0032x + 0,1451 R² = 0,6015
Sample 12 y = 0,0036x + 0,1313 R² = 0,8282
0,2
0,15 y = 0,0028x + 0,1287 R² = 0,6801
0,1 0
5
10
15
20
25
30
Crack extension / mm Fig. 12. Exemplary R-curves from samples reaching full delamination; the evaluation followed MBT with loading block corrections. D for compliance calibration was determined for each propagation curve separately (calibration II).
Calibration method II, as recommended by ASTM standard, gave strongly reduced onset GIc values compared to beam theory, see Fig. 11. It might be argued that only short initial crack length with little crack propagation and few data points were considered, see Fig. 12. That is why an evaluation method additional to test standards was applied. The evaluation method suggested by Tamuzs et al. and Sorensen et al., see Eq. (5), led to the results in Fig. 13. Eq. (5) does not require a crack length and therefore eliminates errors by crack length measurement. Since crack length is not included, it is less sensitive to limitations by beam theory - a compliance calibration is also not necessary. The computed crack length was added in Fig. 13(b) following the work by Brunner et al. [24] in order to show R-curve behaviour in relation with a continuously computed crack length. This crack length value can be used as orientation for the true crack length. However, it does overestimate the real initial and final crack length in the range of about 5 mm. As mentioned before, samples 8, 9 and 12 showed full delamination over the sample length. Sample 34 showed bending failure due to crack deflection in vertical direction. The GIc plateau is therefore not as clear as for the other samples in Fig. 13 (a). Fig. 13(a) and (b) shows that the GIC plateau values are still scattering for the fully delaminated samples. However, the first kink of the GIC-curves is rather constant at average of 0.16 N/mm. Overall the GIC values from Eq. (5) are significantly higher than from MBT with calibration II, see Fig. 12. Alfano et al. made a similar observation when comparing data reduction methods for adhesive joints [11]. Since MBT with calibration I and Eq. (5) gave a similar onset GIc value of about 0.16 N/mm, this value was used for further FEA and beam theory based LEFM computations 5. Numerical results All FE simulations were computed with a linear-elastic material model using a Young’s Modulus of 58 GPa in 0/90° direction. The FE simulations gave a bending stress of about 93 MPa for the initiation loads at all crack lengths in DCB test set-up. The average tensile strength of the investigated material is about 90–120 MPa; the bending strength is 140–190 MPa. Explanations for the difference in tensile and bending strength of C/C-SiC material may be found in [22,23]. The results from 2D-plane strain FEA, experimental results and beam theory based LEFM are compared. Fig. 14 a shows the load–displacement curves from 2D-FEA with and without aluminum loading blocks for sample 9 with a relative short initial crack length of 15.6 mm. The load–displacement curves are similar. It does obviously not affect the results if the load is introduced via loading blocks or directly on the sample surface. Therefore, the subsequent simulations were performed without loading blocks. Fig. 14 b shows the impact of varying critical stress values, rmax, on the loaddisplacement behaviour at constant elastic properties and a constant critical energy release rate of 0.16 N/mm. The critical stress level has only a slight impact on the maximum load. However, the non-linearity before load maximum does significantly increase with decreasing rmax. This effect is also found in publications by Alfano et al. [10,11]. All curves in Fig. 14 b end up following the crack propagation curve from LEFM with E = 58 GPa and GIc = 0.16 N/mm. The crack propagation curves were computed with the equation from LEFM [20]:
u¼
ðbGIc Þ
pffiffiffi 3=2 EI
3F 2
ð9Þ
Fig. 15 shows the width-normalized load-displacement curves for varying initial crack lengths from experiment and FEA as well as the crack propagation curves from beam theory based LEFM for two different Young’s Moduli. The stiffness from FEA and DCB test correspond rather well, except for sample 8 with the largest initial crack length. In this case the FEA is overPlease cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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0,35 Sample 9 Sample 12 Sample 8 Sample 34
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0,3 0,25 0,2 0,15 0,1 0,05 0 0,0
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Sample 12 Sample 8
0,25
Sample 34
0,2 0,15 0,1 0,05 0 20
25
30
35
40
45
50
Computed crack length / mm
(b)
Fig. 13. GIc evolution over displacement (a) and over computed crack length (b) following Brunner et al. [24]; Sample 34 did not show full delamination over sample length but crack deflection caused bending failure. The final GIc drop is therefore differing to Sample 8–12.
6 FEA 8 MPa FEA 5 MPa FEA 3 MPa Experiment Sample 34 LEFM
Normalized Load / N/mm
Sample 9 w/o Blocks Sample 9 w Blocks
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Normalized Load / N/mm
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Displacement / mm
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Displacement / mm
Fig. 14. Comparison of FEA with and without loading blocks (a) for a = 15.6 mm; comparison of FE-results with varying critical stress values for a = 17.8 mm (GIc = 0.16 N/mm and E = 58 GPa were kept constant in FEA and beam theory based LEFM solution) (b).
estimating the stiffness. The explanation is coming from sample preparation: the relatively long grinded pre-crack led to irregular thickness of the cantilever beams. Due to the cubic correlation with thickness, see Eq. (8), overall a softer behaviour is observed in experiment. The beam theory based LEFM curve for the average effective Modulus of 24.5 GPa corresponds well with the experimental load points of crack onset.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Fig. 15. Load-displacement curves from DCB-test, FEA and propagation curves from LEFM for two Young’s Moduli; sample numbers and initial crack lengths are indicated.
The LEFM crack propagation curve for a standard Young’s Modulus of 58 GPa and a GIC of 0.16 N/mm agrees perfectly with the propagation curve from FEA, see Fig. 15. It has to be mentioned here that the onset load value would be strongly overestimated by using beam theory based LEFM with an effective Modulus of 58 GPa. The critical load point, which is the intersection of crack propagation curve, see Eq. (9), and the stiffness from beam theory, see Eq. (7), are shown for varying Young’s Moduli in Fig. 16. The average Young’s Modulus of 24.5 GPa evaluated from the stiffness of the experimental load–displacement curves, see Fig. 16, results in a much better prediction of the critical load point. The initial load-displacement behaviour in Fig. 16 reveals the strong non-linearity from FEA and experiment. The strong non-linearity explains the low effective Young’s Modulus evaluated from DCB test, i.e. 24.5 GPa in average, compare Section 4, Fig. 10. The Young’s Modulus was evaluated after the initial kinking of the load-displacement curve due to the alignment of the test set-up. It has to be assumed that the initial Young’s Modulus was higher, similar to the load-displacement from FEA. Due to the strong non-linearity, the lower Young’s Modulus of 24.5 GPa is more useful to estimate the critical load point by beam theory based LEFM than the initial Young’s Modulus. In order to quantify the non-linearity of the load-displacement curves, the secant Modulus for sample 9 with a = 15.6 mm was evaluated from the FE load-displacement curve: the initial Modulus of 30.87 GPa is decreasing to 18.11 GPa, which is the secant Modulus at maximum load. Since the applied material model itself is linear-elastic, the non-linearity has to be caused by the crack tip, i.e. cohesive zone behaviour. The experimental load-displacement curve shows similar non-linear effects. Similar non-linear effects were also documented by Alfano et al. [10,11] for DCB samples of adhesive joints.
8
LEFM:58 GPa
Normalized Load / N/mm
7 6
Exp.
5
24.5 GPa
FEA
4 3 2 1 0 0,0
0,2
0,4
0,6
0,8
Displacement / mm Fig. 16. Comparison of beam theory based LEFM, FEA and experimental load displacement curves for sample 9 with a = 15.6 mm; different Young’s Moduli were used for the computations by beam theory based LEFM.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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The experimental curves in Figs. 15 and 16 show less decrease of load during crack propagation in contrast to beam theory based LEFM and FEA. Therewith, the comparison of beam theory based LEFM, FEA and experimental crack propagation curves confirmed the results from Section 4, Figs. 12 and 13, showing a strong increase of GIc with crack propagation for C/C-SiC. Of course the simple bi-linear cohesive law within ANSYS could not reproduce the load-displacement plateau caused by the R-curve behaviour of C/C-SiC. Nevertheless, the linear-elastic FE-Analysis, using a constant Young’s Modulus of 58 GPa, a GIc value of 0.16 N/mm with a maximum stress value, rmax, of 5 MPa was predicting the non-linear load-displacement behaviour and estimating the critical load points for varying initial crack lengths successfully. 6. Summary and conclusions The comparison of experimental load-displacement data with solutions from beam theory based LEFM indicated that the strong non-linearity of the load-displacement curves creates the necessity of introducing an effective stiffness. The strong non-linearity of load-displacement curves from DCB-testing was shown earlier by Alfano et al. [10,11]. The effective stiffness can be computed by introducing an effective Modulus, which is significantly lower than the original material Modulus. The effective Modulus allows the computation of critical load points for varying initial crack lengths by LEFM. The evaluation of GIc data reduction methods showed that usual reduction schemes from fibre reinforced polymers have to be considered carefully for CMC materials. Only short straight pre-cracks could be machined in the C/C-SiC-material. The current crack length was difficult to detect. Finally fibre bridging, crack deflection and R-curve behaviour were observed. All those circumstances made the data reduction method by Tamuzs et al. and Sorensen et al. [6,7], which does not require a crack length measurement, much more applicable to C/C-SiC than compliance calibration methods from test standards for fibre reinforced polymers. An alternative approach for compliance calibration, based on varying initial crack length, was presented. Modified beam theory and this calibration approach gave an onset GIC value of about 0.16 N/mm. This result is comparable to the data reduction method presented by Tamuzs and Sorensen [6,7]. However, the approach by Tamuzs and Sorensen is for sure the more convenient way of evaluating GIC values from load-deflection data. The cohesive zone modelling approach with a simple bi-linear traction separation law proved that the non-linear crack onset can be modelled for varying initial crack lengths. This model can now be used for structural design up to crack onset with respective safety factors. As demonstrated, full crack propagation modelling for C/C-SiC would require models which take R-curve behaviour into account see [6,7,25] for fibre reinforced polymers. Interlaminar crack propagation under mode II loading and further CMC materials (SiC/SiC and oxide/oxide composites) will be investigated in the future work. Acknowledgements The financial support by the DLR program directorate for space travel is gratefully acknowledged (grant number 2479101). The author would like to thank Dietmar Koch and the reviewers for their detailed scientific feedback and Harald Kraft for his support and sharing of mechanical test experience. The author thanks Prof. Heinz Voggenreiter from DLR Stuttgart and Prof. Siegfried Schmauder from University Stuttgart, Institute for Materials Testing, Material Science and Strength of Materials for their persistent scientific support. Since the original version of this paper was written during the recovery from a knee surgery, the author would like to deeply thank Irmi and Franz Hofmann for their support during this time. References [1] ASTM Standard D 5528-94a. Standard test method for Mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites; 1994. [2] AITM. Airbus industry test method, 1.0006, Issue 2, Carbon fiber reinforced plastics, Determination of interlaminar fracture; 1994. p. 1–8. [3] ISO Standard 15024. Fibre-reinforced plastic composites – Determination of mode I interlaminar fracture toughness, GIc, for unidirectionally reinforced materials; 2001. [4] ASTM Standard D3433-05. Standard Test Method for Fracture Strength in Cleavage of Adhesives in Bonded Metal Joints, Annual Book of ASTM Standards, Vol. 15.06, ASTM, International, West Conshohocken, PA; 2005. [5] BS7991. Determination of the Mode I Adhesive Fracture Energy, GIC, of Structural Adhesives Using the Double Cantilever Beam (DCB) and Tapered Double Cantilever Beam (TDCB) Specimens,” British Standard Institution, London, United Kingdom; 2001. [6] Tamuzs V, Tarasovs S, Vilks U. Progressive delamination and fiber bridging modelling in double cantilever beam composite specimen. Eng Fract Mech 2001;68:513–25. [7] Sorensen L, Botsis J, Gmür Th, Humbert L. Bridging tractions in mode I delamination: measurements and simulations. Compos Sci Technol 2008;68:2350–8. [8] Dugdale D. Yielding of steel sheets containing slits. J Mech Phys Solids 1960;8. [9] Barenblatt GI. Mathematical theory of equilibrium cracks in brittle fracture. Adv Appl Mech, Academic Press 1962;7. [10] Alfano M, Furgiuele F, Leonardi A, Maletta C, Paulino GH. Mode I fracture of adhesive joints using tailored cohesive zone models. Int J Fract 2009;157:193–204. [11] Alfano M, Furgiuele F, Pagnotta L, Paulino GH. Analysis of fracture in aluminum joints bonded with a bi-component epoxy adhesive. J Test Eval 39(2), Paper ID JTE102753.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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[12] Hofmann S, Oeztürk B, Koch D, Voggenreiter H. Experimental and numerical evaluation of bending and tensile behaviour of carbon fibre reinforced SiC. Composites Part A 2012; 43(11). [13] Mao WG, Chen J, Si MS, Zhang RF, Peng ZB, Dai CY, et al. Study of mechanical properties and cracking extension resistance behavior of C/SiC composites by single edge notched beam and digital image correlation techniques. Mater Sci Engng, A 2016;649:222–8. [14] Kim S, Han IS, Seong Y-H, Kim DK. Mechanical properties of C-SiC composite materials fabricated by the Si–Cr alloy melt-infiltration method. J Compos Mater 2015;49(24):3057–66. [15] Liu Y, Cheng L, Zhang L, Yang W, Zhou S, Zhang W. Fracture behavior and mechanism of 2D C/SiC–BCx composite at room temperature. Mater Sci Engng, A 2011;528:1436–41. [16] Krause T, Tushtev K, Koch D, Grathwohl G. Interlaminar Mode I crack growth energy release rate of carbon/carbon composites. Engng Fract Mech 2013;100:38–51. [17] B. Heidenreich, Manufacturing of CMC by Liquid Silicon Infiltration (LSI), Wiley-VCH, Ed. Krenkel, Ceram Compos Mater 2002; 48-75. ISBN 3-52730529-7. [18] Heidenreich B. ‘‘C/SiC and C/C-SiC Composites”. In: Bansal NP, Lamon J, editors. Ceramic Matrix Composites: Materials, Modeling and Technology, Wiley; 2015. [19] ANSYS Inc., Release 12.1. Documentation for ANSYS; 2009. [20] Allix O, Ladeveze P, Corigliano A. Damage analysis of interlaminar fracture specimens. Compos Struct 1995;31:61–74. [21] Szekrényes A. Overview on the experimental investigations of the fracture toughness in composite materials, HEJ - Mech Eng Transp Sect; 2002.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling S. Hofmann German Aerospace Centre, 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 January 2016 Received in revised form 1 May 2017 Accepted 15 May 2017 Available online xxxx Keywords: Cohesive zone modelling Ceramic matrix composite Carbon fibre reinforced silicon carbide Interlaminar fracture Finite element analysis
a b s t r a c t Delaminations are critical defects to the structural integrity of ceramic matrix composite (CMC) structures. They may be induced during processing due to differing shrinkage of fibres and matrix. The focus of this work is on the crack onset in order to determine conservative load limits for locally delaminated samples and components. Therefore, the critical energy release rate is determined for carbon fibre reinforced silicon carbide, C/C-SiC, at varying initial crack lengths. Double cantilever beam testing and a number of data reduction methods are applied on the fabric-reinforced C/C-SiC material. The resulting critical energy release rates are compared. The validation of the experimental energy release rate is done by cohesive zone modelling, using an average critical energy release rate in combination with a maximum stress criterion and a linear softening rule. A good agreement of numerical and experimental crack onset loads was obtained for varying initial crack lengths. Ó 2017 Published by Elsevier Ltd.
1. Introduction Various data reduction methods exist for the determination of critical energy release rate under mode I loading for fibre reinforced polymers [1–3]. The following work is going to apply the evaluation methods from ASTM [1] and Airbus [2] for a fibre reinforced ceramic material. The ISO [3] definitions are similar to the ASTM standard. Further evaluation methods may be found for fracture of adhesives [4,5]. Since the fracture toughness of adhesives under mode I loading is usually measured by double cantilever beam (DCB) testing, similar equations are applied. The recommendation for adhesives from British standard [5] is identical to ASTM [1] for testing of uni-directional composites, e.g.. The methods recommended by testing standards will be compared by the data reduction method proposed by Tamuzs [6] and Sorensen [7]. Tamuzs and Sorensen applied a simple formula for data reduction which does not require the measurement of crack length and is especially applicable for materials showing fibre bridging effects. In the field of cohesive zone modelling a lot of work is found for fracture of fibre reinforced polymers and fracture of adhesives. The origin of the cohesive zone modelling is going back to Dugdale [8] and Barenblatt [9], who described the debonding of material by the introduction of the cohesive law. The cohesive law determines the behaviour of the elements at the crack tip. In general the cohesive law considers an increase of traction with separation (initial stiffness) followed by the maximum stress (cohesive strength) and then followed by the softening up to complete traction free surfaces.
E-mail address: [email protected] http://dx.doi.org/10.1016/j.engfracmech.2017.05.018 0013-7944/Ó 2017 Published by Elsevier Ltd.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Nomenclature a E GIc h t F b u C D N L’ n A1 A ap
rmax
ss sc Kn dn I
crack length Young’s Modulus critical energy release rate under mode I height of sample vertical distance between load introduction and middle of one cantilever beam load width of sample total load point displacement compliance intersection with x-axis in C(1/3) over crack length plot compliance correction due to loading blocks Half-length of loading blocks slope of log (C) over log (a) plot slope of a/h over C^(1/3) integrated area under load –deflection curve propagation crack length maximum stress value in cohesive zone model, here: interlaminar tensile strength separation at maximum stress value separation at completion of debonding initial contact stiffness debonding or damage parameter geometrical moment of inertia of one cantilever beam
Abbreviations CMC Ceramic Matrix Composite DLR German Aerospace Centre C/C-SiC Carbon fibre reinforced Silicon carbide by LSI LSI Liquid Silicon Infiltration DCB Double Cantilever Beam LEFM Linear Elastic Fracture Mechanics ASTM American Society for Testing and Materials AITM Airbus Industry Test Methods FE Finite Element BT Beam Theory MBT Modified Beam Theory CC Compliance Calibration MCC Modified Compliance Calibration AM Area Method LBC Loading Block Corrected
The shapes of varying cohesive zone laws were investigated in detail e.g. by Alfano et al. [10] who compared the exponential model, the trapezoidal model and the bilinear model on its impact on the load-displacement curves for DCB-samples of adhesive joints. Alfano et al. [10] demonstrated for adhesive joints that the shape of the cohesive law is less decisive and that each approach may be fitted to the experimental load-deflection behaviour by fitting the cohesive zone parameters. That is GIc and cohesive strength. Alfano et al. [10,11] did also show that the numerical load-displacement curves show a strong non-linear behaviour before reaching maximum load, especially when the cohesive strength is relatively low compared to the GIC. This is one effect which will be discussed for the carbon fibre reinforced SiC material later on and is important when comparing with solutions from beam theory based linear-elastic fracture mechanics (LEFM). In contrast to adhesive joints, fibre reinforced composites are showing a tendency for fibre bridging during interlaminar fracture which makes the GIc geometry dependent [6,7]. The fibre bridging phenomena leads to an increase of GIc with crack propagation. In order to model the fibre bridging phenomena Tamuzs as well as Sorensen [6,7] introduced an extended cohesive zone law which is an extension of the bilinear model with a bridging zone. That means traction does get zero only for very high separations. Tamuzs as well as Sorensen [6,7] demonstrated that this bridging model enables modelling of load plateaus in the loaddisplacement curves of DCB samples as well as modelling of R-curve behaviour. Therewith the load-deflection-curves differ
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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strongly from beam theory based LEFM propagation curves where crack extension is accompanied by distinct load decrease. Another effect which will also be discussed for carbon fibre reinforced SiC. Considering carbon fibre reinforced SiC materials (so called C/SiC or C/C-SiC), some literature is found on the crack propagation through the fibre reinforcement [12–15]. In the following, the interlaminar crack propagation (in between the laminate layers) will be investigated in detail for a carbon fibre reinforced SiC material from Liquid Silicon Infiltration (LSI) processing. Since carbon fibre reinforced carbon is the preliminary state of C/C-SiC produced by LSI, the work of Krause et al. [16] is mentioned here. Krause et al. [16] demonstrated the application of DCB standards on carbon fibre reinforced carbon. The load-displacement curves showed a distinct load drop after crack initiation. Further details on processing, microstructure and applications of C/C-SiC from LSI processing can be found at Heidenreich [17,18]. Due to its excellent thermal shock resistance and low density, typical applications of C/C-SiC are thermal protection systems for re-entry, components for rocket engines (nozzles, combustors, jet vanes) and automobile break discs. The present work is adapting DCB test standards for carbon fibre reinforced silicon carbide, C/C-SiC. In addition, the experimentally determined critical energy release rate is used for finite element modelling in order to prove if a reliable prediction of stiffness and crack onset load is possible for varying initial crack lengths. A maximum stress criterion with linear softening rule was applied for the cohesive zone elements within the commercial FE code ANSYS [19]. The experimental load-deflection curves of 0/90°-fabric reinforced C/C‐SiC material are additionally compared with beam theory based fracture mechanical solutions which were applied by Allix et al. and Szekrenyes [20,21] for fibre reinforced polymers. 2. Experimental set-up 2.1. Investigated material The investigated carbon fibre reinforced silicon carbide material, C/C–SiC, was produced from HTA carbon fibre fabrics of twill style 2/2 by LSI processing in house at the German Aerospace Centre. Since delaminations are usually formed during pyrolysis, filled with silicon during melt infiltration and then reopened during desiliconization [22], the following investigations are focused on desiliconized C/C-SiC where delaminations are most critical. The desiliconization step, that is removal of residual free silicon, is done for application temperatures above the melting point of silicon, e.g. in re-entry applications. A C/C-SiC plate of 300 300 5 mm3 was produced. During stacking of the 21 pre-preg layers, the weft and warp direction was switched after each layer to get similar mechanical properties in 0 and 90° direction. After curing in the autoclave, pyrolysis and silicon infiltration, the plate was grinded to a thickness of 4.9 mm to get even surfaces. Finally the desiliconization was conducted. The samples for DCB tests were taken by cut-off grinding. The microstructure of C/C–SiC is built up by a dense SiC-matrix, surrounding blocks of C/C-fibre bundles which contain carbon fibres and residual pyrolytic carbon, see Fig. 1a and b. The desiliconized C/C-SiC material shows almost linear-elastic behaviour in 0/90° tensile loading direction, see Fig. 2a. The following investigations are focusing on this loading direction. The evolution of hysteresis Modulus (secant slope between minimum and maximum stress/strain data points per unloading/reloading cycle) and pseudo-plastic strain for 0/90° tensile tests with unloading cycles are shown in Fig. 2b. A linear degradation of Modulus and a linear increase of plastic strain with reached stress level are observed. The degradation of stiffness is about 20% at 100 MPa of tensile stress. The plastic strain is 11% of the total strain at 100 MPa, which is rather close to tensile strength of about 120 MPa. Due to the almost linear-elastic behaviour under compression load in 0/90° direction, the non-linear effects are further reduced under bending load [12,22,23]. In-plane material non-linearities are neglected in the following computations. However, slight deviations from linearity should be kept in mind when experimental results are compared with computations from linear-elastic frac-
Fig. 1. SEM images of microstructures from C/C-SiC in siliconized (a) and desiliconized state - showing delamination from processing (b); the present work is focusing on the desiliconized state.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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120
Tensile stress / MPa
100 80 60 40 20 0 0,0
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Fig. 2. Tensile-stress strain response from desiliconized C/C-SiC in 0/90° loading direction (a); evolution of hysteresis Modulus and plastic strain with tensile stresses for 0/90° loading (b).
ture mechanics and linear-elastic orthotropic finite element modelling. The C/C-SiC material is approximated linearelastically with a Young’s Modulus of 58 GPa in 0/90° orientation for the following finite element computations. 2.2. DCB test set-up Double cantilever beam tests were performed similar to the ASTM D 5528 – 94a standard [1] for the determination of mode I interlaminar fracture toughness of unidirectional fibre-reinforced polymer matrix composites. An artificial precrack was introduced by abrasive cutting with a diamond plate of 300 µm thickness. A sharp pre-crack was then introduced by opening the sawed crack front with a scalpel. Aluminum blocks with drilled holes were glued with two component adhesive X60 from supplier HBM on the outer surfaces of the two cantilevers. A steel pin connected the aluminum blocks on each side to the load introductions of the Zwick machine (see Fig. 3). The ASTM standard [1] is describing different methods to determine the onset load point for the computation of critical energy release rate under mode I loading, GIc: – Deviation from linearity in load-deflection curve – Visual observation of crack onset – 5%-offset or maximum load A visual resolution of the crack onset was not possible by the used GOM Aramis camera system, due to very fine interlaminar cracks. However the crack position during crack propagation could be estimated by strain evaluations with Aramis software, see Fig. 8. The DCB samples were showing non-linear behaviour from the very beginning, that is why the 5%-offset method was finally applied. In that case the intersection point of the load-deflection curve with a 5%-increased compliance curve was determined to calculate the GIc value. Additionally, the GIc value was determined by an area method following the Airbus Industry Test Method (AITM) [2] and by the data reduction method applied by Tamuzs et al. and Sorensen et al. [6,7]. The samples had a height of 4.9 mm, 10 mm of width and about 50 mm length. So, the test geometry deviated from the ASTM [1] standard geometries (length > 125 mm, width = 20–25 mm) in order to reduce the amount of needed CMC material. Nevertheless the geometrical guidelines of the ASTM standard were fulfilled [1]. Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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F
5
a aartificial asharp
h t
F
L’ Fig. 3. Test geometry for DCB testing.
Loading block corrections from ASTM standard were applied due to the initial crack length a being shorter than 50 mm [1]. The initial crack length was varied between 15 and 30 mm. It was not possible to produce longer cracks by cut-off grinding since the blade tended to drift away, resulting in non-straight notches. The only ASTM guide line which was not fulfilled is that a /h should be larger 10 when loading blocks are used. a /h was varying due to the varying initial crack length between 3 to 6. In order to further reduce loading block effects, typical loading blocks from CFRP-samples were glued with the short side of only 15 mm edge length on the CMC surface. In this way it was assumed that loading blocks would not influence the sample’s load-displacement behaviour – as will be proven later on by FE-modelling, see Section 5, Fig. 14. The test speed was set to 1 mm/min. The deformation by the test set-up and the Zwick machine itself were corrected by subtracting the displacement of two glued aluminum blocks at given loads from the displacement of the DCB samples. During testing the GOM Aramis System for optical strain measurement was used to take images at defined load steps in order to determine the current crack length. Therefore, all samples were painted with black and white speckle pattern before testing. All evaluation methods from ASTM standard were applied to calculate the GIc values for crack onset from 5%-offset loads: First the beam theory was used [1]:
GIc ¼
3Fu 2ba
ð1Þ
with F being the 5%-offset load, u the total load point displacement and b the width of the sample. As described in ASTM standard [1], this value is overestimating the GIc since the beam is usually not perfectly built in and so rotation may occur at the delamination front. This can be corrected by compliance calibration methods. The ASTM standard [1] is recommending the modified beam theory MBT because this method delivered the most conservative results for fibre reinforced polymers during round robin tests. The MBT is considering a slightly longer delamination a + D, where D may be determined by plotting the compliance C1/3 over the crack length a, see exemplary curves in Fig. 4. D is the intersection with the x-axis. The stiffening of the loading blocks had to be corrected, following ASTM standard [1], because the distance between load line and delamination front was less than 50 mm. This was the case in all samples, so the compliance in Fig. 4 was corrected by C/N. N was calculated by the equation from ASTM with the geometrical information from Fig. 3. The GIc was calculated following Modified Beam Theory (MBT) by Eq. (2). The GIc was finally multiplied by 1/N in order to correct the influence of loading blocks:
GIc ¼
3Fu 2bða þ jDjÞ
ð2Þ
Additionally, the Compliance Calibration (CC) and Modified Compliance Calibration methods (MCC), as described by ASTM standard [1], were applied for C/C-SiC. The loading block corrections were done as explained for MBT. The critical energy release rate by CC method is computed similarly as by Beam Theory, but the factor 3 in Eq. (2) is replaced by n. n is the slope of a log (C) over log (a) plot, respectively log (C/N), if loading block correction is done. The GIc was computed by MCC with the following equation:
GIc ¼
3F 2 C 2=3 2A1 bh
ð3Þ
A1 is the slope of the linear curve from a/h over C^1/3. Loading block correction (LBC) was performed as described for the MBT and CC. Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Compliance^1/3 / (mm/N)^1/3
0,4
y = 0,0112x + 0,0393 2 R = 0,954
0,3
y = 0,0116x + 0,023 0,2
2
R = 0,9653
MBT
0,1
MBT corrected 0 14
16
18
20
22
24
26
28
30
Crack length /mm Fig. 4. Determination of D (intersection with x-axis) for Modified Beam Theory (MBT) following ASTM standard with ((C/N)^1/3) and without loading block correction (C^1/3).
Since samples with varying initial crack length were investigated and crack propagation curves were recorded additionally, two different approaches could be considered for the determination of compliance calibration values |D|, n and A1 (necessary for MBT, CC and MCC with and without LBC): Calibration I: |D|, n and A1 were determined once from the initial compliances of all six samples with varying initial crack lengths. One calibration value was determined and applied for the evaluation of all samples. This approach is affected by specimen variability – the compliance calibration must be considered as average calibration of the initial stiffness of all samples. Calibration II: |D|, n and A1 were determined and applied for each crack propagation curve of the three fully delaminated samples separately (this is the recommendation from ASTM standard [1]). Further the Airbus Industry Testing Method [2] was applied to three samples (number 8, 9 and 12) which reached total delamination. Three other samples showed interlaminar crack onset and crack propagation but then the crack started to run upwards leading to bending failure of the upper beam. The Airbus standard is an area method. The determination of onset GIc and R-curves is not possible. The following equation was used:
GIc ¼
A bap
ð4Þ
A is the integrated area under the load-deflection curve, see Fig. 5 and ap is the propagation crack length (final crack length minus initial crack length).
60
3
4
2
50
5 6
Load / N
40
A 30
20
1
10
0 -0,5
0,0
0,5
1,0
1,5
2,0
2,5
Displacement / mm Fig. 5. Exemplary load-deflection curve (sample 9) of DCB test with integrated area (the exceeding work due to offset-corrections was not subtracted) defined as energy to achieve the total propagated crack length following AITM standard [2]; numbers 1–6 refer to Fig. 8a.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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The results from testing standards were finally compared with the data reduction method suggested by Tamuzs [6] and Sorensen [7]. The following Eq. (5) does not require a crack length and therefore eliminates errors by crack length measurement. In addition it was demonstrated that it works even for short crack length and for materials which show fibre bridging effects:
GIc ¼
2=3 F 2 2EIu bEI 2F
ð5Þ
E is the longitudinal Young’s Modulus (58 GPa) and I is the moment of inertia of one cantilever arm, see Eq. (8) below. 3. Finite element set-up The DCB test of desiliconized C/C-SiC was modelled as 2D plane strain state with linear-elastic properties. The elastic properties are summarized in Table 1. The z direction is perpendicular to the fabric plane. The interlaminar fracture of C/C-SiC under mode I loading was modelled with cohesive zone elements. The cohesive zone was introduced in the middle of the sample. The chosen cohesive zone model describes the material with a bi-linear contact stress-separation curve, see Fig. 6. The debonding starts at the maximum stress value rmax followed by linear softening up to the critical separation value sc. The area under the curve is equivalent to the critical fracture energy GIc. dn is the debonding or damage parameter. The debonding parameter for mode I is defined as [19]:
dn ¼
s s s s c s sc ss
ð6Þ
with ss the separation at maximum stress rmax. The initial contact stiffness Kn was defined as 106 MPa/mm which can be considered as ideal rigid contact. The two main input variables for the cohesive zone model are therefore the critical energy release rate as determined from DCB testing, see Section 4, and the cohesive strength rmax. Fig. 7 shows the 2D-FE-set-up with aluminum loading blocks for DCB modelling. The aluminum parts were modelled as isotropic linear-elastic with E = 70 GPa and m = 0.34. Further on the test was also modelled without loading blocks in order to quantify the stiffening effect of loading blocks; then the displacement was directly applied at the sample surface. Failure was modelled by cohesive zone elements, as described in Fig. 6. rmax was defined as 5 MPa, that is the interlaminar tensile strength from an DLR internal data base. The impact of deviating rmax values will be demonstrated in Section 5. The critical fracture energy of 0.16 N/mm was used, as will be determined in Section 4. A convergence study was performed to reach constant load-deflection response which is independent of substep number and mesh size. Linear rectangle elements with
Table 1 Input for linear elastic orthotropic material model in ANSYS; z is perpendicular to the fabric plane. Index
Modulus/GPa
m
xx yy zz xy yz xz
58 58 20 5.14 4 4
– – – 0.01 0.1 0.1
Stress σmax
dn=0 Critical fracture energy
Slope=Kn
Slope=K n (1-dn) dn=1 ss
s c Separation
Fig. 6. Cohesive zone model used with linear softening for interlaminar crack propagation [19].
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Initial crack length Displacement Fixed vertex
Al
Al
Cohesivezone
Fig. 7. FE-model of DCB-test with interlaminar strain distribution under load.
0.5 mm edge length and 1000 substeps for the total displacement of about 1.5 mm were found to give reliable numerical results. 4. Experimental results The test results were evaluated in regard of stiffness, critical energy release rates and compared with crack propagation curves from beam theory based LEFM [20,21]. Fig. 8 a) shows the images and optical strain measurements of sample 9 at varying load points 1–6, see Fig. 5. The optical strain measurements are of qualitative interest to detect the current crack length. The quantitative values are not of significance. It is hardly possible to detect the crack length at the grey-scale images of Fig. 8. The strain measurements indicate the
Fig. 8. Grey-scale images and qualitative interlaminar tensile strain distribution at load steps 1–6, see Fig. 5, from sample 9 (the initial crack length is marked white in image 1); failure surfaces after crack propagation (b).
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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9
crack position at the right end of the high strain areas (green). Fig. 8a and b show strong crack deflection and fibre bridging due to the waviness of the fabric. Overall six samples were investigated in DCB set-up. Fig. 9 shows the width-normalized load-deflection curves of four selected samples with varying initial crack lengths and the 5%-reduced stiffness curves for the determination of crack onset loads. The samples 8, 9 and 12 reached full delamination over the complete sample length (50 mm). The interlaminar crack started to run upwards leading to bending failure of the upper cantilever arm for all other samples. All load deflection curves show a first kink below 1 N/mm width-normalized load. The kink is assumed to be caused by the alignment and rotation of pins together with the aluminum loading blocks. First, the stiffness of the six DCB‐samples with varying initial crack lengths was evaluated. An effective Young’s Modulus was evaluated from the initial load‐deflection curves after the kink (due to set-up alignment) following beam theory, as cited by Allix et al. [20]:
E¼
dF 2a du 3I
ð7Þ
with a, initial crack length, and I, geometrical moment of inertia of one cantilever beam:
I¼b
3 h =12 2
ð8Þ
b is the width, h is the height of the sample. The weakening of the cantilever beams by 0.15 mm on each side due to grinding of the notch was neglected. The average effective Modulus was determined to be 24.5 ± 4.5 GPa. Since this evaluation does not consider crack root deflection and rotation, the ASTM standard for Modified Beam Theory was also applied: same equation as (7) but with (a + |D|)3. MBT gave 33.1 ± 5.3 GPa and 41.1 ± 6.4 GPa with LBC. |D| was determined by Calibration I, see Section 2.2. Fig. 10 shows the width‐normalized stiffness from experiment plus theoretical curves for effective Moduli of 24.5 GPa and 58 GPa. 58 GPa is the hysteresis Modulus of desiliconized C/C-SiC at about 90 MPa maximum stress level, compare Fig. 2 b, Section 2.1. The computed stiffness, Eq. (7), for a Young’s modulus of 24.5 GPa shows a good fit to the experimental results. The strong deviation in bending stiffness indicates limitations of beam theory based methods for the investigated CMCmaterial and sample geometries. Therefore all beam theory based results in Fig. 11 need to be considered carefully (only Area Method is not beam theory based). The FE-computations in Section 5 will reinforce the observed limitations of beam theory based methods. The effective bending stiffness of about 24.5 GPa will be successfully reproduced by a Young’s Modulus of 58 GPa when crack tip behaviour is taken into account by Cohesive Zone Elements. A variety of methods were applied to calculate the GIc values for crack onset load (5%-offset) and propagation load points: The equations from Beam Theory (BT), Modified Beam Theory (MBT), Compliance Calibration (CC) and Modified Compliance Calibration (MCC) with and without loading block corrections (LBC) as well as the Area Method (AM), see equations in section 2.2. Fig. 11 shows the GIc results for the different methods from test standards. Two approaches were considered for the determination of compliance calibration values |D|, n and A1 (necessary for MBT, CC and MCC with and without LBC): Calibration I and Calibration II, as explained in Section 2.2.
Fig. 9. Width-normalized load-deflection curves from selected DCB samples (sample number and initial crack length are indicated) with 95%-slopes for the determination of crack onset load points.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Normalized Stiffness / N/mm²
35 58 GPa
30
24,5 GPa Experiment
25 20 15 10 5 0 14
16
18
20
22
24
26
28
30
Crack length / mm Fig. 10. Width-normalized stiffness from experiment and beam theory computations for different Young’s Moduli at varying initial crack lengths.
Onset, all samples
Propagaon, fully delaminated only
Onset, fully delaminated only
0,30
Calibration I/II
GIc / N/mm
0,25 0,20 0,15 0,10 0,05 0,00 BT
MBT
MBT LBC
CC
CC LBC
MCC
MCC LBC
AM
Fig. 11. Averaged GIc values determined from onset and propagation load points; different calibration approaches are indicated. BT and AM are not compliance calibrated.
Overall Fig. 11 shows that the GIc results lie in a similar range when scatter bars are considered. However, three trends become rather clear: 1. Beam theory (BT) gave the highest GIc value in almost all cases (exception: the MCC LBC average onset value of all samples is similar). 2. The loading block corrections led to slightly (<5%) increased GIc values. 3. Calibration method II is reducing more significantly the average GIc values than calibration method I. After compliance calibration II the average GIC values are much lower than the original values from beam theory (compare BT with MBT, CC and MCC). The reason for the strong reduction of GIc with compliance calibration II may be found in load-deflection curves, see Fig. 9. An increase of GIc with crack propagation is indicated by the load plateaus after crack onset. Since the compliances are deviating from the ideal compliances if the fracture toughness is increasing during crack propagation, the compliance calibration leads to an overall reduction of GIc onset and propagation values. The calibration values |D| and A1 from propagation curves (calibration II) are much higher than from all onset compliances (calibration I). n is respectively lower for the propagation curves (calibration II) than from all onset compliances (calibration I). In that way the calibration values II lead to a strong reduction of original GIc from beam theory, see equations in Section 2.2. The evolution of fracture toughness with crack propagation was computed from MBT with calibration II in Fig. 12. The very low onset GIc values (about 0.12 N/mm) need to be mentioned here. The average GIc from MBT with calibration I was determined to be 0.163 N/mm at onset load; loading block corrected (LBC) MBT gave 0.169 N/mm. The area method (AM) from AITM standard gave a similar value of 0.167 ± 0.04 N/mm. The area method approach is simple; compliance calibration is not possible.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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0,3
Sample 9 Sample 8
GIc / N/mm
0,25
y = 0,0032x + 0,1451 R² = 0,6015
Sample 12 y = 0,0036x + 0,1313 R² = 0,8282
0,2
0,15 y = 0,0028x + 0,1287 R² = 0,6801
0,1 0
5
10
15
20
25
30
Crack extension / mm Fig. 12. Exemplary R-curves from samples reaching full delamination; the evaluation followed MBT with loading block corrections. D for compliance calibration was determined for each propagation curve separately (calibration II).
Calibration method II, as recommended by ASTM standard, gave strongly reduced onset GIc values compared to beam theory, see Fig. 11. It might be argued that only short initial crack length with little crack propagation and few data points were considered, see Fig. 12. That is why an evaluation method additional to test standards was applied. The evaluation method suggested by Tamuzs et al. and Sorensen et al., see Eq. (5), led to the results in Fig. 13. Eq. (5) does not require a crack length and therefore eliminates errors by crack length measurement. Since crack length is not included, it is less sensitive to limitations by beam theory - a compliance calibration is also not necessary. The computed crack length was added in Fig. 13(b) following the work by Brunner et al. [24] in order to show R-curve behaviour in relation with a continuously computed crack length. This crack length value can be used as orientation for the true crack length. However, it does overestimate the real initial and final crack length in the range of about 5 mm. As mentioned before, samples 8, 9 and 12 showed full delamination over the sample length. Sample 34 showed bending failure due to crack deflection in vertical direction. The GIc plateau is therefore not as clear as for the other samples in Fig. 13 (a). Fig. 13(a) and (b) shows that the GIC plateau values are still scattering for the fully delaminated samples. However, the first kink of the GIC-curves is rather constant at average of 0.16 N/mm. Overall the GIC values from Eq. (5) are significantly higher than from MBT with calibration II, see Fig. 12. Alfano et al. made a similar observation when comparing data reduction methods for adhesive joints [11]. Since MBT with calibration I and Eq. (5) gave a similar onset GIc value of about 0.16 N/mm, this value was used for further FEA and beam theory based LEFM computations 5. Numerical results All FE simulations were computed with a linear-elastic material model using a Young’s Modulus of 58 GPa in 0/90° direction. The FE simulations gave a bending stress of about 93 MPa for the initiation loads at all crack lengths in DCB test set-up. The average tensile strength of the investigated material is about 90–120 MPa; the bending strength is 140–190 MPa. Explanations for the difference in tensile and bending strength of C/C-SiC material may be found in [22,23]. The results from 2D-plane strain FEA, experimental results and beam theory based LEFM are compared. Fig. 14 a shows the load–displacement curves from 2D-FEA with and without aluminum loading blocks for sample 9 with a relative short initial crack length of 15.6 mm. The load–displacement curves are similar. It does obviously not affect the results if the load is introduced via loading blocks or directly on the sample surface. Therefore, the subsequent simulations were performed without loading blocks. Fig. 14 b shows the impact of varying critical stress values, rmax, on the loaddisplacement behaviour at constant elastic properties and a constant critical energy release rate of 0.16 N/mm. The critical stress level has only a slight impact on the maximum load. However, the non-linearity before load maximum does significantly increase with decreasing rmax. This effect is also found in publications by Alfano et al. [10,11]. All curves in Fig. 14 b end up following the crack propagation curve from LEFM with E = 58 GPa and GIc = 0.16 N/mm. The crack propagation curves were computed with the equation from LEFM [20]:
u¼
ðbGIc Þ
pffiffiffi 3=2 EI
3F 2
ð9Þ
Fig. 15 shows the width-normalized load-displacement curves for varying initial crack lengths from experiment and FEA as well as the crack propagation curves from beam theory based LEFM for two different Young’s Moduli. The stiffness from FEA and DCB test correspond rather well, except for sample 8 with the largest initial crack length. In this case the FEA is overPlease cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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0,35 Sample 9 Sample 12 Sample 8 Sample 34
GIc / N/mm
0,3 0,25 0,2 0,15 0,1 0,05 0 0,0
0,5
1,0
1,5
2,0
2,5
3,0
Displacement / mm
(a) 0,35
Sample 9
GIc / N/mm
0,3
Sample 12 Sample 8
0,25
Sample 34
0,2 0,15 0,1 0,05 0 20
25
30
35
40
45
50
Computed crack length / mm
(b)
Fig. 13. GIc evolution over displacement (a) and over computed crack length (b) following Brunner et al. [24]; Sample 34 did not show full delamination over sample length but crack deflection caused bending failure. The final GIc drop is therefore differing to Sample 8–12.
6 FEA 8 MPa FEA 5 MPa FEA 3 MPa Experiment Sample 34 LEFM
Normalized Load / N/mm
Sample 9 w/o Blocks Sample 9 w Blocks
4
3
2
1
Normalized Load / N/mm
5 5
4
3
2
1
0
0 0,0
0,5
1,0
Displacement / mm
1,5
2,0
0,0
0,5
1,0
1,5
2,0
2,5
Displacement / mm
Fig. 14. Comparison of FEA with and without loading blocks (a) for a = 15.6 mm; comparison of FE-results with varying critical stress values for a = 17.8 mm (GIc = 0.16 N/mm and E = 58 GPa were kept constant in FEA and beam theory based LEFM solution) (b).
estimating the stiffness. The explanation is coming from sample preparation: the relatively long grinded pre-crack led to irregular thickness of the cantilever beams. Due to the cubic correlation with thickness, see Eq. (8), overall a softer behaviour is observed in experiment. The beam theory based LEFM curve for the average effective Modulus of 24.5 GPa corresponds well with the experimental load points of crack onset.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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Fig. 15. Load-displacement curves from DCB-test, FEA and propagation curves from LEFM for two Young’s Moduli; sample numbers and initial crack lengths are indicated.
The LEFM crack propagation curve for a standard Young’s Modulus of 58 GPa and a GIC of 0.16 N/mm agrees perfectly with the propagation curve from FEA, see Fig. 15. It has to be mentioned here that the onset load value would be strongly overestimated by using beam theory based LEFM with an effective Modulus of 58 GPa. The critical load point, which is the intersection of crack propagation curve, see Eq. (9), and the stiffness from beam theory, see Eq. (7), are shown for varying Young’s Moduli in Fig. 16. The average Young’s Modulus of 24.5 GPa evaluated from the stiffness of the experimental load–displacement curves, see Fig. 16, results in a much better prediction of the critical load point. The initial load-displacement behaviour in Fig. 16 reveals the strong non-linearity from FEA and experiment. The strong non-linearity explains the low effective Young’s Modulus evaluated from DCB test, i.e. 24.5 GPa in average, compare Section 4, Fig. 10. The Young’s Modulus was evaluated after the initial kinking of the load-displacement curve due to the alignment of the test set-up. It has to be assumed that the initial Young’s Modulus was higher, similar to the load-displacement from FEA. Due to the strong non-linearity, the lower Young’s Modulus of 24.5 GPa is more useful to estimate the critical load point by beam theory based LEFM than the initial Young’s Modulus. In order to quantify the non-linearity of the load-displacement curves, the secant Modulus for sample 9 with a = 15.6 mm was evaluated from the FE load-displacement curve: the initial Modulus of 30.87 GPa is decreasing to 18.11 GPa, which is the secant Modulus at maximum load. Since the applied material model itself is linear-elastic, the non-linearity has to be caused by the crack tip, i.e. cohesive zone behaviour. The experimental load-displacement curve shows similar non-linear effects. Similar non-linear effects were also documented by Alfano et al. [10,11] for DCB samples of adhesive joints.
8
LEFM:58 GPa
Normalized Load / N/mm
7 6
Exp.
5
24.5 GPa
FEA
4 3 2 1 0 0,0
0,2
0,4
0,6
0,8
Displacement / mm Fig. 16. Comparison of beam theory based LEFM, FEA and experimental load displacement curves for sample 9 with a = 15.6 mm; different Young’s Moduli were used for the computations by beam theory based LEFM.
Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018
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The experimental curves in Figs. 15 and 16 show less decrease of load during crack propagation in contrast to beam theory based LEFM and FEA. Therewith, the comparison of beam theory based LEFM, FEA and experimental crack propagation curves confirmed the results from Section 4, Figs. 12 and 13, showing a strong increase of GIc with crack propagation for C/C-SiC. Of course the simple bi-linear cohesive law within ANSYS could not reproduce the load-displacement plateau caused by the R-curve behaviour of C/C-SiC. Nevertheless, the linear-elastic FE-Analysis, using a constant Young’s Modulus of 58 GPa, a GIc value of 0.16 N/mm with a maximum stress value, rmax, of 5 MPa was predicting the non-linear load-displacement behaviour and estimating the critical load points for varying initial crack lengths successfully. 6. Summary and conclusions The comparison of experimental load-displacement data with solutions from beam theory based LEFM indicated that the strong non-linearity of the load-displacement curves creates the necessity of introducing an effective stiffness. The strong non-linearity of load-displacement curves from DCB-testing was shown earlier by Alfano et al. [10,11]. The effective stiffness can be computed by introducing an effective Modulus, which is significantly lower than the original material Modulus. The effective Modulus allows the computation of critical load points for varying initial crack lengths by LEFM. The evaluation of GIc data reduction methods showed that usual reduction schemes from fibre reinforced polymers have to be considered carefully for CMC materials. Only short straight pre-cracks could be machined in the C/C-SiC-material. The current crack length was difficult to detect. Finally fibre bridging, crack deflection and R-curve behaviour were observed. All those circumstances made the data reduction method by Tamuzs et al. and Sorensen et al. [6,7], which does not require a crack length measurement, much more applicable to C/C-SiC than compliance calibration methods from test standards for fibre reinforced polymers. An alternative approach for compliance calibration, based on varying initial crack length, was presented. Modified beam theory and this calibration approach gave an onset GIC value of about 0.16 N/mm. This result is comparable to the data reduction method presented by Tamuzs and Sorensen [6,7]. However, the approach by Tamuzs and Sorensen is for sure the more convenient way of evaluating GIC values from load-deflection data. The cohesive zone modelling approach with a simple bi-linear traction separation law proved that the non-linear crack onset can be modelled for varying initial crack lengths. This model can now be used for structural design up to crack onset with respective safety factors. As demonstrated, full crack propagation modelling for C/C-SiC would require models which take R-curve behaviour into account see [6,7,25] for fibre reinforced polymers. Interlaminar crack propagation under mode II loading and further CMC materials (SiC/SiC and oxide/oxide composites) will be investigated in the future work. Acknowledgements The financial support by the DLR program directorate for space travel is gratefully acknowledged (grant number 2479101). The author would like to thank Dietmar Koch and the reviewers for their detailed scientific feedback and Harald Kraft for his support and sharing of mechanical test experience. The author thanks Prof. Heinz Voggenreiter from DLR Stuttgart and Prof. Siegfried Schmauder from University Stuttgart, Institute for Materials Testing, Material Science and Strength of Materials for their persistent scientific support. Since the original version of this paper was written during the recovery from a knee surgery, the author would like to deeply thank Irmi and Franz Hofmann for their support during this time. References [1] ASTM Standard D 5528-94a. Standard test method for Mode I interlaminar fracture toughness of unidirectional fiber-reinforced polymer matrix composites; 1994. [2] AITM. Airbus industry test method, 1.0006, Issue 2, Carbon fiber reinforced plastics, Determination of interlaminar fracture; 1994. p. 1–8. [3] ISO Standard 15024. Fibre-reinforced plastic composites – Determination of mode I interlaminar fracture toughness, GIc, for unidirectionally reinforced materials; 2001. [4] ASTM Standard D3433-05. Standard Test Method for Fracture Strength in Cleavage of Adhesives in Bonded Metal Joints, Annual Book of ASTM Standards, Vol. 15.06, ASTM, International, West Conshohocken, PA; 2005. [5] BS7991. 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Please cite this article in press as: Hofmann S. Mode I delamination onset in carbon fibre reinforced SiC: Double cantilever beam testing and cohesive zone modelling. Engng Fract Mech (2017), http://dx.doi.org/10.1016/j.engfracmech.2017.05.018