Availableonlineatwww.sciencedirect.com COMPOSITES scⅰ enceDirect CIENCE AND TECHNOLOGY ELSEVIER Composites Science and Technology 68 (2008)1165-l1 www.elsevier.com/locate/compscitech Ceramic fiber composites: Experimental analysis and modeling of mechanical properties Dietmar Koch, Kamen Tushtev, Georg Grathwohl Unicersity of Bremen, Department of Ceramic Materials and Components, D-28359 Bremen, Germany Received 16 May 2007: received in revised form 22 June 2007: accepted 29 June 2007 Available online 19 July 2007 Abstract Ceramic fiber reinforced ceramic matrix composites(CMC)are outstanding ceramics with high fracture toughness. This can be real ized if both brittle components of the composite, i.e., fibers and matrix are interacting with each other in an efficient way. Either a weak interface allowing debonding between fiber and matrix controls the fracture processes (WIC-CMC) or the matrix takes this role of a weak and more compliant component(WMC-CMC). An experimental test data base is presented for a WMC-type composite where the materials data are used to establish a model which describes the materials behavior in a macroscopic way. Inelastic deformation and materials damage processes are defined, measured and interpreted on the base of a continuum damage mechanics concept. The elas ic and inelastic response is then predictable up to failure as being dependent on the angle between fiber and loading directions of the specimen o 2007 Elsevier Ltd. All rights reserved Keywords: A. Ceramic-matrix composites( CMCs); B. Stress/strain curves; B. Mechanical properties; C Modelling: C Finite element analysis(FEA) 1. Introduction tolerance of cmc it has to be assured that the fibers remain intact and effective when cracks propagate. For this Continuous fiber reinforced ceramic matrix composites purpose the balance between the strength of the fibers and are mainly developed for specific applications at high tem- the crack resistance of the other microstructural compo- peratures and in oxidative atmosphere. The fibers as rein- nents, i. e. matrix and fiber matrix interface, has to be con forcing components offer higher strength and stiffness trolled in a way that the survival probability of the fibers is compared to the matrices which are, in contrast, character- not too low. This effect can be reached by the adjustment of ized by inferior properties as in most cases their microstruc- the CMC microstructure in two alternative ways tures exhibit microcracks, residual pores and gradients or inhomogeneities caused by the CMC fabrication process. The crack resistance of the fiber matrix interface is low- Typical processing routes are chemical vapor infiltration ered in order to allow debonding between fiber and (CVI), liquid polymer infiltration(LPI), liquid silicon infil matrix CMCs of this type are characterized by a weak tration (Lsi, directed metal oxidation (DiMOx), and cera interface or interphase. They are termed Weak Interface mic slurry impregnation(CSI) which lead to characteristic Composites, WIc properties of the resulting CMC. The interfacial bonding between fiber and matrix is of When CMCs are stressed crack initiation and propaga minor importance as the matrix is weak enough and tion generally originate from the matrix For enhanced flaw susceptible for multiple cracking while the fibers pro- vide strength and crack tolerance of the CMC. These composites are called Weak Matrix Composites, E-mail address: koch(ceramics. uni-bremen de (D. Koch) WMC 02663538/S. see front matter 2007 Elsevier Ltd. All rights reserved doi:10.1016j.compscitech.2007.06.029
Ceramic fiber composites: Experimental analysis and modeling of mechanical properties Dietmar Koch *, Kamen Tushtev, Georg Grathwohl University of Bremen, Department of Ceramic Materials and Components, D-28359 Bremen, Germany Received 16 May 2007; received in revised form 22 June 2007; accepted 29 June 2007 Available online 19 July 2007 Abstract Ceramic fiber reinforced ceramic matrix composites (CMC) are outstanding ceramics with high fracture toughness. This can be realized if both brittle components of the composite, i.e., fibers and matrix are interacting with each other in an efficient way. Either a weak interface allowing debonding between fiber and matrix controls the fracture processes (WIC–CMC) or the matrix takes this role of a weak and more compliant component (WMC–CMC). An experimental test data base is presented for a WMC-type composite where the materials data are used to establish a model which describes the materials behavior in a macroscopic way. Inelastic deformation and materials damage processes are defined, measured and interpreted on the base of a continuum damage mechanics concept. The elastic and inelastic response is then predictable up to failure as being dependent on the angle between fiber and loading directions of the specimens. 2007 Elsevier Ltd. All rights reserved. Keywords: A. Ceramic–matrix composites (CMCs); B. Stress/strain curves; B. Mechanical properties; C. Modelling; C. Finite element analysis (FEA) 1. Introduction Continuous fiber reinforced ceramic matrix composites are mainly developed for specific applications at high temperatures and in oxidative atmosphere. The fibers as reinforcing components offer higher strength and stiffness compared to the matrices which are, in contrast, characterized by inferior properties as in most cases their microstructures exhibit microcracks, residual pores and gradients or inhomogeneities caused by the CMC fabrication process. Typical processing routes are chemical vapor infiltration (CVI), liquid polymer infiltration (LPI), liquid silicon infiltration (LSI), directed metal oxidation (DiMOx), and ceramic slurry impregnation (CSI) which lead to characteristic properties of the resulting CMC. When CMCs are stressed crack initiation and propagation generally originate from the matrix. For enhanced flaw tolerance of CMC it has to be assured that the fibers remain intact and effective when cracks propagate. For this purpose the balance between the strength of the fibers and the crack resistance of the other microstructural components, i.e. matrix and fiber matrix interface, has to be controlled in a way that the survival probability of the fibers is not too low. This effect can be reached by the adjustment of the CMC microstructure in two alternative ways: • The crack resistance of the fiber matrix interface is lowered in order to allow debonding between fiber and matrix. CMCs of this type are characterized by a weak interface or interphase. They are termed Weak Interface Composites, WIC. • The interfacial bonding between fiber and matrix is of minor importance as the matrix is weak enough and susceptible for multiple cracking while the fibers provide strength and crack tolerance of the CMC. These composites are called Weak Matrix Composites, WMC. 0266-3538/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.06.029 * Corresponding author. E-mail address: dkoch@ceramics.uni-bremen.de (D. Koch). www.elsevier.com/locate/compscitech Available online at www.sciencedirect.com Composites Science and Technology 68 (2008) 1165–1172 COMPOSITES SCIENCE AND TECHNOLOGY
D. Koch et al Composites Science and Technology 68(2008)1165-1172 In any case, the composites need a microstructural component which is weak enough to allow deformation, debonding and cracks to propagate without leading to at9oo°C state 0.20 pontaneous failure of the composite 2. Weak interface composites, WIC No debonding The microstructural design of CMC was primarily dri- 0.10 en by the development of a weak interface between fiber Initial and matrix in order to achieve crack deviation at the fiber debond ing matrix interface. With this microstructural model approach the mechanical behaviour of wic can be described accu tely. If an unidirectional composite is loaded in tensile mode the initial cracks propagate first in the matrix as the fibers are stronger and can reach a higher strain to fail- Indenter Displacement /mm Then the matrix crack propagates through the com- Fig. 2. Push-in-curves revealing the prevention of interfacial debonding in posite being bridged by the strong fibers which remain a SiC/DiMOx-CMc due to oxidation and formation of silica at 900C intact as the stress concentration at the interface does not resulting in brittle failure of the composite induce fiber failure but initial interfacial debonding. Using curve allowing slight increase of the critical ratio T/rEif the well-known relationship presented by He and Hutchin- on[I] the required (low) fracture energy of the interface r the relative Young's modulus of the composite deviates for initiation of these debonding processes can be calcu- from zero lated in relation to the fracture energy(surface energy) of If the condition of the critical fracture ratio the fiber rF and a ratio TrTE.krit due to Oxidation WMC 1008-06-04-020.00204060.81.0 Relative Youngs Modulus(EF-EM(E+EM Fig. I. Boundary curve according to He and Hutchinson for realization of non brittle behavior taking into consideration the critical relative fracture energy dependent on the stiffness ratio of fiber and matrix. Additionally, the effects of interfacial oxidation and matrix densification on failure behavior are
In any case, the composites need a microstructural component which is weak enough to allow deformation, debonding and cracks to propagate without leading to spontaneous failure of the composite. 2. Weak interface composites, WIC The microstructural design of CMC was primarily driven by the development of a weak interface between fiber and matrix in order to achieve crack deviation at the fiber matrix interface. With this microstructural model approach the mechanical behaviour of WIC can be described accurately. If an unidirectional composite is loaded in tensile mode the initial cracks propagate first in the matrix as the fibers are stronger and can reach a higher strain to failure. Then the matrix crack propagates through the composite being bridged by the strong fibers which remain intact as the stress concentration at the interface does not induce fiber failure but initial interfacial debonding. Using the well-known relationship presented by He and Hutchinson [1] the required (low) fracture energy of the interface CI for initiation of these debonding processes can be calculated in relation to the fracture energy (surface energy) of the fiber CF and a ratio CI/CF 6 0.25 must not be surpassed for non-brittle behavior if fiber and matrix have similar Young’s moduli EF and EM, respectively. Typical representatives of this type of composites are CMC with dense and crystalline matrices as, e.g., the CVI derived SiC-matrix or the DiMOx derived Al2O3-matrix. These CMCs are typical WIC materials for which fiber coating with the effect of lowering the fracture energy of the interface (e.g., pyrocarbon pyC, SiC, BN, and BCN layers or combinations of these layers) is inevitable. The quantitative result is shown in the He–Hutchinson-diagram (Fig. 1) with the boundary curve allowing slight increase of the critical ratio CI/CF if the relative Young’s modulus of the composite deviates from zero. If the condition of the critical fracture energy ratio according to Fig. 1 is fulfilled, beyond matrix cracking strength WIC ceramics exhibit stress strain curves with a typical nonlinear characteristic arising from debonding, multiple crack initiation, propagation and opening. The load is increasingly transferred to the fibers and fiber failure is initiated sequentially leading to the increase of the fraction of failed fibers due to the statistic scattering of fiber strength up to the final failure of the composite. Applying the composites at high temperatures in oxidative atmosphere the fiber coatings may be attacked due to environmental conditions. If oxidation of the interphase -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Relative Fracture Energy ΓI / ΓF Relative Young's Modulus (EF -EM)/(EF +EM) Non-brittle failure Brittle failure ΓI > ΓF,krit due to reinfiltr . WMC due to Oxidation WIC Fig. 1. Boundary curve according to He and Hutchinson for realization of non brittle behavior taking into consideration the critical relative fracture energy dependent on the stiffness ratio of fiber and matrix. Additionally, the effects of interfacial oxidation and matrix densification on failure behavior are shown. Indenter Displacement / mm 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Indenter Load / N 0.00 0.05 0.10 0.15 0.20 0.25 900˚C RT No debonding Initial debonding After oxidation at 900˚C Original state Silica layer After oxidation at 900°C Original state Silica layer Fig. 2. Push-in-curves revealing the prevention of interfacial debonding in a SiC/DiMOx–CMC due to oxidation and formation of silica at 900 C resulting in brittle failure of the composite. 1166 D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172
D. Koch et al. Composites Science and Technology 68(2008)1165-1172 1167 sic/sic,cⅥ +45°-45° 3 2 0 [12]风闾8]10冈冈图3图3[2 Porous matrix Dense matrix Tensile Test WMC WIC C/C. LPI 0°90° Fig. 4. WMC-WIC classification of different composites according to their stiffness ratio Eoo/E4so. Results from literature [2, 3, 5-8, 10, 12]and own measurements(marked with [x] Size effects are also not really significant and low scattering of strength is observed +45°-45° 3. Weak matrix composites, WMC Infiltration processes as LPl, Lsl, or CSI to provide the matrix in CMCs lead to microstructures which are charac- Strain[°%] terized by a fine porosity and therefore a low stifness. The significantly reduced matrix stifness and strength, com- Fig. 3. Representative stress versus strain curves of (a)WIC(CVI SiC/ pared to WIC, enables debonding processes and thus dam Si[2]and(b)WMC(LPIC/C) from axial(090° and diagonal(±45°) loaded tensile tests age tolerance even in the case of a strong fiber-matrix interface as the cracks which propagate through the matrix easily deviate close to the fiber surface through the matrix. layers occurs, the mechanical properties may change and This concept again corresponds to the theoretical analysis lead to an increase of the relative fracture energy of fiber of He and Hutchinson(Fig. 1)where the large difference and interface. Acceptable changes of the interfacial proper- between the Youngs moduli of the stiff fiber and the weak ties without provoking brittle behavior can be discussed matrix allows a much stronger bonding and a higher ratio using the borderline in Fig. 1. Hence, the interphase has between the fracture energies of interface and fiber. while to accomplish not only mechanical functions in order to the matrix fails at low stresses the composite can still be provide debonding: it also has to fulfill thermal and envi- loaded well above the matrix cracking strength as long as ronmental boundary conditions as e.g. sufficient oxidation the overall load can be transferred to the fibers. However resistance. Fig. 2 shows the effect of oxidation of the fiber as the redistribution of stresses from the fiber to the matrix matrix interface in a SiC/DiMOx composite as manifested does not take place in a significant manner final failure of by single fiber push-in tests. It shows that due to the forma- the composite occurs when the fibers do not fail locally tion of silica at the interface after oxidation at 900C initial restricted but in a large volume of the component. Thus, debonding is no longer possible at sufficiently low stresses. the mechanical behavior can no longer be described by a This leads to brittle failure of the composite as debonding micromechanical approach. Furthermore the mechanical is prevented behavior of WMC is dominated strongly by the properties It can be concluded that WIC with relatively strong and of the fibers and therefore the mechanical performanc stiff matrices and obligatory fiber coating provide high depends on their orientation and the loading direction fracture toughness values. Following the micromechanical As the matrix is not able to carry significant load, low mechanisms of debonding, these CMCs are relatively notch strength will be obtained under compression or loading insensitive with the highest achievable stress being fairly mode with an angle between fiber orientation and loading independent of the fiber and applied stress orientations. direction
layers occurs, the mechanical properties may change and lead to an increase of the relative fracture energy of fiber and interface. Acceptable changes of the interfacial properties without provoking brittle behavior can be discussed using the borderline in Fig. 1. Hence, the interphase has to accomplish not only mechanical functions in order to provide debonding; it also has to fulfill thermal and environmental boundary conditions as e.g. sufficient oxidation resistance. Fig. 2 shows the effect of oxidation of the fiber matrix interface in a SiC/DiMOx composite as manifested by single fiber push-in tests. It shows that due to the formation of silica at the interface after oxidation at 900 C initial debonding is no longer possible at sufficiently low stresses. This leads to brittle failure of the composite as debonding is prevented. It can be concluded that WIC with relatively strong and stiff matrices and obligatory fiber coating provide high fracture toughness values. Following the micromechanical mechanisms of debonding, these CMCs are relatively notch insensitive with the highest achievable stress being fairly independent of the fiber and applied stress orientations. Size effects are also not really significant and low scattering of strength is observed. 3. Weak matrix composites, WMC Infiltration processes as LPI, LSI, or CSI to provide the matrix in CMCs lead to microstructures which are characterized by a fine porosity and therefore a low stiffness. The significantly reduced matrix stiffness and strength, compared to WIC, enables debonding processes and thus damage tolerance even in the case of a strong fiber–matrix interface as the cracks which propagate through the matrix easily deviate close to the fiber surface through the matrix. This concept again corresponds to the theoretical analysis of He and Hutchinson (Fig. 1) where the large difference between the Young’s moduli of the stiff fiber and the weak matrix allows a much stronger bonding and a higher ratio between the fracture energies of interface and fiber. While the matrix fails at low stresses the composite can still be loaded well above the matrix cracking strength as long as the overall load can be transferred to the fibers. However, as the redistribution of stresses from the fiber to the matrix does not take place in a significant manner final failure of the composite occurs when the fibers do not fail locally restricted but in a large volume of the component. Thus, the mechanical behavior can no longer be described by a micromechanical approach. Furthermore the mechanical behavior of WMC is dominated strongly by the properties of the fibers and therefore the mechanical performance depends on their orientation and the loading direction. As the matrix is not able to carry significant load, low strength will be obtained under compression or loading mode with an angle between fiber orientation and loading direction. 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 250 300 350 400 Tensile Test SiC/SiC, CVI +45°/ -45° 0°/ 90° Stress [MPa] Strain [%] 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 50 100 150 200 250 300 350 400 Tensile Test C/C, LPI +45°/ -45° 0°/ 90° Stress [MPa] Strain [%] Fig. 3. Representative stress versus strain curves of (a) WIC (CVI SiC/ SiC) [2] and (b) WMC (LPI C/C) from axial (0/90) and diagonal (±45) loaded tensile tests. 0 1 2 3 4 5 6 E0 / E45 WHIPOX Nicalon/SiC Nicalon/SiC Nicalon/MAS C/SiC Nextel 610 /Mullite + Alumina Nextel 720 /Mullite + Alumina (Al C/C C/C, SIGRABOND C/C O2 3-ZrO2 )mc/Al O3 2 C/C [6] [5] [12] [x] [8] [7] [10] [x] [x] [3] [3] [2] WMC WIC Porous matrix Dense matrix Fig. 4. WMC–WIC classification of different composites according to their stiffness ratio E0/E45. Results from literature [2,3,5–8,10,12] and own measurements (marked with [x]). D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172 1167
D. Koch et aL Composites Science and Technology 68(2008)1165-1172 Table 1 ies of various CMC classified in Fig 4(as far as available from literature Material type Manufacturing method [6] Carbon T-300/Carbon, 16 layers, 0/900 56% fiber content LPL 4 infiltrations [5 Al2O3 Almax/ZrO2 AlO3, 6 layers, 0/900 33% fiber content minicomposite No detailed processing descripti Carbon Torayka M-40/Carbon, 15 layers, 0/90 50% fiber content Preformed yarn method LPI /C Sigrabond 1501 G, 24 layers, 0/90 open porosity 10-12% LPI Preformed 87x图2 [8 Carbon Torayka M40/C, 16 layers, 0 /90 50%fiber cont Nextel 720 /mullite and alumina, 12 layers, 0/900 39% fiber content matrix porosity 38% CSI up to [10] Nextel 610/80% mullite and 20% alumina 0/90 composite porosity 22-25% matrix porosity 38-42% CSI up to T800C/SiC 4 UD 0°/90° WHIPOX0°/90° Nicalon/MAS 12 0/900 40% fiber content fully dense MAS matrix Hot pressing Nicalon Sic/SiC 32% fiber content 8. 6% matrix porosity Nicalon SiC/SiC 0/90 plain weave 35% fiber content residual porosity 10-15% The characteristic difference between on-axis loading are infiltrated with the carbon matrix by several impregna (fibers are oriented in loading direction)and off-axis load- tion cycles with succeeding thermal treatments up to ing is manifested in Fig 3 for both composite types WIC 2000C. Finally, the resulting composite is characterized and WMC. Similar strength is measured for WIC under by an open porosity of 10-12% resulting from shrinkage both loading conditions, as discussed earlier on the base of micromechanical mechanisms On the contrary, in case of WMc the strength is strongly reduced under off-axis conditions because the fibers are not carrying the load sufficiently se I Longitudina The proposed concepts WIC and WMC are typical boundary examples with the real composites being situated somewhere in between. The properties of the composites not only depend on the manufacturing route, but also on d the chosen combination of fiber and matrix. In Fig. I it becomes obvious that with improved mechanical properties gD of the matrix the interfacial fracture energy plays a more important role and must be low enough in order to fulfill C 150/ →+1575 he boundary conditions of non brittle failure. This improvement of matrix properties can be reached by e. g reinfiltration cycles which is the regular case for the LPI process. Thus, if the matrix properties are enhanced the interfacial properties become more important. In Fig. 4 several composites from various sources are ranked con Strain (0%1900 and +45%/-459)as a measure of classification to b o WIC and WMC, respectively. Additionally, their proper- ties which were available from the literature are listed in q Table 1. It turns out that a typical WMC like C/C shows a high Eo/E4se ratio while a typical WIC like Nicalon/ +15%75° SiC is characterized by a Eoo/E4se ratio of 1. Other compos- 2 ites like oxide/oxide composites, LPI derived materials, and Sic fiber reinforced glass matrix composites are situated 090 between these boundary cases 方250 300 4. Materials and experiments ongitudinalTransverse Fundamental tests have been performed using a com- G」 mercially available C/C composite called Sigrabond 1501G(SGL Carbon, Germany). The material consists of 24 bidirectional reinforced layers of carbon fiber mats. Fig. 5. Typical stress strain curves of WMC composite C/C under(a) The composite is designed symmetrically and shows the tension and (b)compression with different angles between fiber orientation same properties in 00 and in 90 orientation. The fiber mats and loading direction
The characteristic difference between on-axis loading (fibers are oriented in loading direction) and off-axis loading is manifested in Fig. 3 for both composite types WIC and WMC. Similar strength is measured for WIC under both loading conditions, as discussed earlier on the base of micromechanical mechanisms. On the contrary, in case of WMC the strength is strongly reduced under off-axis conditions because the fibers are not carrying the load sufficiently. The proposed concepts WIC and WMC are typical boundary examples with the real composites being situated somewhere in between. The properties of the composites not only depend on the manufacturing route, but also on the chosen combination of fiber and matrix. In Fig. 1 it becomes obvious that with improved mechanical properties of the matrix the interfacial fracture energy plays a more important role and must be low enough in order to fulfill the boundary conditions of non brittle failure. This improvement of matrix properties can be reached by e.g. reinfiltration cycles which is the regular case for the LPI process. Thus, if the matrix properties are enhanced the interfacial properties become more important. In Fig. 4 several composites from various sources are ranked concerning their ratio of stiffness in on and off-axis orientation (0/90 and +45/45) as a measure of classification to WIC and WMC, respectively. Additionally, their properties which were available from the literature are listed in Table 1. It turns out that a typical WMC like C/C shows a high E0/E45 ratio while a typical WIC like Nicalon/ SiC is characterized by a E0/E45 ratio of 1. Other composites like oxide/oxide composites, LPI derived materials, and SiC fiber reinforced glass matrix composites are situated between these boundary cases. 4. Materials and experiments Fundamental tests have been performed using a commercially available C/C composite called Sigrabond 1501G (SGL Carbon, Germany). The material consists of 24 bidirectional reinforced layers of carbon fiber mats. The composite is designed symmetrically and shows the same properties in 0 and in 90 orientation. The fiber mats are infiltrated with the carbon matrix by several impregnation cycles with succeeding thermal treatments up to 2000 C. Finally, the resulting composite is characterized by an open porosity of 10–12% resulting from shrinkage Table 1 Properties of various CMC classified in Fig. 4 (as far as available from literature) Material type Manufacturing method [6] Carbon T-300/Carbon, 16 layers, 0/90 56% fiber content LPI, 4 reinfiltrations [5] Al2O3 Almax/ZrO2 + Al2O3, 6 layers, 0/90 33% fiber content minicomposite No detailed processing description [12] Carbon Torayka M-40 /Carbon, 15 layers, 0/90 50% fiber content Preformed yarn method LPI [x] C/C Sigrabond 1501 G, 24 layers, 0/90 open porosity 10–12% LPI [8] Carbon Torayka M40/C, 16 layers, 0/90 50% fiber content Preformed yarn method LPI [7] Nextel 720/mullite and alumina, 12 layers, 0/90 39% fiber content matrix porosity 38% CSI up to 1200 C [10] Nextel 610/80% mullite and 20% alumina 0/90 composite porosity 22–25% matrix porosity 38–42% CSI up to 1200 C [x] T800 C/SiC 4 UD layers 0/90 LPI [x] WHIPOX 0/90 open porosity 34% CSI [3] Nicalon/MAS 12 layers, 0/90 40% fiber content fully dense MAS matrix Hot pressing [3] Nicalon SiC/SiC 0/90 32% fiber content 8.6% matrix porosity CVI [2] Nicalon SiC/SiC 0/90 plain weave 35% fiber content residual porosity 10–15% CVI 0 50 100 150 200 250 300 350 400 -0.4 -0.2 0.0 0.2 0.4 Transverse Longitudinal +45°/-45° +15°/-75° +30°/-70° 0°/90° Strain [%] Stress [MPa] -400 -350 -300 -250 -200 -150 -100 -50 0 -0.4 -0.2 0.0 0.2 0.4 Longitudinal Transverse +45°/-45° +15°/-75° +10°/-80° 0°/90° Strain [%] Stress [MPa] ϕ σ σ σ ϕ σ Fig. 5. Typical stress strain curves of WMC composite C/C under (a) tension and (b) compression with different angles between fiber orientation and loading direction. 1168 D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172
D. Koch et al. Composites Science and Technology 68(2008)1165-1172 Matrix shear failure o Experimental data Fiber buckle failure Fiber tensile failure -100 Fig. 6. Summary of measured strength values and failure modes depending on fiber orientation and loading direction. induced cracks and pores within and between the fiber bun- shear stresses additionally reduce the overall compression dles. Further details are described elsewhere [ll]. trength In+45/45 orientation the specimen fails sim- The mechanical properties were investigated at room ilar to the tensile test with large strain to failure and temperatures under ambient atmosphere. The specimens extended nonlinear stress-strain behavior with various geometries were tested in tension, compres- The results from tension, shear and compression tests sion, and shear modes in a spindle testing machine(Zwick, are summarized in Fig. 6 showing 0I-T12 plane with I Germany)using different angles (0/90, +10/-800, +15/ and 2 representing the fiber orientation in the 2D rein- 75°,+30°/-60°,+45°-45°) between fiber orientation forced material. Depending on load and fiber orientation and loading direction. Strain was measured with strain gauges and with a laser based contactless strain measure- ment system. Complex loaded specimens as DEN (double end notch) coupons are tested in tensile mode in order to Stress- Strain Curve 2500 investigate the influence of stress concentrations on the mechanical behavior of the composites. Amor 2000 hensive description of the tests is found in [4, 9, 11, 13] 1500 5. Experimental results 0cQEu Depending on the angle between fiber orientation and 1000 loading direction significant changes of the stress strain o) 100 curves are observed in tensile as well as in compressive mode(Fig. 5). In 0/90 orientation the material behaves almost linear-elastic up to failure, the fibers that are ori ented in loading direction carry the load. Transversal strain 0.000.050.100.150200.250.300.350 is almost negligible due to the 90 fibers. Failure occurs Axial Strain [% when the fiber strength is reached resulting in large volume damage throughout the total gauge length. With increasing b35 Stress-Strain Curve 6000 angle between fiber orientation and loading direction(off axis loading) strength and Youngs modulus sharply decrease Due to the weak matrix damage occurs already at low stresses resulting in a reduced stifness of the com- 4000 posite. Under off-axis loading shear failure is always observed. The failure processes are not distributed over the whole gauge length but locally restricted. The fracture 2000 surface develops along the fiber axis under similar shear stresses and independent of the ofi-axis angle Under compression mode the material behaves in a sim ilar manner. In 0/90 orientation a linear-elastic behavior is observed, however, the specimens fail at much lower 00000.0020.0040.0060.0080.0100.012 stresses just above 200 MPa which is only half of tensile strength. The weak matrix is not able to prevent fiber buck ling which is also observed at specimens with +10/-80 tensile test and(b) pure shear test with associated acoustic emission and +15/-75 orientation. In these cases superimposed signals
induced cracks and pores within and between the fiber bundles. Further details are described elsewhere [11]. The mechanical properties were investigated at room temperatures under ambient atmosphere. The specimens with various geometries were tested in tension, compression, and shear modes in a spindle testing machine (Zwick, Germany) using different angles (0/90, +10/80, +15/ 75, +30/60, +45/ 45) between fiber orientation and loading direction. Strain was measured with strain gauges and with a laser based contactless strain measurement system. Complex loaded specimens as DEN (double end notch) coupons are tested in tensile mode in order to investigate the influence of stress concentrations on the mechanical behavior of the composites. Amore comprehensive description of the tests is found in [4,9,11,13]. 5. Experimental results Depending on the angle between fiber orientation and loading direction significant changes of the stress strain curves are observed in tensile as well as in compressive mode (Fig. 5). In 0/90 orientation the material behaves almost linear-elastic up to failure, the fibers that are oriented in loading direction carry the load. Transversal strain is almost negligible due to the 90 fibers. Failure occurs when the fiber strength is reached resulting in large volume damage throughout the total gauge length. With increasing angle between fiber orientation and loading direction (off- axis loading) strength and Young’s modulus sharply decrease. Due to the weak matrix damage occurs already at low stresses resulting in a reduced stiffness of the composite. Under off-axis loading shear failure is always observed. The failure processes are not distributed over the whole gauge length but locally restricted. The fracture surface develops along the fiber axis under similar shear stresses and independent of the off-axis angle. Under compression mode the material behaves in a similar manner. In 0/90 orientation a linear-elastic behavior is observed, however, the specimens fail at much lower stresses just above 200 MPa which is only half of tensile strength. The weak matrix is not able to prevent fiber buckling which is also observed at specimens with +10/80 and +15/75 orientation. In these cases superimposed shear stresses additionally reduce the overall compression strength. In +45/45 orientation the specimen fails similar to the tensile test with large strain to failure and extended nonlinear stress–strain behavior. The results from tension, shear and compression tests are summarized in Fig. 6 showing r1–s12 plane with 1 and 2 representing the fiber orientation in the 2D reinforced material. Depending on load and fiber orientation Fig. 6. Summary of measured strength values and failure modes depending on fiber orientation and loading direction. 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0 50 100 150 200 250 300 350 400 Stress-Strain Curve Stress [MPa] Axial Strain [%] 0 500 1000 1500 2000 2500 Sum of Acoustic Emission Signals Acoustic Emission Signals 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0 5 10 15 20 25 30 35 Stress-Strain Curve Shear Stress [MPa] Shear Strain [%] 0 1000 2000 3000 4000 5000 6000 Acoustic Emission Signals Sum of Acoustic Emission Signals Fig. 7. Stress–strain curves with unloading reloading cycles of (a) on-axis tensile test and (b) pure shear test with associated acoustic emission signals. D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172 1169
D. Koch et aL Composites Science and Technology 68(2008)1165-1172 three different failure modes occur. In 0%/90 orientation Unloading-reloading cycles reveal inelastic strain and stiff- high strength at about 400 MPa is measured in tensile ness reduction which are a result of the increasing damage mode and the fracture is characterized by fiber failure. processes in the matrix while the acoustic emission signal Under compression the composite fails due to fiber buck- give a qualitative proof of the ongoing damage processes ling. As the composite is extremely shear sensitive with a The experimentally derived macroscopic properties of shear strength of about 35 MPa shear failure of the com- WMC under tensile, shear, and compressive loading are posite is observed when shear stresses are superimposed. used to establish a finite element model which allows This happens in the off-axis tension and compression tests the prediction of mechanical behavior of these composites with increasing angle between fiber and loading orientation Fig. 8). The model is based on continuum damage resulting in a reduced strength. The experimental data are mechanics and allows the separate calculation of inelastic to be calculated using the failure criterion from Hill with deformation and stifness degradation inducing yield sur the assumption that the composite behaves symmetrically faces and damage surfaces with the assumptions of iso- and using tensile, compressive and pure shear strength val- tropic hardening and associated flaw rules. The ues as input data. Details are described elsewhere [ll] hardening functions, which calculate inelastic deforma tion and stifness reduction are coupled by using the same 6. Modeling of mechanical behavior equivalent stress. The implementation of the model into a finite element code in marc allows the calculation of As already mentioned the damage and failure mecha- the stress-strain behavior of composites loaded with dif- nisms of WMC cannot be described sufficiently by a micro- ferent angles between fiber orientation and loading direc mechanical approach as in general large volume failure tion [9, 11, 14]. occurs. To understand the damage evolution pure tensile and pure shear tests have been performed with unload- 7. Model application for notched specimens ing-reloading cycles. Under these loading conditions the material response is dominated either by the fibers or by The established Fe model is used to predict the fail the matrix(Fig. 7). Under pure tensile mode the stress- behavior of complex shaped samples. As an example strain curve remains almost linear-elastic and almost no DEn (double end notch) specimens are investigated under residual strain is observed during unloading cycles 0/900 and+45/-45 loading conditions( Fig 9).Further (Fig. 7a). However, acoustic emission signals show that more the ligament width and accordingly the notch length at high stresses damage occurs which can be attributed to were varied. With increasing ligament width a slight matrix crack evolution and propagation. Under shear load- decrease of strength is measured in case of on-axis loading ing a strong nonlinear behavior is observed(Fig. 7b).( Fig. 10a). This width effect is also observed at unnotched Experiments Input data Shear Strain [5] FE-Implementatio MARC §直 Prediction of material ' s properties stress-strain curve strength Fig 8. Schematic flow diagram of modeling the inelastic deformation and damage of weak matrix composites
three different failure modes occur. In 0/90 orientation high strength at about 400 MPa is measured in tensile mode and the fracture is characterized by fiber failure. Under compression the composite fails due to fiber buckling. As the composite is extremely shear sensitive with a shear strength of about 35 MPa shear failure of the composite is observed when shear stresses are superimposed. This happens in the off-axis tension and compression tests with increasing angle between fiber and loading orientation resulting in a reduced strength. The experimental data are to be calculated using the failure criterion from Hill with the assumption that the composite behaves symmetrically and using tensile, compressive and pure shear strength values as input data. Details are described elsewhere [11]. 6. Modeling of mechanical behavior As already mentioned the damage and failure mechanisms of WMC cannot be described sufficiently by a micromechanical approach as in general large volume failure occurs. To understand the damage evolution pure tensile and pure shear tests have been performed with unloading–reloading cycles. Under these loading conditions the material response is dominated either by the fibers or by the matrix (Fig. 7). Under pure tensile mode the stress– strain curve remains almost linear-elastic and almost no residual strain is observed during unloading cycles (Fig. 7a). However, acoustic emission signals show that at high stresses damage occurs which can be attributed to matrix crack evolution and propagation. Under shear loading a strong nonlinear behavior is observed (Fig. 7b). Unloading–reloading cycles reveal inelastic strain and stiff- ness reduction which are a result of the increasing damage processes in the matrix while the acoustic emission signals give a qualitative proof of the ongoing damage processes. The experimentally derived macroscopic properties of WMC under tensile, shear, and compressive loading are used to establish a finite element model which allows the prediction of mechanical behavior of these composites (Fig. 8). The model is based on continuum damage mechanics and allows the separate calculation of inelastic deformation and stiffness degradation inducing yield surfaces and damage surfaces with the assumptions of isotropic hardening and associated flaw rules. The hardening functions, which calculate inelastic deformation and stiffness reduction are coupled by using the same equivalent stress. The implementation of the model into a finite element code in MARC allows the calculation of the stress–strain behavior of composites loaded with different angles between fiber orientation and loading direction [9,11,14]. 7. Model application for notched specimens The established FE model is used to predict the failure behavior of complex shaped samples. As an example DEN (double end notch) specimens are investigated under 0/90 and +45/45 loading conditions (Fig. 9). Furthermore the ligament width and accordingly the notch length were varied. With increasing ligament width a slight decrease of strength is measured in case of on-axis loading (Fig. 10a). This width effect is also observed at unnotched Fig. 8. Schematic flow diagram of modeling the inelastic deformation and damage of weak matrix composites. 1170 D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172
D. Koch et al. Composites Science and Technology 68(2008)1165-1172 2W= 30 mm Fig 9. DEN specimen geometry(left) and fracture surfaces of 0/900 and +45/-45 specimens(center and right specimens and is not yet evaluated sufficiently. FE-model ing shows that locally induced stress concentrations at the notch tip and multiaxial loading conditions lead to a600 damage and succeeding stress redistributions. The dam FE- Calculation aged zone evolves in large areas above and below the liga 50 ment (Fig. 9, center) resulting in strength values comparable to unnotched specimens. The FE calculation is in good agreement with the measured values(Figs. 10 and 11 If DEN specimens are loaded in ofi-axis orientation +45/-45) the strength values decrease strongly with 苏 increasing ligament width and decreasing notch length and this can be verified by calculating the strength values +45°-45° using ligament width as reference cross section. The results suggest a notch induced improvement of material proper ties. however if the failure behavior is evaluated Fig. 9, right)it becomes obvious that the load carry 8101214161820 cross section is not the ligament width only but has to be Ligament Width(mm defined as ligament width plus the length of one notch b300 With this calculation the den results are almost identical with respect to the tensile tests performed in +45/-45 considering orientation as it is shown in Fig. 10b real loaded cross section ▲FE· Calculation The elongation close to the ligament was measured using laser extensometer with a gauge length of 25 mm This results in an integral measurement of inhomogenous strain distribution in the damaged area above and below the load carrying cross section. The experimental curves 100 are shown in Fig. lla for 0/90 orientation with ligament tensile strength widths of 3 mm and 12 mm, respectively. As the failure is concentrated in the area where the strain is measured the experiments can be calculated accurately with the FE- model. In case of testing DEN specimens with +45/-44 orientation(Fig. Ilb)the measured strain values do not Ligament Width [mm] fit the calculated curves in the same perfect manner as Fig 10. DEN tests and respective modeling of strength(a) dependent on the occurring failure is not limited to the gauge length of ligament width and fiber orientation and (b)evaluation of data taking into 25 mm ed cross section(o-axis+45°/-45°
specimens and is not yet evaluated sufficiently. FE-modeling shows that locally induced stress concentrations at the notch tip and multiaxial loading conditions lead to damage and succeeding stress redistributions. The damaged zone evolves in large areas above and below the ligament (Fig. 9, center) resulting in strength values comparable to unnotched specimens. The FE calculation is in good agreement with the measured values (Figs. 10 and 11). If DEN specimens are loaded in off-axis orientation (+45/45) the strength values decrease strongly with increasing ligament width and decreasing notch length and this can be verified by calculating the strength values using ligament width as reference cross section. The results suggest a notch induced improvement of material properties, however, if the failure behavior is evaluated (see Fig. 9, right) it becomes obvious that the load carrying cross section is not the ligament width only but has to be defined as ligament width plus the length of one notch. With this calculation the DEN results are almost identical with respect to the tensile tests performed in +45/45 orientation as it is shown in Fig. 10b. The elongation close to the ligament was measured using laser extensometer with a gauge length of 25 mm. This results in an integral measurement of inhomogenous strain distribution in the damaged area above and below the load carrying cross section. The experimental curves are shown in Fig. 11a for 0/90 orientation with ligament widths of 3 mm and 12 mm, respectively. As the failure is concentrated in the area where the strain is measured the experiments can be calculated accurately with the FEmodel. In case of testing DEN specimens with +45/45 orientation (Fig. 11b) the measured strain values do not fit the calculated curves in the same perfect manner as the occurring failure is not limited to the gauge length of 25 mm. 2 4 6 8 10 12 14 16 18 20 0 100 200 300 400 500 600 Experiment +45°/-45° 0°/90° Strength [MPa] Ligament Width [mm] 0 5 10 15 20 0 50 100 150 200 250 300 tensile strength DEN - results from ligament width DEN - results considering real loaded cross section FE - Calculation Strength [MPa] Ligament Width [mm] FE - Calculation Fig. 10. DEN tests and respective modeling of strength (a) dependent on ligament width and fiber orientation and (b) evaluation of data taking into consideration real loaded cross section (off-axis + 45/45). Fig. 9. DEN specimen geometry (left) and fracture surfaces of 0/90 and +45/45 specimens (center and right). D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172 1171
1172 D. Koch et aL Composites Science and Technology 68(2008)1165-1172 Ligament 3 mm in a satisfying manner. The model takes into consideration 排 inelastic deformation and stifness degradation and can be applied to individual WMC materials with a marginal number of basic mechanical tests. It has been shown that the fe based model can also be applied for complex load- gament 12 mm a300 ing conditions as, e.g., DEN samples. The mechanical response of complex shaped CMC components can then be predicted. Acknowledgements Calculation The authors like to thank the german Science Founda- .304 0.5 tion DFG for financial support and the SGL Carbon group Laser Strain( Lo=25mm)[ %] for providing the material References -45°+45° Ligament 3 []He MY, Hutchinson Jw. Kinking of a crack out of an interface. J 豪 Appl Mech 1989: 56: 270-8 2] Camus G. Modelling of the mechanical behavior and damage of fibrous ceramic matrix composites: Application to a 2- iC. Int J Solids Struct 2000: 37: 919-42 方 Ligament 12 JC,Zok FW, Genin GM, Evans AG. Notch-sensitivity of nforced ceramic-matrix composites: Effects of inelastic 吧 straining and volume-dependent strength. J Am Ceram Soc 1999;5:1217-28. 4] Koch D, Tushtev K, Horvath J, Knoche R, Grathwohl G. Evaluation of mechanical properties and comprehens 50 Experiments =· FE-Calculation tiff and weak matrices. Ady sci Technol 2006: 45: 1435-43 5 Mamiya T, Kakisawa H, Liu WH, Zhu SJ, Kagawa Y. Tensile .2 damage evolution and notch sensitivity of Al2O3 fiber-ZrO2 matrix minicomposite-reinforced Al2O3 matrix composites. Mater Sci Eng A Laser Strain( Lo=25mm) [%] 2002;325(1-2:405-13. [6 Neumeister J, Jansson S, Leckie F. The effect of fiber architecture on Fig. ll. Stress-strain curves of DEn tests and respective modeling the mechanical properties of carbon/carbon fiber composites. Acta dependent on ligament width and fiber orientation(a)0°/90°;(b)+45° [7 Carelli EAV, Fujita H, Yang JY, Zok FW. Effects of thermal aging on the mechanical properties of a porous-matrix ceramic composite J The tests of DEN specimens have shown that the estab- Am Ceram Soc2002;8503):595-602. lished model is able to calculate the behavior of complex [8] Goto K, Hatta H, Takahashi H, Kawada H. Effect of shear damage shaped parts under multiaxial loading conditions. The the fracture behavior of carbon-carbon composites. J Am Ceram design of CMC components and the prediction of their fail- Soc2001;84(6:1327-33 ure behavior are then possible by this Fe tool Eng mater20046(8):664 8. Summary and conclusions [10] Heathcote JA, Gong X-Y, Yang JY, Ramamurty U, Zok FW. In- plane mechanical properties of an all-oxide ceramic composite. J Am The concept of weak interface composites WIC and Ceram Soc l'99982(10):2721-30 weak matrix composites WMC has been demonstrated as [ll] Tushtev K, Koch D, Horvath J, Grathwohl G. Mechanismen und Modellierung der verformung und Schadigung keramische Fas- boundary cases in order to evaluate the properties of erverbundwerkstoffe. Int J Mater Res 2006: 97(10): 1460-9 CMCS. If the matrix is stiff and strong the interfacial prop- [12] Hatta H, Denk L, Watanabe T, Shiota I, Aly-Hassan MS.Fracture erties become dominant and have to be adjusted in order to behavior of carbon-carbon composites with cross-ply lamination. J allow matrix cracks to deviate around the fibers Non brit- Compos Mater2004:3817):147994. tle behavior with high fracture toughness can then be [3]Koch D, Tushtev k. Kuntz M, Knoche R, Horvath J, Grathwohl G Modelling of deformation and damage evolution of CMC with achieved and explained by micromechanically based mod strongly anisotropic properties. In: Lara-Curzio E, editor. Proc of int els. If the matrix is weak and porous the material properties conf advanced ceramics and composites, vol. 26(2). Cocoa Beach do not follow these micromechanical approaches anymore Proceed;2005.p.107-14. Therefore a new model has been established which calcu [14Koch D, Kuntz M, Tushtev K, Knoche R, Horvath J, Grathwohl G lates the properties from macromechanical tests. With Keramische Faserverbundwerkstoffe mit schwacher Matrix- Eigens. chaften und Modellierung: Verbundwerkstoffe und Werkstoffverb. input data from three basic tests (tensile, shear and com- unde: Schlimmer M [ Hrsg. I Werkstoff-Informationsgesellschaft pressive) the complex behavior of WMC can be predicted Frankfurt; 2005. P. 133-8
The tests of DEN specimens have shown that the established model is able to calculate the behavior of complex shaped parts under multiaxial loading conditions. The design of CMC components and the prediction of their failure behavior are then possible by this FE tool. 8. Summary and conclusions The concept of weak interface composites WIC and weak matrix composites WMC has been demonstrated as boundary cases in order to evaluate the properties of CMCs. If the matrix is stiff and strong the interfacial properties become dominant and have to be adjusted in order to allow matrix cracks to deviate around the fibers. Non brittle behavior with high fracture toughness can then be achieved and explained by micromechanically based models. If the matrix is weak and porous the material properties do not follow these micromechanical approaches anymore. Therefore a new model has been established which calculates the properties from macromechanical tests. With input data from three basic tests (tensile, shear and compressive) the complex behavior of WMC can be predicted in a satisfying manner. The model takes into consideration inelastic deformation and stiffness degradation and can be applied to individual WMC materials with a marginal number of basic mechanical tests. It has been shown that the FE based model can also be applied for complex loading conditions as, e.g., DEN samples. The mechanical response of complex shaped CMC components can then be predicted. Acknowledgements The authors like to thank the German Science Foundation DFG for financial support and the SGL Carbon group for providing the material. References [1] He MY, Hutchinson JW. Kinking of a crack out of an interface. J Appl Mech 1989;56:270–8. [2] Camus G. Modelling of the mechanical behavior and damage processes of fibrous ceramic matrix composites: Application to a 2- D SiC/SiC. Int J Solids Struct 2000;37:919–42. [3] McNulty JC, Zok FW, Genin GM, Evans AG. Notch-sensitivity of fiber-reinforced ceramic–matrix composites: Effects of inelastic straining and volume-dependent strength. J Am Ceram Soc 1999;5:1217–28. [4] Koch D, Tushtev K, Horvath J, Knoche R, Grathwohl G. Evaluation of mechanical properties and comprehensive modeling of CMC with stiff and weak matrices. Adv Sci Technol 2006;45:1435–43. [5] Mamiya T, Kakisawa H, Liu WH, Zhu SJ, Kagawa Y. Tensile damage evolution and notch sensitivity of Al2O3 fiber-ZrO2 matrix minicomposite-reinforced Al2O3 matrix composites. Mater Sci Eng A 2002;325(1–2):405–13. [6] Neumeister J, Jansson S, Leckie F. The effect of fiber architecture on the mechanical properties of carbon/carbon fiber composites. Acta Mater 1996;44(2):573–85. [7] Carelli EAV, Fujita H, Yang JY, Zok FW. Effects of thermal aging on the mechanical properties of a porous-matrix ceramic composite. J Am Ceram Soc 2002;85(3):595–602. [8] Goto K, Hatta H, Takahashi H, Kawada H. Effect of shear damage on the fracture behavior of carbon–carbon composites. J Am Ceram Soc 2001;84(6):1327–33. [9] Tushtev K, Horvath J, Koch D, Grathwohl G. Deformation and failure modeling of fiber reinforced ceramics with porous matrix. Adv Eng Mater 2004;6(8):664–9. [10] Heathcote JA, Gong X-Y, Yang JY, Ramamurty U, Zok FW. Inplane mechanical properties of an all-oxide ceramic composite. J Am Ceram Soc 1999;82(10):2721–30. [11] Tushtev K, Koch D, Horvath J, Grathwohl G. Mechanismen und Modellierung der Verformung und Scha¨digung keramischer Faserverbundwerkstoffe. Int J Mater Res 2006;97(10):1460–9. [12] Hatta H, Denk L, Watanabe T, Shiota I, Aly-Hassan MS. Fracture behavior of carbon–carbon composites with cross-ply lamination. J Compos Mater 2004;38(17):1479–94. [13] Koch D, Tushtev K, Kuntz M, Knoche R, Horvath J, Grathwohl G. Modelling of deformation and damage evolution of CMC with strongly anisotropic properties. In: Lara-Curzio E, editor. Proc of int conf advanced ceramics and composites, vol. 26(2). Cocoa Beach Proceed; 2005. p. 107–14. [14] Koch D, Kuntz M, Tushtev K, Knoche R, Horvath J, Grathwohl G. Keramische Faserverbundwerkstoffe mit schwacher Matrix – Eigenschaften und Modellierung: Verbundwerkstoffe und Werkstoffverbunde; Schlimmer M [Hrsg.], Werkstoff-Informationsgesellschaft, Frankfurt; 2005. p. 133–8. 0.0 0.1 0.2 0.3 0.4 0.5 0 100 200 300 400 500 Ligament 12 mm Ligament 3 mm Experiments FE - Calculation Nominal Stress [MPa] Laser Strain (Lo=25mm) [%] 0°/90° 0.0 0.2 0.4 0.6 0.8 1.0 0 50 100 150 200 250 300 350 Ligament 3 mm Ligament 12 mm Experiments FE - Calculation Nominal Stress [MPa] Laser Strain (Lo=25mm) [%] -45°/+45° Fig. 11. Stress–strain curves of DEN tests and respective modeling dependent on ligament width and fiber orientation (a) 0/90; (b)+45/ 45. 1172 D. Koch et al. / Composites Science and Technology 68 (2008) 1165–1172