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Multiscale failure assessment of composite laminates
Multiscale failure assessment of composite laminates
15
R. Talreja Texas A&M University, College Station, TX, USA; Luleå University of Technology, Luleå, Sweden
15.1
Introduction
The deformation and failure behavior of composite materials is determined by heterogeneities within their volume. In fiber-reinforced composites, the subject of this chapter, failure initiates in fibers, in the matrix, or at fiberematrix interfaces, depending on the local stress fields generated by interactions between fibers and matrix within the “horizon” called a microscale. This scale depends on the fiber, matrix, and interface properties, and on the fiber size, shape, and distribution within the matrix. It is often as large as a few fiber diameters. In Chapter 3, we discussed initiation of failure events (matrix and fiberematrix interface cracks). For unidirectional (UD) composites not bonded to differently oriented UD composites, the initial failure events often quickly lead to final failure and may be seen as defining the ultimate load-carrying capacity (“strength”). Further progression of the failure process in composite laminates, where the UD composites are the building blocks (laminae), takes the form of distributed multiple cracks within the laminae. These cracks are collectively called damage. The mechanical response of laminates with damage shows deformational response (stiffness) changes measured at a larger scale. A multiscale approach, called synergistic damage mechanics, was described in Chapter 13 to perform durability assessment of stiffness-critical structures. In most cases, a composite structure retains its load-bearing capacity to a significant level, even when its deformational response has become critical from a design point of view. In some structures, retention of the load-bearing capacity to a prespecified level is a design requirement, and loss of this capacity is defined as failure. This chapter is concerned with a multiscale approach to failure assessment for such structures.
15.2
The laminate failure process
Figure 15.1 depicts an overview of the failure process in composite laminates based on a large number of studies, mainly in tensionetension fatigue (Jamison et al., 1984). It summarizes qualitative features of the evolution of failure events and serves to illustrate the nature of the mechanisms that must be analyzed to assess structural failure. As depicted in the figure, the early stage of the failure events consists of multiple cracking within the plies (transverse cracking), which eventually reaches a saturation Modeling Damage, Fatigue and Failure of Composite Materials. http://dx.doi.org/10.1016/B978-1-78242-286-0.00015-7 Copyright © 2016 Elsevier Ltd. All rights reserved.
350
Modeling Damage, Fatigue and Failure of Composite Materials
1. Matrix cracking
3. Delamination 5. Fracture
Damage
0º
0º
0º
0º
CDS (crack saturation)
0º 0º 2. Crack couplinginterfacial debonding
0º
0º
4. Fiber breakage
Percent of life in fatigue (or applied stress)
100
Figure 15.1 Schematic depiction of evolution of failure events (damage) in a broad class of laminates subjected to static or fatigue loading. Jamison et al. (1984).
state labeled as CDS (characteristic damage state) in Figure 15.1. This stage of the failure process affects the deformational response measured as stiffness properties averaged over a representative volume element (RVE), which defines the mesoscale in the multiscale approach described in Chapter 13. Beyond the CDS, the cracks within the plies divert into the ply interfaces, causing linkages of the cracks and delamination. The final stage of total failure involves extensive fiber failures. In assessing the load-bearing capacity of a laminate, the following observations are useful: • • •
The ultimate failure of a laminate involves fiber breakage, which is a statistical process. The load-bearing capacity of a laminate depends on the degree of localization of the failure events. The depletion of the load-bearing capacity is direction-dependent, that is, the laminate “strength” is anisotropic.
15.3
Traditional approaches to composite failure
Historically, the early failure theories (or criteria) for composite materials were developed before the failure mechanisms had been understood. The attempts at developing those theories were concerned with the anisotropy of the ultimate failure load. The first
Multiscale failure assessment of composite laminates
351
such attempt by Azzi and Tsai (1965) resorted to the orthotropic yield criterion proposed by Hill (1948) for a metal sheet rolled in one direction. The AzzieTsai criterion for a UD composite replaced the normal yield constants in the Hill theory by normal strength (ultimate normal failure stress) values in the fiber and transverse directions, and equated the in-plane shear yield stress to the corresponding shear strength of a UD composite. Later, Tsai and Wu (1971) proposed a more general anisotropic failure criterion for UD composites using the tensor polynomial formulation put forth by Gol’denblat and Kopnov (1965). That criterion produced an ellipsoid in the rectangular coordinate axes given by the two normal stresses and the in-plane shear stress. The single, smooth ellipsoid gave certain fundamental inconsistencies, pointed out by Hashin (1980), who proposed ways to mitigate those. Hashin’s proposals retained the strength formulations for anisotropic homogeneous solids, but introduced piecewise smooth failure surfaces instead of one ellipsoid of the TsaieHill criterion. Hashin also proposed separating the fiber failure mode from the matrix failure mode in a UD composite, and introduced the notion of failure on a plane inclined to fibers under combined stresses in the matrix. Hashin did not, however, pursue determining the inclination of the failure plane, other than to suggest an optimization principle for determining it. Puck and associates (Puck, 1992; Puck and Sch€urmann, 1998; Puck et al., 2002) continued where Hashin left off, proposing an elaborate scheme for determining material constants in the resulting formulation. Talreja (2014) has examined in detail the failure theories summarized above, pointing to the fundamental limitations faced in restricting the theories to the macroscale, that is, considering the UD composite as a homogenized solid. In that work, a way forward to overcome the limitations was proposed. Essentially, it consists of conducting a multiscale analysis of failure initiation at the microscale followed by an RVE-based mesoscale analysis of the further progression of failure events. In the proposed approach, a methodology for incorporating manufacturing defects in the multiscale analysis was also proposed. Leaving the treatment of defects to another chapter, the multiscale analysis without the account of defects is discussed next.
15.4
Multiscale failure analysis for load-bearing capacity
In Chapter 3 we discussed microscale analysis of failure initiation. That analysis is a precursor to the mesoscale analysis for deformational response discussed in Chapter 13 and corresponds to the ply-cracking stage until CDS, as depicted in Figure 15.1. Beyond CDS, the failure process involves crack coupling and delamination, resulting in increasing localization of the failure process. The share of the load carried in the local region diminishes as energy dissipated by cracking in the region increases. Final failure comes from unstable progression of the failure events in the region. A mesoscale analysis of the load-bearing capacity requires constructing an RVE within which progression of the energy-dissipating processes can be analyzed. Owing to complexity of the geometry of the crack surfaces formed in the local region,
352
Modeling Damage, Fatigue and Failure of Composite Materials
Interior delaminations Interior delaminations Axial splits
Axial splits
Transverse cracks
Transverse cracks
Figure 15.2 X-ray radiograph (top left) shows interior details of failure events in a cross-ply laminate. The arrows show transverse cracks, axial splits, and interior delaminations. The 3D sketch (top right) depicts the same failure events for clarity. The bottom left image shows transverse cracks with interface cracks at the transverse crack fronts. The bottom right image shows closer detail of coupling of transverse cracks through the interface.
analytical solutions to the crack growth problem are not expected. Instead, a computational scheme for incremental analysis of the advances in crack surfaces is to be devised. For illustration of the complexity involved, Figure 15.2 shows an example of the failure events in a cross-ply laminate subjected to tensionetension fatigue. The X-ray radiograph in the figure shows 2D projections of the interior surfaces (cracks) formed in the stages beyond CDS as indicated in Figure 15.1. The preCDS transverse cracks are marked in the figure, both in the X-ray radiograph and in the 3D sketch alongside. The post-CDS events are also marked. These consist of the axial splits at the 0 /90 ply interfaces of the cross-ply laminate and the interface cracks labeled as interior delaminations to distinguish them from the free-edge delaminations. The figure also shows close-up images of transverse cracks with associated interface cracks and details of coupling of transverse cracks through interfaces. Figure 15.3 illustrates an RVE of the laminate with localized failure events representative of the post-CDS failure process depicted in Figure 15.2. At a given stage in this process, the load-bearing capacity of the laminate can be examined by computationally analyzing the RVE response to selected boundary tractions or displacements, for example, illustrated by the axial displacement u in Figure 15.3. The local stress fields calculated by a numerical method, such as the finite element method, will provide a basis for incrementally advancing the crack surfaces in the RVE by a suitable criterion, such as that based on energy release rate. The loadedisplacement response
Multiscale failure assessment of composite laminates
353
u P
Interior delaminations Transverse cracks
Axial splits
u
u Figure 15.3 An RVE of a laminate containing localized failure events subjected to an axial displacement u (left) and forceedisplacement response (right).
of the RVE is illustrated in Figure 15.3 as well. As indicated there, the load response to the imposed axial displacement shows linearity until the preexisting crack surfaces begin growing. The illustrated load drop in the depicted loadedisplacement curve is indicative of an unstable crack growth. If the unstably growing crack surfaces are not arrested by interfaces or by merger with other crack surfaces, the load will continue to drop. Otherwise, further increase in the imposed displacement may overcome the barriers that arrested the cracks, leading eventually to unstoppable growth of the crack surfaces. A total loss of the load carried by the RVE will involve fiber breakage. The statistical nature of the fiber breakage process will render the ultimate failure load a statistical variable.
15.4.1 Virtual testing in multiscale failure analysis The computational failure analysis of the RVE illustrated in Figure 15.3 may be viewed as a virtual axial tensile test. Although such a test can also be performed as a real, physical test and the loadedisplacement curve of the type in Figure 15.3 can also be plotted, physical tests are generally not easy to perform. For instance, a shear test is significantly more demanding than an axial test, and applying shear in combination with tension and/or compression is a challenge not always possible to tackle. In view of these difficulties, conducting virtual tests seems to be an attractive option. However, while the failure events in a physical test occur according to natural
354
Modeling Damage, Fatigue and Failure of Composite Materials
processes, the outcome of a virtual test depends on the assumed failure models and criteria. If the assumed models and the associated governing failure criteria have been verified and validated, then using these in simulated (virtual) general loading conditions may be acceptable. For instance, growth of traction-free surfaces under brittle fracture conditions using the Griffith criterion (critical energy release rate) is sufficiently validated and has been used extensively in failure simulations. That is not the case with cohesive zone models, advocated, for instance, by Llorca et al. (2011) for interlaminar cracking (delamination). The assumed tractionedisplacement relationship (cohesive failure law) and the associated criticality conditions require numerous material constants (expressing interface “strength” and “toughness”), whose validity for complex failure processes, such as advancement of interconnected intralaminar and interlaminar cracks, cannot be verified. It is advisable to use simpler failure models with fewer material constants than to use seemingly versatile models that require material constants of uncertain interpretation and verification. In any case, the assumed failure models must be scrutinized for a relatively simple case such as the axial test described above. It is important to clarify the nature of depletion of the load-bearing capacity. Referring to Figure 15.3 again, if the failure analysis of the RVE produces multiple load drops in the loadedisplacement curve, then taking the first load drop as an operating failure condition is advisable. Any additional load-bearing capacity beyond the first load drop may not be reliable for practical purposes. Furthermore, the load drop associated with final failure (“breakup”) is governed by fiber breakage, which is inherently a statistical phenomenon. Using this value is only meaningful if appropriate probabilistic methods are used. Cox and Yang (2006) assessed the prospects of virtual testing as a design and certification tool for composite structures using cohesive zone models and multiscale failure analyses. They cautioned that approaching structural failure with a top-down approach, that is, using the sequence of macro-meso-microscales, could lead to many material constants. However, they feared that the bottom-up approach may not fully capture the controlling failure mechanisms. Later, Yang et al. (2011) examined, by elaborate modeling of failure events at different length scales, one particular case of localized damage—a double-notch tension test of a cross-ply laminate. The failure events examined by them are essentially those described in Figures 15.2 and 15.3, which were based on observations of smooth (unnotched) specimens of crossply laminates subjected to tensionetension fatigue (Jamison et al., 1984). The Yang et al. (2011) modeling strategy was not fully validated by experimental observation, and they admitted that major challenges still remained in virtual testing as a viable approach in failure analysis of composite structures.
15.5
Conclusions
This chapter has focused on the load-bearing capacity of a general composite laminate that has undergone localized damage. This type of damage is typically found to occur after the distributed intralaminar cracking has nearly or fully reached its saturation level. The localized damage then takes the form of interconnection of these cracks
Multiscale failure assessment of composite laminates
355
through interlaminar cracking, which on sufficient progression causes delamination and subsequent fiber breakage. The load-bearing capacity of a laminate with interconnected intralaminar cracks has been addressed as a problem of analyzing the response to the boundary loading of a finite volume element containing a representative amount of these cracks. Discontinuities in the loadedisplacement response have been suggested as indicators of instabilities in the progression of the localized damage. These instabilities provide limits to the load-bearing capacity of the laminate, and they have been proposed as criteria for laminate “strength.” The traditional way to determine composite strength properties is to perform physical experiments. The multiscale failure analysis proposed here suggests instead performing computational simulation of the response of a RVE of localized damage to assess the load-bearing capacity. The computational simulation may be viewed as “virtual” tests. It is advisable, however, to validate such tests by selected benchmark tests of the physical type before using them for extensive failure assessment.
References Azzi, V.D., Tsai, S., 1965. Anisotropic strength of composites. Experimental Mechanics 5, 283e288. Cox, B., Yang, Q., 2006. In quest of virtual tests for structural composites. Science 314, 1102e1107. Gol’denblat, I.I., Kopnov, V.A., 1965. Strength criteria for anisotropic materials. Mecchanica IMZGA, Izvestia Academy, Nauk USSR 6, 77e83. Hashin, Z., 1980. Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics 47, 329e334. Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281e297. Jamison, R.D., Schulte, K., Reifsnider, K.L., Stinchcomb, W.W., 1984. Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy laminates. In: Effects of Defects in Composite Materials, ASTM STP, 836, 21e55. LLorca, J., Gonzalez, C., Molina-Aldareguía, J.M., Segurado, J., Seltzer, R., Sket, F., Canal, L.P., 2011. Multiscale modeling of composite materials: a roadmap towards virtual testing. Advanced Materials 23 (44), 5130e5147. Puck, A., 1992. A failure criterion shows the direction. Kunststoffe, German Plastics 82, 29e32 (German text pp. 607e610). Puck, A., Sch€urmann, H., 1998. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology 58, 1045e1067. Puck, A., Kopp, J., Knops, M., 2002. Guidelines for the determination of the parameters in Puck’s action plane strength criterion. Composites Science and Technology 62, 371e378. Talreja, R., 2014. Assessment of the fundamentals of failure theories for composite materials. Composites Science and Technology 105, 190e201. Tsai, S.W., Wu, E.M., 1971. A general theory of strength for anisotropic materials. Journal of Composite Materials 5, 58e80. Yang, Q.D., Cox, B.N., Fang, X.J., Zhou, Z.Q., 2011. Virtual testing for advanced aerospace composites: advances and future needs. Journal of Engineering Materials and Technology 133 (1), 011002.
15
R. Talreja Texas A&M University, College Station, TX, USA; Luleå University of Technology, Luleå, Sweden
15.1
Introduction
The deformation and failure behavior of composite materials is determined by heterogeneities within their volume. In fiber-reinforced composites, the subject of this chapter, failure initiates in fibers, in the matrix, or at fiberematrix interfaces, depending on the local stress fields generated by interactions between fibers and matrix within the “horizon” called a microscale. This scale depends on the fiber, matrix, and interface properties, and on the fiber size, shape, and distribution within the matrix. It is often as large as a few fiber diameters. In Chapter 3, we discussed initiation of failure events (matrix and fiberematrix interface cracks). For unidirectional (UD) composites not bonded to differently oriented UD composites, the initial failure events often quickly lead to final failure and may be seen as defining the ultimate load-carrying capacity (“strength”). Further progression of the failure process in composite laminates, where the UD composites are the building blocks (laminae), takes the form of distributed multiple cracks within the laminae. These cracks are collectively called damage. The mechanical response of laminates with damage shows deformational response (stiffness) changes measured at a larger scale. A multiscale approach, called synergistic damage mechanics, was described in Chapter 13 to perform durability assessment of stiffness-critical structures. In most cases, a composite structure retains its load-bearing capacity to a significant level, even when its deformational response has become critical from a design point of view. In some structures, retention of the load-bearing capacity to a prespecified level is a design requirement, and loss of this capacity is defined as failure. This chapter is concerned with a multiscale approach to failure assessment for such structures.
15.2
The laminate failure process
Figure 15.1 depicts an overview of the failure process in composite laminates based on a large number of studies, mainly in tensionetension fatigue (Jamison et al., 1984). It summarizes qualitative features of the evolution of failure events and serves to illustrate the nature of the mechanisms that must be analyzed to assess structural failure. As depicted in the figure, the early stage of the failure events consists of multiple cracking within the plies (transverse cracking), which eventually reaches a saturation Modeling Damage, Fatigue and Failure of Composite Materials. http://dx.doi.org/10.1016/B978-1-78242-286-0.00015-7 Copyright © 2016 Elsevier Ltd. All rights reserved.
350
Modeling Damage, Fatigue and Failure of Composite Materials
1. Matrix cracking
3. Delamination 5. Fracture
Damage
0º
0º
0º
0º
CDS (crack saturation)
0º 0º 2. Crack couplinginterfacial debonding
0º
0º
4. Fiber breakage
Percent of life in fatigue (or applied stress)
100
Figure 15.1 Schematic depiction of evolution of failure events (damage) in a broad class of laminates subjected to static or fatigue loading. Jamison et al. (1984).
state labeled as CDS (characteristic damage state) in Figure 15.1. This stage of the failure process affects the deformational response measured as stiffness properties averaged over a representative volume element (RVE), which defines the mesoscale in the multiscale approach described in Chapter 13. Beyond the CDS, the cracks within the plies divert into the ply interfaces, causing linkages of the cracks and delamination. The final stage of total failure involves extensive fiber failures. In assessing the load-bearing capacity of a laminate, the following observations are useful: • • •
The ultimate failure of a laminate involves fiber breakage, which is a statistical process. The load-bearing capacity of a laminate depends on the degree of localization of the failure events. The depletion of the load-bearing capacity is direction-dependent, that is, the laminate “strength” is anisotropic.
15.3
Traditional approaches to composite failure
Historically, the early failure theories (or criteria) for composite materials were developed before the failure mechanisms had been understood. The attempts at developing those theories were concerned with the anisotropy of the ultimate failure load. The first
Multiscale failure assessment of composite laminates
351
such attempt by Azzi and Tsai (1965) resorted to the orthotropic yield criterion proposed by Hill (1948) for a metal sheet rolled in one direction. The AzzieTsai criterion for a UD composite replaced the normal yield constants in the Hill theory by normal strength (ultimate normal failure stress) values in the fiber and transverse directions, and equated the in-plane shear yield stress to the corresponding shear strength of a UD composite. Later, Tsai and Wu (1971) proposed a more general anisotropic failure criterion for UD composites using the tensor polynomial formulation put forth by Gol’denblat and Kopnov (1965). That criterion produced an ellipsoid in the rectangular coordinate axes given by the two normal stresses and the in-plane shear stress. The single, smooth ellipsoid gave certain fundamental inconsistencies, pointed out by Hashin (1980), who proposed ways to mitigate those. Hashin’s proposals retained the strength formulations for anisotropic homogeneous solids, but introduced piecewise smooth failure surfaces instead of one ellipsoid of the TsaieHill criterion. Hashin also proposed separating the fiber failure mode from the matrix failure mode in a UD composite, and introduced the notion of failure on a plane inclined to fibers under combined stresses in the matrix. Hashin did not, however, pursue determining the inclination of the failure plane, other than to suggest an optimization principle for determining it. Puck and associates (Puck, 1992; Puck and Sch€urmann, 1998; Puck et al., 2002) continued where Hashin left off, proposing an elaborate scheme for determining material constants in the resulting formulation. Talreja (2014) has examined in detail the failure theories summarized above, pointing to the fundamental limitations faced in restricting the theories to the macroscale, that is, considering the UD composite as a homogenized solid. In that work, a way forward to overcome the limitations was proposed. Essentially, it consists of conducting a multiscale analysis of failure initiation at the microscale followed by an RVE-based mesoscale analysis of the further progression of failure events. In the proposed approach, a methodology for incorporating manufacturing defects in the multiscale analysis was also proposed. Leaving the treatment of defects to another chapter, the multiscale analysis without the account of defects is discussed next.
15.4
Multiscale failure analysis for load-bearing capacity
In Chapter 3 we discussed microscale analysis of failure initiation. That analysis is a precursor to the mesoscale analysis for deformational response discussed in Chapter 13 and corresponds to the ply-cracking stage until CDS, as depicted in Figure 15.1. Beyond CDS, the failure process involves crack coupling and delamination, resulting in increasing localization of the failure process. The share of the load carried in the local region diminishes as energy dissipated by cracking in the region increases. Final failure comes from unstable progression of the failure events in the region. A mesoscale analysis of the load-bearing capacity requires constructing an RVE within which progression of the energy-dissipating processes can be analyzed. Owing to complexity of the geometry of the crack surfaces formed in the local region,
352
Modeling Damage, Fatigue and Failure of Composite Materials
Interior delaminations Interior delaminations Axial splits
Axial splits
Transverse cracks
Transverse cracks
Figure 15.2 X-ray radiograph (top left) shows interior details of failure events in a cross-ply laminate. The arrows show transverse cracks, axial splits, and interior delaminations. The 3D sketch (top right) depicts the same failure events for clarity. The bottom left image shows transverse cracks with interface cracks at the transverse crack fronts. The bottom right image shows closer detail of coupling of transverse cracks through the interface.
analytical solutions to the crack growth problem are not expected. Instead, a computational scheme for incremental analysis of the advances in crack surfaces is to be devised. For illustration of the complexity involved, Figure 15.2 shows an example of the failure events in a cross-ply laminate subjected to tensionetension fatigue. The X-ray radiograph in the figure shows 2D projections of the interior surfaces (cracks) formed in the stages beyond CDS as indicated in Figure 15.1. The preCDS transverse cracks are marked in the figure, both in the X-ray radiograph and in the 3D sketch alongside. The post-CDS events are also marked. These consist of the axial splits at the 0 /90 ply interfaces of the cross-ply laminate and the interface cracks labeled as interior delaminations to distinguish them from the free-edge delaminations. The figure also shows close-up images of transverse cracks with associated interface cracks and details of coupling of transverse cracks through interfaces. Figure 15.3 illustrates an RVE of the laminate with localized failure events representative of the post-CDS failure process depicted in Figure 15.2. At a given stage in this process, the load-bearing capacity of the laminate can be examined by computationally analyzing the RVE response to selected boundary tractions or displacements, for example, illustrated by the axial displacement u in Figure 15.3. The local stress fields calculated by a numerical method, such as the finite element method, will provide a basis for incrementally advancing the crack surfaces in the RVE by a suitable criterion, such as that based on energy release rate. The loadedisplacement response
Multiscale failure assessment of composite laminates
353
u P
Interior delaminations Transverse cracks
Axial splits
u
u Figure 15.3 An RVE of a laminate containing localized failure events subjected to an axial displacement u (left) and forceedisplacement response (right).
of the RVE is illustrated in Figure 15.3 as well. As indicated there, the load response to the imposed axial displacement shows linearity until the preexisting crack surfaces begin growing. The illustrated load drop in the depicted loadedisplacement curve is indicative of an unstable crack growth. If the unstably growing crack surfaces are not arrested by interfaces or by merger with other crack surfaces, the load will continue to drop. Otherwise, further increase in the imposed displacement may overcome the barriers that arrested the cracks, leading eventually to unstoppable growth of the crack surfaces. A total loss of the load carried by the RVE will involve fiber breakage. The statistical nature of the fiber breakage process will render the ultimate failure load a statistical variable.
15.4.1 Virtual testing in multiscale failure analysis The computational failure analysis of the RVE illustrated in Figure 15.3 may be viewed as a virtual axial tensile test. Although such a test can also be performed as a real, physical test and the loadedisplacement curve of the type in Figure 15.3 can also be plotted, physical tests are generally not easy to perform. For instance, a shear test is significantly more demanding than an axial test, and applying shear in combination with tension and/or compression is a challenge not always possible to tackle. In view of these difficulties, conducting virtual tests seems to be an attractive option. However, while the failure events in a physical test occur according to natural
354
Modeling Damage, Fatigue and Failure of Composite Materials
processes, the outcome of a virtual test depends on the assumed failure models and criteria. If the assumed models and the associated governing failure criteria have been verified and validated, then using these in simulated (virtual) general loading conditions may be acceptable. For instance, growth of traction-free surfaces under brittle fracture conditions using the Griffith criterion (critical energy release rate) is sufficiently validated and has been used extensively in failure simulations. That is not the case with cohesive zone models, advocated, for instance, by Llorca et al. (2011) for interlaminar cracking (delamination). The assumed tractionedisplacement relationship (cohesive failure law) and the associated criticality conditions require numerous material constants (expressing interface “strength” and “toughness”), whose validity for complex failure processes, such as advancement of interconnected intralaminar and interlaminar cracks, cannot be verified. It is advisable to use simpler failure models with fewer material constants than to use seemingly versatile models that require material constants of uncertain interpretation and verification. In any case, the assumed failure models must be scrutinized for a relatively simple case such as the axial test described above. It is important to clarify the nature of depletion of the load-bearing capacity. Referring to Figure 15.3 again, if the failure analysis of the RVE produces multiple load drops in the loadedisplacement curve, then taking the first load drop as an operating failure condition is advisable. Any additional load-bearing capacity beyond the first load drop may not be reliable for practical purposes. Furthermore, the load drop associated with final failure (“breakup”) is governed by fiber breakage, which is inherently a statistical phenomenon. Using this value is only meaningful if appropriate probabilistic methods are used. Cox and Yang (2006) assessed the prospects of virtual testing as a design and certification tool for composite structures using cohesive zone models and multiscale failure analyses. They cautioned that approaching structural failure with a top-down approach, that is, using the sequence of macro-meso-microscales, could lead to many material constants. However, they feared that the bottom-up approach may not fully capture the controlling failure mechanisms. Later, Yang et al. (2011) examined, by elaborate modeling of failure events at different length scales, one particular case of localized damage—a double-notch tension test of a cross-ply laminate. The failure events examined by them are essentially those described in Figures 15.2 and 15.3, which were based on observations of smooth (unnotched) specimens of crossply laminates subjected to tensionetension fatigue (Jamison et al., 1984). The Yang et al. (2011) modeling strategy was not fully validated by experimental observation, and they admitted that major challenges still remained in virtual testing as a viable approach in failure analysis of composite structures.
15.5
Conclusions
This chapter has focused on the load-bearing capacity of a general composite laminate that has undergone localized damage. This type of damage is typically found to occur after the distributed intralaminar cracking has nearly or fully reached its saturation level. The localized damage then takes the form of interconnection of these cracks
Multiscale failure assessment of composite laminates
355
through interlaminar cracking, which on sufficient progression causes delamination and subsequent fiber breakage. The load-bearing capacity of a laminate with interconnected intralaminar cracks has been addressed as a problem of analyzing the response to the boundary loading of a finite volume element containing a representative amount of these cracks. Discontinuities in the loadedisplacement response have been suggested as indicators of instabilities in the progression of the localized damage. These instabilities provide limits to the load-bearing capacity of the laminate, and they have been proposed as criteria for laminate “strength.” The traditional way to determine composite strength properties is to perform physical experiments. The multiscale failure analysis proposed here suggests instead performing computational simulation of the response of a RVE of localized damage to assess the load-bearing capacity. The computational simulation may be viewed as “virtual” tests. It is advisable, however, to validate such tests by selected benchmark tests of the physical type before using them for extensive failure assessment.
References Azzi, V.D., Tsai, S., 1965. Anisotropic strength of composites. Experimental Mechanics 5, 283e288. Cox, B., Yang, Q., 2006. In quest of virtual tests for structural composites. Science 314, 1102e1107. Gol’denblat, I.I., Kopnov, V.A., 1965. Strength criteria for anisotropic materials. Mecchanica IMZGA, Izvestia Academy, Nauk USSR 6, 77e83. Hashin, Z., 1980. Failure criteria for unidirectional fiber composites. Journal of Applied Mechanics 47, 329e334. Hill, R., 1948. A theory of the yielding and plastic flow of anisotropic materials. Proceedings of the Royal Society of London A 193, 281e297. Jamison, R.D., Schulte, K., Reifsnider, K.L., Stinchcomb, W.W., 1984. Characterization and analysis of damage mechanisms in tension-tension fatigue of graphite/epoxy laminates. In: Effects of Defects in Composite Materials, ASTM STP, 836, 21e55. LLorca, J., Gonzalez, C., Molina-Aldareguía, J.M., Segurado, J., Seltzer, R., Sket, F., Canal, L.P., 2011. Multiscale modeling of composite materials: a roadmap towards virtual testing. Advanced Materials 23 (44), 5130e5147. Puck, A., 1992. A failure criterion shows the direction. Kunststoffe, German Plastics 82, 29e32 (German text pp. 607e610). Puck, A., Sch€urmann, H., 1998. Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology 58, 1045e1067. Puck, A., Kopp, J., Knops, M., 2002. Guidelines for the determination of the parameters in Puck’s action plane strength criterion. Composites Science and Technology 62, 371e378. Talreja, R., 2014. Assessment of the fundamentals of failure theories for composite materials. Composites Science and Technology 105, 190e201. Tsai, S.W., Wu, E.M., 1971. A general theory of strength for anisotropic materials. Journal of Composite Materials 5, 58e80. Yang, Q.D., Cox, B.N., Fang, X.J., Zhou, Z.Q., 2011. Virtual testing for advanced aerospace composites: advances and future needs. Journal of Engineering Materials and Technology 133 (1), 011002.