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Micromechanical modeling of composites with mechanical interface – Part II: Damage mechanics assessment
COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 66 (2006) 323–332 www.elsevier.com/locate/compscitech
Micromechanical modeling of composites with mechanical interface – Part II: Damage mechanics assessment Nicola Bonora *, Andrew Ruggiero DiMSAT – Department of Mechanics, Structures and Environment, University of Cassino, Via G. Di Biasio 43, I-03043 Cassino, Italy Received 7 March 2005; accepted 12 April 2005 Available online 1 July 2005
Abstract Continuing the work initiated in the Part I [Bonora N, Ruggiero A. Micromechanical modeling of composites with mechanical interface – Part I: unit cell model development and manufacturing process effects. Compos Sci Technol 2003], in this paper the possibility to account for different damage mechanisms, in the unit cell model (UCM), explicitly developed for composites with mechanical interface, is discussed and results for Ti-15-3/SCS-6 composite laminates are presented. Starting from the analysis of the constituent behaviors a probabilistic model based on Weibull statistics is developed for fiber failure, while a ductile damage model which incorporates stress triaxiality effect has been used for predicting metal matrix progressive failure. Fiber–matrix debonding process has been naturally predicted incorporating the material manufacturing process in the stress/strain history. Numerical results performed with the UCM applied to 0 and 90 unidirectional laminates, loaded both in tension and compression, have been compared with experimental results at both macro- and microscopic scale. 2005 Elsevier Ltd. All rights reserved. Keywords: Ductile damage; CDM; MMC; FEM; Plasticity; Multiscale damage
1. Introduction Composite materials find a new field of application almost every day, often as a lighter substitute material to more traditional steels and light alloys, due to their high specific resistance capabilities, low weight, and for the possibility to design components and parts assigning the desired material properties where needed most. Metal matrix composites represent a different generation of materials explicitly designed for high temperature applications. Here, the idea of combining the strength of ceramic reinforcement with the ductility of metal matrix leaded to materials capable to offer high stiffness and strength, as well as fatigue resistance and reduced overall weight, at elevated temperature up to *
Corresponding author. E-mail address: [email protected] (N. Bonora).
0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.04.043
700 C, approximately. The major structural differences of these composites, with respect to polymeric matrix based composites, mainly are: the elastic–plastic behavior of the matrix, which can sustain large plastic deformation, and the nature of interface between the matrix and the reinforcement. Usually, in these composites the interface is mechanical, in the sense that no chemical bond exists between the fiber and the matrix. The joining of the matrix with fiber occurs as a result of the differential shrinkage, due to the mismatch in the a-thermal expansion coefficients, during the cooling down from the assembly temperature to room temperature. A fundamental conceptual difference in the design with composite materials is that the material itself can be designed, at different dimensional scales (constituents, laminate stack sequence, and component) according to the most effectively required behaviors. To this purpose, reliable and practical design tools, capable to accurately
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predict overall material constitutive behavior at different dimensional scales, have been investigated in the last decades. In this framework, micromechanics approach has been widely used by a number of authors [1–3]. Typically, a unit cell model (UCM) is built on the assumption of material periodic microstructure. Finite element technology is used to calculate cell response incorporating complex features, such as constituentsÕ non-linear behavior, contact, combined thermo-mechanical loading, etc. This approach allows predicting the occurrence of failure conditions if the micro-mechanisms of failure, characteristic for the material, are identified and appropriate damage modeling is added in. In this way, damage evolution in the material microstructure can also be followed and the progressive degradation of the composite overall constitutive response accurately predicted. In the part I [5], the authors analyzed the role of the manufacturing process performing an extensive finite element investigation on SCS-6/Ti-15-3 unidirectional laminates. They demonstrated that if the consolidating phase, that is commonly performed by hot isostatic press (HIP) technique, is simulated within the UCM, the composite macroscopic response can be accurately predicted without the need to develop artificial model for the interface strength. In this paper, continuing the previous work, microstructural damage evolution in SCS-6/Ti-15-3 composite laminates has been investigated. The damage mechanisms, which may occur in the microstructure under both tension and compression loading, have been identified and specific damage model for each of them have been developed and implemented in the UCM. Damage evolution during loading has been compared with experimental in situ observations, given by Majumdar and Newaz [23], in order to verify the accuracy of the proposed approach in predicting how, where, and when damage develops. In Section 2, the UCM formulation proposed by the authors is briefly reviewed; in Section 3, the damage mechanisms and the modeling are presented; in Section 4 the description of the material used in the present investigation is given; in Section 5 the finite element results are discussed and compared with experimental data available in the literature.
2. Unit cell model and finite element modeling A UCM can be developed according to the periodicity of the material structure and the dimensional scale of interest. For a composite laminate, the smallest RVE can be taken at single-fiber level if the thickness of the lamina is big enough with respect to the fiber diameter. For closed package fiber layers, the UCM should account for between-fiber distance in order to accurately model local stress concentration and constraint. In metal matrix composites, fibers are quite bigger in diameter with re-
spect to more traditional carbon or glass fibers. For instance, in the case of SCS-6 fiber the average diameter is 140 lm approximately, in contrast to 7 lm diameter of standard T300 carbon fiber. In addition, foil-fibers-foil assembly technique, used for this material, assures a high degree of regularity in the fiber alignment and arrangement, resulting in real periodic microstructure. Since, the fiber diameter is comparable with the distance between two adjacent fibers, both along the lamina in-plane direction and the laminate thickness direction, the choice of the UCM may be influenced by the presence of an horizontal shift in the stacked plies as sketched in Fig. 1, where different choices for the UCM are depicted. Periodic boundary condition can be applied to the cell imposing either ‘‘plane-remains-plane’’ or ‘‘unified’’ periodic boundary conditions. Xia et al. [4] showed that the first is appropriate for in plane symmetric loading but can be over-constraining for shear loading while the latter performs better in all cases. Using plane strain or generalized plane strain elements the same UCM can be used to investigate both 90 and 0 unidirectional laminate response. If expanded in 3D, the same UCM can be used to investigate both cross-ply and angle ply laminate [4]. As far as concern SCS-6/Ti-15-3 composite laminates investigated here, the author observed that, according to microstructural visual observations, the more appropriate UCM is the one given in Fig. 2, due to the presence of a systematic shift in the ply stacking sequence. Under plane strain assumptions, imposing periodic ‘‘plane-remains-plane’’ boundary conditions, the average meso-strain and meso-stress are defined through the displacement and reaction forces at the cell boundaries as follows: u Ex ¼ ln 1 þ 0 ; Lx w ð1Þ Ez ¼ ln 1 þ 0 ; Lz Ey ¼ 0; where u; w; L0x ; L0z are the displacement along x- and z-axes and the cell reference dimensions, respectively. The in-plane meso-stresses are defined as: Fi i; j ¼ x; z; ð2Þ Ri ¼ Lj B where B is the cell thickness, m is Poisson ratio, and Lx and Lz the actual cell dimensions (i.e., Lx ¼ L0x þ u). Under generalized plane strain assumption, also the meso-strain and meso-stress along the normal axis (here, y-axis) can be also given as: v Ey ¼ ln 1 þ ; ð3Þ B Fy . ð4Þ Ri ¼ Lx Lz
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Fig. 1. Fiber arrangement in unidirectional MMC laminate: frontal view and unit cell for SCS/Ti-15-3.
Fig. 2. 90 Unit cell model: contour map of plastic strain in tension: (a) beginning of stage II; (b) beginning of stage III; (c) prior crack initiation in stage III. Black thick line indicates intact interface extension.
The UCM given in Fig. 2 has been implemented in the commercial finite element code MSC/MARC2003. Four node element with bilinear interpolating function has been used. The interface has been obtained modeling the contact between fiber and matrix considered as deformable bodies. In general, contact problems in FE are usually solved by means of Lagrange multipliers or penalty methods. Here, the so-called direct constraint method has been used. In this procedure, the motion of the bodies is tracked and, when contact occurs, direct constraints are placed on the motion using boundary conditions. This procedure can be very accurate if the program can predict when contact occurs. No special
interference elements are required in this procedure and complex changing contact conditions can be simulated since no a priori knowledge of where contact occurs is necessary. The details of the implementation are given in [25]. Elastic–plastic calculations have been performed using large displacement, Lagrangian updating and finite deformation formulation. The cooling down phase has been modeled assuming, for the cell, the stress-free temperature of 815 and accounting for material properties variation with temperature. Details on boundary condition, material properties and finite element cell performance assessment are discussed elsewhere [5].
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3. Damage mechanisms and modeling Damage is indicative of the irreversible modification processes and their mutual interactions, which occur in the material. These processes, even though highly localized in the material microstructure, directly affect the resulting material behavior in the upper dimensional scales (both meso- and macro-scale). Damage in metals is always associated to plastic deformation. Below the brittle–ductile transition temperature, plastic deformation is confined inside the grain and localized in one or few favorable oriented grains. Above the NDT, plastic deformation can spread over large macroscopic volumes, since all grains can plastically deform. Consequently, damage can be an evolutionary process, if plastic deformation occurs at macroscopic scale, or statistical in nature, if plastic deformation are confined to grain level as for brittle fracture. What differentiates ductile damage process from inelastic deformation is the degradation of the material average elastic moduli. This effect can be experimentally observed and monitored in order to quantify the damage evolution. From a dimensional scale point of view, damage considerations always refer to the processes that happen into a single RVE. Failure occurs when the damage variable reaches a critical value characteristic for the material. In ductile metals, the failure of one or few adjacent RVEs usually correspond to the appearance of a macroscopic crack, the growth of which may be eventually followed in the framework of fracture mechanics. Composite materials with mechanical interface are characterized by different damage mechanisms, peculiar of each phase and constituent. Additional micro-/mesomechanism of failure may arise from the reciprocal interaction of the constituents. The basic failure modes may be classified as follows: • reinforcement (fiber, particle or flake) rupture; • matrix damage; • matrix–reinforcement debonding. All these failure modes are driven by strain; take place at the RVE dimensional scale; modify the material overall constitutive response; may be singularly cause of the material catastrophic failure; usually occur in the material according to well defined sequences along with the type of the loading conditions (static, dynamic, impact, etc.). Consequently, for each of these, appropriate modeling is necessary. 3.1. Reinforcement failure The reinforcement in composites are mainly fibers (short or long) or particles (spheroid or flakes). Since the main role of the reinforcement is to sustain loads, its stiffness, as well as the yield and ultimate tensile
strength, is usually much higher than that of the matrix material. Fiber material is typically brittle and failure occurs when the maximum allowable stress is reached. Damage in fiber is a non-progressive process [6], due to the small fiber volume and net resisting section, which practically exclude the presence of potential growing flaws. Consequently, due to the high number of fibers in each layer, fiber failure assessment requires statistical considerations. As far as concern metal matrix composites, Curtin [7] developed an effective fiber breakage model to simulate longitudinal tensile behavior of MMC laminates. According to this, the average stress in the effective fiber bundle is given as: r 1 avg rf ¼ ¼ T 1 qðlf Þ ; ð5Þ f 2 where r is the longitudinal applied stress, f is the fiber volume fraction, T is the stress sustained by the unbroken fibers, q(lf) is the fraction of broken fibers and lf is half of the length of the fiber break region which experiences interfacial slip. The probability function of failure can be given according to Weibull distribution in the following form: m L r pf ¼ 1 exp ; ð6Þ L0 r0 where L0, m and r0 are the reference gauge length, the Weibull distribution exponent, and the average strength at pf = 0.62, respectively. Mahesh et al. [8] investigated strength distribution and size effect in unidirectional composites using Weibull statistics applied to fiber. In their approach, the fiber strength distribution was assumed according to Weibull–Poisson statistics as follows: q r F ðrÞ ¼ 1 exp . ð7Þ rd For 2D unit cell, as the one used here, the information about the fiber length cannot be inferred due to plane strain or generalized plane strain formulation employed. The local approach to failure, initially developed for brittle fracture in metals by Mudry [9], can be modified and used to determine the probability of failure across a reference fiber section. According to this, the fiber probability of failure can be written as follows: m rW pf ¼ 1 exp ; ð8Þ ru where ru and m are the classical stress scale factor and the Weibull exponent. The Weibull stress rW accounts for the maximum principal stress r1 acting over a fiber reference elementary volume V0, Z 1 rmW ¼ rm ðx; y; zÞ dV ; ð9Þ V0 V 1 where, for constant unit thickness, becomes,
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1 rmW P ¼const ¼ S0
Z
rm1 ðx; yÞ dS.
ð10Þ
S
S0 is the reference unit area. The Waibull exponent m is indicative of the variability in the material response associated with a local distribution of material properties and stress state, and, hence, to the extension of volume of the process zone where fracture is initiated. The larger the volume of the process zone the higher the probability to find any possible microstructural variation or flaw. Milella and Bonora [10], investigating the Weibull exponent dependencies in brittle fracture of metals, observed that the process volume depends on the triaxiality of the stress field geometrically induced by a crack or a notch. Superimposed stress triaxiality by external loading over smooth geometry does not change the size of the process zone that remains the entire geometry volume with no effect on m. In fibers, the entire volume participates in the fracture process, since fracture may initiates anywhere from microscopic flaws; consequently S0 in Eq. (10) becomes the entire fiber section. According to Milella and Bonora [10] the expected value of m for the fiber should be in the range of 20–45. These values are consistent with those found by Xia and Curtin [11] and Dutton et al. [12] using similar formulation but based on different considerations. Consequently, this formulation can be implemented in the cell in order to calculate the probability of failure with increasing applied external load. 3.2. Reinforcement–matrix debonding Matrix–reinforcement interface is fundamental for the stress transfer and for the composite material load carrying capability. In polymeric composite, the interface is a narrow region where the fiber, or particle, and the matrix are intimately connected through a chemical bond. Here, a continuous properties transition from those of the matrix to that of the fiber, or particle, is found. In MMC, the interface is mechanical in most of the cases. No chemical bound exists between the fiber, or particle, and the metal matrix. In the literature many papers have been presented in the past years trying to develop an interfaces strength model. A possible procedure to calibrate these models was to manually provide interface release during tensile loading in order to match the slope changes in the macroscopic stress–strain responses, Brust et al. [13]. Lissenden and Herakowich [14] used a traction–displacement, similar to that proposed by Tvergaard [15], that resembles theoretical atomic bond force–displacement response. Still recently, Knight et al. [16] performed a parametric study on the contact stresses associated to perfect interface. Aghdam et al. [17] performed an extensive finite element analysis, investigating the effect associated to perfect or partially debonded interface on the macroscopic stress–strain
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response of metal matrix composites by mean of unit cell model. Bonora and Ruggiero [5], starting from the consideration that the interface strength in MMC is determined by the cooling down ramp from the assembly to the room temperature, demonstrated that no specific interface model is necessary if the manufacturing phase is incorporated in the UCM stress/strain history. Here, the only numerical requirement is that both the fiber, or particle, and the matrix need to be modeled with finite elements as separated deformable bodies which, starting from the zero-contact stress condition at high temperature, may come in reciprocal contact during the cooling phase as a result of the mismatch in the a-thermal expansion coefficients. With this approach, debonding naturally develops in the UCM when the stresses resulting from external applied load exceed contact forces along the interface. As it will be demonstrated later in the paper, if the manufacturing process is simulated, the right time and location of matrix–fiber debonding can be accurately predicted, as well as the resulting macroscopic stiffness reduction due to the extension of the debonding interface. 3.3. Matrix damage Metal matrix material exhibits damage processes typical of metals. According to the operative temperature, with respect to the brittle–ductile transition temperature, metal may fails as a result of cleavage or ductile rupture, respectively. Since metal matrix composites are intended for high temperature applications ductile rupture is the failure mode of interest analyzed here. Ductile damage in metal occurs in the form of nucleation and growth of microvoids that are initiated either by second phase particle inclusions or by dislocations piling-up along the grain boundaries. In the literature, many models have been proposed in order to describe the constitutive response of porous plastic solids [18– 20]. Most of them suffer a number of limitations spreading from an elevated number of damage parameters needed to the lack of their transferability from specimen to structures. In the last decade, Bonora [21] developed a damage model in the framework of continuum damage mechanics (CDM) initially proposed by Lemaitre [20]. In this model, the damage variable accounts for all effects induced by the irreversible modifications that occur in the microstructure under plastic strain accumulation. Since there are clear experimental evidences that ductile damage affects material stiffness, while it is not possible to assess damage effects on yield surface since damage softening cannot be separated from hardening effect in a real tensile test, in this formulation damage accumulation modify material stiffness only. According to this no softening is present in the material flows stress avoiding
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possible localization related problems in the numerical simulations. More details regarding these aspects can be found elsewhere [21,26] Damage evolution, derived from the damage dissipation potential formulation, accounts for stress triaxiality effects, which are responsible for material ductility reduction. The basic set of constitutive equations for the damaged material is summarized as follows: eTij ¼ eeij þ epij ;
ð11Þ
1 þ m rij m rkk dij ; ð12Þ E 1D E 1D ofp 3 s_ ij ¼ k_ ; ð13Þ e_ pij ¼ k_ 2 req orij ofp _ _ ¼ k ¼ p; ð14Þ r_ ¼ k_ oR where Eqs. (13) and (14) are those for standard plasticity while the kinetic law of damage evolution is given by:
eeij ¼
ofD D_ ¼ k_ oY ¼a
1 a1 ðDcr D0 Þa rH p_ f ðDcr DÞ a ; p lnðef =eth Þ req
where 2 P 2 P f ¼ ð1 þ mÞ þ 3 ð1 2mÞ req 3 req
ð15Þ
higher temperature BCC b-phase. An extensive literature is available both as far as concern material experimental characterization and numerical modeling, Gooder and Mall [22], and Majumdar and Newaz [23] performed an extensive study on this material. In particular they performed a detailed investigation on the microstructural modifications occurring in the material during monotonic loading. Using both in situ observations and replica technique, they investigated 0 and 90 laminate response, under both tension and compression, at both RT and high temperature (538 and 700 C). In this paper the experimental observations performed by Majumbdar and Newaz [23] have been used to compare the predicted microstructural modifications and damage development under both tension and compression loading on 0 and 90 unidirectional laminates.
5. Finite element results and discussion The proposed unit cell model is capable to accurately predict macroscopic stress–strain response for both 0 and 90 unidirectional lamina under both tension and compression [5]. In the following, the damage development and the resulting effect on the macroscopic response are given according to the laminate orientation and the load type (tensile or compressive).
ð16Þ
accounts for stress triaxiality. The damage parameters needed are: eth, that is the damage threshold strain at which damage processes initiate; the theoretical strain to failure, ef, under constant triaxiality (stress triaxiality, TF = 0.333); the critical damage, Dcr, at which complete failure occurs, and the damage exponent, a, that defines the shape of damage evolution curve with strain. Detailed discussion on the damage model formulation and damage parameters identification can be found elsewhere [21].
4. Material The proposed approach can be potentially used for any composite, reinforced with either spherical particles or long fibers, with mechanical interface. The unit cell model has been developed with particular reference to a SCS6/Ti-15-3 laminates with 0.34 volume fraction content. This is a Ti–15V–3Cr–3Al–3Sn (Ti-15-3) weight percent alloy reinforced with SiC (SCS6) fibers with a reference diameter of 140 lm. Fibers production is obtained via carbon (C) and silicon (Si) vapor deposition around a soft carbon filament. Fiber are equally spaced by means of a molybdenum interweave. Titanium matrix is an allotropic material containing both lower temperature hexagonal close-packed (HCP) a-phase and
5.1. 90 laminate in tension A peculiar two-slopes feature, before work hardening, characterizes the uniaxial response of 90 unidirectional laminate in tension. According to this, the strain range can be divided in three stages: in stage I the material response is linear elastic, in stage II there is clear knee and a drop of the stress/strain slope, in stage III the stress–strain curve follows a power law which rapidly saturates, as given in Fig. 4. Microscopy investigations and replica analyses revealed that in stage I the matrix and the fiber remains perfectly bonded up to the first knee on the stress–strain curve where debonding starts to occur. In this phase some slip band between fibers initiate to appear. In stage two, extensive fiber debonding takes place together with an intensification of highly concentrated plastic strain band. The extension of the debonding spread up to 45–60 of the fiber circumference. In Fig. 2, the evolution of the debonding as well as the highly concentrated plastic strain band is given. Here, with the black thick line the portion of the intact interface is given. It is worth to note here that, even at fracture, the proposed UCM predicts that the interface is not completely failed. This result finds confirmation in the replica taken at different deformations stage and prior final fracture. In Fig. 3 the replica at 0.5% of overall strain is given together with the finite element
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Fig. 4. Calculated stress–strain curves for 90 laminate: solid line for the UCM without damage in the matrix but accounting for the debonding process; line and symbol for the UCM with the CDM damage modeling [21]; for the Ti matrix, hollow symbols indicate experimental data [10].
Fig. 3. Replica taken under loading at 0.5% strain on 90 laminate: black thick line in FE result indicates the undebonded interface, arrows indicate penetration of replica acetate, i.e., debonded region extension.
results. Black thick line indicates the still bonded interface extension. The break in the symmetry of the contours plots given here and in Figs. 6–8 is caused by small differences in the calculated reaction forces along the two contact surfaces that are magnified by the contour post-processing algorithm. In Fig. 4, the comparison of the predicted stress– strain response is compared with experimental data. During the test, partial unloading at given strain was performed in order to investigate damage development and related effects at the transition of stage II and III. Similarly to the experiments, partial unloading has been also simulated in order to show the possibility to follow, with the proposed unit cell model, non-monotonic load paths. Here, the predicted material response is in a very good agreement with the experimental data showing all the features that characterize the three-stages stress– strain curve for 90 MMC. It is interesting to note that the UCM is capable to predict the changing in slope during the first unloading mainly due to partial back contact between the matrix and the debonded fiber. These results were almost impossible to obtain with the contact techniques used by Brust et al. [13] and Bonora et al. [24]. The lowering of the slope in the third stage of the stress–strain curve is indicative of almost
Fig. 5. Calculated damage contours (white D = 0, black D = Dcr, i.e., crack initiation) compared with micrograph showing highly deformed band and crack initiation [23] in 90 laminate.
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complete debonding and strain accumulation in the ligament. Microscopy analyses reveal that final failure of 90 MMC unidirectional laminate is controlled by the formation of cracks along intense highly concentrated plastic strain band developing between fibers. Simulations performed incorporating the ductile damage model confirm that damage in the matrix develops together with the debonding of the fibers since the early stage II. In the stage III, where the debonding process is almost
completed, damage localizes in a narrow band, as given in Fig. 5, resulting in the formation of a microscopic crack at the interface (black thick line).
Table 1 Damage parameter for Ti-15-3 eth
ef
Dcr
a
0.0005
0.065
0.8
1.0
Fig. 6. (a)–(c) Damage pattern and crack nucleation and growth in 90 laminate. The loading direction is along the cell vertical axis.
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Fig. 7. Contour plots at 0.01 of macroscopic compressive strain in 90 laminate: (a) 0.002 plastic strain contour; (b) equivalent von Mises stress contour in fiber only; radial cracking occurs along horizontal axis.
In Figs. 6(a)–(c) the contour plot showing shear crack initiation and growth is given while in Table 1 the damage parameter set used in this investigation is summarized. It is interesting to note how both the predicted direction and the damage extension are very similar to those experimentally observed as well as the location of crack initiation. Here, the asymmetry in the crack growth is due to the small differences in the plastic strain distribution, as mentioned above. 5.2. 90 laminate in compression In 90 laminates final failure in tension occurs without any fiber fracture since both residual and contact stresses are released after debonding. In compression the debonding process is greatly reduced allowing continuous transversal load transfer to the fibers up to 0.6% in strain. Experimental stress–strain curve with partial unloading shows the disappearing of the characteristic two-slopes features observed in tension [24]. In addition to this, partial unloading after overall yielding shows no changes in the elastic slope confirming the presence of inelastic deformation without damage. Microscopy observations indicate that highly concentrated plastic strain band occurs as for the tension case, while more intense plastic strain accumulation is found between fibers along the compression direction, i.e., along the ply. The stress distribution in the fiber is not uniform showing a concentration region along the
Fig. 8. Plastic strain contour in 0 unidirectional laminates at fracture stress.
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compressive direction. This is exactly the location where radial cracks in the fibers are observed in the experiments, as shown in Figs. 7(a) and (b). 5.3. 0 laminate in tension Tensile test along fiber direction has been simulated using generalized plane strain elements. This model is capable to predict the observed experimental result with fairly good approximation [5]. As far as concern the damage in the cell, the UCM shows that, even close to the failure stress, plasticity in the matrix is localized around the fiber only and limited in value, as shown in Fig. 8 where the plastic strain in the highly concentrated plastic strain band, that inevitably develops, is below 0.2%, while around the fiber the maximum plastic strain value is below 5.0%. The stress in the fiber is uniform with value ranging from 3500 to 4500 MPa, according to the variation in the SCS assumed Young modulus (350–400 GPa). These stress values are consistent with the average fracture stress experimentally measured and reported in the literature.
6. Conclusions In this paper the possibility to accurately predict damage development and macroscopic failure in metal matrix composites with mechanical interface using UCM has been demonstrated. The importance to incorporate appropriate specific damage models for each micromechanism of failure which may occur in the material, according to the constituents behavior and external loading conditions, has been highlighted. The CDM model used to model ductile damage in Ti matrix for a Ti-15-3/SCS6 composite system allowed to accurately predict the exact location of damage initiation as well as the evolution of damage in the formation of inter-fiber cracking. The considerations about the importance of modeling the manufacturing process allow one to get rid of the need to develop more or less complicated model to simulate fiber–matrix interface strength. The UCM proposed model is capable to accurately predict all the fundamental features of the deformation and damage processes in unidirectional MMC laminates loaded both in tension and compression, at both micro- and macroscopic scale.
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[3] Aboudi J. Micromechanical analysis of composite by the method of cells. Appl Mech Rev 1989;42(7):193–221. [4] Xia Z, Zhang Y, Ellyin F. A unified periodical boundary conditions for representative volume elements of composites and applications. Int J Solids Struct 2003;40(8):1907–21. [5] Bonora N, Ruggiero A. Micromechanical modeling of composites with mechanical interface – Part I: unit cell model development and manufacturing process effects. Compos Sci Technol 2006; 66(2):314–22. [6] Talreja RA. Conceptual framework for interpretation of MMC fatigue. Mater Sci Eng A 1995;200:21–8. [7] Curtin WA. Theory of mechanical properties of ceramic–matrix composites. J Am Ceram Soc 1991;76:2837–45. [8] Mahesh S, Phoenix SL, Beyerlein IJ. Strength distributions and size effects for 2D and 3D composites with Weibull fibers in an elastic matrix. Int J Fract 2002;115:41–85. [9] Mudry F. A local approach to fracture. Nucl Eng Des 1987;105:65–76. [10] Milella PP, Bonora N. On the dependence of the Weibull exponent on geometry and loading conditions and its implications on fracture toughness probability curve using a local approach criterion. Int J Fract 2000;104(1):71–87. [11] Xia Z, Curtin WA. Tough-to-brittle transition in ceramic–matrix composites with increasing interfacial shear stress. Acta Mater 2000:4879–92. [12] Dutton RE, Pagano NJ, Kim RY, Parthasarathy TA. J Am Ceram Soc 2000;83:166. [13] Brust FW, Newaz GM, Majumdar BS. In: ASTM spring symposium; 1992. [14] Lissenden CJ, Herakowich CT. Numerical modeling of damage evolution and viscoplasticity in metal matrix composites. Comput Meth Appl Mech Eng 1995;126:289–303. [15] Tvergaard V. Effect of fibre debonding in a whisker-reinforced metal. Mater Sci Eng A 1990;125:203–13. [16] Knight MG, De Lacerda LA, Wrobel LC, Henshall JL. Parametric study of contact stress around spherical and cylindrical inclusions. Comput Mater Sci 2002;25:115–21. [17] Aghdam MM, Pavier MJ, Smith DJ. Micro-mechanics of off-axis loading of metal matrix composites using finite element analysis. Int J Solids Struct 2001;38:3905–25. [18] Gurson AL. Continuum theory of ductile rupture by void nucleation and growth: Part I – yield criteria and flow rules for porous ductile media. J Eng Mater Tech 1977;99:2–15. [19] Rousselier G. Les Mode`les de Rupture Ductile et leurs Possibilite´ Actuelles dans le Cadre de lÕApproche Locale, EDF. Service Re´acteurs Nucle´aires et Echangeurs, Les Renardie`res, BP1, F77250, Moret-Sur-Loing, France; 1986. [20] Lemaitre J. A course in damage mechanics. New York: Springer; 1992. 44-46. [21] Bonora N. A nonlinear CDM model for ductile failure. Eng Fract Mech 1997;58:11–28. [22] Gooder JL, Mall S. Compressive behavior of a titanium metal matrix composite with a circular hole. Compos Struct 1995;31:315–24. [23] Majumdar BS, Newaz GM. Inelastic deformation of metal matrix composites. part I – plasticity and damage mechanisms. NASA Report CR-189095, NASA Lewis Research Center, Contract NAS3-26053, March; 1992. [24] Bonora N, Marchetti M, Costanzi M, Newaz GM. Micro-damage effect on the overall response of long fiber/metal matrix composites. Composites 1994;25:575–82. [25] MSC. Marc. Theory and user information, vol. A, version 2003. [26] Bonora N, Gentile D, Pirondi A, Newaz GM. Ductile damage evolution under triaxial state of stress: theory and experiments. Int J Plasticity 2005;21:981–1007.
Micromechanical modeling of composites with mechanical interface – Part II: Damage mechanics assessment Nicola Bonora *, Andrew Ruggiero DiMSAT – Department of Mechanics, Structures and Environment, University of Cassino, Via G. Di Biasio 43, I-03043 Cassino, Italy Received 7 March 2005; accepted 12 April 2005 Available online 1 July 2005
Abstract Continuing the work initiated in the Part I [Bonora N, Ruggiero A. Micromechanical modeling of composites with mechanical interface – Part I: unit cell model development and manufacturing process effects. Compos Sci Technol 2003], in this paper the possibility to account for different damage mechanisms, in the unit cell model (UCM), explicitly developed for composites with mechanical interface, is discussed and results for Ti-15-3/SCS-6 composite laminates are presented. Starting from the analysis of the constituent behaviors a probabilistic model based on Weibull statistics is developed for fiber failure, while a ductile damage model which incorporates stress triaxiality effect has been used for predicting metal matrix progressive failure. Fiber–matrix debonding process has been naturally predicted incorporating the material manufacturing process in the stress/strain history. Numerical results performed with the UCM applied to 0 and 90 unidirectional laminates, loaded both in tension and compression, have been compared with experimental results at both macro- and microscopic scale. 2005 Elsevier Ltd. All rights reserved. Keywords: Ductile damage; CDM; MMC; FEM; Plasticity; Multiscale damage
1. Introduction Composite materials find a new field of application almost every day, often as a lighter substitute material to more traditional steels and light alloys, due to their high specific resistance capabilities, low weight, and for the possibility to design components and parts assigning the desired material properties where needed most. Metal matrix composites represent a different generation of materials explicitly designed for high temperature applications. Here, the idea of combining the strength of ceramic reinforcement with the ductility of metal matrix leaded to materials capable to offer high stiffness and strength, as well as fatigue resistance and reduced overall weight, at elevated temperature up to *
Corresponding author. E-mail address: [email protected] (N. Bonora).
0266-3538/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2005.04.043
700 C, approximately. The major structural differences of these composites, with respect to polymeric matrix based composites, mainly are: the elastic–plastic behavior of the matrix, which can sustain large plastic deformation, and the nature of interface between the matrix and the reinforcement. Usually, in these composites the interface is mechanical, in the sense that no chemical bond exists between the fiber and the matrix. The joining of the matrix with fiber occurs as a result of the differential shrinkage, due to the mismatch in the a-thermal expansion coefficients, during the cooling down from the assembly temperature to room temperature. A fundamental conceptual difference in the design with composite materials is that the material itself can be designed, at different dimensional scales (constituents, laminate stack sequence, and component) according to the most effectively required behaviors. To this purpose, reliable and practical design tools, capable to accurately
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predict overall material constitutive behavior at different dimensional scales, have been investigated in the last decades. In this framework, micromechanics approach has been widely used by a number of authors [1–3]. Typically, a unit cell model (UCM) is built on the assumption of material periodic microstructure. Finite element technology is used to calculate cell response incorporating complex features, such as constituentsÕ non-linear behavior, contact, combined thermo-mechanical loading, etc. This approach allows predicting the occurrence of failure conditions if the micro-mechanisms of failure, characteristic for the material, are identified and appropriate damage modeling is added in. In this way, damage evolution in the material microstructure can also be followed and the progressive degradation of the composite overall constitutive response accurately predicted. In the part I [5], the authors analyzed the role of the manufacturing process performing an extensive finite element investigation on SCS-6/Ti-15-3 unidirectional laminates. They demonstrated that if the consolidating phase, that is commonly performed by hot isostatic press (HIP) technique, is simulated within the UCM, the composite macroscopic response can be accurately predicted without the need to develop artificial model for the interface strength. In this paper, continuing the previous work, microstructural damage evolution in SCS-6/Ti-15-3 composite laminates has been investigated. The damage mechanisms, which may occur in the microstructure under both tension and compression loading, have been identified and specific damage model for each of them have been developed and implemented in the UCM. Damage evolution during loading has been compared with experimental in situ observations, given by Majumdar and Newaz [23], in order to verify the accuracy of the proposed approach in predicting how, where, and when damage develops. In Section 2, the UCM formulation proposed by the authors is briefly reviewed; in Section 3, the damage mechanisms and the modeling are presented; in Section 4 the description of the material used in the present investigation is given; in Section 5 the finite element results are discussed and compared with experimental data available in the literature.
2. Unit cell model and finite element modeling A UCM can be developed according to the periodicity of the material structure and the dimensional scale of interest. For a composite laminate, the smallest RVE can be taken at single-fiber level if the thickness of the lamina is big enough with respect to the fiber diameter. For closed package fiber layers, the UCM should account for between-fiber distance in order to accurately model local stress concentration and constraint. In metal matrix composites, fibers are quite bigger in diameter with re-
spect to more traditional carbon or glass fibers. For instance, in the case of SCS-6 fiber the average diameter is 140 lm approximately, in contrast to 7 lm diameter of standard T300 carbon fiber. In addition, foil-fibers-foil assembly technique, used for this material, assures a high degree of regularity in the fiber alignment and arrangement, resulting in real periodic microstructure. Since, the fiber diameter is comparable with the distance between two adjacent fibers, both along the lamina in-plane direction and the laminate thickness direction, the choice of the UCM may be influenced by the presence of an horizontal shift in the stacked plies as sketched in Fig. 1, where different choices for the UCM are depicted. Periodic boundary condition can be applied to the cell imposing either ‘‘plane-remains-plane’’ or ‘‘unified’’ periodic boundary conditions. Xia et al. [4] showed that the first is appropriate for in plane symmetric loading but can be over-constraining for shear loading while the latter performs better in all cases. Using plane strain or generalized plane strain elements the same UCM can be used to investigate both 90 and 0 unidirectional laminate response. If expanded in 3D, the same UCM can be used to investigate both cross-ply and angle ply laminate [4]. As far as concern SCS-6/Ti-15-3 composite laminates investigated here, the author observed that, according to microstructural visual observations, the more appropriate UCM is the one given in Fig. 2, due to the presence of a systematic shift in the ply stacking sequence. Under plane strain assumptions, imposing periodic ‘‘plane-remains-plane’’ boundary conditions, the average meso-strain and meso-stress are defined through the displacement and reaction forces at the cell boundaries as follows: u Ex ¼ ln 1 þ 0 ; Lx w ð1Þ Ez ¼ ln 1 þ 0 ; Lz Ey ¼ 0; where u; w; L0x ; L0z are the displacement along x- and z-axes and the cell reference dimensions, respectively. The in-plane meso-stresses are defined as: Fi i; j ¼ x; z; ð2Þ Ri ¼ Lj B where B is the cell thickness, m is Poisson ratio, and Lx and Lz the actual cell dimensions (i.e., Lx ¼ L0x þ u). Under generalized plane strain assumption, also the meso-strain and meso-stress along the normal axis (here, y-axis) can be also given as: v Ey ¼ ln 1 þ ; ð3Þ B Fy . ð4Þ Ri ¼ Lx Lz
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Fig. 1. Fiber arrangement in unidirectional MMC laminate: frontal view and unit cell for SCS/Ti-15-3.
Fig. 2. 90 Unit cell model: contour map of plastic strain in tension: (a) beginning of stage II; (b) beginning of stage III; (c) prior crack initiation in stage III. Black thick line indicates intact interface extension.
The UCM given in Fig. 2 has been implemented in the commercial finite element code MSC/MARC2003. Four node element with bilinear interpolating function has been used. The interface has been obtained modeling the contact between fiber and matrix considered as deformable bodies. In general, contact problems in FE are usually solved by means of Lagrange multipliers or penalty methods. Here, the so-called direct constraint method has been used. In this procedure, the motion of the bodies is tracked and, when contact occurs, direct constraints are placed on the motion using boundary conditions. This procedure can be very accurate if the program can predict when contact occurs. No special
interference elements are required in this procedure and complex changing contact conditions can be simulated since no a priori knowledge of where contact occurs is necessary. The details of the implementation are given in [25]. Elastic–plastic calculations have been performed using large displacement, Lagrangian updating and finite deformation formulation. The cooling down phase has been modeled assuming, for the cell, the stress-free temperature of 815 and accounting for material properties variation with temperature. Details on boundary condition, material properties and finite element cell performance assessment are discussed elsewhere [5].
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3. Damage mechanisms and modeling Damage is indicative of the irreversible modification processes and their mutual interactions, which occur in the material. These processes, even though highly localized in the material microstructure, directly affect the resulting material behavior in the upper dimensional scales (both meso- and macro-scale). Damage in metals is always associated to plastic deformation. Below the brittle–ductile transition temperature, plastic deformation is confined inside the grain and localized in one or few favorable oriented grains. Above the NDT, plastic deformation can spread over large macroscopic volumes, since all grains can plastically deform. Consequently, damage can be an evolutionary process, if plastic deformation occurs at macroscopic scale, or statistical in nature, if plastic deformation are confined to grain level as for brittle fracture. What differentiates ductile damage process from inelastic deformation is the degradation of the material average elastic moduli. This effect can be experimentally observed and monitored in order to quantify the damage evolution. From a dimensional scale point of view, damage considerations always refer to the processes that happen into a single RVE. Failure occurs when the damage variable reaches a critical value characteristic for the material. In ductile metals, the failure of one or few adjacent RVEs usually correspond to the appearance of a macroscopic crack, the growth of which may be eventually followed in the framework of fracture mechanics. Composite materials with mechanical interface are characterized by different damage mechanisms, peculiar of each phase and constituent. Additional micro-/mesomechanism of failure may arise from the reciprocal interaction of the constituents. The basic failure modes may be classified as follows: • reinforcement (fiber, particle or flake) rupture; • matrix damage; • matrix–reinforcement debonding. All these failure modes are driven by strain; take place at the RVE dimensional scale; modify the material overall constitutive response; may be singularly cause of the material catastrophic failure; usually occur in the material according to well defined sequences along with the type of the loading conditions (static, dynamic, impact, etc.). Consequently, for each of these, appropriate modeling is necessary. 3.1. Reinforcement failure The reinforcement in composites are mainly fibers (short or long) or particles (spheroid or flakes). Since the main role of the reinforcement is to sustain loads, its stiffness, as well as the yield and ultimate tensile
strength, is usually much higher than that of the matrix material. Fiber material is typically brittle and failure occurs when the maximum allowable stress is reached. Damage in fiber is a non-progressive process [6], due to the small fiber volume and net resisting section, which practically exclude the presence of potential growing flaws. Consequently, due to the high number of fibers in each layer, fiber failure assessment requires statistical considerations. As far as concern metal matrix composites, Curtin [7] developed an effective fiber breakage model to simulate longitudinal tensile behavior of MMC laminates. According to this, the average stress in the effective fiber bundle is given as: r 1 avg rf ¼ ¼ T 1 qðlf Þ ; ð5Þ f 2 where r is the longitudinal applied stress, f is the fiber volume fraction, T is the stress sustained by the unbroken fibers, q(lf) is the fraction of broken fibers and lf is half of the length of the fiber break region which experiences interfacial slip. The probability function of failure can be given according to Weibull distribution in the following form: m L r pf ¼ 1 exp ; ð6Þ L0 r0 where L0, m and r0 are the reference gauge length, the Weibull distribution exponent, and the average strength at pf = 0.62, respectively. Mahesh et al. [8] investigated strength distribution and size effect in unidirectional composites using Weibull statistics applied to fiber. In their approach, the fiber strength distribution was assumed according to Weibull–Poisson statistics as follows: q r F ðrÞ ¼ 1 exp . ð7Þ rd For 2D unit cell, as the one used here, the information about the fiber length cannot be inferred due to plane strain or generalized plane strain formulation employed. The local approach to failure, initially developed for brittle fracture in metals by Mudry [9], can be modified and used to determine the probability of failure across a reference fiber section. According to this, the fiber probability of failure can be written as follows: m rW pf ¼ 1 exp ; ð8Þ ru where ru and m are the classical stress scale factor and the Weibull exponent. The Weibull stress rW accounts for the maximum principal stress r1 acting over a fiber reference elementary volume V0, Z 1 rmW ¼ rm ðx; y; zÞ dV ; ð9Þ V0 V 1 where, for constant unit thickness, becomes,
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1 rmW P ¼const ¼ S0
Z
rm1 ðx; yÞ dS.
ð10Þ
S
S0 is the reference unit area. The Waibull exponent m is indicative of the variability in the material response associated with a local distribution of material properties and stress state, and, hence, to the extension of volume of the process zone where fracture is initiated. The larger the volume of the process zone the higher the probability to find any possible microstructural variation or flaw. Milella and Bonora [10], investigating the Weibull exponent dependencies in brittle fracture of metals, observed that the process volume depends on the triaxiality of the stress field geometrically induced by a crack or a notch. Superimposed stress triaxiality by external loading over smooth geometry does not change the size of the process zone that remains the entire geometry volume with no effect on m. In fibers, the entire volume participates in the fracture process, since fracture may initiates anywhere from microscopic flaws; consequently S0 in Eq. (10) becomes the entire fiber section. According to Milella and Bonora [10] the expected value of m for the fiber should be in the range of 20–45. These values are consistent with those found by Xia and Curtin [11] and Dutton et al. [12] using similar formulation but based on different considerations. Consequently, this formulation can be implemented in the cell in order to calculate the probability of failure with increasing applied external load. 3.2. Reinforcement–matrix debonding Matrix–reinforcement interface is fundamental for the stress transfer and for the composite material load carrying capability. In polymeric composite, the interface is a narrow region where the fiber, or particle, and the matrix are intimately connected through a chemical bond. Here, a continuous properties transition from those of the matrix to that of the fiber, or particle, is found. In MMC, the interface is mechanical in most of the cases. No chemical bound exists between the fiber, or particle, and the metal matrix. In the literature many papers have been presented in the past years trying to develop an interfaces strength model. A possible procedure to calibrate these models was to manually provide interface release during tensile loading in order to match the slope changes in the macroscopic stress–strain responses, Brust et al. [13]. Lissenden and Herakowich [14] used a traction–displacement, similar to that proposed by Tvergaard [15], that resembles theoretical atomic bond force–displacement response. Still recently, Knight et al. [16] performed a parametric study on the contact stresses associated to perfect interface. Aghdam et al. [17] performed an extensive finite element analysis, investigating the effect associated to perfect or partially debonded interface on the macroscopic stress–strain
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response of metal matrix composites by mean of unit cell model. Bonora and Ruggiero [5], starting from the consideration that the interface strength in MMC is determined by the cooling down ramp from the assembly to the room temperature, demonstrated that no specific interface model is necessary if the manufacturing phase is incorporated in the UCM stress/strain history. Here, the only numerical requirement is that both the fiber, or particle, and the matrix need to be modeled with finite elements as separated deformable bodies which, starting from the zero-contact stress condition at high temperature, may come in reciprocal contact during the cooling phase as a result of the mismatch in the a-thermal expansion coefficients. With this approach, debonding naturally develops in the UCM when the stresses resulting from external applied load exceed contact forces along the interface. As it will be demonstrated later in the paper, if the manufacturing process is simulated, the right time and location of matrix–fiber debonding can be accurately predicted, as well as the resulting macroscopic stiffness reduction due to the extension of the debonding interface. 3.3. Matrix damage Metal matrix material exhibits damage processes typical of metals. According to the operative temperature, with respect to the brittle–ductile transition temperature, metal may fails as a result of cleavage or ductile rupture, respectively. Since metal matrix composites are intended for high temperature applications ductile rupture is the failure mode of interest analyzed here. Ductile damage in metal occurs in the form of nucleation and growth of microvoids that are initiated either by second phase particle inclusions or by dislocations piling-up along the grain boundaries. In the literature, many models have been proposed in order to describe the constitutive response of porous plastic solids [18– 20]. Most of them suffer a number of limitations spreading from an elevated number of damage parameters needed to the lack of their transferability from specimen to structures. In the last decade, Bonora [21] developed a damage model in the framework of continuum damage mechanics (CDM) initially proposed by Lemaitre [20]. In this model, the damage variable accounts for all effects induced by the irreversible modifications that occur in the microstructure under plastic strain accumulation. Since there are clear experimental evidences that ductile damage affects material stiffness, while it is not possible to assess damage effects on yield surface since damage softening cannot be separated from hardening effect in a real tensile test, in this formulation damage accumulation modify material stiffness only. According to this no softening is present in the material flows stress avoiding
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possible localization related problems in the numerical simulations. More details regarding these aspects can be found elsewhere [21,26] Damage evolution, derived from the damage dissipation potential formulation, accounts for stress triaxiality effects, which are responsible for material ductility reduction. The basic set of constitutive equations for the damaged material is summarized as follows: eTij ¼ eeij þ epij ;
ð11Þ
1 þ m rij m rkk dij ; ð12Þ E 1D E 1D ofp 3 s_ ij ¼ k_ ; ð13Þ e_ pij ¼ k_ 2 req orij ofp _ _ ¼ k ¼ p; ð14Þ r_ ¼ k_ oR where Eqs. (13) and (14) are those for standard plasticity while the kinetic law of damage evolution is given by:
eeij ¼
ofD D_ ¼ k_ oY ¼a
1 a1 ðDcr D0 Þa rH p_ f ðDcr DÞ a ; p lnðef =eth Þ req
where 2 P 2 P f ¼ ð1 þ mÞ þ 3 ð1 2mÞ req 3 req
ð15Þ
higher temperature BCC b-phase. An extensive literature is available both as far as concern material experimental characterization and numerical modeling, Gooder and Mall [22], and Majumdar and Newaz [23] performed an extensive study on this material. In particular they performed a detailed investigation on the microstructural modifications occurring in the material during monotonic loading. Using both in situ observations and replica technique, they investigated 0 and 90 laminate response, under both tension and compression, at both RT and high temperature (538 and 700 C). In this paper the experimental observations performed by Majumbdar and Newaz [23] have been used to compare the predicted microstructural modifications and damage development under both tension and compression loading on 0 and 90 unidirectional laminates.
5. Finite element results and discussion The proposed unit cell model is capable to accurately predict macroscopic stress–strain response for both 0 and 90 unidirectional lamina under both tension and compression [5]. In the following, the damage development and the resulting effect on the macroscopic response are given according to the laminate orientation and the load type (tensile or compressive).
ð16Þ
accounts for stress triaxiality. The damage parameters needed are: eth, that is the damage threshold strain at which damage processes initiate; the theoretical strain to failure, ef, under constant triaxiality (stress triaxiality, TF = 0.333); the critical damage, Dcr, at which complete failure occurs, and the damage exponent, a, that defines the shape of damage evolution curve with strain. Detailed discussion on the damage model formulation and damage parameters identification can be found elsewhere [21].
4. Material The proposed approach can be potentially used for any composite, reinforced with either spherical particles or long fibers, with mechanical interface. The unit cell model has been developed with particular reference to a SCS6/Ti-15-3 laminates with 0.34 volume fraction content. This is a Ti–15V–3Cr–3Al–3Sn (Ti-15-3) weight percent alloy reinforced with SiC (SCS6) fibers with a reference diameter of 140 lm. Fibers production is obtained via carbon (C) and silicon (Si) vapor deposition around a soft carbon filament. Fiber are equally spaced by means of a molybdenum interweave. Titanium matrix is an allotropic material containing both lower temperature hexagonal close-packed (HCP) a-phase and
5.1. 90 laminate in tension A peculiar two-slopes feature, before work hardening, characterizes the uniaxial response of 90 unidirectional laminate in tension. According to this, the strain range can be divided in three stages: in stage I the material response is linear elastic, in stage II there is clear knee and a drop of the stress/strain slope, in stage III the stress–strain curve follows a power law which rapidly saturates, as given in Fig. 4. Microscopy investigations and replica analyses revealed that in stage I the matrix and the fiber remains perfectly bonded up to the first knee on the stress–strain curve where debonding starts to occur. In this phase some slip band between fibers initiate to appear. In stage two, extensive fiber debonding takes place together with an intensification of highly concentrated plastic strain band. The extension of the debonding spread up to 45–60 of the fiber circumference. In Fig. 2, the evolution of the debonding as well as the highly concentrated plastic strain band is given. Here, with the black thick line the portion of the intact interface is given. It is worth to note here that, even at fracture, the proposed UCM predicts that the interface is not completely failed. This result finds confirmation in the replica taken at different deformations stage and prior final fracture. In Fig. 3 the replica at 0.5% of overall strain is given together with the finite element
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Fig. 4. Calculated stress–strain curves for 90 laminate: solid line for the UCM without damage in the matrix but accounting for the debonding process; line and symbol for the UCM with the CDM damage modeling [21]; for the Ti matrix, hollow symbols indicate experimental data [10].
Fig. 3. Replica taken under loading at 0.5% strain on 90 laminate: black thick line in FE result indicates the undebonded interface, arrows indicate penetration of replica acetate, i.e., debonded region extension.
results. Black thick line indicates the still bonded interface extension. The break in the symmetry of the contours plots given here and in Figs. 6–8 is caused by small differences in the calculated reaction forces along the two contact surfaces that are magnified by the contour post-processing algorithm. In Fig. 4, the comparison of the predicted stress– strain response is compared with experimental data. During the test, partial unloading at given strain was performed in order to investigate damage development and related effects at the transition of stage II and III. Similarly to the experiments, partial unloading has been also simulated in order to show the possibility to follow, with the proposed unit cell model, non-monotonic load paths. Here, the predicted material response is in a very good agreement with the experimental data showing all the features that characterize the three-stages stress– strain curve for 90 MMC. It is interesting to note that the UCM is capable to predict the changing in slope during the first unloading mainly due to partial back contact between the matrix and the debonded fiber. These results were almost impossible to obtain with the contact techniques used by Brust et al. [13] and Bonora et al. [24]. The lowering of the slope in the third stage of the stress–strain curve is indicative of almost
Fig. 5. Calculated damage contours (white D = 0, black D = Dcr, i.e., crack initiation) compared with micrograph showing highly deformed band and crack initiation [23] in 90 laminate.
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complete debonding and strain accumulation in the ligament. Microscopy analyses reveal that final failure of 90 MMC unidirectional laminate is controlled by the formation of cracks along intense highly concentrated plastic strain band developing between fibers. Simulations performed incorporating the ductile damage model confirm that damage in the matrix develops together with the debonding of the fibers since the early stage II. In the stage III, where the debonding process is almost
completed, damage localizes in a narrow band, as given in Fig. 5, resulting in the formation of a microscopic crack at the interface (black thick line).
Table 1 Damage parameter for Ti-15-3 eth
ef
Dcr
a
0.0005
0.065
0.8
1.0
Fig. 6. (a)–(c) Damage pattern and crack nucleation and growth in 90 laminate. The loading direction is along the cell vertical axis.
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Fig. 7. Contour plots at 0.01 of macroscopic compressive strain in 90 laminate: (a) 0.002 plastic strain contour; (b) equivalent von Mises stress contour in fiber only; radial cracking occurs along horizontal axis.
In Figs. 6(a)–(c) the contour plot showing shear crack initiation and growth is given while in Table 1 the damage parameter set used in this investigation is summarized. It is interesting to note how both the predicted direction and the damage extension are very similar to those experimentally observed as well as the location of crack initiation. Here, the asymmetry in the crack growth is due to the small differences in the plastic strain distribution, as mentioned above. 5.2. 90 laminate in compression In 90 laminates final failure in tension occurs without any fiber fracture since both residual and contact stresses are released after debonding. In compression the debonding process is greatly reduced allowing continuous transversal load transfer to the fibers up to 0.6% in strain. Experimental stress–strain curve with partial unloading shows the disappearing of the characteristic two-slopes features observed in tension [24]. In addition to this, partial unloading after overall yielding shows no changes in the elastic slope confirming the presence of inelastic deformation without damage. Microscopy observations indicate that highly concentrated plastic strain band occurs as for the tension case, while more intense plastic strain accumulation is found between fibers along the compression direction, i.e., along the ply. The stress distribution in the fiber is not uniform showing a concentration region along the
Fig. 8. Plastic strain contour in 0 unidirectional laminates at fracture stress.
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compressive direction. This is exactly the location where radial cracks in the fibers are observed in the experiments, as shown in Figs. 7(a) and (b). 5.3. 0 laminate in tension Tensile test along fiber direction has been simulated using generalized plane strain elements. This model is capable to predict the observed experimental result with fairly good approximation [5]. As far as concern the damage in the cell, the UCM shows that, even close to the failure stress, plasticity in the matrix is localized around the fiber only and limited in value, as shown in Fig. 8 where the plastic strain in the highly concentrated plastic strain band, that inevitably develops, is below 0.2%, while around the fiber the maximum plastic strain value is below 5.0%. The stress in the fiber is uniform with value ranging from 3500 to 4500 MPa, according to the variation in the SCS assumed Young modulus (350–400 GPa). These stress values are consistent with the average fracture stress experimentally measured and reported in the literature.
6. Conclusions In this paper the possibility to accurately predict damage development and macroscopic failure in metal matrix composites with mechanical interface using UCM has been demonstrated. The importance to incorporate appropriate specific damage models for each micromechanism of failure which may occur in the material, according to the constituents behavior and external loading conditions, has been highlighted. The CDM model used to model ductile damage in Ti matrix for a Ti-15-3/SCS6 composite system allowed to accurately predict the exact location of damage initiation as well as the evolution of damage in the formation of inter-fiber cracking. The considerations about the importance of modeling the manufacturing process allow one to get rid of the need to develop more or less complicated model to simulate fiber–matrix interface strength. The UCM proposed model is capable to accurately predict all the fundamental features of the deformation and damage processes in unidirectional MMC laminates loaded both in tension and compression, at both micro- and macroscopic scale.
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