Micromechanical modeling of interface damage of metal matrix composites subjected to off-axis loading

Materials and Design 31 (2010) 829–836

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Micromechanical modeling of interface damage of metal matrix composites subjected to off-axis loading M.J. Mahmoodi *, M.M. Aghdam, M. Shakeri Thermoelasticity Center of Excellency, Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., Tehran, Iran

a r t i c l e

i n f o

Article history: Received 6 May 2009 Accepted 28 July 2009 Available online 6 August 2009 Keywords: Micromechanics Interface damage Off-axis loading Elastoplastic behavior Thermal residual stress

a b s t r a c t A three dimensional micromechanics based analytical model is presented to investigate the effects of initiation and propagation of interface damage on the elastoplastic behavior of unidirectional SiC/Ti metal matrix composites (MMCs) subjected to off-axis loading. Manufacturing process thermal residual stress (RS) is also included in the model. The selected representative volume element (RVE) consists of an r  c unit cells in which a quarter of the fiber is surrounded by matrix sub-cells. The constant compliance interface (CCI) model is modified to model interfacial de-bonding and the successive approximation method together with Von-Mises yield criterion is used to obtain elastic–plastic behavior. Dominance mode of damage including fiber fracture, interfacial de-bonding and matrix yielding and ultimate tensile strength of the SiC/Ti MMC are predicted for various loading directions. The effects of thermal residual stress and fiber volume fraction (FVF) on the stress–strain response of the SiC/Ti MMC are studied. Results revealed that for more realistic predictions both interface damage and thermal residual stress effects should be considered in the analysis. The contribution of interfacial de-bonding and thermal residual stress in the overall behavior of the material is also investigated. Comparison between results of the presented model shows very good agreement with finite element micromechanical analysis and experiment for various off-axis angles. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction There has been an increasing interest in metal matrix composites (MMCs), in particular SiC/Ti MMCs, due to their high specific modulus and strength and temperature resistance. Reliable design of MMCs requires detailed modeling and understanding of the behavior of MMCs under various combined loading conditions. It is well established in the literature [1] that MMCs suffer from existence of a high state of thermal residual stresses during manufacturing process and weak interface between fiber and matrix. Therefore, in order to be more accurate, any presented model for MMCs should include existence of both weak fiber/matrix interface and thermal residual stresses. Furthermore, the strength of composites is usually limited by failure of the fibers, failure of the interface or yield of the matrix, depending on the precise loading condition. Among various loadings, the off-axis loading is the most applicable and complicated loading conditions as combination of various normal and shear loading exist depending on the fiber direction. Many attempts have been made to develop micromechanical models, including finite element (FE) [2–5] and analytical ap-

* Corresponding author. E-mail address: [email protected] (M.J. Mahmoodi). 0261-3069/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.matdes.2009.07.048

proaches, to predict behavior of composite materials subjected to various loading conditions. FE and analytical micromechanical models [5] were also used to predict both initial yield and collapse envelopes for MMCs under different cases of biaxial and shear loading with and without thermal residual stress effects. The analytical model in [5] which was mainly in the category of unit cell models was later [6] called simplified unit cell (SUC) model. Only fully bonded interface was considered in the SUC model [6]. Micromechanical modeling of off-axis loading received relatively less attention in the literature [7–9] due to complexity of loading and boundary conditions. Analytical [9] and finite element micromechanical models were developed to predict behavior of MMCs subjected to off-axis loading with [7] and without [8] effects of thermal residual stress. The model presented in [7] also includes effects of fully de-bonded interface with a level of friction between fiber and matrix. On the other hand, there are different analytical [10–13] and numerical [14–24] studies in which the effects of thermal residual and weak interface on the behavior of MMCs in various loading conditions are investigated. Indeed, experimental techniques have also been employed to measure thermal residual stresses and stress–strain response of the MMC materials [18,19]. Aboudi [10] incorporated the flexible interface (FI) model of Jones and Whittier [25] into the method of cells. Later, the FI model concept was em-

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ployed [26] with an added condition that requires the interfacial compliance to be zero when the interface is in compression. This modification was incorporated [27] into a rate-based formulation of generalized method of cells (GMC). This model has been referred to as the constant compliant interface (CCI) model. Inclusion of a finite interfacial strength is a major improvement since previous work on SiC/Ti composites points to the existence of a weak chemical bond at the fiber–matrix interface [15,28,29]. The CCI model, as implemented in GMC, was later employed by Warrier et al. [30] to model the transverse tensile response of SCS-6/TIMETAL 21S. Also a fiber/matrix de-bonding model was presented for MMCs based on modified Needleman type cohesive zone model under normal and shear loading [11]. However, initiation and propagation of fiber/matrix interfacial de-boning for unidirectional MMCs in the off-axis loading were not found in the literature. In this study, the elasto-plastic response of SiC/Ti unidirectional fiber reinforced composite under off-axis loading is predicted using the SUC model. Nonlinear behavior of the material due to both matrix plastic deformation and interface damage are considered in the model. The model also includes effects of manufacturing process thermal residual stress. In order to provide a more realistic model, the geometry of the RVE in the SUC model is extended to r  c sub-cells mainly to determine more accurately effects of both nonlinearities. The CCI model is also modified to consider interfacial de-bonding. The successive approximation method is used to obtain elastic–plastic behavior of the material. Results for stress– strain response at various off-axis angles show favorably good agreement with experimental data. In the next section, the SUC micromechanical model is described. Material characterizations of the SiC/Ti composite are expressed in Section 3. The fourth section contains the interfacial de-bonding model employed to obtain the behavior of the SiC/Ti composite using the micromechanical model.

RVE shown in Fig. 1 consists of r  c elements with Lc and Lr as the length of the RVE in the x and y directions, respectively, and unit length in the z direction. Each sub-cell labeled as ij in which i and j are considered as counters of the sub-cells in the x and y directions, respectively. Also, the model presented in this study is called extended simplified unit cell (ESUC) model in which the selected RVEs contains more than 10  10 sub-cells to consider circular shape of the fiber as shown in Fig. 2. 2.2. Micromechanical governing equations In off-axis loading, a uniaxial load is applied to a coupon where the fibers are aligned at an angle h to the loading direction as shown in Fig. 3. Two coordinate systems are defined: the (x, y) system, where the x direction is the loading direction, and the (1–2) system, where the 1 axis coincides with the fiber direction and the 2 direction is perpendicular to the fibers. The stress state within the specimen in the material principle axes consists of three stress components: axial S1, transverse S2, and axial shear S12 as indicated in Fig. 3. The symbol S is used for various overall stress components on the RVE. These stress components can be determined from the stress (Sx) applied to the off-axis coupon using the following transformation equations:

9 8 9 8 2 > > > = < S1 = < cos h > 2 ¼  Sx S2 sin h > > > ; > ; : : S12 sin h cos h

ð1Þ

where S1, S2 and S12 are the average normal and shear stresses in the material coordinate system and h denotes off-axis angle.

2. Analysis 2.1. Geometry of the RVE Most analytical and FE models assume regular fiber arrangement. Normally, two types of fiber arrays, square and hexagonal arrays are considered in various analyses. Additional assumption in most analytical models, such as method of cells and SUC is assuming rectangular fibers. However, in order to consider more realistic geometry in the analytical models, one can consider the RVE consisting of r  c rectangular elements in which fiber subcells are surrounded by matrix sub-cells. Geometry of the selected

Fig. 2. RVE in the ESUC model for square array of unidirectional composite materials.

Fig. 1. RVE in the SUC model for unidirectional composite materials.

Fig. 3. Composite coupon under off-axis loading and its corresponding RVE.

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Based on the idea of the unit cell models [8,12,13], displacement components are assumed to be linear functions within each sub-cell. Furthermore, it is assumed that the applied normal stresses on the RVE do not introduce shear stress within the sub-cells and vice versa. The equilibrium conditions between global stresses (Si) over the RVE and local stresses (ri) within each sub-cell are:

8 Pr 1j > > < Pj¼1 bj r2 ¼ S2 Lr c i1 i¼1 ai r3 ¼ 0 > > : Pr Pc a b rij ¼ S L L 1 c r j¼1 i¼1 i j 1

ð2Þ

Equivalence of the normal and shear (skl) stress components along the interfaces of the sub-cells require:

r1j2 ¼ rij2 ði > 1Þ ri13 ¼ rij3 ðj > 1Þ sij23 ¼ 0; sij13 ¼ 0

ð3Þ

While eij is the local strain vector, compatibility of the local strains within each sub-cell yields to:

ei112 ¼ eij12

ð4Þ

Also, compatibility of the displacements within the RVE, assuming a perfectly bonded interface, requires: c X i¼1 r X

ai ei1 2 ¼ bj e1j 3 ¼

c X i¼1 r X

j¼1

ai eij2 ¼ Lc e2

ðj > 1Þ

ð5Þ

bj eij3 ¼ Lr e3

ði > 1Þ

ð6Þ

j¼1

eij1 ¼ e1 ði > 1; j > 1Þ

ð7Þ

It should be noted that these compatibility Eqs. (5) and (6) will be appropriately modified later to consider interface damage using CCI interfacial model. Finally, the three dimensional constitutive equations for sub-cell ij are:

eij ¼ Sij  rij þ aij DT þ eijp þ deijp

ð8Þ

where S is the elastic compliance matrix, a is the thermal expansion coefficient vector, DT is temperature change, ep is the accumulated plastic strain vector, dep is the increment of plastic strain vector and superscript ij refers to the sub-cell. The last two terms ep and dep are zero for elastic analysis and for each fiber cell for elastoplastic analysis. From (4) and using Hook’s law in shear load state the following relation can be derived:

sij12 ¼



si112 2Gi1

þ







Pði1Þ PðijÞ PðijÞ ePði1Þ  e12 þ de12 12 þ de12



 2Gij

ðj – 1Þ ð9Þ

Combination of Eqs. (5)–(8) yields to the following relations: c  X ai

ai 1j mi1 m m m r11 r2  ai ri13 þ ij ai rij3  i1 ai ri11 þ ij ai rij1 þ ai 2  E E E E E Eij i1 ij i1 ij i1 i¼1 h   io Pði1Þ Pði1Þ PðijÞ PðijÞ  e2 þ de2 ¼ ai ðaij  ai1 ÞDT; ðj – 1Þ  e2 þ de2 ð10Þ

r  X m1j mij ij bj 11 bj i1 m1j 1j mij ij  bj r1j bj r2 þ r  r3  bj r1 þ bj r1 þ bj 2 þ E Eij E1j 3 Eij E1j Eij 1j j¼1 h   io Pð1jÞ Pð1jÞ PðijÞ PðijÞ  e2 þ de2 ¼ bj ðaij  a1j ÞDT; ði – 1Þ  e2 þ de2

ð11Þ

Fig. 4. Room temperature stress–strain response of titanium IMI 318.

 1 11 mij ij 1 mij 1 11 1 ij r þ r2  r11 þ rij3 þ r þ r1  E11 2 E11 3 E11 1 Eij Eij Eij h   io Pð1jÞ Pð1jÞ PðijÞ PðijÞ þ e1 þ de1  e1 þ de1 ¼ ðaij  a11 ÞDT;

ði – 1; j – 1Þ

ð12Þ

It should be noted that governing equations of the problem include Eqs. (2) and (9)–(12) which is a system of (r  c + r + 2  c) linear equations with the same number of unknowns can be read as:

½Amm frgm1 þ ½Bmn ðfeP gn1 þ fdeP gn1 Þ ¼ ff gm1 ðwhere m ¼ r  c þ r þ 2  c; n ¼ 4  ðrc  number of fiber cellsÞÞ ð13Þ For solving this iterative system of equations, successive approximation method [31] together with Von-Mises yield criterion is used. 3. Material type The composite system considered in this study includes SiC/Ti MMC consists of a titanium matrix, IMI318 (Ti–6Al–4 V), reinforced by aligned DERA Sigma SM1240, silicon carbide with 33% fiber volume fraction. The fibers are assumed to be elastic up to the fracture point which is about 3240 MPa in tension. The titanium matrix is treated as elastoplastic with yield stress of 910 MPa governed by Von-Mises yield criterion [18]. The strain hardening rate after yielding was taken as 0.5 GPa. The room temperature stress– strain response of the matrix is shown in Fig. 4 [32]. The other mechanical and thermal properties of the constituents of the SiC/ Ti system are tabulated in Table 1. Titanium based fiber reinforced composites is fabricated by diffusion bonding at 850–930 °C [33]. The fiber and matrix were assumed to be strain free at 930 °C and a cool down to room temperature at 20 °C generates the residual stresses. In this study, any relaxation of the residual stresses due to creep of the matrix

Table 1 Materials properties of SiC/Ti [32].

SiC (F) Ti (M)

E (GPa)

m

a (106/°C)

409 107

0.2 0.3

5 10.4

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material was ignored. Therefore, the predicted state of residual stress is likely to be an upper bound.

ez ESUC 50x50 ez Analyitical

exESUC 50x50 ex Analytical

exz ESUC 50x50 exz Analytical

1.6

4. Interfacial de-bonding criterion 1.4

I

I

½u_ n  ¼ Rn r_ n j ; ½u_ t I ¼ Rt r_ t jI ;

I

rn j P rDB j rt jI P rDB jI

I

ð14Þ

I1 ai ei1 2 þ Rn r þ

i¼1

ai ei1 2 ¼

i¼Iþ1

¼ Lc e2 I X

c X

1I bj e1j 3 þ Rn r þ

j¼1

ai eij2 þ Rn rIj þ

c X

i¼1

i¼Iþ1

r X

r X

ai eij2 ð15Þ

bj e1j 3 ¼

j¼Iþ1

¼ Lr e3

c X

ðj > 1Þ r X

bj eij3 þ Rn riI þ

j¼1

bj eij3

j¼Iþ1

ði > 1Þ

ð16Þ

Furthermore, the compatibility equations for overall shear strain become: I X

I1 ai ei1 12 þ Rt s þ

c X

i¼1

i¼Iþ1

I X

r X

j¼1

1I bj e1j 13 þ Rt s þ

 ai ei1 12 ¼ Lc e12 bj e1j 13 ¼

j¼Iþ1

1 0.8 0.6 0.4 0.2

where dots denote time differentiation and Rn and Rt are empirical de-bonding parameters that represent the effective compliance of the interface and r DB is a finite interfacial strength. For the presented SiC/Ti system, Rt = Rn = 104 MPa1 as reported in [26] and rDB = 300 MPa [24] are used. It should be noted that in order to incorporate the CCI model to the de-bonded interface, I, Eqs. (5) and (6) should be modified for normal loading as: I X

1.2

Normalised Strain

In order to include interface damage, the interfacial de-bonding is predicted by constant compliant interface model presented by Wilt and Arnold [28]. This model permits a discontinuity in the normal or tangential displacement component at an interface, I, that is proportional to the appropriate stress component at the interface. Furthermore, the model permits a finite strength for the interface as expressed by the following equations:

r X

Lr e13

ð17Þ ð18Þ

0

0

20

40

60

80

-0.2 -0.4

Off-axis Angle (Degree) Fig. 5. Comparison of various normalized strain components for SiC/Ti composite.

mechanics analysis based on the theory of elasticity [34] with perfectly bonded interface assumption. Fig. 5 depicts predictions of the presented model for axial and shear strain components, based on the governing equations described in Section 2.2. Included in the figure are also results of the theory of elasticity [34]. All strain components are normalized with respect to the strain in the loading direction ex. Results obtained from analytical continuum model and the presented micromechanical ESUC model show good agreement. The off-axis Young’s modulus (Eh) can be determined based on the macro-mechanical model as:

" #1   2 cos4 h 1 tAT sin h 2 2 Eh ¼ þ 2 sin h cos h þ EA GA ET EA

ð19Þ

j¼1

5. Results and discussion Using the ESUC model described in Section 2, elastoplastic stress–strain behavior of the SiC/Ti MMC system is predicted for general off-axis loading. Effects of interfacial damage, thermal residual stress and fiber volume fraction are studied. Interfacial de-bonding is predicted based on the CCI model presented in Section 4 with relevant parameters. To include the influence of manufacturing process thermal residual stresses in the analysis, a decrease from manufacturing temperature (930 °C) to room temperature (25 °C) is considered by the term aDT in Eq. (8). Firstly, elastic behavior and initial damage aspects are investigated to validate the model. Comparison of the predictions for several off-axis angles are presented with finite element analysis and experiment. It should be noted that the effects of the end constraints are not included in all results presented in this study. This is due to the fact that end constraint effects are very small and negligible in the case of SiC/Ti MMC system [7].

where EA and ET denote the axial (h = 0°) and transverse (h = 90°) Young’s modulus and GA and tAT are the axial shear modulus and Poisson’s ratio. The used values in Eq. (15) are: EA = 200.6 GPa, ET = 157.4 GPa, GA = 60.31 GPa, tAT = 0.265 [7]. The analytical prediction for off-axis Young’s modulus of SiC/Ti MMC with a fiber volume fraction of 0.33 was determined. For the presented ESUC model, the off-axis Young’s modulus is determined by applying a unit load in the x direction and measurement of the strain in the load direction based on the governing equations in Section 2.2. Fig. 6 compares predictions of the presented ESUC model for square and hexagonal arrangements with analytical predictions and experimental data [35]. It can be concluded that predictions of the presented micromechanical model for general offaxis Young’s modulus show reasonably good agreement with the analytical macro-mechanical model. However, both macromechanical and ESUC predictions are higher than experimental data. For transverse Young’s modulus, i.e. h = 90° results of ESUC model with hexagonal array are closer to the experiment than square array and analytical macro-mechanical model. 5.2. Initiation of nonlinearity

5.1. Model validation As a first step to validate the presented model, various strain components are compared with results of analytical macro-

Onset of nonlinearity in the stress–strain response of the composite occurs either by matrix yielding or interface failure as the loading direction is varied from directly in line with the fiber axis

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is the main cause for initiation of nonlinearity. It should be noted that matrix yielding happens when the Von-Mises equivalent stress within the most heavily loaded sub-cell reaches the yield stress of the matrix. Furthermore, the onset of nonlinearity in the experiment is obtained from the first point where results are deviated from straight line. Fig. 7 also shows predictions of the presented ESUC model and finite element model for onset of nonlinearity without considering thermal residual stress and interface de-bonding effects. It is interesting to note that in this case initiation of nonlinearity is controlled either by fracture of the fiber or by yielding of the matrix depending on the loading angle. As expected, assuming perfectly bonded interface leads to much higher predictions for initiation of damage in SiC/Ti MMCs. It can be concluded that considering effects of both thermal residual stress and interface damage is necessary to obtain reliable results for metal matrix composites whereas these effects might be ignored in the case of polymer matrix composites.

250 SUC Square array

230

Experiment Analytical

Young's Modulus (GPa)

210

ESUC50x50

190

SUC hexagonal array

170 150 130 110 90 70 50 0

20

40

60

80

Off-axis Angle (Degree) Fig. 6. Comparison of Young’s modulus for SiC/Ti composite under off-axis loading with fiber volume fraction of 33%.

2000 ESUC 50x50 with R.S debonded

Fiber Fracture

Experiment [35]

1800

ESUC 50x50 without R.S perfectly bonded Finite element without R.S perfectly bonded [5] Finite element with R.S debonded [7]

Initial Damage Load (MPa)

1600 Matrix Yielding

1400

1200

1000

Matrix Yielding

800

5.3. On-axis elasto-plastic response The next case is the elasto-plastic response of the SiC/Ti composite during on-axis loading, i.e. loading in the fiber or perpendicular to the fiber direction. A comparison between the predicted longitudinal behavior with experiment [35] and finite element model [7] for 33% fiber volume fraction is presented in Fig. 8. There is an excellent agreement between ESUC, finite element predictions and experimental measurements in case residual stresses are included. It should be noted that a coarse mesh of 10  10 in ESUC model is enough to provide accurate results. Furthermore, stress–strain response of the composite in the case of axial loading is only influenced by residual stresses and interface damage is negligible. Fig. 8 also includes stress–strain behavior of the composite without residual stress effects. It is also worth mentioning that residual stresses are tensile in the matrix and compressive in the fiber. Therefore, ignoring residual stress effects leads to overestimated predictions for material behavior in tension. Moreover, in this case the strength of the MMC is controlled by fiber fracture which is shown in the figure by a diamond sign.

Interfacial De-bonding

600

3000

400 2500

0 0

10

20

30

40

50

60

70

80

90

Off-axis Angle (Degree) Fig. 7. Onset of nonlinearity in stress–strain curve of SiC/Ti composite system versus off-axis angle with 33% fiber volume fraction.

Axial Stress (MPa)

200

2000

Fiber Fracture point 1500

Experiment [35]

1000

to normal to the fiber. Fig. 7 contains the load associated with the initial nonlinearity in the stress–strain curve of the SiC/Ti MMC for various off-axis angles. Effects of both interface damage and thermal residual stresses are considered in the analysis. Included in the figure are also results of experiment [35] and predictions of finite element analysis [7]. As can be seen in the figure, predictions of the ESUC model show very good agreement with experiment [35] for different off-axis angles. It can be concluded from the figure that initiation of the nonlinearity is due to the matrix yielding for off-axis angles up to 22°. Beyond this point, interfacial damage

ESUC 10x10 with R.S Finite elements with R.S [7]

500

ESUC 10x10 without R.S Finite elements without R.S [7]

0

0

0.5

1

1.5

Axial Strain ( %) Fig. 8. Predicted stress–strain curves for longitudinal loading in tension.

2

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M.J. Mahmoodi et al. / Materials and Design 31 (2010) 829–836

600

900 800 700

Overall Stress (MPa)

Transverse Stress (MPa)

500

400

300

200

500 400 300

Experiment [35]

Experiment [35]

Finite element debonded [7]

100

600

200

ESUC 10x10 dedonded

ESUC 50x50 debonded Finite element debonded [7]

ESUC 50x50 dedonded

100

Finite element perfecly bonded [7]

0

ESUC 50x50 perfectly bonded

0

0.2

0.4

0.6

0.8

1

1.2

0

Transverse Strain (%)

0

0.2

0.4

0.6

0.8

Overall Strain (%) Fig. 9. Predicted stress–strain curves for transverse loading in tension.

5.4. Off-axis stress–strain response Figs. 10–12 depict the stress–strain behavior of the SiC/Ti composite with 33% fiber volume fraction for three different off-axis angles of 15°, 30°, 45°, respectively. Effects of residual stress and interfacial damage are included in the analysis. Again, comparison

Fig. 11. Stress–strain response of the SiC/Ti subjected to off-axis loading (h = 30°) with residual stress effects and interfacial de-bonding.

800 700

Overall Stress (MPa)

Fig. 9 shows the prediction of transverse behavior of the SiC/Ti composite with 33% fiber volume fraction in the presence of residual stress and interfacial damage. Figure also includes results of the finite element micromechanical model [7] and experiment [35]. Unlike axial loading, transverse behavior of the composite system is highly affected by both residual stress and weak interface. Interface de-bonding is the first damage mode to occur. Also, results of ESUC model are affected by mesh size as 50  50 mesh provides more accurate results in comparison with 10  10.

600 500 400 300 Experiment [35]

200

ESUC 50x50 debonded

1200

Finite element debonded [7]

100

Finite element perfectly bonded [7] ESUC 50x50 perfectly bonded

1000

0

Overall Stress (MPa)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Overall Strain (%) 800 Fig. 12. Stress–strain response of the SiC/Ti subjected to off-axis loading (h = 45°) with residual stress effects and interfacial de-bonding.

600

400

Experiment [35] ESUC 50x50 debonded Finite element debonded [7]

200

ESUC 50x50 perfectly bonded

of the predictions with experimental data [35] and finite element [7] is quite encouraging. It can be concluded that effects of interfacial de-bonding become more considerable as off-axis angle is increasing. In order to highlight the role of weak interface on the behavior of the material, the fully bonded interface predictions are also included in all figures. As can be seen in the figures, predictions based on the fully bonded assumption are highly overestimated while they are valid only for early stages of the loading.

Finite element perfectly bonded [7]

0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Overall Strain (%) Fig. 10. Stress–strain response of the SiC/Ti subjected to off-axis loading (h = 15°) with residual stress effects and interfacial de-bonding.

5.5. Ultimate tensile strength Comparison between presented ESUC results with finite element micromechanical analysis [7], experimental data [35] and a macro-mechanical failure criterion [36] about the strength predic-

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tion of the SiC/Ti composite system is depicted in Fig. 13 for both perfectly bonded and weak interface. The ESUC results were calculated from the load at which the fiber fracture stress was exceeded or from the load at which Von-Mises stresses of all matrix sub-cells reached to the ultimate tensile strength of the matrix and the stress–strain curve became horizontal. Thermal residual stress was taken into account in the analysis. The macro-mechanical failure criterion presented by Azzi and Tsai based on Hill’s failure criterion to predict the strength of laminates and unidirectional composite subjected to a general loading condition. In an off-axis tension test on a unidirectional composite, this model can be reduced to a simple equation as:

1

rh ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   r

1000

ð20Þ

4

2

h þ s12  r12 sin h cos2 h þ sin r2c90 c c0

800

where rh, rc0, rc90 and sc represent the off-axis, axial, transverse and shear strengths, respectively. The relevant values for the axial and transverse strengths were adapted to the experimental data of 1620 and 498 MPa and for the shear strength was taken to be 287 MPa as reported in [37]. It can be seen in Fig. 13 that while ESUC strength predictions based on perfectly bonded interface are far from experimental data, presented ESUC with considering interfacial de-bonding have very good agreement with experiment, micromechanical finite element and macro-mechanical analysis in various off-axis angles.

Overall Stress (MPa)

cos4 h 2 c0

face on the overall response of 45° coupon of the SiC/Ti composite system is shown. Fig. 14 clearly indicates that considering effects of both parameters are necessary to obtain accurate predictions in comparison with experimental measurements [35]. It is also shown that assuming bonded interface, with or without effects of thermal residual stresses leads to overestimated prediction for strength of the composite up to more than 100% in comparison with experiment. Furthermore, predictions based on ignoring

600

400

Experiment [35]

5.6. Parametric study

ESUC50x50 perfectly bonded without R.S

200

ESUC50x50 perfectly bonded with R.S

In order to study effects of various parameters on the off-axis behavior of the SiC/Ti composite system, the 45° coupon is considered. As reported in other studies [26], the interfacial stress must not only exceed any mechanical clamping due to compressive residual stress, but also rise into the tensile regime to overcome the chemical bond. This physical reality can be cleared in Fig. 14 in which the effect of both thermal residual stress and weak inter-

ESUC50x50 with R.S debonded ESUC50x50 without R.S debonded

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

Overall Strain (%) Fig. 14. Effect of thermal residual stress on the stress–strain response of 45° coupon of the SiC/Ti composite system.

1800 Experiment [35]

1000

Macromechanical criterion [36]

1600

Finite element perfectly bonded [7]

ESUC 50x50 perfectly bonded

1400

ESUC 50x50 debonded

800

1200

Overall Stress (MPa)

Off-Axis Tensile Strength (MPa)

Finite element debonded [7]

1000

800

600

600

400 ESUC 50x50 debonded FVF=33 % ESUC 50x50 debonded FVF=15 %

400

"ESUC 50x50 debonded FVF=50 %"

200

ESUC 50x50 perfectly bonded FVF=50 %

200

ESUC 50x50 perfectly bonded FVF=15 % ESUC 50x50 perfectly bonded FVF=33 %

0

0 0

10

20

30

40

50

60

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Off-Axis Angle (Degree) Fig. 13. Comparison of tensile strength for the SiC/Ti MMC subjected to off-axis loading with thermal residual stress effects.

0

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Overall Strain (%) Fig. 15. Effect of fiber volume fraction on the stress–strain response of the 45° coupon of the SiC/Ti composite system.

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residual stresses with weak interface show that interface damage occurs at stresses lower than reality due to compressive nature of the residual stresses. This creates the first knee in the stress– strain response of the composite in Fig. 14 at a lower stress of about 200 MPa. However, all matrix sub-cells are still in elastic deformation even after total damage of interface which results in higher strength of the composite. Fig. 15 shows effects of fiber volume fraction on the overall stress–strain response of the 45° coupon of the SiC/Ti system for both perfectly bonded and weak interface assumptions. Thermal residual stresses are included in all cases. It is interesting to note that increasing FVF results in increasing or decreasing of strength of the composite in the perfectly bonded or weak interface, respectively. For the fully bonded assumption, increasing strength and stiffness of the composite system is quite reasonable as the contribution of the fibers in the overall behavior is increasing. However, in the case of weak interface, increasing FVF leads to more interface area and less matrix material which in turns results in lower stiffness and strength of the composite. Finally, it should be emphasized that both bonded and weak interface cases show similar stiffness and behavior at very early stages of loading. 6. Conclusion A 3-D micromechanics based analytical SUC model is developed to study effects of interfacial damage on the elasto-plastic response of a fibrous SiC/Ti metal matrix composite (MMC) subjected to general off-axis loading. The model also includes the effect of manufacturing process thermal residual stress. The single CCI model characterizations are calibrated to model interfacial de-bonding in various off-axis angles. The initiation and propagation of damage modes including fiber fracture, interfacial de-bonding and matrix plasticity are reported depending on the loading direction. Results revealed that increasing FVF leads to increasing or decreasing of strength of the composite in the perfectly bonded or interfacial de-bonded cases, respectively. It is shown that while predictions based on perfectly bonded interface are far from reality, the predicted stress–strain behavior in the presence of damaged interface and thermal residual stresses demonstrate very good agreement with experimental data. References [1] Clyne TW, Withers PJ. An introduction to metal matrix composites. Cambridge University Press; 1993. [2] Adams DF. Inelastic analysis of a unidirectional composite subjected to transverse normal loading. J Compos Mater 1970;4:310–28. [3] Sun CT, Vaidya RS. Prediction of composite properties from a representative volume element. Compos Sci Technol 1996;56:171–9. [4] Nedele MR, Wisnom MR. Finite element micromechanical modelling of a unidirectional composite subjected to shear loading. Composites 1994;25:263–72. [5] Aghdam MM, Smith DJ, Pavier MJ. Finite element micromechanical modelling of yield and collapse behaviour of metal matrix composites. J Mech Phys Solids 2000;48(3):499–528. [6] Aghdam MM, Dezhsetan A. Micromechanics based analysis of randomly distributed fiber reinforced composites using simplified unit cell model. Compos Struct 2005;71:327–32. [7] Aghdam MM, Pavier DJ, Smith DJ. Micro-mechanics of off-axis loading of metal matrix composites using finite element analysis. Int J Solids Struct 2001;38:3905–25.

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