Damage analysis of fiber reinforced Ti-alloy subjected to multi-axial loading—A micromechanical approach

Materials Science and Engineering A 528 (2011) 7983–7990

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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea

Damage analysis of fiber reinforced Ti-alloy subjected to multi-axial loading—A micromechanical approach M.J. Mahmoodi a,∗ , M.M. Aghdam b a b

Department of Mechanical and Energy Engineering, Power and Water University of Technology (PWUT), Tehran P.O. Box 16765-1719, Iran Thermoelasticity Center of Excellency, Department of Mechanical Engineering, Amirkabir University of Technology, Hafez Ave., Tehran 15914, Iran

a r t i c l e

i n f o

Article history: Received 13 March 2011 Received in revised form 18 July 2011 Accepted 18 July 2011 Available online 23 July 2011 Keywords: Metal matrix composites Micromechanics Damage analysis Thermal residual stress

a b s t r a c t Effects of initiation and propagation of interface damage on the elastoplastic behavior of a unidirectional SiC/Ti metal matrix composite (MMC) subjected to multi-axial loading are studied using a three-dimensional micromechanics based analytical model. Effects of manufacturing process thermal residual stress (RS) are also included in the analysis. 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 used to model interfacial debonding and the successive approximation method together with Von-Mises yield criterion is used to obtain elastic–plastic behavior. Failure modes during multi-axial tensile/compressive loading in the presence of residual stresses are discussed in details. Results revealed that for more realistic predictions both interface damage and thermal residual stress effects should be considered in the analysis. Comparison between results of the presented model shows very good agreement with available finite element micromechanical analysis and experiment for uniaxial loading. Also, results are extracted and interpreted for equi-biaxial including transverse/transverse and axial/transverse and equi-triaxial loading. © 2011 Elsevier B.V. All rights reserved.

1. Introduction There has been considerable interest in metal matrix composites (MMCs) for their application in various industries. One advantage of MMCs, especially titanium based MMCs, is their high performance in service at elevated temperature [1]. One concern associated with reliability is that MMCs are consolidated at high temperature which induces residual stresses during the cool-down process to room temperature. These residual stresses are produced due to mismatch between the coefficients of thermal expansion of the constituents. The other concern is related to the existence of a weak interface between the fiber and matrix. Reliable use of MMCs requires detailed modeling and understanding of the behavior of MMCs under various combined loading conditions. In order to be more accurate, any presented model for MMCs should include existence of both weak fiber/matrix interface and relatively high manufacturing process thermal residual stresses (RS). 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 conditions. Multi-axial loading seems to be more general and therefore, understanding the manner in which the properties

∗ Corresponding author. Fax: +98 21 6641 9736. E-mail address: [email protected] (M.J. Mahmoodi). 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.07.033

degrade as the loading conditions vary is an essential requirement in the practical use of composites. A considerable number of research work have been undertaken to develop micromechanical models, including finite element (FE) [2–5] and analytical [5–9] approaches, 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 [5,6]. On the other hand, there are different analytical [10–13] and numerical [14–22] 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 [23,24]. Aboudi [10] incorporated the flexible interface (FI) model of Jones and Whittier [25] into the method of cells. Later, the FI model concept was employed [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

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

finite interfacial strength is a major improvement since previous works on SiC/Ti composites point 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 debonding model was presented for MMCs based on modified Needleman type cohesive zone model under normal and shear loading [11]. While there are some studies on damage analysis of MMCs using micromechanical finite elements models for transverse/transverse [20] axial/transverse [21] biaxial loading, initiation and propagation of fiber/matrix interfacial deboning for unidirectional MMCs in general multi-axial loading were not found in the literature. In this study, the elastoplastic response of a SiC/Ti unidirectional fiber reinforced composite under general multi-axial loading is predicted using the modified version of 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. 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 debonding. Results for uniaxial stress–strain response show favorably good agreement with available finite element results and experimental data. Material behavior under transverse/transverse and axial/transverse equi-biaxial loading is also examined. Furthermore, initiation and propagation of damage of the composite is predicted in bi-axial loading depending on the precise loading conditions. Finally, effective response and damage evolution of the MMC are obtained in equi-triaxial loading. 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 sub-cells are surrounded by matrix sub-cells. Geometry of the selected 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. Furthermore, a unit length in the z direction is considered to apply generalized plane strain assumption on the RVE. Each sub-cell labeled as ij in which i

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

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 model circular shape of the fiber as shown in Fig. 2. 2.2. Micromechanical governing equations In order to obtain governing equations, similar to previous unit cell models [9–13], it is assumed that all displacement components within each sub-cell of the RVE varies linearly and therefore, all micro-stress and micro-strain components in sub-cells are constants. Furthermore, it is assumed that the applied normal stress on the RVE does not introduce any shear stress inside the sub-cells and vice versa. In this study, details of the governing equations for multi-axial normal loading are discussed. The equilibrium conditions between global stresses (Si ) over the RVE and local stresses ( i ) within each sub-cell are:

⎧ r  1j ⎪ ⎪ ⎪ bj x = Sx Lr ⎪ ⎪ ⎪ ⎪ j=1 ⎪ ⎪ c ⎨

ai yi1 = Sy Lc

(1)

⎪ ⎪ i=1 ⎪ ⎪ c r ⎪  ⎪ ij ⎪ ⎪ ai bj z = Sz Lc Lr ⎪ ⎩ j=1 i=1

Equivalence of the normal stress components along the interfaces of the sub-cells require: x = x

1j

ij

(i > 1)

yi1

ij y

(j > 1)

=

(2)

Compatibility of the displacements within the RVE, assuming a perfectly bonded interface, requires: c  i=1

ai εi1 x =

c  i=1

ai

ij

εx = Lc ε¯ x

(j > 1)

(3)

M.J. Mahmoodi, M.M. Aghdam / Materials Science and Engineering A 528 (2011) 7983–7990 r 

1j

bj εy =

j=1

r 

ij

bj εy = Lr ε¯ y

(i > 1)

(4)

ij

(i > 1, j > 1)

(5)

It is worth mentioning that these compatibility Eqs. (3)–(5) will be appropriately modified later to consider interface damage using CCI interfacial model. Finally, three-dimensional constitutive equations for sub-cell ij are: ij

ij

␧ij = Sij ␴ij + ␣ij T + ␧p + d␧p

(6)

where S is the elastic compliance matrix, ␣ is the thermal expansion coefficient vector, T is temperature change, ␧p is the accumulated plastic strain vector, d␧p is the increment of plastic strain vector and superscript ij refers to the sub-cell. The last two terms ␧p and d␧p are zero during elastic analysis and for all fiber cells during both elastic and elastoplastic analyses. Substitution of Eqs. (2) and (6) into Eqs. (3)–(5) yield to the following relations: x   ai i=1

Ei1

−

SiC (F) Ti (M)

E (GPa)



˛(10−6 /◦ C)

409 107

0.2 0.3

5 10.4

material was ignored. Therefore, the predicted state of residual stress is likely to be an upper bound. 4. Interfacial debonding criterion In order to include interface damage, the interfacial debonding 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 the interface, I, that is proportional to the appropriate stress component at the



211 −

r   1j j=1

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

j=1

εz = ε¯ z

7985

E1j

ij ij ai 1j i1  P(i1) P(i1) P(ij) P(ij) ij ij  − a  i1 + a  − i1 ai 1i1 + a  + ai × [(ε2 + dε2 ) − (ε2 + dε2 )] Eij i 3 Eij i 1 Eij 2 Ei1 i 3 Ei1

1j

bj 2 +

ij Eij

ij

bj 2 +

bj E1j

311 −

bj Eij

3i1 −

1j E1j

1j

bj 1 +

ij Eij

= ai (˛ij − ˛i1 )T,

(j = / 1)

(7)

 ij

P(1j)

bj 1 + bj × [(ε2

P(1j)

+ dε2

P(ij)

) − (ε2

P(ij)

+ dε2

)]

= bj (˛ij − ˛1j )T,

(i = / 1) (8)





−

1 11 ij ij 1 11 ij ij 1 11 1 ij P(1j) P(1j) P(ij) P(ij)  +  −  +  +  +  + [(ε1 + dε1 ) − (ε1 + dε1 )] E11 2 Eij 2 E11 3 Eij 3 E11 1 Eij 1

It should be noted that governing equations of the problem include Eqs. (1) and (7)–(9) which are a system of (r × c + r + c) linear equations with the same number of unknowns. The system can be read as: [A]m×m {}m×1 + [B]m×n

 P ε

(where m = r × c + r + c,



n×1



 P

+ dε

n×1

=

 f

= (˛ij − ˛11 )T,

(i = / 1, j = / 1)

(9)

interface. Furthermore, the model permits a finite strength for the interface as expressed by the following equations: I

[u˙ n ] = Rn ˙ n |I , I [u˙ t ] = Rt ˙ t |I ,

n |I ≥ DB |I , t |I ≥ DB |I .

(11)

m×1

n = 4 × (rc − number of fiber cells) (10)

In order to solve this iterative system of equations, the successive approximation method [31] together with Von-Mises yield criterion is used.

where dots denote time differentiation and Rn and Rt are empirical debonding parameters that represent the effective compliance of the interface and  DB is a finite interfacial strength. For the presented SiC/Ti system, Rt = Rn = 10−4 MPa−1 as reported in [26] and  DB = 300 MPa [22] are used. It should be noted that in order to incorporate the CCI model to the debonded interface, I, under normal loads Eqs. (3) and (4)

3. Material type The composite system considered in this study includes SiC/Ti MMC consists of a titanium matrix, IMI318 (Ti–6Al–4V), 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 and 9 GPa in compression. The titanium matrix is treated as elastoplastic with yield stress of 910 MPa and ultimate strength of 1012 MPa governed by VonMises yield criterion [23]. The strain hardening rate after yielding was taken as 0.5 GPa. The room temperature stress–strain response of the matrix [32] is shown in Fig. 3. The other mechanical and thermal properties of the constituents of the SiC/Ti system are tabulated in Table 1. In all presented results, a 33% fiber volume fraction specimen is considered. 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 which generates the residual stresses. In this study, any relaxation of the residual stresses due to creep of the matrix

Fig. 3. Room temperature stress–strain response of titanium IMI 318.

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should be modified for normal loading as: I 

c 

I1 ai εi1 x + Rn x +

i=1

ai εi1 x =

i=I+1



c 

ij

ai εx + Rn x Ij

i=1

c

+

ij

ai εx = Lc ε¯ x

(j > 1)

(15)

i=I+1 I 

1j

bj εy + Rn y1I +

j=1

+

r 

1j

bj εy =

j=I+1 r 

ij

bj εy = Lr ε¯ y

r 

ij

bj εy + Rn yiI

j=1

(i > 1)

(16)

j=I+1

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 multi-axial loading including uniaxial, equi-biaxial and equi-triaxial conditions. Interfacial debonding 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 ␣ T in Eq. (8). In all cases of study, when residual stresses are not included the tension and compression curves are identical. Also, the absolute values of stress/strain for tension and compression are plotted in the figures. It should be noted that term “perfectly bonded” and “debonded” indicate results obtained based on Eqs. (3) and (4) and Eqs. (15) and (16), respectively. Also in the all multiaxial results, the loading paths are proportionally increasing static loading which is applied simultaneously on the RVE. 5.1. Model validation (uniaxial response) As a first step to validate the presented ESUC model, the elastoplastic response of the SiC/Ti composite is obtained during uniaxial loading both in the fiber or perpendicular to the fiber direction. A comparison between the predicted longitudinal behavior with experiment [34] and finite element model [14] for the composite is presented in Fig. 4. 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. 4 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. As shown in the figure, when thermal residual stresses were ignored, identical behavior was predicted for tensile and compressive loading. However, considering thermal residual stresses in the analysis leads to different behavior of the MMC in the tensile and compressive loading. Fig. 5 expresses the prediction of the presented ESUC model about the locations of initiation and propagation of matrix plastic deformation within the RVE under tensile/compressive longitudinal loading. Yielding initiates at the points specified by the cross

Fig. 4. Stress–strain curves of the SiC/Ti composite in uniaxial longitudinal loading.

sign and propagates in the whole matrix along the dashed lines which have circumferential symmetry with respect to the diagonal direction. It is obvious that the tensile load causes initiation of yielding at stresses well below the relevant compressive load due to the existence of thermal residual stress as shown in Fig. 4. Moreover, initial yielding occurs at the points with maximum stress concentration which are located along the minimum distance between the fibers over the interface where the genus is changed. Fig. 6 shows the prediction of transverse behavior of the SiC/Ti composite with and without residual stress and interfacial damage. Figure also includes results of the finite element micromechanical model [14] and experiment [34]. Unlike axial loading, transverse behavior of the composite system is highly affected by both residual stress and weak interface. Figure clearly indicates that considering effects of both parameters are necessary to obtain accurate predictions in comparison with experimental measurements. It is also

Fig. 5. Initiation and propagation of matrix plasticity within the RVE under uniaxial longitudinal loading.

M.J. Mahmoodi, M.M. Aghdam / Materials Science and Engineering A 528 (2011) 7983–7990

7987

2000 1800

Transverse Stress (MPa)

1600 1400 1200 1000 800 Tension without R.S

600

Tension with R.S

400

Compression with R.S Tension De-bonded with R.S

200

Tension De-bonded without R.S

0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Transverse Strain (%)

Fig. 8. Stress–strain curves of the SiC/Ti in tensile transverse/transverse equi-biaxial loading. Fig. 6. Stress–strain curves of the SiC/Ti in uniaxial transverse loading.

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. Also, results of ESUC model are affected by mesh size as 50 × 50 mesh provides more accurate results in comparison with 10 × 10. Therefore, unless otherwise stated, in all presented results a 50 × 50 mesh size is considered. It is interesting to note that, different effects of thermal stresses on the yielding of the composite appear in the longitudinal and transverse directions. While residual stresses postponed yielding for compression loading in longitudinal tension, as shown in Fig. 4, residual stresses for compression in the transverse direction promote earlier yielding, as shown in Fig. 6. Fig. 7 describes the location of initiation and propagation of interfacial debonding and matrix plasticity under tensile transverse loading in details. Both of them initiate at the point specified by the cross sign and propagate through the dashed line along the interface and within the whole matrix for interfacial debonding and matrix yielding, respectively. Again, with considering fiber arrangement and loading condition, interfacial debonding and matrix plasticity initiate at the spot with maximum stress concentration.

Fig. 7. Initiation and propagation of interfacial debonding and matrix plasticity within the RVE under tensile uniaxial transverse loading.

5.2. Biaxial loading It was also of interest to examine behavior of the material subjected to equi-biaxial loading by the presented ESUC model. Fig. 8 depicts predictions of the model for transverse/transverse equibiaxial loading in which Sx = Sy are applied simultaneously on the RVE. Effects of residual stress and interfacial damage are included in the analysis. As can be seen in Fig. 8 inclusion of thermal residual stresses changed the behavior of the material in tension and compression differently in the same direction. For uniaxial loading, all curves eventually reached the same limiting √ stress at higher levels of strain. This limiting stress is equal to 20 3 for an elastic perfectly plastic matrix with a yield stress of  0 for bonded interface [35]. However, no limiting stress was observed in the equi-biaxial loading. Furthermore, comparison between uniaxial and transverse/transverse equi-biaxial loading cases revealed that interfacial debonding initiated at higher applied load of 150 MPa and completed at lower strain of 0.3 in the biaxial loading while debonding occurs at 130 MPa in the uniaxial tension. It is also interest to study different modes of failure in transverse/transverse equi-biaxial loading of the MMCs. A study was carried out on the effects of interface bonding and thermal residual stresses on the initiation of deterioration and nonlinear behavior in the transverse/transverse stress space. Initial deterioration which causes nonlinear behavior of the composite is defined either by interface debonding or matrix yielding. Initial yield occurs as the loading on the MMC is increased until the most heavily loaded point within the matrix reaches the yield stress. Results are shown in Fig. 9. As can be seen in the figure, the size of the initial deterioration surface is highly affected and reduced by residual stresses. Moreover, the presence of temperature change shifts initial deterioration envelope in the transverse plane towards positive transverse stresses. Also, tensile behavior of the material with interface is significantly weaker and softer than bonded interface for biaxial tensile stress. However, results demonstrate that in the biaxial compressive loading, both bonded and debonded models show similar behavior. Fig. 10 depicts the location of initiation and propagation manner of interfacial debonding and matrix plasticity under tensile transverse/transverse equi-biaxial loading in details. Both of the interfacial debonding and matrix plasticity initiate at the point

M.J. Mahmoodi, M.M. Aghdam / Materials Science and Engineering A 528 (2011) 7983–7990

Sy (GPa)

7988

2.5

1000

2 800

1

0.5 Sx (GPa)

Overall Stress (MPa)

1.5

600

400

0 -1.5

-1

-0.5

0

0.5

1

1.5

2

Tension bonded without R.S

2.5

Tension bonded with R.S

-0.5

200

Tension De-bonded without R.S Tension De-bonded with R.S

Perfectly bonded Without R.S

-1

Compression with R.S

Perfectly bonded With R.S

0

De- bonded Without R.S

-1.5

0

0.2

0.4

0.6

0.8

Transverse Strain (%)

De- bonded With R.S

Fig. 9. Initial deterioration of the SiC/Ti composite in transverse/transverse equibiaxial loading.

Fig. 11. Stress–transverse strain curves of the SiC/Ti composite in tensile longitudinal/transverse equi-biaxial loading.

specified by the cross sign on the diagonal line and propagate through the dashed lines which are circumferential symmetric with respect to the diagonal direction. Indeed, damages initiate at the point involves highly stress concentration with considering fiber distribution in the matrix and loading conditions. Equi-biaxial loading in axial/transverse direction where Sx = Sz are applied simultaneously on the RVE, was also examined by the presented ESUC model in the presence of thermal residual stresses and interfacial debonding. The predictions were obtained using tension/tension and compression/compression with equal magnitude of load in the fiber and perpendicular to fiber directions. Figs. 11 and 12 depict stress–strain response of the composite in the transverse and axial directions, respectively. Figures include effects

of both weak interface and thermal residual stresses. Comparison between predictions for transverse behavior of the composite in uniaxial and axial/transverse equi-biaxial loading; i.e. Figs. 5 and 11 revealed that damage initiation occurs at higher applied load of 160 MPa in the biaxial loading while propagation of the interface debonding is faster than uniaxial loading. Furthermore, strength of the composite increases from 533 MPa in uniaxial case to 657 MPa in the biaxial loading. Fig. 13 indicates the onset of nonlinearity of the MMC system under axial/transverse equi-biaxial loading including both interface damage and thermal residual stresses effects. As can be seen in the figure, in the presences of thermal residual stresses, initial deterioration surface is shifted towards negative longitudinal

1000

Overall Stress (MPa)

800

600

400

Tension: Bonded without R.S Tension: Bonded with R.S Tension:De- Bonded without R.S

200

Tension:De- Bonded with R.S Compression: With R.S

0 0

0.1

0.2

0.3

0.4

Axial Strain (%) Fig. 10. Initiation and propagation of interfacial debonding and matrix plasticity within the RVE under tensile transverse/transverse equi-biaxial loading.

Fig. 12. Stress–axial strain curves of the SiC/Ti composite in tensile longitudinal/transverse equi-biaxial loading.

Sz (GPa)

M.J. Mahmoodi, M.M. Aghdam / Materials Science and Engineering A 528 (2011) 7983–7990

7989

1200

2.5

2

1000 1.5

0.5 Sx (GPa) 0 -1.5

-1

-0.5

0

0.5

1

1.5

-0.5

-1

Tri-axial Stress (MPa)

1

800

600

400

-1.5

Transverse Strain: Tension De-Bonded without R.S

200 -2

Perfectly bonded without R.S

Transverse Strain: Tension De-Bonded with R.S

Perfectly bonded with R.S

Axial Strain:Tension De-bonded Without R.S

De-bonded without R.S

-2.5

De-bonded with R.S

Fig. 13. Initial deterioration of the SiC/Ti composite in longitudinal//transverse equi-biaxial loading.

Axial Strain: Tension De-bonded With R.S

0 0

0.2

0.4

0.6

0.8

1

Strain (%) Fig. 15. Stress–strain curves of the SiC/Ti composite in tensile equi-triaxial loading.

stress with reduced size. When the composite is subjected to transverse tension, the interfacial bebonding is the main concern of the composite deterioration. On the other hand, under the transverse compressive loads the matrix yields first either for axial tension or compression. Also, in the transverse compression both bonded and the presented models show the same behavior. Investigation of the thermal residual stress effects on the both axial/transverse and transverse/transverse equi-biaxial behavior of the MMC reveals that the effect of a uniform change in temperature is not equivalent to a hydrostatic stress. The effect of a temperature change cannot therefore be replaced by that of a hydrostatic stress, as suggested by Dvorak et al. [36,37].

Fig. 14 depicts the location of initiation and propagation of interface damage and matrix plasticity under tensile axial/transverse equi-biaxial loading in details. Both of the interfacial damage and matrix plasticity initiate at the point specified by the cross sign and propagate through the dashed line along the interface and within the whole matrix for interfacial debonding and matrix yielding, respectively. As a more general case of transverse loading, interfacial debonding and matrix yielding initiate at the point with maximum stress concentration.

5.3. Triaxial loading

Fig. 14. Initiation and propagation of interfacial debonding and matrix plasticity within the RVE under tensile longitudinal/transverse equi-biaxial loading.

As the final loading case, the composite is subjected to equi-triaxial tensile and compressive (hydrostatic) loading where Sx = Sy = Sz are applied on the RVE simultaneously. Fig. 15 depicts the ESUC predicted stress–strain responses of the MMC subjected to tensile equi-triaxial loading with and without effects of thermal residual stresses and weak interface. Fully bonded and compressive predicted stress–strain curves are not included in the figure mainly due to the fact that composite shows linear behavior up to the fiber fracture point which is about 6000 MPa. It should be noted that effects of weak interface is negligible in the case of hydrostatic loading. Comparison between transverse behavior in the uniaxial, biaxial and triaxial loading shows that the trends of curves are the same but interface debonding begins at higher load and is completed at lower strain with increasing load axis. Fig. 16 describes the location of initiation and propagation manner of interfacial debonding and matrix plasticity under tensile equi-triaxial loading. Both of the interfacial damage and matrix plasticity initiate at the point specified by the cross sign on the diagonal line and propagate through the dashed lines which have circumferential symmetry with respect to the diagonal direction. Again, as it can be seen, interfacial debonding and matrix yielding initiate at the point involves highly stress concentration with considering fiber distribution and loading conditions.

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of compressive transverse loading. However, in the tensile transverse loading, composite fails as a result of interface damage either in the axial tension or compression. References

Fig. 16. Initiation and propagation of interfacial debonding and matrix plasticity within the RVE under tensile equi-triaxial loading.

6. Conclusion A 3-D micromechanics based analytical model is developed to study effects of interface damage on the elastoplastic response of a fibrous SiC/Ti metal matrix composite subjected to general multiaxial loading. The model also includes the effect of manufacturing process thermal residual stress. The single CCI model characterizations are calibrated to model interfacial debonding in various loading conditions. The initiation and propagation of deterioration caused by fiber fracture, interfacial debonding and matrix plasticity are reported depending on the loading conditions. Results revealed that while predictions based on perfectly bonded interface are far from reality, the predicted stress–strain behavior of the SiC/Ti composites in the presence of damaged interface and thermal residual stresses demonstrate very good agreement with available finite element results and experimental data in uniaxial loadings. However, residual stress effects are significant in the uniaxial longitudinal loading and cause asymmetric behavior in tension and compression. Initiation of interfacial damage and matrix yielding in the biaxial stress state was different in tensile/compressive transverse loading. Major deterioration mode was matrix yielding in the case

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