Thermal Residual Stress and Interphase Effects on ... - SAGE Journals

Thermal Residual Stress and Interphase Effects on Crack-Inclusion Interactions BRYAN A. CHEESEMAN AND MICHAEL H. SANTARE* Department of Mechanical Engineering University of Delaware 126 Spencer Laboratory Newark, DE 19716-3140, USA (Received April 20, 2000) (Revised November 28, 2000)

ABSTRACT: An analytical investigation is conducted into the problem of a matrix crack interacting with an inclusion possessing an interphase and subjected to a thermal load. The interphase is modeled as a third, homogeneous phase surrounding the circular inclusion. By using a dislocation based approach to model the crack, along with an analytical solution to the thermal stress problem, a set of singular integral equations are formulated and then solved using established numerical techniques. The solution is applied to a glass fiber–epoxy composite system having either radial or circumferential matrix cracks and undergoing a uniform temperature change. This solution gives a quantitative and analytical method to evaluate the influence of interphase properties on the stress intensity factors of nearby, thermally loaded microcracks. KEY WORDS: composite, fracture, interphase, crack-inclusion, residual stress.

INTRODUCTION stresses has generated an enormous amount of interest because of their role in many different phenomena that are detrimental to the integrity of a composite structure. Thermal stresses, termed process-induced stresses, are generated during most composites manufacturing processes. Macroscopically, these stresses are responsible for dimensional change, part warpage, and delamination. Microscopically, they can cause local yielding, microcracking and fiber breakage, all of which affect a composite’s mechanical performance. The current investigation will focus on processinduced microcracking. By identifying and understanding the factors that influence microcrack development, control or prevention strategies can be developed which may lead to improvements in overall composite performance.

T

HE STUDY OF residual

*Author to whom correspondence should be addressed.

Journal of COMPOSITE MATERIALS, Vol. 36, No. 05/2002 0021-9983/02/05 0553–17 $10.00/0 DOI: 10.1106/002199802023489 ß 2002 Sage Publications

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In general, a small ‘‘interphase’’ region forms at the interface between composite constituents, with properties distinct from those of either the fiber or the matrix. The mechanical performance of the composite material can be profoundly influenced by the properties of this interphase. The interphase properties can be controlled, in part, by altering the processing cycle of the composite. Therefore, the mechanical properties of a composite material can be influenced not just by the choice of reinforcement and matrix, but by the choice of the fiber’s coating or chemical finish and modifications to the process cycles as well. However, in attempting to control a composite material’s properties through the modification of the interphase and process cycle, there exist many issues that need to be addressed. The chemical and thermodynamic reactions which occur at the fiber–matrix interface during the process cycle, the thermal stresses induced during cooling and microcracking in the fiber, interphase or matrix are all issues which need to be quantified. There have been several previous and ongoing investigations to study the chemical–thermodynamic reactions and thermal stresses which occur (see, for instance Palmese (1991) and Sottos (1990), respectively); the current investigation will concern itself with microcracking. While previous investigations (Delale, 1988; Muller and Schmauder, 1992, 1993; Cheeseman and Santare, 2000a) have studied the behavior of cracks interacting with inclusions under thermal and mechanical loadings, the current work examines this phenomenon in the presence of the interphase. In the following, we will apply a general analytical technique to investigate the influence of the interphase on specific composite material systems. We base this analysis on the solution of an arbitrarily curved crack interacting with a single circular elastic inclusion with an interphase subjected to a thermal load. No attempt will be made to include interactions with other inclusions.

REVIEW OF PREVIOUS WORK Although there has been substantial research concerning residual stresses around fiber arrays, the current review will only be limited to those studies that have investigated the influence of the interphase. A more extensive review of the research on thermal stresses in composites with interphases can be found in Jayaraman et al. (1993). The influence of the interphase on the stress distribution in a composite is a subject that has recently received much attention. Initial investigations have treated the interphase as a separate region with constant material properties slightly different from those of the matrix. Using a three phase concentric cylinder model as shown in Figure 1, Nairn (1985) solved for the thermal residual stresses in the case of a thick interphase (10% volume fraction), having the same elastic properties as the matrix, but with a different CTE. Modeling a transversely isotropic AS4 carbon fiber in a Hercules 3501-6 epoxy matrix, Nairn found that increasing the CTE of the interphase relative to the matrix decreased the thermal stresses in the matrix. In his study of aluminide-based composites, Misra (1994) also found that an increased CTE of the interphase reduced the radial and circumferential stresses in the matrix but tended to increase the axial stress in the matrix. Similar results were reported by Arnold et al. (1992) in their study of SiC/Ti3 Al þ Nb metal matrix composites. The three-phase model was extended to a four-phase composite cylinder assemblage – fiber, interphase, matrix, macroscopic composite – by Mikata and Taya (1985). They looked at the residual stresses induced in a graphite–aluminum composite where the fibers

Thermal Residual Stress and Interphase Effects on Crack-Inclusion Interactions

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Figure 1. Three-phase composite cylinder assemblage (CCA) model of Nairn (1985).

had either a nickel or silicon carbide coating. The investigators found that the SiC coating gives rise to greater thermal stresses than the nickel coating. Subsequent studies utilizing a three concentric cylinder model were performed by Pagano and Tandon (1988, 1990) who investigated the effects of two different coatings, nickel and carbon, on the residual stress distribution in a Nicalon/BMAS (barium magnesium aluminosilicate) composite. These authors found that individual stress components could be controlled with a proper choice of coating thickness, elastic moduli and CTE. Arnold and Wilt (1993) concluded that the interphase CTE along with its thickness are the dominant factors when it comes to reducing thermal stresses in the fiber and matrix. More specifically, they state that the interphase CTE should be greater than that of the matrix and that the interphase should be as thick as other considerations allow, in order to reduce the in-plane stresses in the fiber and matrix. Benveniste et al. (1989) used the Mori-Tanaka method to investigate thermal stresses in a coated fiber composite. Results are given for a very thin coating (less than 1% volume fraction). Employing this technique, the authors note that results are not much different than for the case of uncoated fiber composites. Mikata and Taya (1985) also investigated the effect of interphase thickness. Their results correlated well with the aforementioned results of Pagano and Tandon (1988, 1990) – a thicker interphase served to decrease the maximum stress in the interphase. Sottos (1990), used a composite cylinder assemblage and a hexagonal fiber array to investigate the effects of spatially varying interphase properties on the residual stress distribution, and found that when the interphase thickness was increased, the magnitude of the interfacial stresses decreased. Misra (1994) found that stresses induced by an interphase with a CTE lower than that of both the fiber and matrix can be reduced by decreasing the thickness of the interfacial layer and choosing an interphase with a lower Young’s modulus. Additionally, Jayaraman and Reifsnider (1992) found that an increase in interphase thickness either increased the radial stress in the interphase for a T300/Ni/Al6061 composite or decreased the radial stress in the interphase for an E-glass/IMHS epoxy composite. In addition to investigating the interphase CTE and thickness, a number of studies have focused on how the modulus of the interphase influences the residual stress distribution. Nairn (1985) investigated the effect of using different (constant) values of the Young’s modulus and Poisson’s ratio of the interphase and found that within a

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reasonable range of variation from that of the matrix values, these material parameters had no significant effect on the matrix thermal stresses. Vedula et al. (1988), using the three phase composite cylinder assemblage in their study of metal matrix composites, found that the tensile hoop stress at the interface of the matrix and interphase could be reduced with the introduction of a compliant interphase. Arnold and Wilt (1993) report that the interphase stiffness has very little effect on the stress state of the fiber or matrix. However, they advise the use of an interphase with a stiffness which is low relative to that of the matrix. In the studies cited, the material properties were assumed to be constant through the thickness of the interphase; however, it is generally acknowledged that these properties are spatially varying. Initial investigations of spatially varying material properties were performed by Sottos (1990), who used a composite cylinder assemblage and a hexagonal fiber array to investigate the effects of heterogeneous interphase properties on the residual stress distribution. Using idealized, linearly decreasing functions for the interphase elastic modulus resulted in a decrease in the thermal stress magnitude when compared with the no interphase case for both assumed fiber arrays. Additional studies were performed by Jayaraman and Reifsnider (1992) who used a three-phase composite cylinder assemblage, with a radially varying Young’s modulus in the interphase. These investigators assumed the Young’s modulus to have alternately, a power, reciprocal or cubic radial variation. For all cases, the spatial variation of the Young’s modulus was seen to have a pronounced effect on the stresses in all three constituents, but particularly in the interphase. Mikata (1994) applied a four-phase composite cylinder assemblage to the current problem and assumed linearly varying elastic constants and CTE’s in the interphase. He found that for a thin interphase, the stress distributions in the fiber and matrix were barely affected by the variability of the elastic constants and CTE’s of the interphase. The stress distribution in the interphase, however, was radically changed due to the variability of the thermoelastic constants. Hiemstra and Sottos (1993) experimentally studied the effect of different interphases and inter-fiber spacing on the thermal microcracking of a square array of silicon carbide fibers in a Shell EPON 828 epoxy matrix. The researchers coated the fibers with different epoxies to create interphases with elastic moduli either higher or lower than that of the EPON 828 epoxy. Applying a change in temperature, Hiemstra and Sottos found that with the higher modulus interphase, microcracks initiated at the fiber–interphase interface, whereas, with the lower modulus interphase, the microcracks developed at the interphase– matrix interface. Additionally, the lower modulus interphase prevented the microcracks from reaching the fiber.

ANALYTICAL SOLUTION Crack Interacting with an Inclusion with an Interphase Under a Thermal Load Previous investigations, cited above, have shown that manipulating the interphase CTE, thickness and modulus affect the residual stresses. The current study will focus on how the properties of the interphase influence a preexisting matrix microcrack in a glass fiber–epoxy composite subjected to a thermal loading. Following the analyses of Cheeseman and Santare (2000a,b) the principle of superposition can be employed to


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represent the problem as the sum of two separate problems: 1. An uncracked, infinite medium containing a circular elastic inclusion surrounded by an interphase subjected to a uniform change in temperature. 2. A cracked, infinite medium containing a circular elastic inclusion with an annular interphase, subjected to a loading on the crack faces which exactly cancels the tractions from the uncracked thermal stress problem. As before, the crack will be represented as some, yet unknown, distribution of dislocations. The complex potentials for the crack-inclusion-interphase problem were developed in Cheeseman and Santare (2000b) and will be used here. However, although the results of the thermal stress problem have already been discussed, a brief review of the applicable portion of Nairn’s (1985) solution will be presented next.

Thermal Stress Solution Nairn’s (1985) thermal stress solution for a three-phase composite cylinder assemblage with a transversely isotropic fiber surrounded by an isotropic interphase and matrix is used in the current study. Since the current investigation is concerned with matrix cracks, only the stress solution for the matrix will be given. When the concentric cylinder assemblage shown in Figure 1 undergoes a change in temperature, the stresses in the matrix can be written as (Nairn, 1985)

rm ¼ A1 þ

A2 r2

ð1aÞ

m ¼ A1

A2 r2

ð1bÞ

zm ¼ A3

ð1cÞ

where the subscripts r, and z designate the radial, circumferential (hoop) and axial direction, respectively, and the subscript m designates the matrix. The coefficients A1, A2 and A3 are determined from the following relations A2 ¼ b2 A1

ð2Þ

9 8 9 8 ði L ÞT > A1 > > > > > = = > < < > ðm i ÞT A3 ¼ ½K ð T ÞT > A > > > > > ; ; > : i : 5> ðm i ÞT A6

ð3Þ

where b is the radius of the matrix, is the coefficient of thermal expansion, T is the change in temperature, the subscript i denotes the interphase and the subscripts L and T represents the longitudinal and transverse directions, respectively. The elements of [K ] are

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given by K11

2Vm a i ¼ 1 Vm EL Ei

K12 ¼

Vm EL Vf

K13

K14

V1 1 ¼ þ EL Vf Ef

K21

ð4bÞ

a Vi i ¼ 2 þ EL Vf Ei

ð4aÞ

ð4cÞ

m Vm i ¼2 þ Em 1 Vm Ei

ð4dÞ ð4eÞ

K22 ¼

1 Em

ð4fÞ

K23 ¼

2i Ei

ð4gÞ

K24 ¼

1 Ei

ð4hÞ

K31 ¼

Vm 1 i 1 T 1 Vm Ei ET

ð4iÞ

K32 ¼

a Vm EL Vf

ð4jÞ

K33 ¼

1 i 1 þ i 1 Vm 1 T Vi þ þ Ei Ei Vf ET Vf

ð4kÞ

K34 ¼

i a Vi þ Ei EL Vf

ð4lÞ

K41 ¼

1 i Vm 1 m 1 þ m 1 þ þ Ei 1 Vm Em Em 1 Vm

ð4mÞ

K42 ¼

m Em

ð4nÞ

K43 ¼

2 Ei

ð4oÞ

K44 ¼

i Ei

ð4pÞ


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where V is the volume fraction of the region, is the Poisson’s ratio and E is the Young’s modulus. The subscript a represents the axial direction and the subscript f represents the fiber. Thus, knowing a change in temperature and the material properties for the fiber, interphase and matrix, the coefficients A1 and A3 can be determined by solving the system of equations given by Equations (3) and (4). Coefficient A2 can then be determined from Equation (2). Once the coefficients are known, the stresses in the matrix can be determined from Equations (1a)–(1c). For a detailed description of the complete stress solution, the authors refer to Nairn (1985).

Crack Interaction Formulation It is well known that when the crack tip resides in a single material region (i.e., not at the interface), the stress has a square root singularity at the tip. This is the case with all the solutions presented herein. Under these conditions, the stress analysis leads to a set of Cauchy singular integral equations. The formulation of these equations for the corresponding crack problem follows the analysis of Cheeseman and Santare (2000b). Recall that the crack interacting with an inclusion is the superposition of the problem of a cracked medium whose crack faces are loaded by tractions which cancel those which occur along the faces in the uncracked problem. This condition along the crack faces can be expressed as ijT þ ijD ¼ 0

ð5Þ

where ijT are the tractions due to the thermal load and ijD are the tractions due to the dislocation distribution. In a general two-dimensional, isotropic elasticity problem, the stresses and displacements can be expressed in terms of two analytic functions of the complex variable z, (z) and (z) (Muskhelishvili, 1953) xx þ yy ¼ 2f0 ðzÞ þ 0 ðzÞg yy xx þ 2ixy ¼ 2fz00 ðzÞ þ

ð6Þ 0

ðzÞg

2ðu þ ivÞ ¼ ðzÞ z0 ðzÞ ðzÞ

ð7Þ ð8Þ

where the prime denotes the derivative, the overbar denotes the conjugate, is the shear modulus and is related to the Poisson’s ratio by ¼ (3 )/(1 þ ) for plane stress and ¼ 3 4 for plane strain. The normal and tangential tractions mn and nt along an arbitrary contour can be expressed, in terms of the complex potentials. 0 ðzÞ þ 0 ðzÞ e2i fz 00 ðzÞ þ

0

ðzÞg ¼ mn int

ð9Þ

The tractions ijD along the crack path shown in Figure 2 and denoted as L, can be computed from the complex stress potentials given by equations 1 ðzÞ ¼

1 X k¼1

A k zk þ

1 X k¼1

hk zk

ð10Þ

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Figure 2. Crack interacting with an inclusion surrounded by an interphase.

1 ðzÞ ¼

1 X

Bk zk þ

1 X

k¼1

Pk zk

ð11Þ

k¼1

where the Laurent series coefficients hk and pk are determined numerically by the procedure detailed in Cheeseman and Santare (2000b) and the coefficients Ak and Bk are given by kzko

ð12Þ

zo 1 k zko zo

ð13Þ

Ak ¼ Bk ¼

where zo is the position of the dislocation and the dislocation density function is given as ¼

m ðbn ibt Þ ð þ 1Þ

ð14Þ

where is the shear modulus of the matrix and bn ibt is the Burgers vector. Substituting Equations (10) and (11) into Equation (9) and integrating along the crack, L, gives the Cauchy singular integral equation of the form Z

Z zf Z zf Z zf dz þ Kðzo , zÞ dz þ dz þ Kðzo , zÞ dz zi ðz zo Þ zi zi ðz zo Þ zi

ðm þ 1Þ T T nn int ¼ , z2L m zf

ð15Þ

where the right hand side of the equation is the thermal traction computed with the previously detailed thermal stress solution. The normal and tangential components are


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determined using standard transformation methods. The authors note that the tractions due to a change in temperature within the three phase composite cylinder assemblage are determined by letting the outer radius in Nairn’s (1985) solution go to infinity (or, for numerical purposes, a very large number). This singular integral equation is solved numerically for the unknown dislocation density using the Lobatto Chebyshev quadrature technique as detailed in Cheeseman and Santare (2000b). Once the dislocation density is determined, the Mode I and Mode II stress intensity factors can be calculated using k1 ðzi Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m lim 2 jz zi j bn ðzÞ ðm þ 1Þ z!zi

ð16Þ

k2 ðzi Þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m lim 2 jz zi j bt ðzÞ ðm þ 1Þ z!zi

ð17Þ

EXAMPLE MATERIAL SYSTEM Thermal Analysis As an example material system of some technological importance, we investigate a glass fiber–IMHS epoxy composite with an interphase with varying coefficients of thermal expansion, thickness and moduli. The uncracked thermal stress profiles for this composite system will be calculated first to ensure that the normal thermal tractions will be tensile when applied to the radial and circumferential matrix cracks. Using the material properties given in Table 1, we calculated the thermal stresses in the matrix as a function of the normalized radial distance for various interphase parameters. (The radial distance is normalized with respect to the fiber radius.) We took T ¼ 1 and the solution for the three phase composite cylinder assemblage with fiber radius equal to one and the outer radius taken to be sufficiently large (in Figure 1, b ¼ 10). Radial and circumferential matrix stresses were calculated for a wide range of interphase moduli, thicknesses and coefficients of thermal expansion. This parametric study produced an enormous quantity of data, not all of which can be shown here. However, several of the general trends can be seen in Figures 3 and 4, for the case where the ratio of the interphase moduli to the matrix moduli is 0.010. Figure 3 shows the radial stress in the matrix as a function of the normalized distance from the fiber for an interphase thickness of 0.10 and various values for the

Table 1. Material properties of glass fiber–epoxy system as reported by Jayaraman, Gao and Reifsnider (1994). Property

E-Glass

IMHS Epoxy

E (GPa) G (GPa)

76.00 31.15

3.50 1.30

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Figure 3. Matrix radial stress versus normalized radial distance for various interphase CTE’s where the Ei/Em ¼ 0.01 and RINTERPHASE/RINCLUSION ¼ 1.10.

Figure 4. Matrix circumferential stress versus normalized radial distance for various interphase CTE’s where the Ei/Em ¼ 0.01 and RINTERPHASE/RINCLUSION ¼ 1.10.

interphase CTE. Figure 4 shows the corresponding circumferential stresses in the matrix as a function of the normalized distance from the fiber for the aforementioned case. From Figure 3 it can be observed that for the modulus ratio shown, the interphases with CTE’s of i ¼ 0.10m and i ¼ 1.0m result in negative radial stresses. These negative stresses would result in the closing of any preexisting circumferential matrix cracks and a subsequent reduction of the stress intensity factor (SIF) to zero. Therefore, the corresponding crack cases will not be investigated here. However, for an interphase with a CTE of i ¼ 10m, the resulting radial stresses are tensile and would cause a circumferential matrix crack to open. Consequently, these circumferentially cracked cases will be examined.


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Figure 5. Matrix radial stress versus normalized radial distance for various interphase thicknesses where the Ei/Em ¼ 100 and i/m ¼ 10.

The circumferential stresses are positive in Figure 4 for interphase CTE’s of i ¼ 0.10m and i ¼ 1.0m. Since these stresses would tend to open radial matrix cracks, the cracked cases of i ¼ 0.10m and i ¼ 1.0m will be studied. However, since these stresses are negative for an interphase with a CTE i ¼ 10m, they would cause a radial crack to close and therefore, these corresponding cases will not be investigated. Figure 3 shows that tensile radial stresses resulted when i ¼ 10m and from further parametric studies (omitted for conciseness) we have seen these stresses increase in magnitude with increasing modulus ratio, Ei/Em. Using the particular case where Ei/Em ¼ 100, the interphase thickness, t, was varied from t ¼ 0.01 through t ¼ 0.20. These thermal stress results are shown in Figure 5. It should be noted that for the thinnest interphase, t ¼ 0.01, the radial stress becomes negative and therefore, the corresponding circumferential crack case will be omitted. Similarly, as shown in Figure 4, the hoop stresses are positive when i ¼ 0.10m and i ¼ 1.0m. Again, further studies show that these stresses increase in magnitude with increasing modulus ratio, Ei/Em. Therefore, the interphase thickness was varied for the case where i ¼ 0.10m and Ei/Em ¼ 100 and a plot of the circumferential thermal stresses as a function of the normalized radial distance is given in Figure 6.

Crack Analysis Once the tensile thermal stress distributions are known, the problem of a radial or circumferential crack interacting with a circular inclusion surrounded by an interphase can be studied. Recalling that the circumferential stresses are positive for interphase CTE’s of i ¼ 0.10m and i ¼ 1.0m and using the material properties for a glass fiber–epoxy system given in Table 1, consider a radial crack located at b/a ¼ 3.5 interacting with a glass fiber which is surrounded by an interphase as shown in Figure 7. Taking the interphase

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Figure 6. Matrix circumferential stress versus normalized radial distance for various interphase thicknesses where the Ei/Em ¼ 100 and i/m ¼ 0.10.

Poisson’s ratio to be the same as the matrix, ¼ 0.35, and taking RINTERPHASE/ RINCLUSION ¼ 1.10, the modulus of the interphase is varied from 0.01 to 100 times that of the matrix and the system is subjected to a T ¼ 1. The Mode I SIFs are then computed at the crack endpoint closest to the inclusion, k12, and normalized by the case of an inclusion with no interphase. These dimensionless SIF’s are plotted as a function of the shear moduli ratio for two CTE ratios are given in Figure 8. From Figure 8, it can be seen that both a stiff and a compliant interphase reduce the stress intensity factors when the interphase has i ¼ 1.0m with respect to the no-interphase case. Therefore, it can be concluded that for this particular composite system, if the interphase CTE is the same as that of the matrix, the interphase shields a thermally loaded crack from the inclusion, regardless of the interphase stiffness. However, when i ¼ 0.10m, the Mode I SIF is reduced when the interphase is very compliant, i.e., when 12 is approximately less than 0.1. The Mode I SIF is increased for slightly compliant and stiff interphases as compared to the uncoated inclusion case. Since i ¼ 10m resulted in tensile radial stresses, using this CTE along with the same mechanical properties as the previous analysis, the case of a circumferential crack interacting with an inclusion surrounded by an interphase subjected to a T ¼ 1 was investigated for various interphase stiffnesses. Shown in Figure 9, for the current analysis, RCRACK ¼ 1.25, ¼ 45 , RINCLUSION ¼ 1.00 and RINTERPHASE ¼ 1.10. As before, the Mode I and II SIFs were then computed at a crack endpoint. However, in this case, they were normalized by the SIF’s calculated when 12 ¼ 1.0 and i ¼ 10m since the no-interphase case results in compressive radial stresses which would cause the crack to close. These dimensionless SIF’s are plotted as a function of the shear moduli ratio and given in Figure 10. Since the no-interphase case results in compressive radial stresses, the presence of the interphase with a high CTE with respect to the matrix (i.e., i ¼ 10m) would be needed to promote circumferential cracking in the current composite system. When the interphase modulus is less than that of the matrix, the influence of the interphase is decreased as


565

Figure 7. Radial crack interacting with an inclusion surrounded by an interphase subjected to a T ¼ 1.

Figure 8. Dimensionless Mode I stress intensity factor versus interphase thickness of 0.10.

12

for radial crack at (1.25, 0), (2.25, 0) with

compared to the case when the interphase modulus is of the order of the matrix. The opposite can be said for stiff interphases. The stiff interphase causes a sharp increase in both normal and shear SIF’s with respect to the case where the interphase and matrix moduli are the same. Therefore, it can be said that a stiff interphase with a large CTE (with respect to the matrix) would encourage circumferential matrix cracks in the current composite system. As an additional study, to account for the interphase thickness, the case of radial crack was again investigated using the same material system as above. However, in this study,

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Figure 9. Circumferential crack interacting with an inclusion surrounded by an interphase subjected to a T ¼ 1.

Figure 10. Normalized stress intensity factors versus 12 for a circumferential crack with a radius ¼ 1.25 interaction with an inclusion RINCLUSION ¼ 1.10 and i ¼ 10 m.

the distance from the interphase to the close crack tip was kept constant while the thickness of the interphase was increased. The Mode I SIF’s were calculated, normalized by the no-interphase case, and plotted in Figure 11 as a function of the normalized interphase radius, RINTERPHASE/RINCLUSION. From this figure, it can be seen for this


567

composite system, where the interphase is stiff and has a low CTE as compared to the matrix, increasing the thickness of the interphase while keeping the distance to the crack constant results in increased Mode I SIF’s at both crack endpoints. Similar studies were conducted for a circumferential crack with ¼ 45 , i ¼ 10m and Ei ¼ 100Em. Using the same material system, the problem was recomputed while keeping the circumferential crack at a constant distance, 0.10, from the interphase while the interphase thickness was increased. As before, the Mode I and II stress intensity factors are calculated. However, in this case they are normalized by 0, where 0 ¼ jEmmT j since the no-interphase case results in compressive radial stresses which would cause the crack to close. The normalized SIF’s are plotted as a function of the normalized interphase radius in Figure 12. Recalling Figure 5, it can be seen that for current

Figure 11. Normalized Mode I stress intensity factor versus normalized interphase radius for Ei/Em ¼ 100 and i/m ¼ 0.10 for a radial crack located at a constant distance from the interphase.

Figure 12. Stress intensity factor versus normalized interphase radius for Ei/Em ¼ 100 and i/m ¼ 0.10 for a circumferential crack located at a constant distance from the interphase.

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composite system, a thin interphase results in compressive radial stresses. Therefore, it is seen that a thin, stiff interphase with a high CTE as compared to the matrix causes the circumferential cracks to close and increasing the interphase thickness while keeping the crack at a constant distance from the interphase causes the SIF’s to increase in magnitude.

CONCLUSION In the current paper, we have analyzed the problem of a matrix crack interacting with an inclusion surrounded by an interphase subjected to a thermal load. Using the dislocation solution derived in Cheeseman and Santare (2000b) along with Nairn’s (1985) solution of the thermal stress in a three-phase composite cylinder assemblage, we solved the resulting singular integral equations using the techniques developed and employed in previous investigations (Cheeseman and Santare, 2000a,b). As an example material system, a glass fiber–epoxy composite was investigated. Prior to solving the thermally loaded crack problems, the thermal stresses in the three-phase composite cylinder assemblage were calculated to ensure that the applied normal stresses were tensile and would promote crack opening. From these results, it was seen that certain combinations of interphase CTE, moduli and thickness produced compressive stresses. These interphases would cause crack closure and would effectively prevent matrix microcracking. However, other interphase material property combinations did produce tensile stresses and the corresponding problems of a radial or circumferential crack were studied. Compliant interphases were seen to shield the inclusion from the crack for the radially cracked case while stiff interphases could either enhance or decrease the Mode I SIF relative to the no-interphase case depending on the interphase CTE. For the circumferentially cracked cases, the presence of the interphase promoted cracking since with no interphase, the resulting stress distribution would be compressive. For this case, a stiff interphase was seen to be more detrimental than a compliant one. Increases in the interphase thickness tended to increase the SIF’s for both the radial and circumferentially cracked cases.

ACKNOWLEDGMENT This work was conducted under a grant from Army Research Office University Research Initiative (ARO/URI) Multidisciplinary Program in Manufacturing Science of Polymeric Composites at the University of Delaware Center for Composite Materials.

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