Modeling of crack density in ceramic matrix composites

Abstract

The feasibility of utilizing the shear lag theory to estimate crack density in fabric reinforced composites was investigated. A geometric model was constructed for the fabric and meshed using a hybrid finite element approach. The small segment of the yarn and the surrounding matrix enclosed within each element were treated as a unidirectional composite and the shear lag theory was used to estimate the crack density. Model results were compared to experimental data for a 5-harness satin melt-infiltrated SiC/SiC composite under tension and showed a pattern similar to experimental data with the model starting to accumulate cracks at a stress corresponding to the point of departure from linearity in the stress–strain curve while cracks were experimentally observed around 60 MPa higher. The model and experimental data had a similar value for the crack density at the saturation level. Sensitivity analysis showed that the crack density was highly sensitive to the fiber volume fraction in the load direction followed by the weave angle of the crimped segments of the yarns and the interfacial shear strength between the fibers and the matrix.

Keywords

Ceramic matrix composites matrix cracking modeling

Introduction

Ceramic matrix composites (CMCs), when compared to monolithic ceramics, exhibit excellent stiffness, strength, and fracture toughness and, therefore, are currently being considered for the manufacture of hot segment components in gas turbine engines. The environment inside these components is harsh and the composite is typically subjected to complex thermo-mechanical loading that can lead to cracking. These cracks form paths for the ingress of the environment oxidizing the fibers and leading to premature failure.

It is important to develop an understanding of the mechanisms responsible for crack formation as the first step towards understanding the oxidation behavior. One of the models used to estimate the crack density in unidirectional composites is the shear lag theory. Aveston et al.¹ were the first to utilize this approach to model matrix cracking in brittle unidirectional ceramic composites using an energy balance approach. The basic assumptions used in the model can be listed as follows: (1) fibers are intact during loading, (2) matrix cracks are formed in planes perpendicular to the load direction and are equally spaced, (3) Poisson’s effects, as well as fiber roughness effects, are neglected, and (4) the value of the interfacial shear stress is constant.²

Pryce and Smith³ developed stress–strain relationships for CMCs with multiple cracks using the shear lag theory. They assumed that in the plane of a matrix crack the fibers carry the entire load, but away from the crack the load is partially shed back to the matrix with a value proportional to its stiffness divided by the stiffness of the whole composite. As the load increases and the distance between cracks decreases, the length of the distance, necessary to shed the load to the matrix, becomes smaller than half the spacing between the cracks limiting the load shedding process to the matrix. This scenario defines three stages of loading: (i) matrix is not cracked, (ii) spacing between matrix cracks is more than twice the distance needed to transfer the load from the fibers to the matrix and (3) spacing between matrix cracks is less than twice the distance needed to transfer the stress from the fibers to the matrix. Pryce and Smith derived explicit equations for unidirectional composites relating stress to strain utilizing material properties, volume of fibers and matrix, fiber diameter, and interfacial shear stress for the last two scenarios. These equations included the effect of thermal residual stresses that are typically present in CMCs. Experimental data for SiC/calcium alumino-silicate composites under quasi-static fatigue were investigated and compared to analytical models to evaluate the effect of the reduction in effective interfacial shear stress on the shape of the hysteresis loops.

Similar equations were derived by Keith and Kedward² to study the stress–strain behavior of porous unidirectional Nicalon®/aluminum phosphate composites. Weibull statistical approach was used to evaluate the strength of the yarns. They reported that the model matched the experimental data reasonably well except for the value of the width of the hysteresis loop, concluding that the nonlinear response of CMCs is predominantly the result of sliding at the fiber/matrix interface.

Sorensen and Holmes⁴ studied thermo-mechanical fatigue response of unidirectional CMCs utilizing a similar approach. They argued that under cyclic load, the fibers will either partially slip or fully slip along its interface with the matrix, and characterized the stress–strain response in fatigue loading as caused by a partial slip between the fibers and the matrix in all cycles, a mixed partial/full slip in the first cycle followed by a partial slip in the following cycles, or a mixed slip in all cycles. To distinguish between these stages, they utilized the equations derived in Pryce and Smith³ to evaluate the modulus of the cracked composites, the composite strain, and the spacing between cracks.

The effect of matrix cracks on damping of laminated CMCs with fibers oriented at different angles was studied using the shear-lag theory.⁵ The authors associated the increase in damping to the matrix crack density in the direction perpendicular to the load direction as well as the dissipation of frictional energy along damaged fiber/matrix interfaces. They also considered the effect of cracks along the length of longitudinal yarn and reported that it had a limited effect on damping.⁶

The agreement between experimental data and model calculations reported by various researchers, some of which are mentioned above, gave credence to the use of the shear-lag theory in evaluating the change of crack density in unidirectional CMCs. In this work, we investigate the efficacy of using the shear-lag theory to model CMCs reinforced with woven fabrics and compare its results to experimental data from archived literature. The sensitivity of crack density to the residual thermal stress in the fibers, the interfacial shear strength between the fibers and the matrix, the relative fiber volume fraction in the load direction, the radius of the fibers and the weave angle of the crimped yarns is assessed.

Numerical model

Shear lag equations were embedded into a hybrid finite element formulation to estimate the density of cracks in CMCs according to the following steps: (1) model the fabric geometry within a repeat unit cell of the composites, (2) mesh the unit cell using hexahedra brick elements, (3) evaluate the average stiffness matrix for each element, (4) calculate the stress in each element, (5) evaluate the effect of defects on the stiffness matrix and stress distribution, and (6) implement the shear lag theory equations within each element. Steps 1 to 3 above were discussed in previous articles (see, e.g. Gowayed et al.⁷). Ensuing sections give details to the steps used to construct the model.

Geometric modeling of fabric preform and calculation of stiffness matrix

The behavior of CMCs reinforced with fabrics is highly dependent on the fabric architecture. It is necessary to identify the fabric geometry before any attempt is made to model the composite behavior. In the current model, an ideal geometric representation of the fabric is constructed by calculating the location of a set of spatial points (knots) that can identify the yarn center-line path within the preform space based on the fabric forming process in a typical textile machine.⁸ A B-spline function is used to approximate a smooth yarn center-line path relative to the knots. A yarn cross-section shape is swept along the center-line path and allowed to change dimensions based on availability of space between yarns and the thickness. A repeat unit cell of the modeled fabric preform is identified from the geometric modeling and used to represent a complete yarn or tow pattern.

A hybrid finite element analysis (FEA) was developed and used to discretize the unit cell into hexahedra brick elements with fiber and matrix around each integration point. Material homogenization is carried out to define the anisotropic material response.⁹ The boundary conditions of the unit cell, dictated by the assumptions of repeatability and continuity, are used to reduce the size of the stiffness matrix. A virtual work technique is used to calculate the stiffness matrix of the unit cell.⁷

Stress in elements

Assumptions of repeatability of the unit cell and the condition of strain continuity in the three directions of a Cartesian coordinate system dictate that mirror nodes on opposite faces of the unit cell of a single layer have similar displacements. For example, for each node on the top and bottom surfaces

u (x, y, 0) = u (x, y, c) v (x, y, 0) = v (x, y, c) w (x, y, 0) = w (x, y, c)

(1)

Similar boundary conditions were used for image points on the sides of the unit cell substituting the variable c with a or b. Here, a, b, and c are the dimensions of the unit cell and u, v, and w are displacements in the x, y, and z directions, respectively. Additionally, the assumptions of repeatability of the unit cell and the equilibrium of forces in the three directions dictate that the forces at mirror nodes on opposite faces of the unit cell are equal in value and opposite in direction.

From the knowledge of the stiffness matrix of each element and the unit cell,⁷ the stress in each element {σ_ele} can be calculated as follows

{σ ele} = [T σ] [C ele] [S unit] {σ ext}

(2)

where [T_σ] is the transformation matrix for stresses between the direction of the external load and the direction of the yarn piece located inside the element, [C_ele] is the average value of the stiffness matrices at the integration points in the element, [S_unit] is the compliance matrix of the unit cell calculated using the virtual work technique, and {σ_ext} is the external stress applied on the composite.

Stress mapping using this approach was verified for a unit cell of a plain weave melt-infiltrated (MI) SiC/SiC composite using ANSYS®. Data from the geometric model mentioned above were used as input for the ANSYS model and yarns were meshed using SOLID45 elements and a sweep mesh with the material properties changing along the yarn path to accommodate the change of angle of fibers with load direction. The total number of elements used in the ANSYS® model was around 16,000 elements. The stress distribution and values from both numerical models were very close.

Effect of defects on stiffness and stress distribution

It is logical to expect that the stiffness matrix of the elements [C_ele], used in the previous section, to be affected by defects that typically exist in ceramic matrix composites. These defects may include: (1) intra-yarn defects such as voids inside the yarns that can lead to fibers not being coated with matrix (dry fibers), and (2) inter-yarn defects such as voids at yarn crossover points, matrix voids away from yarns, shrinkage cracks and interlaminar separation which are elongated voids between layers of fabrics. Figure 1 shows a schematic representation of possible locations and shape of defects. The effect of these defects was included in the evaluation of [C_ele] in equation (2) as follows

[C ele] = [C_u] [D]

(3)

where [C_ele_{_} _u ] is the stiffness matrix of the defect-free composite element as reported by the model in Gowayed et al.⁷ and [D] is a stiffness-depreciation matrix to account for the effect of defects.

Micrographic images, or nondestructive evaluation, can be used to construct a geometric representation of these defects and the matrix [D]. Area projections can be represented in a vector form for each defect ( $A vi$ , where i = x, y, and z are the directions in the global composite axis system). The effect of defects is included by reducing the volume of the matrix in the calculation of the stiffness and representing the geometric effect of area fractions of defects as follows

Further details on this model and micrographic images of defects in MI SiC/SiC composites can be found in Gowayed et al.¹⁰

Crack density

The shear lag theory is formulated to estimate crack density in unidirectional composites. The cracks are assumed to only exist in the direction transverse to the load direction. The effect of cracks that are parallel to the load direction are ignored in accordance with observations of their limited effect.⁶ For each element, half the distance between cracks (s) was calculated using the following equations obtained using the shear lag theory³

m e = \frac{r}{2 τ} [σ k \frac{V m E m}{V f E c} - σ_{f}^{T}]

(5)

For partial fiber slippage (m_e ≤ s)

s = \frac{r}{4 C τ E f σ k} [σ k \frac{V m E m}{V f E c} - σ_{f}^{T}] 2

(6)

For full fiber slippage (m_e>s)

s = \frac{r}{τ} [\frac{σ k}{V f} - \frac{σ i E f}{E k} - σ_{f}^{T}]

(7)

where m_e is the embedded fiber length needed to fully transfer the stress between the fiber and the matrix, s is half the distance between the cracks, τ is the interfacial shear strength between the fiber and the matrix,

σ k

is the element stress in the yarn direction, r is the fiber radius, E_c is the initial modulus of the un-damaged element in the yarn direction, E_k is the slope of the stress–strain curve at different stress levels (i.e. tangent modulus),

σ_{f}^{T}

is the residual thermal stress in the fibers,

C = \frac{1}{E k} - \frac{1}{E c}

and (C > 0), E_f and E_m are the moduli of the fibers and

[D] = (1 - A vx \frac{1}{1 - A vy} \frac{1}{1 - A vz} 000 \frac{1}{1 - A vx} 1 - A vy \frac{1}{1 - A vz} 000 \frac{1}{1 - A vx} \frac{1}{1 - A vy} 1 - A vz 000000 1 - \frac{A vy + A vz}{2} 000000 1 - \frac{A vx + A vz}{2} 000000 1 - \frac{A vx + A vy}{2})

(4)

matrix, and

V f

and

V m

are the volume fractions of fibers and matrix, respectively.

Model equations outlined above were written in C++ and linked to the code developed earlier⁷ to calculate the stiffness matrix of the composite. Numerical convergence studies determined that numerical results reach a plateau at around 10 elements per yarn and this value was used in all subsequent runs.

Stress distribution in 5-harness satin MI SiC/SiC composite

The composite investigated in this study is a MI SiC/SiC composite made by weaving 800-fiber Sylramic® tows into 5-harness satin (5HS) balanced weaves with 20 ends per inch. Manufacturing details of this composite, micrographs of its internal structure, and its stress–strain response and mechanical properties at room temperature and 1204℃ were reported in Gowayed et al.¹¹ Geometric representation of the weave and the FEA mesh is shown in Figure 2 and the properties of the coated fibers and the matrix, as used in the model, are listed in Table 1.

Table 1.

Properties of coated fibers and matrix.¹¹

Tensile modulus of coated fiber (longitudinal) = 332.37 GPa

Tensile modulus of coated fiber (transverse) = 130.18 GPa

Shear modulus of coated fiber = 60.05 GPa

Poisson’s ratio of coated fiber = 0.178

Tensile modulus of matrix = 329.2 GPa

Shear modulus of matrix = 139.1 GPa

Poisson’s ratio of matrix = 0.182

Figure 3 shows the internal stress distribution for a defect-free composite under a 30 MPa applied stress in the warp direction (x-direction), which is in the elastic region of the stress–strain curve. The FEA model used in the study contained 2500 elements (10 elements per yarn in each direction as determined by the convergence study). The variation in stress along the centerline of warp yarns followed the yarn crimp pattern with higher stresses at the crossover segments of the yarn which can be attributed to the locally high fiber volume fraction as compared to the region in-between the crossover points. The stresses between warp yarns were almost constant and higher than those along the center of the warp yarns. This can be due to the high stiffness of the MI–SiC matrix as compared to the modulus of the elements that include the low contribution of the transverse modulus of the coated fibers (see Table 1). The maximum stress value of 37.5 MPa was reported between warp yarns; which is 25% higher than the applied stress.

The effect of defects was evaluated from micrographic images of the composite and listed in Table 2. Figure 3 shows the internal stress distribution in the warp direction for the composite with defects. The stress profile changed at areas of defects, such as the crossover defect and interlaminar separation shown in the figure. The crossover defective area carried around 30% less stress than the nondefective area while the interlaminar separation had a minor impact on the stress in the direction of the warp yarns. The internal stress distribution between warp yarns exhibited peaks in-between the crossover points. This is probably due to the lack of localized defects in the matrix in these regions. The maximum internal stress value increased from 37.5 MPa in nondefective composite to 41.25 MPa (a 10% increase) when defective.

Table 2.

Area fraction $A vi$ (i = x, y, and z) and volume fraction of defects (V_d) for warp and weft directions (in percent).

	Intra-yarn	Inter-yarn
	Dry fiber	Crossover points	Matrix voids	Shrinkage cracks	Interlaminar separation
$A vx$	5.04	3.15	–	–	1.58
$A vy$	4.0	1.24	–	–	1.58
$A vz$	5.04	3.12	–	–	40.0
V_d	5.39	0.40	1.13	–	1.57

The effect of loading the unit cell with 30 MPa in the out-of-plane direction (z-direction) is shown in Figure 4 for cross sections in the nondefective and defective MI SiC/SiC composites. For nondefective composites, the maximum internal stress value is, as should be expected, in-between the crossover points because the modulus in the direction transverse to the fibers is much lower than that of the pure matrix. The maximum stress in the area between the warp yarns of 47.6 MPa is much higher than that for the center of the warp yarns of 30.3 MPa.

When the effect of defects is included according to Table 2, the interlaminar separation and other defective areas, as expected, were not able to carry the internal stress and shed their load to the neighboring areas increasing the maximum internal stress to 78.6 MPa (over 2.6 times higher than applied stress). This situation can have a strong impact on the interlaminar tensile stress experimental values. While the applied stress maybe low, the actual stress sustained by the nondefective segments of the composite can be much higher. This highlights the importance of reducing the interlaminar defects to enhance the interlaminar tensile strength in the composite, and accordingly, the interlaminar shear strength. An interesting implication of this situation can also be seen in the apparent low out-of-plane modulus of this type of composite as highlighted by the work in Gowayed et al.¹¹

Crack density in 5 HS MI SiC/SiC composite

Data reported on crack density (1/2 s) of MI SiC/SiC composites in archived literature¹² were compared to model calculations. The stress–strain diagram for tensile load applied in the warp direction, and shown in Figure 5, was used to find the initial modulus E_cand the percent change in the value of the initial modulus with the increase in stress $(\frac{E c - E k}{E c})$ .

The stress distribution for each element was evaluated, as mentioned above, and the average internal stress value was used to calculate the crack density in each element. The radius of the coated fiber was taken as 5.5 µm (fiber radius = 5 µm and coat thickness = 0.5 µm), the interfacial shear strength considered was 70 MPa,¹³ and the residual thermal stresses was estimated as 64.1 MPa in compression as calculated in the next section. Figure 6 shows the experimental data and model results for crack density in the direction perpendicular to the load direction.

As can be seen from Figure 6, model results showed trends similar to that seen in experimental data. Model results were in conformance with the changes in the stress–strain curve where cracks start appearing at the point of departure from linearity below the knee of the stress–strain curve and continued to increase until it reached a plateau at the crack saturation point at the end of the knee. The estimated value of crack density at the plateau was close to the experimental value of around 10 cracks/mm.¹² The major difference between the model and the experiment is that the model estimated cracks to start accumulating at a stress value around 60 MPa lower than the experimental value. It is interesting to note that the stress–strain of the sample started to deviate from linearity at a stress of around 110 MPa, while cracks were not observed experimentally until 60 MPa higher than this value. The model interpreted the departure from linearity as a point for the initiation of cracks and it is reasonable to assume that the cracks will start at this point. The difference can be due to the use of a stress–strain curve of a sample different than that used for crack detection or due to the inability to visually detect small cracks at low stress levels.

The effect of changing composite parameters used in equations (5) to (7) on the crack density of the MI SiC/SiC composite is presented in the next sections. The composite was loaded with a tensile stress value of 150 MPa in the warp direction which is a point in the knee of the stress–strain curve. All the results are compiled in Figure 7 to not only show the effect of each of these parameters on crack density but also their relative impact.

Effect of residual thermal stresses in fibers

Residual thermal stresses are generated during cooling of the composite from the processing temperature to room temperature due to the difference in the coefficients of thermal expansion (CTE) of the fibers, coat and matrix which is important for CMCs with processing temperatures above 1000℃. If the average residual thermal stress in the composite is known, the longitudinal residual thermal stress in the coated fibers ( $σ_{f}^{T}$ ) can be calculated, assuming isotropic thermal expansion and that the coated fibers and matrix have similar Poisson’s ratio, as follows

σ_{f}^{T} = \frac{V m E f E m}{E m V m + E f V f} (ɛ f - ɛ m)

(8)

and,

ɛ f - ɛ m = \frac{E c}{σ_{c}^{T}}

Or from the equations in Vedula et al.¹⁴ using a composite cylinders model, as follows

σ_{f}^{T} = (\frac{δ E f σ_{c}^{T}}{L 3 E c}) (q \frac{ε f}{L 1} + 1)

(9)

where

δ = 1 - \frac{1}{V f}, q = \frac{2 v f δ}{E f} - \frac{2 v m}{E m}, L 1 = \frac{(1 - v f) δ}{E f} - \frac{(1 - v m)}{E m} - \frac{(1 + v m)}{V f E m}, L 2 = \frac{(v f - v m)}{E m}, L 3 = δ - \frac{E f qL 2}{L 1} - \frac{E f}{E m}

E, σ, ɛ, v

, and V are the modulus, stress, strain, Poisson’s ratio, and volume fraction, respectively, subscript c = composite, f = coated fiber, m = matrix, and superscript T = thermal.

For MI SiC/SiC composite, the value of the residual thermal stress in the composite was measured as 60 MPa.¹⁵ The equations above were utilized giving the residual thermal stress in the coated fibers as 58.1 MPa from equation (8) and 70.1 MPa from equation (9). An average of the two values was used to calculate the crack density shown in Figure 6.

Figure 7 shows that as the residual thermal stress in the fiber increases in compression, the crack density decreases, which is expected. The rate of change in crack density for the compressive stresses is less than that for the tensile stresses.

Effect of fiber volume fraction in load direction

The total fiber volume fraction in the balanced 5HS MI SiC/SiC composite under consideration was kept fixed at 43.2% (volume fraction of the coated fibers) and the ratio between the relative fiber volume fraction in the warp and the weft directions was changed. Figure 7 shows the change in crack density with the increase in the relative fiber volume fraction in the warp direction to a value of 90% of the overall fiber volume fraction. The change is a second-order polynomial, which is in accordance with the equation determining the distance between cracks for partial slip (equation (6)).

Effect of interfacial shear strength

The increase in the interfacial shear strength shows a linear increase in crack density without a limit as seen in Figure 7. Although this is in accordance with the equations for the distance between cracks, it cannot be logically correct because there must be a limit on the fiber ability to carry the stress transferred from the matrix. A condition on the stress in the fiber to be lower than the fiber strength would enhance the model.

Effect of weave angle

In a 5HS weave, yarns crimp to cross one another once every five crossover repeats as can be seen from Figure 2. In the geometric model, dimensions of the cross section of the weft yarns were changed to change the angle of crimp (weave angle) of the warp yarns while maintaining the relative and total fiber volume fractions. The result, as seen in Figure 7, shows a limited increase in crack density with the increase in weave angle following a 1/cos²(weave angle) form which is similar to the major directional cosine value in the stress transformation matrix. The small increase can be attributed to the small volume of crimped regions of the yarns as compared to the total volume of the yarns in the unit cell. At the maximum value of crack density, all the crimped regions become fully cracked.

Effect of fiber radius

The radius of the fiber affects the parameter m_e, as seen in equation (5), needed for transfer the internal stress between the matrix and the fibers. As the value of radius increases and the ratio of the surface area of the fiber to the area of its cross section decreases, the value of $m e$ increases favoring the full-slip scenario (equation (7)) over the partial slip scenario (equation (6)). Also, according to equation (6), there is a linear relationship between the distance between cracks and the fiber radius; as the fiber radius decreases, the distance between the cracks decreases (i.e. crack density increases). The combined effect results in an increase in the crack density with the reduction in the radius of fibers.

As mentioned above, Figure 7 represents the effect of five different variables at the same time. This setup allows for a clear comparison of the impact of each of these variables on the crack density, as they relate to the composite under consideration. Table 3 summarizes some of the data in Figure 7 and shows a possible range of typical values for these variables. It can be concluded from this analysis that, for typical ranges of these variables the crack density is most sensitive to the volume fraction of fibers in the load direction followed by the interfacial shear strength and weave angle and less sensitive to the fiber radius and the residual thermal stresses.

Table 3.

Range of typical values for variables under consideration and their impact on the change in crack density.

Variable	Range of typical values	Change in crack density (%)
Residual thermal stress (MPa)	30–100 (compressive)	224
Fiber volume fraction in load direction (%)	10–35	1115
Interfacial shear strength (MPa)	20–80	396
Weave angle (deg)	3°–15°	467
Fiber radius (µm)	3–8	266

Conclusions

An approach was constructed utilizing geometric modeling of fabric composites, hybrid finite element analysis and the shear lag theory to evaluate the density of cracks in composites under tensile loads. The model accounts for the effect of preform geometry on the distribution of the internal stress in the composite and the effect of defects on such distribution.

Model results were compared to experimental data from archived literature for a 5 harness satin MI SiC/SiC composite under a tensile stress in the warp direction. It was observed that the internal stress distribution and value were sensitive to the weave architecture. The sensitivity increased with the introduction of defects typically witnessed in this type of composites. Of special interest was the effect of interlaminar defects on the value of the stress between layers under load in the out-of-plane direction which can have a drastic effect on the value of the interlaminar tensile and shear strength of the composite.

Estimates of crack density calculated using the proposed model had a similar pattern to that seen in experimental data with the model starting to accumulate cracks at around 60 MPa tensile stress prior to the experimental observations. The value estimated by the model for the crack density at the saturation level was similar to that experimentally observed.

The effects of the residual thermal stresses, interfacial shear strength, relative fiber volume fraction in the load direction, fiber radius, and weave angle on crack density were studied and showed variations similar to those expected from a shear lag model. Model results were more sensitive to the volume fraction of fibers in the load direction more than other variables.

Footnotes

Funding

The Air Force Research Laboratory, Materials & Manufacturing Directorate, Wright-Patterson AFB, OH, USA supported this effort under contract FA8650-12-C-5107and FA8650-12-C-5108.

Conflict of interest

None declared.

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