Random distribution of the coated fibers

Chapter IV. Microstructural Fracture Analysis

4.3 RVE construction process

4.3.2 Random distribution of the coated fibers

fibers. If the virtual spring is in the state of equilibrium, the force at the 𝑗^th fiber can be easily obtained as

𝑓_𝑥^′_,𝑗 = −𝑓_𝑥^′_,𝑖 (71)

Forces in the transverse direction, 𝑓_𝑦^′_,𝑖 and 𝑓_𝑦^′_,𝑗, are zeros under the local coordinate system. When the virtual spring is considered as a one-dimensional rod element in the typical finite element framework, the analogous local matrix system for the spring element shown in Figure 42(b) can be constructed as

{ 𝑓_𝑥^′_,𝑖 𝑓_𝑦^′_,𝑖 𝑓_𝑥^′_,𝑗 𝑓_𝑦^′_,𝑗}

= 𝑘 [

1 0 −1 0

0 0 0 0

−1 0 1 0

0 0 0 0

] {

𝑢_𝑖^′ 𝑣_𝑖^′ 𝑢_𝑗^′ 𝑣_𝑗^′}

(72)

or in a shortened form,

𝐟̅ = 𝐤̅𝐪̅ (73)

where 𝐟̅ is the force vector, 𝐤̅ is the local element stiffness matrix and 𝐪̅ is the displacement vector with respect to the local 𝑥^′𝑦^′–coordinate system. The displacement and the force vectors in Eq. (72) can be related to the corresponding quantities in the global 𝑥𝑦 coordinate system through the angle of 𝜃 and it reads that

{ 𝑢_𝑖^′ 𝑣_𝑖^′ 𝑢_𝑗^′ 𝑣_𝑗^′}

= [

cos 𝜃 sin 𝜃 0 0

− sin 𝜃 cos 𝜃 0 0

0 0 cos 𝜃 sin 𝜃

0 0 − sin 𝜃 cos 𝜃 ] {

𝑢_𝑖 𝑣𝑖

𝑢_𝑗 𝑣_𝑗

} (74)

and

{ 𝑓_𝑥^′_,𝑖 𝑓_𝑦^′_,𝑖 𝑓_𝑥^′_,𝑗 𝑓_𝑦^′_,𝑗}

= [

cos 𝜃 sin 𝜃 0 0

− sin 𝜃 cos 𝜃 0 0

0 0 cos 𝜃 sin 𝜃

0 0 − sin 𝜃 cos 𝜃 ]

{ 𝑓^𝑥,𝑖

𝑓_𝑦,𝑖 𝑓_𝑥,𝑗 𝑓_𝑦,𝑗}

(75)

Eqs. (74) and (75) can be expressed in a shortened form as

𝐪̅ = 𝐓𝐪 (76)

𝐟̅ = 𝐓𝐟 (77)

where 𝐪 is the displacement vector and 𝐟 is the force vector under the global coordinate system. 𝐓 is the transformation matrix. Substituting Eqs. (76) and (77) into Eq. (73) and multiplying both sides by the inverse of the transformation matrix, 𝐓, yields

𝐟 = 𝐓^−𝟏𝐤̅ 𝐓𝐪

= 𝐤𝐪 (78)

where the stiffness matrix in the global coordinate system, 𝐤, is expressed as

𝐤 = 𝑘 [

cos²𝜃 cos 𝜃 sin 𝜃 −cos²𝜃 −cos 𝜃 sin 𝜃 cos 𝜃 sin 𝜃 sin²𝜃 −cos 𝜃 sin 𝜃 −sin²𝜃

−cos²𝜃 −cos 𝜃 sin 𝜃 cos²𝜃 cos 𝜃 sin 𝜃

−cos 𝜃 sin 𝜃 −sin²𝜃 cos 𝜃 sin 𝜃 sin²𝜃

] (79)

Knowing that the transverse forces, 𝑓_𝑦^′_,𝑖 and 𝑓_𝑦^′_,𝑗, are zeros under the local coordinate system, and using Eq. (71), the simplified expression of the force vector with respect to the global coordinate system can be obtained after multiplying both sides of Eq. (75) by the inverse of the transformation matrix;

𝐟 = {

−𝑓^𝑥′,𝑖cos 𝜃

−𝑓_𝑥^′_,𝑖sin 𝜃 𝑓_𝑥^′_,𝑖cos 𝜃 𝑓_𝑥^′_,𝑖sin 𝜃 }

(80)

Eq. (78) exists for any intersecting fiber pairs. Assembling all such local matrices result in the global matrix system that can be written as

𝐅 = 𝐊𝐐 (81)

where 𝐅 is the global force vector, 𝐊 is the global stiffness matrix, and 𝐐 is the global displacement vector. Eq. (81) is first solved for 𝐐. Then 𝐅 and 𝐊 is updated since 𝑑_𝑖𝑗 in Eq. (70) and 𝜃 in Eq.

(79) have been changed due to the displacement of fibers obtained from 𝐐. This process is repeated until the L2-norm of 𝐅 in Eq. (81) is sufficiently close to zero, which means all the fibers are separated without any interference.

While the fibers are not allowed to intersect with each other, intersections between coating layers may exist as observed from the section image in Figure 40(a). When this happens in the RVE model after the fibers are relocated due to the virtual springs, the total area of the coating layers would not be the same as that of the optimal dimension, 𝒕, computed from Eq. (65). Since the intersecting areas result in the reduction of the coating volume fraction, the coating layers in the RVE should be thickened to compensate the areal loss due to the interferences. After the fiber distribution process is completed, the thicknesses of all the coatings are uniformly increased by a factor of 𝛼, defined as the square root of the ratio between the coating area in the RVE and the reference coating area, 𝑉_𝑐𝐻². 𝑉_𝑐 is the measured coating volume fraction and 𝐻 is the side length of the RVE. As the coatings expands, the intersecting areas sequentially grow, which may lead to an additional loss in the entire coating area. Therefore, this compensation process should be iteratively conducted until the area ratio, 𝛼, becomes one, which indicates the equality between the total coating area considering the intersections and the reference coating area. Figure 43 shows the overall process for the construction of an RVE model with the randomly distributed coated fibers.

When determining the size of an RVE model, it is known that the model size should be sufficiently smaller than the characteristic length of composites and sufficiently larger than that of constituents to obtain the credible estimation of effective composite properties [129]. As discussed by Kanit et al. [130], the RVE size determines the precision of the property predictions. In other words, when using small

RVEs, one needs to generate a sufficiently large number of RVEs to obtain reliable computational results. The main purpose of this study does not focus on the precise estimation of homogenized elastic properties, but focuses on the fracture behaviors and post-peak responses, which are greatly affected by the spatial arrangement of fibers. Thus, the size of RVE in this study is set to contain 20 fibers, which is comparable with the previous researchers [94,126,127] who have also studied microstructural fracture behaviors with non-uniform fiber arrangements. Figure 44 shows the RVE models of the same size with the same fiber and coating volume fractions, but different fiber distributions. They will be utilized to study the effect of the fiber distribution on the microstructural behavior of the CMCs.

Figure 43 Overall process of constructing an RVE model.

RVE 1 RVE 2 RVE 3 RVE 4

RVE 5 RVE 6 RVE 7 RVE 8

Figure 44 RVE models having the same fiber and coating volume fractions. The only difference is the fiber distributions.

4.4 Micro-fracture mechanics model of CMCs