1. Short Fiber Composites

A short fiber composite is a material that consists of two main components: a matrix and short fibers. The matrix, which can be a polymer (like plastic), metal, or even ceramic, acts as the base material. Dispersed throughout this matrix are short fibers, typically a few millimeters in length. These fibers can be made from materials like carbon, glass, aramid (Kevlar), or even natural fibers. Together, these two elements form a composite material with enhanced mechanical properties compared to the matrix alone.

Unlike continuous fiber composites, where long, uninterrupted fibers are laid out in a specific direction, the fibers in an SFC are much shorter and often randomly oriented. This random orientation gives SFCs unique properties, including more uniform strength in multiple directions. As industries continue to seek materials that offer high performance at a lower cost, short-fiber composites are expected to play an increasingly prominent role. Advances in fiber materials and manufacturing techniques will likely lead to even better performance, opening the door for SFCs to be used in more demanding applications.

Figure 1: The representative volume element (RVE) of short fiber composite [Ref]

2. Short Fiber Composite Damage Modeling in Abaqus

The main challenge is finding an answer for the heterogeneity of the distribution of short fibers in the entire sample and the progression of damage, which results from complex damage mechanisms such as fiber separation from the resin, fiber breakage, and fiber elongation from the resin. Hence two methods and approaches are used to model short fiber composites in Abaqus:

Phenomenological and Micromechanical approaches. The main difference is that the phenomenological approach requires re-identifying damage parameters whenever any micro-property changes, while the micromechanical approach doesn't have this limitation. We'll explain each of them in the following.

Figure 2: Three well-known short fiber composite damage mechanisms [Ref]

1- The use of damage models proposed in the software library, such as the Hashin damage model, which is a phenomenological approach. The Hashin damage model is a failure criterion specifically designed to predict the onset and progression of damage in fiber-reinforced composites. Named after Zvi Hashin, who developed the theory, it accounts for the unique failure modes of composite materials, where different stresses can cause the fibers and the matrix (the material binding the fibers together) to fail in distinct ways.

Figure 3: How to implement Hashin damage

In this method, a representative volume of the composite should be modeled. And for each specific geometry, properties, and damage parameters should be obtained, which costs a lot of calculations and laboratory tests.

2- A second solution is to use a multiscale micromechanical approach. Hence, the micromechanical approach can help to explain and predict various complex phenomena observed at a larger scale. Specifically, models using mean-field homogenization (MFH) are appropriate for short-fiber composites and provide a good balance between accuracy and computational cost in cases where determining the overall mechanical response involves understanding the specifics of the mechanical field around the individual fibers.

 The multi-scale method of mean field homogenization will be examined more clearly in the following. First, homogenization will be discussed briefly, followed by the damage model used in general and the simulation method in the software.

3. Sequential Multi-Scale Modeling

The model discussed in this blog takes a multi-scale approach, using a macro model based on micromechanics (fig4). This is because purely macro models often lack accuracy, and purely micro models can be inefficient. By combining the two, a multi-scale model can strike a balance between precision and efficiency.

As shown in Figure 4, in multiscale modeling, the larger (macro) model is built using structural relationships from smaller (micro or nano) scales. The macro model is defined first, but some parameters are left open. These parameters are calculated or derived from the micromodel.

This type of multiscale modeling is often called a parameter shifting model because only a few parameters need to move between scales. It works well when only a small amount of information needs to flow between the micro and macro levels. Another method, sequential multiscale modeling, involves connecting a series of calculations at different scales. This allows the results from a smaller scale simulation to set the parameters for the model at the next larger scale.

Overall, this multiscale modeling approach is useful when a material’s behavior can be broken down into multiple distinct scales, with each scale contributing unique properties.

Figure 4: An example of a sequential multiscale model

Multiscale analysis is an integral part of multiscale modeling. Its goal is to create simplified, effective macro models from complex small-scale models. Different techniques are used for this, which can generally be grouped into two categories. The first is local analysis techniques, and the second is averaging methods. These methods help derive a macro-scale model by averaging the behavior of the smaller-scale model. Some common approaches include the identical asymptotes model and the mean-field homogenization method. Below, we'll give a brief explanation of the latter.

3.1. Mean-Field Homogenization (MFH)

Homogenization models are used to predict the overall properties of materials that have a mix of different components. These models rely on details from the material's microstructure, like the shape, orientation, volume fraction, and the properties of each part. The homogenization can be broken down into two main approaches: analytical and computational.

The primary purpose of mean-field homogenization is to determine the stiffness modulus of a composite material. In other words, it aims to homogenize the stiffness moduli of both the matrix and the inclusions to represent the composite as a uniform material. In mean-field homogenization, the composite is limited to the matrix and inclusion parts with complete interfacial bonds between the components and the matrix. In other words, the composite can be divided into two phases: fiber and resin. Generally, the scale for each quantity in the composite can be written as follows.

where  represents the average field of each quantity and represents the volume fraction which is equal to that which represents the total volume. Indices M and I represent the matrix and inclusion phase, respectively. Finally, the homogenization stiffness tensor (C) for the composite including n inclusion models (inclusion shape, orientation) in the representative volume is as follows. 👉👉

where  and  represent stiffness tensor and volume fraction field for  fiber respectively. Also,  and  are the stiffness tensor and the volume fraction field for the  matrix. B is the Ashelby strain concentration tensor. The fourth-order tensor that establishes the relationship between the average strain in the inclusion phase and the uniform strain at the boundary is called the Ashelby concentration tensor. The  symbol represents the average values.

4. Two-Stage Damage Model of Homogenization | Multistep Homogenization

In the previous section, we discussed mean-field homogenization and deriving the constitutive homogenization equations. Here, we present a model based on mean-field homogenization to on short-fiber composites. Although successful theories have been developed to predict the strength of composites with long or short fibers with one-way orientation, the strength of three-dimensional short fiber composites with random orientation using the homogenization damage method is a relatively new and efficient approach compared to previous approaches.

In the two-stage homogenization (multistep homogenization) damage model, with the help of the macro-micro scale, plastic flow and damage in each of the stress and strain components along the fiber and matrix are predicted with a comparison. Considering a hypothetical representative volume of the material (RVE) in which the fibers are randomly with different lengths; The volume is divided into various parts according to the fibers' orientation distribution function (ODF) and the length distribution function (LDF). Each of these divisions is called a seed.

Figure 5: The image on the left is a representative volume with randomly distributed fibers, but in the picture on the right, many of these fibers with the same ODF and LDF are placed in the same family

After granulation, fibers with the same arrangement and the exact size are divided into each grain. Then, the material's response is obtained in two stages homogenization and applying damage (Figure 6). As can be seen in Figure 6, after the fibers are granulated, the damage applied to the field and its numerical equations are first examined. Then, in the first homogenization stage, medium field homogenization was used to distribute the load inside each grain in which fibers with ordinary materials, shape, and orientation are embedded in the damaged matrix phase. In the second stage of homogenization, using Voit's homogenization model, which is a simple model, the grains are homogenized with the overall volume and the overall stiffness modulus, and thus the composite response, which is the result of applying the loading conditions, are obtained.

Figure 6: Overview of the two-step homogenization strategy

4.1. Applying Abaqus Short Fiber Damage Mean Field Homogenization

To use the Abaqus Short Fiber Damage Mean Field Homogenization, you must apply homogenization by adding it to the text file(INP) in Abaqus. This code divides the composite into two phases, inclusion and matrix. First, the method of homogenization must be determined. After determining the homogenization method, it divides the composite into two phases: matrix and inclusions. In the matrix phase, it is enough to assign the name of the phase and its properties. Also, the shape and aspect ratio and its orientation are determined in the inclusion phase. After deciding the shape according to the aspect ratio, the influence of the shape in the equations is determined using the Ashelby tensor, which changes the total stiffness modulus by applying it to the strain concentration tensor.

For example(fig7), the composite material named COMPOSITE should be considered, including an inclusion phase (fibers) whose properties are MAT2 and defined as FIBER_MAT2. The shape of the fibers is elliptical in the matrix material named MATRIX_MAT1 with MAT1 properties. The homogenization method used is Mori-Tanaka. It has a volume fraction vf and an aspect ratio a_r with a fixed orientation defined by the vector (p₁, p₂, p₃). To write the required code, the desired properties are assigned to fibers (MAT1) and resin (MAT2) using the material allocation command (*MATERIAL).

The next step defines a composite material called COMPOSITE with the *MATERIAL command. In the composite definition section, it is necessary to specify which model is used for homogenization. For this purpose, the desired homogenization (Mori-Tanaka(MT)) will be specified with the *MEAN FIEL HOMOGENIZATION command.

In the next step, the constituent components of the composite can be specified with the *CONSTITUENT command. The TYPE command specifies the type of matrix phase (MATRIX) or inclusion (INCLUSION). The NAME command is the name of the phase. The commands SHAPE and DIRECTION specify the shape and direction, respectively. The orientation can be one-way (FIXED) or random (RANDOM).

When the specified phase of the inclusion is selected, for the shape of the inclusion other than spherical and cylindrical, the aspect ratio should be considered, and also the orientation of the fibers should be given; it is necessary to order the volume fraction ν_f, the aspect ratio of the fibers are and the orientation vector (p₁,p₂,p₃) to be specified.

Figure 7: Apply homogenization in Abaqus