What is mean field homogenization method?
Short fiber composites are materials where tiny fibers are embedded in a matrix. Mean field homogenization is a technique used to predict the overall behavior of these composites, even though they’re clearly not uniform on a microscopic level.
Here’s the basic idea:
Imagine the composite is a bunch of average, representative chunks, each containing both fibers and matrix.
We figure out how each chunk would behave under stress based on the properties of the fibers and matrix.
Then, we weight those behaviors according to how much fiber there is overall (fiber volume fraction) to get an average behavior for the entire composite.
This method is appealing because it’s computationally efficient compared to simulating every single fiber. It also provides valuable insights into how fiber content and orientation affect the composite’s stiffness and strength.
There are different variations within the mean field homogenization method, but they all share this core concept of replacing the complex microstructure with a simplified, “mean field” to predict the overall response.
Multistep homogenization | Mean-Field Homogenization Abaqus
For composites with multiple inclusions, a multistep homogenization approach is used, as shown in Figure. The composite is decomposed into “grains,” with each grain containing one inclusion family and the matrix. The inclusions in each family have the same material properties, aspect ratio, and orientation. In the first step homogenization (mean-field homogenization Abaqus) is performed in each grain using the user-specified formulation; in the second step the Voigt formulation is used to compute the properties of the overall composite. An alternative approach is to use the Mori-Tanaka scheme in both the first and the second step, assuming the average strain in the matrix is uniform across all grains. This approach is equivalent to the direct Mori-Tanaka approach proposed by Benveniste. The drawback of this second approach is that it might result in an unsymmetric effective modulus when the inclusions are misaligned and nonisotropic.
Lemaitre damage
The damage of materials is the progressive physical process by which they break. The Lemaitre mechanics of damage is the study, through mechanical variables, of the mechanisms involved in this deterioration when the materials are subjected to loading. At the microscale level this is the accumulation of microstresses in the neighborhood of defects or interfaces and the breaking of bonds, which both damage the material. At the mesoscale level of the representative volume element this is the growth and the coalescence of microcracks or microvoids which together initiate one crack. At the macroscale level this is the growth of that crack. The two first stages may be studied by means of damage variables of the mechanics of continuous media defined at the mesoscale level. The third stage is usually studied using fracture mechanics with variables defined at the macroscale level.
When studying engineering materials such as metals and alloys, polymers and composites, ceramics, rocks, concrete, and wood, it is very surprising to see how such materials, which have different physical structures, are similar in their qualitative mechanical behavior. All show elastic behavior, yielding, some form of plastic or irreversible strain, anisotropy induced by strain, cyclic hysteresis loops, damage by monotonic loading or by fatigue, and crack growth under static or dynamic loads. This means that the common mesoscopic properties can be explained by a few energy mechanisms that are similar for all these materials. This is the main reason it is possible to explain material behavior successfully with the mechanics of continuous media and the thermodynamics of irreversible processes, which model the materials without detailed reference to the complexity of their physical microstructures.
Composites with damage
When damage and failure are defined at the constituent level, the damage in each constituent contributes to the overall damage in an indirect way through stress averaging and strain partitioning. As the stiffness of the constituent decreases, the strain increment in this constituent is likely to increase based on the formulation of the homogenization. In reality, the damage behavior of each constituent is likely to interact with each other and affect the overall damage behavior of the composite; therefore, you might need to specify an additional damage variable, D, for the composite. At any time during the analysis, the stress tensor of the composite is given by.
Model | Abaqus short fiber damage mean field homogenization
The model has been implemented as a user-defined material law (UMAT) in the finite element code ABAQUS. In the two-stage Abaqus short fiber damage mean field homogenization model with the help of macro-micro scale, plastic flow and damage in each of the components of stresses and strains along the fiber and matrix are predicted with a comparison. As can be seen in Figure after the fibers are granulated, the damage applied to the field and its numerical equations are investigated first. Then, in the first stage of homogenization, medium field homogenization with the Mori-Takana method has been used to distribute the load inside each grain in which fibers with the same material, shape, and orientation are embedded in the damaged matrix phase. In the second stage of homogenization, by using the Voight homogenization model, which is a simple model, the grains are homogenized with the overall volume and the overall stiffness modulus and as a result the composite response, which is the result of applying the loading conditions, is obtained.
Case study
In toy-breed dogs (bodyweight < 5 kg), the fractures of the radius and ulna are particularly common and can be caused by minimal trauma. While fracture fixation using metallic plates is a feasible treatment modality, the excessive stiffness of these devices produces the underloading of the bone which may result in the adverse bone remodelling and complications in the healing of the fracture. In this study, we investigated bisphenol A Polyether ether ketone -based carbon fibre reinforced composites as potential alternatives to metals in the devices intended for the fracture fixation of the distal radius in toy-breed dogs.
The dimensions are shown in the figure below:
To check the correctness of the homogenization and damage model, the selected sample has been simulated under the four-point bending test.
Now, what about other methods to calculate the short fiber composite damage?? we have another method here and this one uses the thermodynamic forces. you can learn about the details in “Damage simulation of short fibre composites” package.
this package uses the UMAT subroutine which you can learn it from the Abaqus documentation; but don’t go anywhere else because all you need to know about the UMAT subroutine are in this article Start Writing Your first UMAT Abaqus to begin with.
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