How to obtain the mechanical properties of a composite material?
Composite materials are commonly known by their anisotropic properties. In the case of isotropic materials, the deformation can be obtained using the Young Modulous and Shear Moduli, which are the same in all the orientations. However, in composite material as there are two materials with different properties which work together when a load is applied, a new method is used to to obtain the mechanical properties of the composite materials. That is why rule of mixtures method have been developed.
The Rule of Mixtures (RoM) is a method to predict the composite material mechanical properties.
Simplification of rule of mixtures
Assumptions have to be made in order to simplify the method and obtain good results:
- The fiber is homogeneous, isotropic, linear elastic and continuous.
- The matrix is homogeneous, isotropic, linear elastic and continuous.
- The fiber is ordered in a repetitive sequence.
- In the interface between the matrix and the fiber, the matrix has the same properties as the matrix.
- The interface is perfect, there are no discontinuities.
It is clear that all of this does not occur in the reality, but these simplifications allow to make a model about how the micromechanics of composite materials works.
First, as explain in other posts, it is important to know the volume faction fiber so we can use the following relationship:
However, this is true if it is considered that there are no voids. Thanks to the Rule of Mixtures, it is possible to obtain each mechanical property of the composite material by using the following equation:
Where Pf is the property of the fiber and Pm the property of the matrix.
Young Modulus Calcution (E1)
To obtain the Young Modulus of a composite material in the longitudinal direction by applying the Rule of mixtures, it is necessary to know the volume of fibers and the elastic modulus of the components:
Transverse Young Modulus Calcution (E2)
This estimation gave a good result when obtaining the modulus, only there has an error between 1% to 2%. However, to obtain the Young Modulus in the transverse direction the following equation has to be used:
As it is seen, the elastic modulus of the matrix is the one that dominates the value of the E2. In reality, this formula does not have a good approximation because:
- The matrix does not have a linear behavior as the fibers.
- The deformation between the interface-matrix it is not determined by the model.
- Doing experiments, it is shown that the tension is not constant in the transverse direction.
- Other effects that can appear is the difference in the Poisson coefficient of the matrix and the fiber, the contractions longitudinal of the fiber and the matric are different too, which generate shear loads at the interface.
In the same way the Young Modulus is obtained, Possion’s ratio and Shear modulus can be obtained too.
Rule of Mixtures is not the only method to obtain the mechanical properties of composite materials. Other methods exists such as Förster/Knappe method, Schneider method, Puck method, Tsai method.
Advantages of using rule of mixtures in composite materials
1. Simplicity: Provides a straightforward method to estimate composite properties.
2. Design Flexibility: Helps in tailoring materials with specific properties by adjusting the volume fractions of constituents.
3. Efficiency: Reduces the need for extensive experimental testing by providing a first approximation.
Examples of rule of mixtures
1. Elastic Modulus:
– If a composite is made of 60% glass fibre (modulus = 70 GPa) and 40% epoxy resin (modulus = 3 GPa), the modulus of the composite can be estimated using the upper bound rule:
E_c = V_f E_f + V_m E_m = 0.6 \times 70 + 0.4 \times 3 = 42 + 1.2 = 43.2 \text{ GPa}2. Thermal Conductivity:
– For a composite made of 30% copper (thermal conductivity = 400 W/m·K) and 70% plastic (thermal conductivity = 0.3 W/m·K), using the lower bound rule:
\frac{1}{k_c} = \frac{V_f}{k_f} + \frac{V_m}{k_m} = \frac{0.3}{400} + \frac{0.7}{0.3} = 0.00075 + 2.333 = 2.33375\frac{1}{k_c} = \frac{V_f}{k_f} + \frac{V_m}{k_m} = \frac{0.3}{400} + \frac{0.7}{0.3} = 0.00075 + 2.333 = 2.33375
k_c \approx \frac{1}{2.33375} \approx 0.428 \text{ W/mK}Innovative Solutions
– Hybrid Models: Combining the rule of mixtures with finite element analysis (FEA) to account for more complex behaviours and interactions.
– Machine Learning: Utilizing machine learning algorithms to predict composite properties based on a wider range of variables and more complex interactions than traditional models allow.
– Nanocomposites: Exploring the properties of nanocomposites, where the rule of mixtures can be adapted to consider the unique effects of nanoscale interfaces and interactions.
In conclusion, while the rule of mixtures is a foundational tool in composite material design, ongoing advancements and the integration of new technologies are enhancing its accuracy and applicability, paving the way for more sophisticated and high-performance composite materials.
