Generalized interfacial energy and size effects in composites

TLDR: In this article, a closed form analytical solution is derived to compute the effective interface-enhanced material response, which is in excellent agreement with the numerical results obtained from the finite element method for a broad variety of parameters and dimensions.

Abstract: The objective of this contribution is to explain the size effect in composites due to the interfacial energy between the constituents of the underlying microstructure. The generalized interface energy accounts for both jumps of the deformation as well as the stress across the interface. The cohesive zone and elastic interface are only two limit cases of the general interface model. A closed form analytical solution is derived to compute the effective interface-enhanced material response. Our novel analytical solution is in excellent agreement with the numerical results obtained from the finite element method for a broad variety of parameters and dimensions. A remarkable observation is that the notion of size effect is theoretically bounded verified by numerical examples. Thus, the gain or loss via reducing the dimensions of the microstructure is limited to certain ultimate values, immediately relevant for designing nano-composites.

Figures

Fig. 3. A graphical summary of micro-to-macro transition accounting for general interfaces. The microscale corresponds to the representative volume element. Constitutive laws at the microscale are assumed to be known and the goal is to compute the effective response via homogenizing the response of the underlying microstructure. The zero-thickness interface model at the microscale replaces the finite-thickness interphase between different constituents.

Fig. 12. Effective shear modulus versus size of the RVE for the volume fraction of 28.27%. Increasing the resistance against the change of length along the interface correlates to increasing the interface parameter μ. Different lines on each graph correspond to different orthogonal resistance of the interface against opening, see the legend at the bottom. In the absence of the interface, the size effect is completely ruled out resulting in a constant effective response.

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Generalized interfacial energy and size effects in composites" ?

In this paper, a closed form analytical solution is derived to compute the effective interface-enhanced material response. 

Q2. What is the effective inplane bulk modulus for the perfect interface model?

General interface model allows for a finite resistance along the interface μ as well as a finite resistance orthogonal to the interface k . 

Q3. What is the role of interfaces in nanomaterials?

Emerging applications of nano-materials require better understanding of interfaces since the influence of lower-dimensional media on the overall material response increases with decreasing size. 

Q4. What is the main issue with the elastic interfaces?

The main issue with the elastic interfaces is that the mean field theories can only be applied for reinforcements of constant curvature (spherical particles or cylindrical fibers), due to the fact that in the other cases the Eshelby problem does not provide uniform elastic state inside the inclusion as pointed out by Sharma and Ganti (2004) . 

Q5. What is the RVE for a unidirectional fiber?

For a unidirectional fiber composite, the RVE can be well defined and consists of a cylindrical fiber surrounded by the matrix material. 

Q6. What is the tangent unit of the interface?

The outward unit normal to the curve C but tangent to the interface The authoris denoted ̃ n and plays an important role to derive the balance equations for open interfaces. 

Q7. What is the traction jump across the interface?

The traction jump across the interface is due to the interface stress (see Chen et al., 2006; Javili et al., 2013c , among others). 

Q8. What is the stiffness ratio of a composite?

The stiffness ratio incl. / matr. = 0 . 1 corresponds to the case where the inclusion is 10 times more compliant compared to the matrix.