H. M. Ma

Strain gradient solution for Eshelby’s ellipsoidal inclusion problem

Eshelby’s problem of an ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material and prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is analytically solved. The solution is based on a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The fourth-order Eshelby tensor is obtained in analytical expressions for both the regions inside and outside the inclusion in terms of two line integrals and two surface integrals. This non-classical Eshelby tensor consists of a classical part and a gradient part. The former involves Poisson’s ratio only, while the latter includes the length scale parameter additionally, which enables the newly obtained Eshelby tensor to capture the inclusion size effect, unlike its counterpart based on classical elasticity. The accompanying fifth-order Eshelby-like tensor relates the prescribed eigenstrain gradient to the disturbed strain and has only a gradient part. When the strain gradient effect is not considered, the new Eshelby tensor reduces to the classical Eshelby tensor, and the Eshelby-like tensor vanishes. In addition, the current Eshelby tensor for the ellipsoidal inclusion problem includes those for the spherical and cylindrical inclusion problems based on the SSGET as two limiting cases. The non-classical Eshelby tensor depends on the position and is non-uniform even inside the inclusion, which differ from its classical counterpart. For homogenization applications, the volume average of the new Eshelby tensor over the ellipsoidal inclusion is analytically obtained. The numerical results quantitatively show that the components of the newly derived Eshelby tensor vary with both the position and the inclusion size, unlike their classical counterparts. When the inclusion size is small, it is found that the contribution of the gradient part is significantly large. It is also seen that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. Moreover, these components are observed to approach the values of their classical counterparts from below when the inclusion size becomes sufficiently large.

Analytical Solution for a Pressurized Thick-Walled Spherical Shell Based on a Simplified Strain Gradient Elasticity Theory

The problem of a pressurized thick-walled spherical shell is analytically solved using a simplified strain gradient elasticity theory. The closed-form solution derived contains a material length scale parameter and can account for microstructural effects, which qualitatively differs from Lamé’s solution in classical elasticity. When the strain gradient effect (a measure of the underlying material microstructure) is not considered, the newly derived strain gradient elasticity solution reduces to Lamé’s classical elasticity solution. To illustrate the new solution, a sample problem with specified geometrical parameters, pressure values and material properties is solved. The numerical results reveal that the magnitudes of both the radial and tangential stress components in the shell wall given by the current strain gradient solution are smaller than those given by Lamé’s solution. Also, it is quantitatively shown that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.

Monte Carlo modeling of the fiber curliness effect on percolation of conductive composites

A three-dimensional (3D) Monte Carlo model is developed to study the fiber curliness effect on the percolation threshold of a composite filled with electrically conductive curved fibers. These fibers are simulated as zigzag-shaped fibers that are randomly positioned in the composite, forming a 3D random network. The simulation results show that the fiber curliness can significantly affect the percolation threshold: the more curved the fibers, the higher the threshold. The results also reveal an exponential relationship between the threshold and the fiber aspect ratio: the higher the aspect ratio, the lower the threshold. These predicted trends agree well with existing experimental and simulation results based on straight fibers or curved fibers with simpler shapes.