Model and analysis of size-stiffening in nanoporous cellular solids

Abstract

The size of the struts in nanoporous cellular solids typically has a secondary influence on the stiffness of the solids, but it leads to significant stiffening when it is on the same order as the higher-order material parameter. We examined this size-dependence using the higher-order finite-element method (FEM) in this study. Mathematical analysis showed that the displacement field that satisfies the conventional Lame equation can serve as a displacement field template in higher-order FEM. Benchmarking studies showed that results from simulations of beam bending and rod torsion using this FEM approach were in good agreement with results from analytical solutions and experiments. Using this approach, we showed that the stiffness of cellular solids is strongly affected by the cellular arrangement and the density when the cell size is on the order of the higher-order material parameter and that the stiffening behavior in nanoporous polyimide can be explained using higher-order theory. The FEM results also showed that a porous solid with half the weight can be engineered to become as stiff as a fully dense solid if the porous microarchitecture is tailored to take advantage of higher-order stiffening.

Appendix I: FEM verification

Higher-order strain gradient solutions are available for torsion, pure bending, and cantilever bending. A two-dimensional eight-node high-order element was used to model the higher-order deformation behavior in cantilever bending and pure bending. A three-dimensional 20-node high-order element was used in the torsion case.

Bending of micro-beams

Pure bending and cantilever bending cases are sketched in Fig. 4. For the pure bending case with small deformation, an analytical solution was developed [22]. The ratio of the total moment to the conventional moment in pure bending of a beam with elasticity is

$$ \frac{M}{{M_{0} }} = 1 + 6\left( {1 - v} \right)\left( {\frac{{l_{2} }}{h}} \right)^{2} . $$

(29)

In the absence of higher-order effects, M reverts to the conventional M₀. A comparison of the moment ratio between FEM and the analytical solution in Eq. 29 is shown in Fig. 5.

Fig. 4

Sketches of two kinds of micro-beam bending

Fig. 5

Comparison of normalized moments between FEM results and analytical solutions of pure bending

The bending rigidities for beams with different thicknesses are plotted in Fig. 6. These results show that the FEM results for pure bending and for cantilever bending, where there is shear, are in good agreement with experimentally benchmarked analytical models.

Fig. 6

Comparison of the normalized rigidity from experiments [7] and FEM results

Torsion

The behavior of micro-rod torsion was also examined and its typical mesh is illustrated in Fig. 7. The solution for a couple stress solid rod under torsion was derived in [25] and is given as

$$ Q = Q_{0} \left[ {1 + 6\left( {\frac{{l_{2} }}{r}} \right)^{2} } \right], $$

(30)

where Q and Q₀ are the couple stress torque and conventional torque, respectively, and r is the radius of the micro-rod. The FEM results are consistent with the analytical solution according to Fig. 8.

Fig. 7

A typical micro-rod mesh for FEM

Fig. 8

Comparison of the normalized torque in micro-rod between FEM results and CS theory