Mechanical and Crack Propagating Behavior of Sierpinski Carpet Composites.

Fractals, mathematically defined as "self-similar subsets at different scales", are ubiquitous in nature despite their complexity in assembly and formulation. Fractal geometry formed by simple components has long been applied to many fields, from physics and chemistry to electronics and architecture. The Sierpinski carpet (SC), a fractal with a Hausdorff dimension of approximately 1.8933, has two-dimensional space-filling abilities and therefore provides many structural applications. However, few studies have investigated its mechanical properties and fracture behaviors. Here, utilizing the lattice spring model (LSM), we constructed SC composites with two base materials and simulated tensile tests to show how fractal iterations affect their mechanical properties and crack propagation. From observing the stress-strain responses, we find that, for either the soft-base or stiff-base SC composites, the second iteration has the optimal mechanical performance in the terms of stiffness, strength, and toughness compared to the composites with higher hierarchies. The reason behind this surprising result is that the largest stress intensities occur at the corners of the smallest squares in the middle zone, which consequently induces crack nucleation. We also find that the main crack tends to deflect locally in SC composites with a soft matrix, but in global main crack behavior, SC composites with a stiff matrix have a large equivalent crack deflection. Furthermore, considering the inherent anisotropy of SC composites, we rotated the samples by 45°. The results show that the tensile strength and toughness of rotated SC composites are inferior and the crack propagating behaviors are distinct from the standard SC composites. This finding infers advanced engineering for crack control and deflection by adjusting the orientation of SC composites. Overall, our study opens the door for future engineering applications in stretchable devices, seismic metamaterials, and structural materials with tunable properties and hierarchies.

Predicting pullout strength of pedicle screws in broken bones from X-ray images.

Pedicle screw fixation is one of the most common procedures used in spinal fusion surgery. The screw loosening is a major concern, which may be caused by broken pedicles. In vitro pullout tests or insertion torque are the main approaches for assessing the stability of the screw; however, direct evidence was lacking for clinical human spines. Here, we aim to provide a model that can predict the pullout strengths of pedicle screws in various pedicle conditions from X-ray images. A weighted embedded bone volume (EBV) model is proposed for pullout strengths prediction by considering the bone heterogeneity and confinement of the screw. We showed that the pullout strength is proportional to the EBV for homogeneous bone and the weighted EBV for layered composite bone. The proposed weighted EBV model is validated with in vitro Sawbones® pullout experiments. The results show that the model has better accuracy than the simple EBV model, with a coefficient of determination of 0.94. The proposed weighted EBV model can help assess the stability of a pedicle screw in a broken pedicle by simply examining 2D X-ray images.