Mathematical modeling for stability and instability analysis of carbon nanotube enhanced fibers in clothing design

The current work provides a mathematical model to evaluate the stability and instability of carbon nanotube (CNT)-reinforced fibers as they are employed in clothing design. We model the textile thread as a double-clamped beam, which indicates the mechanical behavior of the fiber. The fibers are multi-scale hybrid nanocomposites (MHC) or combinations of multi-scale composite, integrated with CNTs for added strength and elasticity. We apply Halpin–Tsai model characterizations for composite materials so that we can evaluate the effective properties of the composite substance and incorporate interaction between CNTs and the matrix. The deformation of the thread is modeled by developing a quasi-3D hybrid type (q-3DHT) formula using higher order shear deformation theory (HSDT). The higher order shear deformation theory represents shear deformation and rotational inertia and provides a more accurate representation of the behavior of the material system. A variational method to derive the governing equations of motion for the system exposed to a harmonically induced transverse force is used in the approach. Following this, the stability of the system is assessed using the differential quadrature approach (DQA), which is an adaptation of the Dubner and Abate development for effectively inverting Laplace transforms. This mathematical framework provides insight for dynamic stability of CNT enhanced fibers in the textile industry and provide strategies for optimizing garment design. The model ultimately has potential implications of incorporating CNT reinforcement for developing high-performance fibers that perform well against external force and still have comfort and durability with clothing design.