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Periodic necking of misfit hyperelastic filaments embedded in a soft matrix

The necking instability is a precursor to tensile failure and rupture of materials. A quasistatically loaded free-standing uniaxial specimen typically exhibits necking at a single location, corresponding to a long wavelength bifurcation mode. If confined to a substrate or embedded in a matrix, the same filament can exhibit periodic necking thus creating segments of finite length. While periodic instabilities have been extensively studied in ductile metal filaments and thin sheets, less is known about necking in hyperelastic materials. There is a renewed interest in the role of necking in novel materials to advance fabrication processes and to explain fragmentation phenomena observed in 3D printed active biological matter. In both cases materials are not well described by existing frameworks that employ J2 plasticity, and existing studies ignore the role of misfit stretches that may emerge in these systems through chemical or biological contraction. To address these limitations, we first experimentally demonstrate the role of the matrix on the necking and fragmentation of a compliant embedded filament. Using a strain softening generalized hyperelastic model, our analytical bifurcation analysis explains the experimental observations and agrees with numerical predictions. The analysis reveals 3 distinct bifurcation modes: a long wavelength necking, recovering the Considere criterion; a periodic necking observed in our experiments; and a short wavelength mode characterized by localization along the center cord of the filament and independent of the film-to-matrix stiffness ratio. We find that the softening coefficient and the filament misfit stretch can significantly influence the stability threshold and observed wavelength, respectively. Our results can guide the design and fabrication of composite materials and explain the fragmentation processes observed in active biological materials.