Calculation technique for simulation of wave and fracture dynamics in a reinforced sheet

Mathematical models and computer algorithms are developed to calculate dynamic stress concentration and fracture wave propagation in a reinforced composite sheet. The composite consists of a regular system alternating extensible fibers and pliable adhesive layers. In computer simulations, we derive difference algorithms preventing or minimizing the parasite distortions caused by the mesh dispersion and obtain precise numerical solutions in the plane fracture problem of a pre-stretched sheet along the fibers. Interactive effects of microscale dynamic deformation and multiple damage in fibers and adhesive are studied. Two engineering models of the composite are considered: the first assumes that adhesive can be represented by inertionless bonds of constant stiffness, while in the second one an adhesive is described by inertial medium perceived shear stresses. Comparison of results allows the evaluation of facilities of models in wave and fracture patterns analysis.

💡 Research Summary

This paper presents a novel computational framework for accurately simulating wave propagation and fracture dynamics in a reinforced composite sheet composed of alternating high‑strength fibers and compliant adhesive layers. Recognizing that conventional finite‑difference and finite‑element schemes suffer from amplitude diffusion, phase dispersion, and Gibbs‑type oscillations—collectively termed Mesh Dispersion (MD)—the authors adopt a Mesh Dispersion Minimization (MDM) strategy to suppress these spurious effects.

The mechanical model idealizes the sheet as a periodic array of thin elastic fibers (supporting only axial tension/compression) bonded by adhesive layers that transmit only shear stresses. Two engineering representations of the adhesive are examined: (i) Model 1 treats the adhesive as an inertialess spring network with constant stiffness K = G H, ignoring mass; (ii) Model 2 incorporates the adhesive’s inertia, modeling it as a shear‑wave medium governed by a one‑dimensional wave equation. Both models share the same interfacial shear force expressions, but Model 2 adds a dynamic term reflecting the adhesive’s density.

Mathematically, the fiber dynamics obey a 1‑D wave equation with source terms representing interfacial shear forces, while the adhesive follows a shear‑wave equation. Non‑linear fracture criteria are imposed: a fiber segment fails when its axial stress exceeds a critical value V*; an adhesive segment delaminates when its shear stress exceeds W*. These criteria generate moving free boundaries that alter the governing equations in real time, producing a highly non‑linear hyperbolic system with non‑classical boundary conditions.

To eliminate MD, the authors align the discrete domain of dependence with that of the continuous problem. They select temporal and spatial steps such that c Δt = Δx (and Δy), where c is the wave speed, and employ a five‑point explicit stencil. This choice forces the discrete dispersion relation to match the continuous one, ensuring that phase velocity is wavelength‑independent for both low‑ and high‑frequency components. Consequently, numerical phase errors vanish, and the artificial widening of wave fronts is avoided.

Numerical experiments compare the two adhesive models under identical pre‑stress and initial fracture conditions. Model 1, lacking inertia, yields rapidly attenuated shear waves and limited secondary fiber breakage. Model 2, with inertial shear waves, exhibits long‑range propagation, multiple reflections, and pronounced amplification of stresses in neighboring fibers, leading to cascade fracture. The results demonstrate that the choice of adhesive representation critically influences the predicted damage pattern and overall toughness of the composite.

A side‑by‑side comparison of simulations with and without MDM highlights the method’s effectiveness: conventional schemes produce spurious high‑frequency oscillations and smeared crack fronts, obscuring the true physics; the MDM‑enhanced scheme preserves sharp wave fronts, accurately captures the evolution of the “trauma” zone (the region of permanent deformation after fracture), and yields stress fields that agree with analytical expectations.

While the study confirms that MDM can dramatically improve accuracy for the 2‑D reinforced sheet problem, it also acknowledges that complete elimination of MD in multidimensional settings remains challenging. Nonetheless, the presented approach offers a practical pathway to high‑fidelity simulations of impact‑driven failure in fiber‑reinforced structures, and the authors suggest extending the technique to multi‑scale, multi‑physics problems and validating it against experimental data.