Nonlocal elasticity

Overall code description MATLAB codes for one-dimensional nonlocal elastic wave dispersion and wavefield modeling This code package contains MATLAB scripts used to reproduce the theoretical and numerical results for a one-dimensional lossless nonlocal elastic wave equation with two length-scale parameters, ggg and ddd. The scripts calculate the nonlocal kernel function, solve the analytical dispersion relation, compare exact and numerical phase velocities, and simulate local and nonlocal wave propagation in homogeneous and von Karman random heterogeneous media. The codes generate figures of phase velocity, group velocity, imaginary wavenumber, numerical convergence, stress and velocity snapshots, synthetic seismograms, and random velocity models. These scripts support the analysis of how nonlocal length-scale parameters control frequency-dependent wave velocities and wavefield behavior in microstructured or heterogeneous elastic media. File-level descriptions kenel.mThis script plots Eringen’s one-dimensional nonlocal kernel function,Ψd(∣x−x′∣)=1/(2d)exp⁡(−∣x−x′∣/d)\Psi_d(|x-x'|)=1/(2d)\exp(-|x-x'|/d)Ψd(∣x−x′∣)=1/(2d)exp(−∣x−x′∣/d), using a representative nonlocal length d=10d=10d=10 mm. It visualizes how the contribution from neighboring points decays with distance and is used to illustrate the physical meaning of the nonlocal interaction length. allfig.mThis script computes the analytical dispersion behavior of the lossless two-parameter nonlocal elastic model for four representative cases: g=1,d=2g=1,d=2g=1,d=2, g=2,d=1g=2,d=1g=2,d=1, g=0,d=1g=0,d=1g=0,d=1, and g=1,d=0g=1,d=0g=1,d=0. It solves the quartic dispersion equation and plots phase velocity, imaginary wavenumber, and group velocity as functions of frequency. stable.mThis script performs a convergence test for the one-dimensional lossless numerical scheme. It compares the exact phase velocity with finite-difference phase velocities for different time steps, Δt=10−7\Delta t=10^{-7}Δt=10−7, 10−810^{-8}10−8, 10−910^{-9}10−9, and 10−1010^{-10}10−10 s, over a wide frequency range. The script is used to evaluate the stability and accuracy of the staggered-time numerical formulation. modeling.mThis script implements a one-dimensional elastic nonlocal wave solver without attenuation. Spatial derivatives and the Helmholtz-type inversion are computed using FFT, and time integration is performed using a classical fourth-order Runge–Kutta scheme. The script simulates four nonlocal parameter cases and outputs velocity snapshots, stress snapshots, synthetic seismograms, receiver traces, and a comparison between numerical and theoretical wave speeds. modelingfrac.mThis script simulates one-dimensional local and nonlocal wave propagation in von Karman random heterogeneous media. It generates random velocity models with correlation lengths a=2a=2a=2, 0.2, and 0.02 m, and compares the local case with two nonlocal cases. The output includes random velocity models and normalized particle-velocity seismograms for different heterogeneity scales and nonlocal parameters.