Extended stress gradient elastodynamics: Wave dispersion and micro-macro identification of parameters

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In its original formulation by Forest & Sab (Math. Mech. Solids, 2017), stress gradient elastodynamics incorporate two inner-lengths to account for size effects in continuum theory. Here, an extended one-dimensional stress gradient model is developed by means of Lagrangian formalism, incorporating an additional inner-length and a fourth-order space derivative in the wave equation. Dispersive properties are characterised and hyperbolicity and stability are proven. Group velocity remains bounded in both original and extended models, proving causality is satisfied for both contrary to a usually-accepted postulate. By means of two-scale asymptotic homogenization, the high-order wave equation satisfied by the stress gradient model is shown to stand for an effective description of heterogeneous materials in the low-frequency range. An upscaling method is developed to identify the stress gradient material parameters and bulk forces on the parameters of elastic micro-structures. Application of the micro–macro procedure to periodic multi-laminates demonstrates the accuracy of the stress gradient continuum to account for the dispersive features of wave propagation. Frequency and time-domain simulations illustrate these properties.