A new multi‐scale dispersive gradient elasticity model with micro‐inertia: Formulation and ‐finite element implementation

Self Cite

Summary Computational aspects of a recently developed gradient elasticity model are discussed in this paper. This model includes the (Aifantis) strain gradient term along with two higher‐order acceleration terms (micro‐inertia contributions). It has been demonstrated that the presence of these three gradient terms enables one to capture the dispersive wave propagation with great accuracy. In this paper, the discretisation details of this model are thoroughly investigated, including both discretisation in time and in space. Firstly, the critical time step is derived that is relevant for conditionally stable time integrators. Secondly, recommendations on how to choose the numerical parameters, primarily the element size and time step, are given by comparing the dispersion behaviour of the original higher‐order continuum with that of the discretised medium. In so doing, the accuracy of the discretised model can be assessed a priori depending on the selected discretisation parameters for given length‐scales. A set of guidelines can therefore be established to select optimal discretisation parameters that balance computational efficiency and numerical accuracy. These guidelines are then verified numerically by examining the wave propagation in a one‐dimensional bar as well as in a two‐dimensional example. Copyright © 2016 John Wiley & Sons, Ltd.

“…The model with three parameters formulated in in the hypothesis of homogeneous material (constant density and stiffness tensor) is defined by the following equations of motion:

ρ ((), ü_{i} - α ℓ^{2} ü_{i, nn} + β ℓ^{4} ü_{i, jjnn}) = C_{ijkl} (u_{k, jl} - γ ℓ^{2} u_{k, jlnn}),

Section: Continuum Equations and Spatial Discretisationmentioning

confidence: 99%

Section: Continuum Equations and Spatial Discretisationmentioning

confidence: 99%

C^{1}

‐continuity of the interpolation, that is, continuity of the displacements as well as the much more complicated continuity of the displacement derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…To avoid this, the fourth‐order equations have been split into a set of two (coupled and symmetric) second‐order equations so that

C^{0}

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Computational aspects of a new multi‐scale dispersive gradient elasticity model with micro‐inertia

Domenico

Askes

2016

Self Cite

“…Analytical homogenization entails a full extraction of the intrinsic parameters underlying the effective continuum model based on the properties and volume fractions of the constituent materials, eg, for bilaminate composites . Continuum enrichment using additional variables to represent intrinsic microstructural behavior was discussed in depth by Forest .…”

Section: Introductionmentioning

confidence: 99%

Homogenized enriched model for blast wave propagation in metaconcrete with viscoelastic compliant layer

Tan

Poh

Tkalich

2019

Summary In this contribution, a reduced‐order homogenization approach is adopted and extended to incorporate the linear viscoelasticity effect. A homogenized enriched model emerges from the homogenization framework, which is utilized here for the analysis of blast wave propagation in metaconcrete with linear viscoelastic compliant layer(s). A semidiscrete equation of motion for the unit cell is first extracted for micro‐macro transition, through a series of transformations to effect the periodic boundary conditions and model reduction, respectively. The ensuing microbalance and macrobalance of momenta are next subjected to spatial and temporal discretizations toward a system of equations to be solved numerically. The predictive capability of the proposed model is demonstrated through a series of benchmark examples involving several metaconcrete variants. It is observed that the dual resonator configuration achieves a greater extent of wave attenuation than the single resonator counterpart.

A fully second‐order homogenization formulation for the multi‐scale modeling of heterogeneous materials

Lopes

Pires

2022

A homogenization‐based multi‐scale model at finite strains is proposed linking the micro‐scale to the macro‐scale level through a second‐gradient constitutive theory. It is suitable for modeling localization and size effects in multi‐phase materials. The macroscopic deformation gradient and second gradient are enforced on a microstructural representative volume element (RVE). The coupling between the scales is defined by means of kinematical insertion and homogenization operators, carefully postulated to ensure kinematical conservation in the scale transition, according to the method of multi‐scale virtual power. From the solution of the micro‐scale equilibrium, the macroscopic stress tensors are derived based on the principle of multi‐scale virtual power. The kinematical constraints at the RVE level are enforced with Lagrange multipliers. A mixed finite element approach is adopted for the numerical solution of the micro and macro second gradient equilibrium problems. It conveniently leads to a formulation where the Lagrange multipliers are the homogenized stresses, and the macroscopic consistent tangent operators for FE2 simulations can be directly derived. The Newton–Raphson scheme is employed to solve the non‐linear systems of equations. The numerical results demonstrate the predictive capability of the proposed model. A systematic analysis of the influence of RVE length and the micro‐constituent size is performed.