ON MULTISCALE DAMAGE MODELLING OF HETEROGENEOUS MATERIALS USING NONLOCAL CONTINUUM THEORY

An overview of the modelling of quasi-brittle as well as ductile damage is given. The multiscale procedure employing the nonlocal continuum theory is described in more detail. The softening is introduced at the microlevel in the microstructural volume element and after that the homogenization procedure state variables are mapped at the macrolevel material point via the scale transition approach. In the case of quasi-brittle softening the C1 continuous finite element discretization is applied at both micro- and macrolevel. At the modelling of ductile damage response, the macrolevel is also discretized by the C1 finite element formulation, while the microscale utilizes quadrilateral mixed finite elements employing the nonlocal equivalent plastic strain and gradient-enhanced elastoplasticity. All approaches presented are verified in the standard examples.

Critical Buckling Load of Triple-Walled Carbon Nanotube Based on Nonlocal Elasticity Theory

The present paper investigates the nonlocal buckling of Zigzag Triple-walled carbon nanotubes (TWCNTs) under axial compression with both chirality and small scale effects. Based on the nonlocal continuum theory and the Timoshenko beam model, the governing equations are derived and the critical buckling loads under axial compression are obtained. The TWCNTs are considered as three nanotube shells coupled through the van der Waals interaction between them. The results show that the critical buckling load can be overestimated by the local beam model if the small-scale effect is overlooked for long nanotubes. In addition, a significant dependence of the critical buckling loads on the chirality of zigzag carbon nanotube is confirmed, and these are then compared with: A single-walled carbon nanotubes (SWCNTs); and Double-walled carbon nanotubes (DWCNTs). These findings are important in mechanical design considerations and reinforcement of devices that use carbon nanotubes.

Mathematical modeling of a fractionally damped nonlinear nanobeam via nonlocal continuum approach

Modeling of fractionally damped nanostructure is extremely important because of its inherent ability to capture the memory and hereditary effect of several viscoelastic materials extensively used in nanotechnology. The nonlinear free vibration characteristics of a simply-supported nanobeam with fractional-order derivative damping via nonlocal continuum theory are studied in this article. Using Newton’s second law, the equation of motion for the nanobeam embedded in a viscoelastic matrix is derived. The Galerkin method is employed to transform the integro-partial differential equation of motion into a Duffing-type nonlinear ordinary differential equation. The fractional-order damping term is replaced by a combination of linear damping and linear stiffness term. The approximate analytical solution obtained via method of averaging is found to be in good agreement with solution obtained through numerical scheme. Detailed study of system parameters reveals that the fractional-order derivative damping has significant influence on the time response and effective natural frequency of the nanobeam.

A Peridynamics Model for the Propagation of Hydraulic Fractures in Naturally Fractured Reservoirs

Summary A novel and fully coupled hydraulic-fracturing model derived from a nonlocal continuum theory of peridynamics is presented and applied to the hydraulic-fracture (HF) propagation problem. It is shown that this modeling approach provides an alternative to finite-element and finite-volume methods for solving poroelastic and fracture-propagation problems. In this paper, we specifically investigate the interaction between an HF and natural fractures (NFs). The peridynamics model presented here allows us to simulate the propagation of multiple, nonplanar, interacting fractures and provides a novel approach to simulate the interaction between HFs and NFs. The model predictions in two dimensions have been validated by reproducing published experimental results where the interaction between an HF and an NF is controlled by the principal-stress contrast and the approach angle. A detailed parametric study involving poroelasticity and mechanical properties of the rock is performed to understand why an HF becomes arrested or crosses an NF. This analysis reveals that poroelasticity, resulting from high fracture-fluid leakoff, has a dominant influence on the interaction between an HF and an NF. In addition, the fracture toughness of the rock, the toughness of the NF, and the shear strength of the NF also affect the interaction between an HF and an NF. We also investigate the interaction of multiple completing fractures with NFs in two dimensions and demonstrate the applicability of the approach to simulate complex fracture networks on a field scale. Finally, the 3D interaction study elucidated that the height of the NF, the position of the NF, and the opening resistance of the NF all have a significant effect on the 3D interaction between an HF and an NF.